Three Puzzles on Mathematics, Computation, and Games
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Prizes and Awards Session
PRIZES AND AWARDS SESSION Wednesday, July 12, 2021 9:00 AM EDT 2021 SIAM Annual Meeting July 19 – 23, 2021 Held in Virtual Format 1 Table of Contents AWM-SIAM Sonia Kovalevsky Lecture ................................................................................................... 3 George B. Dantzig Prize ............................................................................................................................. 5 George Pólya Prize for Mathematical Exposition .................................................................................... 7 George Pólya Prize in Applied Combinatorics ......................................................................................... 8 I.E. Block Community Lecture .................................................................................................................. 9 John von Neumann Prize ......................................................................................................................... 11 Lagrange Prize in Continuous Optimization .......................................................................................... 13 Ralph E. Kleinman Prize .......................................................................................................................... 15 SIAM Prize for Distinguished Service to the Profession ....................................................................... 17 SIAM Student Paper Prizes .................................................................................................................... -
Walsh-Like Functions and Their Relations
WaIsh-like functions and their relations B.J. Falkowski S.Rahardja Indexing terms: Discrete transforms, Walshfunctions, Haar functions form except that the absolute value of the output is Abstract: A new discrete transform, the ‘Haar- taken before applying to the next stage. However, the Walsh transform’, has been introduced. Similar to rapid transform is nonorthogonal and does not have an well known Walsh and non-normalised Haar inverse. Another modified transform is based on a transforms, the new transform assumes only +1 hybrid version of the Haar and Walsh transforms [2, and -1 values, hence it is a Walsh-like function 111. This transform is derived from different linear and can be used in different applications of combinations of the basis Haar functions with an digital signal and image processing. In particular, appropriate scaling factor. Such a combinatioii of basis it is extremely well suited to the processing of functions have been found advantageous for feature two-valued binary logic signals. Besides being a selection and pattern recognition. The rationalised ver- discrete transform on its own, the proposed sion of this transform has also been introduced [12]. transform can also convert Haar and Walsh Another family of orthogonal functions related to the spectra uniquely between themselves. Besides the Walsh and Haar functions has been introduced [13-15]1. fast algorithm that can be implemented in the They are called bridge functions and can be generated form of in-place flexible architecture, the new by copy theory originated from work by Swick [16] for transform may be conveniently calculated using Walsh functions. -
The Heisenberg-Weyl Group W (Z2 )
Symmetry and Sequence Design I: m The Heisenberg-Weyl Group W (Z2 ) Robert Calderbank Stephen Howard Bill Moran Princeton University Defence Science & Technology Melbourne University Organization Australia Maximal X - Z = H−1XH Commutative Subgroup Walsh Dirac Orthonormal Basis - Sequences Sequences Supported by Defense Advanced Research Projects Agency and Air Force Office of Scientific Research D4: The Symmetry Group of the Square 0 1 1 0 Generated by matrices x = ( 1 0 ) and z = 0 −1 π xz = 0 −1 anticlockwise rotation by 1 0 2 1 0 z = 0 −1 reflection in the horizontal axis D4 is the set of eight 2 × 2 matrices ε D(a, b) given by 0 1 a 1 0 b ε D(a, b) = ε ( 1 0 ) 0 −1 where ε = ±1 and a, b = 0 or 1. 2 2 x = z = I2 1 1 1 zx = −1 ( 1 ) = −1 # xz = −zx 1 1 −1 xz = ( 1 ) −1 = 1 The Hadamard Transform H = √1 + + reflects the lattice of subgroups across the central 2 2 + − axis of symmetry D4 2 −1 H2 = I2 and H2 = H2 h±xi hxzi h±zi H2xH2 = z hxi h−xi h−I2i h−zi hzi H2zH2 = x hI2i a b a b a b ab b a H2[εx z ]H2 = ε(H2x H2)(H2z H2) = εz x = (−1) x z ab H2[εD(a, b)]H2 = (−1) εD(b, a) Kronecker Products of Matrices Given a p × p matrix X = [xij ] and a q × q matrix Y = [Yij ], the Kronecker products X ⊗ Y is defined by x11Y ... x1pY . -
