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We study the lattice Green's function (LGF) of the screened Poisson equation on a two-dimensional rectangular lattice. This LGF arises in numerical analysis, random walks, solid-state physics, and other fields. Its defining characteristic is the screening term, which defines different regimes. When its coefficient is large, we can accurately approximate the LGF with an exponentially converging asymptotic expansion, and its convergence rate monotonically increases with the coefficient of the screening term. To tabulate the LGF when the coefficient is not large, we derive a one-dimensional integral representation of the LGF. We show that the trapezoidal rule can approximate this integral with exponential convergence, and we propose an efficient algorithm for its evaluation via the Fast Fourier Transform. We discuss applications including computing the LGF of the three-dimensional Poisson equation with one periodic direction and the return probability of a two-dimensional random walk with killing.

This paper introduces a novel methodology for Feature Selection for Functional Classification, FSFC, that addresses the challenge of jointly performing feature selection and classification of functional data in scenarios with categorical responses and multivariate longitudinal features. FSFC tackles a newly defined optimization problem that integrates logistic loss and functional features to identify the most crucial variables for classification. To address the minimization procedure, we employ functional principal components and develop a new adaptive version of the Dual Augmented Lagrangian algorithm. The computational efficiency of FSFC enables handling high-dimensional scenarios where the number of features may considerably exceed the number of statistical units. Simulation experiments demonstrate that FSFC outperforms other machine learning and deep learning methods in computational time and classification accuracy. Furthermore, the FSFC feature selection capability can be leveraged to significantly reduce the problem's dimensionality and enhance the performances of other classification algorithms. The efficacy of FSFC is also demonstrated through a real data application, analyzing relationships between four chronic diseases and other health and demographic factors.

This paper develops a general asymptotic theory of series estimators for spatial data collected at irregularly spaced locations within a sampling region $R_n \subset \mathbb{R}^d$. We employ a stochastic sampling design that can flexibly generate irregularly spaced sampling sites, encompassing both pure increasing and mixed increasing domain frameworks. Specifically, we focus on a spatial trend regression model and a nonparametric regression model with spatially dependent covariates. For these models, we investigate $L^2$-penalized series estimation of the trend and regression functions. We establish uniform and $L^2$ convergence rates and multivariate central limit theorems for general series estimators as main results. Additionally, we show that spline and wavelet series estimators achieve optimal uniform and $L^2$ convergence rates and propose methods for constructing confidence intervals for these estimators. Finally, we demonstrate that our dependence structure conditions on the underlying spatial processes include a broad class of random fields, including L\'evy-driven continuous autoregressive and moving average random fields.

We present our investigation of the study of two variable hypergeometric series, namely Appell $F_{1}$ and $F_{3}$ series, and obtain a comprehensive list of its analytic continuations enough to cover the whole real $(x,y)$ plane, except on their singular loci. We also derive analytic continuations of their 3-variable generalization, the Lauricella $F_{D}^{(3)}$ series and the Lauricella-Saran $F_{S}^{(3)}$ series, leveraging the analytic continuations of $F_{1}$ and $F_{3}$, which ensures that the whole real $(x,y,z)$ space is covered, except on the singular loci of these functions. While these studies are motivated by the frequent occurrence of these multivariable hypergeometric functions in Feynman integral evaluation, they can also be used whenever they appear in other branches of mathematical physics. To facilitate their practical use, we provide four packages: $\texttt{AppellF1.wl}$, $\texttt{AppellF3.wl}$, $\texttt{LauricellaFD.wl}$, and $\texttt{LauricellaSaranFS.wl}$ in $\textit{MATHEMATICA}$. These packages are applicable for generic as well as non-generic values of parameters, keeping in mind their utilities in the evaluation of the Feynman integrals. We explicitly present various physical applications of these packages in the context of Feynman integral evaluation and compare the results using other means as well. Various $\textit{MATHEMATICA}$ notebooks demonstrating different numerical results are also provided along with this paper.

