Spectral Methods for Boolean and Multiple-Valued Input Logic Functions
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Binary Polynomial Transforms for Nonlinear Signal Processing
BINARY POLYNOMIAL TRANSFORMS FOR NONLINEAR SIGNAL PROCESSING Jaakko Astola, Karen Egiazarian, Rugen oktern SOS Agaian Signal Processing Laboratory Electrical Engineering Department Tampere University of Technology University of Texas at San Antonio Tampere, FINLAND San Antonio, Texas, USA ABSTRACT ing computers or systems suitable for simultaneous arith- metical or logical operations. In this paper we introduce parametric binary Rademacher functions of two types. Based on them using different kinds 2. BINARY POLYNOMIAL FUNCTIONS AND of arithmetical and logical operations we generate a set of MATRICES binary polynomial transforms (BPT). Some applications of BPT in nonlinear filtering and in compression of binary 2.1. Rademacher Functions and Matrices images are presented. Let a and b be arbitrary integers. We form the classes r,(t,a, b) and s,(t,a, b) of real periodic functions in the 1. INTRODUCTION following way. Let Spectral analysis is a powerful tool in communication, sig- ro(t.a,b) E so(t,a, b) E 1, (1) nal/ image processing and applied mathematics and there are many reasons behind the usefulness of spectral analysis. First, in many applications it is convenient to transform a a, fortEU~~ol[$,fk++i), problem into another, easier problem; the spectral domain rn+l (t, a, b) = is essentially a steady-state viewpoint; this not only gives b, for t E U~~ol[ $t + 5&l *).2 insight for analysis but also for design. Second, and more (2) important: the spectral approach allows us to treat entire and classes of signals which have similar properties in the spec- tral domain; it has excellent properties such as fast algo- a: for t E LO,&I, n = 0, 1,2, . -
LU Decomposition - Wikipedia, the Free Encyclopedia
LU decomposition - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/LU_decomposition From Wikipedia, the free encyclopedia In linear algebra, LU decomposition (also called LU factorization) is a matrix decomposition which writes a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. This decomposition is used in numerical analysis to solve systems of linear equations or calculate the determinant of a matrix. LU decomposition can be viewed as a matrix form of Gaussian elimination. LU decomposition was introduced by mathematician Alan Turing [1] 1 Definitions 2 Existence and uniqueness 3 Positive definite matrices 4 Explicit formulation 5 Algorithms 5.1 Doolittle algorithm 5.2 Crout and LUP algorithms 5.3 Theoretical complexity 6 Small example 7 Sparse matrix decomposition 8 Applications 8.1 Solving linear equations 8.2 Inverse matrix 8.3 Determinant 9 See also 10 References 11 External links Let A be a square matrix. An LU decomposition is a decomposition of the form where L and U are lower and upper triangular matrices (of the same size), respectively. This means that L has only zeros above the diagonal and U has only zeros below the diagonal. For a matrix, this becomes: An LDU decomposition is a decomposition of the form where D is a diagonal matrix and L and U are unit triangular matrices, meaning that all the entries on the diagonals of L and U LDU decomposition of a Walsh matrix 1 of 7 1/11/2012 5:26 PM LU decomposition - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/LU_decomposition are one. -
Perron Spectratopes and the Real Nonnegative Inverse Eigenvalue Problem
Perron Spectratopes and the Real Nonnegative Inverse Eigenvalue Problem a b, Charles R. Johnson , Pietro Paparella ∗ aDepartment of Mathematics, College of William & Mary, Williamsburg, VA 23187-8795, USA bDivision of Engineering and Mathematics, University of Washington Bothell, Bothell, WA 98011-8246, USA Abstract Call an n-by-n invertible matrix S a Perron similarity if there is a real non- 1 scalar diagonal matrix D such that SDS− is entrywise nonnegative. We give two characterizations of Perron similarities and study the polyhedra (S) := n 1 C x R : SDxS− 0;Dx := diag (x) and (S) := x (S): x1 = 1 , whichf 2 we call the Perron≥ spectracone andgPerronP spectratopef 2, respectively. C Theg set of all normalized real spectra of diagonalizable nonnegative matrices may be covered by Perron spectratopes, so that enumerating them is of interest. The Perron spectracone and spectratope of Hadamard matrices are of par- ticular interest and tend to have large volume. For the canonical Hadamard matrix (as well as other matrices), the Perron spectratope coincides with the convex hull of its rows. In addition, we provide a constructive version of a result due to Fiedler ([9, Theorem 2.4]) for Hadamard orders, and a constructive version of [2, Theorem 5.1] for Sule˘ımanova spectra. Keywords: Perron spectracone, Perron spectratope, real nonnegative inverse eigenvalue problem, Hadamard matrix, association scheme, relative gain array 2010 MSC: 15A18, 15B48, 15A29, 05B20, 05E30 1. Introduction The real nonnegative inverse eigenvalue problem (RNIEP) is to determine arXiv:1508.07400v2 [math.RA] 19 Nov 2015 which sets of n real numbers occur as the spectrum of an n-by-n nonnegative matrix. -
