Convergence of densities of some functionals of Gaussian processes

The aim of this paper is to establish the uniform convergence of the densities of a sequence of random variables, which are functionals of an underlying Gaussian process, to a normal density. Precise estimates for the uniform distance are derived by using the techniques of Malliavin calculus, combined with Stein?s method for normal approximation. We need to assume some non-degeneracy conditions. First, the study is focused on random variables in a fixed Wiener chaos, and later, the results are extended to the uniform convergence of the derivatives of the densities and to the case of random vectors in some fixed chaos, which are uniformly non-degenerate in the sense of Malliavin calculus. Explicit upper bounds for the uniform norm are obtained for random variables in the second Wiener chaos, and an application to the convergence of densities of the least square estimator for the drift parameter in Ornstein–Uhlenbeck processes is discussed.     

Origin of the division of labour and a stochastic mechanism of differentiation

Nonlinear models are introduced to describe the nonequilibrium dynamics of social evolution. The difference between Western and Oriental culture, and their roles in the origin in the division of labour, are described by a behavioural model in information diffusion and learning competition. It shows a tradeoff between stability and diversity. The stochastic mechanism of social differentiation and the empirical evidence for this is discussed in a stochastic model of multi-staged development. It shows the breakdown of the Gaussian distribution during a transition. Finally, an ideal model of social bifurcation is given.     

Analysis of the Stress State of the Surface Layer of a Polymeric Mass upon Filling of Bulky Compression Molds

The stress state of the surface layer of a polymeric mass during filling of bulky compression molds is analyzed. It is shown that, at particular rheological characteristics of the mass, temperature, and filling rates, cracking of the surface layer occurs, which leads to defects in the finished products. A physical analysis of this process makes it possible to conclude that the cracks arise due to the normal stresses operating in the front region of the moving polymeric mass. It is found that, under certain flow conditions, areas with a pressure lower than the atmospheric one appear on the surface of the polymer. If the tensile stresses arising in these local regions are higher than the tensile strength of the mass, the continuity of the composition is broken in the direction determined by the greatest rate of the normal deformation. To confirm the reliability of the crack-formation mechanism proposed, the distribution of the pressure and normal stresses over the free surface is calculated based on a numerical method. These calculations show that, by comparing the stress level achieved in the front region with the tensile-strength characteristics of the polymeric composition, it is possible to predict, with a sufficient accuracy, the possibility of crack formation in the surface layer of such a mass under given flow conditions and thus to solve the question on flawless manufacturing of products.     

The evolution of the concept of homeomorphism

Topology, or analysis situs, has often been regarded as the study of those properties of point sets (in Euclidean space or in abstract spaces) that are invariant under “homeomorphisms.” Besides the modern concept of homeomorphism, at least three other concepts were used in this context during the late 19th and early 20th centuries, and regarded (by various mathematicians) as characterizing topology: deformations, diffeomorphisms, and continuous bijections. Poincaré, in particular, characterized analysis situs in terms of deformations in 1892 but in terms of diffeomorphisms in 1895. Eventually Kuratowski showed in 1921 that in the plane there can be a continuous bijection of P onto Q, and of Q onto P, without P and Q being homeomorphic.     

模糊层次分析法在矩阵论教材评价方面的应用

基础学科的教材直接影响学生的基本功,尤其是几乎每个学科都会涉及的数学类教材,如矩阵论.矩阵论是研究生的基础课程,在对学生以后的学术道路有举足轻重的作用.所以选择一本合适的教材,对学生和教师来说都有不小的帮助.然而,对教材的评价而言,不能单单从一个方面入手,因此将模糊综合分析法与层次分析法结合,在矩阵论教材的评价方面建立评价体系,为高校选择合适的教材提供依据.     

Effect of viscosity of a thermoplastic prepreg and matrix upon winding of rings

The problem of compression of a unidirectional layer and shear of a polymer interlayer during winding of rings is considered. The equations determining the dependence of the layer thickness and stresses on the parameters entering into the power flow law for a prepreg and polymer matrix and on the basic parameters of the winding process—the initial tension of the prepreg, its placement rate, and the radius of a mandrel—are derived. The ring thickness measurements obtained at various temperatures and initial tension forces of plies confirm the adequacy of the model offered. It is found that the viscous properties of the prepreg and matrix upon winding affect the relative change in the layer thickness to a greater extent than the stresses in these layers. With increase in temperature and tension force upon winding, the effect of viscous deformations of the prepreg and matrix increases. A decrease in viscosity and an increase in the tension force of the tape lead to a higher strength of the ring in tension and interlaminar shear; however, the growing percolation of the polymer melt leads to a greater inhomogeneity of the structure of the composite in the ring and to a lower reinforcing effect of the factors mentioned. Presented at the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 3, pp. 419–428, 2000.     

