Papers by Alexander Roytvarf
Thinking in Problems, 2012
The following matrix appears in various problems connected with equidistants and envelopes in ana... more The following matrix appears in various problems connected with equidistants and envelopes in analysis, geometry, calculation of variations, and mathematical physics: $$ A=\left( {\begin{array}{*{20}{c}} {1+{a_1}^2} & {{a_1}{a_2}} & {\ldots } & {{a_1}{a_n}} \\ {{a_2}{a_1}} & {1+{a_2}^2} & {\ldots } & {{a_2}{a_n}} \\ . & . & {\ldots } & . \\ {{a_n}{a_1}} & . & {\ldots } & {1+{a_n}^2} \\ \end{array}} \right) . $$
Thinking in Problems, 2012
Preliminaries. A one-parameter linear group in a vector space L is a continuous map g : t7 !g t ¼... more Preliminaries. A one-parameter linear group in a vector space L is a continuous map g : t7 !g t ¼ gðtÞ on the space of parameters ℝ, taking values in a space of the linear operators on L and having the following properties: (1) g 0 ¼ E and (2) the sum of parameter values corresponds to the composition g sþt ¼ g s g t. Verify that property (1) may be substituted by (1) 0-all g t are invertible operators. Thus, a one-parameter linear group is a group homomorphism g: hℝ,þi ! hGL(L), ∘i, which is continuous as well: t ! t 0) g t ! g t 0. Readers can define this continuity themselves using the usual terminology with which they are familiar. 1 Let us consider the details of the simplest case, where L ¼ ℝ. As readers know, linear operators on a one-dimensional space are multiplications by scalars. (When are these operators invertible?) In particular, we have a natural isomorphism between GL(1,ℝ) and a multiplicative group h ℝ * , i. Thus, one-parameter linear groups in ℝ are the continuous homomorphisms g: hℝ,þi ! hℝ * ,i, or, in other words, the continuous solutions of the functional equations
Thinking in Problems, 2012
For a finite (or countably infinite) sequence an, the generating function (or series) is defined ... more For a finite (or countably infinite) sequence an, the generating function (or series) is defined as the polynomial (resp. formal series) ∑antn . For a multisequence am,n,…, the generating function is a polynomial or series in several variables ∑a m,n,… t m u n Various problems of combinatorial analysis and probability theory are successfully explored with such powerful tools as generating functions. In this chapter readers will encounter several problems related to the combinatorics of binomial coefficients, theory of partitions, and renewal processes in probability theory that can be explored using generating functions. The problems of the first two groups does not require the ability to deal with power series (likewise, the problems of the second group assume no familiarity with the theory of partitions) and can be solved by readers with limited experience.
Thinking in Problems, 2012
As readers know, a polynomial equation of degree n has at most n roots (considering multiplicity)... more As readers know, a polynomial equation of degree n has at most n roots (considering multiplicity) in a number field containing its coefficients. How many roots does a polynomial equation have in a matrix ring? First, how many roots of degree n of 1=E, that is, matrices X, \(X^n = {X \circ \ldots \circ X}_n = E\) are there? And how would one enumerate them? These and related questions that can be answered given a relatively modest level of knowledge on the reader’s part are discussed in this chapter. That is, we will characterize the roots of E in a ring of 2×2 matrices with real entries and show some applications to other mathematical topics: matrix norm estimation and spectral analysis of 3-diagonal Jacobi matrices.
Thinking in Problems, 2012
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.
This short chapter is devoted to a beautiful elementary problem about numerical linear recursive ... more This short chapter is devoted to a beautiful elementary problem about numerical linear recursive sequences that was proposed by the Soviet scientist A.D. Sakharov. We open the chapter with Sakharov’s original formulation of the famous Fibonacci sequence and then ask readers to find its generalizations. These problems assume no familiarity with combinatorial analysis and can be solved by readers with limited experience.
Birkhäuser Boston eBooks, 2013
Concepts of convexity and related techniques are widely used in various branches of mathematics a... more Concepts of convexity and related techniques are widely used in various branches of mathematics and in applications such as functional analysis, calculus of variations, and mathematical physics, geometry, number theory, theory of games, optimization and other computational methods and convex programming, integral geometry (tomography), and many others. In this chapter we will focus mainly on the basic elements of the theory of convex functions, which can be taught to readers with limited experience. The remaining problems are directed toward specific applications such as convex and linear programming, hierarchy of power means, geometric inequalities, the Holder and Minkowski inequalities, Young’s inequality and the Legendre transform, and functions on metric and normed vector spaces. The necessary introductory material is provided in these sections. A specially added section discusses the widely known Cauchy-Schwarz-Bunyakovskii (CSB) inequality and Lagrange-type identities in connection with the hierarchy of the power means.
As readers know, polynomials of degree n, in other words linear combinations of n + 1 monomials 1... more As readers know, polynomials of degree n, in other words linear combinations of n + 1 monomials 1,…, t n , may have at most n real zeros. A far-reaching generalization of this fact raises a fundamental concept of Chebyshev systems, briefly, T-systems. Those systems are defined as follows. For a set (or system) of functions F = {f 0,…}, a linear combination of a finite number of elements, f = ∑c i f i , is called a polynomial on F (considered nonzero when ∃i: c i ≠ 0). A system of n + 1 function F = {f 0,…,f n } on an interval (or a half-interval, or a non-one-point segment) I ⊆ ℝ is referred to as T-system, when nonzero polynomials on F may have at most n distinct zeros in I (zeros of extensions outside I are not counted). The basic ideas of the theory of T-systems may be understood by readers with relatively limited experience. In this chapter we focus both on these ideas and on some nice applications of this theory such as estimation of numbers of zeros and critical points of functions (in analysis and geometry), a real-life problem in tomography, interpolation theory, approximation theory, and least squares.
