We establish a new perturbation theory for orthogonal polynomials using a Riemann-Hilbert approach and consider applications in numerical linear algebra and random matrix theory. This new approach shows that the ortho...
We establish a new perturbation theory for orthogonal polynomials using a Riemann-Hilbert approach and consider applications in numerical linear algebra and random matrix theory. This new approach shows that the orthogonal polynomials with respect to two measures can be effectively compared using the difference of their Stieltjes transforms on a suitably chosen contour. Moreover, when two measures are close and satisfy some regularity conditions, we use the theta functions of a hyperelliptic Riemann surface to derive explicit and accurate expansion formulae for the perturbed orthogonal polynomials. In contrast to other approaches, a key strength of the methodology is that estimates can remain valid as the degree of the polynomial grows. The results are applied to analyze several numerical algorithms from linear algebra, including the Lanczos tridiagonalization procedure, the Cholesky factorization, and the conjugate gradient algorithm. As a case study, we investigate these algorithms applied to a general spiked sample covariance matrix model by considering the eigenvector empirical spectral distribution and its limits. For the first time, we give precise estimates on the output of the algorithms, applied to this wide class of random matrices, as the number of iterations diverges. In this setting, beyond the first order expansion, we also derive a new mesoscopic central limit theorem for the associated orthogonal polynomials and other quantities relevant to numerical algorithms.
In the present paper, the most important aspects of computeralgebra systems applications in complicated calculations for classical queueing theory models and their novel modifications are discussed. We mainly present...
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In the present paper, the most important aspects of computeralgebra systems applications in complicated calculations for classical queueing theory models and their novel modifications are discussed. We mainly present huge computational possibilities of Mathematica environment and effective methods of obtaining symbolic results connected with the most important performance characteristics of queueing systems. First of all, we investigate effective solutions to computational problems appearing in queueing theory such as: finding final probabilities for Markov chains with a huge number of states, calculating derivatives of complicated rational functions of one or many variables with the use of classical and generalized L'Hospital's rules, obtaining exact formulae of Stieltjes convolutions, calculating chosen integral transforms used often in the above-mentioned theory and possible applications of generalized density function of random variables and vectors in these computations. Some exemplary calculations for practical models belonging both to classical models and their generalizations are attached as well.
In this paper, we address the problem of solving the quadratic equation x(2) + bx + c = 0, where b and c are elements of the algebra of coquaternions. The nature of this algebra leads, as expected, to results very dif...
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ISBN:
(数字)9783031651540
ISBN:
(纸本)9783031651533;9783031651540
In this paper, we address the problem of solving the quadratic equation x(2) + bx + c = 0, where b and c are elements of the algebra of coquaternions. The nature of this algebra leads, as expected, to results very different from the ones obtained in the classical complex case, making the root-finding problem much more demanding. In this paper we derive straightforward conditions to characterize the number and form of the non-isolated zeros of monic quadratic coquaternionic equations. Through carefully constructed examples and by leveraging the Mathematica system, we illustrate the core findings and showcase the array of scenarios that may emerge.
In this article, we consider weakly singular Volterra integral equation (WSVIE) with the kernel combining the power-law and logarithmic singularities f(zeta) = Au(zeta) + integral(zeta)(0) log(zeta - eta)u(eta)/(zeta ...
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In this article, we consider weakly singular Volterra integral equation (WSVIE) with the kernel combining the power-law and logarithmic singularities f(zeta) = Au(zeta) + integral(zeta)(0) log(zeta - eta)u(eta)/(zeta - eta)(beta) d eta, zeta is an element of[0,1], beta is an element of[0,1), where, f(zeta) is a smooth function. Equations of this kind, introduced by V. Volterra himself within the framework of the theory of functional compositions, are also directly related to the modern problems of fractional dynamics. Our approach is based on employing the operational matrix technique with the shifted Legendre polynomials (SLP) as a basis. This approach reduces WSVIE to a system of linear algebraic equations with coefficients, which contain closed-form analytical expressions. This allows the use of computeralgebra systems for obtaining approximate solutions based on the finite terms truncation of series. We provided a number of examples highlighting such a workflow in addition to establishing the error bound, convergence analysis and stability analysis.
