Nonoccurrence of the Lavrentiev phenomenon for many nonconvex constrained variational problems

In this paper we study nonoccurrence of the Lavrentiev phenomenon for a large class of nonconvex nonautonomous constrained variational problems. A state variable belongs to a convex subset of a Banach space with nonempty interior. Integrands belong to a complete metric space of functions which satisfy a growth condition common in the literature and are Lipschitzian on bounded sets. In our previous work Zaslavski (Ann. Inst. H. Poincare, Anal. non lineare, 2006) we considered a class of nonconstrained variational problems with integrands belonging to a subset and showed that for any such integrand the infimum on the full admissible class is equal to the infimum on a subclass of Lipschitzian functions with the same Lipschitzian constant. In the present paper we show that if an integrand f belongs to , then this property also holds for any integrand which is contained in a certain neighborhood of f in . Using this result we establish nonoccurrence of the Lavrentiev phenomenon for most elements of in the sense of Baire category.      

The Monge Problem in Banach Spaces

In this paper, we generalize the Kantorovich functional to K?the-spaces for a cost or a profit function. We examine the convergence of probabilities with respect to this functional for some K?the-spaces. We study the Monge problem: Let be a K?the-space, P and Q two Borel probabilities defined on a Polish space M and a cost function . A K?the functional is defined by (P, Q) = inf where is the law of X. If c is a profit function, we note . (P, Q) = sup Under some conditions, we show the existence of a Monge function, φ, such that , or .      

A note on curvature of α-connections of a statistical manifold

The family of α-connections ∇^(α) on a statistical manifold equipped with a pair of conjugate connections and is given as . Here, we develop an expression of curvature R ^(α) for ∇^(α) in relation to those for . Immediately evident from it is that ∇^(α) is equiaffine for any when are dually flat, as previously observed in Takeuchi and Amari (IEEE Transactions on Information Theory 51:1011–1023, 2005). Other related formulae are also developed. The work was conducted when the author was on sabbatical leave as a visiting research scientist at the Mathematical Neuroscience Unit, RIKEN Brain Science Institute, Wako-shi, Saitama 351-0198, Japan.     

Pointwise estimates for relative fundamental solutions of heat equations in $${\mathbb{R} \times \mathbb{C}}$$

Let be a subharmonic, nonharmonic polynomial and a parameter. Define , a closed, densely defined operator on . If and , we solve the heat equations , u(0,z) = f(z) and , . We write the solutions via heat semigroups and show that the solutions can be written as integrals against distributional kernels. We prove that the kernels are C ^∞ off of the diagonal {(s, z, w) : s = 0 and z = w} and find pointwise bounds for the kernels and their derivatives.      

On some homotopical and homological properties of monoid presentations

If a monoid S is given by some finite complete presentation ℘, we construct inductively a chain of CW-complexes

Uniform Comparison of Tails of (Non-Symmetric) Probability Measures and Their Symmetrized Counterparts with Applications

Let (, ) be a separable Banach space and let be a class of probability measures on , and let denote the symmetrization of . We provide two sufficient conditions (one in terms of certain quantiles and the other in terms of certain moments of relative to μ and , ) for the “uniform comparison” of the μ and measure of the complements of the closed balls of centered at zero, for every . As a corollary to these “tail comparison inequalities,” we show that three classical results (the Lévy-type Inequalities, the Kwapień-Contraction Inequality, and a part of the It?–Nisio Theorem) that are valid for the symmetric (but not for the general non-symmetric) independent -valued random vectors do indeed hold for the independent random vectors whose laws belong to any which satisfies one of the two noted conditions and which is closed under convolution. We further point out that these three results (respectively, the tail comparison inequalities) are valid for the centered log-concave, as well as, for the strictly α-stable (or the more general strictly (r, α) -semistable) α ≠ 1 random vectors (respectively, probability measures). We also present several examples which we believe form a valuable part of the paper.      

Auxiliary Problem Principle and Proximal Point Methods

An extension of the auxiliary problem principle to variational inequalities with non-symmetric multi-valued operators in Hilbert spaces is studied. This extension concerns the case that the operator is split into the sum of a single-valued operator , possessing a kind of pseudo Dunn property, and a maximal monotone operator . The current auxiliary problem is k constructed by fixing at the previous iterate, whereas (or its single-valued approximation ^k) k is considered at a variable point. Using auxiliary operators of the form ^k+ , with _k>0, the standard for the auxiliary problem principle assumption of the strong convexity of the function h can be weakened exploiting mutual properties of and h. Convergence of the general scheme is analyzed and some applications are sketched briefly.     

Bases for some reciprocity algebras I

For a complex vector space , let be the algebra of polynomial functions on . In this paper, we construct bases for the algebra of all highest weight vectors in , where and for all , and the algebra of highest weight vectors in .

     

A global approach to fully nonlinear parabolic problems

We consider the general initial-boundary value problem

(1)        
(2)        
(3)        
where is a bounded open set in with sufficiently smooth boundary.  The problem (1)-(3) is first reduced to the analogous problem in the space with zero initial condition and

The resulting problem is then reduced to the problem where the operator satisfies Condition  This reduction is based on a priori estimates which are developed herein for linear parabolic operators with coefficients in Sobolev spaces.  The local and global solvability of the operator equation are achieved via topological methods developed by I. V. Skrypnik. Further applications are also given involving relevant coercive problems, as well as Galerkin approximations.

