Two critical times for the SIR model

We consider the SIR model and study the first time the number of infected individuals begins to decrease and the first time this population is below a given threshold. We interpret these times as functions of the initial susceptible and infected populations and characterize them as solutions of a certain partial differential equation. This allows us to obtain integral representations of these times and in turn to estimate them precisely for large populations.     

Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds

In this paper we establish an SIR model with a standard incidence rate and a nonlinear recovery rate, formulated to consider the impact of available resource of the public health system especially the number of hospital beds. For the three dimensional model with total population regulated by both demographics and diseases incidence, we prove that the model can undergo backward bifurcation, saddle-node bifurcation, Hopf bifurcation and cusp type of Bogdanov–Takens bifurcation of codimension 3. We present the bifurcation diagram near the cusp type of Bogdanov–Takens bifurcation point of codimension 3 and give epidemiological interpretation of the complex dynamical behaviors of endemic due to the variation of the number of hospital beds. This study suggests that maintaining enough number of hospital beds is crucial for the control of the infectious diseases.     

A graphical approach to computing Selmer groups of congruent number curves

Let E _n:y ²=x ³−n ² x denote the family of congruent number elliptic curves. Feng and Xiong (2004) equate the nontriviality of the Selmer groups associated with E _n to the presence of certain types of partitions of graphs associated with the prime factorization of n. In this paper, we extend the ideas of Feng and Xiong in order to compute the Selmer groups of E _n. 2000 Mathematics Subject Classification Primary—11G05; Secondary—14H52, 14H25, 05C90     

可图序列偏序集中极大元的个数

本文确定了某些可图序列偏序集中极大元的个数及其生成函数．     

Stability approach to selecting the number of principal components

Principal component analysis (PCA) is a canonical tool that reduces data dimensionality by finding linear transformations that project the data into a lower dimensional subspace while preserving the variability of the data. Selecting the number of principal components (PC) is essential but challenging for PCA since it represents an unsupervised learning problem without a clear target label at the sample level. In this article, we propose a new method to determine the optimal number of PCs based on the stability of the space spanned by PCs. A series of analyses with both synthetic data and real data demonstrates the superior performance of the proposed method.     

On the dimension of solution spaces of a noncommutative sigma model in the case of uniton number 2

This is primarily an overview article on some results and problems involving the classical Hardy function Z(t):= ζ(1/2 + it)(χ(1/2 + it))^?1/2, ζ(s) = χ(s)ζ(1 ? s). In particular, we discuss the first and third moments of Z(t) (with and without shifts) and the distribution of its positive and negative values. A new result involving the distribution of its values is presented.     

A graphical procedure for an on-condition maintenance policy: imperfect-inspection model and interpretation

This paper considers on-condition maintenance based on periodicinspection and control of a condition parameter which describesthe wear and deterioration of the productive equipment. We developa model based on delay time and imperfect inspection, and builda graphical procedure to choose the best inspection intervalfor different criteria. The interpretation of this graphicalprocedure allows us to emphasize the factors that are relevantfor inspection decision-making.     

A clone-based graphical modeler and mathematical model generator for optimal production planning in process industries

This paper outlines a visually interactive graphical modeling approach for process type production systems, with hidden generation of complex optimization models for production planning. The proposed system lets the users build a graphical model of the production system with one-to-one clones of its production units through its interactive visual interface, accepts production-specific data for its components, and finally, internally generates and solves its mathematical programming model without any interaction from the user. This “clone-based” modeling approach allows the continued use of optimization models with minimal mathematical programming understanding, as generation of mathematical model by clones is hidden and automatic, therefore maintenance-free: Updating graphical production system models is enough for renewing internal optimization models. The concept is demonstrated in this paper with a linear programming prototype developed for a petroleum refinery.     

A comparison between the metric dimension and zero forcing number of trees and unicyclic graphs

The metric dimension dim(G)of a graph G is the minimum number of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices.The zero forcing number Z(G)of a graph G is the minimum cardinality of a set S of black vertices(whereas vertices in V(G)\S are colored white)such that V(G)is turned black after finitely many applications of"the color-change rule":a white vertex is converted black if it is the only white neighbor of a black vertex.We show that dim(T)≤Z(T)for a tree T,and that dim(G)≤Z(G)+1 if G is a unicyclic graph;along the way,we characterize trees T attaining dim(T)=Z(T).For a general graph G,we introduce the"cycle rank conjecture".We conclude with a proof of dim(T)-2≤dim(T+e)≤dim(T)+1 for e∈E(T).     

