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Under appropriate technical assumptions, the simple-loop theory allows to derive various types of asymptotic expansions for the eigenvalues of Toeplitz matrices generated by a function f. Unfortunately, such a theory is not available in the preconditioning setting, that is for matrices of the form with real-valued, g nonnnegative and not identically zero almost everywhere. Independently and under the milder hypothesis that is even and monotonic over , matrix-less algorithms have been developed for the fast eigenvalue computation of large preconditioned matrices of the type above, within a linear complexity in the matrix order: behind the high efficiency of such algorithms there are the expansions as in the case , combined with the extrapolation idea, and hence we conjecture that the simple-loop theory has to be extended in such a new setting, as the numerics strongly suggest. Here we focus our attention on a change of variable, followed by the asymptotic expansion of the new variable, and we consider new matrix-less algorithms ad hoc for the current case. Numerical experiments show a much higher accuracy till machine precision and the same linear computational cost, when compared with the matrix-less procedures already proposed in the literature.

It is known that the generating function of a sequence of Toeplitz matrices may not describe the asymptotic distribution of the eigenvalues of the considered matrix sequence in the non-Hermitian setting. In a recent work, under the assumption that the eigenvalues are real, admitting an asymptotic expansion whose first term is the distribution function, fast algorithms computing all the spectra were proposed in different settings. In the current work, we extend this idea to non-Hermitian Toeplitz matrices with complex eigenvalues, in the case where the range of the generating function does not disconnect the complex field or the limiting set of the spectra, as the matrix-size tends to infinity, has one nonclosed analytic arc. For a generating function having a power singularity, we prove the existence of an asymptotic expansion, that can be used as a theoretical base for the respective numerical algorithm. Different generating functions are explored, highlighting different numerical and theoretical aspects; for example, non-Hermitian and complex symmetric matrix sequences, the reconstruction of the generating function, a consistent eigenvalue ordering, the requirements of high-precision data types. Several numerical experiments are reported and critically discussed, and avenues of possible future research are presented.

The eigenvalues of Toeplitz matrices T n ( f ) with a real-valued generating function f , satisfying some conditions and tracing out a simple loop over the interval [- pi, pi ], are known to admit an asymptotic expansion with the form lambda j ( T n ( f )) = f ( sigma j,n ) + c 1 ( sigma j,n ) h + c 2 ( sigma j,n ) h 2 + O ( h 3 ) , where h = 1 / ( n + 1), sigma j,n = pi jh , and c k are some bounded coefficients depending only on f . The numerical results presented in the literature suggest that the effective conditions for the expansion to hold are weaker and reduce to a fixed smoothness and to having only two intervals of monotonicity over [- pi, pi ]. In this article we investigate the superposition caused over this expansion, when considering the following linear combination lambda j ( T n ( f 0 ) + beta n, 1 T n ( f 1 ) + beta n, 2 T n ( f 2 ) ) , where beta n, 1 , beta n, 2 are certain constants depending on n and the generating functions f 0 , f 1 , f 2 are either simple loop or satisfy the weaker conditions mentioned before. We formally obtain an asymptotic expansion in this setting under simple -loop related assumptions, and we show numerically that there is much more to investigate, opening the door to linear in time algorithms for the computation of eigenvalues of large matrices of this type including a multilevel setting. The problem is of concrete interest, considering spectral features of matrices stemming from the numerical approximation of standard differential operators and distributed order fractional differential equations, via local methods such as Finite Differences, Finite Elements, and Isogeometric Analysis. (c) 2024 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY -NC -ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/).

Exact expressions for the relationship between angles of incidence and reflection from a boundary wall in nonlinear dissipative particle models are extremely rare. Here, we study a particle model of a camphor disk floating on water in a low speed limit, for which the model was derived. We begin with a very rough and inaccurate approximation, based in part on a Hamiltonian limit of the model, and then, using symmetry arguments supported by a series of experiments, show that this rough approximation can likely be repaired by introducing a factor dependent upon model parameters, then extract the full parameter dependence of this factor, and finally conjecture a simple and exact asymptotic relationship between the angles of incidence and reflection. There are reasons to believe that this asymptotic relationship may be universal for such models in their corresponding limits.

