An operator splitting method for variational inequalities with partially unknown mappings

In this paper, we propose a new operator splitting method for solving a class of variational inequality problems in which part of the underlying mappings are unknown. This class of problems arises frequently from engineering, economics and transportation equilibrium problems. At each iteration, by using the information observed from the system, the method solves a system of nonlinear equations, which is well-defined. Under mild assumptions, the global convergence of the method is proved, and its efficiency is demonstrated with numerical examples. The research of D. Han is supported by NSFC grant 10501024 and NSF of Jiangsu Province at Grant No. BK2006214.     

A global solution curve for a class of periodic problems, including the pendulum equation

Using continuation methods and bifurcation theory, we study the exact multiplicity of periodic solutions, and the global solution structure, for a class of periodically forced pendulum-like equations. Our results apply also to the first order equations. We also show that by choosing a forcing term, one can produce periodic solutions with any number of Fourier coefficients arbitrarily prescribed.     

Kneser-type oscillation criteria for self-adjoint two-term differential equations

It is proved that the even-order equationy ⁽²ⁿ⁾ +p(t)y=0 is (n,n) oscillatory at if

Searching for critical angles in a convex cone

The concept of antipodality relative to a closed convex cone has been explored in detail in a recent work of ours. The antipodality problem consists of finding a pair of unit vectors in K achieving the maximal angle of the cone. Our attention now is focused not just in the maximal angle, but in the angular spectrum of the cone. By definition, the angular spectrum of a cone is the set of angles satisfying the stationarity (or criticality) condition associated to the maximization problem involved in the determination of the maximal angle. In the case of a polyhedral cone, the angular spectrum turns out to be a finite set. Among other results, we obtain an upper bound for the cardinality of this set. We also discuss the link between the critical angles of a cone K and the critical angles of its dual cone. Dedicated to Boris Polyak on his 70th Birthday.     

Unbounded solutions of differential equations of second order

In this paper, the existence of unbounded solutions for the following nonlinear asymmetric oscillator

Smoluchowski-Kramers approximation for a general class of SPDEs

We prove that the so-called Smoluchowski-Kramers approximation holds for a class of partial differential equations perturbed by a non-Gaussian noisy term. Namely, we show that the solution of the one-dimensional semi-linear stochastic damped wave equations , u(0) = u₀, u_t (0) = v₀, endowed with Dirichlet boundary conditions, converges as the parameter μ goes to zero to the solution of the semi-linear stochastic heat equation , u(0) = u₀, endowed with Dirichlet boundary conditions. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Remarks on extremal solutions of differential equations with advanced argument

We show the existence of absolutely continuous extremal solutions to the problemx′(t)=f(t, x)h(t)))+g(t)),x(0)=x ₀, whereh is an arbitrary continuous deviated argument. Conditions for the uniqueness of solutions are given. Research partialy supported by grant UG BW 5100 - 5 - 0143 - 4     

A Feasibility Pump for mixed integer nonlinear programs

We present an algorithm for finding a feasible solution to a convex mixed integer nonlinear program. This algorithm, called Feasibility Pump, alternates between solving nonlinear programs and mixed integer linear programs. We also discuss how the algorithm can be iterated so as to improve the first solution it finds, as well as its integration within an outer approximation scheme. We report computational results. P. Bonami is supported in part by a grant from IBM and by ANR grant BLAN06-1-138894. G. Cornuéjols is supported in part by NSF grant CMMI-0653419, ANR grant BLAN06-1-138894 and ONR grant N00014-03-1-0188. Part of this research was carried out when Andrea Lodi was Herman Goldstine Fellow of the IBM T.J. Watson Research Center whose support is gratefully acknowledged. F. Margot is supported in part by a grant from IBM and by ONR grant N00014-03-1-0188.     

On a method to transform algebraic differential equations into universal equations

The paper introduces an algorithm which transforms homogeneous algebraic differential equations into universal differential equations (in the sense of L. A. Rubel) havingC ⁿ (ℝ)-solutions. By applications of the algorithm to different initial equations some new universal differential equations are found, and all the known equations due to R. J. Duffin are rediscovered with this method. Assuming weak conditions one can find Cⁿ(ℝ)-solutionsy of the differential equation close to any continuous function such that 1, with 0 ≤k ₁ <k ₂ < .... <k _s ≤n are linearly independent over the field of real algebraic numbers at the rational points q1,...,qs.     

