Half Space and Quarter Space Dirichlet Problems for the Partial Differential Equation

A customary, heuristic, method, by which the Poisson integral formula for the Dirichlet problem, for the half space, for Laplace's equation is obtained, involves Green's function, and Kelvin's method of images. Although this heuristic method leads one to guess the correct result, this Poisson formula still has to be verified directly, independently of the method by which it was arrived at, in order to be absolutely certain that a solution of the Dirichlet problem for the half space, for Laplace's equation, has been actually obtained. A similar heuristic method, as seems to be generally known, could be followed in solving the Dirichlet problem, for the half space, for the equation where is a real constant. However, in Part 1, a different, labor-saving, method is used to study Dirichlet problems for the equation. This method is essentially based on what Hadamard called the method of descent. Indeed, it is shown that he who has solved the half space Dirichlet problem for Laplace's equation has already solved the half space Dirichlet problem for the equation In Part 2, the solution formula for the quarter space Dirichlet problem for Laplace's equation is obtained from the Poisson integral formula for the half space Dirichlet problem for Laplace's equation. A representation theorem for harmonic functions in the quarter space is deduced. The method of descent is used, in Part 3, to obtain the solution formula for the quarter space Dirichlet problem for the equation by means of the solution formula for the quarter space Dirichlet problem for Laplace's equation. So that, indeed, it is also shown that he who has solved the quarter space Dirichlet problem for Laplace's equation has already solved the quarter space Dirichlet problem for the " equation" For the sake of completeness and clarity, and for the convenience of the reader, the appendix, at the end of Part 3, contains a detailed proof that the Poisson integral formula solves the half space Dirichlet problem for Laplace's equation. The Bibliography for Parts 1,2, 3 is to be found at the end of Part 1.     

Various motivations for managerial (mis)reporting – an experimental study

In the experiment we model all possible consequences from misreporting for both the shareholder and for the manager, since we are interested in patterns in reporting behaviour resulting from different motivations for potential misrepresentation. This allows for examining the stability of the (mis)reporting behaviour in different treatments. Agents are primarily driven by the consequences for themselves rather than by the consequences for the principal, while deciding on misreporting. Participants are willing to sacrifice a small gain for themselves in order to prevent a greater loss for the principal. If agents misreport, they do it in order to generate positive rather than negative consequences for themselves. Reports in favour of the principal, but fruitless or even costly for the agent are very rare. The experiment indicates also that pro-social agents report more truthfully than pro-self agents.     

Asymptotic for the cumulative distribution function of the degrees and homomorphism densities for random graphs sampled from a graphon

We give asymptotics for the cumulative distribution function (CDF) for degrees of large dense random graphs sampled from a graphon. The proof is based on precise asymptotics for binomial random variables. This result is a first step for giving a nonparametric test for identifying the degree function of a large random graph. Replacing the indicator function in the empirical CDF by a smoother function, we get general asymptotic results for functionals of homomorphism densities for partially labeled graphs. This general setting allows to recover recent results on asymptotics for homomorphism densities of sampled graphon.     

Trigonometric polynomial or exponential fitting approach?

The application of a trigonometric polynomial and an exponential fitting approach is compared for a three-point formula for second-order derivatives, for Simpson’s quadrature rule and for Numerov’s scheme for second-order differential equations. The expressions for the occurring parameters are constructed in both the approaches and the behaviour of these parameters with respect to the introduced frequency is studied. The errors for specific problems obtained in both the approaches as a function of the frequency are compared.     

An approximation of stochastic hyperbolic equations: case with Wiener process

In the present paper, the two‐step difference scheme for the Cauchy problem for the stochastic hyperbolic equation is presented. The convergence estimate for the solution of the difference scheme is established. In applications, the convergence estimates for the solution of difference schemes for the numerical solution of four problems for hyperbolic equations are obtained. The theoretical statements for the solution of this difference scheme are supported by the results of the numerical experiment. Copyright © 2012 John Wiley & Sons, Ltd.     