In this paper, the stochastic uctuation of buffer stock level at time t is investigated. Therefore, random walk processes X(t) and Y (t) with two specific barriers have been defined to describe the stochastic uctuation of the product level. Here X(t) = Y (t) - a and the parameter a specifies half capacity of the buffer stock warehouse. Next, the one-dimensional distribution of the process X(t) has calculated. Moreover, the ergodicity of the process X(t) has been proven and the exact formula for the characteristic function has been found. Then, the weak convergence theorem has been proven for the standardized process W(t) = X(t)/a, as a →∞ Additionally, exact and asymptotic expressions for the ergodic moments of the processes X(t) and Y (t) are obtained. © (2024) Isik University, Department of Mathematics, All Rights Reserved.

In this paper, a modification of a renewal-reward process with dependent components is mathematically constructed and the stationary characteristics of this process are studied. Stochastic processes with dependent components have rarely been studied in the literature owing to their complex mathematical structure. We partially fill the gap by studying the effect of the dependence assumption on the stationary properties of the process. To this end, first, we obtain explicit formulas for the ergodic distribution and the stationary moments of the process. Then we analyze the asymptotic behavior of the stationary moments of the process by using the basic results of the renewal theory and the Laplace transform method. Based on the analysis, we obtain two-term asymptotic expansions of the stationary moments. Moreover, we present two-term asymptotic expansions for the expectation, variance, and standard deviation of the process. Finally, the asymptotic results obtained are examined in special cases. © Pleiades Publishing, Ltd. 2024.

We study divide-and-conquer recurrences of the form f(n) = alpha f ([n/2]) + beta f(n/2) + g(n) (n >= 2), with g(n) and f (1) given, where alpha, beta >= 0 with alpha + beta > 0; such recurrences appear often in the analysis of computer algorithms, numeration systems, combinatorial sequences, and related areas. We show under an optimum (iff) condition on g(n) that the solution f always satisfies a simple identity f(n) = n(log2(alpha+beta)) P(log(2) n) - Q(n), where P is a periodic function and Q(n) is of a smaller order than the dominant term. This form is thus not only an identity but also an asymptotic expansion. Explicit forms for the continuity of the periodic function P are provided, together with a few other smoothness properties. We show how our results can be easily applied to many dozens of concrete examples collected from the literature, and how they can be extended in various directions. Our method of proof is surprisingly simple and elementary, but leads to the strongest types of results for all examples to which our theory applies. (c) 2023 Elsevier Inc. All rights reserved.

In this paper, the stochastic fluctuation of buffer stock level at time t is investigated. Therefore, random walk processes X(t) and Y (t) with two specific barriers have been defined to describe the stochastic fluctuation of the product level. Here X(t) equivalent to Y (t) - a and the parameter a specifies half capacity of the buffer stock warehouse. Next, the one-dimensional distribution of the process X(t) has calculated. Moreover, the ergodicity of the process X(t) has been proven and the exact formula for the characteristic function has been found. Then, the weak convergence theorem has been proven for the standardized process W(t) equivalent to X(t)/a, as a -> infinity . Additionally, exact and asymptotic expressions for the ergodic moments of the processes X(t) and Y (t) are obtained., TOBB University of Economics and Technology, Acknowledgement. One of the authors (Aynura Poladova) would like to thank TOBB University of Economics and Technology for its financial support for her scientific studies.

Asymptotic expansion is presented for an estimator of the Hurst coefficient of a fractional Brownian motion. We first derive the expansion formula of the principal term of the error of the estimator using a recently developed theory of asymptotic expansion of the distribution of Wiener functionals, and utilize the perturbation method on the obtained formula in order to calculate the expansion of the estimator. We also discuss some second-order modifications of the estimator. Numerical results show that the asymptotic expansion attains higher accuracy than the normal approximation.

We show in this paper that complete asymptotic expansions exist for a class of holomorphic Dirichlet-Hurwitz-Lerch Eisenstein series (Theorems 1 and 2), which, together with their remainders in exact form (Theorem 3), naturally transfer to several (additive and multiplicative) character analogues of Ramanujan's formula for specific values of the Riemann zeta-function (Theorem 4 and Corollary 4.1), and further to the (quasi) modular relations for similar character analogues of the classical Eisenstein series with integer weights (Corollary 4.2). Prior to the (sketched) derivation of our main formula, we prepare several basic (but new) results on Dirichlet-Hurwitz-Lerch $L$-functions (Theorems 5, 6 and Lemmas 1-3), which play underlying roles in all aspects of the proofs; the detailed version of the proofs will appear in a forthcoming article [16].

