Lithuanian Mathematical Journal最新文献

We prove the existence and uniqueness of solutions to the differential equations of higher order ({x}^{left(lright)}left(sright)+gleft(s,xleft(sright)right)=0,sin left[c,dright],) satisfying three-point boundary conditions that contain a nonhomogeneous term (xleft(cright)=0,{x}{prime}left(cright)=0,{x}^{^{primeprime} }left(cright)=0,dots {x}^{left(l-2right)}left(cright)=0,{x}^{left(l-2right)}left(dright)-{beta x}^{left(l-2right)}left(eta right)=upgamma ,) where (lge mathrm{3,0}le c<eta <d,) the constants (beta ,upgamma ) are real numbers, and (g:left[c,dright]times {mathbb{R}}to {mathbb{R}}) is a continuous function. By using finer bounds on the integral of kernel, the Banach and Rus fixed point theorems on metric spaces are utilized to prove the existence and uniqueness of a solution to the problem.

Abstract

We consider the partial-sum process ({sum }_{k=1}^{left[ntright]}{X}_{k}^{left(nright)},) where (left{{X}_{k}^{left(nright)}={sum }_{j=0}^{infty }{alpha }_{j}^{left(nright)}{xi }_{k-j}left(bleft(nright)right), kin {mathbb{Z}}right},) n ≥ 1, is a series of linear processes with tapered filter ({alpha }_{j}^{left(nright)}={alpha }_{j} {1}_{left{0le jlelambdaleft(nright)right}}) and heavy-tailed tapered innovations ξ_j(b(n)), j ∈ Z. Both tapering parameters b(n) and ⋋ (n) grow to ∞ as n→∞. The limit behavior of the partial-sum process (in the sense of convergence of finite-dimensional distributions) depends on the growth of these two tapering parameters and dependence properties of a linear process with nontapered filter a_i, i ≥ 0, and nontapered innovations. We consider the cases where b(n) grows relatively slowly (soft tapering) and rapidly (hard tapering) and all three cases of growth of ⋋(n) (strong, weak, and moderate tapering).

Abstract

Integral tests are found for the convergence of two Spitzer-type series associated with a class of weighted averages introduced by Jajte [On the strong law of large numbers, Ann. Probab., 31(1):409–412, 2003]. Our main theorems are valid for a large family of dependent random variables that are not necessarily identically distributed. As a byproduct, we improve the Marcinkiewicz–Zygmund strong law of large numbers for asymptotically almost negatively associated sequences due to Chandra and Ghosal [Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund for dependent variables Acta Math. Hung., 71(4):327–336, 1996]. We also complement two limit theorems recently derived by Anh et al. [TheMarcinkiewicz–Zygmund-type strong law of large numbers with general normalizing sequences, J. Theor. Probab., 34(1):331–348, 2021] and Thành [On a new concept of stochastic domination and the laws of large numbers, Test, 32(1):74–106, 2023]. The obtained results are new even when the summands are independent.

In the paper, we establish the complete convergence for weighted sums of random variables satisfying generalized Rosenthal-type inequalities. Our results partially extend some known results and weaken their conditions. As statistical applications, we study the nonparametric regression model and obtain the complete consistency of the weighted regression estimator for the unknown regression functions.

The Hankel determinant ({H}_{mathrm{2,1}}left({F}_{f-1}/2right)) of logarithmic coefficients is defined as

({H}_{mathrm{2,1}}left({F}_{f-1}/2right):=left|begin{array}{cc}{Gamma }_{1}& {Gamma }_{2} {Gamma }_{2}& {Gamma }_{3}end{array}right|={Gamma }_{1}{Gamma }_{3}-{Gamma }_{2}^{2},)

where ({Gamma }_{1},{Gamma }_{2},) and ({Gamma }_{3}) are the first, second, and third logarithmic coefficients of inverse functions belonging to the class (mathcal{S}) of normalized univalent functions. In this paper, we establish sharp inequalities (left|{H}_{mathrm{2,1}}left({F}_{f-1}/2right)right|le 19/288,) (left|{H}_{mathrm{2,1}}left({F}_{f-1}/2right)right|le 1/144,) and (left|{H}_{mathrm{2,1}}left({F}_{f-1}/2right)right|le 1/36) for the logarithmic coefficients of inverse functions, considering starlike and convex functions, as well as functions with bounded turning of order 1/2, respectively.

In this paper, we propose and analyze a spectral collocation method for the numerical solutions of fractional multipantograph delay differential equations. The fractional derivatives are described in the Caputo sense. We present that some suitable variable transformations can convert the equations to a Volterra integral equation defined on the standard interval [−1, 1]. Then the Jacobi–Gauss points are used as collocation nodes, and the Jacobi–Gauss quadrature formula is used to approximate the integral equation. Later, the convergence analysis of the proposed method is investigated in the infinity norm and weighted L² norm. To perform the numerical simulations, some test examples are investigated, and numerical results are presented. Further, we provide the comparative study of the proposed method with some existing numerical methods.

Abstract

We propose and analyze an abstract framework to study the well-posedness for a family of linear degenerate parabolic augmentedmixed equations.We combine the theory for linear degenerate parabolic problems with results about stationary two-fold saddle point equations to deduce sufficient conditions for the existence and uniqueness of a solution for the problem. Finally, we show some applications of the developed theory through examples that come from fluid dynamic and electromagnetic problems.

In this paper, we establish the existence of a ground state solution for a weighted fourth-order equation of Shrödinger type under boundary Dirichlet condition in the unit ball B of ℝ⁴. The potential V is a continuous positive function bounded away from zero in B. The nonlinearity of the equation is assumed to have exponential growth due to Adams-type inequalities combined with polynomial term. We use the constrained minimization in the Nehari set, the quantitative deformation lemma, and degree theory results.

In this work, we set up the distribution function of (mathcal{M}:={mathrm{sup}}_{nge 1}{sum }_{i=1}^{n}left({X}_{i}-1right),) where the random walk ({sum }_{i=1}^{n}{X}_{i},nin {mathbb{N}},) is generated by N periodically occurring distributions, and the integer-valued and nonnegative random variablesX₁,X₂, . . . are independent. The considered random walk generates a so-called multiseasonal discrete-time risk model, and a known distribution of random variable M enables us to calculate the ultimate time ruin or survival probability. Verifying obtained theoretical statements, we demonstrate several computational examples for survival probability P(M < u) when N = 2, 3, or 10.

The International Conference on Number Theory and Probability Theory took place in Palanga from September 11 to 15, 2023, in commemoration of the anniversaries of Lithuanian mathematicians Jonas Kubilius, Donatas Surgailis, Antanas Laurinˇcikas, Eugenijus Manstaviˇcius, K˛estutis Kubilius, and Alfredas Raˇckauskas. This is an interview article with one the jubilarians, D. Surgailis, known for his work in stochastic processes, with interests in long-range dependence, fractionally integrated time series, statistical inference, spatial models, random fields, and their scaling limits. D. Surgailis is the author of the monograph Large Sample Inference for Long Memory Processes, published by Imperial College Press, London, in 2012 (coauthored with Liudas Giraitis and Hira L. Koul), and 150 journal papers. A complete list of his publications is available on http://lma.lt/asmsv/intranetas/index.php?m=profile&user=1577. D. Surgailis has served as the head of the Department of Random Processes at the Institute of Mathematics and Informatics (MII). He is Member Emeritus of the Lithuanian Academy of Sciences (LAS) and Professor Emeritus at Vilnius University (VU). This interview, conducted in October 2023, focuses on personal aspects of life and scientific career of D. Surgailis, that are less widely known but nonetheless captivating.