On the strict monotonicity of spectral radii for classes of bounded positive linear operators

Strict monotonicity of the spectral radii of bounded, positive, ordered linear operators is investigated. It is well-known that under reasonable assumptions, the spectral radii of two ordered positive operators enjoy a non-strict inequality. It is also well-known that a “strict” inequality between operators does not imply strict monotonicity of the spectral radii in general—some additional structure is required. We present a number of sufficient conditions on both the cone and the operators for such a strict ordering to hold which generalise known results in the literature, and have utility in comparison arguments, ubiquitous in positive systems theory.     

Bolzano's Theorems for Holomorphic Mappings(To Ham Brezis,with friendship and admiration)

The existence of a zero for a holomorphic functions on a ball or on a rectangle under some sign conditions on the boundary generalizing Bolzano's ones for real functions on an interval is deduced in a very simple way from Cauchy's theorem for holomorphic functions. A more complicated proof, using Cauchy's argument principle, provides uniqueness of the zero, when the sign conditions on the boundary are strict. Applications are given to corresponding Brouwer fixed point theorems for holomorphic functions. Extensions to holomorphic mappings from C~n to C~n are obtained using Brouwer degree.     

Bolzano's Theorems for Holomorphic Mappings

The existence of a zero for a holomorphic functions on a ball or on a rectangle under some sign conditions on the boundary generalizing Bolzano's ones for real functions on an interval is deduced in a very simple way from Cauchy's theorem for holomorphic functions.A more complicated proof,using Cauchy's argument principle,provides uniqueness of the zero,when the sign conditions on the boundary are strict.Applications are given to corresponding Brouwer fixed point theorems for holomorphic functions.Extensions to holomorphic mappings from Cn to Cn are obtained using Brouwer degree.     

On the functions with pseudoconvex sublevel sets and optimality conditions

A new class of generalized convex functions, called the functions with pseudoconvex sublevel sets, is defined. They include quasiconvex ones. A complete characterization of these functions is derived. Further, it is shown that a continuous function admits pseudoconvex sublevel sets if and only if it is quasiconvex. Optimality conditions for a minimum of the nonsmooth nonlinear programming problem with inequality, equality and a set constraints are obtained in terms of the lower Hadamard directional derivative. In particular sufficient conditions for a strict global minimum are given where the functions have pseudoconvex sublevel sets.     

A Friedrichs–Maz'ya inequality for functions of bounded variation

The aim of this short note is to give an alternative proof, which applies to functions of bounded variation in arbitrary domains, of an inequality by Maz'ya that improves Friedrichs inequality. A remarkable feature of such a proof is that it is rather elementary, if the basic background in the theory of functions of bounded variation is assumed. Nevertheless, it allows to extend all the previously known versions of this fundamental inequality to a completely general version. In fact the inequality presented here is optimal in several respects. As already observed in previous proofs, the crucial step is to provide conditions under which a function of bounded variation on a bounded open set, extended to zero outside, has bounded variation on the whole space. We push such conditions to their limits. In fact, we give a sufficient and necessary condition if the open set has a boundary with σ‐finite surface measure and a sufficient condition if the open set is fully arbitrary. Via a counterexample we show that such a general sufficient condition is sharp.     

Optimality and mixed saddle point criteria in multiobjective optimization

In this paper, higher order strong convexity for Lipschitz functions is introduced and is utilized to derive the optimality conditions for the new concept of strict minimizer of higher order for a multiobjective optimization problem. Variational inequality problem is introduced and its solutions are related to the strict minimizers of higher order for a multiobjective optimization problem. The notion of vector valued partial Lagrangian is also introduced and equivalence of the mixed saddle points of higher order and higher order minima are provided.     

一类反应扩散方程的爆破时间下界估计

该文讨论了一类反应项为非线性非局部热源且热汇具有时间系数的反应扩散方程,分别在Dirichlet、Neumann或Robin边界条件下,在有界区域中的爆破行为.若解可能在有限时间发生爆破,通过构造合适的辅助函数,对时间系数给出适当的条件,利用Sobolev、H?lder不等式及Payne和Schaefer积分不等式等技巧,得出了解的爆破时间下界的估计.     

