Research ArticleAlgebrization of Nonautonomous Differential Equations

Mariacutea Aracelia Alcorta-Garciacutea1 Martiacuten Eduardo Friacuteas-Armenta2

Mariacutea Esther Grimaldo-Reyna1 and Elifalet Loacutepez-Gonzaacutelez3

1UniversidadAutonoma deNuevo Leon AvenidaUniversidad SN CiudadUniversitaria 66451 SanNicolas de los Garza NLMexico2Departamento de Matematicas Universidad de Sonora Boulevard Rosales y Luis Encinas SN Col Centro83000 Hermosillo SON Mexico3Universidad Autonoma de Ciudad Juarez Unidad Multidisciplinaria de la UACJ en CuauhtemocCarretera Cuauhtemoc-Anahuac Col Ejido Anahuac km 35 SN 31600 Municipio de Cuauhtemoc CHIH Mexico

Correspondence should be addressed to Marıa Esther Grimaldo-Reyna mariagrimaldoryuanledumx

Received 4 June 2015 Revised 22 September 2015 Accepted 11 October 2015

Academic Editor Peter G L Leach

Copyright copy 2015 Marıa Aracelia Alcorta-Garcıa et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Given a planar system of nonautonomous ordinary differential equations 119889119908119889119905 = 119865(119905 119908) conditions are given for the existenceof an associative commutative unital algebra A with unit 119890 and a function 119867 Ω sub R2 times R2 rarr R2 on an open set Ω such that119865(119905 119908) = 119867(119905119890 119908) and the maps 119867

1(120591) = 119867(120591 120585) and 119867

2(120585) = 119867(120591 120585) are Lorch differentiable with respect to A for all (120591 120585) isin Ω

where 120591 and 120585 represent variables inA Under these conditions the solutions 120585(120591) of the differential equation 119889120585119889120591 = 119867(120591 120585) overA define solutions (119909(119905) 119910(119905)) = 120585(119905119890) of the planar system

1 Introduction

The theory of analytic functions on algebras is based on Lorchanalyticity see [1ndash5] Results of classical function theory havebeen extended to finite dimensional associative commutativeunital algebras

(i) The Cauchy integral theorem is satisfied for analyticalfunctions in algebras and the Cauchy integral for-mula has an analogous version in algebras

(ii) The classical theorems on Taylor power series areeasily established and Laurent expansion may bedefined in several disjoint regions in each one ofwhich it may define a different analytic function

(iii) Analyticity of functions of variables in algebras ischaracterized by the generalized Cauchy-Riemannequations which is a set of first-order linear partialdifferential equations

This theory allows us to consider differential equations overalgebras which can be used to solve family planar systemshaving the form

119889119909

119889119905= 119891 (119905 119909 119910)

119889119910

119889119905= 119892 (119905 119909 119910)

(1)

For this work and hereinafter any algebra will be assumedto be associative commutative and unital with unit 119890 andA will denote the linear space R2 endowed with an algebrastructure

In this paper a planar vector field 119865 is said to be A-algebrizable orA-differentiable if there exists an algebraA forthe which 119865 is Lorch differentiable (see Section 2 for defi-nitions) In the same way we say that a planar autonomoussystem of ordinary differential equations 119889119908119889119905 = 119865(119908) isalgebrizable if 119865 is A-algebrizable

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2015 Article ID 632150 10 pageshttpdxdoiorg1011552015632150

2 Journal of Applied Mathematics

Definition 1 Let A be an algebra We say that a function119865 119880 sub R3 rarr R2 defined in an open set 119880 has an H(A)-differentiable lifting 119867 Ω sub R2 times R2 rarr R2 if 119867 isa function defined in an open set Ω such that (i) the maps1198671(120591) = 119867(120591 119887) and 119867

2(120585) = 119867(119886 120585) are A-differentiable

functions with respect to A for all (119886 119887) isin Ω where 120591 and 120585

represent variables inA (ii) (119905119890 119909 119910) isin Ω for all (119905 119909 119910) isin 119880and (iii) 119865(119905 119909 119910) = 119867(119905119890 119909 119910) for all (119905 119909 119910) isin 119880

A nonautonomous differential equation over an algebraAis denoted by

119889120585

119889120591= 119867 (120591 120585) (2)

where 119867 Ω sub A times A rarr A is a function defined inan open set Ω For every point (120591

0 1205850) isin Ω a solution to

the equation through (1205910 1205850) consists of an A-differentiable

function 120585 119873(1205910) sub A rarr A defined in a neighborhood

119873(1205910) of 120591

0 with 120585(120591

0) = 120585

0and A-derivative 119889120585119889120591 with

respect to A satisfies 119889120585(120591)119889120591 = 119867(120591 120585(120591)) for all 120591 isin 119873(1205910)

If 119865 = (119891 119892) has anH(A)-differentiable lifting119867 we say thatplanar system (1) is algebrizable and that (2) is an algebrizationof (1) A theorem of existence and uniqueness of solutionsfor differential equations over algebras is proved in [6] anda technique for visualization of solutions is given in [7]

The classical differential equation 119889119908119889119905 = 1199082 has

solutions 119908(119905) = minus(119905 + 119888)minus1 Some differential systems of

autonomous differential equations can be written in this formby using variables in algebras For example the algebrizationof the planar differential system

119889119909

119889119905= 31199092+ 2119909119910 minus 119910

2

119889119910

119889119905= 31199102+ 2119909119910 minus 119909

2

(3)

is the differential equation 119889120585119889120591 = 1205852 over the algebra A

defined by the linear space R2 endowed with the product

(1199091 1199101) (1199092 1199102) = (3119909

11199092

+ (11990911199102+ 11991011199092minus 11991011199102) 311991011199102

+ (11990911199102+ 11991011199092minus 11990911199092))

(4)

The solutions 120585 are given by 120585(120591) = minus(120591 + 119888)minus1 hence the

solutions of the planar system are given by (119909(119905) 119910(119905)) = 120585(119905119890)where 119890 denotes the unit of A Using the first fundamentalrepresentation of A (see Section 2) the following expressionfor the solution (119909(119905) 119910(119905)) of the system is obtained

(119909 (119905) 119910 (119905))

= (minus119905 minus 119910

0

4 (2119905 + 1199090+ 1199100)2

minus119905 minus 1199090

4 (2119905 + 1199090+ 1199100)2)

(119909 (0) 119910 (0)) =minus (1199100 1199090)

4 (1199090+ 1199100)2

(5)

Consider now the planar nonautonomous differentialsystem

119889119909

119889119905= minus

1

1199053minus

2119909

119905+ 1199051199092minus 1199051199102

119889119910

119889119905= minus

2119910

119905+ 2119905119909119910 + 119905119910

2

(6)

whose algebrization is the Riccati differential equation overA

119889120585

119889120591= minus

1

1205913minus

2120585

120591+ 1205911205852 (7)

where A is the algebra defined by R2 and the product(119906 V)(119909 119910) = (119906119909 minus V119910 119909119910 + V119909 + V119910) By the classical Liemethods for solving differential equations (see [8ndash10]) thesolutions of the Riccati equation have the form

120585 (120591) =119862 + 1205912

1205912 (119862 minus 1205912) 119862 = (119886 119887) (8)

The functions (119909(119905) 119910(119905)) = 120585(119905119890) where 119890 is the unit ofA aresolutions of the planar system which can be obtained via thefirst fundamental representation of A

(119909 (119905) 119910 (119905)) = (1198872+ (119886 + 119887 minus 119905

2) (119886 + 119905

2)

(1198862 + 119886119887 + 1198872) 1199052 minus (2119886 + 119887) 1199054 + 1199056

minus2119887

(1198862 + 119886119887 + 1198872) minus (2119886 + 119887) 1199052 + 1199054)

(9)

We consider differential systems having the form (1)where 119891 119892 119880 sub R3 rarr R are 119862

1 functions defined inan open set 119880 The aim of this work is to give a family offunctions 119865 = (119891 119892) having H(A)-differentiable liftings 119867

over some algebra A When they exist the solutions 120585(120591)

of the differential equation 119889120585119889120591 = 119867(120591 120585) over A definesolutions (119909(119905) 119910(119905)) = 120585(119905119890) of system (1) which can beobtained via the first fundamental representation of A

The paper is organized as follows In Section 2 definitionsof algebra algebrizability of planar vector fields and differen-tiability on modules over algebras a characterization of thealgebrizability of planar vector fields and the form of all thequadratic vector fields which are algebrizable are given InSection 3 the definition of algebrizable liftings of functions119901 119868 sub R rarr R2 is presented It is shown that the class ofall of these functions defines an infinite dimensional algebraand the form of a family of these functions 119901 is given InSection 4 it is proved that the solutions of planar systems (1)can be obtained from the solutions of their algebrization atheorem containing conditions under which a planar systemlike (1) which is polynomial is algebrizable is given and itis shown that the class of all the planar systems (1) havingan algebrization (2) defines an infinite dimensional algebraIn Section 5 the case of quadratic systems is considered andtheir algebrizations are given which are Riccati equationsover algebrasThe results presented in Sections 3 4 and 5 arethe main contributions of this paper

Journal of Applied Mathematics 3

2 Algebras and Lorch Analyticity

21 Algebras

Definition 2 (see [11]) An algebra A is a R-linear space E

endowed with a bilinear product A times A rarr A denotedby (119909 119910) 997891rarr 119909119910 which is associative 119909(119910119911) = (119909119910)119911 andcommutative 119909119910 = 119910119909 for all 119909 119910 119911 isin A and has a unit 119890 isin A

satisfying 119890119909 = 119909119890 = 119909 for all 119909 isin A

An element 119886 isin A is called regular if there exists 119886minus1 isin A

called inverse of 119886 such that 119886minus1 sdot 119886 = 119886 sdot119886minus1

= 119890 If 119886 isin A is notregular then 119886 is called singular If 119886 119887 isin A and 119887 is regularthe quotient 119886119887means 119886 sdot 119887

minus1In all the algebras considered in this paper it will be the

case that E = R2 unless otherwise statedConsider an algebra A If 120573 = 119890

1 1198902 is the standard

basis of R2 the product between the elements of 120573 is givenby 119890119894119890119895= sum2

119896=1119888119894119895119896

119890119896 where the coefficients 119888

119894119895119896isin R 119894 119895 119896 isin

1 2 are called structure constants ofAThefirst fundamentalrepresentation of A is the injective linear homomorphism 119877

A rarr 119872(2R) defined by 119877 119890119894997891rarr 119877119894 where 119877

119894is the matrix

whose 119895 119896 entry is 119888119894119896119895 for 119894 = 1 2

22 Differentiability on Algebras In this subsection the defi-nition of Lorch differentiability is recalled which in this paperis calledA-algebrizability orA-differentiability to denote thedependence of the Lorch differential over an algebra A

Let | sdot | be the norm onA defined by |119886| = max119883119877(119886)

119883 isin R2 119883 = 1 (here the vector119883 is represented as a 1times 2

matrix in order for the product119883119877(119886) to make sense) where119877 A rarr 119872(2 119877) is the first fundamental representation ofA and sdot the Euclidean norm in R2 For this norm we have|119886119887| le |119886||119887| for all 119886 119887 isin 119872(2R) Thus we consider in thiswork that every algebraA is a Banach algebra under the norm| sdot |

Definition 3 Let A be an algebra and 119865 119881 sub A rarr A afunction on an open set119881 We say that 119865 isA-algebrizable orA-differentiable on119881 if there exists a function 119865

1015840 119881 sub A rarr

A called the A-derivative of 119865 on 119881 satisfying

limℎrarr0

10038161003816100381610038161003816119865 (119886 + ℎ) minus 119865 (119886) minus 119865

1015840(119886) ℎ

10038161003816100381610038161003816

|ℎ|= 0 (10)

for all 119886 isin 119881 where 1198651015840(119886)ℎ denotes the product inA of 1198651015840(119886)with ℎ

A vector field 119865 119881 sub A rarr A is A-algebrizable on119881 if and only if the Jacobian matrix of 119865 is contained in thefirst fundamental representation of A that is 119869119865(119886) isin 119877(A)

for all 119886 isin 119881 see [5] It can be shown that the notion of A-algebrizability coincides with the holomorphicity when A isthe complex field

A method for determining whether a given planar vectorfield 119865 is algebrizable is the following 119865 is algebrizable ifand only if for some of the following three types of pairs ofmatrices

(I) 1198611= ( 1 0119886 minus1

) 1198612= (0 1

119887 0)

(II) 1198611= ( 1 1198860 minus1

) 1198612= ( 0 01 0

)(III) 119861

1= ( 0 10 0

) 1198612= ( 0 01 0

)

the condition ⟨119861119894 120597119865(119909 119910)120597(119909 119910)⟩ = 0 is satisfied for 119894 = 1 2

and for all (119909 119910) in the domain of definition of 119865 where⟨sdot sdot⟩ denotes the usual inner product in119872(2R) The algebrafor each type of pair of matrices is defined by the followingcorresponding product table of the standard basis vectors1198901 1198902of R2

(I)sdot 1198901

1198902

1198901

1198901

1198902

1198902

1198902

minus1198871198901+ 1198861198902

(II)sdot 1198901

1198902

1198901

minus1198861198901

1198901

1198902

1198901

1198902

(III)sdot 1198901

1198902

1198901

1198901

0

1198902

0 1198902

Consider a planar autonomous system of quadratic ordinarydifferential equations in the variables 119909 and 119910 If this system isalgebrizable for an algebra with Type (I) product then it canbe represented by equations of the form

119889119909

119889119905= 1198860+ (1198872minus 1198861198871) 119909 minus 119887119887

1119910 + (

1

21198874minus 1198861198873)1199092

minus 21198871198873119909119910 minus

1

211988711988741199102

119889119910

119889119905= 1198870+ 1198871119909 + 1198872119910 + 11988731199092+ 1198874119909119910

+ (1

21198861198874minus 1198871198873)1199102

(11)

for some real constants 119886 119887 1198860 1198870 1198871 119887

4 In the case of

algebrizability for an algebra with Type (II) product thesystem is

119889119909

119889119905= 1198860+ 1198861119909 + 1198862119910 minus

1

211988611988631199092+ 1198863119909119910 + 119886

41199102

119889119910

119889119905= 1198870+ (1198861+ 1198861198862) 119910 + (

1

21198863+ 1198861198864)1199102

(12)

for some real constants 119886 1198870 1198860 1198861 1198862 1198863 and 119886

4 For Type

(III) product the system can be represented by equations ofthe form

119889119909

119889119905= 1198860+ 1198861119909 + 11988621199092

119889119910

119889119905= 1198870+ 1198871119910 + 11988721199102

(13)

for some real constants 119886119894 119887119894 119894 = 0 1 2 Moreover conditions

on the components of vector fields 119865 can be given for con-structing scalar functions 120572(119909 119910) which we call algebrizantefactors such that 120572119865 are algebrizable vector fields Inverseintegrating factors (see [12 13]) are constructed for thesevector fields

4 Journal of Applied Mathematics

23 Differentiability onModules over Algebras In this subsec-tion we give the definition of A-differentiability of functionswith domain in A times A and image in A

The Cartesian productAtimesA defines a normedA-modulewith respect to the norm sdot A times A rarr R (119886

1 1198862) =

radic|1198861|2 + |119886

2|2 This norm satisfies

(i) 119883 ge 0 for all 119883 isin A times A and 119883 = 0 if and only if119883 = (0 0)

(ii) 119886119883 le |119886|119883 for all 119886 isin A and119883 isin A times A(iii) 119883 + 119884 le 119883 + 119884 for all119883119884 isin A times A

In the following definition sdot denotes the norm givenabove on the A-module A times A

Definition 4 Let A be an algebra and 119867 Ω sub A times A rarr

A a function where Ω is an open set We say that 119867 isH(A)-differentiable on 119883

0isin Ω if there exists a module

homeomorphism 119872 A times A rarr A which we call thedifferential homomorphism of 119867 at 119883

0 which satisfies the

condition

lim119883rarr1198830

1003817100381710038171003817119867 (119883) minus 119867 (1198830) minus 119872(119883 minus 119883

0)1003817100381710038171003817

1003817100381710038171003817119883 minus 1198830

1003817100381710038171003817

= 0 (14)

We denote 119872 by D119867(1198830) We say that 119867 is H(A)-

differentiable on Ω if 119867 is H(A)-differentiable on all thepoints ofΩ

A function 119867 Ω sub A times A rarr A isH(A)-differentiableat 1198830if and only if the usual Jacobian matrix of 119867 Ω sub

R2 times R2 rarr R2 at 1198830satisfies 119869119867(119883

0) isin 119877(A) times 119877(A)

The differential homomorphismD119867(1198830) is represented by a

matrix in11987212

(A)with respect to the standard basis ofAtimesAwhere119872

12(A) is theA-module of all the matrices of one row

and two columns with entries in A see [14]

3 On Algebrizable Liftings of Functions 119901 119868 sub

R rarr R2

In this section are considered functions 119901 119868 sub R rarr R2

defined in open intervals 119868 and conditions for the existenceof algebrasA andA-algebrizable functions119875 such that 119901(119905) =119875(119905119890) will be determined where 119890 is the unit of A

Definition 5 Let 119901 119868 sub R rarr R2 be a function defined in anopen interval 119868 andA an algebra with unit 119890 We will say that119875 119881 sub R2 rarr R2 is an A-algebrizable lifting of 119901 if (a) 119881 isan open set on which 119875 isA-algebrizable (b) 119905119890 119905 isin 119868 sub 119881and (c) 119901(119905) = 119875(119905119890) for all 119905 isin 119868

As a consequence of the following proposition the familyof all the functions 119901 119868 sub R rarr R2 having algebrizableliftings is an infinite dimensional algebra

Proposition 6 Let A be an algebra and let 119901 119902 119868 sub R rarr

R2 be functions where 119868 is an interval

(a) Every constant function 119901(119905) = 119888 admits the A-algebrizable lifting 119875(120591) = 119888

(b) 119901(119905) = 119905119890 admits the A-algebrizable lifting 119875(120591) = 120591where 119890 isin A is the unit of A

(c) If 119901 and 119902 admit A-algebrizable liftings 119875 and 119876 withrespect to A respectively and 119886 119887 are constants in Athen 119886119901 + 119887119902 and 119901119902 (all products with respect to A)admit algebrizable liftings 119886119875+119887119876 and119875119876 respectively

(d) If 119901 has an A-algebrizable lifting 119875 and 119876 is an A-algebrizable function with Im(119875) sub Dom(119876) then119876 ∘ 119875 is an algebrizable lifting of 119876 ∘ 119901

Proof Identity and constant functions are A-differentiablefor any algebra A Thus (a) and (b) hold

Let 119901 and 119902 be functions with A-algebrizable liftings 119875

being 119876 and 119886 119887 isin A constants The functions 119878 = 119886119875 + 119887119876

and 119879 = 119875119876 are A-algebrizable and satisfy

119878 (119905119890) = 119886119875 (119905119890) + 119887119876 (119905119890) = 119886119901 (119905) + 119887119902 (119905)

119879 (119905119890) = 119875 (119905119890) 119876 (119905119890) = 119901 (119905) 119902 (119905)

(15)

Therefore 119878 and 119879 are algebrizable liftings of 119886119901 + 119887119902 and 119901119902respectively

If 119901 has an A-algebrizable lifting 119875 and 119876 is an A-differentiable function with Im(119875) sub Dom(119876) then 119876 ∘

119875(119905119890) = 119876 ∘ 119901(119905) Therefore 119876 ∘ 119875 is an A-algebrizable liftingof 119876 ∘ 119901

Corollary 7 LetA be an algebraThen the following functionsadmit A-algebrizable liftings polynomial functions rationalfunctions trigonometric functions exponential functions andall of those functions which can be defined by linear com-binations products quotients and compositions of functionsadmitting algebrizable liftings

Every function 119905 997891rarr (ℎ(119905) 119896(119905)) with polynomial com-ponents ℎ(119905) and 119896(119905) has an A-algebrizable lifting Thefollowing proposition gives a wider class of these functions

Proposition 8 Let A be an algebra Any function 119905 997891rarr

(ℎ(119905) 119896(119905)) with components of the form

ℎ (119905) =119886119898

119905119898+

119886119898minus1

119905119898minus1+ sdot sdot sdot +

119886minus1

119905+ 1198860+ 1198861119905 + sdot sdot sdot 119886

119899119905119899

119896 (119905) =119887119898

119905119898+

119887119898minus1

119905119898minus1+ sdot sdot sdot +

119887minus1

119905+ 1198870+ 1198871119905 + sdot sdot sdot 119887

119899119905119899

(16)

119898 119899 isin N has an A-algebrizable lifting An A-algebrizablelifting is given by

120591 997891rarr (119886119898 119887119898)

1

120591119898+ sdot sdot sdot + (119886

minus1 119887minus1)1

120591+ (1198860 1198870)

+ (1198861 1198871) 120591 + sdot sdot sdot + (119886

119899 119887119899) 120591119899

(17)

Proof Consider 119876 given by the above expression then

(119886119896 119887119896) (119905119890)119896= (119886119896 119887119896) 119905119896119890 = (119886

119896119905119896 119887119896119905119896) (18)

holds for 119896 = minus119898 119899 so 119876(119905119890) = (ℎ(119905) 119902(119905)) where 119890 isthe unit ofA Therefore 119876 is anA-algebrizable lifting of 119905 997891rarr(ℎ(119905) 119896(119905))

Journal of Applied Mathematics 5

In particular every function 119901 = (1199011 1199012) 119868 sub R rarr

R2 with quadratic components 1199011(119905) = 119886

0+ 1198861119905 + 11988621199052 and

1199012(119905) = 119887

0+1198871119905+11988721199052 admits theA-algebrizable lifting 119875(120591) =

(1198860 1198870) + (1198861 1198871)120591 + (119886

2 1198872)1205912

4 On Algebrizable Liftings of Planar Systems

Solutions of every algebrizable planar system can be foundby solving an algebrization of the system as it is seen in thefollowing proposition

Proposition 9 If (2) is an algebrization of (1) and 120585(120591) asolution of (2) then (119909(119905) 119910(119905)) = 120585(119905119890) is a solution of (1)where 119890 denotes the unit of the corresponding algebra

Proof Let 120585 be a solution of (2) The derivative of(119909(119905) 119910(119905)) = 120585(119905119890) with respect to 119905 is given by

119889

119889119905(119909 (119905) 119910 (119905)) =

119889

119889119905120585 (119905119890) = [

119889120585 (120591 120585)

119889120591

10038161003816100381610038161003816100381610038161003816120591=(119905119890)

119889120591

119889119905]

= 119867 (120591 120585 (120591))1003816100381610038161003816120591=(119905119890) 119890 = 119867 (119905119890 120585 (119905119890))

= (119891 (119905 119909 (119905) 119910 (119905)) 119892 (119905 119909 (119905) 119910 (119905)))

(19)

Thus (119909(119905) 119910(119905)) is a solution of system (1)

As a consequence of the following proposition the familyof all the functions 119865 Ω sub R3 rarr R2 having algebrizableliftings defines an infinite dimensional algebra

Proposition 10 Let A be an algebra with unit 119890 In thefollowing statements 119865 and119866 denote functions defined on opensets Ω sub R3 and they have values in R2

(a) 119865(119905 119909 119910) = 119888 admits the H(A)-differentiable lifting119867(120591 120585) = 119888

(b) 119865(119905 119909 119910) = 119905119890 admits the H(A)-differentiable lifting119867(120591 120585) = 120591

(c) 119865(119905 119909 119910) = (119909 119910) admits the H(A)-differentiablelifting119867(120591 120585) = 120585

(d) If 119865 and 119866 admit H(A)-differentiable liftings 119867119865and

119867119866 respectively and 119886 119887 are constants inA then 119886119867

119865+

119887119867119866and 119865119866 (all products with respect to A) have

H(A)-differentiable liftings 119886119867119865+ 119887119867119866and 119867

119865119867119866

respectively

(e) Every function 119864 having the form 119864(119905 119909 119910) = 119865(119905 119866(119905

119909 119910)) where 119865 and 119866 admit H(A)-differentiableliftings 119867

119865and 119867

119866 admits an H(A)-differentiable

lifting119867119864given by119867

119864(120591 120585) = 119867

119865(120591119867119866(120591 120585))

(f) Every function 119864 having the form 119864(119905 119909 119910) =

119865(119905 119875(119909 119910)) where 119865 has an H(A)-differentiablelifting119867

119865and 119875 is anA-differentiable function admits

anH(A)-differentiable lifting119867119864(120591 120585) = 119867

119865(120591 119875(120585))

Proof The proofs of (a) (b) and (c) are trivial Let 119867119865and

119867119866be the algebrizable liftings of 119865 and 119866 and then119867

119865+119867119866

119867119865119867119866 and119867

119865119867119866are A-algebrizable and

(i) (119886119867119865+119887119867119866)(119905119890 119909 119910) = 119886119867

119865(119905119890 119909 119910)+119887119867

119866(119905119890 119909 119910) =

119886119865(119905 119909 119910) + 119886119866(119905 119909 119910)

(ii) (119867119865119867119866)(119905119890 119909 119910) = 119867

119865(119905119890 119909 119910)119867

119866(119905119890 119909 119910) =

119865(119905 119909 119910)119866(119905 119909 119910)

(iii) 119867119864(119905119890 119909 119910) = 119867

119865(119905119890119867

119866(119905119890 119909 119910)) = 119865(119905 119866(119905 119909 119910))

(iv) 119867119864(119905119890 119909 119910) = 119867

119865(119905119890 119875(119890 119910)) = 119865(119905 119875(119909 119910))

Thus the proof is complete

Corollary 11 Let A be an algebra The following func-tions admitH(A)-differentiable liftings polynomial functionsrational functions trigonometric functions exponential func-tions and all of those functions which can be defined bylinear combinations products quotients and compositions offunctions admittingH(A)-differentiable liftings

Given a function 119865(119905 119909 119910) = (119891(119905 119909 119910) 119892(119905 119909 119910))where 119891 119892 are polynomial functions of the variables 119909 119910the goal of the paper is to determine if 119865 has an H(A)-differentiable lifting As a consequence of the followingtheorem every function 119865 which is polynomial of thevariables 119905 119909 and 119910 has an H(A)-differentiable liftingwhen (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-differentiable forall 119905

Theorem 12 Let A be an algebra with unit 119890 and 119865

Ω sub R3 rarr R2 119865 = (119891 119892) where 119891 = sum119898

119896=0119891119896

119892 = sum119898

119896=0119892119896 and 119891

119896(119905 119909 119910) 119892

119896(119905 119909 119910) are homogeneous

polynomials of degree 119896 in the variables 119909 119910 and Ω = 119868 times

R2 for an open interval 119868 Then the following statements areequivalent

(a) 119865 has anH(A)-differentiable lifting119867

(b) The map (119909 119910) 997891rarr 119865(119905 119909 119910) is A-algebrizable forall 119905 isin 119868 and the functions ℎ

119896given by ℎ

119896(119905) =

(119891119896(119905 119890) 119892

119896(119905 119890)) have A-algebrizable liftings for 119896 =

0 1 119898

(c) 119867(120591 120585) = sum119898

119896=0119867119896(120591)120585119896 is an H(A)-differentiable

lifting of 119865 where 119867119896(120591) are A-algebrizable liftings of

ℎ119896(119905) = (119891

119896(119905 119890) 119892

119896(119905 119890))

Proof Obviously (a) implies (b) and (c) implies (a) We nowshow that (b) implies (c) Suppose that119891(119905 119909 119910) and119892(119905 119909 119910)

are homogenous polynomials 119891(119905 119909 119910) = sum119899

119896=0119901119896(119905)119909119896119910119899minus119896

and 119892(119905 119909 119910) = sum119899

119896=0119902119896(119905)119909119896119910119899minus119896 of the variables 119909 119910 defined

6 Journal of Applied Mathematics

on the setΩ = 119868 timesR2 as above Taking the partial derivativesof 119865 with respect to 119909 and 119910 we obtain

120597 (119891 119892)

120597 (119909 119910)(119905 119909 119910)

= (

119899

sum

119896=1

119896119901119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119901119896minus1

(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

119896119902119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119902119896minus1

(119905) 119909119896minus1

119910119899minus119896

)

= (

119899

sum

119896=1

119896119901119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119901119896minus1

(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

119896119902119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119902119896minus1

(119905) 119909119896minus1

119910119899minus119896

)

=

119899

sum

119896=1

119909119896minus1

119910119899minus119896

(119896119901119896(119905) (119899 minus 119896 + 1) 119901

119896minus1(119905)

119896119902119896(119905) (119899 minus 119896 + 1) 119902

119896minus1(119905)

)

(20)

Let 119877 be the first fundamental representation of A Since(119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for all 119905 isin 119868then (120597(119891 119892)120597(119909 119910))(119905 119909 119910) isin 119877(A) for all (119905 119909 119910) isin ΩThus

120597 (119891 119892)

120597 (119909 119910)(119905 0 1) = (

1199011(119905) 119899119901

0(119905)

1199021(119905) 119899119902

0(119905)

) isin 119877 (A) (21)

for all 119905 isin 119868 and then

120597 (119891 119892)

120597 (119909 119910)(119905 119909 119910) minus 119910

119899minus1(1199011(119905) 119899119901

0(119905)

1199021(119905) 119899119902

0(119905)

)

=

119899

sum

119896=2

119909119896minus1

119910119899minus119896

(119896119901119896(119905) (119899 minus 119896 + 1) 119901

119896minus1(119905)

119896119902119896(119905) (119899 minus 119896 + 1) 119902

119896minus1(119905)

)

(22)

is in 119877(A) for all (119905 119909 119910) isin Ω If 119909(119904) = 1119904 and 119910(119904) =(119899minus2)radic119904

then

lim119904rarrinfin

[120597 (119891 119892)

120597 (119909 119910)(119905 119909 (119904) 119910 (119904))

minus [119910 (119904)]119899minus1

(1199011(119905) 119899119901

0(119905)

1199021(119905) 119899119902

0(119905)

)]

= (1199012(119905) (119899 minus 1) 119901

1(119905)

1199022(119905) (119899 minus 1) 119902

1(119905)

) isin 119877 (A)

(23)

Following the same idea for 119896 = 1 2 119899

(119896119901119896(119905) (119899 minus 119896 + 1) 119901

119896minus1(119905)

119896119902119896(119905) (119899 minus 119896 + 1) 119902

119896minus1(119905)

) isin 119877 (A) (24)

If (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for analgebra A with Type (I) product (given in Section 2) then

119896119901119896+ 119886119896119902

119896minus (119899 minus 119896 + 1) 119902

119896minus1= 0

119887119896119902119896+ (119899 minus 119896 + 1) 119901

119896minus1= 0

(25)

or equivalently

(119902119896minus1

119901119896minus1

) = (119896119901119896+ 119886119896119902

119896

119899 minus 119896 + 1 minus

119887119896119902119896

119899 minus 119896 + 1) (26)

for 119896 = 1 119899 Thus the functions 119901119896 119902119896

aredetermined by 119901

119899 119902119899

for 119896 = 0 1 119899 minus 1Since (119891(119905 1 0) 119892(119905 1 0)) = (119901

119899(119905) 119902119899(119905)) then

(119891(119905 119909 119910) 119892(119905 119909 119910)) = (119901119899(119905) 119902119899(119905))(119909 119910)

119899Therefore 119865 has anH(A)-differentiable lifting 119867(120591 120585) =

119867119899(120591)120585119899 where 119867

119899(120591) is an A-algebrizable lifting of 119905 997891rarr

(119901119899(119905) 119902119899(119905))

