The Laguerre-Pólya Class
|
|
|
- Leslie Daniel
- 5 years ago
- Views:
Transcription
1 Non-linear operators and the Riemann Hypothesis Department of Mathematics s University of Hawai i at Mānoa lukasz@math.hawaii.edu April 23, 2010
2 The Riemann Hypothesis (1859) The most famous of all unresolved problems. Bernhard Riemann Edmond Laguerre and George Pólya Paul Turán George Csordas, Richard Varga, Timothy Norfolk
3 The Riemann Hypothesis (1859) The most famous of all unresolved problems. ξ(x/2) = 8 0 Bernhard Riemann Edmond Laguerre and George Pólya Paul Turán George Csordas, Richard Varga, Timothy Norfolk Φ(t) cos xt dt,
4 The Riemann Hypothesis (1859) The most famous of all unresolved problems. ξ(x/2) = 8 Φ(t) = n=1 0 Bernhard Riemann Edmond Laguerre and George Pólya Paul Turán George Csordas, Richard Varga, Timothy Norfolk Φ(t) cos xt dt, ( 2n 4 π 2 e 9t 3n 2 πe 5t) e n2 πe 4t.
5 The Riemann Hypothesis (1859) The most famous of all unresolved problems. ξ(x/2) = 8 Φ(t) = n=1 0 Bernhard Riemann Edmond Laguerre and George Pólya Paul Turán George Csordas, Richard Varga, Timothy Norfolk Φ(t) cos xt dt, ( 2n 4 π 2 e 9t 3n 2 πe 5t) e n2 πe 4t....es ist sehr wahrscheinlich, daß alle Wurzeln reell sind.
6 The Riemann Hypothesis (1859) The most famous of all unresolved problems. ξ(x/2) = 8 Φ(t) = n=1 0 Bernhard Riemann Edmond Laguerre and George Pólya Paul Turán George Csordas, Richard Varga, Timothy Norfolk Φ(t) cos xt dt, ( 2n 4 π 2 e 9t 3n 2 πe 5t) e n2 πe 4t....es ist sehr wahrscheinlich, daß alle Wurzeln reell sind. Theorem The Riemann Hypothesis holds if and only if ξ belongs to the Laguerre-Pólya class.
7 (1914) Bernhard Riemann Edmond Laguerre and George Pólya Paul Turán George Csordas, Richard Varga, Timothy Norfolk
8 (1914) Bernhard Riemann Edmond Laguerre and George Pólya Paul Turán George Csordas, Richard Varga, Timothy Norfolk Functions in the Laguerre-Pólya class are uniform limits of polynomials all of whose zeros are real...
9 (1914) Bernhard Riemann Edmond Laguerre and George Pólya Paul Turán George Csordas, Richard Varga, Timothy Norfolk Functions in the Laguerre-Pólya class are uniform limits of polynomials all of whose zeros are real...
10 (1914) Bernhard Riemann Edmond Laguerre and George Pólya Paul Turán George Csordas, Richard Varga, Timothy Norfolk Functions in the Laguerre-Pólya class are uniform limits of polynomials all of whose zeros are real......and only the functions in the Laguerre-Pólya class enjoy this property.
11 The Turán Inequalities (1948) A necessary condition. Bernhard Riemann Edmond Laguerre and George Pólya Paul Turán George Csordas, Richard Varga, Timothy Norfolk
12 The Turán Inequalities (1948) A necessary condition. Bernhard Riemann Edmond Laguerre and George Pólya Paul Turán George Csordas, Richard Varga, Timothy Norfolk Theorem If ϕ(x) = γ k k=0 k! x k is a function in the Laguerre-Pólya class, then γk 2 γ k 1γ k+1 0 for k = 1, 2, 3,....
13 The Riemann ξ Function Bernhard Riemann Edmond Laguerre and George Pólya Paul Turán George Csordas, Richard Varga, Timothy Norfolk ξ(x/2) = 8 Φ(t) = n=1 0 Φ(t) cos xt dt, ( 2n 4 π 2 e 9t 3n 2 πe 5t) e n2 πe 4t. Theorem (Csordas, Varga, Norfolk (1986)) The coefficients of the Riemann ξ function satisfy the Turán inequalities.
14 Non-linear operators. Example Construct a degree 5 polynomial p(x) with zeros x = 1. p(x)
15 Non-linear operators. Example Construct a degree 5 polynomial p(x) with zeros x = 1. p(x) = (x + 1)(x + 1)(x + 1)(x + 1)(x + 1)
16 Non-linear operators. Example Construct a degree 5 polynomial p(x) with zeros x = 1. p(x) = (x + 1)(x + 1)(x + 1)(x + 1)(x + 1) = 1 + 5x + 10x x 3 + 5x 4 + x 5
17 Non-linear operators. Example Construct a degree 5 polynomial p(x) with zeros x = 1. p(x) = (x + 1)(x + 1)(x + 1)(x + 1)(x + 1) = 1 + 5x + 10x x 3 + 5x 4 + x 5 = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 5 + a 5 x 5.
