Numerical calculation of an asymmetric supply function equilibrium with capacity constraints

Producers submit offer curves to a procurement auction, e.g. an electricity auction, before uncertain demand has been realised. In the supply function equilibrium (SFE), every firm commits to the offer curve that maximises its expected profit, given the offer curves of competitors. The equilibrium is given by a system of differential equations. In practice, it has been very difficult to find valid SFE, i.e. non-decreasing solutions, from this system, especially for asymmetric producers. This paper shows that valid SFE can be calculated by means of a shooting algorithm that combines numerical integration with an optimisation procedure that searches for an end-condition. Multiple/parallel shooting is used for ill-conditioned cases.     

Achieving Brouwer’s law with implicit Runge–Kutta methods

In high accuracy long-time integration of differential equations, round-off errors may dominate truncation errors. This article studies the influence of round-off on the conservation of first integrals such as the total energy in Hamiltonian systems. For implicit Runge–Kutta methods, a standard implementation shows an unexpected propagation. We propose a modification that reduces the effect of round-off and shows a qualitative and quantitative improvement for an accurate integration over long times. AMS subject classification (2000)  65L06, 65G50, 65P10     

A SET OF SYMMETRIC QUADRATURE RULES ON TRIANGLES AND TETRAHEDRA

We present a program for computing symmetric quadrature rules on triangles and tetrahedra. A set of rules are obtained by using this program. Quadrature rules up to order 21 on triangles and up to order 14 on tetrahedra have been obtained which are useful for use in finite element computations. All rules presented here have positive weights with points lying within the integration domain.     

信息技术与大学数学课程整合的方式与理论探讨

信息技术与大学数学课程整合正成为当前我国信息技术教育乃至整个教育信息化进程中的一个热点问题,探讨其整合的方式及理论基础是非常必要的.本文首先结合大学数学案例探讨了整合的三种方式:动态的课堂演示型、单机的数学实验型和全交互的网络教学型,其次探讨了整合所依据的四种理论:传播理论、建构主义理论、教学设计理论和系统理论,最后提出了自己对信息技术与大学数学课程整合的一些思考.     

Theory of superfluidity in a polariton system

We consider the superfluidity properties of a two-dimensional system of polaritons in an optical cavity. Deriving an expression for the effective low-energy action for thermodynamic phase fluctuations, we simultaneously obtain the expression for an analogue of superfluid density in the system in terms of the current-current correlation function and also find the expression for the current operator. We describe the Bogoliubov approximation for a polariton system and calculate the superfluid density. We discuss the Berezinskii-Kosterlitz-Thouless transition in the system under study. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 372–384, February, 2008.     

A hybrid method for multidimensional scaling using city-block distances

The problem of multidimensional scaling with city-block distances in the embedding space is reduced to a two level optimization problem consisting of a combinatorial problem at the upper level and a quadratic programming problem at the lower level. A hybrid method is proposed combining randomized search for the upper level problem with a standard quadratic programming algorithm for the lower level problem. Several algorithms for the combinatorial problem have been tested and an evolutionary global search algorithm has been proved most suitable. An experimental code of the proposed hybrid multidimensional scaling algorithm is developed and tested using several test problems of two- and three-dimensional scaling.     

Comparison of ODE methods for laminar reacting gas flow simulations

Two‐dimensional transient simulations are presented of the transport phenomena and multispecies, multireaction chemistry in chemical vapor deposition (CVD). The transient simulations are run until steady state, such that the steady state can be validated against the steady state solutions from literature. We compare various time integration methods in terms of efficiency and robustness. Besides stability, which is important due to the stiffness of the problem, preservation of non‐negativity is crucial. It appears that this latter condition on a time integration method is much more restrictive toward the time step size than stability. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Generalized Widder Theorem via fractional moments

We provide a necessary and sufficient condition for the representability of a function as the classical multidimensional Laplace transform, when the support of the representing measure is contained in some generalized semi-algebraic set. This is done by employing a method of Putinar and Vasilescu [M. Putinar, F.-H. Vasilescu, Solving moment problems by dimensional extension, Ann. of Math. (2) 149 (3) (1999) 1087-1107] for the corresponding multidimensional moment problem.     

Explicit Volume-Preserving Splitting Methods for Linear and Quadratic Divergence-Free Vector Fields

We present new explicit volume-preserving methods based on splitting for polynomial divergence-free vector fields. The methods can be divided in two classes: methods that distinguish between the diagonal part and the off-diagonal part and methods that do not. For the methods in the first class it is possible to combine different treatments of the diagonal and off-diagonal parts, giving rise to a number of possible combinations. This paper is dedicated to Arieh Iserles on the occasion of his 60th anniversary.