It is known that the generating function f of a sequence of Toeplitz matrices {Tn(f)}n may not describe the asymptotic distribution of the eigenvalues of Tn(f) if f is not real. In this paper, we assume as a working hypothesis that, if the eigenvalues of Tn(f) are real for all n, then they admit an asymptotic expansion of the same type as considered in previous works, where the first function, called the eigenvalue symbol f, appearing in this expansion is real and describes the asymptotic distribution of the eigenvalues of Tn(f). This eigenvalue symbol f is in general not known in closed form. After validating this working hypothesis through a number of numerical experiments, we propose a matrix-less algorithm in order to approximate the eigenvalue distribution function f. The proposed algorithm, which opposed to previous versions, does not need any information about neither f nor f is tested on a wide range of numerical examples; in some cases, we are even able to find the analytical expression of f. Future research directions are outlined at the end of the paper.

Bu çalışmada, genelleştirilmiş yansıtan bariyerli bir ödüllü yenileme süreci ele alınmış ve sürecin olasılık karakteristikleri analitik ve asimtotik yöntemlerle hem klasik hem de bulanık mantık çerçevesinde incelenmiştir. Çalışmanın birinci kısmında, süreç matematiksel olarak inşa edilmiş ve sonlu boyutlu dağılımları ele alınmıştır. Ardından sürecin üç önemli sınır fonksiyoneli tanımlanmış ve bu sınır fonksiyonellerinin momentleri hesaplanmıştır. Daha sonra sürecin bazı şartlar altında ergodikliği ispatlanmış ve ergodik dağılımının genel şekli bulunmuştur. Buna ek olarak, ergodik dağılım için zayıf yakınsama teoremi ispat edilmiş ve limit dağılımının açık şekli bulunmuştur. Bunun yanı sıra, bulunan limit dağılımının bir ?Kalan Ömür? dağılımı olduğu gösterilmiştir. Bu bölümün sonunda ise sürecin ergodik dağılımının tüm momentleri için asimtotik açılımlar elde edilmiştir. Ayrıca, bazı önemli durumlarda (Üstel, Erlang ve Rayleigh Dağılımları durumunda) sürecin ilk dört momenti için açık katsayılı asimtotik açılımlar yazılmıştır.Tezin ikinci kısmında ise, talepleri ifade eden rasgele değişkenlerin, bulanık parametreli Üstel, Gama, Erlang ve Weibull dağılımlarına sahip olduğu durumlarda sürecin ergodik dağılımı ele alınmış ve ergodik dağılımın bulanık mantık karakteristikleri incelenmiştir., In this study, a generalized renewal reward process with reflecting barrier is considered and probability characteristics of the process is investigated by analytic and asymptotic methods in both crisp logic and fuzzy logic. In the first part of the study, the process is constructed mathematically and its finite dimensional distributions are examined. Then, the three important boundary functionals are defined and the first four moments of these boundary functionals are calculated. Later, under some conditions the ergodicity of the process is proved and the exact expression of ergodic distribution of the process is obtained. Moreover, the weak convergence theorem for the ergodic distribution of the process is proved and the explicit form of the limit distribution of the process is found. Besides, it is shown that the obtained limit distribution is a residual waiting time distribution. At the end of the first part, the exact form of the moments of the ergodic distribution is obtained. In some important cases (in the cases of Exponential, Erlang and Rayleigh distributions), the asymptotic expansions of the first four moments with the explicit coefficients are written. In the second part of the study, when the random variables expressed demands, having Exponential, Gama, Erlang and Weibull distributions with a fuzzy parameter, the ergodic distribution of the process is examined. After that, the fuzzy characteristics of the ergodic distribution are investigated and the explicit forms of these characteristics are obtained.

In this study, asymptotic expansion for ergodic distribution of an inventory control model of type (s, S) with generalized beta interference of chance is obtained, when S - s -> infinity. Moreover, weak convergence theorem is proved for ergodic distribution. Finally, the accuracy of the asymptotic expansion is examined with Monte Carlo simulation method.

In this study, a semi-Markovian random walk process (X(t)) with a generalized delaying barrier is considered and the ergodic theorem for this process is proved under some weak conditions. Then, the exact expressions and asymptotic expansions for the first four ergodic moments of the process X (t) are obtained., TUBITAK [110T559], This research is supported partially by TUBITAK, under Project 110T559.

