Estimates on the Distribution of the Condition Number of Singular Matrices

We exhibit some new techniques to study volumes of tubes about algebraic varieties in complex projective spaces. We prove the existence of relations between volumes and Intersection Theory in the presence of singularities. In particular, we can exhibit an average Bezout Equality for equidimensional varieties. We also state an upper bound for the volume of a tube about a projective variety. As a main outcome, we prove an upper bound estimate for the volume of the intersection of a tube with an equidimensional projective algebraic variety. We apply these techniques to exhibit upper bounds for the probability distribution of the generalized condition number of singular complex matrices.     

Invariants of Characteristics of Some Systems of Partial Differential Equations

We introduce the notion of an invariant of characteristics for a system of first-order partial differential equations. We prove that the existence of invariants is connected with passiveness of some systems. We describe a few methods for construction of new invariants from those already known. We give a scheme for application of the invariants to reduction and integration of systems of partial differential equations. As an application we consider the equation of gas dynamics.     

Formula for the number of eigenvalues of a three-particle Schrödinger operator on a lattice

We consider a system of three arbitrary quantum particles on a three-dimensional lattice that interact via short-range attractive potentials. We obtain a formula for the number of eigenvalues in an arbitrary interval outside the essential spectrum of the three-particle discrete Schrödinger operator and find a sufficient condition for the discrete spectrum to be finite. We give an example of an application of our results.     

Using a Conic Formulation for Finding Steiner Minimal Trees

We present a new mathematical programming formulation for the Steiner minimal tree problem. We relax the integrality constraints on this formulation and transform the resulting problem (which is convex, but not everywhere differentiable) into a standard convex programming problem in conic form. We consider an efficient computation of an ε-optimal solution for this latter problem using an interior-point algorithm.     

Stabilisation uniforme d'un système hybride

We study the stability of a hybrid system for a family of linear feedback laws. We first give an estimation of energy decay for a usual norm and then give an estimation for a stronger norm.     

Inverse problems for quasi-variational inequalities

In this short note, our aim is to investigate the inverse problem of parameter identification in quasi-variational inequalities. We develop an abstract nonsmooth regularization approach that subsumes the total variation regularization and permits the identification of discontinuous parameters. We study the inverse problem in an optimization setting using the output-least squares formulation. We prove the existence of a global minimizer and give convergence results for the considered optimization problem. We also discretize the identification problem for quasi-variational inequalities and provide the convergence analysis for the discrete problem. We give an application to the gradient obstacle problem.     

On surface radiation conditions for an ellipse

We compare several On Surface Radiation Boundary Conditions in two dimensions, for solving the Helmholtz equation exterior to an ellipse. We also introduce a new boundary condition for an ellipse based on a modal expansion in Mathieu functions. We compare the OSRC to a finite difference method.     

Complex eigenvalues before the Fundamental Theorem of Algebra

We present an elementary proof of the existence of an eigenvalue for an endomorphism of a complex vector space and we derive the Fundamental Theorem of Algebra as a corollary of this existence. We also present new proofs for the corresponding results for endomorphisms of real vector spaces.     

Edgeworth equilibrium in a model of interregional economic relations

We introduce an analog of an Edgeworth equilibrium for a class of multiregional economic systems. We analyze the game-theoretic aspects of the coalition stability of regional development plans and establish a quite general existence theorem for an Edgeworth equilibrium. We discuss the questions of coincidence of the set of these equilibria with the fuzzy core and the set of theWalrasian equilibria of the multiregional systemin question.Our methods rest on a systematic accounting for the polyhedrality of the sets of balanced coalition plans.     

A method for solving boundary value problems for nonlinear control systems in the class of discrete controls

We suggest an algorithm for constructing discrete control functions. This algorithm is sufficiently convenient for numerical implementation and, for a wide class of nonlinear systems of ordinary differential equations, provides the passage from the initial state into an arbitrary given or arbitrarily small neighborhood of a given terminal state. We obtain a constructive criterion for the choice of terminal states and discretization steps for which the passage is possible with regard of the constraints imposed on the control and the phase coordinates. We consider an interorbital flight problem, for which we carry out numerical simulation.     

Local overlaps in special unfoldings of convex polyhedra

We define a notion of local overlaps in polyhedron unfoldings. We use this concept to construct convex polyhedra for which certain classes of edge unfoldings contain overlaps, thereby negatively resolving some open conjectures. In particular, we construct a convex polyhedron for which every shortest path unfolding contains an overlap. We also present a convex polyhedron for which every steepest edge unfolding contains an overlap. We conclude by analyzing a broad class of unfoldings and again find a convex polyhedron for which they all contain overlaps.     

Path regularity and explicit convergence rate for BSDE with truncated quadratic growth

We consider backward stochastic differential equations with drivers of quadratic growth (qgBSDE). We prove several statements concerning path regularity and stochastic smoothness of the solution processes of the qgBSDE, in particular we prove an extension of Zhang’s path regularity theorem to the quadratic growth setting. We give explicit convergence rates for the difference between the solution of a qgBSDE and its truncation, filling an important gap in numerics for qgBSDE. We give an alternative proof of second order Malliavin differentiability for BSDE with drivers that are Lipschitz continuous (and differentiable), and then derive an analogous result for qgBSDE.     