18 Non-linear operators. Example Construct a degree 5 polynomial p(x) with zeros x = 1. p(x) = (x + 1)(x + 1)(x + 1)(x + 1)(x + 1) = 1 + 5x + 10x x 3 + 5x 4 + x 5 = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 5 + a 5 x 5. Replace a k with 3a 2 k 4a k 1a k+1 + a k 2 a k+2.
19 Non-linear operators. Example Construct a degree 5 polynomial p(x) with zeros x = 1. p(x) = (x + 1)(x + 1)(x + 1)(x + 1)(x + 1) = 1 + 5x + 10x x 3 + 5x 4 + x 5 = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 5 + a 5 x 5. Replace a k with 3a 2 k 4a k 1a k+1 + a k 2 a k+2. p(x) becomes x + 105x x x 4 + 3x 5.
20 Non-linear operators. Example Construct a degree 5 polynomial p(x) with zeros x = 1. p(x) = (x + 1)(x + 1)(x + 1)(x + 1)(x + 1) = 1 + 5x + 10x x 3 + 5x 4 + x 5 = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 5 + a 5 x 5. Replace a k with 3a 2 k 4a k 1a k+1 + a k 2 a k+2. p(x) becomes x + 105x x x 4 + 3x 5....and the zeros remain real and negative.
21 Non-linear operators. Example (continued...) p(x) = 1 + 5x + 10x x 3 + 5x 4 + x 5 = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 5 + a 5 x 5. The zeros of p(x) remain real and negative if a k is replaced with: a 2 k a k 1a k+1, 3a 2 k 4a k 1a k+1 + a k 2 a k+2, p(x) becomes x + 105x x x 4 + 3x 5
22 Non-linear operators. Example (continued...) p(x) = 1 + 5x + 10x x 3 + 5x 4 + x 5 = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 5 + a 5 x 5. The zeros of p(x) remain real and negative if a k is replaced with: a 2 k a k 1a k+1, p(x) becomes x + 50x x x 4 + x 5 3a 2 k 4a k 1a k+1 + a k 2 a k+2, p(x) becomes x + 105x x x 4 + 3x 5
23 Non-linear operators. Example (continued...) p(x) = 1 + 5x + 10x x 3 + 5x 4 + x 5 = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 5 + a 5 x 5. The zeros of p(x) remain real and negative if a k is replaced with: a 2 k a k 1a k+1, p(x) becomes x + 50x x x 4 + x 5 3a 2 k 4a k 1a k+1 + a k 2 a k+2, p(x) becomes x + 105x x x 4 + 3x 5 10a 2 k 15a k 1a k+1 + 6a k 2 a k+2 a k 3 a k+3,
24 Non-linear operators. Example (continued...) p(x) = 1 + 5x + 10x x 3 + 5x 4 + x 5 = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 5 + a 5 x 5. The zeros of p(x) remain real and negative if a k is replaced with: a 2 k a k 1a k+1, p(x) becomes x + 50x x x 4 + x 5 3a 2 k 4a k 1a k+1 + a k 2 a k+2, p(x) becomes x + 105x x x 4 + 3x 5 10a 2 k 15a k 1a k+1 + 6a k 2 a k+2 a k 3 a k+3, p(x) becomes x + 280x x x x 5
25 Non-linear operators. Example (continued...) p(x) = 1 + 5x + 10x x 3 + 5x 4 + x 5 = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 5 + a 5 x 5. The zeros of p(x) remain real and negative if a k is replaced with: a 2 k a k 1a k+1, p(x) becomes x + 50x x x 4 + x 5 3a 2 k 4a k 1a k+1 + a k 2 a k+2, p(x) becomes x + 105x x x 4 + 3x 5 10a 2 k 15a k 1a k+1 + 6a k 2 a k+2 a k 3 a k+3, p(x) becomes x + 280x x x x 5 and infinitely many others...