If (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for analgebra A with Type (II) product (given in Section 2) then

119896119901119896+ 119886 (119899 minus 119896 + 1) 119901

119896minus1minus (119899 minus 119896 + 1) 119902

119896minus1= 0

119896119902119896= 0

(27)

or equivalently

(119901119896 119902119896) = (minus

119886 (119899 minus 119896 + 1) 119901119896minus1

119896 0) (28)

for 119896 = 1 119899 Thus 119902119896

= 0 and 119901119896is determined by 119901

0

for 119896 = 1 2 119899 Since (119891(119905 0 1) 119892(119905 0 1)) = (1199010(119905) 1199020(119905))

then (119891(119905 119909 119910) 119892(119905 119909 119910)) = (1199010(119905) 1199020(119905))(119909 119910)

119899Therefore 119865 has anH(A)-differentiable lifting 119867(120591 120585) =

119867119899(120591)120585119899 where 119867

119899(120591) is an A-algebrizable lifting of 119905 997891rarr

(1199010(119905) 1199020(119905))

If (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for analgebra A with Type (III) product (given in Section 2) then(119899 minus 119896 + 1)119901

119896minus1= 0 and 119896119902

119896= 0 that is 119901

119896minus1= 0 119902

119896= 0

for 119896 = 1 119899 Thus 119902119896

= 0 and 119901119896is determined by 119901

0

for 119896 = 1 2 119899 Since (119891(119905 0 1) 119892(119905 0 1)) = (119901119899(119905) 1199020(119905))

then (119891(119905 119909 119910) 119892(119905 119909 119910)) = (119901119899(119905) 1199020(119905))(119909 119910)

119899Therefore 119865 has anH(A)-differentiable lifting 119867(120591 120585) =

119867119899(120591)120585119899 where 119867

119899(120591) is an A-algebrizable lifting of 119905 997891rarr

(119901119899(119905) 1199020(119905))

Thus if 119891 and 119892 are homogenous polynomial of degree119899 in the variables 119909 and 119910 then 119867(120591 120585) = 119867

119899(120591)120585119899 in each

of the cases of algebras defined by products of Types (I)(II) and (III) given in Section 2 Since a polynomial function(119905 119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) in the variables 119909 and 119910 canbe seen as the finite addition of homogenous polynomial inthe variables 119909 and 119910 by (d) of Proposition 10 (b) implies(c)

Example 13 Consider the planar system (6) then

119865 (119905 119909 119910)

= (minus1

1199053minus

2119909

119905+ 1199051199092minus 1199051199102 minus

2119910

119905+ 2119905119909119910 + 119905119910

2)

(29)

The Jacobian 120597119865120597(119909 119910) is given by

120597119865 (119905 119909 119910)

120597 (119909 119910)= (

minus2

119905+ 2119905119909 minus2119905119910

2119905119910 minus2

119905+ 2119905119909 + 2119905119910

) (30)

Journal of Applied Mathematics 7

Thus 120597119865120597(119909 119910) is orthogonal to the matrices

1198611= (

1 0

1 minus1)

1198612= (

0 1

1 0)

(31)

for all (119909 119910) that is ⟨120597119865120597(119909 119910) 119861119894⟩ = 0 for 119894 = 1 2 Themap

(119909 119910) 997891rarr 119865(119905 119909 119910) isA-algebrizable for an algebra of Type (I)with constants 119886 = 1 and 119887 = 1 see Section 2 The function 119865

can be written as

119865 (119905 119909 119910) = (minus1

1199053 0) + (minus

2

119905119909 minus

2

119905119910)

+ (1199051199092minus 1199051199102 1199051199102)

(32)

Thus the functions 119901119894and 119902

119894of Theorem 12 are given by

1199010(119905 119909 119910) = minus1119905

3 1199020(119905 119909 119910) = 0 119901

1(119905 119909 119910) = minus(2119905)119909

1199021(119905 119909 119910) = minus(2119905)119910 119901

2(119905 119909 119910) = 119905119909

2minus 1199051199102 and 119902

2(119905 119909 119910) =

1199051199102The unit 119890 of A is 119890 = (1 0) and then ℎ

0(119905) = (minus1119905

3 0)

ℎ1(119905) = (minus2119905 0) and ℎ

2(119905) = (119905 0) have A-algebrizable

liftings 1198670(120591) = minus1120591

3 1198671(120591) = minus2120591 and 119867

2(120591) = 120591

respectivelyFrom the form of 119865(119905 119909119890) = (minus1119905

3minus (2119905)119909 minus 119905119909

2)119890 119867

must be

119867(120591 120585) = minus1

1205913minus

2

120591120585 + 120591120585

2 (33)

5 The Case of Second-Degree Polynomials inthe Variables 119905 119909 and 119910

If 119891 and 119892 in (1) are quadratic polynomials in three variables119905 119909 and 119910 andA is an algebra with respect to which the map(119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable it will beshowed that the H(A)-differentiable lifting 119867 of 119865 = (119891 119892)

is a polynomial in two variables ofA Under these conditions(1) has an algebrization which is a Riccati equation over Ahaving the form

119889120585

119889120591= 119875 (120591) + 119876 (120591) 120585 + 119877 (120591) 120585

2 (34)

where119875119876 and119877 are polynomials overA of degrees two oneand zero respectively

Consider system (1) where 119891 119892 Ω sub R3 rarr R2 aresecond-degree polynomials of three variables 119905 119909 and 119910 thatis

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 1198863119910 + 11988641199052+ 1198865119905119909

+ 1198866119905119910 + 119886

71199092+ 1198868119909119910 + 119886

91199102

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198872119909 + 1198873119910 + 11988741199052+ 1198875119905119909 + 119887

6119905119910

+ 11988771199092+ 1198878119909119910 + 119887

91199102

(35)

All the quadratic vector fields which are algebrizable withrespect to algebras with Type (I) products have the form

119889119909

119889119905= 1198600+ (1198612minus 1198861198611) 119909 minus 119887119861

1119910 + (

1

21198614minus 1198861198613)1199092

minus 21198871198613119909119910 minus

1

211988711986141199102

119889119910

119889119905= 1198610+ 1198611119909 + 1198612119910 + 11986131199092+ 1198614119909119910

+ (1

21198861198614minus 1198871198613)1199102

(36)

where 119886 119887 1198600 1198610 1198611 119861

5are real constants

see Section 22 The algebrizability of (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) with respect to algebras with Type(I) products can be verified by considering 119905 as a constant

The following theorems give conditions that characterizethe algebrizability of planar systems like (1) when 119891 and119892 are quadratic polynomials The algebrizability of nonau-tonomous quadratic systems with respect to algebras withType (I) products is given in the following theorem

Theorem14 LetA be an algebrawith Type (I) product definedby constants 119886 and 119887 and 119891 119892 the polynomials (35) Thefollowing statements are equivalent

(1) The map 119865 (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 have the form

119891 (119905 119909 119910) = 1198860+ 1198861119905 + (1198873minus 1198861198872) 119909 minus 119887119887

2119910 + 11988641199052

+ (1198876minus 1198861198875) 119905119909 minus 119887119887

5119905119910

+ (1198878

2minus 1198861198877)1199092minus 21198871198877119909119910 minus

1198871198878

21199102

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198872119909 + 1198873119910 + 11988741199052+ 1198875119905119909 + 119887

6119905119910

+ 11988771199092+ 1198878119909119910 + (

1198861198878

2minus 1198871198877)1199102

(37)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (38)

is anH(A)-differentiable lifting of 119865 where

1198600= (1198860 1198870)

1198601= (1198861 1198871)

1198602= (1198873minus 1198861198872 1198872)

1198603= (1198864 1198874)

1198604= (1198876minus 1198861198875 1198875)

1198605= (

1198878

2minus 1198861198877 1198877)

(39)

8 Journal of Applied Mathematics

Proof Writing119892(119905 119909 119910) = 1198610+1198611119909+1198612119910+11986131199092+1198614119909119910+119861

51199102

yields

1198610= 1198870+ 1198871119905 + 11988741199052

1198611= 1198872+ 1198875119905

1198612= 1198873+ 1198876119905

1198613= 1198877

1198614= 1198878

1198615= 1198879

(40)

The function (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) isA-algebrizableif and only if

119891 (119905 119909 119910) = 1198600+ (1198612minus 1198861198611) 119909 minus 119887119861

1119910

+ (1

21198614minus 1198861198613)1199092minus 2119887119861

3119909119910 minus

1

211988711986141199102

(41)

where 1198600

= 1198860+ 1198861119905 + 11988641199052 and 119887

9= 11988611988782 minus 119887119887

7 Thus

statements (1) and (2) are equivalentThe function 119867 is polynomial in the variables 120591 and 120585 of

A hence 119867 is H(A)-differentiable 119867 satisfies 119867(119905119890 119909 119910) =

(119891(119905 119909 119910) 119892(119905 119909 119910)) where 119890 is the unit of A So 119867 is anH(A)-differentiable lifting of 119865 Thus statement (2) impliesstatement (3)

Since 119867 is H(A)-differentiable and 119865(119905 119909 119910) =

119867(119905119890 119909 119910) then (119909 119910) 997891rarr 119865(119905 119909 119910) is A-algebrizablefor all 119905 Thus statement (3) implies statement (1)

All the quadratic vector fields which are algebrizable withrespect to algebras with Type (II) products have the form

= 1198600+ 1198601119909 + 119860

2119910 minus

1

211988611986031199092+ 1198603119909119910 + 119860

41199102

119910 = 1198610+ (1198601+ 1198861198602) 119910 + (

1

21198603+ 1198861198604)1199102

(42)

where 119886 1198600 119860

4 1198610are real constants see Section 22

The algebrizability of (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) withrespect to algebras with Type (II) products can be verifiedby considering 119905 as a constant

The algebrizability of nonautonomous quadratic systemswith respect to algebras with Type (II) products is given in thefollowing theorem

Theorem 15 Let A be an algebra with Type (II) productdefined by the constant 119886 and let 119891 119892 be the polynomials (35)The following statements are equivalent

(1) The map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 are given by

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 1198863119910 + 11988641199052+ 1198865119905119909

+ 1198866119905119910 minus

1

211988611988681199092+ 1198868119909119910 + 119886

91199102

119892 (119905 119909 119910) = 1198870+ 1198871119905 + (1198862+ 1198861198863) 119910 + 119887

41199052

+ (1198865+ 1198861198866) 119905119910 + (

1198868

2+ 1198861198869)1199102

(43)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (44)

is anH(A)-differentiable lifting of 119865 where

1198600= (1198860 1198870)

1198601= (1198861 1198871)

1198602= (1198863 1198862+ 1198861198863)

1198603= (1198864 1198874)

1198604= (1198866 1198865+ 1198861198866)

1198605= (11988691198868

2+ 1198861198869)

(45)

Proof Writing 119891(119905 119909 119910) = 1198600+ 1198601119909 + 119860

2119910 minus (12)119886119860

31199092+

1198603119909119910 + 119860

41199102 yields

1198600= 1198860+ 1198861119905 + 11988641199052

1198601= 1198862+ 1198865119905

1198602= 1198863+ 1198866119905

1198603= 1198868

1198604= 1198869

(46)

and the map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizableif and only if

119892 (119905 119909 119910) = 1198610+ (1198601+ 1198861198602) 119910 + (

1

21198603+ 1198861198604)1199102 (47)

where 1198610= 1198870+ 1198871119905 + 11988741199052 Thus statements (1) and (2) are

equivalentThe rest of the proof is similar to that of Theorem 14

All the quadratic vector fields which are differentiablewith respect to algebras with Type (III) products have theform

= 1198600+ 1198601119909 + 119860

21199092

119910 = 1198610+ 1198611119910 + 11986121199102

(48)

Journal of Applied Mathematics 9

where 1198600 1198610 1198601 1198611 1198602 1198612

are real constantssee Section 22 The algebrizability of (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) with respect to algebras with Type(III) products can be verified by considering 119905 as a constant

The algebrizability of nonautonomous quadratic systemswith respect to algebras with Type (III) products is given inthe following theorem

Theorem 16 Let A be an algebra with Type (III) productdefined by the constant 119886 and let 119891 119892 be the polynomials (35)The following statements are equivalent

(1) The map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 are given by

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 11988641199052+ 1198865119905119909 + 119886

71199092

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198873119910 + 11988741199052+ 1198876119905119910 + 119887

91199102

(49)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (50)

is an H(A)-differentiable lifting of 119865 where 1198600

=

(1198860 1198870) 1198601

= (1198861 1198871) 1198602

= (1198862 1198873) 1198603

= (1198864 1198874)

1198604= (1198865 1198876) and 119860

5= (1198867 1198879)

Proof Function (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable if and only if

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 11988641199052+ 1198865119905119909 + 119886

71199092

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198873119910 + 11988741199052+ 1198876119905119910 + 119887

91199102

(51)

Thus first and second statements are equivalents The func-tion119867 given by119867(120591 120585) = 119860

0+1198601120591+1198602120585+11986031205912+1198604120591120585+119860

51205852

where1198600= (1198860 1198870)1198601= (1198861 1198871)1198602= (1198862 1198873)1198603= (1198864 1198874)

1198604= (1198865 1198876) 1198605= (1198867 1198879) 120591 = (119903 119904) and 120585 = (119909 119910) satisfies

119867(119905119890 119909 119910) = (119891(119905 119909 119910) 119892(119905 119909 119910)) Thus second and thirdstatements are equivalents

By Theorems 14 15 and 16 an algebrization of quadraticsystems is a Riccati equation over an algebra 119889120585119889120591 = 119875(120591) +

119876(120591)120585 + 119877(120591)1205852 where 119875(120591) = 119860

0+ 1198601120591 + 119860

31205912 119876(120591) =

1198602+ 1198604120591 and 119877(120591) = 119860

5 In the following example is

given a nonautonomous quadratic system for the which analgebrization is found by usingTheorem 14

Example 17 Consider system (1) given by

119891 (119905 119909 119910) = 1 + 7119905 minus 119909 minus 3119910 + 51199052minus 119905119909 minus 3119905119910 minus 6119909119910

minus 61199102

119892 (119905 119909 119910) = 1 + 119905 + 119909 + 119910 + 1199052+ 119905119909 + 119905119910 + 119909

2+ 4119909119910

+ 1199102

(52)

The matrices

1198611= (

1 0

2 minus1)

1198612= (

0 1

3 0)

(53)

satisfy ⟨119861119894 120597(119891 119892)120597(119909 119910)⟩ = 0 for 119894 = 1 2Thus 119865 (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for an algebra A withType (I) product defined by the constants 119886 = 2 and 119887 = 3

The conditions of Theorem 14 are satisfied then theH(A)-differentiable lifting119867 of 119865 can be written as

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (54)

where 1198600= (1 1) 119860

1= (7 1) 119860

2= (minus1 1) 119860

3= (5 1)

1198604= (minus1 1) and 119860

5= (0 1)

Disclosure

The authors declare having no financial affiliation with anyorganization regarding the material discussed here

Conflict of Interests

The authors declare that there is no conflict of interestsconcerning this textThe issues discussed in this paper do nothave any secondary interest for any of the authors

Acknowledgments

The authors wish to acknowledge the support of GrantsPromep103513 and CB-2010150532 Conacyt

References

[1] E K Blum ldquoA theory of analytic functions in Banach algebrasrdquoTransactions of the American Mathematical Society vol 78 no2 pp 343ndash370 1955

[2] P W Ketchum ldquoAnalytic functions of hypercomplex variablesrdquoTransactions of the American Mathematical Society vol 30 no4 pp 641ndash641 1928

[3] E R Lorch ldquoThe theory of analytic functions in normed abelianvector ringsrdquo Transactions of the American Mathematical Soci-ety vol 54 no 3 pp 414ndash425 1943

[4] J A Ward ldquoA theory of analytic functions in linear associativealgebrasrdquo Duke Mathematical Journal vol 7 no 1 pp 233ndash2481940

[5] J A Ward ldquoFrom generalized Cauchy-Riemann equationsto linear algebrardquo Proceedings of the American MathematicalSociety vol 4 no 3 pp 456ndash461 1953

[6] E Lopez-Gonzalez ldquoDifferential equations over algebrasrdquoAdvances and Applications in Mathematical Sciences vol 8 no2 pp 189ndash214 2011

[7] A Alvarez-Parrilla M E Frıas-Armenta E Lopez-Gonzalezand C Yee-Romero ldquoOn solving systems of autonomousordinary differential equations by reduction to a variable of analgebrardquo International Journal of Mathematics and Mathemati-cal Sciences vol 2012 Article ID 753916 21 pages 2012

10 Journal of Applied Mathematics

[8] E J Wilczynski ldquoReview Abraham Cohen an introductionto the lie theory of one-parameter groups with applicationsto the solution of differential equationsrdquo Bulletin of the Amer-ican Mathematical Society vol 18 no 10 pp 514ndash515 1912httpprojecteuclidorgeuclidbams1183421829

[9] J M PageOrdinary Differential Equations with an Introductionto Liersquos Theory or the Group of One Parameter MacmillanPublishers London UK 1897

[10] R A Steinhour The truth about lie symmetries solvingdifferential equations with symmetry methods [SeniorIndependent StudyTheses] 2013 httpopenworkswoostereduindependentstudy949

[11] R Pierce Associative Algebras Springer New York NY USA1982

[12] K I T Al-Dosary ldquoInverse integrating factor for classes of pla-nar differential systemsrdquo International Journal of MathematicalAnalysis vol 4 no 29ndash32 pp 1433ndash1446 2010

[13] I A Garcıa and M Grau ldquoA Survey on the inverse integratingfactorrdquo Qualitative Theory of Dynamical Systems vol 9 no 1-2pp 115ndash166 2010

[14] W Brown Matrices over Commutative Rings Monographs andTextbooks in Pure and Applied Mathematics Marcel Dekker1992

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

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Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

Definition 1 Let A be an algebra We say that a function119865 119880 sub R3 rarr R2 defined in an open set 119880 has an H(A)-differentiable lifting 119867 Ω sub R2 times R2 rarr R2 if 119867 isa function defined in an open set Ω such that (i) the maps1198671(120591) = 119867(120591 119887) and 119867

2(120585) = 119867(119886 120585) are A-differentiable

functions with respect to A for all (119886 119887) isin Ω where 120591 and 120585

represent variables inA (ii) (119905119890 119909 119910) isin Ω for all (119905 119909 119910) isin 119880and (iii) 119865(119905 119909 119910) = 119867(119905119890 119909 119910) for all (119905 119909 119910) isin 119880

A nonautonomous differential equation over an algebraAis denoted by

119889120585

119889120591= 119867 (120591 120585) (2)

where 119867 Ω sub A times A rarr A is a function defined inan open set Ω For every point (120591

0 1205850) isin Ω a solution to

the equation through (1205910 1205850) consists of an A-differentiable

function 120585 119873(1205910) sub A rarr A defined in a neighborhood

119873(1205910) of 120591

0 with 120585(120591

0) = 120585

0and A-derivative 119889120585119889120591 with

respect to A satisfies 119889120585(120591)119889120591 = 119867(120591 120585(120591)) for all 120591 isin 119873(1205910)

If 119865 = (119891 119892) has anH(A)-differentiable lifting119867 we say thatplanar system (1) is algebrizable and that (2) is an algebrizationof (1) A theorem of existence and uniqueness of solutionsfor differential equations over algebras is proved in [6] anda technique for visualization of solutions is given in [7]

The classical differential equation 119889119908119889119905 = 1199082 has

solutions 119908(119905) = minus(119905 + 119888)minus1 Some differential systems of

autonomous differential equations can be written in this formby using variables in algebras For example the algebrizationof the planar differential system

119889119909

119889119905= 31199092+ 2119909119910 minus 119910

2

119889119910

119889119905= 31199102+ 2119909119910 minus 119909

2

(3)

is the differential equation 119889120585119889120591 = 1205852 over the algebra A

defined by the linear space R2 endowed with the product

(1199091 1199101) (1199092 1199102) = (3119909

11199092

+ (11990911199102+ 11991011199092minus 11991011199102) 311991011199102

+ (11990911199102+ 11991011199092minus 11990911199092))

(4)

The solutions 120585 are given by 120585(120591) = minus(120591 + 119888)minus1 hence the

solutions of the planar system are given by (119909(119905) 119910(119905)) = 120585(119905119890)where 119890 denotes the unit of A Using the first fundamentalrepresentation of A (see Section 2) the following expressionfor the solution (119909(119905) 119910(119905)) of the system is obtained

(119909 (119905) 119910 (119905))

= (minus119905 minus 119910

0

4 (2119905 + 1199090+ 1199100)2

minus119905 minus 1199090

4 (2119905 + 1199090+ 1199100)2)

(119909 (0) 119910 (0)) =minus (1199100 1199090)

4 (1199090+ 1199100)2

(5)

Consider now the planar nonautonomous differentialsystem

119889119909

119889119905= minus

1

1199053minus

2119909

119905+ 1199051199092minus 1199051199102

119889119910

119889119905= minus

2119910

119905+ 2119905119909119910 + 119905119910

2

(6)

whose algebrization is the Riccati differential equation overA

119889120585

119889120591= minus

1

1205913minus

2120585

120591+ 1205911205852 (7)

where A is the algebra defined by R2 and the product(119906 V)(119909 119910) = (119906119909 minus V119910 119909119910 + V119909 + V119910) By the classical Liemethods for solving differential equations (see [8ndash10]) thesolutions of the Riccati equation have the form

120585 (120591) =119862 + 1205912

1205912 (119862 minus 1205912) 119862 = (119886 119887) (8)

The functions (119909(119905) 119910(119905)) = 120585(119905119890) where 119890 is the unit ofA aresolutions of the planar system which can be obtained via thefirst fundamental representation of A

(119909 (119905) 119910 (119905)) = (1198872+ (119886 + 119887 minus 119905

2) (119886 + 119905

2)

(1198862 + 119886119887 + 1198872) 1199052 minus (2119886 + 119887) 1199054 + 1199056

minus2119887

(1198862 + 119886119887 + 1198872) minus (2119886 + 119887) 1199052 + 1199054)

(9)

We consider differential systems having the form (1)where 119891 119892 119880 sub R3 rarr R are 119862

1 functions defined inan open set 119880 The aim of this work is to give a family offunctions 119865 = (119891 119892) having H(A)-differentiable liftings 119867

over some algebra A When they exist the solutions 120585(120591)

of the differential equation 119889120585119889120591 = 119867(120591 120585) over A definesolutions (119909(119905) 119910(119905)) = 120585(119905119890) of system (1) which can beobtained via the first fundamental representation of A

The paper is organized as follows In Section 2 definitionsof algebra algebrizability of planar vector fields and differen-tiability on modules over algebras a characterization of thealgebrizability of planar vector fields and the form of all thequadratic vector fields which are algebrizable are given InSection 3 the definition of algebrizable liftings of functions119901 119868 sub R rarr R2 is presented It is shown that the class ofall of these functions defines an infinite dimensional algebraand the form of a family of these functions 119901 is given InSection 4 it is proved that the solutions of planar systems (1)can be obtained from the solutions of their algebrization atheorem containing conditions under which a planar systemlike (1) which is polynomial is algebrizable is given and itis shown that the class of all the planar systems (1) havingan algebrization (2) defines an infinite dimensional algebraIn Section 5 the case of quadratic systems is considered andtheir algebrizations are given which are Riccati equationsover algebrasThe results presented in Sections 3 4 and 5 arethe main contributions of this paper

Journal of Applied Mathematics 3

2 Algebras and Lorch Analyticity

21 Algebras

Definition 2 (see [11]) An algebra A is a R-linear space E

endowed with a bilinear product A times A rarr A denotedby (119909 119910) 997891rarr 119909119910 which is associative 119909(119910119911) = (119909119910)119911 andcommutative 119909119910 = 119910119909 for all 119909 119910 119911 isin A and has a unit 119890 isin A

satisfying 119890119909 = 119909119890 = 119909 for all 119909 isin A

An element 119886 isin A is called regular if there exists 119886minus1 isin A

called inverse of 119886 such that 119886minus1 sdot 119886 = 119886 sdot119886minus1

= 119890 If 119886 isin A is notregular then 119886 is called singular If 119886 119887 isin A and 119887 is regularthe quotient 119886119887means 119886 sdot 119887

minus1In all the algebras considered in this paper it will be the

case that E = R2 unless otherwise statedConsider an algebra A If 120573 = 119890

1 1198902 is the standard

basis of R2 the product between the elements of 120573 is givenby 119890119894119890119895= sum2

119896=1119888119894119895119896

119890119896 where the coefficients 119888

119894119895119896isin R 119894 119895 119896 isin

1 2 are called structure constants ofAThefirst fundamentalrepresentation of A is the injective linear homomorphism 119877

A rarr 119872(2R) defined by 119877 119890119894997891rarr 119877119894 where 119877

119894is the matrix

whose 119895 119896 entry is 119888119894119896119895 for 119894 = 1 2

22 Differentiability on Algebras In this subsection the defi-nition of Lorch differentiability is recalled which in this paperis calledA-algebrizability orA-differentiability to denote thedependence of the Lorch differential over an algebra A

Let | sdot | be the norm onA defined by |119886| = max119883119877(119886)

119883 isin R2 119883 = 1 (here the vector119883 is represented as a 1times 2

matrix in order for the product119883119877(119886) to make sense) where119877 A rarr 119872(2 119877) is the first fundamental representation ofA and sdot the Euclidean norm in R2 For this norm we have|119886119887| le |119886||119887| for all 119886 119887 isin 119872(2R) Thus we consider in thiswork that every algebraA is a Banach algebra under the norm| sdot |

Definition 3 Let A be an algebra and 119865 119881 sub A rarr A afunction on an open set119881 We say that 119865 isA-algebrizable orA-differentiable on119881 if there exists a function 119865

1015840 119881 sub A rarr

A called the A-derivative of 119865 on 119881 satisfying

limℎrarr0

10038161003816100381610038161003816119865 (119886 + ℎ) minus 119865 (119886) minus 119865

1015840(119886) ℎ

10038161003816100381610038161003816

|ℎ|= 0 (10)

for all 119886 isin 119881 where 1198651015840(119886)ℎ denotes the product inA of 1198651015840(119886)with ℎ

A vector field 119865 119881 sub A rarr A is A-algebrizable on119881 if and only if the Jacobian matrix of 119865 is contained in thefirst fundamental representation of A that is 119869119865(119886) isin 119877(A)

for all 119886 isin 119881 see [5] It can be shown that the notion of A-algebrizability coincides with the holomorphicity when A isthe complex field

A method for determining whether a given planar vectorfield 119865 is algebrizable is the following 119865 is algebrizable ifand only if for some of the following three types of pairs ofmatrices

(I) 1198611= ( 1 0119886 minus1

) 1198612= (0 1

119887 0)

(II) 1198611= ( 1 1198860 minus1

) 1198612= ( 0 01 0

)(III) 119861

1= ( 0 10 0

) 1198612= ( 0 01 0

)

the condition ⟨119861119894 120597119865(119909 119910)120597(119909 119910)⟩ = 0 is satisfied for 119894 = 1 2

and for all (119909 119910) in the domain of definition of 119865 where⟨sdot sdot⟩ denotes the usual inner product in119872(2R) The algebrafor each type of pair of matrices is defined by the followingcorresponding product table of the standard basis vectors1198901 1198902of R2

(I)sdot 1198901

1198902

1198901

1198901

1198902

1198902

1198902

minus1198871198901+ 1198861198902

(II)sdot 1198901

1198902

1198901

minus1198861198901

1198901

1198902

1198901

1198902

(III)sdot 1198901

1198902

1198901

1198901

0

1198902

0 1198902

Consider a planar autonomous system of quadratic ordinarydifferential equations in the variables 119909 and 119910 If this system isalgebrizable for an algebra with Type (I) product then it canbe represented by equations of the form

119889119909

119889119905= 1198860+ (1198872minus 1198861198871) 119909 minus 119887119887

1119910 + (

1

21198874minus 1198861198873)1199092

minus 21198871198873119909119910 minus

1

211988711988741199102

119889119910

119889119905= 1198870+ 1198871119909 + 1198872119910 + 11988731199092+ 1198874119909119910

+ (1

21198861198874minus 1198871198873)1199102

(11)

for some real constants 119886 119887 1198860 1198870 1198871 119887

4 In the case of

algebrizability for an algebra with Type (II) product thesystem is

119889119909

119889119905= 1198860+ 1198861119909 + 1198862119910 minus

1

211988611988631199092+ 1198863119909119910 + 119886

41199102

119889119910

119889119905= 1198870+ (1198861+ 1198861198862) 119910 + (

1

21198863+ 1198861198864)1199102

(12)

for some real constants 119886 1198870 1198860 1198861 1198862 1198863 and 119886

4 For Type

(III) product the system can be represented by equations ofthe form

119889119909

119889119905= 1198860+ 1198861119909 + 11988621199092

119889119910

119889119905= 1198870+ 1198871119910 + 11988721199102

(13)

for some real constants 119886119894 119887119894 119894 = 0 1 2 Moreover conditions

on the components of vector fields 119865 can be given for con-structing scalar functions 120572(119909 119910) which we call algebrizantefactors such that 120572119865 are algebrizable vector fields Inverseintegrating factors (see [12 13]) are constructed for thesevector fields

4 Journal of Applied Mathematics

23 Differentiability onModules over Algebras In this subsec-tion we give the definition of A-differentiability of functionswith domain in A times A and image in A

The Cartesian productAtimesA defines a normedA-modulewith respect to the norm sdot A times A rarr R (119886

1 1198862) =

radic|1198861|2 + |119886

2|2 This norm satisfies

(i) 119883 ge 0 for all 119883 isin A times A and 119883 = 0 if and only if119883 = (0 0)

(ii) 119886119883 le |119886|119883 for all 119886 isin A and119883 isin A times A(iii) 119883 + 119884 le 119883 + 119884 for all119883119884 isin A times A

In the following definition sdot denotes the norm givenabove on the A-module A times A

Definition 4 Let A be an algebra and 119867 Ω sub A times A rarr

A a function where Ω is an open set We say that 119867 isH(A)-differentiable on 119883

0isin Ω if there exists a module

homeomorphism 119872 A times A rarr A which we call thedifferential homomorphism of 119867 at 119883

0 which satisfies the

condition

lim119883rarr1198830

1003817100381710038171003817119867 (119883) minus 119867 (1198830) minus 119872(119883 minus 119883

0)1003817100381710038171003817

1003817100381710038171003817119883 minus 1198830

1003817100381710038171003817

= 0 (14)

We denote 119872 by D119867(1198830) We say that 119867 is H(A)-

differentiable on Ω if 119867 is H(A)-differentiable on all thepoints ofΩ

A function 119867 Ω sub A times A rarr A isH(A)-differentiableat 1198830if and only if the usual Jacobian matrix of 119867 Ω sub

R2 times R2 rarr R2 at 1198830satisfies 119869119867(119883

0) isin 119877(A) times 119877(A)

The differential homomorphismD119867(1198830) is represented by a

matrix in11987212

(A)with respect to the standard basis ofAtimesAwhere119872

12(A) is theA-module of all the matrices of one row

and two columns with entries in A see [14]

3 On Algebrizable Liftings of Functions 119901 119868 sub

R rarr R2

In this section are considered functions 119901 119868 sub R rarr R2

defined in open intervals 119868 and conditions for the existenceof algebrasA andA-algebrizable functions119875 such that 119901(119905) =119875(119905119890) will be determined where 119890 is the unit of A