Three Puzzles on Mathematics Computations, and Games
January 5, 2019 17:43 icm-961x669 main-pr page 551 P. I. C. M. – 2018 Rio de Janeiro, Vol. 1 (551–606) THREE PUZZLES ON MATHEMATICS, COMPUTATION, AND GAMES G K Abstract In this lecture I will talk about three mathematical puzzles involving mathemat- ics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible? 1 Introduction The theory of computing and computer science as a whole are precious resources for mathematicians. They bring up new questions, profound new ideas, and new perspec- tives on classical mathematical objects, and serve as new areas for applications of math- ematics and mathematical reasoning. In my lecture I will talk about three mathematical puzzles involving mathematics and computation (and, at times, other fields) that have preoccupied me over the years. The connection between mathematics and computing is especially strong in my field of combinatorics, and I believe that being able to person- ally experience the scientific developments described here over the past three decades may give my description some added value. For all three puzzles I will try to describe in some detail both the large picture at hand, and zoom in on topics related to my own work. Puzzle 1: What can explain the success of the simplex algorithm? Linear program- ming is the problem of maximizing a linear function subject to a system of linear inequalities. -
A Multiscale Model of Nucleic Acid Imaging
Scientific Visualization, 2020, volume 12, number 3, pages 61 - 78, DOI: 10.26583/sv.12.3.06 A multiscale model of nucleic acid imaging I.V. Stepanyan1 Institute of Machine Science named after A.A.Blagonravov of the RAS 1 ORCID: 0000-0003-3176-5279, [email protected] Abstract The paper describes new results in the field of algebraic biology, where matrix methods are used [Petukhov, 2008, 2012, 2013; Petuhov, He, 2010] with the transition from matrix algebra to discrete geometry and computer visualization of the genetic code. The algorithms allow to display the composition of sequences of nitrogenous bases in parametric spaces of various dimensions. Examples of visualization of the nucleotide composition of genetic se- quences of various species of living organisms are given. The analysis was carried out in the spaces of binary orthogonal Walsh functions taking into account the physical and chemical parameters of the nitrogen bases. The results are compared with the rules of Erwin Chargaff concerning genetic sequences in the composition of DNA molecules. The developed method makes it possible to substantiate the relationship between DNA and RNA molecules with fractal and other geometric mosaics, reveals the orderliness and symmetries of polynucleotide chains of nitrogen bases and the noise immunity of their visual representations in the orthog- onal coordinate system. The proposed methods can serve to simplify the researchers' percep- tion of long chains of nitrogenous bases through their geometrical visualization in parametric spaces of various dimensions, and also serve as an additional criterion for classifying and identifying interspecific relationships. Keywords: visualization algorithms, Walsh functions, Chargaff’s rules, multidimension- al analysis, nucleotide composition, fractals, bioinformatics, DNA, chromosomes, symme- tries. -
Recent Developments in the Theory of Walsh Series