On bounds for the mode and median of the generalized hyperbolic and related distributions

Except for certain parameter values, a closed form formula for the mode of the generalized hyperbolic (GH) distribution is not available. In this paper, we exploit results from the literature on modified Bessel functions and their ratios to obtain simple but tight two-sided inequalities for the mode of the GH distribution for general parameter values. As a special case, we deduce tight two-sided inequalities for the mode of the variance-gamma (VG) distribution, and through a similar approach we also obtain tight two-sided inequalities for the mode of the McKay Type I distribution. The analogous problem for the median is more challenging, but we conjecture some monotonicity results for the median of the VG and McKay Type I distributions, from we which we conjecture some tight two-sided inequalities for their medians. Numerical experiments support these conjectures and also lead us to a conjectured tight lower bound for the median of the GH distribution.     

Superconvergence of mixed finite element semi-discretizations of two time-dependent problems

We will show that some of the superconvergence properties for the mixed finite element method for elliptic problems are preserved in the mixed semi-discretizations for a diffusion equation and for a Maxwell equation in two space dimensions. With the help of mixed elliptic projection we will present estimates global and pointwise in time. The results for the Maxwell equations form an extension of existing results. For both problems, our results imply that post-processing and a posteriori error estimation for the error in the space discretization can be performed in the same way as for the underlying elliptic problem.     

Pricing strategies for perishable products: the case of Vienna and the hotel reservation system hrs.com

Consider a retailer who sells perishable products for which there is uncertain demand. Yield management with dynamic pricing is a standard practice that firms use for revenue management. For perishable products, recent analysis has focused on the distribution of flight capacity, referred to as ticket sales. Other non- storable, non-transportable, immaterial hospitality products include hotel capacity. The article discusses the extent to which hotel pricing strategies vary within the internet distribution system hrs.com. This study focuses on the distribution of hotel rooms available for booking on the internet for Vienna and gives an outlook to Euroland capitals. The main research interests are the underlying pricing models and the setting of the end price. Data was taken from hrs.com, which is the most important specialist for hotel room internet distribution in Germany according to recent studies by KMPG and others. The results include the identification of different pricing strategy clusters with regard to hotel category and hotel availability over a 22-day period for Vienna and one city from all Euroland countries (the capitals were studied for all cases except for the Netherlands, for which data was collected for Amsterdam). The study took the arrival days Mondays, Tuesdays, Wednesdays and Thursdays into account, and used data for all these days from the 11th of July, 2005, to the 10th of October, 2005, for Vienna, and the first and the last of these dates as a comparison base for the other Euroland cities.     

非强制混合向量变分不等式解的存在性研究

本文利用例外簇方法研究非强制混合向量变分不等式的弱有效解的存在性:首先证明若混合向量变分不等式问题不存在例外簇,则混合向量变分不等式问题的弱有效解集为非空集合:利用向量值映射的渐近映射给出自反Banach空间中非强制混合向量变分不等式的弱有效解集不存在例外簇的充分条件,从而得到混合向量变分不等式问题的弱有效解的存在性结果;我们研究了当算子为余正仿射算子时,给出混合仿射向量变分不等式不存在例外簇的充分条件,得到混合仿射向量变分不等式弱有效解的存在性,给出了混合仿射向量变分不等式的弱有效解集为非空紧致集的充分条件.将Iusem等人(2019)在有限维空间中标量混合变分不等式解的存在性结果推广到自反Banach空间中混合向量变分不等式.     

Determinants of matrices over semirings

In this paper, the concept of determinants for the matrices over a commutative semiring is introduced, and a development of determinantal identities is presented. This includes a generalization of the Laplace and Binet–Cauchy Theorems, as well as on adjoint matrices. Also, the determinants and the adjoint matrices over a commutative difference-ordered semiring are discussed and some inequalities for the determinants and for the adjoint matrices are obtained. The main results in this paper generalize the corresponding results for matrices over commutative rings, for fuzzy matrices, for lattice matrices and for incline matrices.     

Representation for the Green’s function of the Dirichlet problem for polyharmonic equations in a ball

We explicitly construct the Green’s function for the Dirichlet problem for polyharmonic equations in a ball in a space of arbitrary dimension. The formulas for the Green’s function are of interest in their own right. In particular, the explicit representations for a solution to the Dirichlet problem for the biharmonic equation are important in elasticity.     