Some homogenization problems for the system of elasticity with nonlinear boundary conditions in perforated domains

The system of linear elasticity is considered in a perforated domain with an ε-periodic structure. External forces nonlinearly depending on the displacements are applied to the surface of the cavities (or channels), while the body is fixed along the outer portion of its boundary. We investigate the asymptotic behavior of solutions to such boundary value problems asε→0 and construct the limit problem, according to the external surface forces and their dependence on the parameter ε. In some cases, this dependence results in the homogenized problem having the form of a variational inequality over a certain closed convex cone in a Sobolev space. This cone is described in terms of the functions involved in the nonlinear boundary conditions on the perforated boundary. A homogenization theorem is also proved for some unilateral problems with boundary conditions of Signorini type for the system of elasticity in a perforated domain. We discuss some cases when the homogenized tensor may depend on the functions specifying the boundary conditions.     

The Minkowski problem for polytopes

The traditional solution to the Minkowski problem for polytopes involves two steps. First, the existence of a polytope satisfying given boundary data is demonstrated. In the second step, the uniqueness of that polytope (up to translation) is then shown to follow from the equality conditions of Minkowski's inequality, a generalized isoperimetric inequality for mixed volumes that is typically proved in a separate context. In this article we adapt the classical argument to prove both the existence theorem of Minkowski and his mixed volume inequality simultaneously, thereby providing a new proof of Minkowski's inequality that demonstrates the equiprimordial relationship between these two fundamental theorems of convex geometry.     

A Poincar�� Inequality for Orlicz�CSobolev Functions with Zero Boundary Values on Metric Spaces

We prove a Poincaré inequality for Orlicz–Sobolev functions with zero boundary values in bounded open subsets of a metric measure space. This result generalizes the (p, p)-Poincaré inequality for Newtonian functions with zero boundary values in metric measure spaces, as well as a Poincaré inequality for Orlicz–Sobolev functions on a Euclidean space, proved by Fuchs and Osmolovski (J Anal Appl (Z.A.A.) 17(2):393–415, 1998). Using the Poincaré inequality for Orlicz–Sobolev functions with zero boundary values we prove the existence and uniqueness of a solution to an obstacle problem for a variational integral with nonstandard growth.     

Penalized complementarity functions on symmetric cones

We show that penalized functions of the Fischer–Burmeister and the natural residual functions defined on symmetric cones are complementarity functions. Boundedness of the solution set of a symmetric cone complementarity problem, based on the penalized natural residual function, is proved under monotonicity and strict feasibility. The proof relies on a trace inequality on Euclidean Jordan algebras.     

The Poincaré Inequality and <Emphasis Type="Italic">p</Emphasis>-Connectedness of a Stratified Set

We extend the Poincaré inequality to functions of Sobolev type on a stratified set. The integrability exponents in these analogs depend on the geometric characteristics of the stratified set which show to what extent their strata are connected with each other and the boundary. We apply the results to proving the solvability of boundary value problems for the p-Laplacian with boundary conditions of Neumann or Wentzel type.     

On poset similarity

Let P be a poset, and let A be an element of its strict incidence algebra. Saks (SIAM J. Algebraic Discrete Methods 1 (1980) 211–215; Discrete Math. 59 (1986) 135–166) and Gansner (SIAM J. Algebraic Discrete Methods 2 (1981) 429–440) proved that the kth Dilworth number of P is less than or equal to the dimension of the nullspace of A^k, and that there is some member of the strict incidence algebra of P for which equality is attained (for all k simultaneously). In this paper we focus attention on the question of when equality is attained with the strict zeta matrix, and proceed under a particular random poset model. We provide an invariant depending only on two measures of nonunimodality of the level structure for the poset that, with probability tending to 1 as the smallest level tends to infinity, takes on the same value as the inequality gap between the width of P and the dimension of the nullspace of its strict zeta matrix. In particular, we characterize the level structures for which the width of P is, with probability tending to 1, equal to the dimension of the nullspace of its strict zeta matrix. As a consequence, by the Kleitman–Rothschild Theorem 5, almost all posets in the Uniform random poset model have width equal to the dimension of the nullspace of their zeta matrices. We hope this is a first step toward a complete characterization of when equality holds in Saks’ and Gansner's inequality for the strict zeta matrix and for all k. New to this paper are also the canonical representatives of the poset similarity classes (where two posets are said to be similar if their strict zeta matrices are similar in the matrix-theoretic sense), and these form the setting for our work on Saks’ and Gansner's inequalities. (Also new are two functions that measure the nonunimodality of a sequence of real numbers.)     