心理状态数的估计方法在选拔选手中的应用

本文通过使用心理状态数的极大似然估计方法，对参赛学生进行选拔，保证有真正实力的选手入选，充分体现公正性原则．     

Cosmology in one dimension: A two-component model

We investigate structure formation in a one dimensional model of a matter-dominated universe using a quasi-Newtonian formulation. In addition to dissipation-free dark matter, dissipative luminous matter is introduced to examine the potential bias in the distributions. We use multifractal analysis techniques to identify scale-dependent structures, including clusters and voids. Both dark matter and luminous matter exhibit multifractal geometry over a finite range as the universe evolves in time. We present the results for the generalized dimensions computed on various scales for each matter distribution which clearly supports the bottom-up structure formation scenario. We compare and contrast the multifractal dimensions of two types of matter for the first time and show how dynamical considerations cause them to differ.     

A delay SIR epidemic model with pulse vaccination and incubation times

In this paper, a new delay SIR epidemic model with pulse vaccination and incubation times is considered. We obtain an infection-free semi-trivial periodic solution and establish the sufficient conditions for the global attractivity of the semi-trivial periodic solution. By use of new computational techniques for impulsive differential equations with delay, we prove that the system is permanent under appropriate conditions. The results show that time delay, pulse vaccination and nonlinear incidence have significant effects on the dynamics behaviors of the model. Our results are illustrated and corroborated with some numerical experiments.     

Global dynamics and bifurcation in delayed SIR epidemic model

An SIR epidemic model with time delay, information variable and saturated incidence rate, where the susceptibles are assumed to satisfy the logistic equation and the incidence term, is of saturated form with the susceptibles. This model exhibits two bifurcations, one is transcritical bifurcation and the other is Hopf bifurcation. The local and global stability of endemic equilibrium is also discussed. Finally, numerical simulations are carried out to explain the mathematical conclusions.     

Estimates for the fractal dimension and the number of determining modes for invariant sets of dynamical systems

Majorants of the fractal dimension and of the number of determining modes for unbounded sets, invariant with respect to operators of semigroups of classes 1 and 2, are obtained. They are computed for the Navier-Stokes equations (two- and three-dimensional) under the first boundary condition and under periodicity conditions in the spaces and .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 105–129, 1987.     

A model structure approach to the finitistic dimension conjectures

We explore the interlacing between model category structures attained to classes of modules of finite $\mathcal {X}$‐dimension, for certain classes of modules $\mathcal {X}$. As an application we give a model structure approach to the finitistic dimension conjectures and present a new conceptual framework in which these conjectures can be studied.     

On selecting a process with the smallest number of unfortunate events

Managers often desire to assign resources to minimize balking in service systems. Discrete event simulations are often used to study alternative assignments. When the distribution of the number of balking events in a business day is approximated by a Poisson distribution, the objective becomes that of selecting a population corresponding to the smallest mean number of unfortunate events. A procedure for selecting a Poisson population with the smallest mean is described in which the selection is carried out based on a random sample of size n from each population. Examples of a bank lobby and manufacturing process are used to illustrate this procedure.     

Chromatic number of Hasse diagrams,eyebrows and dimension

We construct posets of dimension 2 with highly chromatic Hasse diagrams. This solves a previous problem by Nesetril and Trotter.     

A low-Mach number model for time-harmonic acoustics in arbitrary flows

This paper concerns the finite element simulation of the diffraction of a time-harmonic acoustic wave in the presence of an arbitrary mean flow. Considering the equation for the perturbation of displacement (due to Galbrun), we derive a low-Mach number formulation of the problem which is proved to be of Fredholm type and is therefore well suited for discretization by classical Lagrange finite elements. Numerical experiments are done in the case of a potential flow for which an exact approach is available, and a good agreement is observed.