Lectures 8 & 9: The Z-transform.
|
|
|
- Clyde Warren
- 6 years ago
- Views:
Transcription
1 Lectures 8 & 9: The Z-transfor. 1. Definitions. The Z-transfor is defined as a function series (a series in which each ter is a function of one or ore variables: Z[] where is a C valued function f : N C. This series is defined for those C for which the series is absolutely convergent. We have the following result about convergence: Theore 1.1. Suppose that one of the following liits exists. Then the series R f li 1/, R f li f( + 1 Z[] is defined and absolutely convergent for all with > R f. Furtherore this series is analytic for these : this eans that the function F ( is infinitely differentiable with respect to with > R f and we also have F ( d d (i.e. the series is differentiable terwise for > R f. Proof: The series is a power series in ρ 1/. A power series a ρ is, as is well-nown, absolutely convergent for ρ < K where 1/K is one of the liits li a 1/, a +1 li a (reeber that we have K if one of these liits is, and we have K if one of the liits is. Then put a and R f 1/K and the first part of the result follows easily. Differentiability is ore coplicated to prove, and we give this proof in an appendix at the end of these notes. Exaple 1. 1,, 1, 2,... We have 1
2 2 c 27 P. Basarab-Horwath F ( for > 1 because then we have 1/ < 1 and the series is a geoetric series with quotient 1/ and first ter 1. Exaple 2.,, 1, 2,... In this case we find that for > 1 and then we see that F ( d d 1 Z[] d d Z[1] ( 1 2. The Z-transfor is a atheatical tool for solving so-called difference equations. Here is an exaple: f( , f(,, 1, 2,... In order to transfor this equation, we ust now what Z[f( + 1] is when we now F ( Z[]. We have Z[f( + 1] 1 ( F ( f( + 1 f( because f(. Transforing the equation then yields F ( F ( + by Exaple 1. A little algebra then gives 1 f( F ( ( 1 Z[] 2 fro which it follows that. This last step follows fro the following result: Theore 1.2. (Uniqueness Theore. If F ( and G( are Z-transfors of the functions and g( respectively, and if F ( G( for all > R for soe R >, then g( for all, 1, 2,...
3 c 27 P. Basarab-Horwath 3 Proof: If we have a power series and a w a w for all w with w < K for soe R >, then a,, 1, 2,.... Thus, if F ( G( for all > R for soe R >, then we have g( ( g( for all > R. Now put w 1/ and a g( and we find that for w < K 1/R a w whence a,, 1,.... Hence, g(,, 1,.... As you can see fro the arguent, this result is fundaental in the solution: we are able to associate a unique function to a given series in 1/. Another useful result is the following: Theore 1.3. If we are given a function F ( which is analytic for all with > R for soe R >, and if F ( A as where A C, then there is a unique function f : N C so that F ( for all with > R. That is, F ( is the Z-transfor of a unique function f : N C. The proof of this result requires coplex analysis, so we oit it here. However, we note one crucial property of F (: in the theore we have the condition F ( A as where A C. If this fails to be satisfied, then F ( is not the Z-transfor of any function. This is a useful test to apply, in order to chec whether your calculations are reasonable. For instance, + 1, satisfy the test (and are definitely analytic for > 1 so they are Z-transfors of functions, whereas does not satisfy the test. The result we use here is forulated as follows: Proposition 1.1. If F ( Z[] for all with > R f for soe R f >, then F ( f( as. a
4 4 c 27 P. Basarab-Horwath Proof: Put w 1/, then w when. We then have F ( G(w w for w < R f. Now, this series converges uniforly for all w a for any a < R f, and consequently G(w is a continuous function of w for w < R f. Hence G(w G( f( as w. That is, F ( f( as. Exaple 3: 1, 1, 2,..., f(. In this case we find that F ( 1 log(1 1 för > 1. Here we have used the Maclaurin expansion for t < 1. log(1 + t ( 1 t 1 Exaple 4: 1! for, 1, 2,.... Then för >. F ( (1/ e 1/! Exaple 5: We define δ(, ; δ( 1. A siple calculation gives Z[δ(] 1. Further, we define for fixed N functions δ(, ; δ( 1,. This gives Z[δ( ] Rules of calculation and anipulation. To handle the Z-transfor we need soe basic properties. First is the following: Theore 2.1. If the functions f and g on N have Z-transfors F (, G( for > R f and > R g respectively, then the following results hold: Z[c]( cf (, c C, > R f Z[ + g(]( F ( + G(, > ax(r f, R g. Proof: The proofs are eleentary calculations using the definition of the Z-transfor. In the following Theore we prove soe siple rules of calculation which are used to siplify calculations. Theore 2.2. If F ( Z[] and G( Z[g(] then the following hold: (1 For g( a with a > one has G( F ( a. (2 For g( f( + with a positive integer, one has
5 c 27 P. Basarab-Horwath 5 G( F ( f( 1 f(1 f( 1. (3 For g( f( with a positive integer, one has G( F ( + +1 f( f( f(. (4 For g( with a positive integer, one has G( F (. (5 If g( χ( f( then G( F ( Proof: If g( a then we have G( For g( f( + one has a ( a F ( a. G( f( + ( 1 F ( f( 1 f(1 f( 1. The case of g( f( is dealt with in a siilar anner. Case (4 is left to you as an exercise. If g( χ( f( then we have G( f( χ( f( + F (. Proposition 2.1. If g( then we have G( d d F (.