Definition 5 Let 119901 119868 sub R rarr R2 be a function defined in anopen interval 119868 andA an algebra with unit 119890 We will say that119875 119881 sub R2 rarr R2 is an A-algebrizable lifting of 119901 if (a) 119881 isan open set on which 119875 isA-algebrizable (b) 119905119890 119905 isin 119868 sub 119881and (c) 119901(119905) = 119875(119905119890) for all 119905 isin 119868

As a consequence of the following proposition the familyof all the functions 119901 119868 sub R rarr R2 having algebrizableliftings is an infinite dimensional algebra

Proposition 6 Let A be an algebra and let 119901 119902 119868 sub R rarr

R2 be functions where 119868 is an interval

(a) Every constant function 119901(119905) = 119888 admits the A-algebrizable lifting 119875(120591) = 119888

(b) 119901(119905) = 119905119890 admits the A-algebrizable lifting 119875(120591) = 120591where 119890 isin A is the unit of A

(c) If 119901 and 119902 admit A-algebrizable liftings 119875 and 119876 withrespect to A respectively and 119886 119887 are constants in Athen 119886119901 + 119887119902 and 119901119902 (all products with respect to A)admit algebrizable liftings 119886119875+119887119876 and119875119876 respectively

(d) If 119901 has an A-algebrizable lifting 119875 and 119876 is an A-algebrizable function with Im(119875) sub Dom(119876) then119876 ∘ 119875 is an algebrizable lifting of 119876 ∘ 119901

Proof Identity and constant functions are A-differentiablefor any algebra A Thus (a) and (b) hold

Let 119901 and 119902 be functions with A-algebrizable liftings 119875

being 119876 and 119886 119887 isin A constants The functions 119878 = 119886119875 + 119887119876

and 119879 = 119875119876 are A-algebrizable and satisfy

119878 (119905119890) = 119886119875 (119905119890) + 119887119876 (119905119890) = 119886119901 (119905) + 119887119902 (119905)

119879 (119905119890) = 119875 (119905119890) 119876 (119905119890) = 119901 (119905) 119902 (119905)

(15)

Therefore 119878 and 119879 are algebrizable liftings of 119886119901 + 119887119902 and 119901119902respectively

If 119901 has an A-algebrizable lifting 119875 and 119876 is an A-differentiable function with Im(119875) sub Dom(119876) then 119876 ∘

119875(119905119890) = 119876 ∘ 119901(119905) Therefore 119876 ∘ 119875 is an A-algebrizable liftingof 119876 ∘ 119901

Corollary 7 LetA be an algebraThen the following functionsadmit A-algebrizable liftings polynomial functions rationalfunctions trigonometric functions exponential functions andall of those functions which can be defined by linear com-binations products quotients and compositions of functionsadmitting algebrizable liftings

Every function 119905 997891rarr (ℎ(119905) 119896(119905)) with polynomial com-ponents ℎ(119905) and 119896(119905) has an A-algebrizable lifting Thefollowing proposition gives a wider class of these functions

Proposition 8 Let A be an algebra Any function 119905 997891rarr

(ℎ(119905) 119896(119905)) with components of the form

ℎ (119905) =119886119898

119905119898+

119886119898minus1

119905119898minus1+ sdot sdot sdot +

119886minus1

119905+ 1198860+ 1198861119905 + sdot sdot sdot 119886

119899119905119899

119896 (119905) =119887119898

119905119898+

119887119898minus1

119905119898minus1+ sdot sdot sdot +

119887minus1

119905+ 1198870+ 1198871119905 + sdot sdot sdot 119887

119899119905119899

(16)

119898 119899 isin N has an A-algebrizable lifting An A-algebrizablelifting is given by

120591 997891rarr (119886119898 119887119898)

1

120591119898+ sdot sdot sdot + (119886

minus1 119887minus1)1

120591+ (1198860 1198870)

+ (1198861 1198871) 120591 + sdot sdot sdot + (119886

119899 119887119899) 120591119899

(17)

Proof Consider 119876 given by the above expression then

(119886119896 119887119896) (119905119890)119896= (119886119896 119887119896) 119905119896119890 = (119886

119896119905119896 119887119896119905119896) (18)

holds for 119896 = minus119898 119899 so 119876(119905119890) = (ℎ(119905) 119902(119905)) where 119890 isthe unit ofA Therefore 119876 is anA-algebrizable lifting of 119905 997891rarr(ℎ(119905) 119896(119905))

Journal of Applied Mathematics 5

In particular every function 119901 = (1199011 1199012) 119868 sub R rarr

R2 with quadratic components 1199011(119905) = 119886

0+ 1198861119905 + 11988621199052 and

1199012(119905) = 119887

0+1198871119905+11988721199052 admits theA-algebrizable lifting 119875(120591) =

(1198860 1198870) + (1198861 1198871)120591 + (119886

2 1198872)1205912

4 On Algebrizable Liftings of Planar Systems

Solutions of every algebrizable planar system can be foundby solving an algebrization of the system as it is seen in thefollowing proposition

Proposition 9 If (2) is an algebrization of (1) and 120585(120591) asolution of (2) then (119909(119905) 119910(119905)) = 120585(119905119890) is a solution of (1)where 119890 denotes the unit of the corresponding algebra

Proof Let 120585 be a solution of (2) The derivative of(119909(119905) 119910(119905)) = 120585(119905119890) with respect to 119905 is given by

119889

119889119905(119909 (119905) 119910 (119905)) =

119889

119889119905120585 (119905119890) = [

119889120585 (120591 120585)

119889120591

10038161003816100381610038161003816100381610038161003816120591=(119905119890)

119889120591

119889119905]

= 119867 (120591 120585 (120591))1003816100381610038161003816120591=(119905119890) 119890 = 119867 (119905119890 120585 (119905119890))

= (119891 (119905 119909 (119905) 119910 (119905)) 119892 (119905 119909 (119905) 119910 (119905)))

(19)

Thus (119909(119905) 119910(119905)) is a solution of system (1)

As a consequence of the following proposition the familyof all the functions 119865 Ω sub R3 rarr R2 having algebrizableliftings defines an infinite dimensional algebra

Proposition 10 Let A be an algebra with unit 119890 In thefollowing statements 119865 and119866 denote functions defined on opensets Ω sub R3 and they have values in R2

(a) 119865(119905 119909 119910) = 119888 admits the H(A)-differentiable lifting119867(120591 120585) = 119888

(b) 119865(119905 119909 119910) = 119905119890 admits the H(A)-differentiable lifting119867(120591 120585) = 120591

(c) 119865(119905 119909 119910) = (119909 119910) admits the H(A)-differentiablelifting119867(120591 120585) = 120585

(d) If 119865 and 119866 admit H(A)-differentiable liftings 119867119865and

119867119866 respectively and 119886 119887 are constants inA then 119886119867

119865+

119887119867119866and 119865119866 (all products with respect to A) have

H(A)-differentiable liftings 119886119867119865+ 119887119867119866and 119867

119865119867119866

respectively

(e) Every function 119864 having the form 119864(119905 119909 119910) = 119865(119905 119866(119905

119909 119910)) where 119865 and 119866 admit H(A)-differentiableliftings 119867

119865and 119867

119866 admits an H(A)-differentiable

lifting119867119864given by119867

119864(120591 120585) = 119867

119865(120591119867119866(120591 120585))

(f) Every function 119864 having the form 119864(119905 119909 119910) =

119865(119905 119875(119909 119910)) where 119865 has an H(A)-differentiablelifting119867

119865and 119875 is anA-differentiable function admits

anH(A)-differentiable lifting119867119864(120591 120585) = 119867

119865(120591 119875(120585))

Proof The proofs of (a) (b) and (c) are trivial Let 119867119865and

119867119866be the algebrizable liftings of 119865 and 119866 and then119867

119865+119867119866

119867119865119867119866 and119867

119865119867119866are A-algebrizable and

(i) (119886119867119865+119887119867119866)(119905119890 119909 119910) = 119886119867

119865(119905119890 119909 119910)+119887119867

119866(119905119890 119909 119910) =

119886119865(119905 119909 119910) + 119886119866(119905 119909 119910)

(ii) (119867119865119867119866)(119905119890 119909 119910) = 119867

119865(119905119890 119909 119910)119867

119866(119905119890 119909 119910) =

119865(119905 119909 119910)119866(119905 119909 119910)

(iii) 119867119864(119905119890 119909 119910) = 119867

119865(119905119890119867

119866(119905119890 119909 119910)) = 119865(119905 119866(119905 119909 119910))

(iv) 119867119864(119905119890 119909 119910) = 119867

119865(119905119890 119875(119890 119910)) = 119865(119905 119875(119909 119910))

Thus the proof is complete

Corollary 11 Let A be an algebra The following func-tions admitH(A)-differentiable liftings polynomial functionsrational functions trigonometric functions exponential func-tions and all of those functions which can be defined bylinear combinations products quotients and compositions offunctions admittingH(A)-differentiable liftings

Given a function 119865(119905 119909 119910) = (119891(119905 119909 119910) 119892(119905 119909 119910))where 119891 119892 are polynomial functions of the variables 119909 119910the goal of the paper is to determine if 119865 has an H(A)-differentiable lifting As a consequence of the followingtheorem every function 119865 which is polynomial of thevariables 119905 119909 and 119910 has an H(A)-differentiable liftingwhen (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-differentiable forall 119905

Theorem 12 Let A be an algebra with unit 119890 and 119865

Ω sub R3 rarr R2 119865 = (119891 119892) where 119891 = sum119898

119896=0119891119896

119892 = sum119898

119896=0119892119896 and 119891

119896(119905 119909 119910) 119892

119896(119905 119909 119910) are homogeneous

polynomials of degree 119896 in the variables 119909 119910 and Ω = 119868 times

R2 for an open interval 119868 Then the following statements areequivalent

(a) 119865 has anH(A)-differentiable lifting119867

(b) The map (119909 119910) 997891rarr 119865(119905 119909 119910) is A-algebrizable forall 119905 isin 119868 and the functions ℎ

119896given by ℎ

119896(119905) =

(119891119896(119905 119890) 119892

119896(119905 119890)) have A-algebrizable liftings for 119896 =

0 1 119898

(c) 119867(120591 120585) = sum119898

119896=0119867119896(120591)120585119896 is an H(A)-differentiable

lifting of 119865 where 119867119896(120591) are A-algebrizable liftings of

ℎ119896(119905) = (119891

119896(119905 119890) 119892

119896(119905 119890))

Proof Obviously (a) implies (b) and (c) implies (a) We nowshow that (b) implies (c) Suppose that119891(119905 119909 119910) and119892(119905 119909 119910)

are homogenous polynomials 119891(119905 119909 119910) = sum119899

119896=0119901119896(119905)119909119896119910119899minus119896

and 119892(119905 119909 119910) = sum119899

119896=0119902119896(119905)119909119896119910119899minus119896 of the variables 119909 119910 defined

6 Journal of Applied Mathematics

on the setΩ = 119868 timesR2 as above Taking the partial derivativesof 119865 with respect to 119909 and 119910 we obtain

120597 (119891 119892)

120597 (119909 119910)(119905 119909 119910)

= (

119899

sum

119896=1

119896119901119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119901119896minus1

(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

119896119902119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119902119896minus1

(119905) 119909119896minus1

119910119899minus119896

)

= (

119899

sum

119896=1

119896119901119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119901119896minus1

(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

119896119902119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119902119896minus1

(119905) 119909119896minus1

119910119899minus119896

)

=

119899

sum

119896=1

119909119896minus1

119910119899minus119896

(119896119901119896(119905) (119899 minus 119896 + 1) 119901

119896minus1(119905)

119896119902119896(119905) (119899 minus 119896 + 1) 119902

119896minus1(119905)

)

(20)

Let 119877 be the first fundamental representation of A Since(119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for all 119905 isin 119868then (120597(119891 119892)120597(119909 119910))(119905 119909 119910) isin 119877(A) for all (119905 119909 119910) isin ΩThus

120597 (119891 119892)

120597 (119909 119910)(119905 0 1) = (

1199011(119905) 119899119901

0(119905)

1199021(119905) 119899119902

0(119905)

) isin 119877 (A) (21)

for all 119905 isin 119868 and then

120597 (119891 119892)

120597 (119909 119910)(119905 119909 119910) minus 119910

119899minus1(1199011(119905) 119899119901

0(119905)

1199021(119905) 119899119902

0(119905)

)

=

119899

sum

119896=2

119909119896minus1

119910119899minus119896

(119896119901119896(119905) (119899 minus 119896 + 1) 119901

119896minus1(119905)

119896119902119896(119905) (119899 minus 119896 + 1) 119902

119896minus1(119905)

)

(22)

is in 119877(A) for all (119905 119909 119910) isin Ω If 119909(119904) = 1119904 and 119910(119904) =(119899minus2)radic119904

then

lim119904rarrinfin

[120597 (119891 119892)

120597 (119909 119910)(119905 119909 (119904) 119910 (119904))

minus [119910 (119904)]119899minus1

(1199011(119905) 119899119901

0(119905)

1199021(119905) 119899119902

0(119905)

)]

= (1199012(119905) (119899 minus 1) 119901

1(119905)

1199022(119905) (119899 minus 1) 119902

1(119905)

) isin 119877 (A)

(23)

Following the same idea for 119896 = 1 2 119899

(119896119901119896(119905) (119899 minus 119896 + 1) 119901

119896minus1(119905)

119896119902119896(119905) (119899 minus 119896 + 1) 119902

119896minus1(119905)

) isin 119877 (A) (24)

If (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for analgebra A with Type (I) product (given in Section 2) then

119896119901119896+ 119886119896119902

119896minus (119899 minus 119896 + 1) 119902

119896minus1= 0

119887119896119902119896+ (119899 minus 119896 + 1) 119901

119896minus1= 0

(25)

or equivalently

(119902119896minus1

119901119896minus1

) = (119896119901119896+ 119886119896119902

119896

119899 minus 119896 + 1 minus

119887119896119902119896

119899 minus 119896 + 1) (26)

for 119896 = 1 119899 Thus the functions 119901119896 119902119896

aredetermined by 119901

119899 119902119899

for 119896 = 0 1 119899 minus 1Since (119891(119905 1 0) 119892(119905 1 0)) = (119901

119899(119905) 119902119899(119905)) then

(119891(119905 119909 119910) 119892(119905 119909 119910)) = (119901119899(119905) 119902119899(119905))(119909 119910)

119899Therefore 119865 has anH(A)-differentiable lifting 119867(120591 120585) =

119867119899(120591)120585119899 where 119867

119899(120591) is an A-algebrizable lifting of 119905 997891rarr

(119901119899(119905) 119902119899(119905))

If (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for analgebra A with Type (II) product (given in Section 2) then

119896119901119896+ 119886 (119899 minus 119896 + 1) 119901

119896minus1minus (119899 minus 119896 + 1) 119902

119896minus1= 0

119896119902119896= 0

(27)

or equivalently

(119901119896 119902119896) = (minus

119886 (119899 minus 119896 + 1) 119901119896minus1

119896 0) (28)

for 119896 = 1 119899 Thus 119902119896

= 0 and 119901119896is determined by 119901

0

for 119896 = 1 2 119899 Since (119891(119905 0 1) 119892(119905 0 1)) = (1199010(119905) 1199020(119905))

then (119891(119905 119909 119910) 119892(119905 119909 119910)) = (1199010(119905) 1199020(119905))(119909 119910)

119899Therefore 119865 has anH(A)-differentiable lifting 119867(120591 120585) =

119867119899(120591)120585119899 where 119867

119899(120591) is an A-algebrizable lifting of 119905 997891rarr

(1199010(119905) 1199020(119905))

If (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for analgebra A with Type (III) product (given in Section 2) then(119899 minus 119896 + 1)119901

119896minus1= 0 and 119896119902

119896= 0 that is 119901

119896minus1= 0 119902

119896= 0

for 119896 = 1 119899 Thus 119902119896

= 0 and 119901119896is determined by 119901

0

for 119896 = 1 2 119899 Since (119891(119905 0 1) 119892(119905 0 1)) = (119901119899(119905) 1199020(119905))

then (119891(119905 119909 119910) 119892(119905 119909 119910)) = (119901119899(119905) 1199020(119905))(119909 119910)

119899Therefore 119865 has anH(A)-differentiable lifting 119867(120591 120585) =

119867119899(120591)120585119899 where 119867

119899(120591) is an A-algebrizable lifting of 119905 997891rarr

(119901119899(119905) 1199020(119905))

Thus if 119891 and 119892 are homogenous polynomial of degree119899 in the variables 119909 and 119910 then 119867(120591 120585) = 119867

119899(120591)120585119899 in each

of the cases of algebras defined by products of Types (I)(II) and (III) given in Section 2 Since a polynomial function(119905 119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) in the variables 119909 and 119910 canbe seen as the finite addition of homogenous polynomial inthe variables 119909 and 119910 by (d) of Proposition 10 (b) implies(c)

Example 13 Consider the planar system (6) then

119865 (119905 119909 119910)

= (minus1

1199053minus

2119909

119905+ 1199051199092minus 1199051199102 minus

2119910

119905+ 2119905119909119910 + 119905119910

2)

(29)

The Jacobian 120597119865120597(119909 119910) is given by

120597119865 (119905 119909 119910)

120597 (119909 119910)= (

minus2

119905+ 2119905119909 minus2119905119910

2119905119910 minus2

119905+ 2119905119909 + 2119905119910

) (30)

Journal of Applied Mathematics 7

Thus 120597119865120597(119909 119910) is orthogonal to the matrices

1198611= (

1 0

1 minus1)

1198612= (

0 1

1 0)

(31)

for all (119909 119910) that is ⟨120597119865120597(119909 119910) 119861119894⟩ = 0 for 119894 = 1 2 Themap

(119909 119910) 997891rarr 119865(119905 119909 119910) isA-algebrizable for an algebra of Type (I)with constants 119886 = 1 and 119887 = 1 see Section 2 The function 119865

can be written as

119865 (119905 119909 119910) = (minus1

1199053 0) + (minus

2

119905119909 minus

2

119905119910)

+ (1199051199092minus 1199051199102 1199051199102)

(32)

Thus the functions 119901119894and 119902

119894of Theorem 12 are given by

1199010(119905 119909 119910) = minus1119905

3 1199020(119905 119909 119910) = 0 119901

1(119905 119909 119910) = minus(2119905)119909

1199021(119905 119909 119910) = minus(2119905)119910 119901

2(119905 119909 119910) = 119905119909

2minus 1199051199102 and 119902

2(119905 119909 119910) =

1199051199102The unit 119890 of A is 119890 = (1 0) and then ℎ

0(119905) = (minus1119905

3 0)

ℎ1(119905) = (minus2119905 0) and ℎ

2(119905) = (119905 0) have A-algebrizable

liftings 1198670(120591) = minus1120591

3 1198671(120591) = minus2120591 and 119867

2(120591) = 120591

respectivelyFrom the form of 119865(119905 119909119890) = (minus1119905

3minus (2119905)119909 minus 119905119909

2)119890 119867

must be

119867(120591 120585) = minus1

1205913minus

2

120591120585 + 120591120585

2 (33)

5 The Case of Second-Degree Polynomials inthe Variables 119905 119909 and 119910

If 119891 and 119892 in (1) are quadratic polynomials in three variables119905 119909 and 119910 andA is an algebra with respect to which the map(119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable it will beshowed that the H(A)-differentiable lifting 119867 of 119865 = (119891 119892)

is a polynomial in two variables ofA Under these conditions(1) has an algebrization which is a Riccati equation over Ahaving the form

119889120585

119889120591= 119875 (120591) + 119876 (120591) 120585 + 119877 (120591) 120585

2 (34)

where119875119876 and119877 are polynomials overA of degrees two oneand zero respectively

Consider system (1) where 119891 119892 Ω sub R3 rarr R2 aresecond-degree polynomials of three variables 119905 119909 and 119910 thatis

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 1198863119910 + 11988641199052+ 1198865119905119909

+ 1198866119905119910 + 119886

71199092+ 1198868119909119910 + 119886

91199102

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198872119909 + 1198873119910 + 11988741199052+ 1198875119905119909 + 119887

6119905119910

+ 11988771199092+ 1198878119909119910 + 119887

91199102

(35)

All the quadratic vector fields which are algebrizable withrespect to algebras with Type (I) products have the form

119889119909

119889119905= 1198600+ (1198612minus 1198861198611) 119909 minus 119887119861

1119910 + (

1

21198614minus 1198861198613)1199092

minus 21198871198613119909119910 minus

1

211988711986141199102

119889119910

119889119905= 1198610+ 1198611119909 + 1198612119910 + 11986131199092+ 1198614119909119910

+ (1

21198861198614minus 1198871198613)1199102

(36)

where 119886 119887 1198600 1198610 1198611 119861

5are real constants

see Section 22 The algebrizability of (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) with respect to algebras with Type(I) products can be verified by considering 119905 as a constant

The following theorems give conditions that characterizethe algebrizability of planar systems like (1) when 119891 and119892 are quadratic polynomials The algebrizability of nonau-tonomous quadratic systems with respect to algebras withType (I) products is given in the following theorem

Theorem14 LetA be an algebrawith Type (I) product definedby constants 119886 and 119887 and 119891 119892 the polynomials (35) Thefollowing statements are equivalent

(1) The map 119865 (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 have the form

119891 (119905 119909 119910) = 1198860+ 1198861119905 + (1198873minus 1198861198872) 119909 minus 119887119887

2119910 + 11988641199052

+ (1198876minus 1198861198875) 119905119909 minus 119887119887

5119905119910

+ (1198878

2minus 1198861198877)1199092minus 21198871198877119909119910 minus

1198871198878

21199102

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198872119909 + 1198873119910 + 11988741199052+ 1198875119905119909 + 119887

6119905119910

+ 11988771199092+ 1198878119909119910 + (

1198861198878

2minus 1198871198877)1199102

(37)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (38)

is anH(A)-differentiable lifting of 119865 where

1198600= (1198860 1198870)

1198601= (1198861 1198871)

1198602= (1198873minus 1198861198872 1198872)

1198603= (1198864 1198874)

1198604= (1198876minus 1198861198875 1198875)

1198605= (

1198878

2minus 1198861198877 1198877)

(39)

8 Journal of Applied Mathematics

Proof Writing119892(119905 119909 119910) = 1198610+1198611119909+1198612119910+11986131199092+1198614119909119910+119861

51199102

yields

1198610= 1198870+ 1198871119905 + 11988741199052

1198611= 1198872+ 1198875119905

1198612= 1198873+ 1198876119905

1198613= 1198877

1198614= 1198878

1198615= 1198879

(40)

The function (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) isA-algebrizableif and only if

119891 (119905 119909 119910) = 1198600+ (1198612minus 1198861198611) 119909 minus 119887119861

1119910

+ (1

21198614minus 1198861198613)1199092minus 2119887119861

3119909119910 minus

1

211988711986141199102

(41)

where 1198600

= 1198860+ 1198861119905 + 11988641199052 and 119887

9= 11988611988782 minus 119887119887

7 Thus

statements (1) and (2) are equivalentThe function 119867 is polynomial in the variables 120591 and 120585 of

A hence 119867 is H(A)-differentiable 119867 satisfies 119867(119905119890 119909 119910) =

(119891(119905 119909 119910) 119892(119905 119909 119910)) where 119890 is the unit of A So 119867 is anH(A)-differentiable lifting of 119865 Thus statement (2) impliesstatement (3)

Since 119867 is H(A)-differentiable and 119865(119905 119909 119910) =

119867(119905119890 119909 119910) then (119909 119910) 997891rarr 119865(119905 119909 119910) is A-algebrizablefor all 119905 Thus statement (3) implies statement (1)

All the quadratic vector fields which are algebrizable withrespect to algebras with Type (II) products have the form

= 1198600+ 1198601119909 + 119860

2119910 minus

1

211988611986031199092+ 1198603119909119910 + 119860

41199102

119910 = 1198610+ (1198601+ 1198861198602) 119910 + (

1

21198603+ 1198861198604)1199102

(42)

where 119886 1198600 119860

4 1198610are real constants see Section 22

The algebrizability of (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) withrespect to algebras with Type (II) products can be verifiedby considering 119905 as a constant

The algebrizability of nonautonomous quadratic systemswith respect to algebras with Type (II) products is given in thefollowing theorem

Theorem 15 Let A be an algebra with Type (II) productdefined by the constant 119886 and let 119891 119892 be the polynomials (35)The following statements are equivalent

(1) The map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 are given by

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 1198863119910 + 11988641199052+ 1198865119905119909

+ 1198866119905119910 minus

1

211988611988681199092+ 1198868119909119910 + 119886

91199102

119892 (119905 119909 119910) = 1198870+ 1198871119905 + (1198862+ 1198861198863) 119910 + 119887

41199052

+ (1198865+ 1198861198866) 119905119910 + (

1198868

2+ 1198861198869)1199102

(43)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (44)

is anH(A)-differentiable lifting of 119865 where

1198600= (1198860 1198870)

1198601= (1198861 1198871)

1198602= (1198863 1198862+ 1198861198863)

1198603= (1198864 1198874)

1198604= (1198866 1198865+ 1198861198866)

1198605= (11988691198868

2+ 1198861198869)

(45)

Proof Writing 119891(119905 119909 119910) = 1198600+ 1198601119909 + 119860

2119910 minus (12)119886119860

31199092+

1198603119909119910 + 119860

41199102 yields

1198600= 1198860+ 1198861119905 + 11988641199052

1198601= 1198862+ 1198865119905

1198602= 1198863+ 1198866119905

1198603= 1198868

1198604= 1198869

(46)

and the map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizableif and only if

119892 (119905 119909 119910) = 1198610+ (1198601+ 1198861198602) 119910 + (

1

21198603+ 1198861198604)1199102 (47)

where 1198610= 1198870+ 1198871119905 + 11988741199052 Thus statements (1) and (2) are

equivalentThe rest of the proof is similar to that of Theorem 14

All the quadratic vector fields which are differentiablewith respect to algebras with Type (III) products have theform

= 1198600+ 1198601119909 + 119860

21199092

119910 = 1198610+ 1198611119910 + 11986121199102

(48)

Journal of Applied Mathematics 9

where 1198600 1198610 1198601 1198611 1198602 1198612

are real constantssee Section 22 The algebrizability of (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) with respect to algebras with Type(III) products can be verified by considering 119905 as a constant

The algebrizability of nonautonomous quadratic systemswith respect to algebras with Type (III) products is given inthe following theorem

Theorem 16 Let A be an algebra with Type (III) productdefined by the constant 119886 and let 119891 119892 be the polynomials (35)The following statements are equivalent

(1) The map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 are given by

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 11988641199052+ 1198865119905119909 + 119886

71199092

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198873119910 + 11988741199052+ 1198876119905119910 + 119887

91199102

(49)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (50)

is an H(A)-differentiable lifting of 119865 where 1198600

=

(1198860 1198870) 1198601

= (1198861 1198871) 1198602

= (1198862 1198873) 1198603

= (1198864 1198874)

1198604= (1198865 1198876) and 119860

5= (1198867 1198879)

Proof Function (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable if and only if

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 11988641199052+ 1198865119905119909 + 119886

71199092

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198873119910 + 11988741199052+ 1198876119905119910 + 119887

91199102

(51)

Thus first and second statements are equivalents The func-tion119867 given by119867(120591 120585) = 119860

0+1198601120591+1198602120585+11986031205912+1198604120591120585+119860

51205852

where1198600= (1198860 1198870)1198601= (1198861 1198871)1198602= (1198862 1198873)1198603= (1198864 1198874)

1198604= (1198865 1198876) 1198605= (1198867 1198879) 120591 = (119903 119904) and 120585 = (119909 119910) satisfies

119867(119905119890 119909 119910) = (119891(119905 119909 119910) 119892(119905 119909 119910)) Thus second and thirdstatements are equivalents

By Theorems 14 15 and 16 an algebrization of quadraticsystems is a Riccati equation over an algebra 119889120585119889120591 = 119875(120591) +

119876(120591)120585 + 119877(120591)1205852 where 119875(120591) = 119860

0+ 1198601120591 + 119860

31205912 119876(120591) =

1198602+ 1198604120591 and 119877(120591) = 119860

5 In the following example is

given a nonautonomous quadratic system for the which analgebrization is found by usingTheorem 14

Example 17 Consider system (1) given by

119891 (119905 119909 119910) = 1 + 7119905 minus 119909 minus 3119910 + 51199052minus 119905119909 minus 3119905119910 minus 6119909119910

minus 61199102

119892 (119905 119909 119910) = 1 + 119905 + 119909 + 119910 + 1199052+ 119905119909 + 119905119910 + 119909

2+ 4119909119910

+ 1199102

(52)

The matrices

1198611= (

1 0

2 minus1)

1198612= (

0 1

3 0)

(53)

satisfy ⟨119861119894 120597(119891 119892)120597(119909 119910)⟩ = 0 for 119894 = 1 2Thus 119865 (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for an algebra A withType (I) product defined by the constants 119886 = 2 and 119887 = 3

The conditions of Theorem 14 are satisfied then theH(A)-differentiable lifting119867 of 119865 can be written as

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (54)

where 1198600= (1 1) 119860

1= (7 1) 119860

2= (minus1 1) 119860

3= (5 1)

1198604= (minus1 1) and 119860

5= (0 1)

Disclosure

The authors declare having no financial affiliation with anyorganization regarding the material discussed here

Conflict of Interests

The authors declare that there is no conflict of interestsconcerning this textThe issues discussed in this paper do nothave any secondary interest for any of the authors

Acknowledgments

The authors wish to acknowledge the support of GrantsPromep103513 and CB-2010150532 Conacyt

References

[1] E K Blum ldquoA theory of analytic functions in Banach algebrasrdquoTransactions of the American Mathematical Society vol 78 no2 pp 343ndash370 1955

[2] P W Ketchum ldquoAnalytic functions of hypercomplex variablesrdquoTransactions of the American Mathematical Society vol 30 no4 pp 641ndash641 1928

[3] E R Lorch ldquoThe theory of analytic functions in normed abelianvector ringsrdquo Transactions of the American Mathematical Soci-ety vol 54 no 3 pp 414ndash425 1943

[4] J A Ward ldquoA theory of analytic functions in linear associativealgebrasrdquo Duke Mathematical Journal vol 7 no 1 pp 233ndash2481940

[5] J A Ward ldquoFrom generalized Cauchy-Riemann equationsto linear algebrardquo Proceedings of the American MathematicalSociety vol 4 no 3 pp 456ndash461 1953

[6] E Lopez-Gonzalez ldquoDifferential equations over algebrasrdquoAdvances and Applications in Mathematical Sciences vol 8 no2 pp 189ndash214 2011

[7] A Alvarez-Parrilla M E Frıas-Armenta E Lopez-Gonzalezand C Yee-Romero ldquoOn solving systems of autonomousordinary differential equations by reduction to a variable of analgebrardquo International Journal of Mathematics and Mathemati-cal Sciences vol 2012 Article ID 753916 21 pages 2012

10 Journal of Applied Mathematics

[8] E J Wilczynski ldquoReview Abraham Cohen an introductionto the lie theory of one-parameter groups with applicationsto the solution of differential equationsrdquo Bulletin of the Amer-ican Mathematical Society vol 18 no 10 pp 514ndash515 1912httpprojecteuclidorgeuclidbams1183421829

[9] J M PageOrdinary Differential Equations with an Introductionto Liersquos Theory or the Group of One Parameter MacmillanPublishers London UK 1897

[10] R A Steinhour The truth about lie symmetries solvingdifferential equations with symmetry methods [SeniorIndependent StudyTheses] 2013 httpopenworkswoostereduindependentstudy949

