Elliptic Apostol sums and their reciprocity laws

We introduce an elliptic analogue of the Apostol sums, which we call elliptic Apostol sums. These sums are defined by means of certain elliptic functions with a complex parameter having positive imaginary part. When , these elliptic Apostol sums represent the well-known Apostol generalized Dedekind sums. Also these elliptic Apostol sums are modular forms in the variable . We obtain a reciprocity law for these sums, which gives rise to new relations between certain modular forms (of one variable).

     

Stability of <Emphasis Type="Italic">C</Emphasis>-regularized Semigroups

Let T = (T(t))t≥0 be a bounded C-regularized semigroup generated by A on a Banach space X and R(C) be dense in X. We show that if there is a dense subspace Y of X such that for every x ∈ Y, σu(A, Cx), the set of all points λ ∈ iR to which (λ - A)^-1 Cx can not be extended holomorphically, is at most countable and σr(A) N iR = Ф, then T is stable. A stability result for the case of R(C) being non-dense is also given. Our results generalize the work on the stability of strongly continuous senfigroups.     

Robust Algorithms for the Stable Set Problem

The stable set problem is to find in a simple graph a maximum subset of pairwise non-adjacent vertices. The problem is known to be NP-hard in general and can be solved in polynomial time on some special classes, like cographs or claw-free graphs. Usually, efficient algorithms assume membership of a given graph in a special class. Robust algorithms apply to any graph G and either solve the problem for G or find in it special forbidden configurations. In the present paper we describe several efficient robust algorithms, extending some known results.     

On 2x2 conservation laws with large data

We deal with a strictly hyperbolic system of two conservation laws in one spatial dimension. One of the eigenvalues of the system is of Temple type (rarefaction and shock curves coincide), the other eigenvalue is only required to be genuinely nonlinear.We consider the initial value problem for data of the following kind: the total variation of the Temple component is bounded, possibly large, while the total variation of the other component is small. For such data we prove global existence, uniqueness and L⊃-Lipschitz continuous dependence of solutions.AMS Subject Classification: Primary 35L65; Secondary 35D05, 35L45.     

BV estimates of Lax-Friedrichs' scheme for a class of nonlinear hyperbolic conservation laws

We give uniform BV estimates and -stability of Lax-Friedrichs' scheme for a class of systems of strictly hyperbolic conservation laws whose integral curves of the eigenvector fields are straight lines, i.e., Temple class, under the assumption of small total variation. This implies that the approximate solutions generated via the Lax-Friedrichs' scheme converge to the solution given by the method of vanishing viscosity or the Godunov scheme, and then the Glimm scheme or the wave front tracking method.

     

THE SPACE—FRACTIONAL TELEGRAPH EQUATION AND THE RELATED FRACTIONAL TELEGRAPH PROCESS

The space-fractional telegraph equation is analyzed and the Fourier transform of its fundamental solution is obtained and discussed.A symmetric process with discontinuous trajectories, whose transition function satisfies the space-fractional telegraph equation, is presented. Its limiting behaviour and the connection with symmetric stable processes is also examined.     

Entropy Production and Admissibility of Shocks

In shock wave theory there are two considerations in selecting the physically relevant shock waves.There is the admissibility criterion for the well-posedness of hyperbolic conservation laws.Another consideraztion concerns the entropy production across the shochs.The latter is natural from the physical point of view,but is not sufficient in its straightforward formulation,if the system is not genuinely nonlinear.In this paper we propose the principles of increasing entropy production and that of the superposition of shocks.These principles arc shown to be equivalent to the admissibility criterion.     

Equilibrium schemes for scalar conservation laws with stiff sources

We consider a simple model case of stiff source terms in hyperbolic conservation laws, namely, the case of scalar conservation laws with a zeroth order source with low regularity. It is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced CFL condition and refined meshes required because of the lack of accuracy on equilibrium states. The source term should be taken into account in the upwinding and discretized at the nodes of the grid. In order to solve numerically the problem, we introduce a so-called equilibrium schemes with the properties that (i) the maximum principle holds true; (ii) discrete entropy inequalities are satisfied; (iii) steady state solutions of the problem are maintained. One of the difficulties in studying the convergence is that there are no estimates for this problem. We therefore introduce a kinetic interpretation of upwinding taking into account the source terms. Based on the kinetic formulation we give a new convergence proof that only uses property (ii) in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions. The computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that, in comparison with an usual upwind scheme, the corresponding equilibrium version is far more accurate. Furthermore, numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients.

     

On Uniform Laws of Large Numbers for Ergodic Diffusions and Consistency of Estimators

Consider a regular diffusion process X with finite speed measure m. Denote the normalized speed measure by μ. We prove that the uniform law of large numbers holds if the class has an envelope function that is μ-integrable, or if is bounded in L _p(μ) for some p>1. In contrast with uniform laws of large numbers for i.i.d. random variables, we do not need conditions on the ‘size’ of the class in terms of bracketing or covering numbers. The result is a consequence of a number of asymptotic properties of diffusion local time that we derive. We apply our abstract results to improve consistency results for the local time estimator (LTE) and to prove consistency for a class of simple M-estimators. This revised version was published online in June 2006 with corrections to the Cover Date.     

非齐次守恒律方程分片光滑解的粘性方法

1　引　　言考虑非齐次守恒律方程ut+f(u) x =g(u) ,　　 -∞ 0 ,(1 .1 )u(x,0 ) =u0 (x) ,　　 -∞ 0 , (1 .5)g∈ C3且 g是 Lipschitz连续的 ,Lipschitz系数为 L . (1 .6 )对于一般守恒律齐次方程 ,粘性解逼近熵解的收敛阶为 O(ε ) [1 ] .在 f严格凸的条件下 ,其收敛速度可以提高到 O(ε|lnε|+ε) [2 ] ,[3] .本文考虑具有条件 (1 .5) (1 .6 )的非齐次方程(1 .1 ) ,在较广泛的一类初值条件下…