6 6 c 27 P. Basarab-Horwath Proof: This is a siple calculation: G( d d ( d d d d F (. The penultiate step (replacing the su of derivatives by the derivative of the su is allowed for > R f (where the Z-transfor of is defined and analytic and absolutely convergent. This result relates the ultiplication by the variable to the derivative of the Z- transfor of, and it is the analogy of the corresponding results for the Fourier and Laplace transfors. We have the following useful results Proposition 2.2. Z[a ] a Z[a a ] ( a 2 Z[ 2 a ] a2 + a 2 ( a 3. Proof: These are left as exercises using Proposition 2.1. We also have the following rules for calculation: Proposition 2.3. [( ] + Z a +1 2, 3,... ( a +1 [( ] Z a a, 1, 2,... ( a +1 [( ] + n Z a a n n+1 2, 3,... ; n 1,...,. ( a +1 Proof: We first prove [( Z ] and leave the other calculations ( as exercises. We have, noting that when <, ( 1 +1
7 c 27 P. Basarab-Horwath 7 [( Z ] ( 1 [w 1/] ( w 1! 1! ( w! (!! w ( 1... ( + 1w w 1! w 1 d w! dw d dw (w d 1 d w! dw 1!! w dw (w ( w ( 1 1 w (1 w +1 w (1 w +1 [w 1/] ( In the above calculation we have used the following facts, which are easily verified using atheatical induction: d dw (w ( 1... ( + 1w, ( Next, use the rule Z[a ]( F to obtain a [( ] Z a The other forulas are proved in the sae way. Finally we have d ( 1 dw 1 w a ( a +1. 1! (1 w. +1
8 8 c 27 P. Basarab-Horwath Proposition 2.4. Z[cos α] 2 cos α 2 2 cos α + 1 sin α Z[sin α] 2 2 cos α + 1 Z[cos(α + β] 2 cos β cos(α β 2 2 cos α + 1 Z[sin(α + β] 2 sin β + sin(α β. 2 2 cos α + 1 Proof: First calculate Z[e iα ] and then apply this to cos α, sin α. The rest is left as an exercise. 3. Inverse Z-transfor. The result we wish to record here is the following: Theore 3.1. If, g( are two C-valued functions on N such that their Z- transfors satisfy F ( G( for all > R for soe R >, then g( for all N. Proof: This follows fro Theore Convolution. We define the convolution of two functions, g( are two functions on N as follows: Then we have: Theore 4.1. Proof: h( (f g( f( g(. H( Z[(f g]( F (G(. H( h( l+n f( g( l+n f(ng(l f(n n g(l l f(n n n l F (G(. g(l l
4 = (0.02) 3 13, = 0.25 because = 25. Simi-
Theore. Let b and be integers greater than. If = (. a a 2 a i ) b,then for any t N, in base (b + t), the fraction has the digital representation = (. a a 2 a i ) b+t, where a i = a i + tk i with k i =
3.8 Three Types of Convergence
3.8 Three Types of Convergence 3.8 Three Types of Convergence 93 Suppose that we are given a sequence functions {f k } k N on a set X and another function f on X. What does it ean for f k to converge to
A RECURRENCE RELATION FOR BERNOULLI NUMBERS. Mümün Can, Mehmet Cenkci, and Veli Kurt
Bull Korean Math Soc 42 2005, No 3, pp 67 622 A RECURRENCE RELATION FOR BERNOULLI NUMBERS Müün Can, Mehet Cenci, and Veli Kurt Abstract In this paper, using Gauss ultiplication forula, a recurrence relation
ON SEQUENCES OF NUMBERS IN GENERALIZED ARITHMETIC AND GEOMETRIC PROGRESSIONS
Palestine Journal of Matheatics Vol 4) 05), 70 76 Palestine Polytechnic University-PPU 05 ON SEQUENCES OF NUMBERS IN GENERALIZED ARITHMETIC AND GEOMETRIC PROGRESSIONS Julius Fergy T Rabago Counicated by
Fourier Series Summary (From Salivahanan et al, 2002)
Fourier Series Suary (Fro Salivahanan et al, ) A periodic continuous signal f(t), - < t
Hermite s Rule Surpasses Simpson s: in Mathematics Curricula Simpson s Rule. Should be Replaced by Hermite s
International Matheatical Foru, 4, 9, no. 34, 663-686 Herite s Rule Surpasses Sipson s: in Matheatics Curricula Sipson s Rule Should be Replaced by Herite s Vito Lapret University of Lublana Faculty of
Chapter 6 1-D Continuous Groups
Chapter 6 1-D Continuous Groups Continuous groups consist of group eleents labelled by one or ore continuous variables, say a 1, a 2,, a r, where each variable has a well- defined range. This chapter explores:
The Weierstrass Approximation Theorem
36 The Weierstrass Approxiation Theore Recall that the fundaental idea underlying the construction of the real nubers is approxiation by the sipler rational nubers. Firstly, nubers are often deterined
Closed-form evaluations of Fibonacci Lucas reciprocal sums with three factors
Notes on Nuber Theory Discrete Matheatics Print ISSN 30-32 Online ISSN 2367-827 Vol. 23 207 No. 2 04 6 Closed-for evaluations of Fibonacci Lucas reciprocal sus with three factors Robert Frontczak Lesbank
Solutions 1. Introduction to Coding Theory - Spring 2010 Solutions 1. Exercise 1.1. See Examples 1.2 and 1.11 in the course notes.