     

Sharp estimates for intrinsic ultracontractivity on C 1,α-domains

Let be the first Dirichlet eigenfunction on a connected bounded C ^1,α-domain in and the corresponding Dirichlet heat kernel. It is proved that where λ₂ > λ₁ are the first two Dirichlet eigenvalues. This estimate is sharp for both short and long times. Bounded Lipschitz domains, elliptic operators on manifolds, and a general framework are also discussed. Supported in part by Creative Research Group Fund of the National Foundation of China (no. 10121101), the 973-Project in China and RFDP(20040027009).     

Unstable extremal surfaces of the “Shiffman functional” spanning rectifiable boundary curves

In this paper we derive a sufficient condition for the existence of extremal surfaces of a parametric functional with a dominant area term, which do not furnish global minima of within the class of H ^1,2-surfaces spanning an arbitrary closed rectifiable Jordan curve that merely has to satisfy a chord-arc condition. The proof is based on the “mountain pass result” of (Jakob in Calc Var 21:401–427, 2004) which yields an unstable -extremal surface bounded by an arbitrary simple closed polygon and Heinz’ ”approximation method” in (Arch Rat Mech Anal 38:257–267, 1970). Hence, we give a precise proof of a partial result of the mountain pass theorem claimed by Shiffman in (Ann Math 45:543–576, 1944) who only outlined a very sketchy and partially incorrect proof.     

Poincaré inequality and Palais–Smale condition for the <Emphasis Type="Italic">p</Emphasis>-Laplacian

An improved Poincaré inequality and validity of the Palais-Smale condition are investigated for the energy functional on , 1 < p < ∞, where Ω is a bounded domain in , is a spectral (control) parameter, and is a given function, in Ω. Analysis is focused on the case λ = λ₁, where −λ₁ is the first eigenvalue of the Dirichlet p-Laplacian Δ _p on , λ₁ > 0, and on the “quadratization” of within an arbitrarily small cone in around the axis spanned by , where stands for the first eigenfunction of Δ _p associated with −λ₁.     

Sums of random symmetric matrices and quadratic optimization under orthogonality constraints

Let B _i be deterministic real symmetric m × m matrices, and ξ _i be independent random scalars with zero mean and “of order of one” (e.g., ). We are interested to know under what conditions “typical norm” of the random matrix is of order of 1. An evident necessary condition is , which, essentially, translates to ; a natural conjecture is that the latter condition is sufficient as well. In the paper, we prove a relaxed version of this conjecture, specifically, that under the above condition the typical norm of S _N is : for all Ω > 0 We outline some applications of this result, primarily in investigating the quality of semidefinite relaxations of a general quadratic optimization problem with orthogonality constraints , where F is quadratic in X = (X ₁,... ,X _k ). We show that when F is convex in every one of X _j , a natural semidefinite relaxation of the problem is tight within a factor slowly growing with the size m of the matrices . Research was partly supported by the Binational Science Foundation grant #2002038.     

On the Optimal Value Function of a Linearly Perturbed Quadratic Program

The optimal value function of the quadratic program , where is a given symmetric matrix, a given matrix, and are the linear perturbations, is considered. It is proved that is directionally differentiable at any point in its effective domain . Formulae for computing the directional derivative of at in a direction are obtained. We also present an example showing that, in general, is not piecewise linear-quadratic on W. The preceding (unpublished) example of Klatte is also discussed.     

A numerical method for computing the Hamiltonian Schur form

We derive a new numerical method for computing the Hamiltonian Schur form of a Hamiltonian matrix that has no purely imaginary eigenvalues. We demonstrate the properties of the new method by showing its performance for the benchmark collection of continuous-time algebraic Riccati equations. Despite the fact that no complete error analysis for the method is yet available, the numerical results indicate that if no eigenvalues of are close to the imaginary axis then the method computes the exact Hamiltonian Schur form of a nearby Hamiltonian matrix and thus is numerically strongly backward stable. The new method is of complexity and hence it solves a long-standing open problem in numerical analysis. Volker Mehrmann was supported by Deutsche Forschungsgemeinschaft, Research Grant Me 790/11-3.     

Homotopy Methods for Solving Variational Inequalities in Unbounded Sets

In this paper, for solving the finite-dimensional variational inequality problem

On Transversally Holomorphic Maps of Kählerian Foliations

Any transversally holomorphic foliated map of Kählerianfoliations with harmonic, is shown to be a transversallyharmonic map and an absolute minimum of the energy functional inits foliated homotopy class.     

New bounds on the unconstrained quadratic integer programming problem

We consider the maximization for a given symmetric . It was shown recently, using properties of zonotopes and hyperplane arrangements, that in the special case that A has a small rank, so that A = VV ^T where with m < n, then there exists a polynomial time algorithm (polynomial in n for a given m) to solve the problem . In this paper, we use this result, as well as a spectral decomposition of A to obtain a sequence of non-increasing upper bounds on γ with no constraints on the rank of A. We also give easily computable necessary and sufficient conditions for the absence of a gap between the solution and its upper bound. Finally, we incorporate the semidefinite relaxation upper bound into our scheme and give an illustrative example.     

Differential inclusions for differential forms

We study necessary and sufficient conditions for the existence of solutions in of the problem where is a given set. Special attention is given to the case of the curl (i.e. k = 1), particularly in dimension 3. Some applications to the calculus of variations are also stated.     

On a singular perturbation problem related to optimal lifting in BV-space

We prove a Γ-convergence result for the family of functionals defined on H ¹(Ω) by for a given and a parameter . We show that in either of the two cases, p = 2 or , any limit of the minimizers is an optimal lifting.