Simulation of the effect of learning on the topology of the functional connectivity of neural networks

We introduce a procedure for simulating adaptive learning in neural networks and the effect this learning has on the way in which the functional connections between the nodes of the network are established. The procedure combines two mechanisms: firstly, the gradual dilution of the network through the elimination of synaptic weights in increasing order of magnitude, thus reducing the costs of the network structure. Secondly, to train the network as it is diluted so as not to compromise its performance pursuant to the proposed task. Considering different levels of learning difficulty, we compare the topology of the functional connectivities that result from the application of this procedure with those obtained using fMRI in healthy volunteers. According to our results, the topology of functional connectivities in healthy subjects can be interpreted as the product of a learning process with a specific degree of difficulty.     

Robust counterparts of inequalities containing sums of maxima of linear functions

This paper addresses the robust counterparts of optimization problems containing sums of maxima of linear functions. These problems include many practical problems, e.g. problems with sums of absolute values, and arise when taking the robust counterpart of a linear inequality that is affine in the decision variables, affine in a parameter with box uncertainty, and affine in a parameter with general uncertainty.     

Variational analysis of functions of the roots of polynomials

The Gauss-Lucas Theorem on the roots of polynomials nicely simplifies the computation of the subderivative and regular subdifferential of the abscissa mapping on polynomials (the maximum of the real parts of the roots). This paper extends this approach to more general functions of the roots. By combining the Gauss-Lucas methodology with an analysis of the splitting behavior of the roots, we obtain characterizations of the subderivative and regular subdifferential for these functions as well. In particular, we completely characterize the subderivative and regular subdifferential of the radius mapping (the maximum of the moduli of the roots). The abscissa and radius mappings are important for the study of continuous and discrete time linear dynamical systems. Dedicated to R. Tyrrell Rockafellar on the occasion of his 70th birthday. Terry is one of those rare individuals who combine a broad vision, deep insight, and the outstanding writing and lecturing skills crucial for engaging others in his subject. With these qualities he has won universal respect as a founding father of our discipline. We, and the broader mathematical community, owe Terry a great deal. But most of all we are personally thankful to Terry for his friendship and guidance. Research supported in part by the National Science Foundation Grant DMS-0203175. Research supported in part by the Natural Sciences and Engineering Research Council of Canada. Research supported in part by the National Science Foundation Grant DMS-0412049.     

Sensitivity of solutions of linear DAE to perturbations of the system matrices

This paper studies the effect of perturbations in the system matrices of linear Differential Algebraic Equations (DAE) onto the solutions. It turns out that these may result in a more complicated perturbation pattern for higher index problems than in the case for (standard) additive perturbations. We give an analysis here for linear index-1 and index-2 problems, which, however, has clear ramifications in nonlinear problems (e.g., via the Newton process). This analysis is sustained by a number of examples. This revised version was published online in June 2006 with corrections to the Cover Date.     

Unitarizability of a certain class of irreducible representations of classical groups

We prove that a certain class of irreducible representations of the classical p-adic groups is unitarizable and in general, can be isolated in the unitary dual. These representations are Aubert duals of a certain class of square-integrable representations, thus, in this case, Bernstein’s conjecture, which states that the Aubert involution preserves unitarizability, is confirmed.     

Modelling and analysis of dynamics of viral infection of cells and of interferon resistance

Interferons are active biomolecules, which help fight viral infections by spreading from infected to uninfected cells and activate effector molecules, which confer resistance from the virus on cells. We propose a new model of dynamics of viral infection, including endocytosis, cell death, production of interferon and development of resistance. The novel element is a specific biologically justified mechanism of interferon action, which results in dynamics different from other infection models. The model reflects conditions prevailing in liquid cultures (ideal mixing), and the absence of cells or virus influx from outside. The basic model is a nonlinear system of five ordinary differential equations. For this variant, it is possible to characterise global behaviour, using a conservation law. Analytic results are supplemented by computational studies. The second variant of the model includes age-of-infection structure of infected cells, which is described by a transport-type partial differential equation for infected cells. The conclusions are: (i) If virus mortality is included, the virus becomes eventually extinct and subpopulations of uninfected and resistant cells are established. (ii) If virus mortality is not included, the dynamics may lead to extinction of uninfected cells. (iii) Switching off the interferon defense results in a decrease of the sum total of uninfected and resistant cells. (iv) Infection-age structure of infected cells may result in stabilisation or destabilisation of the system, depending on detailed assumptions. Our work seems to constitute the first comprehensive mathematical analysis of the cell-virus-interferon system based on biologically plausible hypotheses.     