Solutions 1 Exercise 1.1. See Exaples 1.2 and 1.11 in the course notes. Exercise 1.2. Observe that the Haing distance of two vectors is the iniu nuber of bit flips required to transfor one into the other.
Uniform Approximation and Bernstein Polynomials with Coefficients in the Unit Interval
Unifor Approxiation and Bernstein Polynoials with Coefficients in the Unit Interval Weiang Qian and Marc D. Riedel Electrical and Coputer Engineering, University of Minnesota 200 Union St. S.E. Minneapolis,
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 11 10/15/2008 ABSTRACT INTEGRATION I
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 11 10/15/2008 ABSTRACT INTEGRATION I Contents 1. Preliinaries 2. The ain result 3. The Rieann integral 4. The integral of a nonnegative
Introduction to Robotics (CS223A) (Winter 2006/2007) Homework #5 solutions
Introduction to Robotics (CS3A) Handout (Winter 6/7) Hoework #5 solutions. (a) Derive a forula that transfors an inertia tensor given in soe frae {C} into a new frae {A}. The frae {A} can differ fro frae
2 Q 10. Likewise, in case of multiple particles, the corresponding density in 2 must be averaged over all
Lecture 6 Introduction to kinetic theory of plasa waves Introduction to kinetic theory So far we have been odeling plasa dynaics using fluid equations. The assuption has been that the pressure can be either
1 Proof of learning bounds
COS 511: Theoretical Machine Learning Lecturer: Rob Schapire Lecture #4 Scribe: Akshay Mittal February 13, 2013 1 Proof of learning bounds For intuition of the following theore, suppose there exists a
Four-vector, Dirac spinor representation and Lorentz Transformations
Available online at www.pelagiaresearchlibrary.co Advances in Applied Science Research, 2012, 3 (2):749-756 Four-vector, Dirac spinor representation and Lorentz Transforations S. B. Khasare 1, J. N. Rateke
The Euler-Maclaurin Formula and Sums of Powers
DRAFT VOL 79, NO 1, FEBRUARY 26 1 The Euler-Maclaurin Forula and Sus of Powers Michael Z Spivey University of Puget Sound Tacoa, WA 98416 spivey@upsedu Matheaticians have long been intrigued by the su
The Fundamental Basis Theorem of Geometry from an algebraic point of view
Journal of Physics: Conference Series PAPER OPEN ACCESS The Fundaental Basis Theore of Geoetry fro an algebraic point of view To cite this article: U Bekbaev 2017 J Phys: Conf Ser 819 012013 View the article
The Möbius inversion formula for Fourier series applied to Bernoulli and Euler polynomials
The Möbius inversion forula for Fourier series applied to Bernoulli and Euler polynoials Luis M Navas a, Francisco J Ruiz b,, Juan L Varona c,, a Departaento de Mateáticas, Universidad de Salaanca, Plaza
MA304 Differential Geometry
MA304 Differential Geoetry Hoework 4 solutions Spring 018 6% of the final ark 1. The paraeterised curve αt = t cosh t for t R is called the catenary. Find the curvature of αt. Solution. Fro hoework question
Question 1. Question 3. Question 4. Graduate Analysis I Exercise 4
Graduate Analysis I Exercise 4 Question 1 If f is easurable and λ is any real nuber, f + λ and λf are easurable. Proof. Since {f > a λ} is easurable, {f + λ > a} = {f > a λ} is easurable, then f + λ is
LATTICE POINT SOLUTION OF THE GENERALIZED PROBLEM OF TERQUEi. AND AN EXTENSION OF FIBONACCI NUMBERS.