[11] R Pierce Associative Algebras Springer New York NY USA1982

[12] K I T Al-Dosary ldquoInverse integrating factor for classes of pla-nar differential systemsrdquo International Journal of MathematicalAnalysis vol 4 no 29ndash32 pp 1433ndash1446 2010

[13] I A Garcıa and M Grau ldquoA Survey on the inverse integratingfactorrdquo Qualitative Theory of Dynamical Systems vol 9 no 1-2pp 115ndash166 2010

[14] W Brown Matrices over Commutative Rings Monographs andTextbooks in Pure and Applied Mathematics Marcel Dekker1992

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

2 Algebras and Lorch Analyticity

21 Algebras

Definition 2 (see [11]) An algebra A is a R-linear space E

endowed with a bilinear product A times A rarr A denotedby (119909 119910) 997891rarr 119909119910 which is associative 119909(119910119911) = (119909119910)119911 andcommutative 119909119910 = 119910119909 for all 119909 119910 119911 isin A and has a unit 119890 isin A

satisfying 119890119909 = 119909119890 = 119909 for all 119909 isin A

An element 119886 isin A is called regular if there exists 119886minus1 isin A

called inverse of 119886 such that 119886minus1 sdot 119886 = 119886 sdot119886minus1

= 119890 If 119886 isin A is notregular then 119886 is called singular If 119886 119887 isin A and 119887 is regularthe quotient 119886119887means 119886 sdot 119887

minus1In all the algebras considered in this paper it will be the

case that E = R2 unless otherwise statedConsider an algebra A If 120573 = 119890

1 1198902 is the standard

basis of R2 the product between the elements of 120573 is givenby 119890119894119890119895= sum2

119896=1119888119894119895119896

119890119896 where the coefficients 119888

119894119895119896isin R 119894 119895 119896 isin

1 2 are called structure constants ofAThefirst fundamentalrepresentation of A is the injective linear homomorphism 119877

A rarr 119872(2R) defined by 119877 119890119894997891rarr 119877119894 where 119877

119894is the matrix

whose 119895 119896 entry is 119888119894119896119895 for 119894 = 1 2

22 Differentiability on Algebras In this subsection the defi-nition of Lorch differentiability is recalled which in this paperis calledA-algebrizability orA-differentiability to denote thedependence of the Lorch differential over an algebra A

Let | sdot | be the norm onA defined by |119886| = max119883119877(119886)

119883 isin R2 119883 = 1 (here the vector119883 is represented as a 1times 2

matrix in order for the product119883119877(119886) to make sense) where119877 A rarr 119872(2 119877) is the first fundamental representation ofA and sdot the Euclidean norm in R2 For this norm we have|119886119887| le |119886||119887| for all 119886 119887 isin 119872(2R) Thus we consider in thiswork that every algebraA is a Banach algebra under the norm| sdot |

Definition 3 Let A be an algebra and 119865 119881 sub A rarr A afunction on an open set119881 We say that 119865 isA-algebrizable orA-differentiable on119881 if there exists a function 119865

1015840 119881 sub A rarr

A called the A-derivative of 119865 on 119881 satisfying

limℎrarr0

10038161003816100381610038161003816119865 (119886 + ℎ) minus 119865 (119886) minus 119865

1015840(119886) ℎ

10038161003816100381610038161003816

|ℎ|= 0 (10)

for all 119886 isin 119881 where 1198651015840(119886)ℎ denotes the product inA of 1198651015840(119886)with ℎ

A vector field 119865 119881 sub A rarr A is A-algebrizable on119881 if and only if the Jacobian matrix of 119865 is contained in thefirst fundamental representation of A that is 119869119865(119886) isin 119877(A)

for all 119886 isin 119881 see [5] It can be shown that the notion of A-algebrizability coincides with the holomorphicity when A isthe complex field

A method for determining whether a given planar vectorfield 119865 is algebrizable is the following 119865 is algebrizable ifand only if for some of the following three types of pairs ofmatrices

(I) 1198611= ( 1 0119886 minus1

) 1198612= (0 1

119887 0)

(II) 1198611= ( 1 1198860 minus1

) 1198612= ( 0 01 0

)(III) 119861

1= ( 0 10 0

) 1198612= ( 0 01 0

)

the condition ⟨119861119894 120597119865(119909 119910)120597(119909 119910)⟩ = 0 is satisfied for 119894 = 1 2

and for all (119909 119910) in the domain of definition of 119865 where⟨sdot sdot⟩ denotes the usual inner product in119872(2R) The algebrafor each type of pair of matrices is defined by the followingcorresponding product table of the standard basis vectors1198901 1198902of R2

(I)sdot 1198901

1198902

1198901

1198901

1198902

1198902

1198902

minus1198871198901+ 1198861198902

(II)sdot 1198901

1198902

1198901

minus1198861198901

1198901

1198902

1198901

1198902

(III)sdot 1198901

1198902

1198901

1198901

0

1198902

0 1198902

Consider a planar autonomous system of quadratic ordinarydifferential equations in the variables 119909 and 119910 If this system isalgebrizable for an algebra with Type (I) product then it canbe represented by equations of the form

119889119909

119889119905= 1198860+ (1198872minus 1198861198871) 119909 minus 119887119887

1119910 + (

1

21198874minus 1198861198873)1199092

minus 21198871198873119909119910 minus

1

211988711988741199102

119889119910

119889119905= 1198870+ 1198871119909 + 1198872119910 + 11988731199092+ 1198874119909119910

+ (1

21198861198874minus 1198871198873)1199102

(11)

for some real constants 119886 119887 1198860 1198870 1198871 119887

4 In the case of

algebrizability for an algebra with Type (II) product thesystem is

119889119909

119889119905= 1198860+ 1198861119909 + 1198862119910 minus

1

211988611988631199092+ 1198863119909119910 + 119886

41199102

119889119910

119889119905= 1198870+ (1198861+ 1198861198862) 119910 + (

1

21198863+ 1198861198864)1199102

(12)

for some real constants 119886 1198870 1198860 1198861 1198862 1198863 and 119886

4 For Type

(III) product the system can be represented by equations ofthe form

119889119909

119889119905= 1198860+ 1198861119909 + 11988621199092

119889119910

119889119905= 1198870+ 1198871119910 + 11988721199102

(13)

for some real constants 119886119894 119887119894 119894 = 0 1 2 Moreover conditions

on the components of vector fields 119865 can be given for con-structing scalar functions 120572(119909 119910) which we call algebrizantefactors such that 120572119865 are algebrizable vector fields Inverseintegrating factors (see [12 13]) are constructed for thesevector fields

4 Journal of Applied Mathematics

23 Differentiability onModules over Algebras In this subsec-tion we give the definition of A-differentiability of functionswith domain in A times A and image in A

The Cartesian productAtimesA defines a normedA-modulewith respect to the norm sdot A times A rarr R (119886

1 1198862) =

radic|1198861|2 + |119886

2|2 This norm satisfies

(i) 119883 ge 0 for all 119883 isin A times A and 119883 = 0 if and only if119883 = (0 0)

(ii) 119886119883 le |119886|119883 for all 119886 isin A and119883 isin A times A(iii) 119883 + 119884 le 119883 + 119884 for all119883119884 isin A times A

In the following definition sdot denotes the norm givenabove on the A-module A times A

Definition 4 Let A be an algebra and 119867 Ω sub A times A rarr

A a function where Ω is an open set We say that 119867 isH(A)-differentiable on 119883

0isin Ω if there exists a module

homeomorphism 119872 A times A rarr A which we call thedifferential homomorphism of 119867 at 119883

0 which satisfies the

condition

lim119883rarr1198830

1003817100381710038171003817119867 (119883) minus 119867 (1198830) minus 119872(119883 minus 119883

0)1003817100381710038171003817

1003817100381710038171003817119883 minus 1198830

1003817100381710038171003817

= 0 (14)

We denote 119872 by D119867(1198830) We say that 119867 is H(A)-

differentiable on Ω if 119867 is H(A)-differentiable on all thepoints ofΩ

A function 119867 Ω sub A times A rarr A isH(A)-differentiableat 1198830if and only if the usual Jacobian matrix of 119867 Ω sub

R2 times R2 rarr R2 at 1198830satisfies 119869119867(119883

0) isin 119877(A) times 119877(A)

The differential homomorphismD119867(1198830) is represented by a

matrix in11987212

(A)with respect to the standard basis ofAtimesAwhere119872

12(A) is theA-module of all the matrices of one row

and two columns with entries in A see [14]

3 On Algebrizable Liftings of Functions 119901 119868 sub

R rarr R2

In this section are considered functions 119901 119868 sub R rarr R2

defined in open intervals 119868 and conditions for the existenceof algebrasA andA-algebrizable functions119875 such that 119901(119905) =119875(119905119890) will be determined where 119890 is the unit of A

Definition 5 Let 119901 119868 sub R rarr R2 be a function defined in anopen interval 119868 andA an algebra with unit 119890 We will say that119875 119881 sub R2 rarr R2 is an A-algebrizable lifting of 119901 if (a) 119881 isan open set on which 119875 isA-algebrizable (b) 119905119890 119905 isin 119868 sub 119881and (c) 119901(119905) = 119875(119905119890) for all 119905 isin 119868

As a consequence of the following proposition the familyof all the functions 119901 119868 sub R rarr R2 having algebrizableliftings is an infinite dimensional algebra

Proposition 6 Let A be an algebra and let 119901 119902 119868 sub R rarr

R2 be functions where 119868 is an interval

(a) Every constant function 119901(119905) = 119888 admits the A-algebrizable lifting 119875(120591) = 119888

(b) 119901(119905) = 119905119890 admits the A-algebrizable lifting 119875(120591) = 120591where 119890 isin A is the unit of A

(c) If 119901 and 119902 admit A-algebrizable liftings 119875 and 119876 withrespect to A respectively and 119886 119887 are constants in Athen 119886119901 + 119887119902 and 119901119902 (all products with respect to A)admit algebrizable liftings 119886119875+119887119876 and119875119876 respectively

(d) If 119901 has an A-algebrizable lifting 119875 and 119876 is an A-algebrizable function with Im(119875) sub Dom(119876) then119876 ∘ 119875 is an algebrizable lifting of 119876 ∘ 119901

Proof Identity and constant functions are A-differentiablefor any algebra A Thus (a) and (b) hold

Let 119901 and 119902 be functions with A-algebrizable liftings 119875

being 119876 and 119886 119887 isin A constants The functions 119878 = 119886119875 + 119887119876

and 119879 = 119875119876 are A-algebrizable and satisfy

119878 (119905119890) = 119886119875 (119905119890) + 119887119876 (119905119890) = 119886119901 (119905) + 119887119902 (119905)

119879 (119905119890) = 119875 (119905119890) 119876 (119905119890) = 119901 (119905) 119902 (119905)

(15)

Therefore 119878 and 119879 are algebrizable liftings of 119886119901 + 119887119902 and 119901119902respectively

If 119901 has an A-algebrizable lifting 119875 and 119876 is an A-differentiable function with Im(119875) sub Dom(119876) then 119876 ∘

119875(119905119890) = 119876 ∘ 119901(119905) Therefore 119876 ∘ 119875 is an A-algebrizable liftingof 119876 ∘ 119901

Corollary 7 LetA be an algebraThen the following functionsadmit A-algebrizable liftings polynomial functions rationalfunctions trigonometric functions exponential functions andall of those functions which can be defined by linear com-binations products quotients and compositions of functionsadmitting algebrizable liftings

Every function 119905 997891rarr (ℎ(119905) 119896(119905)) with polynomial com-ponents ℎ(119905) and 119896(119905) has an A-algebrizable lifting Thefollowing proposition gives a wider class of these functions

Proposition 8 Let A be an algebra Any function 119905 997891rarr

(ℎ(119905) 119896(119905)) with components of the form

ℎ (119905) =119886119898

119905119898+

119886119898minus1

119905119898minus1+ sdot sdot sdot +

119886minus1

119905+ 1198860+ 1198861119905 + sdot sdot sdot 119886

119899119905119899

119896 (119905) =119887119898

119905119898+

119887119898minus1

119905119898minus1+ sdot sdot sdot +

119887minus1

119905+ 1198870+ 1198871119905 + sdot sdot sdot 119887

119899119905119899

(16)

119898 119899 isin N has an A-algebrizable lifting An A-algebrizablelifting is given by

120591 997891rarr (119886119898 119887119898)

1

120591119898+ sdot sdot sdot + (119886

minus1 119887minus1)1

120591+ (1198860 1198870)

+ (1198861 1198871) 120591 + sdot sdot sdot + (119886

119899 119887119899) 120591119899

(17)

Proof Consider 119876 given by the above expression then

(119886119896 119887119896) (119905119890)119896= (119886119896 119887119896) 119905119896119890 = (119886

119896119905119896 119887119896119905119896) (18)

holds for 119896 = minus119898 119899 so 119876(119905119890) = (ℎ(119905) 119902(119905)) where 119890 isthe unit ofA Therefore 119876 is anA-algebrizable lifting of 119905 997891rarr(ℎ(119905) 119896(119905))

Journal of Applied Mathematics 5

In particular every function 119901 = (1199011 1199012) 119868 sub R rarr

R2 with quadratic components 1199011(119905) = 119886

0+ 1198861119905 + 11988621199052 and

1199012(119905) = 119887

0+1198871119905+11988721199052 admits theA-algebrizable lifting 119875(120591) =

(1198860 1198870) + (1198861 1198871)120591 + (119886

2 1198872)1205912

4 On Algebrizable Liftings of Planar Systems

Solutions of every algebrizable planar system can be foundby solving an algebrization of the system as it is seen in thefollowing proposition

Proposition 9 If (2) is an algebrization of (1) and 120585(120591) asolution of (2) then (119909(119905) 119910(119905)) = 120585(119905119890) is a solution of (1)where 119890 denotes the unit of the corresponding algebra

Proof Let 120585 be a solution of (2) The derivative of(119909(119905) 119910(119905)) = 120585(119905119890) with respect to 119905 is given by

119889

119889119905(119909 (119905) 119910 (119905)) =

119889

119889119905120585 (119905119890) = [

119889120585 (120591 120585)

119889120591

10038161003816100381610038161003816100381610038161003816120591=(119905119890)

119889120591

119889119905]

= 119867 (120591 120585 (120591))1003816100381610038161003816120591=(119905119890) 119890 = 119867 (119905119890 120585 (119905119890))

= (119891 (119905 119909 (119905) 119910 (119905)) 119892 (119905 119909 (119905) 119910 (119905)))

(19)

Thus (119909(119905) 119910(119905)) is a solution of system (1)

As a consequence of the following proposition the familyof all the functions 119865 Ω sub R3 rarr R2 having algebrizableliftings defines an infinite dimensional algebra

Proposition 10 Let A be an algebra with unit 119890 In thefollowing statements 119865 and119866 denote functions defined on opensets Ω sub R3 and they have values in R2

(a) 119865(119905 119909 119910) = 119888 admits the H(A)-differentiable lifting119867(120591 120585) = 119888

(b) 119865(119905 119909 119910) = 119905119890 admits the H(A)-differentiable lifting119867(120591 120585) = 120591

(c) 119865(119905 119909 119910) = (119909 119910) admits the H(A)-differentiablelifting119867(120591 120585) = 120585

(d) If 119865 and 119866 admit H(A)-differentiable liftings 119867119865and

119867119866 respectively and 119886 119887 are constants inA then 119886119867

119865+

119887119867119866and 119865119866 (all products with respect to A) have

H(A)-differentiable liftings 119886119867119865+ 119887119867119866and 119867

119865119867119866

respectively

(e) Every function 119864 having the form 119864(119905 119909 119910) = 119865(119905 119866(119905

119909 119910)) where 119865 and 119866 admit H(A)-differentiableliftings 119867

119865and 119867

119866 admits an H(A)-differentiable

lifting119867119864given by119867

119864(120591 120585) = 119867

119865(120591119867119866(120591 120585))

(f) Every function 119864 having the form 119864(119905 119909 119910) =

119865(119905 119875(119909 119910)) where 119865 has an H(A)-differentiablelifting119867

119865and 119875 is anA-differentiable function admits

anH(A)-differentiable lifting119867119864(120591 120585) = 119867

119865(120591 119875(120585))

Proof The proofs of (a) (b) and (c) are trivial Let 119867119865and

119867119866be the algebrizable liftings of 119865 and 119866 and then119867

119865+119867119866

119867119865119867119866 and119867

119865119867119866are A-algebrizable and

(i) (119886119867119865+119887119867119866)(119905119890 119909 119910) = 119886119867

119865(119905119890 119909 119910)+119887119867

119866(119905119890 119909 119910) =

119886119865(119905 119909 119910) + 119886119866(119905 119909 119910)

(ii) (119867119865119867119866)(119905119890 119909 119910) = 119867

119865(119905119890 119909 119910)119867

119866(119905119890 119909 119910) =

119865(119905 119909 119910)119866(119905 119909 119910)

(iii) 119867119864(119905119890 119909 119910) = 119867

119865(119905119890119867

119866(119905119890 119909 119910)) = 119865(119905 119866(119905 119909 119910))

(iv) 119867119864(119905119890 119909 119910) = 119867

119865(119905119890 119875(119890 119910)) = 119865(119905 119875(119909 119910))

Thus the proof is complete

Corollary 11 Let A be an algebra The following func-tions admitH(A)-differentiable liftings polynomial functionsrational functions trigonometric functions exponential func-tions and all of those functions which can be defined bylinear combinations products quotients and compositions offunctions admittingH(A)-differentiable liftings

Given a function 119865(119905 119909 119910) = (119891(119905 119909 119910) 119892(119905 119909 119910))where 119891 119892 are polynomial functions of the variables 119909 119910the goal of the paper is to determine if 119865 has an H(A)-differentiable lifting As a consequence of the followingtheorem every function 119865 which is polynomial of thevariables 119905 119909 and 119910 has an H(A)-differentiable liftingwhen (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-differentiable forall 119905

Theorem 12 Let A be an algebra with unit 119890 and 119865

Ω sub R3 rarr R2 119865 = (119891 119892) where 119891 = sum119898

119896=0119891119896

119892 = sum119898

119896=0119892119896 and 119891

119896(119905 119909 119910) 119892

119896(119905 119909 119910) are homogeneous

polynomials of degree 119896 in the variables 119909 119910 and Ω = 119868 times

R2 for an open interval 119868 Then the following statements areequivalent

(a) 119865 has anH(A)-differentiable lifting119867

(b) The map (119909 119910) 997891rarr 119865(119905 119909 119910) is A-algebrizable forall 119905 isin 119868 and the functions ℎ

119896given by ℎ

119896(119905) =

(119891119896(119905 119890) 119892

119896(119905 119890)) have A-algebrizable liftings for 119896 =

0 1 119898

(c) 119867(120591 120585) = sum119898

119896=0119867119896(120591)120585119896 is an H(A)-differentiable

lifting of 119865 where 119867119896(120591) are A-algebrizable liftings of

ℎ119896(119905) = (119891

119896(119905 119890) 119892

119896(119905 119890))

Proof Obviously (a) implies (b) and (c) implies (a) We nowshow that (b) implies (c) Suppose that119891(119905 119909 119910) and119892(119905 119909 119910)

are homogenous polynomials 119891(119905 119909 119910) = sum119899

119896=0119901119896(119905)119909119896119910119899minus119896

and 119892(119905 119909 119910) = sum119899

119896=0119902119896(119905)119909119896119910119899minus119896 of the variables 119909 119910 defined

6 Journal of Applied Mathematics

on the setΩ = 119868 timesR2 as above Taking the partial derivativesof 119865 with respect to 119909 and 119910 we obtain

120597 (119891 119892)

120597 (119909 119910)(119905 119909 119910)

= (

119899

sum

119896=1

119896119901119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119901119896minus1

(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

119896119902119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119902119896minus1

(119905) 119909119896minus1

119910119899minus119896

)

= (

119899

sum

119896=1

119896119901119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119901119896minus1

(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

119896119902119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119902119896minus1

(119905) 119909119896minus1

119910119899minus119896

)

=

119899

sum

119896=1

119909119896minus1

119910119899minus119896

(119896119901119896(119905) (119899 minus 119896 + 1) 119901

119896minus1(119905)

119896119902119896(119905) (119899 minus 119896 + 1) 119902

119896minus1(119905)

)

(20)

Let 119877 be the first fundamental representation of A Since(119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for all 119905 isin 119868then (120597(119891 119892)120597(119909 119910))(119905 119909 119910) isin 119877(A) for all (119905 119909 119910) isin ΩThus

120597 (119891 119892)

120597 (119909 119910)(119905 0 1) = (

1199011(119905) 119899119901

0(119905)

1199021(119905) 119899119902

0(119905)

) isin 119877 (A) (21)

for all 119905 isin 119868 and then

120597 (119891 119892)

120597 (119909 119910)(119905 119909 119910) minus 119910

119899minus1(1199011(119905) 119899119901

0(119905)

1199021(119905) 119899119902

0(119905)

)

=

119899

sum

119896=2

119909119896minus1

119910119899minus119896

(119896119901119896(119905) (119899 minus 119896 + 1) 119901

119896minus1(119905)

119896119902119896(119905) (119899 minus 119896 + 1) 119902

119896minus1(119905)

)

(22)

is in 119877(A) for all (119905 119909 119910) isin Ω If 119909(119904) = 1119904 and 119910(119904) =(119899minus2)radic119904

then

lim119904rarrinfin

[120597 (119891 119892)

120597 (119909 119910)(119905 119909 (119904) 119910 (119904))

minus [119910 (119904)]119899minus1

(1199011(119905) 119899119901

0(119905)

1199021(119905) 119899119902

0(119905)

)]

= (1199012(119905) (119899 minus 1) 119901

1(119905)

1199022(119905) (119899 minus 1) 119902

1(119905)

) isin 119877 (A)

(23)

Following the same idea for 119896 = 1 2 119899

(119896119901119896(119905) (119899 minus 119896 + 1) 119901

119896minus1(119905)

119896119902119896(119905) (119899 minus 119896 + 1) 119902

119896minus1(119905)

) isin 119877 (A) (24)

If (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for analgebra A with Type (I) product (given in Section 2) then

119896119901119896+ 119886119896119902

119896minus (119899 minus 119896 + 1) 119902

119896minus1= 0

119887119896119902119896+ (119899 minus 119896 + 1) 119901

119896minus1= 0

(25)

or equivalently

(119902119896minus1

119901119896minus1

) = (119896119901119896+ 119886119896119902

119896

119899 minus 119896 + 1 minus

119887119896119902119896

119899 minus 119896 + 1) (26)

for 119896 = 1 119899 Thus the functions 119901119896 119902119896

aredetermined by 119901

119899 119902119899

for 119896 = 0 1 119899 minus 1Since (119891(119905 1 0) 119892(119905 1 0)) = (119901

119899(119905) 119902119899(119905)) then

(119891(119905 119909 119910) 119892(119905 119909 119910)) = (119901119899(119905) 119902119899(119905))(119909 119910)

119899Therefore 119865 has anH(A)-differentiable lifting 119867(120591 120585) =

119867119899(120591)120585119899 where 119867

119899(120591) is an A-algebrizable lifting of 119905 997891rarr

(119901119899(119905) 119902119899(119905))

If (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for analgebra A with Type (II) product (given in Section 2) then

119896119901119896+ 119886 (119899 minus 119896 + 1) 119901

119896minus1minus (119899 minus 119896 + 1) 119902

119896minus1= 0

119896119902119896= 0

(27)

or equivalently

(119901119896 119902119896) = (minus

119886 (119899 minus 119896 + 1) 119901119896minus1

119896 0) (28)

for 119896 = 1 119899 Thus 119902119896

= 0 and 119901119896is determined by 119901

0

for 119896 = 1 2 119899 Since (119891(119905 0 1) 119892(119905 0 1)) = (1199010(119905) 1199020(119905))

then (119891(119905 119909 119910) 119892(119905 119909 119910)) = (1199010(119905) 1199020(119905))(119909 119910)

119899Therefore 119865 has anH(A)-differentiable lifting 119867(120591 120585) =

119867119899(120591)120585119899 where 119867

119899(120591) is an A-algebrizable lifting of 119905 997891rarr

(1199010(119905) 1199020(119905))

If (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for analgebra A with Type (III) product (given in Section 2) then(119899 minus 119896 + 1)119901

119896minus1= 0 and 119896119902

119896= 0 that is 119901

119896minus1= 0 119902

119896= 0

for 119896 = 1 119899 Thus 119902119896

= 0 and 119901119896is determined by 119901

0

for 119896 = 1 2 119899 Since (119891(119905 0 1) 119892(119905 0 1)) = (119901119899(119905) 1199020(119905))

then (119891(119905 119909 119910) 119892(119905 119909 119910)) = (119901119899(119905) 1199020(119905))(119909 119910)

119899Therefore 119865 has anH(A)-differentiable lifting 119867(120591 120585) =

119867119899(120591)120585119899 where 119867

119899(120591) is an A-algebrizable lifting of 119905 997891rarr

(119901119899(119905) 1199020(119905))

Thus if 119891 and 119892 are homogenous polynomial of degree119899 in the variables 119909 and 119910 then 119867(120591 120585) = 119867

119899(120591)120585119899 in each

of the cases of algebras defined by products of Types (I)(II) and (III) given in Section 2 Since a polynomial function(119905 119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) in the variables 119909 and 119910 canbe seen as the finite addition of homogenous polynomial inthe variables 119909 and 119910 by (d) of Proposition 10 (b) implies(c)

Example 13 Consider the planar system (6) then

119865 (119905 119909 119910)

= (minus1

1199053minus

2119909

119905+ 1199051199092minus 1199051199102 minus

2119910

119905+ 2119905119909119910 + 119905119910

2)

(29)

The Jacobian 120597119865120597(119909 119910) is given by

120597119865 (119905 119909 119910)

120597 (119909 119910)= (

minus2

119905+ 2119905119909 minus2119905119910

2119905119910 minus2

119905+ 2119905119909 + 2119905119910

) (30)

Journal of Applied Mathematics 7

Thus 120597119865120597(119909 119910) is orthogonal to the matrices

1198611= (

1 0

1 minus1)

1198612= (

0 1

1 0)

(31)

for all (119909 119910) that is ⟨120597119865120597(119909 119910) 119861119894⟩ = 0 for 119894 = 1 2 Themap

(119909 119910) 997891rarr 119865(119905 119909 119910) isA-algebrizable for an algebra of Type (I)with constants 119886 = 1 and 119887 = 1 see Section 2 The function 119865

can be written as

119865 (119905 119909 119910) = (minus1

1199053 0) + (minus

2

119905119909 minus

2

119905119910)

+ (1199051199092minus 1199051199102 1199051199102)

(32)

Thus the functions 119901119894and 119902

119894of Theorem 12 are given by

1199010(119905 119909 119910) = minus1119905

3 1199020(119905 119909 119910) = 0 119901

1(119905 119909 119910) = minus(2119905)119909

1199021(119905 119909 119910) = minus(2119905)119910 119901

2(119905 119909 119910) = 119905119909

2minus 1199051199102 and 119902

2(119905 119909 119910) =

1199051199102The unit 119890 of A is 119890 = (1 0) and then ℎ

0(119905) = (minus1119905

3 0)

ℎ1(119905) = (minus2119905 0) and ℎ

2(119905) = (119905 0) have A-algebrizable

liftings 1198670(120591) = minus1120591

3 1198671(120591) = minus2120591 and 119867

2(120591) = 120591

respectivelyFrom the form of 119865(119905 119909119890) = (minus1119905

3minus (2119905)119909 minus 119905119909

2)119890 119867

must be

119867(120591 120585) = minus1

1205913minus

2

120591120585 + 120591120585

2 (33)

5 The Case of Second-Degree Polynomials inthe Variables 119905 119909 and 119910

If 119891 and 119892 in (1) are quadratic polynomials in three variables119905 119909 and 119910 andA is an algebra with respect to which the map(119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable it will beshowed that the H(A)-differentiable lifting 119867 of 119865 = (119891 119892)

is a polynomial in two variables ofA Under these conditions(1) has an algebrization which is a Riccati equation over Ahaving the form

119889120585

119889120591= 119875 (120591) + 119876 (120591) 120585 + 119877 (120591) 120585

2 (34)

where119875119876 and119877 are polynomials overA of degrees two oneand zero respectively

Consider system (1) where 119891 119892 Ω sub R3 rarr R2 aresecond-degree polynomials of three variables 119905 119909 and 119910 thatis

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 1198863119910 + 11988641199052+ 1198865119905119909

+ 1198866119905119910 + 119886

71199092+ 1198868119909119910 + 119886

91199102

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198872119909 + 1198873119910 + 11988741199052+ 1198875119905119909 + 119887

6119905119910

+ 11988771199092+ 1198878119909119910 + 119887

91199102

(35)

All the quadratic vector fields which are algebrizable withrespect to algebras with Type (I) products have the form

119889119909

119889119905= 1198600+ (1198612minus 1198861198611) 119909 minus 119887119861

1119910 + (

1

21198614minus 1198861198613)1199092

minus 21198871198613119909119910 minus

1

211988711986141199102

119889119910

119889119905= 1198610+ 1198611119909 + 1198612119910 + 11986131199092+ 1198614119909119910

+ (1

21198861198614minus 1198871198613)1199102

(36)

where 119886 119887 1198600 1198610 1198611 119861

5are real constants

see Section 22 The algebrizability of (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) with respect to algebras with Type(I) products can be verified by considering 119905 as a constant

The following theorems give conditions that characterizethe algebrizability of planar systems like (1) when 119891 and119892 are quadratic polynomials The algebrizability of nonau-tonomous quadratic systems with respect to algebras withType (I) products is given in the following theorem

Theorem14 LetA be an algebrawith Type (I) product definedby constants 119886 and 119887 and 119891 119892 the polynomials (35) Thefollowing statements are equivalent

(1) The map 119865 (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 have the form

119891 (119905 119909 119910) = 1198860+ 1198861119905 + (1198873minus 1198861198872) 119909 minus 119887119887

2119910 + 11988641199052

+ (1198876minus 1198861198875) 119905119909 minus 119887119887

5119905119910

+ (1198878

2minus 1198861198877)1199092minus 21198871198877119909119910 minus

1198871198878

21199102

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198872119909 + 1198873119910 + 11988741199052+ 1198875119905119909 + 119887

6119905119910

+ 11988771199092+ 1198878119909119910 + (

1198861198878

2minus 1198871198877)1199102

(37)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (38)

is anH(A)-differentiable lifting of 119865 where

1198600= (1198860 1198870)

1198601= (1198861 1198871)

1198602= (1198873minus 1198861198872 1198872)

1198603= (1198864 1198874)

1198604= (1198876minus 1198861198875 1198875)

1198605= (

1198878

2minus 1198861198877 1198877)

(39)

8 Journal of Applied Mathematics

Proof Writing119892(119905 119909 119910) = 1198610+1198611119909+1198612119910+11986131199092+1198614119909119910+119861

51199102

yields

1198610= 1198870+ 1198871119905 + 11988741199052

1198611= 1198872+ 1198875119905

1198612= 1198873+ 1198876119905

1198613= 1198877

1198614= 1198878

1198615= 1198879

(40)

The function (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) isA-algebrizableif and only if

119891 (119905 119909 119910) = 1198600+ (1198612minus 1198861198611) 119909 minus 119887119861

1119910

+ (1

21198614minus 1198861198613)1199092minus 2119887119861

3119909119910 minus

1

211988711986141199102

(41)

where 1198600

= 1198860+ 1198861119905 + 11988641199052 and 119887

9= 11988611988782 minus 119887119887

7 Thus

statements (1) and (2) are equivalentThe function 119867 is polynomial in the variables 120591 and 120585 of

