Tight Lower Bounds on the Size of a Maximum Matching in a Regular Graph

In this paper we study tight lower bounds on the size of a maximum matching in a regular graph. For k ≥3, let G be a connected k-regular graph of order n and let α′(G) be the size of a maximum matching in G. We show that if k is even, then , while if k is odd, then . We show that both bounds are tight. Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.     

Confidence intervals and ellipsoids for estimable functions and estimable vectors in models with orthogonal block structure

Sub-models of mixed linear models are considered. The independence of these sub-models leads to sufficient statistics for the parameters relevant for their densities. Using pivot variables, confidence regions are obtained as well hypothesis testing for variance components, estimable functions, and estimable vectors. In addition, to compare the estimators and the models, we present the histograms with the empirical joint densities for positive and negative parts of the estimators. The figures, for the two-dimensional charts, contain the corresponding UMVUE and are all unimodal with the UMVUE near the mode. The nearness of the estimators and the modes validates the presented methodology and allows the safe use of induced densities. A numerical example applied to real data is presented.     

K3 surfaces,Picard numbers and Siegel disks

If a K3 surface admits an automorphism with a Siegel disk, then its Picard number is an even integer between 0 and 18. Conversely, using the method of hypergeometric groups, we are able to construct K3 surface automorphisms with Siegel disks that realize all possible Picard numbers. The constructions involve extensive computer searches for appropriate Salem numbers and computations of algebraic numbers arising from holomorphic Lefschetz-type fixed point formulas and related Grothendieck residues.     

Mixed radial-angular integrability for rough maximal singular integrals and Marcinkiewicz integrals with mixed homogeneity

This paper is devoted to studying maximal singular integrals and Marcinkiewicz integrals with rough kernels in a mixed homogeneity setting. Assuming that the kernels satisfy certain rather weak size conditions, the boundedness of such operators on the mixed radial-angular spaces are established, respectively. Meanwhile, the corresponding vector-valued versions are also given.     

A Low Order Nonconforming Mixed Finite Element Method for Non-Stationary Incompressible Magnetohydrodynamics System

A low order nonconforming mixed finite element method (FEM) is established for the fully coupled non-stationary incompressible magnetohydrodynamics (MHD) problem in a bounded domain in 3D. The lowest order finite elements on tetrahedra or hexahedra are chosen to approximate the pressure, the velocity field and the magnetic field, in which the hydrodynamic unknowns are approximated by inf-sup stable finite element pairs and the magnetic field by $H^1(\Omega)$-conforming finite elements, respectively. The existence and uniqueness of the approximate solutions are shown. Optimal order error estimates of $L^2(H^1)$-norm for the velocity field, $L^2(L^2)$-norm for the pressure and the broken $L^2(H^1)$-norm for the magnetic field are derived.     

Finite Integral Transform Solutions for Free Vibrations of Rectangular Thin Plates With Mixed Boundary Constraints北大核心CSCD

Analytical solutions, with unique research value, can serve as benchmarks for empirical formulas and numerical methods, a tool for rapid parameter analysis and optimization, and a theoretical basis for experimental designs. Conventional analytical methods, e.g., the Lévy solution method, are only applicable to mechanical problems of plates and shells with opposite simply-supported edges, which, however, may fail to obtain analytical solutions for the issues with complex boundary constraints. In recent years, the finite integral transform method for plate and shell problems was developed to deal with non-Lévy-type plates and shells, but it is still infeasible to solve the mixed boundary constrains-induced complex boundary value problems of higher-order partial differential equations. Herein, for the first time, the finite integral transform method was combined with the sub-domain decomposition technique to solve the free vibrations of rectangular thin plates with mixed boundary constraints. The rectangular plate was first divided into 2 sub-domains according to the mixed boundary constraints, and the 2 sub-domains were solved analytically with the finite integral transform method. Finally, the continuity conditions were introduced to obtain the analytical solution of the original problem. Based on the side spot-welded cantilever plates commonly used in engineering, the free vibration problem of a rectangular thin plate with 1 edge subjected to clamped-simply supported constraints and the other 3 edges free, was analyzed. The obtained natural frequencies and mode shapes are in good agreement with those from the finite element method as well as the solutions in literature, thus verifying the accuracy of the proposed method. The solution procedure of the finite integral transform method can be implemented based on the governing equations without any assumption of the solution form. Therefore, this strict analytical method is widely applicable to complex boundary value problems of higher-order partial differential equations for such mechanical problems of plates and shells. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Effect of atmospheric pressure pulsed discharge spatial inhomogeneity on Ar I and H I lines: Electron number density study