i LATTICE POINT SOLUTION OF THE GENERALIZED PROBLEM OF TERQUEi. AND AN EXTENSION OF FIBONACCI NUMBERS. C. A. CHURCH, Jr. and H. W. GOULD, W. Virginia University, Morgantown, W. V a. In this paper we give
A GENERAL FORM FOR THE ELECTRIC FIELD LINES EQUATION CONCERNING AN AXIALLY SYMMETRIC CONTINUOUS CHARGE DISTRIBUTION
A GENEAL FOM FO THE ELECTIC FIELD LINES EQUATION CONCENING AN AXIALLY SYMMETIC CONTINUOUS CHAGE DISTIBUTION BY MUGU B. ăuţ Abstract..By using an unexpected approach it results a general for for the electric
arxiv: v1 [math.nt] 14 Sep 2014
![arxiv: v1 [math.nt] 14 Sep 2014](/thumbs/89/98762755.jpg "arxiv: v1 [math.nt] 14 Sep 2014")
ROTATION REMAINDERS P. JAMESON GRABER, WASHINGTON AND LEE UNIVERSITY 08 arxiv:1409.411v1 [ath.nt] 14 Sep 014 Abstract. We study properties of an array of nubers, called the triangle, in which each row
On Conditions for Linearity of Optimal Estimation
On Conditions for Linearity of Optial Estiation Erah Akyol, Kuar Viswanatha and Kenneth Rose {eakyol, kuar, rose}@ece.ucsb.edu Departent of Electrical and Coputer Engineering University of California at
ADVANCES ON THE BESSIS- MOUSSA-VILLANI TRACE CONJECTURE
ADVANCES ON THE BESSIS- MOUSSA-VILLANI TRACE CONJECTURE CHRISTOPHER J. HILLAR Abstract. A long-standing conjecture asserts that the polynoial p(t = Tr(A + tb ] has nonnegative coefficients whenever is
Generalized eigenfunctions and a Borel Theorem on the Sierpinski Gasket.
Generalized eigenfunctions and a Borel Theore on the Sierpinski Gasket. Kasso A. Okoudjou, Luke G. Rogers, and Robert S. Strichartz May 26, 2006 1 Introduction There is a well developed theory (see [5,
Beyond Mere Convergence
Beyond Mere Convergence Jaes A. Sellers Departent of Matheatics The Pennsylvania State University 07 Whitore Laboratory University Park, PA 680 sellers@ath.psu.edu February 5, 00 REVISED Abstract In this
Chaotic Coupled Map Lattices
Chaotic Coupled Map Lattices Author: Dustin Keys Advisors: Dr. Robert Indik, Dr. Kevin Lin 1 Introduction When a syste of chaotic aps is coupled in a way that allows the to share inforation about each
Electromagnetic scattering. Graduate Course Electrical Engineering (Communications) 1 st Semester, Sharif University of Technology
Electroagnetic scattering Graduate Course Electrical Engineering (Counications) 1 st Seester, 1388-1389 Sharif University of Technology Contents of lecture 5 Contents of lecture 5: Scattering fro a conductive
Physics 215 Winter The Density Matrix
Physics 215 Winter 2018 The Density Matrix The quantu space of states is a Hilbert space H. Any state vector ψ H is a pure state. Since any linear cobination of eleents of H are also an eleent of H, it
Curious Bounds for Floor Function Sums
1 47 6 11 Journal of Integer Sequences, Vol. 1 (018), Article 18.1.8 Curious Bounds for Floor Function Sus Thotsaporn Thanatipanonda and Elaine Wong 1 Science Division Mahidol University International
13.2 Fully Polynomial Randomized Approximation Scheme for Permanent of Random 0-1 Matrices
CS71 Randoness & Coputation Spring 018 Instructor: Alistair Sinclair Lecture 13: February 7 Disclaier: These notes have not been subjected to the usual scrutiny accorded to foral publications. They ay
Entangling characterization of (SWAP) 1/m and Controlled unitary gates
Entangling characterization of (SWAP) / and Controlled unitary gates S.Balakrishnan and R.Sankaranarayanan Departent of Physics, National Institute of Technology, Tiruchirappalli 65, India. We study the
Linear recurrences and asymptotic behavior of exponential sums of symmetric boolean functions
Linear recurrences and asyptotic behavior of exponential sus of syetric boolean functions Francis N. Castro Departent of Matheatics University of Puerto Rico, San Juan, PR 00931 francis.castro@upr.edu
IN modern society that various systems have become more
Developent of Reliability Function in -Coponent Standby Redundant Syste with Priority Based on Maxiu Entropy Principle Ryosuke Hirata, Ikuo Arizono, Ryosuke Toohiro, Satoshi Oigawa, and Yasuhiko Takeoto
Physical interpretation of the Riemann hypothesis
Physical interpretation of the Rieann hypothesis Ditry Pozdnyaov Faculty of Radiophysics and Coputer Technologies of Belarusian State University Nezavisiosty av4 Mins Belarus E-ail: pozdnyaov@tutby Keywords:
A DISCRETE ZAK TRANSFORM. Christopher Heil. The MITRE Corporation McLean, Virginia Technical Report MTR-89W00128.