26 Non-linear operators. The Main Result Theorem (Grabarek (2010)) Let ϕ(x) = ω k=0 a kx k, 0 ω, be a function in the Laguerre-Pólya class. If the zeros of ϕ(x) are real and negative, then the zeros remain real and negative after replacing a k with ( ) 2p 1 p ( ) 2p ak 2 p + ( 1) j a k j a k+j (p = 1, 2, 3,...). p j j=1
EXTENDED LAGUERRE INEQUALITIES AND A CRITERION FOR REAL ZEROS
EXTENDED LAGUERRE INEQUALITIES AND A CRITERION FOR REAL ZEROS DAVID A. CARDON Abstract. Let fz) = e bz2 f z) where b 0 and f z) is a real entire function of genus 0 or. We give a necessary and sufficient
WRONSKIANS AND THE RIEMANN HYPOTHESIS
WRONSKIANS AND THE RIEMANN HYPOTHESIS JOHN NUTTALL Abstract. We have previously proposed that the sign-regularity properties of a function Φ(t) related to the transform of the ζ-function could be the key
TURÁN INEQUALITIES AND SUBTRACTION-FREE EXPRESSIONS
TURÁN INEQUALITIES AND SUBTRACTION-FREE EXPRESSIONS DAVID A. CARDON AND ADAM RICH Abstract. By using subtraction-free expressions, we are able to provide a new proof of the Turán inequalities for the Taylor
Conjectures and Theorems in the Theory of Entire Functions
Numerical Algorithms (2) 1 14 1 Conjectures and Theorems in the Theory of Entire Functions George Csordas Department of Mathematics, University of Hawaii, Honolulu, HI 96822 E-mail: george@math.hawaii.edu
The Riemann Hypothesis
University of Hawai i at Mānoa January 26, 2016 The distribution of primes Euclid In ancient Greek times, Euclid s Elements already answered the question: Q: How many primes are there? Euclid In ancient
DETERMINANTAL APPROACH TO A PROOF OF THE RIEMANN HYPOTHESIS
DETERMINANTAL APPROACH TO A PROOF OF THE RIEMANN HYPOTHESIS JOHN NUTTALL Abstract. We discuss the application of the determinantal method to the proof of the Riemann Hypothesis. We start from the fact
THE LAGUERRE INEQUALITY AND THE DISTRIBUTION OF ZEROS OF ENTIRE FUNCTIONS. 0. Introduction
THE LAGUERRE INEQUALITY AND THE DISTRIBUTION OF ZEROS OF ENTIRE FUNCTIONS BRANDON MURANAKA Abstract. The purpose of this paper is fourfold: () to survey some classical and recent results in the theory
mat.haus project Zurab Janelidze s Lectures on UNIVERSE OF SETS last updated 18 May 2017 Stellenbosch University 2017
Zurab Janelidze s Lectures on UNIVERSE OF SETS last updated 18 May 2017 Stellenbosch University 2017 Contents 1. Axiom-free universe of sets 1 2. Equality of sets and the empty set 2 3. Comprehension and
On the Nicolas inequality involving primorial numbers
On the Nicolas inequality involving primorial numbers Tatenda Isaac Kubalalika March 4, 017 ABSTRACT. In 1983, J.L. Nicholas demonstrated that the Riemann Hypothesis (RH) is equivalent to the statement
ITERATED LAGUERRE AND TURÁN INEQUALITIES. Thomas Craven and George Csordas
ITERATED LAGUERRE AND TURÁN INEQUALITIES Thomas Craven and George Csordas Abstract. New inequalities are investigated for real entire functions in the Laguerre-Pólya class. These are generalizations of
Proof of Lagarias s Elementary Version of the Riemann Hypothesis.
Proof of Lagarias s Elementary Version of the Riemann Hypothesis. Stephen Marshall 27 Nov 2018 Abstract The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative
Journal of Mathematical Analysis and Applications
J. Math. Anal. Appl. 35 29) 276 282 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa A Turán-type inequality for the gamma function
MULTIPLIER SEQUENCES AND LOGARITHMIC MESH
MULTIPLIER SEQUENCES AND LOGARITHMIC MESH OLGA KATKOVA, BORIS SHAPIRO, AND ANNA VISHNYAKOVA Abstract. In this note we prove a new result about (finite) multiplier sequences, i.e. linear operators acting
Dedekind zeta function and BDS conjecture
arxiv:1003.4813v3 [math.gm] 16 Jan 2017 Dedekind zeta function and BDS conjecture Abstract. Keywords: Dang Vu Giang Hanoi Institute of Mathematics Vietnam Academy of Science and Technology 18 Hoang Quoc
CUMULANTS, THE RIEMANN HYPOTHESIS, AND SIMILAR PROBLEMS
CUMULANTS, THE RIEMANN HYPOTHESIS, AND SIMILAR PROBLEMS JOHN NUTTALL Abstract. In 1994 [ Conrey ] and Ghosh studied a function that may be written as Θ 1 (u) = exp u 4 πeu 12 m=1 (1 e 2πmeu ). It is thought
PRINTABLE VERSION. Practice Final. Question 1. Find the coordinates of the y-intercept for 5x 9y + 6 = 0. 2 (0, ) 3 3 (0, ) 2 2 (0, ) 3 6 (0, ) 5
PRINTABLE VERSION Practice Final Question Find the coordinates of the y-intercept for 5x 9y + 6 = 0. (0, ) (0, ) (0, ) 6 (0, ) 5 6 (0, ) 5 Question Find the slope of the line: 7x 4y = 0 7 4 4 4 7 7 4 4
(x k ) sequence in F, lim x k = x x F. If F : R n R is a function, level sets and sublevel sets of F are any sets of the form (respectively);
STABILITY OF EQUILIBRIA AND LIAPUNOV FUNCTIONS. By topological properties in general we mean qualitative geometric properties (of subsets of R n or of functions in R n ), that is, those that don t depend
FOURIER TRANSFORMS HAVING ONLY REAL ZEROS
PROCEEDINGS OF THE MERICN MTHEMTICL SOCIETY Volume, Number, Pages S 2-9939(XX- FOURIER TRNSFORMS HVING ONLY REL ZEROS DVID CRDON (Communicated by Joseph Ball bstract Let G(z be a real entire function of
Quadratic and Polynomial Inequalities in one variable have look like the example below.
