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2024, Vol. 44 No. 1   Published date: 28 March 2024
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• On the Anti-concentration Functions of Some Familiar Families of Distributions

  Hu Zechun, Song Renming, Tan Yuan

  2024, 44(1): 1-15. doi: 10.3969/j.issn.1006-8074.2024.01.001

  Abstract ( 986 )   PDF (194KB) ( 189 )   　　

  Let $\{X_{\alpha}\}$ be a family of random variables following a certain type of distributions with finite expectation $\E[X_{\alpha}]$

  and finite variance $\Var(X_{\alpha})$, where $\alpha$ is a parameter. Motivated by the recent paper of Hollom and Portier (arXiv: 2306.07811v1), we study the anti-concentration function

  $(0, \infty)\ni y\to \inf_{\alpha}\P\left(|X_{\alpha}-\E[X_{\alpha}]|\geq y \sqrt{\Var(X_{\alpha})}\right)$ and find its explicit expression.

  We show that, for certain familiar families of distributions, including the uniform, exponential, non-degenerate Gaussian and student's $t$-distributions, the anti-concentration function is not identically zero, which means that the corresponding families of random variables

  have some sort of anti-concentration property; while for some other familiar families of distributions, including the binomial, Poisson, negative binomial, hypergeometric, Gamma, Pareto, Weibull, log-normal and Beta distributions, the anti-concentration function is identically zero.

• Reflected Stochastic Partial Differential Equations with Elliptic Operators

  Qian Hongchao, Li Ruizhi, Gui Yewei, Peng Jun

  2024, 44(1): 16-30. doi: 10.3969/j.issn.1006-8074.2024.01.002

  Abstract ( 569 )   PDF (214KB) ( 172 )   　　

  This paper is concerned with a class of

  multi-dimensional reflected stochastic partial differential equations with elliptic operators, whose solutions are constrained on a bounded convex domain.

  The aim of this paper is to establish the existence and uniqueness theorem of solutions for the reflected stochastic partial differential equations with the penalization method.

• Oscillations in a Predator-prey Model with Cooperative Breeding of Predators

  Zhang Yun, Su Juan, Zou Lan

  2024, 44(1): 31-44. doi: 10.3969/j.issn.1006-8074.2024.01.003

  Abstract ( 517 )   PDF (1077KB) ( 149 )   　　

  The behavior of cooperation is common in animals. A lot of work showed that cooperative predation in predator-prey systems will induce complicated dynamics such as complicated coexistence of equilibria and population oscillations. In this paper, we consider a predator-prey system with cooperative breeding of predators, and focus on the oscillations related to the Hopf bifurcations of the model. Firstly, we discuss the number of internal equilibrium and the qualitative properties of internal equilibrium, and show that the equilibrium ~$E_{1}$~ is of center type under certain parameter conditions. Secondly, we analyze the Hopf bifurcation at $E_{1}$, and give the corresponding bifurcation conditions. Finally, we give an example of a weak focus of order 3, and study the complex oscillation behaviors of the model through this example. Because of the coexistence of multiple limit cycles, even under the same parameter conditions, the periods and amplitudes vary with initial values. It also reflects the high sensitivity of solutions on initial values.

• Embeddedness, Compactness and Uniform Convexity of Anisotropic Banach Space Valued Musielak-Orlicz Spaces

  He Yal, Xu Jingshi

  2024, 44(1): 45-61. doi: 10.3969/j.issn.1006-8074.2024.01.004

  Abstract ( 519 )   PDF (185KB) ( 164 )   　　

  In this paper we give the necessary and sufficient conditions for the embedding of an anisotropic Banach space valued Musielak-Orlicz space into another anisotropic Banach space valued Musielak-Orlicz space, and the necessary and sufficient conditions for a subset of an anisotropic Banach space valued Musielak-Orlicz space to be relatively compact, and characterize the uniform convexity of anisotropic Banach space valued Musielak-Orlicz space.

• Inclusion of Musielak-Orlicz Type Spaces

  Liu Kaituo, Yu Qian

  2024, 44(1): 62-77. doi: 10.3969/j.issn.1006-8074.2024.01.005

  Abstract ( 462 )   PDF (220KB) ( 131 )   　　

  In this paper, the inclusion of Musielak-Orlicz spaces is proved, which is a generalization of the inclusion of Orlicz spaces. Due to the fact that a variable exponent space is a special case of the Musielak-Orlicz space, we can obtain the equivalent relations of inclusion conditions between the spaces $L^{p(\cdot)}$ and $L^{q( \cdot)}$. At the same time, we get the inclusion of weighted Orlicz spaces. In addition, we also prove the inclusion of weak Musielak-Orlicz spaces, which is a generalization of weak variable exponent spaces. As applications, the equivalent relations of inclusion conditions between weak variable exponent spaces $wL^{p(\cdot)}$ and $wL^{q(\cdot)}$ and the inclusion of weak Orlicz spaces are established.

• Identifying and Locating Codes on Some  Cayley  Graphs

  Lu Qiming, Song Shujiao

  2024, 44(1): 78-92. doi: 10.3969/j.issn.1006-8074.2024.01.006

  Abstract ( 510 )   PDF (2576KB) ( 156 )   　　

  In 2019, Junnila, Laihonen and Paris studied the identifying codes and locating codes on the circulant graphs $C_{n}(1,d)$, $C_{n}(1,d-1,d)$ and $C_{n}(1,d-1,d,d+1)$. In this paper we study the identifying codes and locating codes of Cayley graphs on the Abelian groups of order $p^{2}$ and order $2n$ within 8 degrees, determine their optimal bounds, and give some examples that reach the optimal bounds. Our results generalize multiple results on identifying and locating codes.

• The Transition Probability and Instantaneous Distribution of the Output Process of  M/M/1  Queueing System

  Li Junping, Cheng Lan

  2024, 44(1): 93-108. doi: 10.3969/j.issn.1006-8074.2024.01.007

  Abstract ( 488 )   PDF (209KB) ( 132 )   　　

  In this paper, we study the output process of M/M/1 queueing system, obtain its transition probability at arrival times and instantaneous distribution at any time.

• The Gerschgorin Disk Theorem and Regularity Conditions for Complex Interval Matrices

  Cheng Long, Xia Dandan, Li Yaotang

  2024, 44(1): 109-121. doi: 10.3969/j.issn.1006-8074.2024.01.008

  Abstract ( 521 )   PDF (252KB) ( 205 )   　　

  In this paper, the Gerschgorin disk theorem on eigenvalues of complex matrices is generalized to complex interval matrices, in which the Gerschgorin disk regions of eigenvalues of complex interval matrices are presented. It is showed that the Gerschgorin disk regions are contained in the Gerschgorin square regions for eigenvalues of complex interval matrices. Then, two new sufficient conditions for the regularity of complex interval matrices are obtained by applying the Gerschgorin disk theorem of complex interval matrices.