I nternat. J. Math. & Math. Sci. 625 Vol. 5 No. 4 (I982) 625-673 RECENT DEVELOPMENTS IN THE THEORY OF WALSH SERIES WILLIAM R. WADE Department of Mathematics University of Tennessee Knoxville, Tennessee 37916 U.S.A. (Received May 20, 1982) ABSTRACT. We survey research done on the theory of Walsh series during the decade 1971-1981. Particular attention is given to convergence of Walsh-Fourier series, gap Walsh series, growth of Walsh-Fourier coefficients, dyadic differentiation, and uniqueness of Walsh series. KEY WORDS AND PHRASES. Walsh-Paley system, dyadic group, maximal functns, dyadic Hp space, BMO, Vilenkin gups, weak type (p,p) dyadic differentiation. 980 MATHEMATICS SUBJECT CLASSIFICATION CODES. 42C10, 43A75. I. INTRODUCTION. This article surveys recent results on Walsh series. To avoid duplication of material appearing in Balaov and Rubintein [1970], a decision was made to concen- trate on the decade 1971-1981. References to earlier work will be made when neces- sary to relate what is herein reported to that which preceeded it. Discussion of the relationships between this material and the general theory of orthogonal series has been left to those more qualified for this task (e.g., Ul'janov [1972], Olevskii [1975] and Bokarev [1978b])o In addition to this introductory section, there remain five sections: II. Walsh-Fourier Series, III. Approximation by Walsh Series, IV. Walsh-Fourier Coefficients, V. Dyadic Differentiation, and VI. Uniqueness. These sections have been further divided into consecutively numbered subsections, each dealing with a 626 W.R. WADE particular facet of the subject and each carrying a descriptive title to help the reader quickly find those parts which interest him most. -
Three Puzzles on Mathematics, Computation, and Games
THREE PUZZLES ON MATHEMATICS, COMPUTATION, AND GAMES GIL KALAI HEBREW UNIVERSITY OF JERUSALEM AND YALE UNIVERSITY ABSTRACT. In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible? 1. INTRODUCTION The theory of computing and computer science as a whole are precious resources for mathematicians. They bring new questions, new profound ideas, and new perspectives on classical mathematical ob- jects, and serve as new areas for applications of mathematics and of mathematical reasoning. In my lecture I will talk about three mathematical puzzles involving mathematics and computation (and, at times, other fields) that have preoccupied me over the years. The connection between mathematics and computing is especially strong in my field of combinatorics, and I believe that being able to per- sonally experience the scientific developments described here over the last three decades may give my description some added value. For all three puzzles I will try to describe with some detail both the large picture at hand, and zoom in on topics related to my own work. Puzzle 1: What can explain the success of the simplex algorithm? Linear programming is the problem of maximizing a linear function φ subject to a system of linear inequalities. The set of solutions for the linear inequalities is a convex polyhedron P . The simplex algorithm was developed by George Dantzig. -
A Demonstration of the Use of Walsh Functions for Multiplexed Imaging
Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1990-12 A demonstration of the use of Walsh functions for multiplexed imaging McKenzie, Robert Hugh, III Monterey, California: Naval Postgraduate School http://hdl.handle.net/10945/27636 NAVAL POSTGRADUATE SCHOOL " Monterey, California AD-A246 138 ft LECTE i R ADt' FEB 2 0, 199 2 U THESIS A DEMONSTRATION OF THE USE OF WALSH FUNCTIONS FOR MULTIPLEXED IMAGING by Robert Hugh McKenzie III DECEMBER 1990 Thesis Advisor: David S. Davis Approved for public release: Distribution is unlimited 92-03974 92 2 NCLASSIFIED CURITY CLASS!FICATION OF THIS PAGE i i i iiN I Form Approved REPORT DOCUMENTATION PAGE OMB No 0704-0188 a REPORT SECURITY CLASSIFiCATION lb RESTRICT:VE MARK!NGS Unclassified a SECURITY CLASSIFICATION AUTHORITY 3 DISTRIBUTION ,AVAI.ABILITY OF EPORT Approved for pubi c release: b DECLASSIFICATION! DOWNGRADING SCHEDULE Distribution is unlimited PERFORMING ORGANIZATION REPORT NUMBER(S) S MONITORING ORGANIZATION REPORT NUMBER(S) a NAME OF PERFORMING ORGANIZATION r6b OFFICE SYMBOL 7a NAME OF MONITORING ORGANiZATION 4aval Postgraduate School I6Dv(If applicable) Naval Postgraduate School ,c.ADDRESS (City, State, and ZIPCode) 7b ADDRESS (City, State, and