A DISCRETE ZAK TRANSFORM Christopher Heil The MITRE Corporation McLean, Virginia 22102 Technical Report MTR-89W00128 August 1989 Abstract. A discrete version of the Zak transfor is defined and used to
ORIGAMI CONSTRUCTIONS OF RINGS OF INTEGERS OF IMAGINARY QUADRATIC FIELDS
#A34 INTEGERS 17 (017) ORIGAMI CONSTRUCTIONS OF RINGS OF INTEGERS OF IMAGINARY QUADRATIC FIELDS Jürgen Kritschgau Departent of Matheatics, Iowa State University, Aes, Iowa jkritsch@iastateedu Adriana Salerno
THE AVERAGE NORM OF POLYNOMIALS OF FIXED HEIGHT
THE AVERAGE NORM OF POLYNOMIALS OF FIXED HEIGHT PETER BORWEIN AND KWOK-KWONG STEPHEN CHOI Abstract. Let n be any integer and ( n ) X F n : a i z i : a i, ± i be the set of all polynoials of height and
a a a a a a a m a b a b
Algebra / Trig Final Exa Study Guide (Fall Seester) Moncada/Dunphy Inforation About the Final Exa The final exa is cuulative, covering Appendix A (A.1-A.5) and Chapter 1. All probles will be ultiple choice
i ij j ( ) sin cos x y z x x x interchangeably.)
Tensor Operators Michael Fowler,2/3/12 Introduction: Cartesian Vectors and Tensors Physics is full of vectors: x, L, S and so on Classically, a (three-diensional) vector is defined by its properties under
Feature Extraction Techniques
Feature Extraction Techniques Unsupervised Learning II Feature Extraction Unsupervised ethods can also be used to find features which can be useful for categorization. There are unsupervised ethods that
Standard & Canonical Forms
Standard & Canonical Fors CHAPTER OBJECTIVES Learn Binary Logic and BOOLEAN AlgebraLearn How to ap a Boolean Expression into Logic Circuit Ipleentation Learn How To anipulate Boolean Expressions and Siplify
E0 370 Statistical Learning Theory Lecture 6 (Aug 30, 2011) Margin Analysis
E0 370 tatistical Learning Theory Lecture 6 (Aug 30, 20) Margin Analysis Lecturer: hivani Agarwal cribe: Narasihan R Introduction In the last few lectures we have seen how to obtain high confidence bounds
New upper bound for the B-spline basis condition number II. K. Scherer. Institut fur Angewandte Mathematik, Universitat Bonn, Bonn, Germany.
New upper bound for the B-spline basis condition nuber II. A proof of de Boor's 2 -conjecture K. Scherer Institut fur Angewandte Matheati, Universitat Bonn, 535 Bonn, Gerany and A. Yu. Shadrin Coputing
Research Article Some Formulae of Products of the Apostol-Bernoulli and Apostol-Euler Polynomials
Discrete Dynaics in Nature and Society Volue 202, Article ID 927953, pages doi:055/202/927953 Research Article Soe Forulae of Products of the Apostol-Bernoulli and Apostol-Euler Polynoials Yuan He and
A symbolic operator approach to several summation formulas for power series II
A sybolic operator approach to several suation forulas for power series II T. X. He, L. C. Hsu 2, and P. J.-S. Shiue 3 Departent of Matheatics and Coputer Science Illinois Wesleyan University Blooington,
On the Navier Stokes equations
On the Navier Stokes equations Daniel Thoas Hayes April 26, 2018 The proble on the existence and soothness of the Navier Stokes equations is resolved. 1. Proble description The Navier Stokes equations
Algebraic Montgomery-Yang problem: the log del Pezzo surface case
c 2014 The Matheatical Society of Japan J. Math. Soc. Japan Vol. 66, No. 4 (2014) pp. 1073 1089 doi: 10.2969/jsj/06641073 Algebraic Montgoery-Yang proble: the log del Pezzo surface case By DongSeon Hwang
Prerequisites. We recall: Theorem 2 A subset of a countably innite set is countable.
Prerequisites 1 Set Theory We recall the basic facts about countable and uncountable sets, union and intersection of sets and iages and preiages of functions. 1.1 Countable and uncountable sets We can
Probability Distributions
Probability Distributions In Chapter, we ephasized the central role played by probability theory in the solution of pattern recognition probles. We turn now to an exploration of soe particular exaples
The accelerated expansion of the universe is explained by quantum field theory.
The accelerated expansion of the universe is explained by quantu field theory. Abstract. Forulas describing interactions, in fact, use the liiting speed of inforation transfer, and not the speed of light.