L－fuzzy集网的R－收敛性质

本文在ＬＦ拓扑空间中建立了Ｌ－ｆｕｚｚｙ集网的弱收敛（Ｒ－收敛）概念，应用文［４］中的Ｒ－闭包，系统讨论了它们的性质，证明了等式ＲｌｉｍＡ＿ｎ＝∧（∨Ａ＿ｍ）＿Ｒ和ＲｌｉｍＡ＿ｎ＝Ａ＿ｎ＝∧（∨Ａ＿ｍ）＿Ｒ并且给出了Ｌ－ｆｕｚｚｙ集网与其子网之间的关系。     

Colength of varieties of linear algebras

In this paper, an example of a variety of nonassociative algebras is constructed in which the system of Capelli identities of small rank is satisfied, whereas the colength function has exponential growth. As is well known, in the associative case and in the Lie case, the validity of Capelli identities implies the polynomial growth of the colength.     

Choice of the best alternative in case of a continuous set of states of nature-application of fuzzy numbers

On the basis of a known application of an order weighted averaging operator to the decision making in the case of a discrete set of states of nature, a general approach to the case of a continuous set of states of nature is proposed. The general approach encompasses various types of attitudes of the decision maker, expressed in the form of fuzzy numbers.     

Convergence of random series and the rate of convergence of the strong law of large numbers in game-theoretic probability

We give a unified treatment of the convergence of random series and the rate of convergence of the strong law of large numbers in the framework of game-theoretic probability of Shafer and Vovk (2001) [24]. We consider games with the quadratic hedge as well as more general weaker hedges. The latter corresponds to the existence of an absolute moment of order smaller than 2 in the measure-theoretic framework. We prove some precise relations between the convergence of centered random series and the convergence of the series of prices of the hedges. When interpreted in the measure-theoretic framework, these results characterize the convergence of a martingale in terms of the convergence of the series of conditional absolute moments. In order to prove these results we derive some fundamental results on deterministic strategies of Reality, who is a player in a protocol of game-theoretic probability. It is of particular interest, since Reality’s strategies do not have any counterparts in the measure-theoretic framework, ant yet they can be used to prove results which can be interpreted in the measure-theoretic framework.     

Student achievement effects of technology-supported remediation of understanding of fractions

Students have difficulty learning fractions, and problems in understanding fractions persist into adulthood, with moderate to severe consequences for everyday and occupational decision-making. Remediation of student misconceptions is hampered by deficiencies in teachers’ knowledge of the discipline and pedagogical content knowledge. We theorized that a technology resource could provide the sequencing and scaffolding that teachers might have difficulty providing. Five sets of learning objects, called CLIPS (Critical Learning Instructional Paths Supports), were developed to provide remediation on fraction concepts. In this article, we describe one stage in a research program to develop, implement and evaluate CLIPS. Two studies were conducted. In Study 1, 14 grade 7–10 classrooms were randomly assigned, within schools, to early and late treatment conditions. A pre-post, delayed treatment design found that CLIPS had no effect on achievement for the Early Treatment group due to unforeseen implementation problems. These hardware and software issues were mitigated in the late treatment in which CLIPS contributed to student achievement (Cohen's d = 0.30). Study 2 was a pre-post, single group replication involving 18 grade 7 classrooms. The independent variable was the number of CLIPS completed. Completion of all five CLIPS contributed to higher student achievement: Cohen's d = 0.53, compared to students who completed none (d = 0.00) or 1–4 CLIPS (d = 0.02). The two studies indicate that a research-based set of learning objects is effective when the full program is implemented. Incomplete sequences deprive students of instruction in one or more constructs linked to other key ideas in the conceptual map and reduce the amount of practice required to remediate student misconceptions.     

Application of Gray codes to the study of the theory of symbolic dynamics of unimodal maps

In this paper we provide a closed mathematical formulation of our previous results in the field of symbolic dynamics of unimodal maps. This being the case, we discuss the classical theory of applied symbolic dynamics for unimodal maps and its reinterpretation using Gray codes. This connection was previously emphasized but no explicit mathematical proof was provided. The work described in this paper not only contributes to the integration of the different interpretations of symbolic dynamics of unimodal maps, it also points out some inaccuracies that exist in previous works.     

Probability distribution of the number of initiated cells of carcinogenesis under prevention

In this paper, I have shown that under some mild conditions, the number of initiated cells in an extended two-stage model of carcinogenesis can be approximated by a diffusion process. By using this approximation, I have derived the probability distribution for the number of initiated cells in terms of Laguerre polynomials under normal prevention conditions. This follows from the fact that many of the dietary components are antioxidants which would neutralize the hydroxyl free radicals and hence, reduce the proliferation rates of initiated cells to interrupt or slow down the promotion stage in carcinogenesis.