Birkhäuser Boston eBooks, 2013
There exists a very powerful set of techniques for dealing with sets of equations or matrices tha... more There exists a very powerful set of techniques for dealing with sets of equations or matrices that are either singular or numerically very close to singular. In many cases where Gaussian elimination and triangle decomposition fail to give satisfactory results, this set of techniques, known as singular value decomposition (SVD) will solve it, in the sense of giving you a useful numerical answer. In this chapter, the reader will see that SVD of any matrix is always realizable. Applications of SVD [and a related tool known as polar decomposition (PD), or canonical factorization, generalizing a complex number’s factorization by its modulus and phase factor] are not restricted to numerical analysis; other application examples are presented in this chapter. More advanced readers who wish to familiarize themselves with infinite-dimensional versions of SVD and PD and their various applications may consult the guide to the literature that is provided.
Thinking in Problems, 2012
The derivatives of a quadratic trinomial F = ax 2 + bx + c at its roots x 1, x 2 add up to zero (... more The derivatives of a quadratic trinomial F = ax 2 + bx + c at its roots x 1, x 2 add up to zero (Vieta’s theorem). Alternatively, we can write this as 1/F′(x 1) + 1/F′(x 2) = 0 or, equivalently, 1/(x 1 − x 2) + 1/(x 2 − x 1) = 0. Also, there is an obvious identity: x 1/(x 1 − x 2) + x 2/(x 2 − x 1) = 1. C.G.J. Jacobi’s concern with questions of dynamics and geometry led him to derive far-reaching generalizations of these identities for polynomials of arbitrary degree, which enabled him to solve several complex problems. For example, he squared an ellipsoid’s surface, determined the geodesics on this surface, and described the dynamics with two motionless centers of gravity. In this chapter, readers will become acquainted with the Jacobi identities and related formulas and their close ties with complex analysis; in addition, they will learn about some less traditional applications of the Jacobi identities (to linear differential equations). The chapter also contains notes about further generalizations and applications and a short guide to the literature.
Birkhäuser Boston eBooks, 2013
Deciphering, or multiexponential analysis of signals in a nuclear magnetic resonance (NMR) machin... more Deciphering, or multiexponential analysis of signals in a nuclear magnetic resonance (NMR) machine, is necessary for a composite analysis (e.g., for separating oil fractions in well logging, detecting affected tissue in medicine). NMR signals are modeled by functions of the form \( f(x)=\sum {{A_i}{e^{{{\omega_i}x}}}} \), which are linear combinations of exponential terms; in these f, the parameters ω i correspond to different components, and A i characterize their relative weights in the composite. The number, and value of the parameters A i and ω i are not known, and so the multiexponential analysis consists in determining them (within some tolerance); this analysis relates to so-called inverse problems, usually characterized by a high degree of instability. This chapter contains real-life algebraic and combinatorial problems that arose in the development of a stable algorithm for multiexponential analysis.
Thinking in Problems, 2012
This chapter contains real-life problems that can arise in the analysis of the dynamic behavior o... more This chapter contains real-life problems that can arise in the analysis of the dynamic behavior of a gradient algorithm for an iterative reconstruction in computerized tomography (CT). In most practical cases, the gradient algorithm apparently converges. However, the theory cannot exclude situations where its behavior becomes very complicated, even chaotic, depending on the parameters. Even in a simplified situation this algorithm may show some striking behavior! The simplified situation is related to the recursion formula $$ {x_0}>0, \quad \quad \quad \quad {x_{n+1 }}={x_n}\cdot {e^{{-({x_n}-\xi )}}} $$
Choice Reviews Online, 2013
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.
IEEE Transactions on Biomedical Engineering, 2008
The energetic balance of forces in the cardiovascular system is vital to the stability of blood f... more The energetic balance of forces in the cardiovascular system is vital to the stability of blood flow to all physiological systems in mammals. Yet, a large-scale, theoretical model, summarizing the energetic balance of major forces in a single, mathematically closed system has not been described. Although a number of computer simulations have been successfully performed with the use of analog models, the analysis of energetic balance of forces in such models is obscured by a big number of interacting elements. Hence, the goal of our study was to develop a theoretical model that represents large-scale, energetic balance in the cardiovascular system, including the energies of arterial pressure wave, blood flow, and the smooth muscle tone of arterial walls. Because the emphasis of our study was on tracking beat-to-beat changes in the balance of forces, we used a simplified representation of the blood pressure wave as a trapezoidal pressure-pulse with a strong-discontinuity leading front. This allowed significant reduction in the number of required parameters. Our approach has been validated using theoretical analysis, and its accuracy has been confirmed experimentally. The model predicted the dynamics of arterial pressure in human subjects undergoing physiological tests and provided insights into the relationships between arterial pressure and pressure wave velocity.
Physica D: Nonlinear Phenomena, 1994
... NH Eq ELSEVIER Physica D 73 (1994) 189204 On the dynamics of a onedimensional selfgravitating... more ... NH Eq ELSEVIER Physica D 73 (1994) 189204 On the dynamics of a onedimensional selfgravitating medium Alexander Roytvarf' The Weizmann Institute of ... with the velocity field (more precisely, with its restriction on the closed domain in the spacetime covered thrice by the ...
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Papers by Alexander Roytvarf