PROJECT ABSTRACT

The Project Abstract shall include a statement of objectives, methods to be employed, and the significance of the proposed activity tothe advancement of knowledge or education. Avoid use of the first person to complete this summary. DO NOT EXCEED ONE PAGE.The abstract should be suitable for release under the Freedom of Information Act, 5 U.S.C. 552, as amended.

AbstractOver the last century electrical engineers have had a significant impact on the evolution of mathematics. Some ofthe best examples are the Gibbs/Heaviside system of vector calculus and information theory. These mathematicalframeworks arose from application-driven research efforts. Because the tools used by engineers must be usefuland efficient, the techniques they develop are different than those of pure mathematicians. Both pure and appliedapproaches have merit, and the most fruitful results come from the union of the two. Since working to create andrefine models for physical systems has proven to be a successful method for advancing mathematical frameworks,we propose to develop a wave dynamics framework by modernizing the techniques used in microwave engineering.

Microwave Engineering lays at the boundary between electromagnetics and circuit theory, and plays a vitalrole in the precise manipulation of electromagnetic energy needed for countless defense applications. While thetools and concepts used in microwave engineering are highly refined, the mathematics employed by these tools arefragmented and limiting. Reformulating microwave theory in the language of Geometric Algebra is an importantstep in modernizing and integrating these techniques into other domains. Geometric Algebra (GA) is a universalalgebra which subsumes complex algebra, quaternions, linear algebra and several other independent mathematicalsystems. It provides a clear and coherent approach to higher geometry and calculus by introducing additionalobjects and multiplication rules. GA has recently received considerable interest from engineering and appliedsciences which has provided important advances needed for practical applications. Much of this interest has beengenerated by Conformal Geometric Algebra, a tool which provides an object-oriented, operator-based approach togeometry. Problem solving with GA is fundamentally different from the conventional approaches because of theadditional objects and operators it introduces.

The foundations for this project have been introduced in the paper “Applications of Conformal GeometricAlgebra to Transmission line Theory”, which was invited to the journal IEEE Access. We plan to extend this workinto the following areas:

1. Unify models used in electromagnetic, network, and transmission line theory using Space-Time Algebra.

2. Provide worked examples with the unified framework for problems in antenna analysis, filter design, and VNAcalibration.

3. Provide computational support and translation methods to/from existing formulations

4. Integrate and extend the model to support other wave mechanical systems such as quantum mechanics, optics,and acoustics.

Once the models used by microwave engineering have been consolidated with GA, the integration of other wavedynamics systems into the framework can begin.

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PROJECT DESCRIPTION

IntroductionElectrical engineering has developed some of the most powerful and widely applied problem solving techniques.This success is due to the continued, application-driven research stimulus over the last century. While electricalphenomenon where originally understood and explained through mechanical analogues[13], the process was quicklyreversed, using circuits to model and understand mechanical systems[14]. Thus, the tools and models of electricalengineering have become the tools and models of systems, generally[1]. Such widespread use justifies efforts toimprove and refine the models used in electrical engineering, and makes these efforts relevant to several other areas.

While microwave theory may appear to be a highly specialized subject within electrical engineering, it’s abilitiesto engineer complex wave dynamics is unquestionable. As the field has developed, several different models have beencreated. Electromagnetics uses vector calculus, transmission line theory uses complex algebra, while network theoryuses linear algebra. Unifying the diverse set of models within the discipline is an import step toward constructing amore efficient and generalized framework for wave dynamics. For this to happen, the fragmented set of frameworkscurrently employed in microwave theory have to be integrated using Geometric Algebra.