A hence 119867 is H(A)-differentiable 119867 satisfies 119867(119905119890 119909 119910) =

(119891(119905 119909 119910) 119892(119905 119909 119910)) where 119890 is the unit of A So 119867 is anH(A)-differentiable lifting of 119865 Thus statement (2) impliesstatement (3)

Since 119867 is H(A)-differentiable and 119865(119905 119909 119910) =

119867(119905119890 119909 119910) then (119909 119910) 997891rarr 119865(119905 119909 119910) is A-algebrizablefor all 119905 Thus statement (3) implies statement (1)

All the quadratic vector fields which are algebrizable withrespect to algebras with Type (II) products have the form

= 1198600+ 1198601119909 + 119860

2119910 minus

1

211988611986031199092+ 1198603119909119910 + 119860

41199102

119910 = 1198610+ (1198601+ 1198861198602) 119910 + (

1

21198603+ 1198861198604)1199102

(42)

where 119886 1198600 119860

4 1198610are real constants see Section 22

The algebrizability of (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) withrespect to algebras with Type (II) products can be verifiedby considering 119905 as a constant

The algebrizability of nonautonomous quadratic systemswith respect to algebras with Type (II) products is given in thefollowing theorem

Theorem 15 Let A be an algebra with Type (II) productdefined by the constant 119886 and let 119891 119892 be the polynomials (35)The following statements are equivalent

(1) The map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 are given by

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 1198863119910 + 11988641199052+ 1198865119905119909

+ 1198866119905119910 minus

1

211988611988681199092+ 1198868119909119910 + 119886

91199102

119892 (119905 119909 119910) = 1198870+ 1198871119905 + (1198862+ 1198861198863) 119910 + 119887

41199052

+ (1198865+ 1198861198866) 119905119910 + (

1198868

2+ 1198861198869)1199102

(43)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (44)

is anH(A)-differentiable lifting of 119865 where

1198600= (1198860 1198870)

1198601= (1198861 1198871)

1198602= (1198863 1198862+ 1198861198863)

1198603= (1198864 1198874)

1198604= (1198866 1198865+ 1198861198866)

1198605= (11988691198868

2+ 1198861198869)

(45)

Proof Writing 119891(119905 119909 119910) = 1198600+ 1198601119909 + 119860

2119910 minus (12)119886119860

31199092+

1198603119909119910 + 119860

41199102 yields

1198600= 1198860+ 1198861119905 + 11988641199052

1198601= 1198862+ 1198865119905

1198602= 1198863+ 1198866119905

1198603= 1198868

1198604= 1198869

(46)

and the map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizableif and only if

119892 (119905 119909 119910) = 1198610+ (1198601+ 1198861198602) 119910 + (

1

21198603+ 1198861198604)1199102 (47)

where 1198610= 1198870+ 1198871119905 + 11988741199052 Thus statements (1) and (2) are

equivalentThe rest of the proof is similar to that of Theorem 14

All the quadratic vector fields which are differentiablewith respect to algebras with Type (III) products have theform

= 1198600+ 1198601119909 + 119860

21199092

119910 = 1198610+ 1198611119910 + 11986121199102

(48)

Journal of Applied Mathematics 9

where 1198600 1198610 1198601 1198611 1198602 1198612

are real constantssee Section 22 The algebrizability of (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) with respect to algebras with Type(III) products can be verified by considering 119905 as a constant

The algebrizability of nonautonomous quadratic systemswith respect to algebras with Type (III) products is given inthe following theorem

Theorem 16 Let A be an algebra with Type (III) productdefined by the constant 119886 and let 119891 119892 be the polynomials (35)The following statements are equivalent

(1) The map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 are given by

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 11988641199052+ 1198865119905119909 + 119886

71199092

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198873119910 + 11988741199052+ 1198876119905119910 + 119887

91199102

(49)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (50)

is an H(A)-differentiable lifting of 119865 where 1198600

=

(1198860 1198870) 1198601

= (1198861 1198871) 1198602

= (1198862 1198873) 1198603

= (1198864 1198874)

1198604= (1198865 1198876) and 119860

5= (1198867 1198879)

Proof Function (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable if and only if

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 11988641199052+ 1198865119905119909 + 119886

71199092

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198873119910 + 11988741199052+ 1198876119905119910 + 119887

91199102

(51)

Thus first and second statements are equivalents The func-tion119867 given by119867(120591 120585) = 119860

0+1198601120591+1198602120585+11986031205912+1198604120591120585+119860

51205852

where1198600= (1198860 1198870)1198601= (1198861 1198871)1198602= (1198862 1198873)1198603= (1198864 1198874)

1198604= (1198865 1198876) 1198605= (1198867 1198879) 120591 = (119903 119904) and 120585 = (119909 119910) satisfies

119867(119905119890 119909 119910) = (119891(119905 119909 119910) 119892(119905 119909 119910)) Thus second and thirdstatements are equivalents

By Theorems 14 15 and 16 an algebrization of quadraticsystems is a Riccati equation over an algebra 119889120585119889120591 = 119875(120591) +

119876(120591)120585 + 119877(120591)1205852 where 119875(120591) = 119860

0+ 1198601120591 + 119860

31205912 119876(120591) =

1198602+ 1198604120591 and 119877(120591) = 119860

5 In the following example is

given a nonautonomous quadratic system for the which analgebrization is found by usingTheorem 14

Example 17 Consider system (1) given by

119891 (119905 119909 119910) = 1 + 7119905 minus 119909 minus 3119910 + 51199052minus 119905119909 minus 3119905119910 minus 6119909119910

minus 61199102

119892 (119905 119909 119910) = 1 + 119905 + 119909 + 119910 + 1199052+ 119905119909 + 119905119910 + 119909

2+ 4119909119910

+ 1199102

(52)

The matrices

1198611= (

1 0

2 minus1)

1198612= (

0 1

3 0)

(53)

satisfy ⟨119861119894 120597(119891 119892)120597(119909 119910)⟩ = 0 for 119894 = 1 2Thus 119865 (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for an algebra A withType (I) product defined by the constants 119886 = 2 and 119887 = 3

The conditions of Theorem 14 are satisfied then theH(A)-differentiable lifting119867 of 119865 can be written as

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (54)

where 1198600= (1 1) 119860

1= (7 1) 119860

2= (minus1 1) 119860

3= (5 1)

1198604= (minus1 1) and 119860

5= (0 1)

Disclosure

The authors declare having no financial affiliation with anyorganization regarding the material discussed here

Conflict of Interests

The authors declare that there is no conflict of interestsconcerning this textThe issues discussed in this paper do nothave any secondary interest for any of the authors

Acknowledgments

The authors wish to acknowledge the support of GrantsPromep103513 and CB-2010150532 Conacyt

References

[1] E K Blum ldquoA theory of analytic functions in Banach algebrasrdquoTransactions of the American Mathematical Society vol 78 no2 pp 343ndash370 1955

[2] P W Ketchum ldquoAnalytic functions of hypercomplex variablesrdquoTransactions of the American Mathematical Society vol 30 no4 pp 641ndash641 1928

[3] E R Lorch ldquoThe theory of analytic functions in normed abelianvector ringsrdquo Transactions of the American Mathematical Soci-ety vol 54 no 3 pp 414ndash425 1943

[4] J A Ward ldquoA theory of analytic functions in linear associativealgebrasrdquo Duke Mathematical Journal vol 7 no 1 pp 233ndash2481940

[5] J A Ward ldquoFrom generalized Cauchy-Riemann equationsto linear algebrardquo Proceedings of the American MathematicalSociety vol 4 no 3 pp 456ndash461 1953

[6] E Lopez-Gonzalez ldquoDifferential equations over algebrasrdquoAdvances and Applications in Mathematical Sciences vol 8 no2 pp 189ndash214 2011

[7] A Alvarez-Parrilla M E Frıas-Armenta E Lopez-Gonzalezand C Yee-Romero ldquoOn solving systems of autonomousordinary differential equations by reduction to a variable of analgebrardquo International Journal of Mathematics and Mathemati-cal Sciences vol 2012 Article ID 753916 21 pages 2012

10 Journal of Applied Mathematics

[8] E J Wilczynski ldquoReview Abraham Cohen an introductionto the lie theory of one-parameter groups with applicationsto the solution of differential equationsrdquo Bulletin of the Amer-ican Mathematical Society vol 18 no 10 pp 514ndash515 1912httpprojecteuclidorgeuclidbams1183421829

[9] J M PageOrdinary Differential Equations with an Introductionto Liersquos Theory or the Group of One Parameter MacmillanPublishers London UK 1897

[10] R A Steinhour The truth about lie symmetries solvingdifferential equations with symmetry methods [SeniorIndependent StudyTheses] 2013 httpopenworkswoostereduindependentstudy949

[11] R Pierce Associative Algebras Springer New York NY USA1982

[12] K I T Al-Dosary ldquoInverse integrating factor for classes of pla-nar differential systemsrdquo International Journal of MathematicalAnalysis vol 4 no 29ndash32 pp 1433ndash1446 2010

[13] I A Garcıa and M Grau ldquoA Survey on the inverse integratingfactorrdquo Qualitative Theory of Dynamical Systems vol 9 no 1-2pp 115ndash166 2010

[14] W Brown Matrices over Commutative Rings Monographs andTextbooks in Pure and Applied Mathematics Marcel Dekker1992

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

23 Differentiability onModules over Algebras In this subsec-tion we give the definition of A-differentiability of functionswith domain in A times A and image in A

The Cartesian productAtimesA defines a normedA-modulewith respect to the norm sdot A times A rarr R (119886

1 1198862) =

radic|1198861|2 + |119886

2|2 This norm satisfies

(i) 119883 ge 0 for all 119883 isin A times A and 119883 = 0 if and only if119883 = (0 0)

(ii) 119886119883 le |119886|119883 for all 119886 isin A and119883 isin A times A(iii) 119883 + 119884 le 119883 + 119884 for all119883119884 isin A times A

In the following definition sdot denotes the norm givenabove on the A-module A times A

Definition 4 Let A be an algebra and 119867 Ω sub A times A rarr

A a function where Ω is an open set We say that 119867 isH(A)-differentiable on 119883

0isin Ω if there exists a module

homeomorphism 119872 A times A rarr A which we call thedifferential homomorphism of 119867 at 119883

0 which satisfies the

condition

lim119883rarr1198830

1003817100381710038171003817119867 (119883) minus 119867 (1198830) minus 119872(119883 minus 119883

0)1003817100381710038171003817

1003817100381710038171003817119883 minus 1198830

1003817100381710038171003817

= 0 (14)

We denote 119872 by D119867(1198830) We say that 119867 is H(A)-

differentiable on Ω if 119867 is H(A)-differentiable on all thepoints ofΩ

A function 119867 Ω sub A times A rarr A isH(A)-differentiableat 1198830if and only if the usual Jacobian matrix of 119867 Ω sub

R2 times R2 rarr R2 at 1198830satisfies 119869119867(119883

0) isin 119877(A) times 119877(A)

The differential homomorphismD119867(1198830) is represented by a

matrix in11987212

(A)with respect to the standard basis ofAtimesAwhere119872

12(A) is theA-module of all the matrices of one row

and two columns with entries in A see [14]

3 On Algebrizable Liftings of Functions 119901 119868 sub

R rarr R2

In this section are considered functions 119901 119868 sub R rarr R2

defined in open intervals 119868 and conditions for the existenceof algebrasA andA-algebrizable functions119875 such that 119901(119905) =119875(119905119890) will be determined where 119890 is the unit of A

Definition 5 Let 119901 119868 sub R rarr R2 be a function defined in anopen interval 119868 andA an algebra with unit 119890 We will say that119875 119881 sub R2 rarr R2 is an A-algebrizable lifting of 119901 if (a) 119881 isan open set on which 119875 isA-algebrizable (b) 119905119890 119905 isin 119868 sub 119881and (c) 119901(119905) = 119875(119905119890) for all 119905 isin 119868

As a consequence of the following proposition the familyof all the functions 119901 119868 sub R rarr R2 having algebrizableliftings is an infinite dimensional algebra

Proposition 6 Let A be an algebra and let 119901 119902 119868 sub R rarr

R2 be functions where 119868 is an interval

(a) Every constant function 119901(119905) = 119888 admits the A-algebrizable lifting 119875(120591) = 119888

(b) 119901(119905) = 119905119890 admits the A-algebrizable lifting 119875(120591) = 120591where 119890 isin A is the unit of A

(c) If 119901 and 119902 admit A-algebrizable liftings 119875 and 119876 withrespect to A respectively and 119886 119887 are constants in Athen 119886119901 + 119887119902 and 119901119902 (all products with respect to A)admit algebrizable liftings 119886119875+119887119876 and119875119876 respectively

(d) If 119901 has an A-algebrizable lifting 119875 and 119876 is an A-algebrizable function with Im(119875) sub Dom(119876) then119876 ∘ 119875 is an algebrizable lifting of 119876 ∘ 119901

Proof Identity and constant functions are A-differentiablefor any algebra A Thus (a) and (b) hold

Let 119901 and 119902 be functions with A-algebrizable liftings 119875

being 119876 and 119886 119887 isin A constants The functions 119878 = 119886119875 + 119887119876

and 119879 = 119875119876 are A-algebrizable and satisfy

119878 (119905119890) = 119886119875 (119905119890) + 119887119876 (119905119890) = 119886119901 (119905) + 119887119902 (119905)

119879 (119905119890) = 119875 (119905119890) 119876 (119905119890) = 119901 (119905) 119902 (119905)

(15)

Therefore 119878 and 119879 are algebrizable liftings of 119886119901 + 119887119902 and 119901119902respectively

If 119901 has an A-algebrizable lifting 119875 and 119876 is an A-differentiable function with Im(119875) sub Dom(119876) then 119876 ∘

119875(119905119890) = 119876 ∘ 119901(119905) Therefore 119876 ∘ 119875 is an A-algebrizable liftingof 119876 ∘ 119901

Corollary 7 LetA be an algebraThen the following functionsadmit A-algebrizable liftings polynomial functions rationalfunctions trigonometric functions exponential functions andall of those functions which can be defined by linear com-binations products quotients and compositions of functionsadmitting algebrizable liftings

Every function 119905 997891rarr (ℎ(119905) 119896(119905)) with polynomial com-ponents ℎ(119905) and 119896(119905) has an A-algebrizable lifting Thefollowing proposition gives a wider class of these functions

Proposition 8 Let A be an algebra Any function 119905 997891rarr

(ℎ(119905) 119896(119905)) with components of the form

ℎ (119905) =119886119898

119905119898+

119886119898minus1

119905119898minus1+ sdot sdot sdot +

119886minus1

119905+ 1198860+ 1198861119905 + sdot sdot sdot 119886

119899119905119899

119896 (119905) =119887119898

119905119898+

119887119898minus1

119905119898minus1+ sdot sdot sdot +

119887minus1

119905+ 1198870+ 1198871119905 + sdot sdot sdot 119887

119899119905119899

(16)

119898 119899 isin N has an A-algebrizable lifting An A-algebrizablelifting is given by

120591 997891rarr (119886119898 119887119898)

1

120591119898+ sdot sdot sdot + (119886

minus1 119887minus1)1

120591+ (1198860 1198870)

+ (1198861 1198871) 120591 + sdot sdot sdot + (119886

119899 119887119899) 120591119899

(17)

Proof Consider 119876 given by the above expression then

(119886119896 119887119896) (119905119890)119896= (119886119896 119887119896) 119905119896119890 = (119886

119896119905119896 119887119896119905119896) (18)

holds for 119896 = minus119898 119899 so 119876(119905119890) = (ℎ(119905) 119902(119905)) where 119890 isthe unit ofA Therefore 119876 is anA-algebrizable lifting of 119905 997891rarr(ℎ(119905) 119896(119905))

Journal of Applied Mathematics 5

In particular every function 119901 = (1199011 1199012) 119868 sub R rarr

R2 with quadratic components 1199011(119905) = 119886

0+ 1198861119905 + 11988621199052 and

1199012(119905) = 119887

0+1198871119905+11988721199052 admits theA-algebrizable lifting 119875(120591) =

(1198860 1198870) + (1198861 1198871)120591 + (119886

2 1198872)1205912

4 On Algebrizable Liftings of Planar Systems

Solutions of every algebrizable planar system can be foundby solving an algebrization of the system as it is seen in thefollowing proposition

Proposition 9 If (2) is an algebrization of (1) and 120585(120591) asolution of (2) then (119909(119905) 119910(119905)) = 120585(119905119890) is a solution of (1)where 119890 denotes the unit of the corresponding algebra

Proof Let 120585 be a solution of (2) The derivative of(119909(119905) 119910(119905)) = 120585(119905119890) with respect to 119905 is given by

119889

119889119905(119909 (119905) 119910 (119905)) =

119889

119889119905120585 (119905119890) = [

119889120585 (120591 120585)

119889120591

10038161003816100381610038161003816100381610038161003816120591=(119905119890)

119889120591

119889119905]

= 119867 (120591 120585 (120591))1003816100381610038161003816120591=(119905119890) 119890 = 119867 (119905119890 120585 (119905119890))

= (119891 (119905 119909 (119905) 119910 (119905)) 119892 (119905 119909 (119905) 119910 (119905)))

(19)

Thus (119909(119905) 119910(119905)) is a solution of system (1)

As a consequence of the following proposition the familyof all the functions 119865 Ω sub R3 rarr R2 having algebrizableliftings defines an infinite dimensional algebra

Proposition 10 Let A be an algebra with unit 119890 In thefollowing statements 119865 and119866 denote functions defined on opensets Ω sub R3 and they have values in R2

(a) 119865(119905 119909 119910) = 119888 admits the H(A)-differentiable lifting119867(120591 120585) = 119888

(b) 119865(119905 119909 119910) = 119905119890 admits the H(A)-differentiable lifting119867(120591 120585) = 120591

(c) 119865(119905 119909 119910) = (119909 119910) admits the H(A)-differentiablelifting119867(120591 120585) = 120585

(d) If 119865 and 119866 admit H(A)-differentiable liftings 119867119865and

119867119866 respectively and 119886 119887 are constants inA then 119886119867

119865+

119887119867119866and 119865119866 (all products with respect to A) have

H(A)-differentiable liftings 119886119867119865+ 119887119867119866and 119867

119865119867119866

respectively

(e) Every function 119864 having the form 119864(119905 119909 119910) = 119865(119905 119866(119905

119909 119910)) where 119865 and 119866 admit H(A)-differentiableliftings 119867

119865and 119867

119866 admits an H(A)-differentiable

lifting119867119864given by119867

119864(120591 120585) = 119867

119865(120591119867119866(120591 120585))

(f) Every function 119864 having the form 119864(119905 119909 119910) =

119865(119905 119875(119909 119910)) where 119865 has an H(A)-differentiablelifting119867

119865and 119875 is anA-differentiable function admits

anH(A)-differentiable lifting119867119864(120591 120585) = 119867

119865(120591 119875(120585))

Proof The proofs of (a) (b) and (c) are trivial Let 119867119865and

119867119866be the algebrizable liftings of 119865 and 119866 and then119867

119865+119867119866

119867119865119867119866 and119867

119865119867119866are A-algebrizable and

(i) (119886119867119865+119887119867119866)(119905119890 119909 119910) = 119886119867

119865(119905119890 119909 119910)+119887119867

119866(119905119890 119909 119910) =

119886119865(119905 119909 119910) + 119886119866(119905 119909 119910)

(ii) (119867119865119867119866)(119905119890 119909 119910) = 119867

119865(119905119890 119909 119910)119867

119866(119905119890 119909 119910) =

119865(119905 119909 119910)119866(119905 119909 119910)

(iii) 119867119864(119905119890 119909 119910) = 119867

119865(119905119890119867

119866(119905119890 119909 119910)) = 119865(119905 119866(119905 119909 119910))

(iv) 119867119864(119905119890 119909 119910) = 119867

119865(119905119890 119875(119890 119910)) = 119865(119905 119875(119909 119910))

Thus the proof is complete

Corollary 11 Let A be an algebra The following func-tions admitH(A)-differentiable liftings polynomial functionsrational functions trigonometric functions exponential func-tions and all of those functions which can be defined bylinear combinations products quotients and compositions offunctions admittingH(A)-differentiable liftings

Given a function 119865(119905 119909 119910) = (119891(119905 119909 119910) 119892(119905 119909 119910))where 119891 119892 are polynomial functions of the variables 119909 119910the goal of the paper is to determine if 119865 has an H(A)-differentiable lifting As a consequence of the followingtheorem every function 119865 which is polynomial of thevariables 119905 119909 and 119910 has an H(A)-differentiable liftingwhen (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-differentiable forall 119905

Theorem 12 Let A be an algebra with unit 119890 and 119865

Ω sub R3 rarr R2 119865 = (119891 119892) where 119891 = sum119898

119896=0119891119896

119892 = sum119898

119896=0119892119896 and 119891

119896(119905 119909 119910) 119892

119896(119905 119909 119910) are homogeneous

polynomials of degree 119896 in the variables 119909 119910 and Ω = 119868 times

R2 for an open interval 119868 Then the following statements areequivalent

(a) 119865 has anH(A)-differentiable lifting119867

(b) The map (119909 119910) 997891rarr 119865(119905 119909 119910) is A-algebrizable forall 119905 isin 119868 and the functions ℎ

119896given by ℎ

119896(119905) =

(119891119896(119905 119890) 119892

119896(119905 119890)) have A-algebrizable liftings for 119896 =

0 1 119898

(c) 119867(120591 120585) = sum119898

119896=0119867119896(120591)120585119896 is an H(A)-differentiable

lifting of 119865 where 119867119896(120591) are A-algebrizable liftings of

ℎ119896(119905) = (119891

119896(119905 119890) 119892

119896(119905 119890))

Proof Obviously (a) implies (b) and (c) implies (a) We nowshow that (b) implies (c) Suppose that119891(119905 119909 119910) and119892(119905 119909 119910)

are homogenous polynomials 119891(119905 119909 119910) = sum119899

119896=0119901119896(119905)119909119896119910119899minus119896

and 119892(119905 119909 119910) = sum119899

119896=0119902119896(119905)119909119896119910119899minus119896 of the variables 119909 119910 defined

6 Journal of Applied Mathematics

on the setΩ = 119868 timesR2 as above Taking the partial derivativesof 119865 with respect to 119909 and 119910 we obtain

120597 (119891 119892)

120597 (119909 119910)(119905 119909 119910)

= (

119899

sum

119896=1

119896119901119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119901119896minus1

(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

119896119902119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119902119896minus1

(119905) 119909119896minus1

119910119899minus119896

)

= (

119899

sum

119896=1

119896119901119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119901119896minus1

(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

119896119902119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119902119896minus1

(119905) 119909119896minus1

119910119899minus119896

)

=

119899

sum

119896=1

119909119896minus1

119910119899minus119896

(119896119901119896(119905) (119899 minus 119896 + 1) 119901

119896minus1(119905)

119896119902119896(119905) (119899 minus 119896 + 1) 119902

119896minus1(119905)

)

(20)

Let 119877 be the first fundamental representation of A Since(119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for all 119905 isin 119868then (120597(119891 119892)120597(119909 119910))(119905 119909 119910) isin 119877(A) for all (119905 119909 119910) isin ΩThus

120597 (119891 119892)

120597 (119909 119910)(119905 0 1) = (

1199011(119905) 119899119901

0(119905)

1199021(119905) 119899119902

0(119905)

) isin 119877 (A) (21)

for all 119905 isin 119868 and then

120597 (119891 119892)

120597 (119909 119910)(119905 119909 119910) minus 119910

119899minus1(1199011(119905) 119899119901

0(119905)

1199021(119905) 119899119902

0(119905)

)

=

119899

sum

119896=2

119909119896minus1

119910119899minus119896

(119896119901119896(119905) (119899 minus 119896 + 1) 119901

119896minus1(119905)

119896119902119896(119905) (119899 minus 119896 + 1) 119902

119896minus1(119905)

)

(22)

is in 119877(A) for all (119905 119909 119910) isin Ω If 119909(119904) = 1119904 and 119910(119904) =(119899minus2)radic119904

then

lim119904rarrinfin

[120597 (119891 119892)

120597 (119909 119910)(119905 119909 (119904) 119910 (119904))

minus [119910 (119904)]119899minus1

(1199011(119905) 119899119901

0(119905)

1199021(119905) 119899119902

0(119905)

)]

= (1199012(119905) (119899 minus 1) 119901

1(119905)

1199022(119905) (119899 minus 1) 119902

1(119905)

) isin 119877 (A)

(23)

Following the same idea for 119896 = 1 2 119899

(119896119901119896(119905) (119899 minus 119896 + 1) 119901

119896minus1(119905)

119896119902119896(119905) (119899 minus 119896 + 1) 119902

119896minus1(119905)

) isin 119877 (A) (24)

If (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for analgebra A with Type (I) product (given in Section 2) then

119896119901119896+ 119886119896119902

119896minus (119899 minus 119896 + 1) 119902

119896minus1= 0

119887119896119902119896+ (119899 minus 119896 + 1) 119901

119896minus1= 0

(25)

or equivalently

(119902119896minus1

119901119896minus1

) = (119896119901119896+ 119886119896119902

119896

119899 minus 119896 + 1 minus

119887119896119902119896

119899 minus 119896 + 1) (26)

for 119896 = 1 119899 Thus the functions 119901119896 119902119896

aredetermined by 119901

119899 119902119899

for 119896 = 0 1 119899 minus 1Since (119891(119905 1 0) 119892(119905 1 0)) = (119901

119899(119905) 119902119899(119905)) then

(119891(119905 119909 119910) 119892(119905 119909 119910)) = (119901119899(119905) 119902119899(119905))(119909 119910)

119899Therefore 119865 has anH(A)-differentiable lifting 119867(120591 120585) =

119867119899(120591)120585119899 where 119867

119899(120591) is an A-algebrizable lifting of 119905 997891rarr

(119901119899(119905) 119902119899(119905))

If (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for analgebra A with Type (II) product (given in Section 2) then

119896119901119896+ 119886 (119899 minus 119896 + 1) 119901

119896minus1minus (119899 minus 119896 + 1) 119902

119896minus1= 0

119896119902119896= 0

(27)

or equivalently

(119901119896 119902119896) = (minus

119886 (119899 minus 119896 + 1) 119901119896minus1

119896 0) (28)

for 119896 = 1 119899 Thus 119902119896

= 0 and 119901119896is determined by 119901

0

for 119896 = 1 2 119899 Since (119891(119905 0 1) 119892(119905 0 1)) = (1199010(119905) 1199020(119905))

then (119891(119905 119909 119910) 119892(119905 119909 119910)) = (1199010(119905) 1199020(119905))(119909 119910)

119899Therefore 119865 has anH(A)-differentiable lifting 119867(120591 120585) =

119867119899(120591)120585119899 where 119867

119899(120591) is an A-algebrizable lifting of 119905 997891rarr

(1199010(119905) 1199020(119905))

If (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for analgebra A with Type (III) product (given in Section 2) then(119899 minus 119896 + 1)119901

119896minus1= 0 and 119896119902

119896= 0 that is 119901

119896minus1= 0 119902

119896= 0

for 119896 = 1 119899 Thus 119902119896

= 0 and 119901119896is determined by 119901

0

for 119896 = 1 2 119899 Since (119891(119905 0 1) 119892(119905 0 1)) = (119901119899(119905) 1199020(119905))

then (119891(119905 119909 119910) 119892(119905 119909 119910)) = (119901119899(119905) 1199020(119905))(119909 119910)

119899Therefore 119865 has anH(A)-differentiable lifting 119867(120591 120585) =

119867119899(120591)120585119899 where 119867

119899(120591) is an A-algebrizable lifting of 119905 997891rarr

(119901119899(119905) 1199020(119905))

Thus if 119891 and 119892 are homogenous polynomial of degree119899 in the variables 119909 and 119910 then 119867(120591 120585) = 119867

119899(120591)120585119899 in each

of the cases of algebras defined by products of Types (I)(II) and (III) given in Section 2 Since a polynomial function(119905 119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) in the variables 119909 and 119910 canbe seen as the finite addition of homogenous polynomial inthe variables 119909 and 119910 by (d) of Proposition 10 (b) implies(c)

Example 13 Consider the planar system (6) then

119865 (119905 119909 119910)

= (minus1

1199053minus

2119909

119905+ 1199051199092minus 1199051199102 minus

2119910

119905+ 2119905119909119910 + 119905119910

2)

(29)

The Jacobian 120597119865120597(119909 119910) is given by

120597119865 (119905 119909 119910)

120597 (119909 119910)= (

minus2

119905+ 2119905119909 minus2119905119910

2119905119910 minus2

119905+ 2119905119909 + 2119905119910

) (30)

Journal of Applied Mathematics 7

Thus 120597119865120597(119909 119910) is orthogonal to the matrices

1198611= (

1 0

1 minus1)

1198612= (

0 1

1 0)

(31)

for all (119909 119910) that is ⟨120597119865120597(119909 119910) 119861119894⟩ = 0 for 119894 = 1 2 Themap

(119909 119910) 997891rarr 119865(119905 119909 119910) isA-algebrizable for an algebra of Type (I)with constants 119886 = 1 and 119887 = 1 see Section 2 The function 119865

can be written as

119865 (119905 119909 119910) = (minus1

1199053 0) + (minus

2

119905119909 minus

2

119905119910)

+ (1199051199092minus 1199051199102 1199051199102)

(32)

Thus the functions 119901119894and 119902

119894of Theorem 12 are given by

1199010(119905 119909 119910) = minus1119905

3 1199020(119905 119909 119910) = 0 119901

1(119905 119909 119910) = minus(2119905)119909

1199021(119905 119909 119910) = minus(2119905)119910 119901

2(119905 119909 119910) = 119905119909

2minus 1199051199102 and 119902

2(119905 119909 119910) =

1199051199102The unit 119890 of A is 119890 = (1 0) and then ℎ

0(119905) = (minus1119905

3 0)

ℎ1(119905) = (minus2119905 0) and ℎ

2(119905) = (119905 0) have A-algebrizable

liftings 1198670(120591) = minus1120591

3 1198671(120591) = minus2120591 and 119867

2(120591) = 120591

respectivelyFrom the form of 119865(119905 119909119890) = (minus1119905

3minus (2119905)119909 minus 119905119909

2)119890 119867

must be

119867(120591 120585) = minus1

1205913minus

2

120591120585 + 120591120585

2 (33)

5 The Case of Second-Degree Polynomials inthe Variables 119905 119909 and 119910

If 119891 and 119892 in (1) are quadratic polynomials in three variables119905 119909 and 119910 andA is an algebra with respect to which the map(119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable it will beshowed that the H(A)-differentiable lifting 119867 of 119865 = (119891 119892)

is a polynomial in two variables ofA Under these conditions(1) has an algebrization which is a Riccati equation over Ahaving the form

119889120585

119889120591= 119875 (120591) + 119876 (120591) 120585 + 119877 (120591) 120585

2 (34)

where119875119876 and119877 are polynomials overA of degrees two oneand zero respectively

Consider system (1) where 119891 119892 Ω sub R3 rarr R2 aresecond-degree polynomials of three variables 119905 119909 and 119910 thatis

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 1198863119910 + 11988641199052+ 1198865119905119909

+ 1198866119905119910 + 119886

71199092+ 1198868119909119910 + 119886

91199102

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198872119909 + 1198873119910 + 11988741199052+ 1198875119905119909 + 119887