Section 8 4: Polynomial Inequalities in One Variable Quadratic and Polynomial Inequalities in one variable have look like the example below. x 2 5x 6 0 (x 2) (x + 4) > 0 x 2 (x 3) > 0 (x 2) 2 (x + 4) 0
Question 1. Find the coordinates of the y-intercept for. f) None of the above. Question 2. Find the slope of the line:
of 4 4/4/017 8:44 AM Question 1 Find the coordinates of the y-intercept for. Question Find the slope of the line: of 4 4/4/017 8:44 AM Question 3 Solve the following equation for x : Question 4 Paul has
THE ZEROS OF FOURIER TRANSFORMS. Haseo Ki. s(s 1) 2
THE ZEROS OF FOURIER TRANSFORMS Haseo Ki 1. Introduction We denote the Riemann Ξ-function Ξ(z) by Ξ(z) = s(s 1) π s/2 Γ(s/2)ζ(s), 2 where ζ(s) is the Riemann zeta function and s = 1/2 + iz. Then Ξ(z) is
KARLIN S CONJECTURE AND A QUESTION OF PÓLYA
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 35, Number 1, 2005 KARLIN S CONJECTURE AND A QUESTION OF PÓLYA THOMAS CRAVEN AND GEORGE CSORDAS ABSTRACT The paper answers an old question of Pólya involving
Best constants in Markov-type inequalities with mixed Hermite weights
Best constants in Markov-type inequalities with mixed Hermite weights Holger Langenau Technische Universität Chemnitz IWOTA August 15, 2017 Introduction We consider the inequalities of the form D ν f C
UNIFORM BOUNDS FOR BESSEL FUNCTIONS
Journal of Applied Analysis Vol. 1, No. 1 (006), pp. 83 91 UNIFORM BOUNDS FOR BESSEL FUNCTIONS I. KRASIKOV Received October 8, 001 and, in revised form, July 6, 004 Abstract. For ν > 1/ and x real we shall
Chapter 11. Taylor Series. Josef Leydold Mathematical Methods WS 2018/19 11 Taylor Series 1 / 27
Chapter 11 Taylor Series Josef Leydold Mathematical Methods WS 2018/19 11 Taylor Series 1 / 27 First-Order Approximation We want to approximate function f by some simple function. Best possible approximation
Boundary Conditions associated with the Left-Definite Theory for Differential Operators
Boundary Conditions associated with the Left-Definite Theory for Differential Operators Baylor University IWOTA August 15th, 2017 Overview of Left-Definite Theory A general framework for Left-Definite
On an uniqueness theorem for characteristic functions
ISSN 392-53 Nonlinear Analysis: Modelling and Control, 207, Vol. 22, No. 3, 42 420 https://doi.org/0.5388/na.207.3.9 On an uniqueness theorem for characteristic functions Saulius Norvidas Institute of
PROOF OF RIEMANN S HYPOTHESIS. MSC 2010 : 11A41; 11M06; 11M26 Keywords: Euler; Riemann; Hilbert; Polya; conjecture
PROOF OF RIEMANN S HYPOTHESIS ROBERT DELOIN Abstract. Riemann s hypothesis (859) is the conjecture stating that: The real part of every non trivial zero of Riemann s zeta function is /. The main contribution
Analysis Part 1. 1 Chapter Q1(q) 1.2 Q1(r) Book: Measure and Integral by Wheeden and Zygmund
Analysis Part 1 www.mathtuition88.com Book: Measure and Integral by Wheeden and Zygmund 1 Chapter 1 1.1 Q1(q) Let {T x k } be a sequence of points of T E. Since E is compact, {x k } has a subsequence {x
Section Taylor and Maclaurin Series
Section.0 Taylor and Maclaurin Series Ruipeng Shen Feb 5 Taylor and Maclaurin Series Main Goal: How to find a power series representation for a smooth function us assume that a smooth function has a power
FOURIER TRANSFORMS. 1. Fourier series 1.1. The trigonometric system. The sequence of functions
FOURIER TRANSFORMS. Fourier series.. The trigonometric system. The sequence of functions, cos x, sin x,..., cos nx, sin nx,... is called the trigonometric system. These functions have period π. The trigonometric
Scattered Data Interpolation with Polynomial Precision and Conditionally Positive Definite Functions
Chapter 3 Scattered Data Interpolation with Polynomial Precision and Conditionally Positive Definite Functions 3.1 Scattered Data Interpolation with Polynomial Precision Sometimes the assumption on the
Continuous Functions on Metric Spaces
Continuous Functions on Metric Spaces Math 201A, Fall 2016 1 Continuous functions Definition 1. Let (X, d X ) and (Y, d Y ) be metric spaces. A function f : X Y is continuous at a X if for every ɛ > 0
1. Find the solution of the following uncontrolled linear system. 2 α 1 1
Appendix B Revision Problems 1. Find the solution of the following uncontrolled linear system 0 1 1 ẋ = x, x(0) =. 2 3 1 Class test, August 1998 2. Given the linear system described by 2 α 1 1 ẋ = x +