What is the problem?A substantial amount of electrical engineering is currently expressed in the language of complex numbers. Despitetheir usefulness, complex numbers are limited to rotations in two-dimensions. This is a serious drawback giventhat many problems in electrical engineering are fundamentally multi-dimensional. High-dimensional problems aresolved using matrices of complex numbers, effectively modeling a N-dimensional space of as sets of 2D sub-spaces.Within transmission line theory alone, complex numbers are used to model the following concepts:

• reflection coefficient

• input impedance/admittance

• characteristic impedance

• propagation constant

• distributed impedance

• distributed admittance

While it is possible that representing all of these different concepts with a single mathematical structure isoptimal in some sense, it seems more likely that the model has conformed to the mathematical tools which wereavailable to the architects of the theory, namely Oliver Heaviside[13]. If all you have is a hammer, everything lookslike a nail. In fact, most electrical engineers don’t understand complex numbers. This is because complex algebraconfuses the concept of a vector with a rotation/dilation operator, or put more abstractly, the distinction betweenquantity and operator. The same problem plagues quaternion analysis as well, and it is what drove Heaviside tocreate his vector calculus. The importance of the quantity/operator dichotomy was emphasized by W. K. Cliffordin his original article on Geometric Algebra[6]. He even explicitly warned that in three dimensions the distinctionbetween a vector and it’s dual plane “...which for some purposes it is so convenient to ignore, has to be re-introducedinto physics.”

Another source of inefficiency is the variety of different mathematical models employed. As stated earlier,microwave theory requires three different frameworks; 1) electromagnetics uses vector calculus, 2) network theoryuses complex linear algebra, and 3) transmission lines use complex algebra with mobius transformations. Studentsare required to learn all three models, and practitioners are forced to constantly switch between models to solveproblems. Additionally, both transmission line and network theory make use of hyperbolic functions, but neitherincorporates non-euclidean geometry. If relativistic or quantum effects need to be taken into account, a host ofadditional machinery is needed.

Superfluous models are a natural result of framework evolution and not unique to microwave engineering. Itcan be seen in subjects as fundamental as rotations. For example, there are several methods currently in use forhandling rotations,

• 2D: matrix, complex numbers

• 3D: matrix, quaternion, Euler angles, Pauli spinors

• 4D: matrix, Lorentz transformation, Dirac spinors

• in ND: matrix

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This is redundant, discontinuous, and confusing. Almost no one learns all of the domain-specific models, whichmakes sharing results difficult, and translating results expensive[12]. Geometric Algebra subsumes all of thesemodels, and presents a unified way to express rotations which works in all dimensions as well as in non-euclideangeometry[8].

Confusion generated by model fragmentation can be seen in several aspects of microwave engineering. Intransmission line theory, power dissipation is represented by the magnitude of the reflection coefficient, while innetwork theory this quantity becomes the “unitary-ness of a network’s scattering matrix”[7]. This discontinuity isdue to the non-scalability of complex algebra and uninterpretable nature of matrices. Fragmentation also increasesthe requirements for those who want to use the models. An introductory microwave engineering course requires ata minimum: vector calculus, complex algebra, differential equations, linear algebra, and Fourier analysis.

Finally, from an applications perspective using discontinuous mathematical frameworks leads to algorithmswhich are more complicated, less robust, and more computationally expensive. Nowhere is this more evidentthan in the sub-field of Vector Network Analyzer (VNA) calibration, a discipline at the foundation of microwavemetrology[11]. Vector calibration currently uses a handful of what could be considered standard algorithms. How-ever, as new measurement challenges arise, these algorithms must be modified and adapted to suit new mediumand uncertainties[3, 4]. The lack of an intuitive geometric model for two-port networks makes extending calibrationalgorithms difficult.

So far we have argued that microwave theory has importance and applications beyond microwaves, and thisjustifies refining it’s models. While the theory is unquestionably useful, it has several flaws such as limited dimen-sionality, model fragmentation, and lack of intuitive geometric models. Therefore, if the current models used bymicrowave theory are to be challenged, the next question is; what other structures might be employed to create abetter model? Furthermore, what properties should a good theory possess?

Proposed SolutionGeometric Algebra (GA) is a universal algebra which subsumes complex algebra, quaternions, linear algebra andseveral other independent mathematical systems[12]. Problem solving with GA is fundamentally different from theconventional complex-based approach because of the additional objects and operators it introduces. Additionally,GA scales to arbitrary dimensions and supports non-euclidean geometries, making it an ideal tool for applicationsin microwave and other wave-mechanics disciplines. In recent years there has been a surge of interest in usingGA for engineering and computer science, which has generated many important results and algorithms neededfor practical computation[9, 10, 8, 5]. Many of these advances are contained in the journal “Advances in AppliedClifford Algebras”.

