Journal of Mathematics

Let be an -dimensional lattice. For any -dimensional vector and positive real number , let and denote the continuous Gaussian distribution and the discrete Gaussian distribution over , respectively. In this paper, we establish the exact relationship between the second and fourth moments centered around of the discrete Gaussian distribution and those of the continuous Gaussian distribution , respectively. This provides a quantization form of the result obtained by Micciancio and Regev on the second and fourth moments of discrete Gaussian distribution. Using the relationship, we also derive an uncertainty principle for Gaussian functions, which extend the result of Zheng, Zhao, and Xu. Our proof is based on combination of the idea of Micciancio and Regev and the idea of Zheng, Zhao, and Xu, where the main tool is high-dimensional Fourier transform.

In this study, we present a novel framework for the sixtic B-spline collocation approach in n-dimensions. Building upon previous research that focused on developing B-spline functions in n-dimensions to solve mathematical models, this work represents an extension of those efforts. We provide formulations of the sixtic B-spline collocation algorithm in one-dimensional, two-dimensional, and three-dimensional settings. These structures play a crucial role in solving mathematical models across diverse fields of study. To showcase the efficacy and accuracy of the proposed method, we employ a range of test problems in two and three dimensions. These examples serve as demonstrations of the effectiveness and precision of the suggested approach.

The main purpose of this article is using the elementary methods and the properties of the Fubini polynomials to study the congruence properties of a signless Stirling number of the first kind and solve a conjecture proposed by J. H. Zhao and Z. Y. Chen. Without a doubt, the novel approach employed in this work provides a useful reference for researching the congruence properties of other nonlinear binary recursive sequences.

In this paper, we study the existence and uniqueness of solutions for impulsive Atangana-Baleanu-Caputo fractional integro-differential equations with boundary conditions. Schaefer’s fixed point theorem and Banach contraction principle are used to prove the existence and uniqueness results. An example is presented to illustrate the results.

The minimum average variance estimation (MAVE) method has proven to be an effective approach to sufficient dimension reduction. In this study, we apply the computationally efficient optimization algorithm named alternating direction method of multipliers (ADMM) to a particular approach (MAVE or minimum average variance estimation) to the problem of sufficient dimension reduction (SDR). Under some assumptions, we prove that the iterative sequence generated by ADMM converges to some point of the associated augmented Lagrangian function. Moreover, that point is stationary. It also presents some numerical simulations on synthetic data to demonstrate the computational efficiency of the algorithm.

Due to its potential applications in image restoration and deep convolutional neural networks, the study of irregular frames has interested some researchers. This paper addresses irregular wavelet systems (IWSs) and irregular Gabor systems (IGSs) in Sobolev space . We obtain the sufficient and necessary conditions for IWS and IGS to be frames. By applying these conditions, we also derive the characterizations of IWS and IGS to be frames. Finally, we discuss the perturbation theorem of irregular wavelet frames (IWFs) and irregular Gabor frames (IGFs). We also provided some examples to support our results.