ZIP Code) Monterey, CA 93943-5000 Monterey, CA 93943-5000 Ia NAME OF FUNDING/SPONSORING r8b OFFICE SYMBOL 9 PROCUREMENT INSTRUMENT IDENTIFICATION NU.VBER ORGANIZATIONj (ifapplicable) c. ADDRESS (City, State, and ZIP Code) 10 SOURCE OF FUNDING NUMBERS PROGRAM IPROjECT ITASK I VORo, UNIT ELEMENT NO NO NO ACCESSiO% NO 1 1 TITLE (Include Security Classification) A DEMONSTRATION OF THE USE OF W 7 SH FUNCTIONS FOR MULTIPLEXED IMAGING "2 P - MCKENZIE III 13a TYPE OF REPORT 13b TIME COVERED 14 D REPORT ~onth.0Day) :5 PAGE CO AJT Master's Thesis FROM _ TO _____ 131 16 SUPPLEMENTARY NOTATION The iews expressed in this thesis are those of the author and do not reflect the official policy or position of the Department of Defense or the U.S. -
Arxiv:1909.01143V2 [Math.NA] 30 Mar 2021
NON-UNIFORM RECOVERY GUARANTEES FOR BINARY MEASUREMENTS AND INFINITE-DIMENSIONAL COMPRESSED SENSING L. THESING AND A. C. HANSEN ABSTRACT. Due to the many applications in Magnetic Resonance Imaging (MRI), Nuclear Magnetic Reso- nance (NMR), radio interferometry, helium atom scattering etc., the theory of compressed sensing with Fourier transform measurements has reached a mature level. However, for binary measurements via the Walsh trans- form, the theory has long been merely non-existent, despite the large number of applications such as fluorescence microscopy, single pixel cameras, lensless cameras, compressive holography, laser-based failure-analysis etc. Bi- nary measurements are a mainstay in signal and image processing and can be modelled by the Walsh transform and Walsh series that are binary cousins of the respective Fourier counterparts. We help bridging the theoreti- cal gap by providing non-uniform recovery guarantees for infinite-dimensional compressed sensing with Walsh samples and wavelet reconstruction. The theoretical results demonstrate that compressed sensing with Walsh samples, as long as the sampling strategy is highly structured and follows the structured sparsity of the signal, is as effective as in the Fourier case. However, there is a fundamental difference in the asymptotic results when the smoothness and vanishing moments of the wavelet increase. In the Fourier case, this changes the optimal sampling patterns, whereas this is not the case in the Walsh setting. 1. INTRODUCTION Since Shannon’s classical sampling theorem [59, 64], sampling theory has been a widely studied field in signal and image processing. Infinite-dimensional compressed sensing [7, 9, 18, 43, 44, 56, 57] is part of this rich theory and offers a method that allows for infinite-dimensional signals to be recovered from undersampled linear measurements. -
Arxiv:1902.02406V2 [Math.FA] 8 Sep 2020 Where Def 1 X Fb(A) = F(Δ)Wa(Δ) ∈ X
POLYNOMIAL INEQUALITIES ON THE HAMMING CUBE ALEXANDROS ESKENAZIS AND PAATA IVANISVILI Abstract. Let (X; k · kX ) be a Banach space. The purpose of this article is to systematically investigate dimension independent properties of vector valued functions f : {−1; 1gn ! X on the Hamming cube whose spectrum is bounded above or below. Our proofs exploit contractivity properties of the heat flow, induced by the geometry of the target space (X; k · kX ), combined with duality arguments and suitable tools from approximation theory and complex analysis. We obtain a series of improvements of various well-studied estimates for functions with bounded spectrum, including moment comparison results for low degree Walsh polynomials and Bernstein{Markov type inequalities, which constitute discrete vector valued analogues of Freud's inequality in Gauss space (1971). Many of these inequalities are new even for scalar valued functions. Furthermore, we provide a short proof of Mendel and Naor's heat smoothing theorem (2014) for functions in tail spaces with values in spaces of nontrivial type and we also prove a dual lower bound on the decay of the heat semigroup acting on functions with spectrum bounded from above. Finally, we improve the reverse Bernstein{Markov inequalities of Meyer (1984) and Mendel and Naor (2014) for functions with narrow enough spectrum and improve the bounds of Filmus, Hatami, Keller and 4 Lifshitz (2016) on the `p sums of influences of bounded functions for p 2 1; 3 . 2010 Mathematics Subject Classification. Primary: 42C10; Secondary: 41A17, 41A63, 46B07. Key words. Hamming cube, heat semigroup, hypercontractivity, Bernstein{Markov inequality, moment comparison. 1. -