If successful, the techniques of microwave theory will be consolidated into a single framework, producing asimpler and more efficient system. In addition, by building theory from Space-time Algebra, relativity and quantummechanics can be both understood and integrated much easier. This will substantially reduce the redundancy presentin the current theories and greatly expand the abilities of a microwave engineer. Finally, the universality of GA willallow the integration of of other wave-mechanics based disciplines, thereby promoting a more unified framework forwave dynamics.

Preliminary WorkConformal Geometric Algebra

The foundations for this project have been introduced in the paper “Applications of Conformal Geometric Algebrato Transmission line Theory”, which was invited to the journal IEEE Access[2]. This models allows the fundamentalnetwork operations such as adding impedance, admittance, and changing line impedance can be implemented withrotations, and are shown to form a group. The group structure goes unnoticed when using the conventional theory.CGA also allows the traditional bilinear transformations relating impedance, admittance, and reflection coefficientto be translated into rotations as well, thus, the majority of relationships in transmission line theory are linearized.This allows for geometric interpretability. For example, the normalized input impedance of a load zL as seenthrough a lossless transmission line of electrical length θ is conventionally described by the following formula,

zin = zL + j tan (θ)1 + zLj tan (θ) .

This formula has no geometric interpretation! In contrast, CGA allows this same relationship to be representedas a simple rotation in three dimensional space, which is expressed in the same way all rotations are expressed inGA,

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zin = RθzLR̃θ.

Thus, complex relationships can be visualized and problems can be solved using geometric reasoning. We havefound that representing networks with domain-invariant, operator-based framework is more amenable to algorithmdesign and this substantially narrows the gap between theory and implementation. In summary, the benefits ofthe CGA technique for transmission line theory are that it incorporates group theory, provides an operator-based,domain invariant approach and allows for geometrical reasoning.

Technical StrategyBelow is a narrative description of our technical strategy followed by the same material in a time-line list formatas well as a gantt chart.

Year 1

The first year of this project will focus on the foundations of microwave network theory. Currently, the conformaltransmission line model used in [2] is built from several results of conventional theory. Developing the entire theoryunder one mathematical system is a priority for this project. Therefore, the foundations for transmission line physicswill be derived from the Space Time Algebra formulation of Maxwell’s equations as pioneered by Hestenes. Thiswill enable topics such as antenna arrays, quasi-optics, and multi-moded waveguides to be incorporated into thesystem.

We plan to apply the group theory results from section 6 of [2] to classify the distributed and discrete ele-ments groups and the expression of equivalent circuits. The classes of electrical/mechanical analogies used suchas impedance, mobility, and ’through and across’ should be expressible as duality relationships within the grouptheory [1]. Another important result shown in section 6 of [2] is that by analyzing transmission lines through CGA,the coupled differential equations known as the telegraphers equations, have been circumvented. In their placeis a simpler rotor derivative, which yields a geometrical classification system of specific transmission lines types.Working out the implications of this exchange with mechanical systems will also be explored.

The models developed CGA transmission line theory will be extended into two-port networks. Two-port networksplay an important role in many fundamental techniques within microwave engineering, such as filter design andVector calibration. Currently, such networks are represented by various matrix formats, such as the scattering(S), impedance (Z), and admittance (Y) matrices. Choosing a given format is equivalent to choosing a basis inwhich to frame a transformation. We plan to unify these network representations in a similar way as differentloads representations were unified with the CGA model. Once this is done, the actions and operations performedon two-ports such as flipping and cascaded will be translated, followed by classifications of symmetry, reciprocity,and losslessness. We expect the existing filter and impedance matching techniques can be developed much moreefficiently and intuitively with the geometrically-based model. Perhaps more importantly, extension and exportationof these models will be easier and more intuitive.

Finally, if time permits in the first year, we would like to explore extending the two-port network model intohigher rank N-ports. The process for this exploration must be guided by the results from the topics mentioned thusfar. It is no doubt possible and an exciting avenue for advancing the theory.