Sub-Riemannian geometry in groups of diffeomorphisms and shape spaces
Sub-Riemannian geometry in groups of diffeomorphisms and shape spaces Sylvain Arguillère, Emmanuel Trélat (Paris 6), Alain Trouvé (ENS Cachan), Laurent May 2013 Plan 1 Sub-Riemannian geometry 2 Right-invariant
Tonelli Full-Regularity in the Calculus of Variations and Optimal Control
Tonelli Full-Regularity in the Calculus of Variations and Optimal Control Delfim F. M. Torres delfim@mat.ua.pt Department of Mathematics University of Aveiro 3810 193 Aveiro, Portugal http://www.mat.ua.pt/delfim
Objective Mathematics
Multiple choice questions with ONE correct answer : ( Questions No. 1-5 ) 1. If the equation x n = (x + ) is having exactly three distinct real solutions, then exhaustive set of values of 'n' is given
LINEAR PRESERVERS AND ENTIRE FUNCTIONS WITH RESTRICTED ZERO LOCI
LINEAR PRESERVERS AND ENTIRE FUNCTIONS WITH RESTRICTED ZERO LOCI A DISSERTATION SUBMITTED TO THE GRADUATE DIVISION OF THE UNIVERSITY OF HAWAI I IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE
Alexei F. Cheviakov. University of Saskatchewan, Saskatoon, Canada. INPL seminar June 09, 2011
Direct Method of Construction of Conservation Laws for Nonlinear Differential Equations, its Relation with Noether s Theorem, Applications, and Symbolic Software Alexei F. Cheviakov University of Saskatchewan,
ζ(u) z du du Since γ does not pass through z, f is defined and continuous on [a, b]. Furthermore, for all t such that dζ
![ζ(u) z du du Since γ does not pass through z, f is defined and continuous on [a, b]. Furthermore, for all t such that dζ](/thumbs/92/109116539.jpg "ζ(u) z du du Since γ does not pass through z, f is defined and continuous on [a, b]. Furthermore, for all t such that dζ")
Lecture 6 Consequences of Cauchy s Theorem MATH-GA 45.00 Complex Variables Cauchy s Integral Formula. Index of a point with respect to a closed curve Let z C, and a piecewise differentiable closed curve
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS. Special Functions GEORGE E. ANDREWS RICHARD ASKEY RANJAN ROY CAMBRIDGE UNIVERSITY PRESS
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS Special Functions GEORGE E. ANDREWS RICHARD ASKEY RANJAN ROY CAMBRIDGE UNIVERSITY PRESS Preface page xiii 1 The Gamma and Beta Functions 1 1.1 The Gamma
Newman s Conjecture for function field L-functions
Newman s Conjecture for function field L-functions David Mehrle Presenters: t < Λ t Λ Λ Joseph Stahl September 28, 2014 Joint work with: Tomer Reiter, Dylan Yott Advisors: Steve Miller, Alan Chang The
Topological Vector Spaces III: Finite Dimensional Spaces
TVS III c Gabriel Nagy Topological Vector Spaces III: Finite Dimensional Spaces Notes from the Functional Analysis Course (Fall 07 - Spring 08) Convention. Throughout this note K will be one of the fields
4 Expectation & the Lebesgue Theorems
STA 205: Probability & Measure Theory Robert L. Wolpert 4 Expectation & the Lebesgue Theorems Let X and {X n : n N} be random variables on a probability space (Ω,F,P). If X n (ω) X(ω) for each ω Ω, does
Lecture 4: Numerical solution of ordinary differential equations
Lecture 4: Numerical solution of ordinary differential equations Department of Mathematics, ETH Zürich General explicit one-step method: Consistency; Stability; Convergence. High-order methods: Taylor
Kernel Method: Data Analysis with Positive Definite Kernels
Kernel Method: Data Analysis with Positive Definite Kernels 2. Positive Definite Kernel and Reproducing Kernel Hilbert Space Kenji Fukumizu The Institute of Statistical Mathematics. Graduate University
DI. N i AUlRS) REPORT DOCUMENTATION PAGE. 7 i. a "
a " F7 REPORT DOCUMENTATION PAGE ftdfft flw maj of @4bvMse to awwq I W 'emw. me mo W.-OW" P -M raq eu Iowa 7 i t of 04 Ma,"lWrt sag qlt em&om Se"Wel "wnl (?7. I. I Wmm. 0C 0ISL 1. A4E- UM ONLY (Lw' bwsek)
Introduction to Signal Spaces
Introduction to Signal Spaces Selin Aviyente Department of Electrical and Computer Engineering Michigan State University January 12, 2010 Motivation Outline 1 Motivation 2 Vector Space 3 Inner Product
On the Bilateral Laplace Transform of the positive even functions and proof of the Riemann Hypothesis. Seong Won Cha Ph.D.
On the Bilateral Laplace Transform of the positive even functions and proof of the Riemann Hypothesis Seong Won Cha Ph.D. Seongwon.cha@gmail.com Remark This is not an official paper, rather a brief report.