The Gatheral model is a three factor model with mean-reverting stochastic volatility that reverts to a stochastic long run mean. This chapter reviews previous analytical results on the first and second order implied volatility expansions under this model. Using the Monte Carlo simulation as the benchmark method, numerical studies are conducted to investigate the accuracy and properties of these analytical expansions. The classical Black–Scholes option pricing model assumes that the underlying asset follows a geometric Brownian motion with constant volatility. The chapter discusses partial calibration procedure is proposed and synthetic and real data calibration. If a full calibration is desired, we can use the results from the partial calibration as inputs for the final local optimization over all model parameters. In implementing the calibration procedure, the effect of the Covid-19 pandemic on the model calibration is high.

We consider a long-term portfolio choice problem with two illiquid and correlated assets, which is associated with an eigenvalue problem in the form of a variational inequality. The eigenvalue and the free boundaries implied by the variational inequality correspond to the portfolio's optimal long-term growth rate and the optimal trading strategy, respectively. After proving the existence and uniqueness of viscosity solutions for the eigenvalue problem, we perform an asymptotic expansion in terms of small correlations and obtain semi-analytical approximations of the free boundaries and the optimal growth rate. Our leading order expansion implies that the free boundaries are orthogonal to each other at four corners and have C1 regularity. We propose an efficient numerical algorithm based on the expansion, which proves to be accurate even for large correlations and transaction costs. Moreover, following the approximate trading strategy, the resulting growth rate is very close to the optimal one. © 2022 Wiley Periodicals LLC.

In this paper, we derive asymptotic expansion of the wavelet transform for small values of the dilation parameter a by using Lopez and Pagola technique. Asymptotic expansion of Morlet wavelet, Mexican Hat wavelet and Haar wavelet transform are obtained as a special cases.

In this paper, a semi-Markovian random walk process (X(t)) with a discrete interference of chance is constructed and the ergodicity of this process is discussed. Some exact formulas for the first- and second-order moments of the ergodic distribution of the process X(t) are obtained, when the random variable zeta(1) has an exponential distribution with the parameter lambda > 0. Here zeta(1) expresses the quantity of a discrete interference of chance. Based on these results, the third-order asymptotic expansions for mathematical expectation and variance of the ergodic distribution of the process X(t) are derived, when lambda -> 0. (c) 2007 Elsevier B.V. All rights reserved.

The Bloch-Floquet theory are popular tools for the investigation of materials with periodic structures. For example, we can show that the spectrum of periodic Schrodinger operators have band structures. Here we shall extend the Bloch-Floquet theory, which is appicable to abelian groups, to the Heisenberg group. Our method is based on a combination of the representations of the discrete Heisenberg groups and of the Heisenberg Lie group. We apply this method to asymptotic problems for heat kernels and for counting prime closed geodesics. In this application, we need additional ingradients, the semi-classical analysis and the Chen's iterated integrals. As a by-product, we give another mathematical explanation of the semi-classical asymptotic expansion formula for the Hofstadter butterfly of Wilkinson, which is originally due to Helffer-Sjostrand.

Asymptotic approximations of Jacobi polynomials are given in terms of elementary functions for large degree n and parameters α and β. From these new results, asymptotic expansions of the zeros are derived and methods are given to obtain the coefficients in the expansions. These approximations can be used as initial values in iterative methods for computing the nodes of Gauss–Jacobi quadrature for large degree and parameters. The performance of the asymptotic approximations for computing the nodes and weights of these Gaussian quadratures is illustrated with numerical examples.

The paper is devoted to studies of regularly and singularly perturbed Markov chains with damping component. In such models, a matrix of transition probabilities is regularised by adding a special damping matrix multiplied by a small damping (perturbation) parameter epsilon. We perform a detailed perturbation analysis for such Markov chains, particularly, give effective upper bounds for the rate of approximation for stationary distributions of unperturbed Markov chains by stationary distributions of perturbed Markov chains with regularised matrices of transition probabilities, asymptotic expansions for approximating stationary distributions with respect to damping parameter, explicit coupling type upper bounds for the rate of convergence in ergodic theorems for n-step transition probabilities, as well as ergodic theorems in triangular array mode.

