Decompositions of semigroups induced by identities

In this paper we consider decompositions of semigroups induced by identities. Here we give some new characterizations of a semilattice of Archimedean semigroups and, using this, we describe all identities which induce decompositions into a semilattice of Archimedean semigroups. Also, we give a solution for one problem ofШеврин andСуханов [27]. Supported by Grant 0401A of RFNS through Math. Inst. SANU     

Maximum and minimum solutions for nonlinear parabolic problems with discontinuities

In this paper we examine nonlinear parabolic problems with a discontinuous right hand side. Assuming the existence of an upper solution φ and a lower solution ψ such that ψ ≤ φ, we establish the existence of a maximum and a minimum solution in the order interval [ψ, φ]. Our approach does not consider the multivalued interpretation of the problem, but a weak one side “Lipschitz” condition on the discontinuous term. By employing a fixed point theorem for nondecreasing maps, we prove the existence of extremal solutions in [ψ, φ for the original single valued version of the problem.     

Refined common knowledge logics or logics of common information

 In terms of formal deductive systems and multi-dimensional Kripke frames we study logical operations know, informed, common knowledge and common information. Based on [6] we introduce formal axiomatic systems for common information logics and prove that these systems are sound and complete. Analyzing the common information operation we show that it can be understood as greatest open fixed points for knowledge formulas. Using obtained results we explore monotonicity, omniscience problem, and inward monotonocity, describe their connections and give dividing examples. Also we find algorithms recognizing these properties for some particular cases. Received: 21 October 2000 / Published online: 2 September 2002 Key words or phrases: Multi-agent systems – Non-standard logic – Knowledge representation – Common knowledge – Common information – Fixed points, Kripke models – Modal logic     

Homogenization of eigenvalue problems in perforated domains

In this paper, we treat some eigenvalue problems in periodically perforated domains and study the asymptotic behaviour of the eigenvalues and the eigenvectors when the number of holes in the domain increases to infinity Using the method of asymptotic expansion, we give explicit formula for the homogenized coefficients and expansion for eigenvalues and eigenvectors. If we denote by ε the size of each hole in the domain, then we obtain the following aysmptotic expansion for the eigenvalues: Dirichlet: λ_ε = ε⁻² λ + λ₀ +O (ε), Stekloff: λ_ε = ελ₁ +O (ε²), Neumann: λ_ε = λ₀ + ελ₁ +O (ε²). Using the method of energy, we prove a theorem of convergence in each case considered here. We briefly study correctors in the case of Neumann eigenvalue problem.     

Unitary Orbits in a Full Matrix Algebra

The Hilbert manifold ∑ _∞ consisting of positive invertible (unitized) Hilbert-Schmidt operators has a rich structure and geometry. The geometry of unitary orbits Ω⊂∑ _∞ is studied from the topological and metric viewpoints: we seek for conditions that ensure the existence of a smooth local structure for the set Ω, and we study the convexity of this set for the geodesic structures that arise when we give ∑ _∞ different Riemannian metrics.     

A new algorithm for computing the Geronimus transformation with large shifts

A monic Jacobi matrix is a tridiagonal matrix which contains the parameters of the three-term recurrence relation satisfied by the sequence of monic polynomials orthogonal with respect to a measure. The basic Geronimus transformation with shift α transforms the monic Jacobi matrix associated with a measure dμ into the monic Jacobi matrix associated with dμ/(x − α) + Cδ(x − α), for some constant C. In this paper we examine the algorithms available to compute this transformation and we propose a more accurate algorithm, estimate its forward errors, and prove that it is forward stable. In particular, we show that for C = 0 the problem is very ill-conditioned, and we present a new algorithm that uses extended precision.     

Time-Frequency Analysis of Pseudodifferential Operators

 In this paper we apply a time-frequency approach to the study of pseudodifferential operators. Both the Weyl and the Kohn–Nirenberg correspondences are considered. In order to quantify the time-frequency content of a function or distribution, we use certain function spaces called modulation spaces. We deduce a time-frequency characterization of the twisted product of two symbols σ and τ, and we show that modulation spaces provide the natural setting to exactly control the time-frequency content of from the time-frequency content of σ and τ. As a consequence, we discuss some boundedness and spectral properties of the corresponding operator with symbol .     