6119905119910

+ 11988771199092+ 1198878119909119910 + 119887

91199102

(35)

All the quadratic vector fields which are algebrizable withrespect to algebras with Type (I) products have the form

119889119909

119889119905= 1198600+ (1198612minus 1198861198611) 119909 minus 119887119861

1119910 + (

1

21198614minus 1198861198613)1199092

minus 21198871198613119909119910 minus

1

211988711986141199102

119889119910

119889119905= 1198610+ 1198611119909 + 1198612119910 + 11986131199092+ 1198614119909119910

+ (1

21198861198614minus 1198871198613)1199102

(36)

where 119886 119887 1198600 1198610 1198611 119861

5are real constants

see Section 22 The algebrizability of (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) with respect to algebras with Type(I) products can be verified by considering 119905 as a constant

The following theorems give conditions that characterizethe algebrizability of planar systems like (1) when 119891 and119892 are quadratic polynomials The algebrizability of nonau-tonomous quadratic systems with respect to algebras withType (I) products is given in the following theorem

Theorem14 LetA be an algebrawith Type (I) product definedby constants 119886 and 119887 and 119891 119892 the polynomials (35) Thefollowing statements are equivalent

(1) The map 119865 (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 have the form

119891 (119905 119909 119910) = 1198860+ 1198861119905 + (1198873minus 1198861198872) 119909 minus 119887119887

2119910 + 11988641199052

+ (1198876minus 1198861198875) 119905119909 minus 119887119887

5119905119910

+ (1198878

2minus 1198861198877)1199092minus 21198871198877119909119910 minus

1198871198878

21199102

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198872119909 + 1198873119910 + 11988741199052+ 1198875119905119909 + 119887

6119905119910

+ 11988771199092+ 1198878119909119910 + (

1198861198878

2minus 1198871198877)1199102

(37)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (38)

is anH(A)-differentiable lifting of 119865 where

1198600= (1198860 1198870)

1198601= (1198861 1198871)

1198602= (1198873minus 1198861198872 1198872)

1198603= (1198864 1198874)

1198604= (1198876minus 1198861198875 1198875)

1198605= (

1198878

2minus 1198861198877 1198877)

(39)

8 Journal of Applied Mathematics

Proof Writing119892(119905 119909 119910) = 1198610+1198611119909+1198612119910+11986131199092+1198614119909119910+119861

51199102

yields

1198610= 1198870+ 1198871119905 + 11988741199052

1198611= 1198872+ 1198875119905

1198612= 1198873+ 1198876119905

1198613= 1198877

1198614= 1198878

1198615= 1198879

(40)

The function (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) isA-algebrizableif and only if

119891 (119905 119909 119910) = 1198600+ (1198612minus 1198861198611) 119909 minus 119887119861

1119910

+ (1

21198614minus 1198861198613)1199092minus 2119887119861

3119909119910 minus

1

211988711986141199102

(41)

where 1198600

= 1198860+ 1198861119905 + 11988641199052 and 119887

9= 11988611988782 minus 119887119887

7 Thus

statements (1) and (2) are equivalentThe function 119867 is polynomial in the variables 120591 and 120585 of

A hence 119867 is H(A)-differentiable 119867 satisfies 119867(119905119890 119909 119910) =

(119891(119905 119909 119910) 119892(119905 119909 119910)) where 119890 is the unit of A So 119867 is anH(A)-differentiable lifting of 119865 Thus statement (2) impliesstatement (3)

Since 119867 is H(A)-differentiable and 119865(119905 119909 119910) =

119867(119905119890 119909 119910) then (119909 119910) 997891rarr 119865(119905 119909 119910) is A-algebrizablefor all 119905 Thus statement (3) implies statement (1)

All the quadratic vector fields which are algebrizable withrespect to algebras with Type (II) products have the form

= 1198600+ 1198601119909 + 119860

2119910 minus

1

211988611986031199092+ 1198603119909119910 + 119860

41199102

119910 = 1198610+ (1198601+ 1198861198602) 119910 + (

1

21198603+ 1198861198604)1199102

(42)

where 119886 1198600 119860

4 1198610are real constants see Section 22

The algebrizability of (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) withrespect to algebras with Type (II) products can be verifiedby considering 119905 as a constant

The algebrizability of nonautonomous quadratic systemswith respect to algebras with Type (II) products is given in thefollowing theorem

Theorem 15 Let A be an algebra with Type (II) productdefined by the constant 119886 and let 119891 119892 be the polynomials (35)The following statements are equivalent

(1) The map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 are given by

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 1198863119910 + 11988641199052+ 1198865119905119909

+ 1198866119905119910 minus

1

211988611988681199092+ 1198868119909119910 + 119886

91199102

119892 (119905 119909 119910) = 1198870+ 1198871119905 + (1198862+ 1198861198863) 119910 + 119887

41199052

+ (1198865+ 1198861198866) 119905119910 + (

1198868

2+ 1198861198869)1199102

(43)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (44)

is anH(A)-differentiable lifting of 119865 where

1198600= (1198860 1198870)

1198601= (1198861 1198871)

1198602= (1198863 1198862+ 1198861198863)

1198603= (1198864 1198874)

1198604= (1198866 1198865+ 1198861198866)

1198605= (11988691198868

2+ 1198861198869)

(45)

Proof Writing 119891(119905 119909 119910) = 1198600+ 1198601119909 + 119860

2119910 minus (12)119886119860

31199092+

1198603119909119910 + 119860

41199102 yields

1198600= 1198860+ 1198861119905 + 11988641199052

1198601= 1198862+ 1198865119905

1198602= 1198863+ 1198866119905

1198603= 1198868

1198604= 1198869

(46)

and the map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizableif and only if

119892 (119905 119909 119910) = 1198610+ (1198601+ 1198861198602) 119910 + (

1

21198603+ 1198861198604)1199102 (47)

where 1198610= 1198870+ 1198871119905 + 11988741199052 Thus statements (1) and (2) are

equivalentThe rest of the proof is similar to that of Theorem 14

All the quadratic vector fields which are differentiablewith respect to algebras with Type (III) products have theform

= 1198600+ 1198601119909 + 119860

21199092

119910 = 1198610+ 1198611119910 + 11986121199102

(48)

Journal of Applied Mathematics 9

where 1198600 1198610 1198601 1198611 1198602 1198612

are real constantssee Section 22 The algebrizability of (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) with respect to algebras with Type(III) products can be verified by considering 119905 as a constant

The algebrizability of nonautonomous quadratic systemswith respect to algebras with Type (III) products is given inthe following theorem

Theorem 16 Let A be an algebra with Type (III) productdefined by the constant 119886 and let 119891 119892 be the polynomials (35)The following statements are equivalent

(1) The map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 are given by

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 11988641199052+ 1198865119905119909 + 119886

71199092

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198873119910 + 11988741199052+ 1198876119905119910 + 119887

91199102

(49)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (50)

is an H(A)-differentiable lifting of 119865 where 1198600

=

(1198860 1198870) 1198601

= (1198861 1198871) 1198602

= (1198862 1198873) 1198603

= (1198864 1198874)

1198604= (1198865 1198876) and 119860

5= (1198867 1198879)

Proof Function (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable if and only if

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 11988641199052+ 1198865119905119909 + 119886

71199092

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198873119910 + 11988741199052+ 1198876119905119910 + 119887

91199102

(51)

Thus first and second statements are equivalents The func-tion119867 given by119867(120591 120585) = 119860

0+1198601120591+1198602120585+11986031205912+1198604120591120585+119860

51205852

where1198600= (1198860 1198870)1198601= (1198861 1198871)1198602= (1198862 1198873)1198603= (1198864 1198874)

1198604= (1198865 1198876) 1198605= (1198867 1198879) 120591 = (119903 119904) and 120585 = (119909 119910) satisfies

119867(119905119890 119909 119910) = (119891(119905 119909 119910) 119892(119905 119909 119910)) Thus second and thirdstatements are equivalents

By Theorems 14 15 and 16 an algebrization of quadraticsystems is a Riccati equation over an algebra 119889120585119889120591 = 119875(120591) +

119876(120591)120585 + 119877(120591)1205852 where 119875(120591) = 119860

0+ 1198601120591 + 119860

31205912 119876(120591) =

1198602+ 1198604120591 and 119877(120591) = 119860

5 In the following example is

given a nonautonomous quadratic system for the which analgebrization is found by usingTheorem 14

Example 17 Consider system (1) given by

119891 (119905 119909 119910) = 1 + 7119905 minus 119909 minus 3119910 + 51199052minus 119905119909 minus 3119905119910 minus 6119909119910

minus 61199102

119892 (119905 119909 119910) = 1 + 119905 + 119909 + 119910 + 1199052+ 119905119909 + 119905119910 + 119909

2+ 4119909119910

+ 1199102

(52)

The matrices

1198611= (

1 0

2 minus1)

1198612= (

0 1

3 0)

(53)

satisfy ⟨119861119894 120597(119891 119892)120597(119909 119910)⟩ = 0 for 119894 = 1 2Thus 119865 (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for an algebra A withType (I) product defined by the constants 119886 = 2 and 119887 = 3

The conditions of Theorem 14 are satisfied then theH(A)-differentiable lifting119867 of 119865 can be written as

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (54)

where 1198600= (1 1) 119860

1= (7 1) 119860

2= (minus1 1) 119860

3= (5 1)

1198604= (minus1 1) and 119860

5= (0 1)

Disclosure

The authors declare having no financial affiliation with anyorganization regarding the material discussed here

Conflict of Interests

The authors declare that there is no conflict of interestsconcerning this textThe issues discussed in this paper do nothave any secondary interest for any of the authors

Acknowledgments

The authors wish to acknowledge the support of GrantsPromep103513 and CB-2010150532 Conacyt

References

[1] E K Blum ldquoA theory of analytic functions in Banach algebrasrdquoTransactions of the American Mathematical Society vol 78 no2 pp 343ndash370 1955

[2] P W Ketchum ldquoAnalytic functions of hypercomplex variablesrdquoTransactions of the American Mathematical Society vol 30 no4 pp 641ndash641 1928

[3] E R Lorch ldquoThe theory of analytic functions in normed abelianvector ringsrdquo Transactions of the American Mathematical Soci-ety vol 54 no 3 pp 414ndash425 1943

[4] J A Ward ldquoA theory of analytic functions in linear associativealgebrasrdquo Duke Mathematical Journal vol 7 no 1 pp 233ndash2481940

[5] J A Ward ldquoFrom generalized Cauchy-Riemann equationsto linear algebrardquo Proceedings of the American MathematicalSociety vol 4 no 3 pp 456ndash461 1953

[6] E Lopez-Gonzalez ldquoDifferential equations over algebrasrdquoAdvances and Applications in Mathematical Sciences vol 8 no2 pp 189ndash214 2011

[7] A Alvarez-Parrilla M E Frıas-Armenta E Lopez-Gonzalezand C Yee-Romero ldquoOn solving systems of autonomousordinary differential equations by reduction to a variable of analgebrardquo International Journal of Mathematics and Mathemati-cal Sciences vol 2012 Article ID 753916 21 pages 2012

10 Journal of Applied Mathematics

[8] E J Wilczynski ldquoReview Abraham Cohen an introductionto the lie theory of one-parameter groups with applicationsto the solution of differential equationsrdquo Bulletin of the Amer-ican Mathematical Society vol 18 no 10 pp 514ndash515 1912httpprojecteuclidorgeuclidbams1183421829

[9] J M PageOrdinary Differential Equations with an Introductionto Liersquos Theory or the Group of One Parameter MacmillanPublishers London UK 1897

[10] R A Steinhour The truth about lie symmetries solvingdifferential equations with symmetry methods [SeniorIndependent StudyTheses] 2013 httpopenworkswoostereduindependentstudy949

[11] R Pierce Associative Algebras Springer New York NY USA1982

[12] K I T Al-Dosary ldquoInverse integrating factor for classes of pla-nar differential systemsrdquo International Journal of MathematicalAnalysis vol 4 no 29ndash32 pp 1433ndash1446 2010

[13] I A Garcıa and M Grau ldquoA Survey on the inverse integratingfactorrdquo Qualitative Theory of Dynamical Systems vol 9 no 1-2pp 115ndash166 2010

[14] W Brown Matrices over Commutative Rings Monographs andTextbooks in Pure and Applied Mathematics Marcel Dekker1992

Submit your manuscripts athttpwwwhindawicom

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

In particular every function 119901 = (1199011 1199012) 119868 sub R rarr

R2 with quadratic components 1199011(119905) = 119886

0+ 1198861119905 + 11988621199052 and

1199012(119905) = 119887

0+1198871119905+11988721199052 admits theA-algebrizable lifting 119875(120591) =

(1198860 1198870) + (1198861 1198871)120591 + (119886

2 1198872)1205912

4 On Algebrizable Liftings of Planar Systems

Solutions of every algebrizable planar system can be foundby solving an algebrization of the system as it is seen in thefollowing proposition

Proposition 9 If (2) is an algebrization of (1) and 120585(120591) asolution of (2) then (119909(119905) 119910(119905)) = 120585(119905119890) is a solution of (1)where 119890 denotes the unit of the corresponding algebra

Proof Let 120585 be a solution of (2) The derivative of(119909(119905) 119910(119905)) = 120585(119905119890) with respect to 119905 is given by

119889

119889119905(119909 (119905) 119910 (119905)) =

119889

119889119905120585 (119905119890) = [

119889120585 (120591 120585)

119889120591

10038161003816100381610038161003816100381610038161003816120591=(119905119890)

119889120591

119889119905]

= 119867 (120591 120585 (120591))1003816100381610038161003816120591=(119905119890) 119890 = 119867 (119905119890 120585 (119905119890))

= (119891 (119905 119909 (119905) 119910 (119905)) 119892 (119905 119909 (119905) 119910 (119905)))

(19)

Thus (119909(119905) 119910(119905)) is a solution of system (1)

As a consequence of the following proposition the familyof all the functions 119865 Ω sub R3 rarr R2 having algebrizableliftings defines an infinite dimensional algebra

Proposition 10 Let A be an algebra with unit 119890 In thefollowing statements 119865 and119866 denote functions defined on opensets Ω sub R3 and they have values in R2

(a) 119865(119905 119909 119910) = 119888 admits the H(A)-differentiable lifting119867(120591 120585) = 119888

(b) 119865(119905 119909 119910) = 119905119890 admits the H(A)-differentiable lifting119867(120591 120585) = 120591

(c) 119865(119905 119909 119910) = (119909 119910) admits the H(A)-differentiablelifting119867(120591 120585) = 120585

(d) If 119865 and 119866 admit H(A)-differentiable liftings 119867119865and

119867119866 respectively and 119886 119887 are constants inA then 119886119867

119865+

119887119867119866and 119865119866 (all products with respect to A) have

H(A)-differentiable liftings 119886119867119865+ 119887119867119866and 119867

119865119867119866

respectively

(e) Every function 119864 having the form 119864(119905 119909 119910) = 119865(119905 119866(119905

119909 119910)) where 119865 and 119866 admit H(A)-differentiableliftings 119867

119865and 119867

119866 admits an H(A)-differentiable

lifting119867119864given by119867

119864(120591 120585) = 119867

119865(120591119867119866(120591 120585))

(f) Every function 119864 having the form 119864(119905 119909 119910) =

119865(119905 119875(119909 119910)) where 119865 has an H(A)-differentiablelifting119867

119865and 119875 is anA-differentiable function admits

anH(A)-differentiable lifting119867119864(120591 120585) = 119867

119865(120591 119875(120585))

Proof The proofs of (a) (b) and (c) are trivial Let 119867119865and

119867119866be the algebrizable liftings of 119865 and 119866 and then119867

119865+119867119866

119867119865119867119866 and119867

119865119867119866are A-algebrizable and

(i) (119886119867119865+119887119867119866)(119905119890 119909 119910) = 119886119867

119865(119905119890 119909 119910)+119887119867

119866(119905119890 119909 119910) =

119886119865(119905 119909 119910) + 119886119866(119905 119909 119910)

(ii) (119867119865119867119866)(119905119890 119909 119910) = 119867

119865(119905119890 119909 119910)119867

119866(119905119890 119909 119910) =

119865(119905 119909 119910)119866(119905 119909 119910)

(iii) 119867119864(119905119890 119909 119910) = 119867

119865(119905119890119867

119866(119905119890 119909 119910)) = 119865(119905 119866(119905 119909 119910))

(iv) 119867119864(119905119890 119909 119910) = 119867

119865(119905119890 119875(119890 119910)) = 119865(119905 119875(119909 119910))

Thus the proof is complete

Corollary 11 Let A be an algebra The following func-tions admitH(A)-differentiable liftings polynomial functionsrational functions trigonometric functions exponential func-tions and all of those functions which can be defined bylinear combinations products quotients and compositions offunctions admittingH(A)-differentiable liftings

Given a function 119865(119905 119909 119910) = (119891(119905 119909 119910) 119892(119905 119909 119910))where 119891 119892 are polynomial functions of the variables 119909 119910the goal of the paper is to determine if 119865 has an H(A)-differentiable lifting As a consequence of the followingtheorem every function 119865 which is polynomial of thevariables 119905 119909 and 119910 has an H(A)-differentiable liftingwhen (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-differentiable forall 119905

Theorem 12 Let A be an algebra with unit 119890 and 119865

Ω sub R3 rarr R2 119865 = (119891 119892) where 119891 = sum119898

119896=0119891119896

119892 = sum119898

119896=0119892119896 and 119891

119896(119905 119909 119910) 119892

119896(119905 119909 119910) are homogeneous

polynomials of degree 119896 in the variables 119909 119910 and Ω = 119868 times

R2 for an open interval 119868 Then the following statements areequivalent

(a) 119865 has anH(A)-differentiable lifting119867

(b) The map (119909 119910) 997891rarr 119865(119905 119909 119910) is A-algebrizable forall 119905 isin 119868 and the functions ℎ

119896given by ℎ

119896(119905) =

(119891119896(119905 119890) 119892

119896(119905 119890)) have A-algebrizable liftings for 119896 =

0 1 119898

(c) 119867(120591 120585) = sum119898

119896=0119867119896(120591)120585119896 is an H(A)-differentiable

lifting of 119865 where 119867119896(120591) are A-algebrizable liftings of

ℎ119896(119905) = (119891

119896(119905 119890) 119892

119896(119905 119890))

Proof Obviously (a) implies (b) and (c) implies (a) We nowshow that (b) implies (c) Suppose that119891(119905 119909 119910) and119892(119905 119909 119910)

are homogenous polynomials 119891(119905 119909 119910) = sum119899

119896=0119901119896(119905)119909119896119910119899minus119896

and 119892(119905 119909 119910) = sum119899

119896=0119902119896(119905)119909119896119910119899minus119896 of the variables 119909 119910 defined

6 Journal of Applied Mathematics

on the setΩ = 119868 timesR2 as above Taking the partial derivativesof 119865 with respect to 119909 and 119910 we obtain

120597 (119891 119892)

120597 (119909 119910)(119905 119909 119910)

= (

119899

sum

119896=1

119896119901119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119901119896minus1

(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

119896119902119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119902119896minus1

(119905) 119909119896minus1

119910119899minus119896

)

= (

119899

sum

119896=1

119896119901119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119901119896minus1

(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

119896119902119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119902119896minus1

(119905) 119909119896minus1

119910119899minus119896

)

=

119899

sum

119896=1

119909119896minus1

119910119899minus119896

(119896119901119896(119905) (119899 minus 119896 + 1) 119901

119896minus1(119905)

119896119902119896(119905) (119899 minus 119896 + 1) 119902

119896minus1(119905)

)

(20)

Let 119877 be the first fundamental representation of A Since(119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for all 119905 isin 119868then (120597(119891 119892)120597(119909 119910))(119905 119909 119910) isin 119877(A) for all (119905 119909 119910) isin ΩThus

120597 (119891 119892)

120597 (119909 119910)(119905 0 1) = (

1199011(119905) 119899119901

0(119905)

1199021(119905) 119899119902

0(119905)

) isin 119877 (A) (21)

for all 119905 isin 119868 and then

120597 (119891 119892)

120597 (119909 119910)(119905 119909 119910) minus 119910

119899minus1(1199011(119905) 119899119901

0(119905)

1199021(119905) 119899119902

0(119905)

)

=

119899

sum

119896=2

119909119896minus1

119910119899minus119896

(119896119901119896(119905) (119899 minus 119896 + 1) 119901

119896minus1(119905)

119896119902119896(119905) (119899 minus 119896 + 1) 119902

119896minus1(119905)

)

(22)

is in 119877(A) for all (119905 119909 119910) isin Ω If 119909(119904) = 1119904 and 119910(119904) =(119899minus2)radic119904

then

lim119904rarrinfin

[120597 (119891 119892)

120597 (119909 119910)(119905 119909 (119904) 119910 (119904))

minus [119910 (119904)]119899minus1

(1199011(119905) 119899119901

0(119905)

1199021(119905) 119899119902

0(119905)

)]

= (1199012(119905) (119899 minus 1) 119901

1(119905)

1199022(119905) (119899 minus 1) 119902

1(119905)

) isin 119877 (A)

(23)

Following the same idea for 119896 = 1 2 119899

(119896119901119896(119905) (119899 minus 119896 + 1) 119901

119896minus1(119905)

119896119902119896(119905) (119899 minus 119896 + 1) 119902

119896minus1(119905)

) isin 119877 (A) (24)

If (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for analgebra A with Type (I) product (given in Section 2) then

119896119901119896+ 119886119896119902

119896minus (119899 minus 119896 + 1) 119902

119896minus1= 0

119887119896119902119896+ (119899 minus 119896 + 1) 119901

119896minus1= 0

(25)

or equivalently

(119902119896minus1

119901119896minus1

) = (119896119901119896+ 119886119896119902

119896

119899 minus 119896 + 1 minus

119887119896119902119896

119899 minus 119896 + 1) (26)

for 119896 = 1 119899 Thus the functions 119901119896 119902119896

aredetermined by 119901

119899 119902119899

for 119896 = 0 1 119899 minus 1Since (119891(119905 1 0) 119892(119905 1 0)) = (119901

119899(119905) 119902119899(119905)) then

(119891(119905 119909 119910) 119892(119905 119909 119910)) = (119901119899(119905) 119902119899(119905))(119909 119910)

119899Therefore 119865 has anH(A)-differentiable lifting 119867(120591 120585) =

119867119899(120591)120585119899 where 119867

119899(120591) is an A-algebrizable lifting of 119905 997891rarr

(119901119899(119905) 119902119899(119905))

If (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for analgebra A with Type (II) product (given in Section 2) then

119896119901119896+ 119886 (119899 minus 119896 + 1) 119901

119896minus1minus (119899 minus 119896 + 1) 119902

119896minus1= 0

119896119902119896= 0

(27)

or equivalently

(119901119896 119902119896) = (minus

119886 (119899 minus 119896 + 1) 119901119896minus1

119896 0) (28)

for 119896 = 1 119899 Thus 119902119896

= 0 and 119901119896is determined by 119901

0

for 119896 = 1 2 119899 Since (119891(119905 0 1) 119892(119905 0 1)) = (1199010(119905) 1199020(119905))

then (119891(119905 119909 119910) 119892(119905 119909 119910)) = (1199010(119905) 1199020(119905))(119909 119910)

119899Therefore 119865 has anH(A)-differentiable lifting 119867(120591 120585) =

119867119899(120591)120585119899 where 119867

119899(120591) is an A-algebrizable lifting of 119905 997891rarr

(1199010(119905) 1199020(119905))

If (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for analgebra A with Type (III) product (given in Section 2) then(119899 minus 119896 + 1)119901

119896minus1= 0 and 119896119902

119896= 0 that is 119901

119896minus1= 0 119902

119896= 0

for 119896 = 1 119899 Thus 119902119896

= 0 and 119901119896is determined by 119901

0

for 119896 = 1 2 119899 Since (119891(119905 0 1) 119892(119905 0 1)) = (119901119899(119905) 1199020(119905))

then (119891(119905 119909 119910) 119892(119905 119909 119910)) = (119901119899(119905) 1199020(119905))(119909 119910)

119899Therefore 119865 has anH(A)-differentiable lifting 119867(120591 120585) =

119867119899(120591)120585119899 where 119867

119899(120591) is an A-algebrizable lifting of 119905 997891rarr

(119901119899(119905) 1199020(119905))

Thus if 119891 and 119892 are homogenous polynomial of degree119899 in the variables 119909 and 119910 then 119867(120591 120585) = 119867

119899(120591)120585119899 in each

of the cases of algebras defined by products of Types (I)(II) and (III) given in Section 2 Since a polynomial function(119905 119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) in the variables 119909 and 119910 canbe seen as the finite addition of homogenous polynomial inthe variables 119909 and 119910 by (d) of Proposition 10 (b) implies(c)

Example 13 Consider the planar system (6) then

119865 (119905 119909 119910)

= (minus1

1199053minus

2119909

119905+ 1199051199092minus 1199051199102 minus

2119910

119905+ 2119905119909119910 + 119905119910

2)

(29)

The Jacobian 120597119865120597(119909 119910) is given by

120597119865 (119905 119909 119910)

120597 (119909 119910)= (

minus2

119905+ 2119905119909 minus2119905119910

2119905119910 minus2

119905+ 2119905119909 + 2119905119910

) (30)

Journal of Applied Mathematics 7

Thus 120597119865120597(119909 119910) is orthogonal to the matrices

1198611= (

1 0

1 minus1)

1198612= (

0 1

1 0)

(31)

for all (119909 119910) that is ⟨120597119865120597(119909 119910) 119861119894⟩ = 0 for 119894 = 1 2 Themap

(119909 119910) 997891rarr 119865(119905 119909 119910) isA-algebrizable for an algebra of Type (I)with constants 119886 = 1 and 119887 = 1 see Section 2 The function 119865

can be written as

119865 (119905 119909 119910) = (minus1

1199053 0) + (minus

2

119905119909 minus

2

119905119910)

+ (1199051199092minus 1199051199102 1199051199102)

(32)

Thus the functions 119901119894and 119902

119894of Theorem 12 are given by

1199010(119905 119909 119910) = minus1119905

3 1199020(119905 119909 119910) = 0 119901

1(119905 119909 119910) = minus(2119905)119909

1199021(119905 119909 119910) = minus(2119905)119910 119901

2(119905 119909 119910) = 119905119909

2minus 1199051199102 and 119902

2(119905 119909 119910) =

1199051199102The unit 119890 of A is 119890 = (1 0) and then ℎ

0(119905) = (minus1119905

3 0)

ℎ1(119905) = (minus2119905 0) and ℎ

2(119905) = (119905 0) have A-algebrizable

liftings 1198670(120591) = minus1120591

3 1198671(120591) = minus2120591 and 119867

2(120591) = 120591

respectivelyFrom the form of 119865(119905 119909119890) = (minus1119905

3minus (2119905)119909 minus 119905119909

2)119890 119867

must be

119867(120591 120585) = minus1

1205913minus

2

120591120585 + 120591120585

2 (33)

5 The Case of Second-Degree Polynomials inthe Variables 119905 119909 and 119910

If 119891 and 119892 in (1) are quadratic polynomials in three variables119905 119909 and 119910 andA is an algebra with respect to which the map(119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable it will beshowed that the H(A)-differentiable lifting 119867 of 119865 = (119891 119892)

is a polynomial in two variables ofA Under these conditions(1) has an algebrization which is a Riccati equation over Ahaving the form

119889120585

119889120591= 119875 (120591) + 119876 (120591) 120585 + 119877 (120591) 120585

2 (34)

where119875119876 and119877 are polynomials overA of degrees two oneand zero respectively

Consider system (1) where 119891 119892 Ω sub R3 rarr R2 aresecond-degree polynomials of three variables 119905 119909 and 119910 thatis

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 1198863119910 + 11988641199052+ 1198865119905119909

+ 1198866119905119910 + 119886

71199092+ 1198868119909119910 + 119886

91199102

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198872119909 + 1198873119910 + 11988741199052+ 1198875119905119909 + 119887

6119905119910

+ 11988771199092+ 1198878119909119910 + 119887

91199102

(35)

All the quadratic vector fields which are algebrizable withrespect to algebras with Type (I) products have the form

119889119909

119889119905= 1198600+ (1198612minus 1198861198611) 119909 minus 119887119861

1119910 + (

1

21198614minus 1198861198613)1199092

minus 21198871198613119909119910 minus

1

211988711986141199102

119889119910

119889119905= 1198610+ 1198611119909 + 1198612119910 + 11986131199092+ 1198614119909119910

+ (1

21198861198614minus 1198871198613)1199102

(36)

where 119886 119887 1198600 1198610 1198611 119861

5are real constants

see Section 22 The algebrizability of (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) with respect to algebras with Type(I) products can be verified by considering 119905 as a constant

The following theorems give conditions that characterizethe algebrizability of planar systems like (1) when 119891 and119892 are quadratic polynomials The algebrizability of nonau-tonomous quadratic systems with respect to algebras withType (I) products is given in the following theorem

Theorem14 LetA be an algebrawith Type (I) product definedby constants 119886 and 119887 and 119891 119892 the polynomials (35) Thefollowing statements are equivalent

(1) The map 119865 (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 have the form

119891 (119905 119909 119910) = 1198860+ 1198861119905 + (1198873minus 1198861198872) 119909 minus 119887119887

2119910 + 11988641199052

+ (1198876minus 1198861198875) 119905119909 minus 119887119887

5119905119910

+ (1198878

2minus 1198861198877)1199092minus 21198871198877119909119910 minus

1198871198878

21199102

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198872119909 + 1198873119910 + 11988741199052+ 1198875119905119909 + 119887

6119905119910

+ 11988771199092+ 1198878119909119910 + (

1198861198878

2minus 1198871198877)1199102

(37)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (38)

is anH(A)-differentiable lifting of 119865 where

1198600= (1198860 1198870)

1198601= (1198861 1198871)

1198602= (1198873minus 1198861198872 1198872)

1198603= (1198864 1198874)

1198604= (1198876minus 1198861198875 1198875)

1198605= (

1198878

2minus 1198861198877 1198877)

(39)

8 Journal of Applied Mathematics

Proof Writing119892(119905 119909 119910) = 1198610+1198611119909+1198612119910+11986131199092+1198614119909119910+119861

51199102

yields

1198610= 1198870+ 1198871119905 + 11988741199052

1198611= 1198872+ 1198875119905

1198612= 1198873+ 1198876119905

1198613= 1198877

1198614= 1198878

1198615= 1198879

(40)

The function (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) isA-algebrizableif and only if

119891 (119905 119909 119910) = 1198600+ (1198612minus 1198861198611) 119909 minus 119887119861

1119910

+ (1

21198614minus 1198861198613)1199092minus 2119887119861

3119909119910 minus

1

211988711986141199102

(41)

where 1198600

= 1198860+ 1198861119905 + 11988641199052 and 119887

9= 11988611988782 minus 119887119887

7 Thus

statements (1) and (2) are equivalentThe function 119867 is polynomial in the variables 120591 and 120585 of

A hence 119867 is H(A)-differentiable 119867 satisfies 119867(119905119890 119909 119910) =

(119891(119905 119909 119910) 119892(119905 119909 119910)) where 119890 is the unit of A So 119867 is anH(A)-differentiable lifting of 119865 Thus statement (2) impliesstatement (3)

Since 119867 is H(A)-differentiable and 119865(119905 119909 119910) =

119867(119905119890 119909 119910) then (119909 119910) 997891rarr 119865(119905 119909 119910) is A-algebrizablefor all 119905 Thus statement (3) implies statement (1)