Multiphase, compressible and viscous flows are of crucial importance in a wide range of scientific and engineering problems. Despite the large effort paid in the last decades to develop accurate and efficient numerical techniques to address this kind of problems, current models need to be further improved to address realistic applications. In this context, we propose a numerical approach to the simulation of multiphase, viscous flows where a compressible and an incompressible phase interact in the low-Mach number regime. In this frame, acoustics are neglected but large density variations of the compressible phase can be accounted for as well as heat transfer, convection and diffusion processes. The problem is addressed in a fully Eulerian framework exploiting a low-Mach number asymptotic expansion of the Navier-Stokes equations. A Volume of Fluid approach (VOF) is used to capture the liquid-gas interface, built on top of a massive parallel solver, second order accurate both in time and space. The second-order-pressure term is treated implicitly and the resulting pressure equation is solved with the eigenexpansion method employing a robust and novel formulation. We provide a detailed and complete description of the theoretical approach together with information about the numerical technique and implementation details. Results of benchmarking tests are provided for five different test cases., QC 20210409

This paper studies the large time behavior of a small solution to the generalized Benjamin-Ono equation with a short range nonlinearity, i.e., with the power of nonlinearity greater than 3. In this case, it is well-known that the solution is asymptotically free as . We are interested in the asymptotic expansion of the solution, and determine the second asymptotic term. In order to specify the second asymptotic term, we will apply a technique of Fourier series expansion.

In this study, asymptotic expansions of the moments of the maximum (M(beta)) of Gaussian random walk with negative drift ( -aEuro parts per thousand beta), beta > 0, are established by using Bell Polynomials. In addition, the weak convergence theorem for the distribution of the random variable Y(beta) a parts per thousand aEuro parts per thousand 2 beta M(beta) is proved, and the explicit form of the limit distribution is derived. Moreover, the approximation formulas for the first four moments of the maximum of a Gaussian random walk are obtained for the parameter beta aaEuro parts per thousand(0.5, 3.2] using meta-modeling.

In this paper, a semi-Markovian random walk with a discrete interference of chance (X(t)) is considered. In this study, it is assumed that the sequence of random variables {zeta(n)}, n = 1,2, ... , which describes the discrete interference of chance, forms an ergodic Markov chain with the Weibull stationary distribution. Under this assumption, the ergodic theorem for the process X(t) is discussed. Then the weak convergence theorem is proved for the ergodic distribution of the process X(t) and the limit form of the ergodic distribution is derived.

A stochastic process is considered that describes the so-called (s, S) inventory model. The asymptotic behavior of the ergodic distribution of the process is investigated. It is established that the ergodic distribution of the process on the interval [s, S] tends to the uniform distribution for sufficiently large values of the parameter ?=S-s and the convergence rate is estimated. © 2012 Springer Science+Business Media, Inc.

In this study, a semi-Markovian random walk with a discrete interference of chance (X(t)) is considered and under some weak assumptions the ergodicity of this process is discussed. The exact formulas for the first four moments of ergodic distribution of the process X(t) are obtained when the random variable which is describing a discrete interference of chance, has a triangular distribution in the interval Is, SI with center (S + s)/2. Based on these results, the asymptotic expansions with three-term are obtained for the first four moments of the ergodic distribution of X(t), as a (S - s)/2 -> infinity. Furthermore, the asymptotic expansions for the variance, skewness and kurtosis of the ergodic distribution of the process X(t) are established. Finally, by using Monte Carlo experiments it is shown that the given approximating formulas provide high accuracy even for small values of parameter a. (c) 2010 Elsevier Inc. All rights reserved., Michigan State University, We would like to express our regards to Professor A.V. Skorohod, Michigan State University, for his support and valuable advices. Also we are grateful to the Editor and Reviewers for their helpful comments.

In this study, a renewal - reward process with a discrete interference of chance is constructed. This process describes in particular a semi-Markovian inventory model of type (s, S). The ergodic distribution of this process is expressed by a renewal function, and a second-order approximation for the ergodic distribution of the process is obtained as S - s -> infinity when the interference has a triangular distribution. Then, the weak convergence theorem is proved for the ergodic distribution and the limit distribution is derived. Finally, the accuracy of the approximation formula is tested by the Monte Carlo simulation method.

In this study, two boundary functionals N-1 and tau(1) of the renewal reward process with a discrete interference of chance ( X(t)) are investigated. A relation between the moment generating function (Psi(N)(z)) of the boundary functional N-1 and the Laplace transform (Phi(tau)(mu)) of the boundary functional tau(1) is obtained. Using this relation, the exact formulas for the first four moments of the boundary functional tau(1) are expressed by means of the first four moments of the boundary functional N-1. Moreover, the asymptotic expansions for the first four moments of these boundary functionals are established when the random variables {zeta(n)}, n >= 0, which describe a discrete interference of chance, have an exponential distribution with parameter lambda > 0. Finally, the accuracy of the approximation formulas for the moments (EN1k) of the boundary functional N-1 are tested by Monte Carlo simulation method., Michigan State University, The authors express their thanks to Professor A.V. Skorohod, Michigan State University, for his supports and valuable advices.