Arithmetic height functions over finitely generated fields

In this paper, we propose a new height function for a variety defined over a finitely generated field over ℚ. For this height function, we prove Northcott’s theorem and Bogomolov’s conjecture, so that we can recover the original Raynaud’s theorem (Manin-Mumford’s conjecture). Oblatum 7-VI-1999 & 21-IX-1999 / Published online: 24 January 2000     

Partial algebras for Łukasiewicz logics and its extensions

It is a well-known fact that MV-algebras, the algebraic counterpart of Łukasiewicz logic, correspond to a certain type of partial algebras: lattice-ordered effect algebras fulfilling the Riesz decomposition property. The latter are based on a partial, but cancellative addition, and we may construct from them the representing ℓ-groups in a straightforward manner. In this paper, we consider several logics differing from Łukasiewicz logics in that they contain further connectives: the PŁ-, PŁ'-, PŁ'_△-, and ŁΠ-logics. For all their algebraic counterparts, we characterise the corresponding type of partial algebras. We moreover consider the representing f-rings. All in all, we get three-fold correspondences: the total algebras - the partial algebras - the representing rings.     

Boundary Asymptotic Analysis for an Incompressible Viscous Flow: Navier Wall Laws

We consider a new way of establishing Navier wall laws. Considering a bounded domain Ω of R ^N , N=2,3, surrounded by a thin layer Σ _ε , along a part Γ₂ of its boundary ∂Ω, we consider a Navier-Stokes flow in Ω∪∂Ω∪Σ _ε with Reynolds’ number of order 1/ε in Σ _ε . Using Γ-convergence arguments, we describe the asymptotic behaviour of the solution of this problem and get a general Navier law involving a matrix of Borel measures having the same support contained in the interface Γ₂. We then consider two special cases where we characterize this matrix of measures. As a further application, we consider an optimal control problem within this context.     

Variable metric bundle methods: From conceptual to implementable forms

To minimize a convex function, we combine Moreau-Yosida regularizations, quasi-Newton matrices and bundling mechanisms. First we develop conceptual forms using “reversal” quasi-Newton formulae and we state their global and local convergence. Then, to produce implementable versions, we incorporate a bundle strategy together with a “curve-search”. No convergence results are given for the implementable versions; however some numerical illustrations show their good behaviour even for large-scale problems.     

P-distances, q-distances and a generalized Ekeland’s variational principle in uniform spaces

In this paper, we attempt to give a unified approach to the existing several versions of Ekeland’s variational principle. In the framework of uniform spaces, we introduce p-distances and more generally, q-distances. Then we introduce a new type of completeness for uniform spaces, i.e., sequential completeness with respect to a q-distance (particularly, a p-distance), which is a very extensive concept of completeness. By using q-distances and the new type of completeness, we prove a generalized Takahashi’s nonconvex minimization theorem, a generalized Ekeland’s variational principle and a generalized Caristi’s fixed point theorem. Moreover, we show that the above three theorems are equivalent to each other. From the generalized Ekeland’s variational principle, we deduce a number of particular versions of Ekeland’s principle, which include many known versions of the principle and their improvements.     

On Fefferman and Burkholder–Davis–Gundy inequalities for ℰ-martingales

In a previous paper we introduced a new concept, the notion of ℰ-martingales and we extended the well-known Doob inequality (for 1 < p < + ∞) and the Burkholder–Davis–Gundy inequalities (for p = 2) to ℰ-martingales. After showing new Fefferman-type inequalities that involve sharp brackets as well as the space bmo _q , we extend the Burkholder–Davis–Gundy inequalities (for 1 < p < + ∞) to ℰ-martingales. By means of these inequalities we give sufficient conditions for the closedness in L ^p of a space of stochastic integrals with respect to a fixed ℝ^d-valued semimartingale, a question which arises naturally in the applications to financial mathematics. Finally we investigate the relation between uniform convergence in probability and semimartingale topology. Received: 22 July 1997 / Revised version: 3 July 1998     

Invariant (α, β)-metrics on homogeneous manifolds

In this paper we study invariant (α, β)-metrics on homogeneous spaces. We first give a method to construct invariant (α, β)-metrics on homogeneous spaces. Then we obtain some conditions for some special type of (α, β)-metrics to be of Berwald type and Douglas type. At last, we give a rigidity result concerning the Randers metrics and Matsumoto metrics of Berwald type on homogeneous spaces.     