Year 2

The second year of the project will dedicated to absorbing and integrating models from other fields. A first start willbe to translate the various matrix technique used in optics for light polarization, such as Jones, Stokes, and Muellercalculus. Given that polarized light can be treated as a a pair of 2-port or full 4-port network, the integration ofthese systems should be straight forward. Additionally, the vector calibration methods of microwave metrology willlikely be applicable to such areas as well.

The next most obvious system to integrate is that of scattering in quantum mechanics. Spin 1/2 systems canbe treated as polarized light with a 4-port, and simple reflection problems with potentials in half-space will bemodeled.

SoftwareFor engineers and scientists to apply the results produced by this project, they will require computational supportfor the new data-types and operations GA introduces. In recent years Python has become a dominant language inthe field of scientific and engineering software development, making it a suitable language for a GA implementation.

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Therefore, we will be employing the Geometric Algebra Python module ‘clifford’ as a proof-of-concept computa-tional backend. This module already provides a large amount of GA objects and operations, and will be developedas needed for this project. Additionally, the Co-PI currently maintains the open-source project ‘scikit-rf’, themost popular Python module for RF/Microwave applications. Given our ability to develop and integrate both ofthese projects, we are in a unique position to rapidly translate the results of this project into real world applications.

Cost and ScheduleThe project’s budget is $280,000, delivered as $140,000 each year over two years. The entire cost is associated withthe research and software development labor. Both of these tasks are equally shared between the PI and Co-PI.

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Name Begin date End dateFoundations 1/1/19 3/22/19

Network with STA (Hestenesto Harrington)

1/1/19 2/11/19

EM to TL (Marcuvitz) 2/12/19 3/22/19

Transmission Lines 3/22/19 5/1/19Discrete/Distributed Elements 3/22/19 4/11/19Equivalent circuits/Duality 4/11/19 5/1/19

Two-port Network 5/1/19 9/4/19Basic Model 5/1/19 6/11/19Matrix Conversions 6/11/19 7/8/19Physical Properties 7/8/19 8/16/19Actions 8/16/19 9/4/19

Two-port Applications 9/5/19 11/18/19Periodic Structures 9/5/19 10/2/19Impedance Matching 10/3/19 10/22/19Vector Calibration 10/22/19 11/18/19

Misc 11/18/19 12/31/19Extensions to N-ports 11/18/19 12/31/19

Optics 1/1/20 2/3/20Stokes/Jones/Mueller 1/1/20 1/28/20Transfer/Abeles 1/28/20 2/3/20

Quantum 2/4/20 3/27/20Spin 1/2-Network Model 2/4/20 3/2/20Shrodinger and Scattering 3/2/20 3/27/20

Mechanical/Electrical Analogies 3/27/20 4/23/20Impedance/Mobility/Transfer 3/27/20 4/23/20

Topics Dependent on Progress 4/23/20 12/30/20N-port 4/23/20 6/3/20N-port Vector Calibration 4/23/20 6/3/20Open 4/23/20 12/30/20

Jun 11, 2018

Tasks 2

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Jun 11, 2018

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BIOGRAPHICAL SKETCHProvide the following information for the senior personnel on the project. Begin with the Principal Investigator/Project Director.

DO NOT EXCEED THREE PAGES PER PERSON

NAME: Robert Weikle II POSITIONAL TITLE: Proffesor, CTO at Dominion MicroprobesRESEARCH AND PROFFESIONAL EXPERIENCE:

Robert M. Weikle, II received his B.S. in electrical engineering and physics from Rice University in 1986, and hisM.S. and Ph.D. in electrical engineering from the California Institute of Technology in 1987 and 1992, respectively.At Caltech, he developed a variety of new techniques for realizing and modeling arrays of coupled nonlinear activedevices for microwave/millimeter-wave power combining. For this work, he shared the 1993 IEEE Microwave Prize.During 1992, Dr. Weikle was a postdoctoral research associate with the Department of Applied Electron Physicsat Chalmers Tekniska Hogskola in Goteborg, Sweden where he worked on millimeter-wave amplifiers based on highelectron mobility transistors and low-noise terahertz mixers using superconducting hot electron bolometers.