In this study, we constructed a stochastic process (X(t)) that expresses a semi-Markovian inventory model of type (s, S) and it is shown that this process is ergodic under some weak conditions. Moreover, we obtained exact and asymptotic expressions for the n(th) order moments (n=1,2,3,...) of ergodic distribution of the process X(t), as S-s. Finally, we tested how close the obtained approximation formulas are to the exact expressions. Copyright (c) 2012 John Wiley ; Sons, Ltd.

In this paper, a semi-Markovian random walk with a discrete interference of chance (X(t)) is considered. In this study, it is assumed that the sequence of random variables {zeta(n)}, n = 1,2, ... , which describes the discrete interference of chance, forms an ergodic Markov chain with the Weibull stationary distribution. Under this assumption, the ergodic theorem for the process X(t) is discussed. Then the weak convergence theorem is proved for the ergodic distribution of the process X(t) and the limit form of the ergodic distribution is derived.

In this study, asymptotic expansions of the moments of the maximum (M(beta)) of Gaussian random walk with negative drift ( -aEuro parts per thousand beta), beta > 0, are established by using Bell Polynomials. In addition, the weak convergence theorem for the distribution of the random variable Y(beta) a parts per thousand aEuro parts per thousand 2 beta M(beta) is proved, and the explicit form of the limit distribution is derived. Moreover, the approximation formulas for the first four moments of the maximum of a Gaussian random walk are obtained for the parameter beta aaEuro parts per thousand(0.5, 3.2] using meta-modeling.

A stochastic process is considered that describes the so-called (s, S) inventory model. The asymptotic behavior of the ergodic distribution of the process is investigated. It is established that the ergodic distribution of the process on the interval [s, S] tends to the uniform distribution for sufficiently large values of the parameter ?=S-s and the convergence rate is estimated. © 2012 Springer Science+Business Media, Inc.

In this study, two boundary functionals N-1 and tau(1) of the renewal reward process with a discrete interference of chance ( X(t)) are investigated. A relation between the moment generating function (Psi(N)(z)) of the boundary functional N-1 and the Laplace transform (Phi(tau)(mu)) of the boundary functional tau(1) is obtained. Using this relation, the exact formulas for the first four moments of the boundary functional tau(1) are expressed by means of the first four moments of the boundary functional N-1. Moreover, the asymptotic expansions for the first four moments of these boundary functionals are established when the random variables {zeta(n)}, n >= 0, which describe a discrete interference of chance, have an exponential distribution with parameter lambda > 0. Finally, the accuracy of the approximation formulas for the moments (EN1k) of the boundary functional N-1 are tested by Monte Carlo simulation method., Michigan State University, The authors express their thanks to Professor A.V. Skorohod, Michigan State University, for his supports and valuable advices.

In this study, a renewal - reward process with a discrete interference of chance is constructed. This process describes in particular a semi-Markovian inventory model of type (s, S). The ergodic distribution of this process is expressed by a renewal function, and a second-order approximation for the ergodic distribution of the process is obtained as S - s -> infinity when the interference has a triangular distribution. Then, the weak convergence theorem is proved for the ergodic distribution and the limit distribution is derived. Finally, the accuracy of the approximation formula is tested by the Monte Carlo simulation method.

We introduce a regularisation process for a sequence of real numbers computing the constant coefficient in an asymptotic expansion associated with the original sequence. We give explicit computations in several interesting particular cases. This approach originates in a geometric context.

We consider special Lambert series as generating functions of divisor sums and determine their complete transseries expansion near rational roots of unity. Our methods also yield new insights into the Laurent expansions and modularity properties of iterated Eisenstein integrals that have recently attracted attention in the context of certain period integrals and string theory scattering amplitudes., SCOPUS: ar.j, DecretOANoAutActif, info:eu-repo/semantics/published

It is shown that complete asymptotic expansions exist for the double nonholomorphic Eisenstein series∫z2(s;z) having two variables s ∈ C2 and two parameters Z = (z1, z2) ∈N+×N―, when both Iz1-z2|→＋∞and lz1 -z2|→ 0 (Theorems 1 and 3).