Chapter 10: Sinusoidal Steady-State Analysis
Chapter 0: Sinusoidal Steady-State Analysis Sinusoidal Sources If a circuit is driven by a sinusoidal source, after 5 tie constants, the circuit reaches a steady-state (reeber the RC lab with t = τ). Consequently,
lecture 36: Linear Multistep Mehods: Zero Stability
95 lecture 36: Linear Multistep Mehods: Zero Stability 5.6 Linear ultistep ethods: zero stability Does consistency iply convergence for linear ultistep ethods? This is always the case for one-step ethods,
#A52 INTEGERS 10 (2010), COMBINATORIAL INTERPRETATIONS OF BINOMIAL COEFFICIENT ANALOGUES RELATED TO LUCAS SEQUENCES
#A5 INTEGERS 10 (010), 697-703 COMBINATORIAL INTERPRETATIONS OF BINOMIAL COEFFICIENT ANALOGUES RELATED TO LUCAS SEQUENCES Bruce E Sagan 1 Departent of Matheatics, Michigan State University, East Lansing,
. The univariate situation. It is well-known for a long tie that denoinators of Pade approxiants can be considered as orthogonal polynoials with respe
PROPERTIES OF MULTIVARIATE HOMOGENEOUS ORTHOGONAL POLYNOMIALS Brahi Benouahane y Annie Cuyt? Keywords Abstract It is well-known that the denoinators of Pade approxiants can be considered as orthogonal
arxiv: v3 [math.nt] 14 Nov 2016
![arxiv: v3 [math.nt] 14 Nov 2016](/thumbs/80/80794762.jpg "arxiv: v3 [math.nt] 14 Nov 2016")
A new integral-series identity of ultiple zeta values and regularizations Masanobu Kaneko and Shuji Yaaoto Noveber 15, 2016 arxiv:1605.03117v3 [ath.nt] 14 Nov 2016 Abstract We present a new integral =
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.323: Relativistic Quantum Field Theory I
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Departent 8.33: Relativistic Quantu Field Theory I INFORMAL NOTES DISTRIBUTIONS AND THE FOURIER TRANSFORM Basic idea: In QFT it is coon to encounter integrals
Lecture #8-3 Oscillations, Simple Harmonic Motion
Lecture #8-3 Oscillations Siple Haronic Motion So far we have considered two basic types of otion: translation and rotation. But these are not the only two types of otion we can observe in every day life.
Model Fitting. CURM Background Material, Fall 2014 Dr. Doreen De Leon
Model Fitting CURM Background Material, Fall 014 Dr. Doreen De Leon 1 Introduction Given a set of data points, we often want to fit a selected odel or type to the data (e.g., we suspect an exponential
CSE525: Randomized Algorithms and Probabilistic Analysis May 16, Lecture 13
CSE55: Randoied Algoriths and obabilistic Analysis May 6, Lecture Lecturer: Anna Karlin Scribe: Noah Siegel, Jonathan Shi Rando walks and Markov chains This lecture discusses Markov chains, which capture
Introductions to JacobiND
Introductions to JacobiND Introduction to the Jacobi elliptic functions General Historical rearks Jacobi functions are naed for the faous atheatician C. G. J. Jacobi. In 87 he introduced the elliptic aplitude
On the summations involving Wigner rotation matrix elements
Journal of Matheatical Cheistry 24 (1998 123 132 123 On the suations involving Wigner rotation atrix eleents Shan-Tao Lai a, Pancracio Palting b, Ying-Nan Chiu b and Harris J. Silverstone c a Vitreous
Calculus with the TI-89. Sample Activity: Exploration 6. Brendan Kelly
Calculus with the TI-89 Saple Activity: Exploration 6 Brendan Kelly EXPLORATION 6 The Derivative as a Liit THE BETTMANN ARCHIVE Isaac Newton, Karl Gauss, and Archiedes are rated by historians of atheatics
EE5900 Spring Lecture 4 IC interconnect modeling methods Zhuo Feng
EE59 Spring Parallel LSI AD Algoriths Lecture I interconnect odeling ethods Zhuo Feng. Z. Feng MTU EE59 So far we ve considered only tie doain analyses We ll soon see that it is soeties preferable to odel
Generalized r-modes of the Maclaurin spheroids
PHYSICAL REVIEW D, VOLUME 59, 044009 Generalized r-odes of the Maclaurin spheroids Lee Lindblo Theoretical Astrophysics 130-33, California Institute of Technology, Pasadena, California 9115 Jaes R. Ipser
G G G G G. Spec k G. G Spec k G G. G G m G. G Spec k. Spec k