In January 1993, Dr. Weikle joined the faculty of the University of Virginia where he is currently Professor in theCharles L. Brown Department of Electrical and Computer Engineering. During this time, he has built a laboratoryfor millimeter and submillimeter-wave device characterization, circuit design, prototyping, and metrology and haspursued research on millimeter-wave and submillimeter-wave electronics, devices, and systems. Among his groups’research efforts are design and fabrication techniques for submillimeter-wave integrated circuits, heterogenous inte-graton of III-V semiconductor devices with micromachined silicon, investigation of measurement instrumentationand calibration techniques for terahertz device and circuit characterization (including micromachined probes forsubmillimeter-wave on-wafer measurements), and research on planar antennas and quasi-optical components formillimeter-wave imaging and power-combining.

In 2011, Dr. Weikle co-founded Dominion Microprobes, Inc., with colleagues Scott Barker and Arthur Licht-enberger, to develop on-wafer probe technologies for terahertz measurements. He currently serves at its ChiefTechnology Officer.

THE FOLLOWING ADDITIONAL INFORMATION IS REQUIRED:A. List up to 5 publications most closely related to the proposed project and up to 5 other significant publications, including thosebeing printed. Patents, copyrights, or software systems developed may be substituted for publications. Do not include additional listof publications, invited lectures, etc.B. List of persons, other than those cited in the publication list, who have collaborated on a project or a book, article, report or paperwithin the last 4 years. Negative reports should be indicated.C. Names of graduate and post graduate advisors and advisees.The information in B. and C. is used to help identify potential conflicts or bias in the selection of reviewers.

A) Publications• “Design and characterization of integrated submillimeter-wave quasi-vertical diodes,” IEEE Transactions onTerahertz Sci. and Tech., vol. 5, no. 1, pp. 73-80, January 2015. N. ALIJABBARI, M.F. BAUWENS, ANDR.M. WEIKLE, II

• Characterization of micromachined on-wafer probes for the 600-900 GHz band,” IEEE Trans. Terahertz Sci.and Tech., vol. 4, no. 4, pp. 527-529, July 2014. M.F. BAUWENS, L. CHEN, C. ZHANG, A. ARSENOVIC,A.W. LICHTENBERGER, N.S. BARKER, AND R.M. WEIKLE, II

• “An experimental technique for calibration uncertainty analysis,” IEEE Trans. Microwave Theory Tech., vol.61, no. 1, pp. 263-269, January 2013. A. ARSENOVIC, L. CHEN, M.F. BAUWENS, H. LI, N.S. BARKER,AND R.M. WEIKLE, II

• “Terahertz micromachined on-wafer probes: repeatability and reliability,” IEEE Trans. Microwave Theory andTech., vol. 60, no. 9, pp. 2894-2902, September 2012. L. CHEN, C. ZHANG, T.J. RECK, A. ARSENOVIC,M. BAUWENS, C. GROPPI, A.W. LICHTENBERGER, R.M. WEIKLE, II, AND N. SCOTT BARKER

• “Micromachined probes for submillimeter-wave on-wafer measurements part II - RF design and characteri-zation,” IEEE Trans. Terahertz Sci. and Tech., vol. 1, no. 2, pp. 349-356, November 2011. T.J. RECK,L. CHEN, C. ZHANG, A. ARSENOVIC, C. GROPPI, A. LICHTENBERGER, R.M.WEIKLE II, AND N.S.BARKER

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B) Collaborators outside publication listC) Graduate and post graduate advisors and advisees

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BIOGRAPHICAL SKETCHProvide the following information for the senior personnel on the project. Begin with the Principal Investigator/Project Director.

DO NOT EXCEED THREE PAGES PER PERSON

NAME: Alexander Arsenovic POSITIONAL TITLE: CEO, Eight Ten Labs LLCRESEARCH AND PROFFESIONAL EXPERIENCE:

Alex Arsenovic received a B.S and Ph.D in Electrical Engineering from the University of Virginia in 2007 and2012, respectively. Alex has worked as an independent consultant in Central Virginia for the past 6 years, withclients such as Virginia Diodes Inc, Nuvotronics, and Plotly. He continues to collaborate with the University ofVirginia, and has authored and co-authored over 15 technical papers in the field of microwave metrology andgeometric algebra. In 2016, he created Eight Ten Labs LLC to continue providing the services for microwavemetrology, software development, and applied mathematics. His chief interest is in modernizing the theoretical andcomputation tools used by electrical engineers and scientists.