流动稳定性弱非线性理论的重新考虑*

弱非性理论已被广泛用于流动稳定性理论及其它领域.然而其应用对某些问题虽是成功的,但对另一些问题,其结果却常不令人满意,特别是对转捩或自由剪切流中涡的演化这类问题,这时理论研究的目的不是寻找稳态解,而是预测演化过程.在本文中,我们将研究不成功的原因并建议一些改进的办法.     

Determinant formulas with applications to designing when the observations are correlated

In the general linear model consider the designing problem for the Gauß-Markov estimator or for the least squares estimator when the observations are correlated. Determinant formulas are proved being useful for theD-criterion. They allow, for example, a (nearly) elementary proof and a generalization of recent results for an important linear model with multiple response. In the second part of the paper the determinant formulas are used for deriving lower bounds for the efficiency of a design. These bounds are applied in examples for tridiagonal covariance matrices. For these examples maximin designs are determined.Parts of the paper are based on a part of the author's Habilitationsschrift Bischoff (1993a).     

Fourier-Laplace analysis of the multigrid waveform relaxation method for hyperbolic equations

The multigrid waveform relaxation (WR) algorithm has been fairly studied and implemented for parabolic equations. It has been found that the performance of the multigrid WR method for a parabolic equation is practically the same as that of multigrid iteration for the associated steady state elliptic equation. However, the properties of the multigrid WR method for hyperbolic problems are relatively unknown. This paper studies the multigrid acceleration to the WR iteration for hyperbolic problems, with a focus on the convergence comparison between the multigrid WR iteration and the multigrid iteration for the corresponding steady state equations. Using a Fourier-Laplace analysis in two case studies, it is found that the multigrid performance on hyperbolic problems no longer shares the close resemblance in convergence factors between the WR iteration for parabolic equations and the iteration for the associated steady state equations.     

Geometric control of hybrid systems

In this paper, we present a geometric approach for computing controlled invariant sets for hybrid control systems. While the problem is well studied in the ellipsoidal case, this family is quite conservative for constrained or switched linear systems. We reformulate the invariance of a set as an inequality for its support function that is valid for any convex set. This produces novel algebraic conditions for the invariance of sets with polynomial or piecewise quadratic support functions.     

Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture

We prove two-weight norm inequalities for Calderón-Zygmund singular integrals that are sharp for the Hilbert transform and for the Riesz transforms. In addition, we give results for the dyadic square function and for commutators of singular integrals. As an application we give new results for the Sarason conjecture on the product of unbounded Toeplitz operators on Hardy spaces.     

Sparre Andersen模型中破产概率的边界

研究在Andersen Spaxre模型中,当破产概率的初始边界已知的时候,根据更新方程和更新方程中函数的单调性来改进破产概率的边界,并进一步改进了严重损失函数G（x,y）的边界．     

Stabilized Nonconforming Mixed Finite Element Method for Linear Elasticity on Rectangular or Cubic Meshes

Based on the primal mixed variational formulation, a stabilized nonconforming mixed finite element method is proposed for the linear elasticity on rectangular and cubic meshes. Two kinds of penalty terms are introduced in the stabilized mixed formulation, which are the jump penalty term for the displacement and the divergence penalty term for the stress. We use the classical nonconforming rectangular and cubic elements for the displacement and the discontinuous piecewise polynomial space for the stress, where the discrete space for stress are carefully chosen to guarantee the well-posedness of discrete formulation. The stabilized mixed method is locking-free. The optimal convergence order is derived in the $L^2$-norm for stress and in the broken $H^1$-norm and $L^2$-norm for displacement. A numerical test is carried out to verify the optimal convergence of the stabilized method.     

重建极性连续统理论的基本定律和原理(Ⅺ)——兼容性问题

对现有连续统场论中存在若干有关兼容性问题进行了简略的评论．根据重建的极性的连续统基本定律阐述三类兼容性问题．第一类是讨论各基本定律间的兼容性问题．第二类是讨论微极连续统基本定律兼容一般连续统基本定律的问题．第三类是讨论微极连续统基本定律兼容刚体动力学基本方程的问题．推导出的结果有助于深刻理解各种连续统理论的基本定律体系的结构和它们之间的联系．明确地指出,在传统的连续统场论的基本均衡定律的框架下,相容性问题是不可能自然地得到解决．