Linear Differential Equations. Problems
Chapter 1 Linear Differential Equations. Problems 1.1 Introduction 1.1.1 Show that the function ϕ : R R, given by the expression ϕ(t) = 2e 3t for all t R, is a solution of the Initial Value Problem x =
Differential equations
Differential equations Math 7 Spring Practice problems for April Exam Problem Use the method of elimination to find the x-component of the general solution of x y = 6x 9x + y = x 6y 9y Soln: The system
Lecture Notes of EE 714
Lecture Notes of EE 714 Lecture 1 Motivation Systems theory that we have studied so far deals with the notion of specified input and output spaces. But there are systems which do not have a clear demarcation
arxiv: v22 [math.gm] 20 Sep 2018
![arxiv: v22 [math.gm] 20 Sep 2018](/thumbs/86/94725284.jpg "arxiv: v22 [math.gm] 20 Sep 2018")
O RIEMA HYPOTHESIS arxiv:96.464v [math.gm] Sep 8 RUIMIG ZHAG Abstract. In this work we present a proof to the celebrated Riemann hypothesis. In it we first apply the Plancherel theorem properties of the
Error Bounds in Power Series
Error Bounds in Power Series When using a power series to approximate a function (usually a Taylor Series), we often want to know how large any potential error is on a specified interval. There are two
TEST CODE: MIII (Objective type) 2010 SYLLABUS
TEST CODE: MIII (Objective type) 200 SYLLABUS Algebra Permutations and combinations. Binomial theorem. Theory of equations. Inequalities. Complex numbers and De Moivre s theorem. Elementary set theory.
MATH 215/255 Solutions to Additional Practice Problems April dy dt
. For the nonlinear system MATH 5/55 Solutions to Additional Practice Problems April 08 dx dt = x( x y, dy dt = y(.5 y x, x 0, y 0, (a Show that if x(0 > 0 and y(0 = 0, then the solution (x(t, y(t of the
Inner Product Spaces An inner product on a complex linear space X is a function x y from X X C such that. (1) (2) (3) x x > 0 for x 0.
Inner Product Spaces An inner product on a complex linear space X is a function x y from X X C such that (1) () () (4) x 1 + x y = x 1 y + x y y x = x y x αy = α x y x x > 0 for x 0 Consequently, (5) (6)
Exam 3. MA 114 Exam 3 Fall Multiple Choice Questions. 1. Find the average value of the function f (x) = 2 sin x sin 2x on 0 x π. C. 0 D. 4 E.
Exam 3 Multiple Choice Questions 1. Find the average value of the function f (x) = sin x sin x on x π. A. π 5 π C. E. 5. Find the volume of the solid S whose base is the disk bounded by the circle x +
2 Chance constrained programming
2 Chance constrained programming In this Chapter we give a brief introduction to chance constrained programming. The goals are to motivate the subject and to give the reader an idea of the related difficulties.
Nonlinear Control. Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems
Nonlinear Control Lecture # 8 Time Varying and Perturbed Systems Time-varying Systems ẋ = f(t,x) f(t,x) is piecewise continuous in t and locally Lipschitz in x for all t 0 and all x D, (0 D). The origin
Elements of Positive Definite Kernel and Reproducing Kernel Hilbert Space
Elements of Positive Definite Kernel and Reproducing Kernel Hilbert Space Statistical Inference with Reproducing Kernel Hilbert Space Kenji Fukumizu Institute of Statistical Mathematics, ROIS Department
COMPUTER ALGEBRA AND POWER SERIES WITH POSITIVE COEFFICIENTS. 1. Introduction
COMPUTER ALGEBRA AND POWER SERIES WITH POSITIVE COEFFICIENTS MANUEL KAUERS Abstract. We consider the question whether all the coefficients in the series expansions of some specific rational functions are
The one-dimensional equations for the fluid dynamics of a gas can be written in conservation form as follows:
Topic 7 Fluid Dynamics Lecture The Riemann Problem and Shock Tube Problem A simple one dimensional model of a gas was introduced by G.A. Sod, J. Computational Physics 7, 1 (1978), to test various algorithms
A lower estimate for the Lebesgue constants of linear means of Laguerre expansions
A lower estimate for the Lebesgue constants of linear means of Laguerre expansions George Gasper 1 and Walter Trebels 2 Dedicated to P. L. Butzer on the occasion of his 70th birthday in gratitude (Oct.