12 VICTORIA HOSKINS 3. Algebraic group actions and quotients In this section we consider group actions on algebraic varieties and also describe what type of quotients we would like to have for such group
1 Bounding the Margin
COS 511: Theoretical Machine Learning Lecturer: Rob Schapire Lecture #12 Scribe: Jian Min Si March 14, 2013 1 Bounding the Margin We are continuing the proof of a bound on the generalization error of AdaBoost
Singularity Extraction for Reflected Sommerfeld Integrals over Multilayered Media
Telfor Journal, Vol. 6, No., 4. 7 Singularity Extraction for Reflected Soerfeld Integrals over Multilayered Media Vladiir V. Petrovic, Senior Meber, IEEE, Aleksandra J. Krneta, and Branko M. Kolundzija,
Topic 5a Introduction to Curve Fitting & Linear Regression
/7/08 Course Instructor Dr. Rayond C. Rup Oice: A 337 Phone: (95) 747 6958 E ail: rcrup@utep.edu opic 5a Introduction to Curve Fitting & Linear Regression EE 4386/530 Coputational ethods in EE Outline
Physics 2107 Oscillations using Springs Experiment 2
PY07 Oscillations using Springs Experient Physics 07 Oscillations using Springs Experient Prelab Read the following bacground/setup and ensure you are failiar with the concepts and theory required for
SIMPLE HARMONIC MOTION: NEWTON S LAW
SIMPLE HARMONIC MOTION: NEWTON S LAW siple not siple PRIOR READING: Main 1.1, 2.1 Taylor 5.1, 5.2 http://www.yoops.org/twocw/it/nr/rdonlyres/physics/8-012fall-2005/7cce46ac-405d-4652-a724-64f831e70388/0/chp_physi_pndul.jpg
On the Sliding-Window Representation in Digital Signal Processing
868 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, SIGNAL AND PROCESSING, VOL. ASSP-33, Nb. 4, AUGUST 1985 On the Sliding-Window Representation in Digital Signal Processing MARTIN J. BASTIAANS Abstract-The short-tie
The Frobenius problem, sums of powers of integers, and recurrences for the Bernoulli numbers
Journal of Nuber Theory 117 (2006 376 386 www.elsevier.co/locate/jnt The Frobenius proble, sus of powers of integers, and recurrences for the Bernoulli nubers Hans J.H. Tuenter Schulich School of Business,
On a Multisection Style Binomial Summation Identity for Fibonacci Numbers
Int J Contep Math Sciences, Vol 9, 04, no 4, 75-86 HIKARI Ltd, www-hiarico http://dxdoiorg/0988/ics0447 On a Multisection Style Binoial Suation Identity for Fibonacci Nubers Bernhard A Moser Software Copetence
1 Identical Parallel Machines
FB3: Matheatik/Inforatik Dr. Syaantak Das Winter 2017/18 Optiizing under Uncertainty Lecture Notes 3: Scheduling to Miniize Makespan In any standard scheduling proble, we are given a set of jobs J = {j
Chapter 10: Sinusoidal Steady-State Analysis
Chapter 0: Sinusoidal Steady-State Analysis Sinusoidal Sources If a circuit is driven by a sinusoidal source, after 5 tie constants, the circuit reaches a steady-state (reeber the RC lab with t τ). Consequently,
POWER SUM IDENTITIES WITH GENERALIZED STIRLING NUMBERS
POWER SUM IDENTITIES WITH GENERALIZED STIRLING NUMBERS KHRISTO N. BOYADZHIEV Abstract. We prove several cobinatorial identities involving Stirling functions of the second ind with a coplex variable. The
VINOGRADOV S THREE PRIME THEOREM
VINOGRADOV S THREE PRIME THEOREM NICHOLAS ROUSE Abstract I sketch Vinogradov s 937 proof that every sufficiently large odd integer is the su of three prie nubers The result is dependent on nuerous interediate
In this lecture... Axial flow turbine Impulse and reaction turbine stages Work and stage dynamics Turbine blade cascade
Lect- 0 1 Lect-0 In this lecture... Axial flow turbine Ipulse and reaction turbine stages Work and stage dynaics Turbine blade cascade Lect-0 Axial flow turbines Axial turbines like axial copressors usually
The Transactional Nature of Quantum Information
The Transactional Nature of Quantu Inforation Subhash Kak Departent of Coputer Science Oklahoa State University Stillwater, OK 7478 ABSTRACT Inforation, in its counications sense, is a transactional property.