1.1 Logic 1.1.1
Section 1.1 Logic 1.1.1 1.1 LOGIC Mathematics is used to predict empirical reality, and is therefore the foundation of engineering. Logic gives precise meaning to mathematical statements. PROPOSITIONS def: A proposition is a statement that is either true (T) or false (F), but not both. Example 1.1.1: 1+1=2.(T) 2+2=5.(F) Example 1.1.2: A fact-based declaration is a proposition, even if no one knows whether it is true. 11213 is prime. 1isprime. There exists an odd perfect number. Coursenotes by Prof. Jonathan L. Gross for use with Rosen: Discrete Math and Its Applic., 5th Ed. Chapter 1 FOUNDATIONS 1.1.2 Example 1.1.3: Logical analysis of rhetoric begins with the modeling of fact-based natural language declarations by propositions. Portland is the capital of Oregon. Columbia University was founded in 1754 by Romulus and Remus. If 2+2 = 5, then you are the pope. (a conditional fact-based declaration). Example 1.1.4: A statement cannot be true or false unless it is declarative. This excludes commands and questions. Go directly to jail. What time is it? Example 1.1.5: Declarations about semantic tokens of non-constant value are NOT proposi- tions. x+2=5. Coursenotes by Prof. Jonathan L. Gross for use with Rosen: Discrete Math and Its Applic., 5th Ed. Section 1.1 Logic 1.1.3 TRUTH TABLES def: The boolean domain is the set {T,F}. Either of its elements is called a boolean value. An n-tuple (p1,...,pn)ofboolean values is called a boolean n-tuple. -
Fast Random Projections Using Lean Walsh Transforms
Fast Random Projections using Lean Walsh Transforms Abstract We present a random projection matrix that is applicable to vectors x 2 Rd in O(d) operations if 0 d ¸ k2+± for arbitrary small ±0 and k = O(log(n)="2). The projection succeeds with probability 1 ¡ 1=n ¡(1+±) and it preserves lengths up to distortion " for all vectors such that kxk1 · kxk2k for arbitrary small ±. Previous results are either not applicable in linear time or require a bound on kxk1 that depends on d. 1 Introduction We are interested in producing a random projection procedure such that applying it to a vector in Rd requires O(d) operation. The idea is to ¯rst project an incoming vector x 2 Rd into an intermediate dimension d~ = d® (0 < ® < 1) and then use a known construction to project it further down to the minimal Johnson 2 Lindenstrauss (JL) dimension k = O(log(n)=" )). The mapping is composed of three matrices x 7! RADsx. ~ The ¯rst, Ds, is a random §1 diagonal matrix. The second, A, is a dense matrix of size d £ d termed the Lean Walsh Matrix. The third, R, is a k £ d~ Fast Johnson Lindenstrauss matrix. As R, we use the fast JL construction by Ailon and Liberty [2]. Their matrix, R, can be applied ~ ~ 2+±00 00 (2+±00)=® to ADsx in O(d log(k)) operations as long as d ¸ k for arbitrary small ± . If d > k this 1 condition is met . Therefore, for such d, applying R to ADsx is length preserving w.p and accomplishable in O(d~log(k)) = O(d® log(k)) = O(d) operations. -
Exclusive Or from Wikipedia, the Free Encyclopedia
New features Log in / create account Article Discussion Read Edit View history Exclusive or From Wikipedia, the free encyclopedia "XOR" redirects here. For other uses, see XOR (disambiguation), XOR gate. Navigation "Either or" redirects here. For Kierkegaard's philosophical work, see Either/Or. Main page The logical operation exclusive disjunction, also called exclusive or (symbolized XOR, EOR, Contents EXOR, ⊻ or ⊕, pronounced either / ks / or /z /), is a type of logical disjunction on two Featured content operands that results in a value of true if exactly one of the operands has a value of true.[1] A Current events simple way to state this is "one or the other but not both." Random article Donate Put differently, exclusive disjunction is a logical operation on two logical values, typically the values of two propositions, that produces a value of true only in cases where the truth value of the operands differ. Interaction Contents About Wikipedia Venn diagram of Community portal 1 Truth table Recent changes 2 Equivalencies, elimination, and introduction but not is Contact Wikipedia 3 Relation to modern algebra Help 4 Exclusive “or” in natural language 5 Alternative symbols Toolbox 6 Properties 6.1 Associativity and commutativity What links here 6.2 Other properties Related changes 7 Computer science Upload file 7.1 Bitwise operation Special pages 8 See also Permanent link 9 Notes Cite this page 10 External links 4, 2010 November Print/export Truth table on [edit] archived The truth table of (also written as or ) is as follows: Venn diagram of Create a book 08-17094 Download as PDF No. -
A Novel 4-Point Discrete Fourier Transforms Circuit Based on Product of Rademacher Functions