Comparing the Walsh Domain to the Fourier Domain with a Labview-Based Communication Systems Toolkit
AC 2007-1038: COMPARING THE WALSH DOMAIN TO THE FOURIER DOMAIN WITH A LABVIEW-BASED COMMUNICATION SYSTEMS TOOLKIT Murat Tanyel, Geneva College Murat Tanyel is a professor of engineering at Geneva College. He teaches upper level electrical engineering courses. Prior to Geneva College, Dr. Tanyel taught at Dordt College, Sioux Center, IA from Aug. 1995 to Aug. 2003. Prior to 1995, he was at Drexel University, Philadelphia, PA where he worked for the Enhanced Educational Experience for Engineering Students (E4) project, setting up and teaching laboratory and hands-on computer experiments for engineering freshmen and sophomores. For one semester, he was also a visiting professor at the United Arab Emirates University in Al-Ain, UAE where he helped set up an innovative introductory engineering curriculum. Dr. Tanyel received his B. S. degree in electrical engineering from Bo?aziçi University, Istanbul, Turkey in 1981, his M. S. degree in electrical engineering from Bucknell University, Lewisburg, PA in 1985 and his Ph. D. in biomedical engineering from Drexel University, Philadelphia, PA in 1990. Page 12.382.1 Page © American Society for Engineering Education, 2007 Comparing the Walsh Domain to the Fourier Domain with a LabVIEW Based Communication Systems Toolkit Abstract Although the Fourier Transform is the traditional frequency domain analysis tool in communications systems, other transforms are pointed out in the context of orthogonal series representation of periodic signals. Last year, we became interested in the Walsh Transform and developed virtual instruments (VIs) to compute the Walsh transform, to generate the Walsh basis functions and modified LabVIEW’s natural ordered fast Walsh Transform (FWT) routine to provide Walsh ordered Walsh transforms and a recent publication 1 reported on this expansion of the Communication Systems Toolkit into the Walsh domain. -
International Centre for Theoretical Physics
IC/95/242 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS CERTAIN PROBLEMS CONCERNING WAVELETS AND WAVELET PACKETS A.H. Siddiqi INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION MIRAMARE-TRIESTE IC/95/242 International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS CERTAIN PROBLEMS CONCERNING WAVELETS AND WAVELET PACKETS A.H. Siddiqi1 International Centre for Theoretical Physics, Trieste, Italy. ABSTRACT Wavelet analysis is the outcome of the synthesis of ideas that have emerged in different branches of science and technology, mainly, in the last decade. The concept of wavelet packets, which are superpositions of wavelets, has been introduced a couple of years ago. They form bases which retain many properties of wavelets like orthogonality, smoothness and localization. The Walsh orthonormal system is a special case of wavelet packet. The wavelet packets provide at our disposal a library of orthonormal bases, each of which can be used to analyse a given signal of finite energy. The optimal choice is decided by the entropy criterion. In the present paper we discuss results concerning convergence, coefficients, and approximation of wavelet packets series in general and wavelets series in particular. Wavelet packet techniques for solutions of differential equations are also mentioned. MIRAMARE - TRIESTE September 1995 'Permanent address: Department of Mathematics, Aligarh Muslim University, Aligarh • 202002, India. 1 Introduction Wavelet analysis is the outcome of the synthesis of ideas that have emerged in different branches of Mathematics, Physics and Engineering. Since the days of Fourier, scientists and engineers, besides mathematicians themselves, have made vigorous efforts to represent square integrable functions (signals having finite energy) as linear combination of functions having some nice properties.