The concepts of differentiation and integration for matrices are known. As far as each matrix is differentiable, it is not clear a priori whether a given matrix is integrable or not. Recently some progress was obtaine...
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The concepts of differentiation and integration for matrices are known. As far as each matrix is differentiable, it is not clear a priori whether a given matrix is integrable or not. Recently some progress was obtained for diagonalizable matrices, however general problem remained open. In this paper, we present a full solution of the integrability problem. Namely, we provide necessary and sufficient conditions for a given matrix to be integrable in terms of its characteristic polynomial. Furthermore, we find necessary and sufficient conditions for the existence of integrable and non-integrable matrices with given geometric multiplicities of eigenvalues. Our approach relies on properties of some special classes of polynomials, namely, Shabat polynomials and conservative polynomials, arising in numbertheory and dynamics. (c) 2023 Elsevier Inc. All rights reserved.
This paper introduces Riordan-Krylov matrices. These matrices naturally generalize Riordan matrices by using Krylov matrices and a more general class of algebras in place of formal power series. Fundamental properties...
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This paper introduces Riordan-Krylov matrices. These matrices naturally generalize Riordan matrices by using Krylov matrices and a more general class of algebras in place of formal power series. Fundamental properties of RiordanKrylov matrices that are analogous to those of Riordan matrices are developed. These results and techniques are used to study incidence algebras of a poset as well as the chain and zeta-multichain polynomials of the poset. Throughout the paper applications to enumeration problems in combinatorics and numbertheory are provided.(c) 2021 Elsevier Inc. All rights reserved.
We give an explicit formula for the value of the Bernoulli polynomial B2k(t) when t is a rational number in the interval (0, 1). When t = 12, 31, 32, 41, 34, 61, 65 the value of B2k(t) is known explicitly. In 1938 Emm...
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We give an explicit formula for the value of the Bernoulli polynomial B2k(t) when t is a rational number in the interval (0, 1). When t = 12, 31, 32, 41, 34, 61, 65 the value of B2k(t) is known explicitly. In 1938 Emma Lehmer asked for the value of B2k(t) when the denominator of t is 5, 8, 10, or 12. We apply our formula to determine B2k(t) when the denominator of t is 5, 8, 10, and 12 thereby answering Lehmer's 87 year old question. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
The k-generalized Fibonacci and Pell polynomials are the polynomials X-k - Xk-1 - Xk-2 -center dot center dot center dot - 1 and X-k - 2X(k-1) - Xk-2 - center dot center dot center dot - 1, respectively. Here, k >=...
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The k-generalized Fibonacci and Pell polynomials are the polynomials X-k - Xk-1 - Xk-2 -center dot center dot center dot - 1 and X-k - 2X(k-1) - Xk-2 - center dot center dot center dot - 1, respectively. Here, k >= 2 is any integer. In this paper, we show that any two roots of some generalized Fibonacci and Pell polynomials are multiplicatively independent confirming a conjecture from [Bravo, Herrera and Luca, Common values of generalized Fibonacci and Pell sequences, J. numbertheory 226 (2021) 51-71].
Computations on a computer with a floating point arithmetic are always approximate. Conversely, computations with the rational arithmetic (in a computeralgebra system, for example) are always absolutely exact and rep...
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Computations on a computer with a floating point arithmetic are always approximate. Conversely, computations with the rational arithmetic (in a computeralgebra system, for example) are always absolutely exact and reproducible both on other computers and (theoretically) by hand. Consequently, these computations can be demonstrative in a sense that a proof obtained with their help is no different from a traditional one (computer assisted proof). However, usually such computations are impossible in a sufficiently complicated problem due to limitations on resources of memory and time. We propose a mechanism of rounding off rational numbers in computations with rational arithmetic, which solves this problem (of resources), i.e., computations can still be demonstrative but do not require unbounded resources. We give some examples of implementation of standard numerical algorithms with this arithmetic. The results have applications to analytical numbertheory.
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