The 5th International Conference on Electrical Engineering and Informatics 2015 August 10-11, 2015, Bali, Indonesia A Novel 4-Point Discrete Fourier Transforms Circuit based on Product of Rademacher Functions Zulfikar1, Hubbul Walidainy2 Department of Electrical Engineering Syiah Kuala University Banda Aceh 23111, Indonesia. E-mail: [email protected], [email protected] Abstract—This paper presents a new circuit design for algorithms have been developed for certain purposes and each implementing 4-point DFT algorithm based on product of of them comes with advantages and drawbacks. Rademacher functions. The circuit has been derived from the similarity of how Fourier transforms and Walsh transforms are Walsh transforms algorithm is simpler than Fourier implemented. Walsh matrices contain numbers either +1 or -1 transform, but this transformation model is rarely used in except for first row. Similarly, the 4-point DFT matrix contain application and almost forgotten. Walsh transforms has a few numbers either positive or negative except for first row. This similarity to the Fourier transforms [2]-[5]. Based on this, some similarity has been taken into the case of how to implement the researchers have adopted this algorithm for developing the DFT circuit based on how Walsh transforms is generated. Since more efficient Fourier transforms [6]-[8]. An algorithm for Walsh transforms is derived based on product of Rademacher calculating DFT using Walsh transforms is developed through functions, the proposed 4-point DFT circuit is designed according the factorization of intermediate transform T [6]. Monir T et al to product Rademacher functions. The circuit consist of negative proposed an efficient combination of Walsh and DFT circuit, multiplexers, accumulator (real and imaginary), buffers, calculation. -
Technology Mapping for Vlsi Circuits Exploiting Boolean Properties and Operations
TECHNOLOGY MAPPING FOR VLSI CIRCUITS EXPLOITING BOOLEAN PROPERTIES AND OPERATIONS disserttion sumi ttedtothe deprtmentofeletrilengi neering ndtheommitteeongrdutestudies ofstnforduniversi ty in prtilfulfillmentoftherequirements forthedegreeof dotorofphilosophy By Fr´ed´eric Mailhot December 1991 c Copyright 1991 by Fr´ed´eric Mailhot ii I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a dissertation for the degree of Doctor of Philosophy. Giovanni De Micheli (Principal Advisor) I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a dissertation for the degree of Doctor of Philosophy. Mark A. Horowitz I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a dissertation for the degree of Doctor of Philosophy. Robert W. Dutton Approved for the University Committee on Graduate Stud- ies: Dean of Graduate Studies & Research iii Abstract Automatic synthesis of digital circuits has gained increasing importance. The synthesis process consists of transforming an abstract representation of a system into an implemen- tation in a target technology. The set of transformations has traditionally been broken into three steps: high-level synthesis, logic synthesis and physical design. This dissertation is concerned with logic synthesis. More specifically, we study tech- nology mapping, which is the link between logic synthesis and physical design. The object of technology mapping is to transform a technology-independent logic description into an implementation in a target technology. One of the key operations during tech- nology mapping is to recognize logic equivalence between a portion of the initial logic description and an element of the target technology. -
Boolean.Py Documentation Release 3.2
boolean.py Documentation Release 3.2 Sebastian Krämer Aug 06, 2018 Contents 1 User Guide 1 1.1 Introduction...............................................1 1.2 Installation................................................2 1.3 Creating boolean expressions......................................2 1.4 Evaluation of expressions........................................2 1.5 Equality of expressions..........................................3 1.6 Analyzing a boolean expression.....................................4 1.7 Using boolean.py to define your own boolean algebra.........................5 2 Concepts and Definitions 7 2.1 Basic Definitions.............................................7 2.2 Laws...................................................9 3 Development Guide 13 3.1 Testing.................................................. 13 3.2 Classes Hierarchy............................................ 14 3.3 Class creation............................................... 14 3.4 Class initialization............................................ 14 3.5 Ordering................................................. 14 3.6 Parsing.................................................. 15 4 Acknowledgments 17 i ii CHAPTER 1 User Guide This document provides an introduction on boolean.py usage. It requires that you are already familiar with Python and know a little bit about boolean algebra. All definitions and laws are stated in Concepts and Definitions. Contents • User Guide – Introduction – Installation – Creating boolean expressions – Evaluation of expressions – Equality -
Accelerated Dense Random Projections
