High-dimensional sample correlation matrices are a crucial class of random matrices in multivariate statistical analysis. The central limit theorem (CLT) provides a theoretical foundation for statistical inference. In this paper, assuming that the data dimension increases proportionally with the sample size, we derive the limiting spectral distribution of the matrix $\widehat{\mathbf{R}}_n\mathbf{M}$ and establish the CLTs for the linear spectral statistics (LSS) of $\widehat{\mathbf{R}}_n\mathbf{M}$ in two structures: linear independent component structure and elliptical structure. In contrast to existing literature, our proposed spectral properties do not require $\mathbf{M}$ to be an identity matrix. Moreover, we also derive the joint limiting distribution of LSSs of $\widehat{\mathbf{R}}_n \mathbf{M}_1,\ldots,\widehat{\mathbf{R}}_n \mathbf{M}_K$. As an illustration, an application is given for the CLT.
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We explore a linear inhomogeneous elasticity equation with random Lam\'e parameters. The latter are parameterized by a countably infinite number of terms in separated expansions. The main aim of this work is to estimate expected values (considered as an infinite dimensional integral on the parametric space corresponding to the random coefficients) of linear functionals acting on the solution of the elasticity equation. To achieve this, the expansions of the random parameters are truncated, a high-order quasi-Monte Carlo (QMC) is combined with a sparse grid approach to approximate the high dimensional integral, and a Galerkin finite element method (FEM) is introduced to approximate the solution of the elasticity equation over the physical domain. The error estimates from (1) truncating the infinite expansion, (2) the Galerkin FEM, and (3) the QMC sparse grid quadrature rule are all studied. For this purpose, we show certain required regularity properties of the continuous solution with respect to both the parametric and physical variables. To achieve our theoretical regularity and convergence results, some reasonable assumptions on the expansions of the random coefficients are imposed. Finally, some numerical results are delivered.
The aim of this study is to establish a general transformation matrix between B-spline surfaces and ANCF surface elements. This study is a further study of the conversion between the ANCF and B-spline surfaces. In this paper, a general transformation matrix between the Bezier surfaces and ANCF surface element is established. This general transformation matrix essentially describes the linear relationship between ANCF and Bezier surfaces. Moreover, the general transformation matrix can help to improve the efficiency of the process to transfer the distorted configuration in the CAA back to the CAD, an urgent requirement in engineering practice. In addition, a special Bezier surface control polygon is given in this study. The Bezier surface described with this control polygon can be converted to an ANCF surface element with fewer d.o.f.. And the converted ANCF surface element with 36 d.o.f. was once addressed by Dufva and Shabana. So the special control polygon can be regarded as the geometric condition in conversion to an ANCF surface element with 36 d.o.f. Based on the fact that a B-spline surface can be seen as a set of Bezier surfaces connected together, the method to establish a general transformation matrix between the ANCF and lower-order B-spline surfaces is given. Specially, the general transformation is not in a recursive form, but in a simplified form.
Neutron noise analysis is a predominant technique for fissile matter identification with passive methods. Quantifying the uncertainties associated with the estimated nuclear parameters is crucial for decision-making. A conservative uncertainty quantification procedure is possible by solving a Bayesian inverse problem with the help of statistical surrogate models but generally leads to large uncertainties due to the surrogate models' errors. In this work, we develop two methods for robust uncertainty quantification in neutron and gamma noise analysis based on the resolution of Bayesian inverse problems. We show that the uncertainties can be reduced by including information on gamma correlations. The investigation of a joint analysis of the neutron and gamma observations is also conducted with the help of active learning strategies to fine-tune surrogate models. We test our methods on a model of the SILENE reactor core, using simulated and real-world measurements.
We propose a fast scheme for approximating the Mittag-Leffler function by an efficient sum-of-exponentials (SOE), and apply the scheme to the viscoelastic model of wave propagation with mixed finite element methods for the spatial discretization and the Newmark-beta scheme for the second-order temporal derivative. Compared with traditional L1 scheme for fractional derivative, our fast scheme reduces the memory complexity from $\mathcal O(N_sN) $ to $\mathcal O(N_sN_{exp})$ and the computation complexity from $\mathcal O(N_sN^2)$ to $\mathcal O(N_sN_{exp}N)$, where $N$ denotes the total number of temporal grid points, $N_{exp}$ is the number of exponentials in SOE, and $N_s$ represents the complexity of memory and computation related to the spatial discretization. Numerical experiments are provided to verify the theoretical results.
In the analysis of spatially resolved transcriptomics data, detecting spatially variable genes (SVGs) is crucial. Numerous computational methods exist, but varying SVG definitions and methodologies lead to incomparable results. We review \rv{33} state-of-the-art methods, categorizing SVGs into three types: overall, cell-type-specific, and spatial-domain-marker SVGs. Our review explains the intuitions underlying these methods, summarizes their applications, and categorizes the hypothesis tests they use in the trade-off between generality and specificity for SVG detection. We discuss challenges in SVG detection and propose future directions for improvement. Our review offers insights for method developers and users, advocating for category-specific benchmarking.
The fundamental functional summary statistics used for studying spatial point patterns are developed for marked homogeneous and inhomogeneous point processes on the surface of a sphere. These are extended to point processes on the surface of three dimensional convex shapes given the bijective mapping from the shape to the sphere is known. These functional summary statistics are used to test for independence between the marginals of multi-type spatial point processes with methods for sampling the null distribution proposed and discussed. This is illustrated on both simulated data and the RNGC galaxy point pattern, revealing attractive dependencies between different galaxy types.
We construct an interpolatory high-order cubature rule to compute integrals of smooth functions over self-affine sets with respect to an invariant measure. The main difficulty is the computation of the cubature weights, which we characterize algebraically, by exploiting a self-similarity property of the integral. We propose an $h$-version and a $p$-version of the cubature, present an error analysis and conduct numerical experiments.
We study combinatorial properties of plateaued functions. All quadratic functions, bent functions and most known APN functions are plateaued, so many cryptographic primitives rely on plateaued functions as building blocks. The main focus of our study is the interplay of the Walsh transform and linearity of a plateaued function, its differential properties, and their value distributions, i.e., the sizes of image and preimage sets. In particular, we study the special case of ``almost balanced'' plateaued functions, which only have two nonzero preimage set sizes, generalizing for instance all monomial functions. We achieve several direct connections and (non)existence conditions for these functions, showing for instance that plateaued $d$-to-$1$ functions (and thus plateaued monomials) only exist for a very select choice of $d$, and we derive for all these functions their linearity as well as bounds on their differential uniformity. We also specifically study the Walsh transform of plateaued APN functions and their relation to their value distribution.
We discuss the second-order differential uniformity of vectorial Boolean functions. The closely related notion of second-order zero differential uniformity has recently been studied in connection to resistance to the boomerang attack. We prove that monomial functions with univariate form $x^d$ where $d=2^{2k}+2^k+1$ and $\gcd(k,n)=1$ have optimal second-order differential uniformity. Computational results suggest that, up to affine equivalence, these might be the only optimal cubic power functions. We begin work towards generalising such conditions to all monomial functions of algebraic degree 3. We also discuss further questions arising from computational results.
Finding eigenvalue distributions for a number of sparse random matrix ensembles can be reduced to solving nonlinear integral equations of the Hammerstein type. While a systematic mathematical theory of such equations exists, it has not been previously applied to sparse matrix problems. We close this gap in the literature by showing how one can employ numerical solutions of Hammerstein equations to accurately recover the spectra of adjacency matrices and Laplacians of random graphs. While our treatment focuses on random graphs for concreteness, the methodology has broad applications to more general sparse random matrices.
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