The spatial inhomogeneity of pulsed atmospheric pressure discharge in argon is investigated using the electron number density N_e diagnostics procedure applied to asymmetrically broadened Ar I lines. A dedicated fitting procedure is used for describing Ar I 703.0 nm line shape recorded from argon gas discharge and H I (at 486.13 and 656.28 nm) lines recorded from Ar-H₂ gas mixture discharge. The results revealed the change in N_e in both axial and radial directions. The additional Ar I lines at 614.5, 710.7, 731.2, and 731.6 nm, recorded from integral spatial radiation, are analysed as well to confirm the results from the plasma column region. The possibility of using AlO (B²∑⁺–X²∑⁺) and CN (B²∑⁺–X²∑⁺) molecular bands for gas temperature T_g measurements in this type of gas discharge source is demonstrated and T_g used as an input parameter for the N_e diagnostics procedure. For the proper identification of molecular band spectral lines, the Fortrat parabolas are constructed. The results obtained from Ar I 703.0 nm line indicate three different N_e values, with N_e1 ≈ 0.6 × 10¹⁶ cm⁻³, N_e2 ≈ 3.6 × 10¹⁶ cm⁻³, and N_e3 ≈ 19 × 10¹⁶ cm⁻³ measured from the plasma column. These N_e values increase in the cathode and anode region.     

图的邻点全和可区别全染色

设f:V(G)∪E(G)→{1,2,…,k}是图G的一个正常k-全染色。令■其中N(x)={y∈V(G)|xy∈E(G)}。对任意的边uv∈E(C),若有Φ(u)≠Φ(v)成立,则称f是图G的一个邻点全和可区别k-全染色。图G的邻点全和可区别全染色中最小的颜色数k叫做G的邻点全和可区别全色数,记为f tndi∑(G)。本文确定了路、圈、星、轮、完全二部图、完全图以及树的邻点全和可区别全色数,同时猜想:简单图G(≠K2)的邻点全和可区别全色数不超过△(G)+2。     

符号圈拼接图的零度

令$\eta(\Gamma)$和$c(\Gamma)$是符号图$\Gamma$的零度和基本圈数. 一个符号圈拼接图是指每个块都是圈的连通符号图. 本文证明了对任意符号拼接图$\eta(\Gamma)\le c(\Gamma)+1$成立, 并且刻画了等号成立的极图, 推广了王登银等人(2022)在简单圈拼接图上的结果. 此外, 我们证明了任意的符号拼接图$\eta(\Gamma)\neq c(\Gamma)$, 给出了满足$\eta(\Gamma)=c(\Gamma)-1$的符号拼接图的一些性质并刻画处$\eta(\Gamma)=c(\Gamma)-1$的二部符号拼接图.     

Semiconductor Emitters in Entropy Sources for Quantum Random Number Generation

Random number generation (RNG) is needed for a myriad of applications ranging from secure communication encryption to numerical simulations to sports and games. However, generating truly random numbers can be elusive. Pseudorandom bit generation using computer algorithms provides a high random bit generation rate. Nevertheless, the reliance on predefined algorithms makes it deterministic and predictable once initial conditions are known. Relying on physical phenomena (such as measuring electrical noise or even rolling dice) can achieve a less predictable sequence of bits. Furthermore, if the physical phenomena originate from quantum effects, they can be truly random and completely unpredictable due to quantum indeterminacy. Traditionally, physical RNG is significantly slower than pseudorandom techniques. To meet the demand for high-speed RNG with perfect unpredictability, semiconductor light sources are adopted as parts of the sources of randomness, i.e., entropy sources, in quantum RNG (QRNG) systems. The high speed of their noise, the high efficiency, and the small scale of these devices make them ideal for chip-scale QRNG. Here, the applications and recent advances of QRNG are reviewed using semiconductor emitters. Finally, the performance of these emitters is compared and discuss their potential in future technologies.