Accelerated Dense Random Projections A Dissertation Presented to the faculty of the Graduate School of Yale University in Candidacy for the degree of Doctor of Philosophy by Edo Liberty Dissertation Director: Steven Zucker May 2009 Accelerated Dense Random Projections Edo Liberty May 2009 ABSTRACT In dimensionality reduction, a set of points in Rd is mapped into Rk, with the target dimension k smaller than the original dimension d, while distances between all pairs of points are approximately preserved. Currently popular methods for achieving this involve random projection, or choosing a linear mapping (a k £ d matrix) from a distribution that is independent of the input points. Applying the mapping (chosen according to this distribution) is shown to give the desired property with at least constant probability. The contributions in this thesis are twofold. First, we provide a framework for designing such distributions. Second, we derive e±cient random projection algorithms using this framework. Our results achieve performance exceeding other existing approaches. When the target dimension is signi¯cantly smaller than the original dimension we gain signi¯cant improvement by designing e±cient algorithms for applying certain linear algebraic transforms. To demonstrate practicality, we supplement the theoretical derivations with experimental results. Acknowledgments: Special thanks to Steven Zucker, Nir Ailon, Daniel Spielman, Ronald Coifman, Amit Singer and Mark Tygert. Their work, ideas and guidance can be seen throughout this manuscript. 1. LIST OF COMMON NOTATIONS d . .input vectors' original dimension Rd ...............................................................................real d dimensional space d x or xi . .input vector(s) in R n . number of input vectors or a constant polynomial in that number k . -
New Perspectives on Non-Negative Matrix Factorization for Grouped Topic Models
New Perspectives on Non-negative Matrix Factorization for Grouped Topic Models by Gabriel C. Phelan B.Sc., Rochester Institute of Technology, 2017 Project Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in the Department of Statistics and Actuarial Science Faculty of Science c Gabriel C. Phelan 2020 SIMON FRASER UNIVERSITY Summer 2020 Copyright in this work rests with the author. Please ensure that any reproduction or re-use is done in accordance with the relevant national copyright legislation. Approval Name: Gabriel C. Phelan Degree: Master of Science (Statistics) Title: New Perspectives on Non-negative Matrix Factorization for Grouped Topic Models Examining Committee: Chair: Jinko Graham Professor David Campbell Senior Supervisor Professor School of Mathematics and Statistics Carleton University Lloyd T. Elliott Supervisor Assistant Professor Thomas M. Loughin Internal Examiner Professor Date Defended: August 18, 2020 ii Abstract Probabilistic topic models (PTM’s) have become a ubiquitous approach for finding a set of latent themes (“topics”) in collections of unstructured text. A simpler, linear algebraic technique for the same problem is non-negative matrix factorization: we are given a matrix with non-negative entries and asked to find a pair of low-rank matrices, also non-negative, whose product is approximately the original matrix. A drawback of NMF is the non-convex nature of the optimization problem it poses. Recent work by the theoretical computer sci- ence community addresses this issue, utilizing NMF’s inherent structure to find conditions under which the objective function admits convexity. With convexity comes tractability, and the central theme of this thesis is the exploitation of this tractability to ally NMF with resampling-based nonparametrics. -
Human Identification Using Compressed ECG Signals
This is a postprint version of the following published document: C. Camara, P. Peris-Lopez, and J. E. Tapiador (2015). Human Identification Using Compressed ECG Signals. Journal of Medical Systems, 39: 148. Avalaible in https://doi.org/10.1007/s10916-015-0323-2 © Springer Science+Business Media New York 2015 Human ldentification Using Compressed ECG Signals Carmen Camara1 • Pedro Peris-Lopez1 • Juan E. Tapiador1 Abstract As a result of the increased demand for improved Introduction life styles and the increment of senior citizens over the age of 65, new home care services are demanded. Simul According to [3], the medica! sector is the area that has taneously, the medica! sector is increasingly becoming the suffered the major number of hacking incidents over the new target of cybercrirninals due the potential value of last two year�3 % of the data breaches in US. Medica! users' medica! information. Toe use of biometrics seems companies and hospitals have begun to introduce biometric an effective tool as a deterrent for many of such attacks. solutions to rnitigate attacks and reduce costs. Further In this paper, we propase the use of electrocardiograms more, the proper identification of patients when they walk (ECGs) for the identification of individuals. For instance, through the door is a major issue nowadays for all the for a telecare service, a user could be authenticated using hospitals around the world. Errors in medica} records, or the information extracted from her ECG signa!. Toe major even incorrect treatrnents, are very costly for the medica! ity of ECG-based biometrics systems extract information centres and harmful for the patient.