Perturbed Markov chains are popular models for description of information networks. In such models, the transition matrix P0 of an information Markov chain is usually approximated by matrix Pε = (1 - ε) P0 + ε D, where D is a so-called damping stochastic matrix with identical rows and all positive elements, while ε is a damping (perturbation) parameter. We perform a detailed perturbation analysis for stationary distributions of such Markov chains, in particular get effective explicit series representations for the corresponding stationary distributions πε, upper bounds for the deviation |πε- π0 |, and asymptotic expansions for πε with respect to the perturbation parameter ε.

The paper deals with the problem of optimal control with a Boltz–type quality index over a finite time interval for a linear steady–state control system in the class of piecewise continuous controls with smooth control constraints. In particular, we study the problem of controlling the motion of a system of small mass points under the action of a bounded force. The terminal part of the convex integral quality index additively depends on slow and fast variables, and the integral term is a strictly convex function of control variable. If the system is completely controllable, then the Pontryagin maximum principle is a necessary and sufficient condition for optimality. The main difference between this study and previous works is that the equation contains the zero matrix of fast variables and, thus, the results of A. B. Vasilieva on the asymptotic of the fundamental matrix of a control system are not valid. However, the linear steady–state system satisfies the condition of complete controllability. The article shows that problems of optimal control with a convex integral quality index are more regular than time–optimal problems. © 2020 Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta. All rights reserved.

The paper continues the authors’ previous studies. We consider a time-optimal control problem for a singularly perturbed linear autonomous system with two independent small parameters and smooth geometric constraints on the control in the form of a ball cases formulae-sequence superscript R2 superscript R2 missing-subexpressionmissing-subexpressionsuperscript 3 formulae-sequencenorm 1formulae-sequence0 much-less-than 1missing-subexpressionmissing-subexpressionformulae-sequence 0subscript 0 superscriptsubscript 01superscript 3 0subscript 0missing-subexpressionmissing-subexpressionmissing-subexpressionformulae-sequence subscript 0formulae-sequence subscript 0 subscript missing-subexpressionmissing-subexpressionmissing-subexpression where 0100 The main difference of this case from the systems with fast and slow variables studied earlier is that here the matrix at the fast variables is the second-order Jordan block with zero eigenvalue and, thus, does not satisfy the standard asymptotic stability condition. Continuing the research, we consider initial conditions depending on the second small parameter. We derive and justify a complete asymptotic expansion in the sense of Erdelyi of the optimal time and optimal control with respect to the asymptotic sequence superscript superscript superscript, 01. © 2020, Pleiades Publishing, Ltd.

In this study, a random walk process with general interference of chance is investigated. The main feature of the examined process is that general formulas are obtained, except for some special cases. The process is mathematically constructed, after that the general ergodic theorem is proved and the ergodic distribution of the process is obtained. The exact form of the characteristic function of this distribution is found with the help of the basic identity of random walk. Then, exact expressions and three-term asymptotic expansions are found for the moments of the boundary function of the process. Two term asymptotic expansion for the characteristic function of the standardized process is found and the weak convergence theorem is proved by using this expansion. It is been proved that the ergodic distribution of the standardized process according to the continuity theorem for characteristic functions converges to the limit distribution of the residual waiting time of the renewal process generated by a sequence of random variables expressing the discrete interference of chance. Exact expressions are found for the first five moments of the ergodic distribution. Three term asymptotic expansions for the first five ergodic moments and four term asymptotic expansions for the first four ergodic moments of the process are obtained with the help of asymptotic expansions for the boundary functionals of the process. Approximate formulas are found for the characteristics of the process such as coefficient of variation, kurtosis and skewness with obtained asymptotic expansions of ergodic moments. In addition, the results obtained in the literature for special cases are calculated for the discrete interference of chances with different distributions (Weibull, symmetric triangular, exponential, Gamma) with the help of found general asymptotic expressions in this study., Bu çalışmada, genel müdahaleli rasgele yürüyüş süreci ele alınmıştır. İncelenen sürecin temel özelliği bazı durumlar dışında genel ifadeler elde edilmesidir. Süreç matematiksel olarak inşa edildikten sonra süreç için genel ergodik teorem ispat edilmiş ve sürecin ergodik dağılımı elde edilmiştir. Bu dağılıma ait karakteristik fonksiyonunun aşikâr şekli rasgele yürüyüşün temel özdeşliği yardımıyla bulunmuştur. Daha sonra sürecin sınır fonksiyonelinin momentleri için kesin ifadeler ve üç terimli asimtotik açılımlar bulunmuştur. Standartlaştırılmış sürecin karakteristik fonksiyonu için iki terimli asimtotik açılımı elde edilmiş ve bulunan açılım yardımıyla zayıf yakınsama teoremi ispatlanmıştır. Karakteristik fonksiyonlar için süreklilik teoremine göre standartlaştırılmış sürecin ergodik dağılımının müdahaleye karşılık gelen rasgele değişkenler dizisi tarafından üretilen yenileme sürecinin kalan ömrünün limit dağılımına yakınsadığı ispatlanmıştır. Ergodik dağılımın ilk beş momenti için kesin ifadeler bulunmuştur. Sürecin ilk beş ergodik momenti için üç terimli, ilk dört ergodik momenti için ise dört terimli asimtotik açılımlar sürecin sınır fonksiyonelleri için bulunan asimtotik açılımlar yardımıyla elde edilmiştir. Ergodik momentler için elde edilen asimtotik açılımlar yardımıyla sürecin değişim katsayısı, basıklık ve çarpıklık gibi karakteristikleri için yaklaşık formüller bulunmuştur. Ayrıca, literatürde özel durumlar için ulaşılan sonuçlar bu çalışmada bulunan genel asimtotik ifadeler yardımıyla farklı dağılımlara sahip müdahaleler (Weibull, simetrik üçgensel, üstel, Gamma) için hesaplanmıştır.