THE FOLLOWING ADDITIONAL INFORMATION IS REQUIRED:A. List up to 5 publications most closely related to the proposed project and up to 5 other significant publications, including thosebeing printed. Patents, copyrights, or software systems developed may be substituted for publications. Do not include additional listof publications, invited lectures, etc.B. List of persons, other than those cited in the publication list, who have collaborated on a project or a book, article, report or paperwithin the last 4 years. Negative reports should be indicated.C. Names of graduate and post graduate advisors and advisees.The information in B. and C. is used to help identify potential conflicts or bias in the selection of reviewers.

A) PublicationsPatents

• “A Method for Interpolating Transmissive, Two-port Networks”, A. Arsenovic; pending United States Provi-sional Patent Application Serial No. 62663575 (Attorney Docket No. 1135-05), filed April 27, 2018

• “Methods for Removal of Effects of Local Oscillator Drift from Vector Network Analyzer Measurements” , A.Arsenovic; pending United States Provisional Patent Application Serial No. 62332550 (Attorney Docket No.1135-01), filed May 6, 2016

Publications

• “Applications of Conformal Geometric Algebra to Transmission Line Theory”, IEEE Access , 5:19920–19941,2017. A. Arsenovic

• “Reflectometer calibration with a pair of partially known standards”, A. Arsenovic, R. M. Weikle, and J. L.Hesler; in Microwave Conference (EuMC), 2015 European, 2015, pp. 327–330.

• “Two port calibration insensitive to flange misalignment”, Microwave Measurement Conference (ARFTG),2014 84th ARFTG, 2014, pp. 1–7. A. Arsenovic, R. M. Weikle, and J. Hesler

• “An Experimentally-Based Technique for Calibration Uncertainty Analysis”, Microwave Theory and Tech-niques, IEEE Transactions on , January 15, 2013. Alexander Arsenovic and Chen, L. and Bauwens, M. andLi, H. and Barker, N. S. and Weikle, R. M.

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• “A monostatic coded aperture reflectometer for imaging at submillimeter-wavelengths” M. B. Eller, N. D.Sauber, A. Arsenovic, S. Nadri, L. Xie, and R. M. Weikle, in 2017 IEEE MTT-S International MicrowaveSymposium (IMS) , pp. 1207-1209, June 2017

• “Micromachined Probes for Submillimeter-Wave On-Wafer Measurements-Part II: RF Design and Character-ization” Reck, T.J.; Lihan Chen; Chunhu Zhang; Arsenovic, A.; Groppi, C.; Lichtenberger, A.; Weikle, R.M.;Barker, N.S.; , Terahertz Science and Technology, IEEE Transactions on , vol.1, no.2, pp.357-363, Nov. 2011.

• “Micromachined Probes for Submillimeter-Wave On-Wafer Measurements- Part I: Mechanical Design andCharacterization” Reck, T.J.; Lihan Chen; Chunhu Zhang; Arsenovic, A.; Groppi, C.; Lichtenberger, A.W.;Weikle, R.M.; Barker, N.S.; , , Terahertz Science and Technology, IEEE Transactions on , vol.1, no.2, pp.349-356, Nov. 2011.

Software

• Original author and maintainer of the Python module scikit-rf; www.scikit-rf.org.

• Maintainer of the python module clifford; www.clifford.readthedocs.io

Awards

• Nominee for 2017 Clifford Prize (wkcliffordprize.org)

B) Collaborators outside publication listRichard Clawson, David Hestenes, Allan Cortzen

C) Graduate and post graduate advisors and adviseesRobert Weikle.

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NOTABLE ADDITIONAL PERSONNEL

Dr. David Hestenes

David Hestenes has graciously agreed to provide technical assistance upon request as an unpaid consultant. DavidHestenes is Emeritus Professor of Physics at Arizona State University. He is a Fellow of the APS and OverseasFellow of Churchill College, Cambridge. He has also been UCLA University Fellow, NSF Postdoctoral Fellow,NASA Faculty Fellow and Senior Fulbright Research Fellow. His scientific research has focused on development andapplication of Geometric Algebra as a unified mathematical language for physics and engineering. In recognitionof this work he was designated Foundations of Physics Honoree in 1993. For contributions to physics education hewas awarded the 2002 Oersted Medal by the AAPT, the 2003 Education Research Award by the National Councilof Scientific Society Presidents, and the 2014 Excellence in Physics Education Award by the American PhysicalSociety.