Solvability of systems of volterra integral equations of the first kind with piecewise continuous kernels

We construct an asymptotic approximation for solutions of systems of Volterra integral equations of the first kind with piecewise continuous kernels. We use the asymptotics as an initial approximation in the proposed method of successive approximations to the desired solutions. We prove the existence of a continuous solution depending on free parameters and establish sufficient conditions for the existence of a unique continuous solution. We illustrate the proved existence theorems with examples.     

Application of nonclassical separation of variables to a nonlocal heat problem

We consider an initial-boundary value problem for the heat equation with a nonlocal two-point boundary condition containing a parameter. By separating the variables in an auxiliary function system, we construct a regular solution. We obtain sufficient conditions for the absolute and uniform convergence of the series in the auxiliary system. We prove conditions close to necessary ones for the existence of a regular solution of the initial-boundary value problem.     

On digraphs with the odd cycle property

We say that a digraph D has the odd cycle property if there exists an edge subset S such that every cycle of D has an odd number of edges from S. We give necessary and sufficient conditions for a digraph to have the odd cycle property. We also consider the analogous problem for graphs.     

Performance of first-order methods for smooth convex minimization: a novel approach

We introduce a novel approach for analyzing the worst-case performance of first-order black-box optimization methods. We focus on smooth unconstrained convex minimization over the Euclidean space. Our approach relies on the observation that by definition, the worst-case behavior of a black-box optimization method is by itself an optimization problem, which we call the performance estimation problem (PEP). We formulate and analyze the PEP for two classes of first-order algorithms. We first apply this approach on the classical gradient method and derive a new and tight analytical bound on its performance. We then consider a broader class of first-order black-box methods, which among others, include the so-called heavy-ball method and the fast gradient schemes. We show that for this broader class, it is possible to derive new bounds on the performance of these methods by solving an adequately relaxed convex semidefinite PEP. Finally, we show an efficient procedure for finding optimal step sizes which results in a first-order black-box method that achieves best worst-case performance.     

Explicit Construction and Uniqueness for Universal Operator Algebras of Directed Graphs

Given a directed graph, there exist a universal operator algebraand universal C*-algebra associated to the directed graph. Inthis paper we give intrinsic constructions for these objects.We also provide an explicit construction for the maximal C*-algebraof an operator algebra. We discuss uniqueness of the universalalgebras for finite graphs, showing that for finite graphs thegraph is an isomorphism invariant for the universal operatoralgebra of a directed graph. We show that the underlying undirectedgraph is a Banach algebra isomorphism invariant for the universalC*-algebra of a directed graph.     

Cutting and packing problems for irregular objects with continuous rotations: mathematical modelling and non-linear optimization

We further improve our methodology for solving irregular packing and cutting problems. We deal with an accurate representation of objects bounded by circular arcs and line segments and allow their continuous rotations and translations within rectangular and circular containers. We formulate a basic irregular placement problem which covers a wide spectrum of packing and cutting problems. We provide an exact non-linear programming (NLP) model of the problem, employing ready-to-use phi-functions. We develop an efficient solution algorithm to search for local optimal solutions for the problem in a reasonable time. The algorithm reduces our problem to a sequence of NLP subproblems and employs optimization procedures to generate starting feasible points and feasible subregions. Our algorithm allows us to considerably reduce the number of inequalities in NLP subproblems. To show the benefits of our methodology we give computational results for a number of new challenger and the best known benchmark instances.     

A new proof of Suzuki's formula

We give a different proof of a formula of Suzuki and its strengthening by Zaidenberg for the topological Euler characteristic of an affine surface fibered over a curve. We deduce this formula using the ideas for the proof of an analogous formula for a proper morphism.     

Tensor factorization and Spin construction for Kac-Moody algebras

In this paper we discuss the “Factorization phenomenon” which occurs when a representation of a Lie algebra is restricted to a subalgebra, and the result factors into a tensor product of smaller representations of the subalgebra. We analyze this phenomenon for symmetrizable Kac-Moody algebras (including finite-dimensional, semi-simple Lie algebras). We present a few factorization results for a general embedding of a symmetrizable Kac-Moody algebra into another and provide an algebraic explanation for such a phenomenon using Spin construction. We also give some application of these results for semi-simple, finite-dimensional Lie algebras.We extend the notion of Spin functor from finite-dimensional to symmetrizable Kac-Moody algebras, which requires a very delicate treatment. We introduce a certain category of orthogonal g-representations for which, surprisingly, the Spin functor gives a g-representation in Bernstein-Gelfand-Gelfand category O. Also, for an integrable representation, Spin produces an integrable representation. We give the formula for the character of Spin representation for the above category and work out the factorization results for an embedding of a finite-dimensional, semi-simple Lie algebra into its untwisted affine Lie algebra. Finally, we discuss the classification of those representations for which Spin is irreducible.