Gap functions for constrained vector variational inequalities with applications

In this paper, the image space analysis is applied to investigate scalar-valued gap functions and their applications for a (parametric)-constrained vector variational inequality. Firstly, using a non-linear regular weak separation function in image space, a gap function of a constrained vector variational inequality is obtained without any assumptions. Then, as an application of the gap function, two error bounds for the constrained vector variational inequality are derived by means of the gap function under some mild assumptions. Further, a parametric gap function of a parametric constrained vector variational inequality is presented. As an application of the parametric gap function, a sufficient condition for the continuity of the solution map of the parametric constrained vector variational inequality is established within the continuity and strict convexity of the parametric gap function. These assumptions do not include any information on the solution set of the parametric constrained vector variational inequality.     

Jensen's inequality for a convex vector-valued function on an infinite-dimensional space

Jensen's inequality f(EX) ≤ Ef(X) for the expectation of a convex function of a random variable is extended to a generalized class of convex functions f whose domain and range are subsets of (possibly) infinite-dimensional linear topological spaces. Convexity of f is defined with respect to closed cone partial orderings, or more general binary relations, on the range of f. Two different methods of proof are given, one based on geometric properties of convex sets and the other based on the Strong Law of Large Numbers. Various conditions under which Jensen's inequality becomes strict are studied. The relation between Jensen's inequality and Fatou's Lemma is examined.     

Some remarks on a three critical points theorem

Some remarks on a strict minimax inequality, which plays a fundamental role in Ricceri's three critical points theorem, are presented. As a consequence, some recent applications of Ricceri's theorem to nonlinear boundary value problems are revisited by obtaining more precise conclusions.     

具有分片线性NCP函数的QP-Free可行域方法

In this paper, a QP-free feasible method with piecewise NCP functions is proposed for nonlinear inequality constrained optimization problems. The new NCP functions are piecewise linear-rational, regular pseudo-smooth and have nice properties. This method is based on the solutions of linear systems of equation reformulation of KKT optimality conditions, by using the piecewise NCP functions. This method is implementable and globally convergent without assuming the strict complementarity condition, the isolatedness of accumulation points. Furthermore, the gradients of active constraints are not requested to be linearly independent. The submatrix which may be obtained by quasi-Newton methods, is not requested to be uniformly positive definite. Preliminary numerical results indicate that this new QP-free method is quite promising.     

PIECEWISE LINEAR NCP FUNCTION FOR QP FREE FEASIBLE METHOD

In this paper,a QP-free feasible method with piecewise NCP functions is proposed for nonlinear inequality constrained optimization problems.The new NCP functions are piece- wise linear-rational,regular pseudo-smooth and have nice properties.This method is based on the solutions of linear systems of equation reformulation of KKT optimality conditions,by using the piecewise NCP functions.This method is implementable and globally convergent without assuming the strict complementarity condition,the isolatedness of accumulation points.Fur- thermore,the gradients of active constraints are not requested to be linearly independent.The submatrix which may be obtained by quasi-Newton methods,is not requested to be uniformly positive definite.Preliminary numerical results indicate that this new QP-free method is quite promising.     

Generalized Lyapunov inequalities involving critical Sobolev exponents

We consider Lyapunov-type inequalities generalizing the famous inequality that gives a necessary condition for the existence of solutions to a boundary value problem for a second order ordinary differential equation. For certain critical cases, when the inequalities are strict, we study the asymptotic behavior of minimizing sequences.     

Green functions of the spectral ball and symmetrized polydisk

The Green function of the spectral ball is constant over the isospectral varieties, is never less than the pullback of its counterpart on the symmetrized polydisk, and is equal to it in the generic case where the pole is a cyclic (non-derogatory) matrix. When the pole matrix is derogatory, the inequality is always strict, and the difference between the two functions depends on the multiplicity of the eigenvalues as roots of the minimal polynomial of that matrix. In particular, the Green function of the spectral ball is not symmetric in its arguments. Additionally, some estimates are given for invariant functions in the symmetrized polydisc, e.g. (infinitesimal versions of) the Carathéodory distance and the Green function, that show that they are distinct in dimension greater or equal to 3.