On manifolds satisfying stable systolic inequalities

We show that for closed orientable manifolds the k-dimensional stable systole admits a metric-independent volume bound if and only if there are cohomology classes of degree k that generate cohomology in top-degree. Moreover, it turns out that in the nonorientable case such a bound does not exist for stable systoles of dimension at least two. Additionally, we prove that the stable systolic constant depends only on the image of the fundamental class in a suitable Eilenberg–Mac Lane space. Consequently, the stable k-systolic constant is completely determined by the multilinear intersection form on k-dimensional cohomology.     

Computing the integer programming gap

We determine the maximal gap between the optimal values of an integer program and its linear programming relaxation, where the matrix and cost function are fixed but the right hand side is unspecified. Our formula involves irreducible decomposition of monomial ideals. The gap can be computed in polynomial time when the dimension is fixed. Partially supported by the National Science Foundation (DMS-0200729).     

The Existence of a Positive Solution of D α[x(t) − x(0)] = x(t)f(t,x t )

This paper is devoted to the problem of existence of positive solution for a delay differential equation with fractional order. Some sufficient conditions for its existence of positive solution are given. This work was supported in part by the Key Project of the National Nature Science Foundation of China (No. 60534020), the National Nature Science Foundation of China (No. 60474037), and Program for New Century Excellent Talents in University (No. NCET-04-415).     

Connected but not path-connected subspaces of infinite graphs

Solving a problem of Diestel [9] relevant to the theory of cycle spaces of infinite graphs, we show that the Freudenthal compactification of a locally finite graph can have connected subsets that are not path-connected. However we prove that connectedness and path-connectedness to coincide for all but a few sets, which have a complicated structure.     

On surfaces of prescribed weighted mean curvature

Utilising a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and von der Mosel. Next we study graphs of prescribed weighted mean curvature, for which a quasilinear elliptic equation is proved. Using this equation, we can show height and boundary gradient estimates. Finally, we solve the Dirichlet problem for graphs of prescribed weighted mean curvature.     

The first Hochschild cohomology group of a schurian cluster-tilted algebra

Given a cluster-tilted algebra B we study its first Hochschild cohomology group HH¹(B) with coefficients in the B-B-bimodule B. We find several consequences when B is representation-finite, and also in the case where B is cluster-tilted of type . M. J. Redondo is a researcher from CONICET, Argentina.     

The inflation class operator

We consider the inflation class operator, denoted by F, where for any class K of algebras, F(K) is the class of all inflations of algebras in K. We study the interaction of this operator with the usual algebraic operators H, S andP, and describe the partially-ordered monoid generated by H, S, P andF (with the isomorphism operator I as an identity). Received February 3, 2004; accepted in final form January 3, 2006.     

Constant mean curvature spheres in Riemannian manifolds

We prove the existence of embedded spheres with large constant mean curvature in any compact Riemannian manifold (M, g). This result partially generalizes a result of R. Ye which handles the case where the scalar curvature function of the ambient manifold (M, g) has non-degenerate critical points.     

A class of simple proper Bol loops

The existence of finite simple non-Moufang Bol loops has long been considered to be one of the main open problems in the theory of loops and quasigroups. In this paper, we present a class of simple proper Bol loops. This class contains finite and new infinite simple proper Bol loops. This paper was written during the author’s Marie Curie Fellowship MEIF-CT-2006-041105 at the University of Würzburg (Germany).     

Blossom V: a new implementation of a minimum cost perfect matching algorithm

We describe a new implementation of the Edmonds’s algorithm for computing a perfect matching of minimum cost, to which we refer as Blossom V. A key feature of our implementation is a combination of two ideas that were shown to be effective for this problem: the “variable dual updates” approach of Cook and Rohe (INFORMS J Comput 11(2):138–148, 1999) and the use of priority queues. We achieve this by maintaining an auxiliary graph whose nodes correspond to alternating trees in the Edmonds’s algorithm. While our use of priority queues does not improve the worst-case complexity, it appears to lead to an efficient technique. In the majority of our tests Blossom V outperformed previous implementations of Cook and Rohe (INFORMS J Comput 11(2):138–148, 1999) and Mehlhorn and Schäfer (J Algorithmics Exp (JEA) 7:4, 2002), sometimes by an order of magnitude. We also show that for large VLSI instances it is beneficial to update duals by solving a linear program, contrary to a conjecture by Cook and Rohe.     

Existence of positive solutions for a class of second-order two-point boundary value problem

By using different convex functionals to compute fixed point index, the existence of positive solutions for a class of second-order two-point boundary value problem