Awards

• 2014 Excellence in Physics Education Award from the American Physical Society

• 2003 Award for excellence in educational research by the Council of Scientific Society Presidents

• 2002 Oersted Medal, awarded by the American Association of Physics Teachers for notable contributions tothe teaching of physics

• Fellow of the American Physical Society

• Overseas Fellow of Churchill College, Cambridge

• Foundations of Physics Honoree (Sept.–Nov. issues, 1993)

• Fulbright Research Scholar (England) 1987–1988

• NASA Faculty Fellow (Jet Propulsion Laboratory) 1980, 1981

• NSF Postdoctoral Fellow (Princeton) 1964–1966

• University Fellow (UCLA) 1958–1959

Selected Publications

• D. Hestenes: A Unified Language for Mathematics and Physics. In: J.S.R. Chisholm/A.K. Common (eds.):Clifford Algebras and their Applications in Mathematical Physics (Reidel: Dordrecht/Boston, 1986), p. 1–23.

• D. Hestenes, Space-Time Algebra (Gordon & Breach: New York, 1966).

• D. Hestenes, New Foundations for Classical Mechanics (Kluwer: Dordrecht/Boston, 1986), Second Edition(1999).

• D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus, a unified language for mathematics andphysics (Kluwer: Dordrecht/Boston, 1984).

• D. Hestenes & J. Holt, The Crystallographic Space Groups in Geometric Algebra, Journal of MathematicalPhysics 48: 023514 (2007)

Full info

• Bio: https://en.wikipedia.org/wiki/David_Hestenes

• Publications: https://www.researchgate.net/profile/David_Hestenes

• Website: http://geocalc.clas.asu.edu/

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BIBLIOGRAPHY

References[1] Mechanical-electrical analogies, April 2018. Page Version ID: 837870487.

[2] A. Arsenovic. Applications of Conformal Geometric Algebra to Transmission Line Theory. IEEE Access,5:19920–19941, 2017.

[3] A. Arsenovic, R. M. Weikle, and J. L. Hesler. Reflectometer calibration with a pair of partially known standards.In Microwave Conference (EuMC), 2015 European, pages 327–330, September 2015.

[4] A. Arsenovic, R.M. Weikle, and J. Hesler. Two port calibration insensitive to flange misalignment. In MicrowaveMeasurement Conference (ARFTG), 2014 84th ARFTG, pages 1–7, December 2014.

[5] D. Hestenes C. Doran. Lie groups as spin groups. Journal of Mathematical Physics, 34(8), 1993.

[6] Professor Clifford. Applications of Grassmann’s Extensive Algebra. American Journal of Mathematics,1(4):350–358, 1878.

[7] Robert E. Collin. Foundations for Microwave Engineering - 2nd edition. Wiley-IEEE Press, 2000.

[8] Chris Doran. Geometric Algebra for Physicists. Cambridge University Press, Cambridge; New York, 1st pbk.ed. with corr edition edition, November 2007.

[9] L. Dorst and S. Mann. Geometric Algebra: A computational framework for geometrical applications Part 1.IEEE Computer Graphics and Applications, 22(3):24–31, May 2002.

[10] Leo Dorst and Joan Lasenby, editors. Guide to Geometric Algebra in Practice. Springer, London ; New York,2011 edition edition, August 2011.

[11] Joel Philipp Dunsmore. Handbook of microwave component measurements: with advanced vna techniques.Wiley, Hoboken, N.J, 2012.

[12] David Hestenes. Oersted Medal Lecture 2002: Reforming the mathematical language of physics. Am. J. Phys,71:104–121, 2003.

[13] Bruce J. Hunt. The Maxwellians. Cornell University Press, Ithaca, September 1994.

[14] W. P. Mason. Electrical and mechanical analogies. In Bell System Technical Journal, volume 20, pages 405–414.1941.

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