We obtain an asymptotic series ∑j=0∞Ijnj for the integral ∫01[xn+(1-x)n]1ndx as n→ ∞, and compute Ij in terms of alternating (or “colored”) multiple zeta values. We also show that Ij is a rational polynomial in the ordinary zeta values, and give explicit formulas for j≤ 12. As a by-product, we obtain precise results about the convergence of norms of random variables and their moments. We study Zn= ‖ (U, 1 - U) ‖ n as n tends to infinity and we also discuss Wn=‖(U1,U2,⋯,Ur)‖n for standard uniformly distributed random variables., SCOPUS: ar.j, DecretOANoAutActif, info:eu-repo/semantics/published

© 2020 Elsevier Inc. In this paper we study unfoldings of planar vector fields in a neighbourhood of a hyperbolic resonant saddle. We give a structure theorem for the asymptotic expansion of the local Dulac time (as well as the local Dulac map) with the remainder uniformly flat with respect to the unfolding parameters. Here local means close enough to the saddle in order that the normalizing coordinates provided by a suitable normal form can be used. The principal part of the asymptotic expansion is given in a monomial scale containing a deformation of the logarithm, the so-called Roussarie-Ecalle compensator. Especial attention is paid to the remainder's properties concerning the derivation with respect to the unfolding parameters.

In 1669 and 1671, Isaac Newton resolved an algebraic equation f(x, y)=0 by expressing y as an infinite series of x . In this paper, we formulate Newton's resolution as a contemporary algorithm along the line of his original text and prove that the series converges asymptotically to the implicit function or one of the branches under certain conditions.

We consider the initial value problem for the generalized Allen-Cahn equation, = Φ- ε-2 x (ΦtΦ-) Ω t > 0, where is an n X n real matrix-valued field, is a two-dimensional square with periodic boundary conditions, and > 0. This equation is the gradient flow for the energy, EΦ= f \\ ΦtΦ where \\ • \\f denotes the Frobenius norm. The primary contribution of this paper is to use asymptotic methods to describe the solution of this initial value problem. If the initial condition has a single-signe d determinant, at each point of the domain, at a fast O(s~2t) time scale, the solution evolves towards the closest orthogonal matrix. Then, at the O(t) time scale, the solution evolves according to the On diffusion equation. Stationary solutions to the On diffusion equation are analyzed for n = 2. If the initial condition has regions where the determinant is positive and negative, a free interface develops. Away from the interface, in each region, the matrix-valued field behaves as in the single-signed determinant case. At the O(t) time scale, the interface evolves in the normal direction by curvature. At a slow O (et) time scale, the interface is driven by curvature and the surface diffusion of the matrix-valued field. For n =2, the interface is driven by curvature and the jump in the squar ed tangental derivative of the phase across the interface. In particular, we emphasize that the interface when n > 2 is driven by surface diffusion, while for n =1, the original Allen-Cahn equation, the interface is only driven by mean curvature. A variety of numerical experiments are performed to verify, support, and illustrate our analytical results.. © 2019 Society for Industrial and Applied Mathematics Publications. All rights reserved.