Study of some equivalence classes of primes
Notes on Number Theory and Discrete Mathematics Print ISSN 3-532, Online ISSN 2367-8275 Vol 23, 27, No 2, 2 29 Study of some equivalence classes of primes Sadani Idir Department of Mathematics University
ERRATA AND SUGGESTION SHEETS Advanced Calculus, Second Edition
ERRATA AND SUGGESTION SHEETS Advanced Calculus, Second Edition Prof. Patrick Fitzpatrick February 6, 2013 Page 11, line 38: b 2 < r should be b 2 < c Page 13, line 1: for... number a and b, write... numbers
NOTES ON SCHAUDER ESTIMATES. r 2 x y 2
NOTES ON SCHAUDER ESTIMATES CRISTIAN E GUTIÉRREZ JULY 26, 2005 Lemma 1 If u f in B r y), then ux) u + r2 x y 2 B r y) B r y) f, x B r y) Proof Let gx) = ux) Br y) u r2 x y 2 Br y) f We have g = u + Br
On The Weights of Binary Irreducible Cyclic Codes
On The Weights of Binary Irreducible Cyclic Codes Yves Aubry and Philippe Langevin Université du Sud Toulon-Var, Laboratoire GRIM F-83270 La Garde, France, {langevin,yaubry}@univ-tln.fr, WWW home page:
LINEAR DISPERSIVE WAVES
LINEAR DISPERSIVE WAVES Nathaniel Egwu December 16, 2009 Outline 1 INTRODUCTION Dispersion Relations Definition of Dispersive Waves 2 SOLUTION AND ASYMPTOTIC ANALYSIS The Beam Equation General Solution
MASTERS EXAMINATION IN MATHEMATICS SOLUTIONS
MASTERS EXAMINATION IN MATHEMATICS PURE MATHEMATICS OPTION SPRING 010 SOLUTIONS Algebra A1. Let F be a finite field. Prove that F [x] contains infinitely many prime ideals. Solution: The ring F [x] of
1 Strict local optimality in unconstrained optimization
ORF 53 Lecture 14 Spring 016, Princeton University Instructor: A.A. Ahmadi Scribe: G. Hall Thursday, April 14, 016 When in doubt on the accuracy of these notes, please cross check with the instructor s
A Symbolic Operator Approach to Power Series Transformation-Expansion Formulas
A Symbolic Operator Approach to Power Series Transformation-Expansion Formulas Tian- Xiao He Department of Mathematics and Computer Science Illinois Wesleyan University Bloomington, IL 61702-2900, USA
Surprising Sinc Sums and Integrals
David Borwein University of Western Ontario Peter Borwein Conference May 12-16, 28 1 Motivation and preliminaries. This talk is based on material in a paper to appear shortly in MAA MONTHLY with the above
Quantization of Scalar Field
Quantization of Scalar Field Wei Wang 2017.10.12 Wei Wang(SJTU) Lectures on QFT 2017.10.12 1 / 41 Contents 1 From classical theory to quantum theory 2 Quantization of real scalar field 3 Quantization of
Analysis Comprehensive Exam Questions Fall 2008
Analysis Comprehensive xam Questions Fall 28. (a) Let R be measurable with finite Lebesgue measure. Suppose that {f n } n N is a bounded sequence in L 2 () and there exists a function f such that f n (x)
Solutions for Problem Set #4 due October 10, 2003 Dustin Cartwright
Solutions for Problem Set #4 due October 1, 3 Dustin Cartwright (B&N 4.3) Evaluate C f where f(z) 1/z as in Example, and C is given by z(t) sin t + i cos t, t π. Why is the result different from that of
ON PERIODIC SOLUTIONS TO SOME LAGRANGIAN SYSTEM WITH TWO DEGREES OF FREEDOM
ON PERIODIC SOLUTIONS TO SOME LAGRANGIAN SYSTEM WITH TWO DEGREES OF FREEDOM OLEG ZUBELEVICH DEPT. OF THEORETICAL MECHANICS, MECHANICS AND MATHEMATICS FACULTY, M. V. LOMONOSOV MOSCOW STATE UNIVERSITY RUSSIA,
Generalized divisor problem
Generalized divisor problem C. Calderón * M.J. Zárate The authors study the generalized divisor problem by means of the residue theorem of Cauchy and the properties of the Riemann Zeta function. By an
NEW MULTIPLIER SEQUENCES VIA DISCRIMINANT AMOEBAE
NEW MULTIPLIER SEQUENCES VIA DISCRIMINANT AMOEBAE MIKAEL PASSARE, J. MAURICE ROJAS, AND BORIS SHAPIRO To Vladimir Igorevich Arnold who left us too early. Abstract. In their classic 1914 paper, Polýa and
Verona Course April Lecture 1. Review of probability
Verona Course April 215. Lecture 1. Review of probability Viorel Barbu Al.I. Cuza University of Iaşi and the Romanian Academy A probability space is a triple (Ω, F, P) where Ω is an abstract set, F is
MATH 31BH Homework 1 Solutions
MATH 3BH Homework Solutions January 0, 04 Problem.5. (a) (x, y)-plane in R 3 is closed and not open. To see that this plane is not open, notice that any ball around the origin (0, 0, 0) will contain points
Sequences and Series of Functions
Chapter 13 Sequences and Series of Functions These notes are based on the notes A Teacher s Guide to Calculus by Dr. Louis Talman. The treatment of power series that we find in most of today s elementary
Math Ordinary Differential Equations
