Lemma A.1 (a) A real-valued convex function is also 0-convex and hence k-convex for all k ≥ 0. A k1-convex function is also a k2-convex function for k1 ≤ k2. (b) If f1(y) and f2(y) are k1-convex and k2-convex respectively, then for α, β ≥ 0, αf1(y) + βf2(y) is (αk1 + βk2)-convex. (c) If f(y) is k-convex and w is a random variable, then E{f(y − w)} is also k-convex, provided E{|f(y − w)|} < ∞ fo...

 We consider generalized semi-infinite programming problems in which the index set of the inequality constraints depends on the decision vector and all emerging functions are assumed to be convex. Considering a lower level constraint qualification, we derive a formula for estimating the subdifferential of the value function. Finally, we establish the Fritz-John necessary optimality con...

In this article, we introduce a new class of ideal convergent sequence spaces using an infinite matrix, Musielak-Orlicz function and a new generalized difference matrix in locally convex spaces. We investigate some linear topological structures and algebraic properties of these spaces. We also give some relations related to these sequence spaces.

Self-assembled monolayers (SAMs) of a mu 3-eta 2:eta 2:eta 2-C60 triosmium cluster complex Os3(CO)8(CN(CH2)3Si(OEt)3)(mu 3-eta 2:eta 2:eta 2-C60) (2) on ITO or Au surface exhibit ideal, well-defined electrochemical responses and remarkable electrochemical stability being reducible up to tetranionic species in their cyclic voltammograms.

The familiar bijections between the representations of permutations as words and as products of cycles have a natural class of “data driven” extensions that permit us to use purely combinatorial means to obtain precise probabilistic information about the geometry of random walks. In particular, we show that the algorithmic bijection of Bohnenblust and Spitzer can be used to obtain means, varian...

The glueballs lead to gluon and QCD monopole condensations as by-products of color confinement. A dielectric function $G(|\phi|)$ coupled with Abelian gauge field is properly defined mediate the glueball interactions at confining regime after {\it spontaneous symmetry breaking} (SSB) symmetry. particles are expected form through quark plasma (QGP) hadronization phase where free quarks gluons st...

Khrapchenko’s classical lower bound n on the formula size of the parity function f can be interpreted as designing a suitable measure of subrectangles of the combinatorial rectangle f−1(0)× f−1(1). Trying to generalize this approach we arrived at the concept of convex measures. We prove the negative result that convex measures are bounded by O(n) and show that several measures considered for pr...

The problem of minimizing f̃ = f +p over some convex subset of a Euclidean space is investigated, where f(x) = x Ax + b x is a strictly convex quadratic function and |p| is only assumed to be bounded by some positive number s. It is shown that the function f̃ is strictly outer γ-convex for any γ > γ∗, where γ∗ is determined by s and the smallest eigenvalue of A. As consequence, a γ∗-local minimal...