Regularity of a free boundary problem

Let u be the Newtonian potential of a real analytic distribution in an open set Ω. In this paper we assume u is analytically continuable from the complement of Ω into some neighborhood of a point x₀ ∈ ∂Ω, and we study conditions under which the analytic continuation implies that ∂Ω is a real analytic hypersurface in some neighborhood of x₀.     

The Stochastic Wave Equation with Multiplicative Fractional Noise: A Malliavin Calculus Approach

We consider the stochastic wave equation with multiplicative noise, which is fractional in time with index H > 1/2, and has a homogeneous spatial covariance structure given by the Riesz kernel of order α. The solution is interpreted using the Skorohod integral. We show that the sufficient condition for the existence of the solution is α > d − 2, which coincides with the condition obtained in Dalang (Electr J Probab 4(6):29, 1999), when the noise is white in time. Under this condition, we obtain estimates for the p-th moments of the solution, we deduce its H?lder continuity, and we show that the solution is Malliavin differentiable of any order. When d ≤ 2, we prove that the first-order Malliavin derivative of the solution satisfies a certain integral equation.     

Complete settling of the multiplier conjecture for the case of n=3pr

In this paper we improve the character approach to the multiplier conjecture that we presented after 1992, and thus we have made considerable progress in the case of n = 3n₁. We prove that in the case of n = 3n₁ Second multiplier theorem remains true if the assumption “n₁ > λ” is replaced by “(n₁, λ) = 1”. Consequentially we prove that if we let D be a (v, k, λ)-difference set in an abelian group G, and n = 3p^r for some prime p, (p,v) = 1, then p is a numerical multiplier of D.     

Extended formulations in combinatorial optimization

This survey is concerned with the size of perfect formulations for combinatorial optimization problems. By “perfect formulation”, we mean a system of linear inequalities that describes the convex hull of feasible solutions, viewed as vectors. Natural perfect formulations often have a number of inequalities that is exponential in the size of the data needed to describe the problem. Here we are particularly interested in situations where the addition of a polynomial number of extra variables allows a formulation with a polynomial number of inequalities. Such formulations are called “compact extended formulations”. We survey various tools for deriving and studying extended formulations, such as Fourier’s procedure for projection, Minkowski–Weyl’s theorem, Balas’ theorem for the union of polyhedra, Yannakakis’ theorem on the size of an extended formulation, dynamic programming, and variable discretization. For each tool that we introduce, we present one or several examples of how this tool is applied. In particular, we present compact extended formulations for several graph problems involving cuts, trees, cycles and matchings, and for the mixing set. We also present Bienstock’s approximate compact extended formulation for the knapsack problem, Goemans’ result on the size of an extended formulation for the permutahedron, and the Faenza-Kaibel extended formulation for orbitopes.     

On algebraic automorphisms and their rational invariants

Let X be an affine irreducible variety over an algebraically closed field k of characteristic zero. Given an automorphism Φ, we denote by k(X)^Φ its field of invariants, i.e., the set of rational functions f on X such that f o Φ = f. Let n(Φ) be the transcendence degree of k(X)^Φ over k. In this paper we study the class of automorphisms Φ of X for which n(Φ) = dim X - 1. More precisely, we show that under some conditions on X, every such automorphism is of the form Φ = ϕ_g, where ϕ is an algebraic action of a linear algebraic group G of dimension 1 on X, and where g belongs to G. As an application, we determine the conjugacy classes of automorphisms of the plane for which n(Φ) = 1.     

On the homology of mapping spaces

Following a Bendersky-Gitler idea, we construct an isomorphism between Anderson’s and Arone’s complexes modelling the chain complex of a mapping space. This allows us to apply Shipley’s convergence theorem to Arone’s model. As a corollary, we reduce the problem of homotopy equivalence for certain “toy” spaces to a problem in homological algebra.