Math 411 - Ordinary Differential Equations Review Notes - 1 1 - Basic Theory A first order ordinary differential equation has the form x = f(t, x) (11) Here x = dx/dt Given an initial data x(t 0 ) = x
Mahler measure and special values of L-functions
Mahler measure and special values of L-functions Matilde N. Laĺın University of Alberta mlalin@math.ualberta.ca http://www.math.ualberta.ca/~mlalin October 24, 2008 Matilde N. Laĺın (U of A) Mahler measure
Kernel methods, kernel SVM and ridge regression
Kernel methods, kernel SVM and ridge regression Le Song Machine Learning II: Advanced Topics CSE 8803ML, Spring 2012 Collaborative Filtering 2 Collaborative Filtering R: rating matrix; U: user factor;
Lecture Notes 7 Stationary Random Processes. Strict-Sense and Wide-Sense Stationarity. Autocorrelation Function of a Stationary Process
Lecture Notes 7 Stationary Random Processes Strict-Sense and Wide-Sense Stationarity Autocorrelation Function of a Stationary Process Power Spectral Density Continuity and Integration of Random Processes
Algebraic number theory Solutions to exercise sheet for chapter 4
Algebraic number theory Solutions to exercise sheet for chapter 4 Nicolas Mascot n.a.v.mascot@warwick.ac.uk), Aurel Page a.r.page@warwick.ac.uk) TAs: Chris Birkbeck c.d.birkbeck@warwick.ac.uk), George
Rational numbers with purely periodic beta-expansion. Boris Adamczeswki, C. Frougny, A. Siegel, W.Steiner
Rational numbers with purely periodic beta-expansion Boris Adamczeswki, C. Frougny, A. Siegel, W.Steiner Fractals, Tilings, and Things? Fractals, Tilings, and Things? Number theory : expansions in several
TEST CODE: MMA (Objective type) 2015 SYLLABUS
TEST CODE: MMA (Objective type) 2015 SYLLABUS Analytical Reasoning Algebra Arithmetic, geometric and harmonic progression. Continued fractions. Elementary combinatorics: Permutations and combinations,
Introduction to Nonlinear Control Lecture # 3 Time-Varying and Perturbed Systems
p. 1/5 Introduction to Nonlinear Control Lecture # 3 Time-Varying and Perturbed Systems p. 2/5 Time-varying Systems ẋ = f(t, x) f(t, x) is piecewise continuous in t and locally Lipschitz in x for all t
SUMS OF ENTIRE FUNCTIONS HAVING ONLY REAL ZEROS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 SUMS OF ENTIRE FUNCTIONS HAVING ONLY REAL ZEROS STEVEN R. ADAMS AND DAVID A. CARDON (Communicated
Honors Differential Equations
MIT OpenCourseWare http://ocw.mit.edu 8.034 Honors Differential Equations Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. LECTURE 20. TRANSFORM
1. Stochastic Processes and filtrations
1. Stochastic Processes and 1. Stoch. pr., A stochastic process (X t ) t T is a collection of random variables on (Ω, F) with values in a measurable space (S, S), i.e., for all t, In our case X t : Ω S
Conservation Laws: Systematic Construction, Noether s Theorem, Applications, and Symbolic Computations.
Conservation Laws: Systematic Construction, Noether s Theorem, Applications, and Symbolic Computations. Alexey Shevyakov Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon,
en-. כ Some recent results on the distribution of zeros of real, 1822 Fourier. 1937, Ålander, Pólya, Wiman
Commun. Korean Math. Soc. 17 (2002), No. 1, pp. 1 14, en-. כ Some recent results on the distribution of zeros of real tire functions are surveyed. 1 1930 Fourier Pólya, 1822 Fourier כ [19]., Fourier de
Introduction to Fourier Transforms. Lecture 7 ELE 301: Signals and Systems. Fourier Series. Rect Example
Introduction to Fourier ransforms Lecture 7 ELE 3: Signals and Systems Fourier transform as a limit of the Fourier series Inverse Fourier transform: he Fourier integral theorem Prof. Paul Cuff Princeton
Analysis III. Exam 1
Analysis III Math 414 Spring 27 Professor Ben Richert Exam 1 Solutions Problem 1 Let X be the set of all continuous real valued functions on [, 1], and let ρ : X X R be the function ρ(f, g) = sup f g (1)
Introduction to Orthogonal Polynomials: Definition and basic properties
Introduction to Orthogonal Polynomials: Definition and basic properties Prof. Dr. Mama Foupouagnigni African Institute for Mathematical Sciences, Limbe, Cameroon and Department of Mathematics, Higher Teachers
Tests of Profile Stiffness Using Modulated Electron Cyclotron Heating
Tests of Profile Stiffness Using Modulated Electron Cyclotron Heating by T.C. Luce in collaboration with J.C. DeBoo, C.C. Petty, J. Pino, J.M. Nelson, and J.C.M. dehaas Presented at 9th EU-US Transport
Binet-Cauchy Kernerls on Dynamical Systems and its Application to the Analysis of Dynamic Scenes
on Dynamical Systems and its Application to the Analysis of Dynamic Scenes Dynamical Smola Alexander J. Vidal René presented by Tron Roberto July 31, 2006 Support Vector Machines (SVMs) Classification