All the quadratic vector fields which are algebrizable withrespect to algebras with Type (II) products have the form

= 1198600+ 1198601119909 + 119860

2119910 minus

1

211988611986031199092+ 1198603119909119910 + 119860

41199102

119910 = 1198610+ (1198601+ 1198861198602) 119910 + (

1

21198603+ 1198861198604)1199102

(42)

where 119886 1198600 119860

4 1198610are real constants see Section 22

The algebrizability of (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) withrespect to algebras with Type (II) products can be verifiedby considering 119905 as a constant

The algebrizability of nonautonomous quadratic systemswith respect to algebras with Type (II) products is given in thefollowing theorem

Theorem 15 Let A be an algebra with Type (II) productdefined by the constant 119886 and let 119891 119892 be the polynomials (35)The following statements are equivalent

(1) The map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 are given by

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 1198863119910 + 11988641199052+ 1198865119905119909

+ 1198866119905119910 minus

1

211988611988681199092+ 1198868119909119910 + 119886

91199102

119892 (119905 119909 119910) = 1198870+ 1198871119905 + (1198862+ 1198861198863) 119910 + 119887

41199052

+ (1198865+ 1198861198866) 119905119910 + (

1198868

2+ 1198861198869)1199102

(43)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (44)

is anH(A)-differentiable lifting of 119865 where

1198600= (1198860 1198870)

1198601= (1198861 1198871)

1198602= (1198863 1198862+ 1198861198863)

1198603= (1198864 1198874)

1198604= (1198866 1198865+ 1198861198866)

1198605= (11988691198868

2+ 1198861198869)

(45)

Proof Writing 119891(119905 119909 119910) = 1198600+ 1198601119909 + 119860

2119910 minus (12)119886119860

31199092+

1198603119909119910 + 119860

41199102 yields

1198600= 1198860+ 1198861119905 + 11988641199052

1198601= 1198862+ 1198865119905

1198602= 1198863+ 1198866119905

1198603= 1198868

1198604= 1198869

(46)

and the map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizableif and only if

119892 (119905 119909 119910) = 1198610+ (1198601+ 1198861198602) 119910 + (

1

21198603+ 1198861198604)1199102 (47)

where 1198610= 1198870+ 1198871119905 + 11988741199052 Thus statements (1) and (2) are

equivalentThe rest of the proof is similar to that of Theorem 14

All the quadratic vector fields which are differentiablewith respect to algebras with Type (III) products have theform

= 1198600+ 1198601119909 + 119860

21199092

119910 = 1198610+ 1198611119910 + 11986121199102

(48)

Journal of Applied Mathematics 9

where 1198600 1198610 1198601 1198611 1198602 1198612

are real constantssee Section 22 The algebrizability of (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) with respect to algebras with Type(III) products can be verified by considering 119905 as a constant

The algebrizability of nonautonomous quadratic systemswith respect to algebras with Type (III) products is given inthe following theorem

Theorem 16 Let A be an algebra with Type (III) productdefined by the constant 119886 and let 119891 119892 be the polynomials (35)The following statements are equivalent

(1) The map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 are given by

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 11988641199052+ 1198865119905119909 + 119886

71199092

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198873119910 + 11988741199052+ 1198876119905119910 + 119887

91199102

(49)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (50)

is an H(A)-differentiable lifting of 119865 where 1198600

=

(1198860 1198870) 1198601

= (1198861 1198871) 1198602

= (1198862 1198873) 1198603

= (1198864 1198874)

1198604= (1198865 1198876) and 119860

5= (1198867 1198879)

Proof Function (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable if and only if

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 11988641199052+ 1198865119905119909 + 119886

71199092

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198873119910 + 11988741199052+ 1198876119905119910 + 119887

91199102

(51)

Thus first and second statements are equivalents The func-tion119867 given by119867(120591 120585) = 119860

0+1198601120591+1198602120585+11986031205912+1198604120591120585+119860

51205852

where1198600= (1198860 1198870)1198601= (1198861 1198871)1198602= (1198862 1198873)1198603= (1198864 1198874)

1198604= (1198865 1198876) 1198605= (1198867 1198879) 120591 = (119903 119904) and 120585 = (119909 119910) satisfies

119867(119905119890 119909 119910) = (119891(119905 119909 119910) 119892(119905 119909 119910)) Thus second and thirdstatements are equivalents

By Theorems 14 15 and 16 an algebrization of quadraticsystems is a Riccati equation over an algebra 119889120585119889120591 = 119875(120591) +

119876(120591)120585 + 119877(120591)1205852 where 119875(120591) = 119860

0+ 1198601120591 + 119860

31205912 119876(120591) =

1198602+ 1198604120591 and 119877(120591) = 119860

5 In the following example is

given a nonautonomous quadratic system for the which analgebrization is found by usingTheorem 14

Example 17 Consider system (1) given by

119891 (119905 119909 119910) = 1 + 7119905 minus 119909 minus 3119910 + 51199052minus 119905119909 minus 3119905119910 minus 6119909119910

minus 61199102

119892 (119905 119909 119910) = 1 + 119905 + 119909 + 119910 + 1199052+ 119905119909 + 119905119910 + 119909

2+ 4119909119910

+ 1199102

(52)

The matrices

1198611= (

1 0

2 minus1)

1198612= (

0 1

3 0)

(53)

satisfy ⟨119861119894 120597(119891 119892)120597(119909 119910)⟩ = 0 for 119894 = 1 2Thus 119865 (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for an algebra A withType (I) product defined by the constants 119886 = 2 and 119887 = 3

The conditions of Theorem 14 are satisfied then theH(A)-differentiable lifting119867 of 119865 can be written as

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (54)

where 1198600= (1 1) 119860

1= (7 1) 119860

2= (minus1 1) 119860

3= (5 1)

1198604= (minus1 1) and 119860

5= (0 1)

Disclosure

The authors declare having no financial affiliation with anyorganization regarding the material discussed here

Conflict of Interests

The authors declare that there is no conflict of interestsconcerning this textThe issues discussed in this paper do nothave any secondary interest for any of the authors

Acknowledgments

The authors wish to acknowledge the support of GrantsPromep103513 and CB-2010150532 Conacyt

References

[1] E K Blum ldquoA theory of analytic functions in Banach algebrasrdquoTransactions of the American Mathematical Society vol 78 no2 pp 343ndash370 1955

[2] P W Ketchum ldquoAnalytic functions of hypercomplex variablesrdquoTransactions of the American Mathematical Society vol 30 no4 pp 641ndash641 1928

[3] E R Lorch ldquoThe theory of analytic functions in normed abelianvector ringsrdquo Transactions of the American Mathematical Soci-ety vol 54 no 3 pp 414ndash425 1943

[4] J A Ward ldquoA theory of analytic functions in linear associativealgebrasrdquo Duke Mathematical Journal vol 7 no 1 pp 233ndash2481940

[5] J A Ward ldquoFrom generalized Cauchy-Riemann equationsto linear algebrardquo Proceedings of the American MathematicalSociety vol 4 no 3 pp 456ndash461 1953

[6] E Lopez-Gonzalez ldquoDifferential equations over algebrasrdquoAdvances and Applications in Mathematical Sciences vol 8 no2 pp 189ndash214 2011

[7] A Alvarez-Parrilla M E Frıas-Armenta E Lopez-Gonzalezand C Yee-Romero ldquoOn solving systems of autonomousordinary differential equations by reduction to a variable of analgebrardquo International Journal of Mathematics and Mathemati-cal Sciences vol 2012 Article ID 753916 21 pages 2012

10 Journal of Applied Mathematics

[8] E J Wilczynski ldquoReview Abraham Cohen an introductionto the lie theory of one-parameter groups with applicationsto the solution of differential equationsrdquo Bulletin of the Amer-ican Mathematical Society vol 18 no 10 pp 514ndash515 1912httpprojecteuclidorgeuclidbams1183421829

[9] J M PageOrdinary Differential Equations with an Introductionto Liersquos Theory or the Group of One Parameter MacmillanPublishers London UK 1897

[10] R A Steinhour The truth about lie symmetries solvingdifferential equations with symmetry methods [SeniorIndependent StudyTheses] 2013 httpopenworkswoostereduindependentstudy949

[11] R Pierce Associative Algebras Springer New York NY USA1982

[12] K I T Al-Dosary ldquoInverse integrating factor for classes of pla-nar differential systemsrdquo International Journal of MathematicalAnalysis vol 4 no 29ndash32 pp 1433ndash1446 2010

[13] I A Garcıa and M Grau ldquoA Survey on the inverse integratingfactorrdquo Qualitative Theory of Dynamical Systems vol 9 no 1-2pp 115ndash166 2010

[14] W Brown Matrices over Commutative Rings Monographs andTextbooks in Pure and Applied Mathematics Marcel Dekker1992

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Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

on the setΩ = 119868 timesR2 as above Taking the partial derivativesof 119865 with respect to 119909 and 119910 we obtain

120597 (119891 119892)

120597 (119909 119910)(119905 119909 119910)

= (

119899

sum

119896=1

119896119901119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119901119896minus1

(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

119896119902119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119902119896minus1

(119905) 119909119896minus1

119910119899minus119896

)

= (

119899

sum

119896=1

119896119901119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119901119896minus1

(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

119896119902119896(119905) 119909119896minus1

119910119899minus119896

119899

sum

119896=1

(119899 minus 119896 + 1) 119902119896minus1

(119905) 119909119896minus1

119910119899minus119896

)

=

119899

sum

119896=1

119909119896minus1

119910119899minus119896

(119896119901119896(119905) (119899 minus 119896 + 1) 119901

119896minus1(119905)

119896119902119896(119905) (119899 minus 119896 + 1) 119902

119896minus1(119905)

)

(20)

Let 119877 be the first fundamental representation of A Since(119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for all 119905 isin 119868then (120597(119891 119892)120597(119909 119910))(119905 119909 119910) isin 119877(A) for all (119905 119909 119910) isin ΩThus

120597 (119891 119892)

120597 (119909 119910)(119905 0 1) = (

1199011(119905) 119899119901

0(119905)

1199021(119905) 119899119902

0(119905)

) isin 119877 (A) (21)

for all 119905 isin 119868 and then

120597 (119891 119892)

120597 (119909 119910)(119905 119909 119910) minus 119910

119899minus1(1199011(119905) 119899119901

0(119905)

1199021(119905) 119899119902

0(119905)

)

=

119899

sum

119896=2

119909119896minus1

119910119899minus119896

(119896119901119896(119905) (119899 minus 119896 + 1) 119901

119896minus1(119905)

119896119902119896(119905) (119899 minus 119896 + 1) 119902

119896minus1(119905)

)

(22)

is in 119877(A) for all (119905 119909 119910) isin Ω If 119909(119904) = 1119904 and 119910(119904) =(119899minus2)radic119904

then

lim119904rarrinfin

[120597 (119891 119892)

120597 (119909 119910)(119905 119909 (119904) 119910 (119904))

minus [119910 (119904)]119899minus1

(1199011(119905) 119899119901

0(119905)

1199021(119905) 119899119902

0(119905)

)]

= (1199012(119905) (119899 minus 1) 119901

1(119905)

1199022(119905) (119899 minus 1) 119902

1(119905)

) isin 119877 (A)

(23)

Following the same idea for 119896 = 1 2 119899

(119896119901119896(119905) (119899 minus 119896 + 1) 119901

119896minus1(119905)

119896119902119896(119905) (119899 minus 119896 + 1) 119902

119896minus1(119905)

) isin 119877 (A) (24)

If (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for analgebra A with Type (I) product (given in Section 2) then

119896119901119896+ 119886119896119902

119896minus (119899 minus 119896 + 1) 119902

119896minus1= 0

119887119896119902119896+ (119899 minus 119896 + 1) 119901

119896minus1= 0

(25)

or equivalently

(119902119896minus1

119901119896minus1

) = (119896119901119896+ 119886119896119902

119896

119899 minus 119896 + 1 minus

119887119896119902119896

119899 minus 119896 + 1) (26)

for 119896 = 1 119899 Thus the functions 119901119896 119902119896

aredetermined by 119901

119899 119902119899

for 119896 = 0 1 119899 minus 1Since (119891(119905 1 0) 119892(119905 1 0)) = (119901

119899(119905) 119902119899(119905)) then

(119891(119905 119909 119910) 119892(119905 119909 119910)) = (119901119899(119905) 119902119899(119905))(119909 119910)

119899Therefore 119865 has anH(A)-differentiable lifting 119867(120591 120585) =

119867119899(120591)120585119899 where 119867

119899(120591) is an A-algebrizable lifting of 119905 997891rarr

(119901119899(119905) 119902119899(119905))

If (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for analgebra A with Type (II) product (given in Section 2) then

119896119901119896+ 119886 (119899 minus 119896 + 1) 119901

119896minus1minus (119899 minus 119896 + 1) 119902

119896minus1= 0

119896119902119896= 0

(27)

or equivalently

(119901119896 119902119896) = (minus

119886 (119899 minus 119896 + 1) 119901119896minus1

119896 0) (28)

for 119896 = 1 119899 Thus 119902119896

= 0 and 119901119896is determined by 119901

0

for 119896 = 1 2 119899 Since (119891(119905 0 1) 119892(119905 0 1)) = (1199010(119905) 1199020(119905))

then (119891(119905 119909 119910) 119892(119905 119909 119910)) = (1199010(119905) 1199020(119905))(119909 119910)

119899Therefore 119865 has anH(A)-differentiable lifting 119867(120591 120585) =

119867119899(120591)120585119899 where 119867

119899(120591) is an A-algebrizable lifting of 119905 997891rarr

(1199010(119905) 1199020(119905))

If (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for analgebra A with Type (III) product (given in Section 2) then(119899 minus 119896 + 1)119901

119896minus1= 0 and 119896119902

119896= 0 that is 119901

119896minus1= 0 119902

119896= 0

for 119896 = 1 119899 Thus 119902119896

= 0 and 119901119896is determined by 119901

0

for 119896 = 1 2 119899 Since (119891(119905 0 1) 119892(119905 0 1)) = (119901119899(119905) 1199020(119905))

then (119891(119905 119909 119910) 119892(119905 119909 119910)) = (119901119899(119905) 1199020(119905))(119909 119910)

119899Therefore 119865 has anH(A)-differentiable lifting 119867(120591 120585) =

119867119899(120591)120585119899 where 119867

119899(120591) is an A-algebrizable lifting of 119905 997891rarr

(119901119899(119905) 1199020(119905))

Thus if 119891 and 119892 are homogenous polynomial of degree119899 in the variables 119909 and 119910 then 119867(120591 120585) = 119867

119899(120591)120585119899 in each

of the cases of algebras defined by products of Types (I)(II) and (III) given in Section 2 Since a polynomial function(119905 119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) in the variables 119909 and 119910 canbe seen as the finite addition of homogenous polynomial inthe variables 119909 and 119910 by (d) of Proposition 10 (b) implies(c)

Example 13 Consider the planar system (6) then

119865 (119905 119909 119910)

= (minus1

1199053minus

2119909

119905+ 1199051199092minus 1199051199102 minus

2119910

119905+ 2119905119909119910 + 119905119910

2)

(29)

The Jacobian 120597119865120597(119909 119910) is given by

120597119865 (119905 119909 119910)

120597 (119909 119910)= (

minus2

119905+ 2119905119909 minus2119905119910

2119905119910 minus2

119905+ 2119905119909 + 2119905119910

) (30)

Journal of Applied Mathematics 7

Thus 120597119865120597(119909 119910) is orthogonal to the matrices

1198611= (

1 0

1 minus1)

1198612= (

0 1

1 0)

(31)

for all (119909 119910) that is ⟨120597119865120597(119909 119910) 119861119894⟩ = 0 for 119894 = 1 2 Themap

(119909 119910) 997891rarr 119865(119905 119909 119910) isA-algebrizable for an algebra of Type (I)with constants 119886 = 1 and 119887 = 1 see Section 2 The function 119865

can be written as

119865 (119905 119909 119910) = (minus1

1199053 0) + (minus

2

119905119909 minus

2

119905119910)

+ (1199051199092minus 1199051199102 1199051199102)

(32)

Thus the functions 119901119894and 119902

119894of Theorem 12 are given by

1199010(119905 119909 119910) = minus1119905

3 1199020(119905 119909 119910) = 0 119901

1(119905 119909 119910) = minus(2119905)119909

1199021(119905 119909 119910) = minus(2119905)119910 119901

2(119905 119909 119910) = 119905119909

2minus 1199051199102 and 119902

2(119905 119909 119910) =

1199051199102The unit 119890 of A is 119890 = (1 0) and then ℎ

0(119905) = (minus1119905

3 0)

ℎ1(119905) = (minus2119905 0) and ℎ

2(119905) = (119905 0) have A-algebrizable

liftings 1198670(120591) = minus1120591

3 1198671(120591) = minus2120591 and 119867

2(120591) = 120591

respectivelyFrom the form of 119865(119905 119909119890) = (minus1119905

3minus (2119905)119909 minus 119905119909

2)119890 119867

must be

119867(120591 120585) = minus1

1205913minus

2

120591120585 + 120591120585

2 (33)

5 The Case of Second-Degree Polynomials inthe Variables 119905 119909 and 119910

If 119891 and 119892 in (1) are quadratic polynomials in three variables119905 119909 and 119910 andA is an algebra with respect to which the map(119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable it will beshowed that the H(A)-differentiable lifting 119867 of 119865 = (119891 119892)

is a polynomial in two variables ofA Under these conditions(1) has an algebrization which is a Riccati equation over Ahaving the form

119889120585

119889120591= 119875 (120591) + 119876 (120591) 120585 + 119877 (120591) 120585

2 (34)

where119875119876 and119877 are polynomials overA of degrees two oneand zero respectively

Consider system (1) where 119891 119892 Ω sub R3 rarr R2 aresecond-degree polynomials of three variables 119905 119909 and 119910 thatis

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 1198863119910 + 11988641199052+ 1198865119905119909

+ 1198866119905119910 + 119886

71199092+ 1198868119909119910 + 119886

91199102

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198872119909 + 1198873119910 + 11988741199052+ 1198875119905119909 + 119887

6119905119910

+ 11988771199092+ 1198878119909119910 + 119887

91199102

(35)

All the quadratic vector fields which are algebrizable withrespect to algebras with Type (I) products have the form

119889119909

119889119905= 1198600+ (1198612minus 1198861198611) 119909 minus 119887119861

1119910 + (

1

21198614minus 1198861198613)1199092

minus 21198871198613119909119910 minus

1

211988711986141199102

119889119910

119889119905= 1198610+ 1198611119909 + 1198612119910 + 11986131199092+ 1198614119909119910

+ (1

21198861198614minus 1198871198613)1199102

(36)

where 119886 119887 1198600 1198610 1198611 119861

5are real constants

see Section 22 The algebrizability of (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) with respect to algebras with Type(I) products can be verified by considering 119905 as a constant

The following theorems give conditions that characterizethe algebrizability of planar systems like (1) when 119891 and119892 are quadratic polynomials The algebrizability of nonau-tonomous quadratic systems with respect to algebras withType (I) products is given in the following theorem

Theorem14 LetA be an algebrawith Type (I) product definedby constants 119886 and 119887 and 119891 119892 the polynomials (35) Thefollowing statements are equivalent

(1) The map 119865 (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 have the form

119891 (119905 119909 119910) = 1198860+ 1198861119905 + (1198873minus 1198861198872) 119909 minus 119887119887

2119910 + 11988641199052

+ (1198876minus 1198861198875) 119905119909 minus 119887119887

5119905119910

+ (1198878

2minus 1198861198877)1199092minus 21198871198877119909119910 minus

1198871198878

21199102

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198872119909 + 1198873119910 + 11988741199052+ 1198875119905119909 + 119887

6119905119910

+ 11988771199092+ 1198878119909119910 + (

1198861198878

2minus 1198871198877)1199102

(37)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (38)

is anH(A)-differentiable lifting of 119865 where

1198600= (1198860 1198870)

1198601= (1198861 1198871)

1198602= (1198873minus 1198861198872 1198872)

1198603= (1198864 1198874)

1198604= (1198876minus 1198861198875 1198875)

1198605= (

1198878

2minus 1198861198877 1198877)

(39)

8 Journal of Applied Mathematics

Proof Writing119892(119905 119909 119910) = 1198610+1198611119909+1198612119910+11986131199092+1198614119909119910+119861

51199102

yields

1198610= 1198870+ 1198871119905 + 11988741199052

1198611= 1198872+ 1198875119905

1198612= 1198873+ 1198876119905

1198613= 1198877

1198614= 1198878

1198615= 1198879

(40)

The function (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) isA-algebrizableif and only if

119891 (119905 119909 119910) = 1198600+ (1198612minus 1198861198611) 119909 minus 119887119861

1119910

+ (1

21198614minus 1198861198613)1199092minus 2119887119861

3119909119910 minus

1

211988711986141199102

(41)

where 1198600

= 1198860+ 1198861119905 + 11988641199052 and 119887

9= 11988611988782 minus 119887119887

7 Thus

statements (1) and (2) are equivalentThe function 119867 is polynomial in the variables 120591 and 120585 of

A hence 119867 is H(A)-differentiable 119867 satisfies 119867(119905119890 119909 119910) =

(119891(119905 119909 119910) 119892(119905 119909 119910)) where 119890 is the unit of A So 119867 is anH(A)-differentiable lifting of 119865 Thus statement (2) impliesstatement (3)

Since 119867 is H(A)-differentiable and 119865(119905 119909 119910) =

119867(119905119890 119909 119910) then (119909 119910) 997891rarr 119865(119905 119909 119910) is A-algebrizablefor all 119905 Thus statement (3) implies statement (1)

All the quadratic vector fields which are algebrizable withrespect to algebras with Type (II) products have the form

= 1198600+ 1198601119909 + 119860

2119910 minus

1

211988611986031199092+ 1198603119909119910 + 119860

41199102

119910 = 1198610+ (1198601+ 1198861198602) 119910 + (

1

21198603+ 1198861198604)1199102

(42)

where 119886 1198600 119860

4 1198610are real constants see Section 22

The algebrizability of (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) withrespect to algebras with Type (II) products can be verifiedby considering 119905 as a constant

The algebrizability of nonautonomous quadratic systemswith respect to algebras with Type (II) products is given in thefollowing theorem

Theorem 15 Let A be an algebra with Type (II) productdefined by the constant 119886 and let 119891 119892 be the polynomials (35)The following statements are equivalent

(1) The map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 are given by

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 1198863119910 + 11988641199052+ 1198865119905119909

+ 1198866119905119910 minus

1

211988611988681199092+ 1198868119909119910 + 119886

91199102

119892 (119905 119909 119910) = 1198870+ 1198871119905 + (1198862+ 1198861198863) 119910 + 119887

41199052

+ (1198865+ 1198861198866) 119905119910 + (

1198868

2+ 1198861198869)1199102

(43)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (44)

is anH(A)-differentiable lifting of 119865 where

1198600= (1198860 1198870)

1198601= (1198861 1198871)

1198602= (1198863 1198862+ 1198861198863)

1198603= (1198864 1198874)

1198604= (1198866 1198865+ 1198861198866)

1198605= (11988691198868

2+ 1198861198869)

(45)

Proof Writing 119891(119905 119909 119910) = 1198600+ 1198601119909 + 119860

2119910 minus (12)119886119860

31199092+

1198603119909119910 + 119860

41199102 yields

1198600= 1198860+ 1198861119905 + 11988641199052

1198601= 1198862+ 1198865119905

1198602= 1198863+ 1198866119905

1198603= 1198868

1198604= 1198869

(46)

and the map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizableif and only if

119892 (119905 119909 119910) = 1198610+ (1198601+ 1198861198602) 119910 + (

1

21198603+ 1198861198604)1199102 (47)

where 1198610= 1198870+ 1198871119905 + 11988741199052 Thus statements (1) and (2) are

equivalentThe rest of the proof is similar to that of Theorem 14

All the quadratic vector fields which are differentiablewith respect to algebras with Type (III) products have theform

= 1198600+ 1198601119909 + 119860

21199092

119910 = 1198610+ 1198611119910 + 11986121199102

(48)

Journal of Applied Mathematics 9

where 1198600 1198610 1198601 1198611 1198602 1198612

are real constantssee Section 22 The algebrizability of (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) with respect to algebras with Type(III) products can be verified by considering 119905 as a constant

The algebrizability of nonautonomous quadratic systemswith respect to algebras with Type (III) products is given inthe following theorem

Theorem 16 Let A be an algebra with Type (III) productdefined by the constant 119886 and let 119891 119892 be the polynomials (35)The following statements are equivalent

(1) The map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 are given by

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 11988641199052+ 1198865119905119909 + 119886

71199092

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198873119910 + 11988741199052+ 1198876119905119910 + 119887

91199102

(49)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (50)

is an H(A)-differentiable lifting of 119865 where 1198600

=

(1198860 1198870) 1198601

= (1198861 1198871) 1198602

= (1198862 1198873) 1198603

= (1198864 1198874)

1198604= (1198865 1198876) and 119860

5= (1198867 1198879)

Proof Function (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable if and only if

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 11988641199052+ 1198865119905119909 + 119886

71199092

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198873119910 + 11988741199052+ 1198876119905119910 + 119887

91199102

(51)

Thus first and second statements are equivalents The func-tion119867 given by119867(120591 120585) = 119860

0+1198601120591+1198602120585+11986031205912+1198604120591120585+119860

51205852

where1198600= (1198860 1198870)1198601= (1198861 1198871)1198602= (1198862 1198873)1198603= (1198864 1198874)

1198604= (1198865 1198876) 1198605= (1198867 1198879) 120591 = (119903 119904) and 120585 = (119909 119910) satisfies

119867(119905119890 119909 119910) = (119891(119905 119909 119910) 119892(119905 119909 119910)) Thus second and thirdstatements are equivalents

By Theorems 14 15 and 16 an algebrization of quadraticsystems is a Riccati equation over an algebra 119889120585119889120591 = 119875(120591) +

119876(120591)120585 + 119877(120591)1205852 where 119875(120591) = 119860

0+ 1198601120591 + 119860

31205912 119876(120591) =

1198602+ 1198604120591 and 119877(120591) = 119860

5 In the following example is

given a nonautonomous quadratic system for the which analgebrization is found by usingTheorem 14

Example 17 Consider system (1) given by

119891 (119905 119909 119910) = 1 + 7119905 minus 119909 minus 3119910 + 51199052minus 119905119909 minus 3119905119910 minus 6119909119910

minus 61199102

119892 (119905 119909 119910) = 1 + 119905 + 119909 + 119910 + 1199052+ 119905119909 + 119905119910 + 119909

2+ 4119909119910

+ 1199102

(52)

The matrices

1198611= (

1 0

2 minus1)

1198612= (

0 1

3 0)

(53)

satisfy ⟨119861119894 120597(119891 119892)120597(119909 119910)⟩ = 0 for 119894 = 1 2Thus 119865 (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for an algebra A withType (I) product defined by the constants 119886 = 2 and 119887 = 3

The conditions of Theorem 14 are satisfied then theH(A)-differentiable lifting119867 of 119865 can be written as

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (54)

where 1198600= (1 1) 119860

1= (7 1) 119860

2= (minus1 1) 119860

3= (5 1)

1198604= (minus1 1) and 119860

5= (0 1)

Disclosure

The authors declare having no financial affiliation with anyorganization regarding the material discussed here

Conflict of Interests

The authors declare that there is no conflict of interestsconcerning this textThe issues discussed in this paper do nothave any secondary interest for any of the authors

Acknowledgments

The authors wish to acknowledge the support of GrantsPromep103513 and CB-2010150532 Conacyt

References

[1] E K Blum ldquoA theory of analytic functions in Banach algebrasrdquoTransactions of the American Mathematical Society vol 78 no2 pp 343ndash370 1955

[2] P W Ketchum ldquoAnalytic functions of hypercomplex variablesrdquoTransactions of the American Mathematical Society vol 30 no4 pp 641ndash641 1928

[3] E R Lorch ldquoThe theory of analytic functions in normed abelianvector ringsrdquo Transactions of the American Mathematical Soci-ety vol 54 no 3 pp 414ndash425 1943

[4] J A Ward ldquoA theory of analytic functions in linear associativealgebrasrdquo Duke Mathematical Journal vol 7 no 1 pp 233ndash2481940

[5] J A Ward ldquoFrom generalized Cauchy-Riemann equationsto linear algebrardquo Proceedings of the American MathematicalSociety vol 4 no 3 pp 456ndash461 1953

[6] E Lopez-Gonzalez ldquoDifferential equations over algebrasrdquoAdvances and Applications in Mathematical Sciences vol 8 no2 pp 189ndash214 2011

[7] A Alvarez-Parrilla M E Frıas-Armenta E Lopez-Gonzalezand C Yee-Romero ldquoOn solving systems of autonomousordinary differential equations by reduction to a variable of analgebrardquo International Journal of Mathematics and Mathemati-cal Sciences vol 2012 Article ID 753916 21 pages 2012

10 Journal of Applied Mathematics

[8] E J Wilczynski ldquoReview Abraham Cohen an introductionto the lie theory of one-parameter groups with applicationsto the solution of differential equationsrdquo Bulletin of the Amer-ican Mathematical Society vol 18 no 10 pp 514ndash515 1912httpprojecteuclidorgeuclidbams1183421829

[9] J M PageOrdinary Differential Equations with an Introductionto Liersquos Theory or the Group of One Parameter MacmillanPublishers London UK 1897

[10] R A Steinhour The truth about lie symmetries solvingdifferential equations with symmetry methods [SeniorIndependent StudyTheses] 2013 httpopenworkswoostereduindependentstudy949

[11] R Pierce Associative Algebras Springer New York NY USA1982

[12] K I T Al-Dosary ldquoInverse integrating factor for classes of pla-nar differential systemsrdquo International Journal of MathematicalAnalysis vol 4 no 29ndash32 pp 1433ndash1446 2010

[13] I A Garcıa and M Grau ldquoA Survey on the inverse integratingfactorrdquo Qualitative Theory of Dynamical Systems vol 9 no 1-2pp 115ndash166 2010

[14] W Brown Matrices over Commutative Rings Monographs andTextbooks in Pure and Applied Mathematics Marcel Dekker1992

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Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

Thus 120597119865120597(119909 119910) is orthogonal to the matrices

1198611= (

1 0

1 minus1)

1198612= (

0 1

1 0)

(31)

for all (119909 119910) that is ⟨120597119865120597(119909 119910) 119861119894⟩ = 0 for 119894 = 1 2 Themap

(119909 119910) 997891rarr 119865(119905 119909 119910) isA-algebrizable for an algebra of Type (I)with constants 119886 = 1 and 119887 = 1 see Section 2 The function 119865

can be written as

119865 (119905 119909 119910) = (minus1

1199053 0) + (minus

2

119905119909 minus

2

119905119910)

+ (1199051199092minus 1199051199102 1199051199102)

(32)

Thus the functions 119901119894and 119902

119894of Theorem 12 are given by

1199010(119905 119909 119910) = minus1119905

3 1199020(119905 119909 119910) = 0 119901

1(119905 119909 119910) = minus(2119905)119909

1199021(119905 119909 119910) = minus(2119905)119910 119901

2(119905 119909 119910) = 119905119909

2minus 1199051199102 and 119902

2(119905 119909 119910) =

1199051199102The unit 119890 of A is 119890 = (1 0) and then ℎ

0(119905) = (minus1119905

3 0)

