Proof of a Lin-Peng-Toh's conjecture on an Andrews-Beck type congruence

Recently, Andrews proved two conjectures of Beck related to ranks of partitions. Very recently, Chern established some results on weighted rank and crank moments and proved many Andrews-Beck type congruences. Motivated by Andrews and Chern's work, Lin, Peng and Toh proved a number of Andrews-Beck type congruences for k-colored partitions. At the end of their paper, Lin, Peng and Toh posed several conjectures on Andrews-Beck type congruences. In this paper, we confirm one of those conjectures based on some q-series identities.     

On an inhomogeneous Tricomi problem for a parabolic-hyperbolic equation

We consider an inhomogeneous Tricomi problem for a parabolic-hyperbolic equation with noncharacteristic type change line and with Frankl type matching condition for the normal derivatives on the type change line. The auxiliary function method is used to establish an a priori estimate of the solution. The existence of the solution is proved by the spectral method.     

抛物线型结点组上的插值

本文讨论了R~2上的弧线型结点组之一:抛物线型结点组插值的适定性     

黎曼流形上一类指数调和型热方程的梯度估计

本文我们推到了黎曼流形上指数调和型热方程的一个Hamilton-Souplet-Zhang型梯度估计. 利用这个估计, 我们得到了一个Liouville型定理.     

Perturbation of an abstract differential equation containing fractional Riemann-Liouville derivatives

We consider a Cauchy type problem in a Banach space. Under the assumption that the corresponding Cauchy type problem with the operator A is uniformly well-posed and the operator B(t) is subordinate to A in some sense, we prove the unique solvability of the considered problem and its continuous dependence on initial data.     

Self-organized criticality and partial synchronization in an evolving network

We describe an evolving network type model of stock market, and present numerical results. Recent works on Self-organized criticality in pulse coupled relaxation oscillators have shown SOC to be related to frustrated attempts of the system to synchronise. Other works have shown the emergence of a self-organized control parameter which feeds back onto an order parameter, causing avalanches which separate periods of stasis. We define an all-to-all network of connected spins which also have an attached fitness. We find a self-organized fitness threshold given by the mean fitness, which separates a solid type state from a gaseous type state. We also find a self-organized control parameter given by the fitness deviation. Approach of this parameter to zero causes `avalanches' of partial synchronizations which occur on all sizes, separated by periods of stasis, and re-order the system.     

On an approximation family of discrete polynomial operators

We present a new family of linear discrete polynomial operators giving a Timan type approximation theorem for functions of arbitrary smoothness. Using this we construct two families of operators of this kind to extend Freud type approximation results to functions of higher smoothness.     

Hall polynomials and the fixed space of an automorphism of a finite abelian p-group

Recent work on Hall polynomials is used to study the fixed space of a random automorphism of a finite abelian p-group. An expression is found for the chance that an automorphism of an abelian p-group of type l\lambda fixes only the identity. A formula is obtained for the chance that a given subgroup H of type n\nu is the fixed space of an automorphism of an abelian p-group of type (k^l). There results generalize work of Rudvalis and Shinoda on the fixed space of an element of GL (n, q).     

Time discretization via Laplace transformation of an integro-differential equation of parabolic type

We consider the discretization in time of an inhomogeneous parabolic integro-differential equation, with a memory term of convolution type, in a Banach space setting. The method is based on representing the solution as an integral along a smooth curve in the complex plane which is evaluated to high accuracy by quadrature, using the approach in recent work of López-Fernández and Palencia. This reduces the problem to a finite set of elliptic equations with complex coefficients, which may be solved in parallel. The method is combined with finite element discretization in the spatial variables to yield a fully discrete method. The paper is a further development of earlier work by the authors, which on the one hand treated purely parabolic equations and, on the other, an evolution equation with a positive type memory term. The authors acknowledge the support of the Australian Research Council.     

Stable determination of an inclusion in a layered medium

We consider the inverse problem of detecting an inclusion in conductor through electrostatic measurements taken on the boundary. We analyze in particular the stability issue showing that the solution depends from the available data with a rate of continuity of logarithmic type.     

Eigenfunctions of the Tricomi problem with an inclined type change line

We construct the eigenfunctions of the Tricomi problem for the case in which the type change line of the elliptic-hyperbolic equation is inclined and forms an arbitrary angle α with the x-axis. These eigenfunctions form a basis in the elliptic domain. In addition, we find an integral constraint on the inclined type change line.     