THE POLYNOMIAL REPRESENTATION OF THE TYPE A n 1 RATIONAL CHEREDNIK ALGEBRA IN CHARACTERISTIC p n
THE POLYNOMIAL REPRESENTATION OF THE TYPE A n RATIONAL CHEREDNIK ALGEBRA IN CHARACTERISTIC p n SHEELA DEVADAS AND YI SUN Abstract. We study the polynoial representation of the rational Cherednik algebra
Lecture 21 Principle of Inclusion and Exclusion
Lecture 21 Principle of Inclusion and Exclusion Holden Lee and Yoni Miller 5/6/11 1 Introduction and first exaples We start off with an exaple Exaple 11: At Sunnydale High School there are 28 students
Characterization of the Line Complexity of Cellular Automata Generated by Polynomial Transition Rules. Bertrand Stone
Characterization of the Line Coplexity of Cellular Autoata Generated by Polynoial Transition Rules Bertrand Stone Abstract Cellular autoata are discrete dynaical systes which consist of changing patterns
Jacobi symbols. p 1. Note: The Jacobi symbol does not necessarily distinguish between quadratic residues and nonresidues. That is, we could have ( a
Jacobi sybols efiitio Let be a odd positive iteger If 1, the Jacobi sybol : Z C is the costat fuctio 1 1 If > 1, it has a decopositio ( as ) a product of (ot ecessarily distict) pries p 1 p r The Jacobi
Descent polynomials. Mohamed Omar Department of Mathematics, Harvey Mudd College, 301 Platt Boulevard, Claremont, CA , USA,
Descent polynoials arxiv:1710.11033v2 [ath.co] 13 Nov 2017 Alexander Diaz-Lopez Departent of Matheatics and Statistics, Villanova University, 800 Lancaster Avenue, Villanova, PA 19085, USA, alexander.diaz-lopez@villanova.edu
Symbolic Analysis as Universal Tool for Deriving Properties of Non-linear Algorithms Case study of EM Algorithm
Acta Polytechnica Hungarica Vol., No., 04 Sybolic Analysis as Universal Tool for Deriving Properties of Non-linear Algoriths Case study of EM Algorith Vladiir Mladenović, Miroslav Lutovac, Dana Porrat
arxiv:math/ v1 [math.co] 22 Jul 2005
![arxiv:math/ v1 [math.co] 22 Jul 2005](/thumbs/76/73505817.jpg "arxiv:math/ v1 [math.co] 22 Jul 2005")
Distances between the winning nubers in Lottery Konstantinos Drakakis arxiv:ath/0507469v1 [ath.co] 22 Jul 2005 16 March 2005 Abstract We prove an interesting fact about Lottery: the winning 6 nubers (out
LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH
LORENTZ SPACES AND REAL INTERPOLATION THE KEEL-TAO APPROACH GUILLERMO REY. Introduction If an operator T is bounded on two Lebesgue spaces, the theory of coplex interpolation allows us to deduce the boundedness
A Bernstein-Markov Theorem for Normed Spaces
A Bernstein-Markov Theore for Nored Spaces Lawrence A. Harris Departent of Matheatics, University of Kentucky Lexington, Kentucky 40506-0027 Abstract Let X and Y be real nored linear spaces and let φ :
Lecture 3: October 2, 2017
Inforation and Coding Theory Autun 2017 Lecturer: Madhur Tulsiani Lecture 3: October 2, 2017 1 Shearer s lea and alications In the revious lecture, we saw the following stateent of Shearer s lea. Lea 1.1
Generalized Alignment Chain: Improved Converse Results for Index Coding
Generalized Alignent Chain: Iproved Converse Results for Index Coding Yucheng Liu and Parastoo Sadeghi Research School of Electrical, Energy and Materials Engineering Australian National University, Canberra,
Waves & Normal Modes. Matt Jarvis
Waves & Noral Modes Matt Jarvis January 19, 016 Contents 1 Oscillations 1.0.1 Siple Haronic Motion - revision................... Noral Modes 5.1 The coupled pendulu.............................. 6.1.1
DISCRETE LOGARITHMS IN FREE GROUPS YIANNIS N. PETRIDIS AND MORTEN S. RISAGER. (Communicated by Wen-Ching Winnie Li)
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volue 134, Nuber 4, Pages 1003 1012 S 0002-9939(05)08074-3 Article electronically published on October 5, 2005 DISCRETE LOGARITHMS IN FREE GROUPS YIANNIS
Explicit solution of the polynomial least-squares approximation problem on Chebyshev extrema nodes
Explicit solution of the polynoial least-squares approxiation proble on Chebyshev extrea nodes Alfredo Eisinberg, Giuseppe Fedele Dipartiento di Elettronica Inforatica e Sisteistica, Università degli Studi
On Constant Power Water-filling
On Constant Power Water-filling Wei Yu and John M. Cioffi Electrical Engineering Departent Stanford University, Stanford, CA94305, U.S.A. eails: {weiyu,cioffi}@stanford.edu Abstract This paper derives
USEFUL HINTS FOR SOLVING PHYSICS OLYMPIAD PROBLEMS. By: Ian Blokland, Augustana Campus, University of Alberta
1 USEFUL HINTS FOR SOLVING PHYSICS OLYMPIAD PROBLEMS By: Ian Bloland, Augustana Capus, University of Alberta For: Physics Olypiad Weeend, April 6, 008, UofA Introduction: Physicists often attept to solve
27 Oscillations: Introduction, Mass on a Spring
Chapter 7 Oscillations: Introduction, Mass on a Spring 7 Oscillations: Introduction, Mass on a Spring If a siple haronic oscillation proble does not involve the tie, you should probably be using conservation
Jordan Journal of Physics
Volue 5, Nuber 3, 212. pp. 113-118 ARTILE Jordan Journal of Physics Networks of Identical apacitors with a Substitutional apacitor Departent of Physics, Al-Hussein Bin Talal University, Ma an, 2, 71111,