ℎ1(119905) = (minus2119905 0) and ℎ

2(119905) = (119905 0) have A-algebrizable

liftings 1198670(120591) = minus1120591

3 1198671(120591) = minus2120591 and 119867

2(120591) = 120591

respectivelyFrom the form of 119865(119905 119909119890) = (minus1119905

3minus (2119905)119909 minus 119905119909

2)119890 119867

must be

119867(120591 120585) = minus1

1205913minus

2

120591120585 + 120591120585

2 (33)

5 The Case of Second-Degree Polynomials inthe Variables 119905 119909 and 119910

If 119891 and 119892 in (1) are quadratic polynomials in three variables119905 119909 and 119910 andA is an algebra with respect to which the map(119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable it will beshowed that the H(A)-differentiable lifting 119867 of 119865 = (119891 119892)

is a polynomial in two variables ofA Under these conditions(1) has an algebrization which is a Riccati equation over Ahaving the form

119889120585

119889120591= 119875 (120591) + 119876 (120591) 120585 + 119877 (120591) 120585

2 (34)

where119875119876 and119877 are polynomials overA of degrees two oneand zero respectively

Consider system (1) where 119891 119892 Ω sub R3 rarr R2 aresecond-degree polynomials of three variables 119905 119909 and 119910 thatis

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 1198863119910 + 11988641199052+ 1198865119905119909

+ 1198866119905119910 + 119886

71199092+ 1198868119909119910 + 119886

91199102

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198872119909 + 1198873119910 + 11988741199052+ 1198875119905119909 + 119887

6119905119910

+ 11988771199092+ 1198878119909119910 + 119887

91199102

(35)

All the quadratic vector fields which are algebrizable withrespect to algebras with Type (I) products have the form

119889119909

119889119905= 1198600+ (1198612minus 1198861198611) 119909 minus 119887119861

1119910 + (

1

21198614minus 1198861198613)1199092

minus 21198871198613119909119910 minus

1

211988711986141199102

119889119910

119889119905= 1198610+ 1198611119909 + 1198612119910 + 11986131199092+ 1198614119909119910

+ (1

21198861198614minus 1198871198613)1199102

(36)

where 119886 119887 1198600 1198610 1198611 119861

5are real constants

see Section 22 The algebrizability of (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) with respect to algebras with Type(I) products can be verified by considering 119905 as a constant

The following theorems give conditions that characterizethe algebrizability of planar systems like (1) when 119891 and119892 are quadratic polynomials The algebrizability of nonau-tonomous quadratic systems with respect to algebras withType (I) products is given in the following theorem

Theorem14 LetA be an algebrawith Type (I) product definedby constants 119886 and 119887 and 119891 119892 the polynomials (35) Thefollowing statements are equivalent

(1) The map 119865 (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 have the form

119891 (119905 119909 119910) = 1198860+ 1198861119905 + (1198873minus 1198861198872) 119909 minus 119887119887

2119910 + 11988641199052

+ (1198876minus 1198861198875) 119905119909 minus 119887119887

5119905119910

+ (1198878

2minus 1198861198877)1199092minus 21198871198877119909119910 minus

1198871198878

21199102

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198872119909 + 1198873119910 + 11988741199052+ 1198875119905119909 + 119887

6119905119910

+ 11988771199092+ 1198878119909119910 + (

1198861198878

2minus 1198871198877)1199102

(37)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (38)

is anH(A)-differentiable lifting of 119865 where

1198600= (1198860 1198870)

1198601= (1198861 1198871)

1198602= (1198873minus 1198861198872 1198872)

1198603= (1198864 1198874)

1198604= (1198876minus 1198861198875 1198875)

1198605= (

1198878

2minus 1198861198877 1198877)

(39)

8 Journal of Applied Mathematics

Proof Writing119892(119905 119909 119910) = 1198610+1198611119909+1198612119910+11986131199092+1198614119909119910+119861

51199102

yields

1198610= 1198870+ 1198871119905 + 11988741199052

1198611= 1198872+ 1198875119905

1198612= 1198873+ 1198876119905

1198613= 1198877

1198614= 1198878

1198615= 1198879

(40)

The function (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) isA-algebrizableif and only if

119891 (119905 119909 119910) = 1198600+ (1198612minus 1198861198611) 119909 minus 119887119861

1119910

+ (1

21198614minus 1198861198613)1199092minus 2119887119861

3119909119910 minus

1

211988711986141199102

(41)

where 1198600

= 1198860+ 1198861119905 + 11988641199052 and 119887

9= 11988611988782 minus 119887119887

7 Thus

statements (1) and (2) are equivalentThe function 119867 is polynomial in the variables 120591 and 120585 of

A hence 119867 is H(A)-differentiable 119867 satisfies 119867(119905119890 119909 119910) =

(119891(119905 119909 119910) 119892(119905 119909 119910)) where 119890 is the unit of A So 119867 is anH(A)-differentiable lifting of 119865 Thus statement (2) impliesstatement (3)

Since 119867 is H(A)-differentiable and 119865(119905 119909 119910) =

119867(119905119890 119909 119910) then (119909 119910) 997891rarr 119865(119905 119909 119910) is A-algebrizablefor all 119905 Thus statement (3) implies statement (1)

All the quadratic vector fields which are algebrizable withrespect to algebras with Type (II) products have the form

= 1198600+ 1198601119909 + 119860

2119910 minus

1

211988611986031199092+ 1198603119909119910 + 119860

41199102

119910 = 1198610+ (1198601+ 1198861198602) 119910 + (

1

21198603+ 1198861198604)1199102

(42)

where 119886 1198600 119860

4 1198610are real constants see Section 22

The algebrizability of (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) withrespect to algebras with Type (II) products can be verifiedby considering 119905 as a constant

The algebrizability of nonautonomous quadratic systemswith respect to algebras with Type (II) products is given in thefollowing theorem

Theorem 15 Let A be an algebra with Type (II) productdefined by the constant 119886 and let 119891 119892 be the polynomials (35)The following statements are equivalent

(1) The map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 are given by

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 1198863119910 + 11988641199052+ 1198865119905119909

+ 1198866119905119910 minus

1

211988611988681199092+ 1198868119909119910 + 119886

91199102

119892 (119905 119909 119910) = 1198870+ 1198871119905 + (1198862+ 1198861198863) 119910 + 119887

41199052

+ (1198865+ 1198861198866) 119905119910 + (

1198868

2+ 1198861198869)1199102

(43)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (44)

is anH(A)-differentiable lifting of 119865 where

1198600= (1198860 1198870)

1198601= (1198861 1198871)

1198602= (1198863 1198862+ 1198861198863)

1198603= (1198864 1198874)

1198604= (1198866 1198865+ 1198861198866)

1198605= (11988691198868

2+ 1198861198869)

(45)

Proof Writing 119891(119905 119909 119910) = 1198600+ 1198601119909 + 119860

2119910 minus (12)119886119860

31199092+

1198603119909119910 + 119860

41199102 yields

1198600= 1198860+ 1198861119905 + 11988641199052

1198601= 1198862+ 1198865119905

1198602= 1198863+ 1198866119905

1198603= 1198868

1198604= 1198869

(46)

and the map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizableif and only if

119892 (119905 119909 119910) = 1198610+ (1198601+ 1198861198602) 119910 + (

1

21198603+ 1198861198604)1199102 (47)

where 1198610= 1198870+ 1198871119905 + 11988741199052 Thus statements (1) and (2) are

equivalentThe rest of the proof is similar to that of Theorem 14

All the quadratic vector fields which are differentiablewith respect to algebras with Type (III) products have theform

= 1198600+ 1198601119909 + 119860

21199092

119910 = 1198610+ 1198611119910 + 11986121199102

(48)

Journal of Applied Mathematics 9

where 1198600 1198610 1198601 1198611 1198602 1198612

are real constantssee Section 22 The algebrizability of (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) with respect to algebras with Type(III) products can be verified by considering 119905 as a constant

The algebrizability of nonautonomous quadratic systemswith respect to algebras with Type (III) products is given inthe following theorem

Theorem 16 Let A be an algebra with Type (III) productdefined by the constant 119886 and let 119891 119892 be the polynomials (35)The following statements are equivalent

(1) The map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 are given by

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 11988641199052+ 1198865119905119909 + 119886

71199092

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198873119910 + 11988741199052+ 1198876119905119910 + 119887

91199102

(49)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (50)

is an H(A)-differentiable lifting of 119865 where 1198600

=

(1198860 1198870) 1198601

= (1198861 1198871) 1198602

= (1198862 1198873) 1198603

= (1198864 1198874)

1198604= (1198865 1198876) and 119860

5= (1198867 1198879)

Proof Function (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable if and only if

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 11988641199052+ 1198865119905119909 + 119886

71199092

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198873119910 + 11988741199052+ 1198876119905119910 + 119887

91199102

(51)

Thus first and second statements are equivalents The func-tion119867 given by119867(120591 120585) = 119860

0+1198601120591+1198602120585+11986031205912+1198604120591120585+119860

51205852

where1198600= (1198860 1198870)1198601= (1198861 1198871)1198602= (1198862 1198873)1198603= (1198864 1198874)

1198604= (1198865 1198876) 1198605= (1198867 1198879) 120591 = (119903 119904) and 120585 = (119909 119910) satisfies

119867(119905119890 119909 119910) = (119891(119905 119909 119910) 119892(119905 119909 119910)) Thus second and thirdstatements are equivalents

By Theorems 14 15 and 16 an algebrization of quadraticsystems is a Riccati equation over an algebra 119889120585119889120591 = 119875(120591) +

119876(120591)120585 + 119877(120591)1205852 where 119875(120591) = 119860

0+ 1198601120591 + 119860

31205912 119876(120591) =

1198602+ 1198604120591 and 119877(120591) = 119860

5 In the following example is

given a nonautonomous quadratic system for the which analgebrization is found by usingTheorem 14

Example 17 Consider system (1) given by

119891 (119905 119909 119910) = 1 + 7119905 minus 119909 minus 3119910 + 51199052minus 119905119909 minus 3119905119910 minus 6119909119910

minus 61199102

119892 (119905 119909 119910) = 1 + 119905 + 119909 + 119910 + 1199052+ 119905119909 + 119905119910 + 119909

2+ 4119909119910

+ 1199102

(52)

The matrices

1198611= (

1 0

2 minus1)

1198612= (

0 1

3 0)

(53)

satisfy ⟨119861119894 120597(119891 119892)120597(119909 119910)⟩ = 0 for 119894 = 1 2Thus 119865 (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for an algebra A withType (I) product defined by the constants 119886 = 2 and 119887 = 3

The conditions of Theorem 14 are satisfied then theH(A)-differentiable lifting119867 of 119865 can be written as

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (54)

where 1198600= (1 1) 119860

1= (7 1) 119860

2= (minus1 1) 119860

3= (5 1)

1198604= (minus1 1) and 119860

5= (0 1)

Disclosure

The authors declare having no financial affiliation with anyorganization regarding the material discussed here

Conflict of Interests

The authors declare that there is no conflict of interestsconcerning this textThe issues discussed in this paper do nothave any secondary interest for any of the authors

Acknowledgments

The authors wish to acknowledge the support of GrantsPromep103513 and CB-2010150532 Conacyt

References

[1] E K Blum ldquoA theory of analytic functions in Banach algebrasrdquoTransactions of the American Mathematical Society vol 78 no2 pp 343ndash370 1955

[2] P W Ketchum ldquoAnalytic functions of hypercomplex variablesrdquoTransactions of the American Mathematical Society vol 30 no4 pp 641ndash641 1928

[3] E R Lorch ldquoThe theory of analytic functions in normed abelianvector ringsrdquo Transactions of the American Mathematical Soci-ety vol 54 no 3 pp 414ndash425 1943

[4] J A Ward ldquoA theory of analytic functions in linear associativealgebrasrdquo Duke Mathematical Journal vol 7 no 1 pp 233ndash2481940

[5] J A Ward ldquoFrom generalized Cauchy-Riemann equationsto linear algebrardquo Proceedings of the American MathematicalSociety vol 4 no 3 pp 456ndash461 1953

[6] E Lopez-Gonzalez ldquoDifferential equations over algebrasrdquoAdvances and Applications in Mathematical Sciences vol 8 no2 pp 189ndash214 2011

[7] A Alvarez-Parrilla M E Frıas-Armenta E Lopez-Gonzalezand C Yee-Romero ldquoOn solving systems of autonomousordinary differential equations by reduction to a variable of analgebrardquo International Journal of Mathematics and Mathemati-cal Sciences vol 2012 Article ID 753916 21 pages 2012

10 Journal of Applied Mathematics

[8] E J Wilczynski ldquoReview Abraham Cohen an introductionto the lie theory of one-parameter groups with applicationsto the solution of differential equationsrdquo Bulletin of the Amer-ican Mathematical Society vol 18 no 10 pp 514ndash515 1912httpprojecteuclidorgeuclidbams1183421829

[9] J M PageOrdinary Differential Equations with an Introductionto Liersquos Theory or the Group of One Parameter MacmillanPublishers London UK 1897

[10] R A Steinhour The truth about lie symmetries solvingdifferential equations with symmetry methods [SeniorIndependent StudyTheses] 2013 httpopenworkswoostereduindependentstudy949

[11] R Pierce Associative Algebras Springer New York NY USA1982

[12] K I T Al-Dosary ldquoInverse integrating factor for classes of pla-nar differential systemsrdquo International Journal of MathematicalAnalysis vol 4 no 29ndash32 pp 1433ndash1446 2010

[13] I A Garcıa and M Grau ldquoA Survey on the inverse integratingfactorrdquo Qualitative Theory of Dynamical Systems vol 9 no 1-2pp 115ndash166 2010

[14] W Brown Matrices over Commutative Rings Monographs andTextbooks in Pure and Applied Mathematics Marcel Dekker1992

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

Proof Writing119892(119905 119909 119910) = 1198610+1198611119909+1198612119910+11986131199092+1198614119909119910+119861

51199102

yields

1198610= 1198870+ 1198871119905 + 11988741199052

1198611= 1198872+ 1198875119905

1198612= 1198873+ 1198876119905

1198613= 1198877

1198614= 1198878

1198615= 1198879

(40)

The function (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) isA-algebrizableif and only if

119891 (119905 119909 119910) = 1198600+ (1198612minus 1198861198611) 119909 minus 119887119861

1119910

+ (1

21198614minus 1198861198613)1199092minus 2119887119861

3119909119910 minus

1

211988711986141199102

(41)

where 1198600

= 1198860+ 1198861119905 + 11988641199052 and 119887

9= 11988611988782 minus 119887119887

7 Thus

statements (1) and (2) are equivalentThe function 119867 is polynomial in the variables 120591 and 120585 of

A hence 119867 is H(A)-differentiable 119867 satisfies 119867(119905119890 119909 119910) =

(119891(119905 119909 119910) 119892(119905 119909 119910)) where 119890 is the unit of A So 119867 is anH(A)-differentiable lifting of 119865 Thus statement (2) impliesstatement (3)

Since 119867 is H(A)-differentiable and 119865(119905 119909 119910) =

119867(119905119890 119909 119910) then (119909 119910) 997891rarr 119865(119905 119909 119910) is A-algebrizablefor all 119905 Thus statement (3) implies statement (1)

All the quadratic vector fields which are algebrizable withrespect to algebras with Type (II) products have the form

= 1198600+ 1198601119909 + 119860

2119910 minus

1

211988611986031199092+ 1198603119909119910 + 119860

41199102

119910 = 1198610+ (1198601+ 1198861198602) 119910 + (

1

21198603+ 1198861198604)1199102

(42)

where 119886 1198600 119860

4 1198610are real constants see Section 22

The algebrizability of (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) withrespect to algebras with Type (II) products can be verifiedby considering 119905 as a constant

The algebrizability of nonautonomous quadratic systemswith respect to algebras with Type (II) products is given in thefollowing theorem

Theorem 15 Let A be an algebra with Type (II) productdefined by the constant 119886 and let 119891 119892 be the polynomials (35)The following statements are equivalent

(1) The map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 are given by

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 1198863119910 + 11988641199052+ 1198865119905119909

+ 1198866119905119910 minus

1

211988611988681199092+ 1198868119909119910 + 119886

91199102

119892 (119905 119909 119910) = 1198870+ 1198871119905 + (1198862+ 1198861198863) 119910 + 119887

41199052

+ (1198865+ 1198861198866) 119905119910 + (

1198868

2+ 1198861198869)1199102

(43)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (44)

is anH(A)-differentiable lifting of 119865 where

1198600= (1198860 1198870)

1198601= (1198861 1198871)

1198602= (1198863 1198862+ 1198861198863)

1198603= (1198864 1198874)

1198604= (1198866 1198865+ 1198861198866)

1198605= (11988691198868

2+ 1198861198869)

(45)

Proof Writing 119891(119905 119909 119910) = 1198600+ 1198601119909 + 119860

2119910 minus (12)119886119860

31199092+

1198603119909119910 + 119860

41199102 yields

1198600= 1198860+ 1198861119905 + 11988641199052

1198601= 1198862+ 1198865119905

1198602= 1198863+ 1198866119905

1198603= 1198868

1198604= 1198869

(46)

and the map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizableif and only if

119892 (119905 119909 119910) = 1198610+ (1198601+ 1198861198602) 119910 + (

1

21198603+ 1198861198604)1199102 (47)

where 1198610= 1198870+ 1198871119905 + 11988741199052 Thus statements (1) and (2) are

equivalentThe rest of the proof is similar to that of Theorem 14

All the quadratic vector fields which are differentiablewith respect to algebras with Type (III) products have theform

= 1198600+ 1198601119909 + 119860

21199092

119910 = 1198610+ 1198611119910 + 11986121199102

(48)

Journal of Applied Mathematics 9

where 1198600 1198610 1198601 1198611 1198602 1198612

are real constantssee Section 22 The algebrizability of (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) with respect to algebras with Type(III) products can be verified by considering 119905 as a constant

The algebrizability of nonautonomous quadratic systemswith respect to algebras with Type (III) products is given inthe following theorem

Theorem 16 Let A be an algebra with Type (III) productdefined by the constant 119886 and let 119891 119892 be the polynomials (35)The following statements are equivalent

(1) The map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 are given by

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 11988641199052+ 1198865119905119909 + 119886

71199092

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198873119910 + 11988741199052+ 1198876119905119910 + 119887

91199102

(49)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (50)

is an H(A)-differentiable lifting of 119865 where 1198600

=

(1198860 1198870) 1198601

= (1198861 1198871) 1198602

= (1198862 1198873) 1198603

= (1198864 1198874)

1198604= (1198865 1198876) and 119860

5= (1198867 1198879)

Proof Function (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable if and only if

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 11988641199052+ 1198865119905119909 + 119886

71199092

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198873119910 + 11988741199052+ 1198876119905119910 + 119887

91199102

(51)

Thus first and second statements are equivalents The func-tion119867 given by119867(120591 120585) = 119860

0+1198601120591+1198602120585+11986031205912+1198604120591120585+119860

51205852

where1198600= (1198860 1198870)1198601= (1198861 1198871)1198602= (1198862 1198873)1198603= (1198864 1198874)

1198604= (1198865 1198876) 1198605= (1198867 1198879) 120591 = (119903 119904) and 120585 = (119909 119910) satisfies

119867(119905119890 119909 119910) = (119891(119905 119909 119910) 119892(119905 119909 119910)) Thus second and thirdstatements are equivalents

By Theorems 14 15 and 16 an algebrization of quadraticsystems is a Riccati equation over an algebra 119889120585119889120591 = 119875(120591) +

119876(120591)120585 + 119877(120591)1205852 where 119875(120591) = 119860

0+ 1198601120591 + 119860

31205912 119876(120591) =

1198602+ 1198604120591 and 119877(120591) = 119860

5 In the following example is

given a nonautonomous quadratic system for the which analgebrization is found by usingTheorem 14

Example 17 Consider system (1) given by

119891 (119905 119909 119910) = 1 + 7119905 minus 119909 minus 3119910 + 51199052minus 119905119909 minus 3119905119910 minus 6119909119910

minus 61199102

119892 (119905 119909 119910) = 1 + 119905 + 119909 + 119910 + 1199052+ 119905119909 + 119905119910 + 119909

2+ 4119909119910

+ 1199102

(52)

The matrices

1198611= (

1 0

2 minus1)

1198612= (

0 1

3 0)

(53)

satisfy ⟨119861119894 120597(119891 119892)120597(119909 119910)⟩ = 0 for 119894 = 1 2Thus 119865 (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for an algebra A withType (I) product defined by the constants 119886 = 2 and 119887 = 3

The conditions of Theorem 14 are satisfied then theH(A)-differentiable lifting119867 of 119865 can be written as

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (54)

where 1198600= (1 1) 119860

1= (7 1) 119860

2= (minus1 1) 119860

3= (5 1)

1198604= (minus1 1) and 119860

5= (0 1)

Disclosure

The authors declare having no financial affiliation with anyorganization regarding the material discussed here

Conflict of Interests

The authors declare that there is no conflict of interestsconcerning this textThe issues discussed in this paper do nothave any secondary interest for any of the authors

Acknowledgments

The authors wish to acknowledge the support of GrantsPromep103513 and CB-2010150532 Conacyt

References

[1] E K Blum ldquoA theory of analytic functions in Banach algebrasrdquoTransactions of the American Mathematical Society vol 78 no2 pp 343ndash370 1955

[2] P W Ketchum ldquoAnalytic functions of hypercomplex variablesrdquoTransactions of the American Mathematical Society vol 30 no4 pp 641ndash641 1928

[3] E R Lorch ldquoThe theory of analytic functions in normed abelianvector ringsrdquo Transactions of the American Mathematical Soci-ety vol 54 no 3 pp 414ndash425 1943

[4] J A Ward ldquoA theory of analytic functions in linear associativealgebrasrdquo Duke Mathematical Journal vol 7 no 1 pp 233ndash2481940

[5] J A Ward ldquoFrom generalized Cauchy-Riemann equationsto linear algebrardquo Proceedings of the American MathematicalSociety vol 4 no 3 pp 456ndash461 1953

[6] E Lopez-Gonzalez ldquoDifferential equations over algebrasrdquoAdvances and Applications in Mathematical Sciences vol 8 no2 pp 189ndash214 2011

[7] A Alvarez-Parrilla M E Frıas-Armenta E Lopez-Gonzalezand C Yee-Romero ldquoOn solving systems of autonomousordinary differential equations by reduction to a variable of analgebrardquo International Journal of Mathematics and Mathemati-cal Sciences vol 2012 Article ID 753916 21 pages 2012

10 Journal of Applied Mathematics

[8] E J Wilczynski ldquoReview Abraham Cohen an introductionto the lie theory of one-parameter groups with applicationsto the solution of differential equationsrdquo Bulletin of the Amer-ican Mathematical Society vol 18 no 10 pp 514ndash515 1912httpprojecteuclidorgeuclidbams1183421829

[9] J M PageOrdinary Differential Equations with an Introductionto Liersquos Theory or the Group of One Parameter MacmillanPublishers London UK 1897

[10] R A Steinhour The truth about lie symmetries solvingdifferential equations with symmetry methods [SeniorIndependent StudyTheses] 2013 httpopenworkswoostereduindependentstudy949

[11] R Pierce Associative Algebras Springer New York NY USA1982

[12] K I T Al-Dosary ldquoInverse integrating factor for classes of pla-nar differential systemsrdquo International Journal of MathematicalAnalysis vol 4 no 29ndash32 pp 1433ndash1446 2010

[13] I A Garcıa and M Grau ldquoA Survey on the inverse integratingfactorrdquo Qualitative Theory of Dynamical Systems vol 9 no 1-2pp 115ndash166 2010

[14] W Brown Matrices over Commutative Rings Monographs andTextbooks in Pure and Applied Mathematics Marcel Dekker1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

where 1198600 1198610 1198601 1198611 1198602 1198612

are real constantssee Section 22 The algebrizability of (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) with respect to algebras with Type(III) products can be verified by considering 119905 as a constant

The algebrizability of nonautonomous quadratic systemswith respect to algebras with Type (III) products is given inthe following theorem

Theorem 16 Let A be an algebra with Type (III) productdefined by the constant 119886 and let 119891 119892 be the polynomials (35)The following statements are equivalent

(1) The map (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable

(2) The functions 119891 and 119892 are given by

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 11988641199052+ 1198865119905119909 + 119886

71199092

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198873119910 + 11988741199052+ 1198876119905119910 + 119887

91199102

(49)

(3) The function119867 AtimesA rarr A of the variables 120591 = (119905 119904)

and 120585 = (119909 119910) defined by

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (50)

is an H(A)-differentiable lifting of 119865 where 1198600

=

(1198860 1198870) 1198601

= (1198861 1198871) 1198602

= (1198862 1198873) 1198603

= (1198864 1198874)

1198604= (1198865 1198876) and 119860

5= (1198867 1198879)

Proof Function (119909 119910) 997891rarr (119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable if and only if

119891 (119905 119909 119910) = 1198860+ 1198861119905 + 1198862119909 + 11988641199052+ 1198865119905119909 + 119886

71199092

119892 (119905 119909 119910) = 1198870+ 1198871119905 + 1198873119910 + 11988741199052+ 1198876119905119910 + 119887

91199102

(51)

Thus first and second statements are equivalents The func-tion119867 given by119867(120591 120585) = 119860

0+1198601120591+1198602120585+11986031205912+1198604120591120585+119860

51205852

where1198600= (1198860 1198870)1198601= (1198861 1198871)1198602= (1198862 1198873)1198603= (1198864 1198874)

1198604= (1198865 1198876) 1198605= (1198867 1198879) 120591 = (119903 119904) and 120585 = (119909 119910) satisfies

119867(119905119890 119909 119910) = (119891(119905 119909 119910) 119892(119905 119909 119910)) Thus second and thirdstatements are equivalents

By Theorems 14 15 and 16 an algebrization of quadraticsystems is a Riccati equation over an algebra 119889120585119889120591 = 119875(120591) +

119876(120591)120585 + 119877(120591)1205852 where 119875(120591) = 119860

0+ 1198601120591 + 119860

31205912 119876(120591) =

1198602+ 1198604120591 and 119877(120591) = 119860

5 In the following example is

given a nonautonomous quadratic system for the which analgebrization is found by usingTheorem 14

Example 17 Consider system (1) given by

119891 (119905 119909 119910) = 1 + 7119905 minus 119909 minus 3119910 + 51199052minus 119905119909 minus 3119905119910 minus 6119909119910

minus 61199102

119892 (119905 119909 119910) = 1 + 119905 + 119909 + 119910 + 1199052+ 119905119909 + 119905119910 + 119909

2+ 4119909119910

+ 1199102

(52)

The matrices

1198611= (

1 0

2 minus1)

1198612= (

0 1

3 0)

(53)

satisfy ⟨119861119894 120597(119891 119892)120597(119909 119910)⟩ = 0 for 119894 = 1 2Thus 119865 (119909 119910) 997891rarr

(119891(119905 119909 119910) 119892(119905 119909 119910)) is A-algebrizable for an algebra A withType (I) product defined by the constants 119886 = 2 and 119887 = 3

The conditions of Theorem 14 are satisfied then theH(A)-differentiable lifting119867 of 119865 can be written as

119867(120591 120585) = 1198600+ 1198601120591 + 119860

2120585 + 119860

31205912+ 1198604120591120585 + 119860

51205852 (54)

where 1198600= (1 1) 119860

1= (7 1) 119860

2= (minus1 1) 119860

3= (5 1)

1198604= (minus1 1) and 119860

5= (0 1)

Disclosure

The authors declare having no financial affiliation with anyorganization regarding the material discussed here

Conflict of Interests

The authors declare that there is no conflict of interestsconcerning this textThe issues discussed in this paper do nothave any secondary interest for any of the authors

Acknowledgments

The authors wish to acknowledge the support of GrantsPromep103513 and CB-2010150532 Conacyt

References

[1] E K Blum ldquoA theory of analytic functions in Banach algebrasrdquoTransactions of the American Mathematical Society vol 78 no2 pp 343ndash370 1955

[2] P W Ketchum ldquoAnalytic functions of hypercomplex variablesrdquoTransactions of the American Mathematical Society vol 30 no4 pp 641ndash641 1928

[3] E R Lorch ldquoThe theory of analytic functions in normed abelianvector ringsrdquo Transactions of the American Mathematical Soci-ety vol 54 no 3 pp 414ndash425 1943

[4] J A Ward ldquoA theory of analytic functions in linear associativealgebrasrdquo Duke Mathematical Journal vol 7 no 1 pp 233ndash2481940

[5] J A Ward ldquoFrom generalized Cauchy-Riemann equationsto linear algebrardquo Proceedings of the American MathematicalSociety vol 4 no 3 pp 456ndash461 1953

[6] E Lopez-Gonzalez ldquoDifferential equations over algebrasrdquoAdvances and Applications in Mathematical Sciences vol 8 no2 pp 189ndash214 2011

[7] A Alvarez-Parrilla M E Frıas-Armenta E Lopez-Gonzalezand C Yee-Romero ldquoOn solving systems of autonomousordinary differential equations by reduction to a variable of analgebrardquo International Journal of Mathematics and Mathemati-cal Sciences vol 2012 Article ID 753916 21 pages 2012

10 Journal of Applied Mathematics

[8] E J Wilczynski ldquoReview Abraham Cohen an introductionto the lie theory of one-parameter groups with applicationsto the solution of differential equationsrdquo Bulletin of the Amer-ican Mathematical Society vol 18 no 10 pp 514ndash515 1912httpprojecteuclidorgeuclidbams1183421829

[9] J M PageOrdinary Differential Equations with an Introductionto Liersquos Theory or the Group of One Parameter MacmillanPublishers London UK 1897

[10] R A Steinhour The truth about lie symmetries solvingdifferential equations with symmetry methods [SeniorIndependent StudyTheses] 2013 httpopenworkswoostereduindependentstudy949

[11] R Pierce Associative Algebras Springer New York NY USA1982

[12] K I T Al-Dosary ldquoInverse integrating factor for classes of pla-nar differential systemsrdquo International Journal of MathematicalAnalysis vol 4 no 29ndash32 pp 1433ndash1446 2010

[13] I A Garcıa and M Grau ldquoA Survey on the inverse integratingfactorrdquo Qualitative Theory of Dynamical Systems vol 9 no 1-2pp 115ndash166 2010

[14] W Brown Matrices over Commutative Rings Monographs andTextbooks in Pure and Applied Mathematics Marcel Dekker1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Applied Mathematics

[8] E J Wilczynski ldquoReview Abraham Cohen an introductionto the lie theory of one-parameter groups with applicationsto the solution of differential equationsrdquo Bulletin of the Amer-ican Mathematical Society vol 18 no 10 pp 514ndash515 1912httpprojecteuclidorgeuclidbams1183421829

[9] J M PageOrdinary Differential Equations with an Introductionto Liersquos Theory or the Group of One Parameter MacmillanPublishers London UK 1897

[10] R A Steinhour The truth about lie symmetries solvingdifferential equations with symmetry methods [SeniorIndependent StudyTheses] 2013 httpopenworkswoostereduindependentstudy949

[11] R Pierce Associative Algebras Springer New York NY USA1982

[12] K I T Al-Dosary ldquoInverse integrating factor for classes of pla-nar differential systemsrdquo International Journal of MathematicalAnalysis vol 4 no 29ndash32 pp 1433ndash1446 2010

[13] I A Garcıa and M Grau ldquoA Survey on the inverse integratingfactorrdquo Qualitative Theory of Dynamical Systems vol 9 no 1-2pp 115ndash166 2010

[14] W Brown Matrices over Commutative Rings Monographs andTextbooks in Pure and Applied Mathematics Marcel Dekker1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of