Time Discretization Schemes for an Integro-Differential Equation of Parabolic Type

1.IntroductionWeconsiderthetimediscretizationoftheequationwhereAisanunboundedpositivedefiniteselfadjointoperatorwithdensedomainD(A)inaHilbertspaceHandBisanotheroperatorwithdomainD(B)DD(A),Thekernelb(t,s)isassumedtobeasmoothreaJ-vaJuedfunctionofbothtandsforoSsStandf(t)EHisasmoothfunction-ThistrpeofproblemoccursinaPplicationssuchasheatconductioninmaterialwithmemorycompressiollofporthviscoelasticmedia,nuclearreactordynamics,etc.ThenumericaJsolutionbymeansofspatialdiscretizationbyfinitedif…     

Asymptotic type for sectorial operators and an integral of fractional powers

There is a standard notion of type for a sectorial linear operator acting in a Banach space. We introduce a notion of asymptotic type for a linear operator V, involving estimates on the resolvent ⁻¹(λI+V) as λ→0. We show, for example, that if V is sectorial and of asymptotic type ω then the fractional power Vα is of asymptotic type αω for a suitable range of positive α. Moreover, we establish various properties of the operator ; in particular, this operator is of asymptotic type 0, for a sectorial operator V. This result has an application to the construction of operators satisfying the well-known Ritt resolvent condition.     

Uniform Stability of Local Extrema of an Integral Curve of an ODE of Second Order

A second-order equation can have singular sets of first and second type, S₁ and S₂ (see the introduction), where the integral curve x(y) does not exist in the ordinary sense but where it can be extended by using the first integral [1–-5]. Denote by Y the Cartesian axis y=0. If the function x(y) has a derivative at a point of local extremum of this function, then this point belongs to S ₁∪Y. The extrema at which y'(x) does not exist can be placed on S ₂. In [5–-8], the stability and instability of extrema on S ₁∪ S ₂ under small perturbations of the equation were considered, and the stability of the mutual arrangement of the maxima and minima of x(y) on the singular set was studied (locally as a rule, i.e., in small neighborhoods of singular points). In the present paper, sufficient conditions for the preservation of type of a local extremum on the finite part of S ₁ or S ₂ are found for the case in which the perturbation on all of this part does not exceed some explicitly indicated quantity which is the same on the entire singular set.     

Adaptive estimation of an ergodic diffusion process based on sampled data

We consider adaptive maximum likelihood type estimation of both drift and diffusion coefficient parameters for an ergodic diffusion process based on discrete observations. Two kinds of adaptive maximum likelihood type estimators are proposed and asymptotic properties of the adaptive estimators, including convergence of moments, are obtained.     

The attractor of an equation of Tricomi's type

Consider the attractor A of an equation of pendulum type with friction driven by a constant torque. The results of M. Levi and also obtained independently by Q. Min, S. Xian, and Z. Jinyan show that if the friction coefficient is larger than a certain bound then A is homeomorphic to the circle. We shall study the bifurcation diagram of a particular class of equations of pendulum type and show that the bounds obtained before are optimal.     

Half of an antipodal spherical design

We investigate several antipodal spherical designs on which we can choose half of the points, one from each antipodal pair, such that they are balanced at the origin. In particular, root systems of type A, D and E, minimal points of the Leech lattice, and the unique tight 7-design on \(S^{22}\) are studied. We also study a half of an antipodal spherical design from the viewpoint of association schemes and spherical designs of harmonic index T.     

Stochastic reaction diffusion equations on an infinite interval with reflection

We consider a time evolution of random fields with non-negative values on the real line. Such evolution is described by an infinite dimensional stochastic differential equation of Skorokhod's type, which is a stochastic partial differential equation (SPDE) of parabolic type with reflection. We shall show the existence of the solution, and its uniqueness when the diffusion coefficient is constant.     

Well posedness of an integrodifferential kinetic model of Fokker–Planck type for angiogenesis

Tumor induced angiogenesis processes including the effect of stochastic motion and branching of blood vessels can be described coupling a (nonlocal in time) integrodifferential kinetic equation of Fokker–Planck type with a diffusion equation for the tumor induced angiogenic factor. The chemotactic force field depends on the flux of blood vessels through the angiogenic factor. We develop an existence and uniqueness theory for this system under natural assumptions on the initial data. The proof combines the construction of fundamental solutions for associated linearized problems with comparison principles, sharp estimates of the velocity integrals and compactness results for this type of kinetic and parabolic operators.     

Consistency of an estimator of doubly nondecreasing regression functions

Consistency of a well known isotonic type estimator of doubly nondecreasing regression functions is obtained under hypotheses suggested by anr ^th order Kolmogorov condition, when finite moments of order greater than two are assumed, and under hypotheses that include uniformly bounded second moments. Observations are taken at points on a rectangular grid.