In this paper we study two dimensional minimal linear code over the ring $\mathbb{Z}_{p^n}$(where $p$ is prime). We show that if the generator matrix $G$ of the two dimensional linear code $M$ contains $p^n+p^{n-1}$ column vector of the following type {\scriptsize{$u_{l_1}\begin{pmatrix} 1\\ 0 \end{pmatrix}$, $u_{l_2}\begin{pmatrix} 0\\1 \end{pmatrix}$, $u_{l_3}\begin{pmatrix} 1\\u_1 \end{pmatrix}$, $u_{l_4}\begin{pmatrix} 1\\u_2 \end{pmatrix}$,...,$u_{l_{p^n-p^{n-1}+2}} \begin{pmatrix} 1\\u_{p^n-p^{n-1}} \end{pmatrix}$, $u_{l_{p^n-p^{n-1}+3}}\begin{pmatrix} d_1 \\ 1 \end{pmatrix}$, $u_{l_{p^n-p^{n-1}+4}}\begin{pmatrix} d_2\\ 1 \end{pmatrix}$,..., $u_{l_{p^n+1}}\begin{pmatrix} d_{p^{n-1}-1}\\1 \end{pmatrix}$, $u_{l_{p^n+2}}\begin{pmatrix} 1\\d_1 \end{pmatrix}$, $u_{l_{p^n+3}}\begin{pmatrix} 1\\d_2 \end{pmatrix}$,...,$u_{l_{p^n+p^{n-1}}}\begin{pmatrix} 1 \\d_{p^{n-1}-1} \end{pmatrix}$}}, where $u_i$ and $d_j$ are distinct units and zero divisors respectively in the ring $\mathbb{Z}_{p^n}$ for $1\leq i \leq p^n+p^{n-1}$, $1\leq j \leq p^{n-1}-1$ and additionally, denote $u_{l_i}$ as units in $\mathbb{Z}_{p^n}$, then the module generated by $G$ is a minimal linear code. Also we show that if any one column vector of the above types are not present entirely in $G$, then the generated module is not a minimal linear code.

In this paper, we discuss tests for mean vector of high-dimensional data when the dimension $p$ is a function of sample size $n$. One of the tests, called the decomposite $T^{2}$-test, in the high-dimensional testing problem is constructed based on the estimation work of Ledoit and Wolf (2018), which is an optimal orthogonally equivariant estimator of the inverse of population covariance matrix under Stein loss function. The asymptotic distribution function of the test statistic is investigated under a sequence of local alternatives. The asymptotic relative efficiency is used to see whether a test is optimal and to perform the power comparisons of tests. An application of the decomposite $T^{2}$-test is in testing significance for the effect of monthly unlimited transport policy on public transportation, in which the data are taken from Taipei Metro System.

This paper introduces $\infty$-Diff, a generative diffusion model defined in an infinite-dimensional Hilbert space, which can model infinite resolution data. By training on randomly sampled subsets of coordinates and denoising content only at those locations, we learn a continuous function for arbitrary resolution sampling. Unlike prior neural field-based infinite-dimensional models, which use point-wise functions requiring latent compression, our method employs non-local integral operators to map between Hilbert spaces, allowing spatial context aggregation. This is achieved with an efficient multi-scale function-space architecture that operates directly on raw sparse coordinates, coupled with a mollified diffusion process that smooths out irregularities. Through experiments on high-resolution datasets, we found that even at an $8\times$ subsampling rate, our model retains high-quality diffusion. This leads to significant run-time and memory savings, delivers samples with lower FID scores, and scales beyond the training resolution while retaining detail.

We establish a theoretical framework of the particle relaxation method for uniform particle generation of Smoothed Particle Hydrodynamics. We achieve this by reformulating the particle relaxation as an optimization problem. The objective function is an integral difference between discrete particle-based and smoothed-analytical volume fractions. The analysis demonstrates that the particle relaxation method in the domain interior is essentially equivalent to employing a gradient descent approach to solve this optimization problem, and we can extend such an equivalence to the bounded domain by introducing a proper boundary term. Additionally, each periodic particle distribution has a spatially uniform particle volume, denoted as characteristic volume. The relaxed particle distribution has the largest characteristic volume, and the kernel cut-off radius determines this volume. This insight enables us to control the relaxed particle distribution by selecting the target kernel cut-off radius for a given kernel function.

Object-oriented programming (OOP) is one of the most popular paradigms used for building software systems. However, despite its industrial and academic popularity, OOP is still missing a formal apparatus similar to $\lambda$-calculus, which functional programming is based on. There were a number of attempts to formalize OOP, but none of them managed to cover all the features available in modern OO programming languages, such as C++ or Java. We have made yet another attempt and created $\varphi$-calculus. We also created EOLANG (also called EO), an experimental programming language based on $\varphi$-calculus.

We derive sharp-interface models for one-dimensional brittle fracture via the inverse-deformation approach. Methods of Gamma-convergence are employed to obtain the singular limits of previously proposed models. The latter feature a local, non-convex stored energy of inverse strain, augmented by small interfacial energy, formulated in terms of the inverse-strain gradient. They predict spontaneous fracture with exact crack-opening discontinuities, without the use of damage (phase) fields or pre-existing cracks; crack faces are endowed with a thin layer of surface energy. The models obtained herewith inherit the same properties, except that surface energy is now concentrated at the crack faces. Accordingly, we construct energy-minimizing configurations. For a composite bar with a breakable layer, our results predict a pattern of equally spaced cracks whose number is given as an increasing function of applied load.