Let $S$ be a regular semigroup and $E(S)$ be the set of its idempotents. We call the sets $S(e,f)f$ and $eS(e,f)$ one-sided sandwich sets and characterize them abstractly where $e,f \in E(S)$. For $a, a^{\prime } \in S$ such that $a=aa^{\prime }a$, $a^{\prime }=a^{\prime }aa^{\prime }$, we call $S(a)=S(a^{\prime }a, aa^{\prime })$ the sandwich set of $a$. We characterize regular semigroups $S$ in which all $S(e,f)$ (or all $S(a))$ are right zero semigroups (respectively are trivial) in several ways including weak versions of compatibility of the natural order. For every $a \in S$, we also define $E(a)$ as the set of all idempotets $e$ such that, for any congruence $\rho $ on $S$, $a \rho a^2$ implies that $a \rho e$. We study the restrictions on $S$ in order that $S(a)$ or $E(a)\cap D_{a^2}$ be trivial. For $\mathcal F \in \lbrace \mathcal S, \mathcal E\rbrace $, we define $\mathcal F$ on $S$ by $a \mathrel {\mathcal F}b$ if $F(a) \cap F (b)\ne \emptyset $. We establish for which $S$ are $\mathcal S$ or $\mathcal E$ congruences.
Studying a critical value function \vi in parametric nonlinear programming, we recall conditions guaranteeing that \vi is a C1,1 function and derive second order Taylor expansion formulas including second-order terms in the form of certain generalized derivatives of D\vi. Several specializations and applications are discussed. These results are understood as supplements to the well-developed theory of first- and second-order directional differentiability of the optimal value function in parametric optimization.
Let $f$ be a function defined on the set ${\mathbf M}^{2\times 2}$ of all $2$ by $2$ matrices that is invariant with respect to left and right multiplications of its argument by proper orthogonal matrices. The function $f$ can be represented as a function $\tilde{f}$ of the signed singular values of its matrix argument. The paper expresses the ordinary convexity, polyconvexity, and rank 1 convexity of $f$ in terms of its representation $\tilde{f}.$
We propose a sequential monitoring scheme for detecting a change in scale. We consider a stable historical period of length m. The goal is to propose a test with asymptotically small probability of false alarm and power 1 as the length of the historical period tends to infinity. The asymptotic distribution under the null hypothesis and consistency under the alternative hypothesis is derived. A small simulation study illustrates the finite sample performance of the monitoring scheme.
We show that a conjunction of Mazur and Gelfand-Phillips properties of a Banach space $E$ can be naturally expressed in terms of {\it weak}* continuity of seminorms on the unit ball of $E^*$. \endgraf We attempt to carry out a construction of a Banach space of the form $C(K)$ which has the Mazur property but does not have the Gelfand-Phillips property. For this purpose we analyze the compact spaces on which all regular measures lie in the {\it weak}* sequential closure of atomic measures, and the set-theoretic properties of generalized densities on the natural numbers.
The information contained in the measure of all shifts of two or three given points contained in an observed compact subset of $\mathbb{R}^d $ is studied. In particular, the connection of the first order directional derivatives of the described characteristic with the oriented and the unoriented normal measure of a set representable as a finite union of sets with positive reach is established. For smooth convex bodies with positive curvatures, the second and the third order directional derivatives of the characteristic is computed.
We observe that each set from the system A˜ (or even C˜) is Γ-null; consequently, the version of Rademacher’s theorem (on Gˆateaux differentiability of Lipschitz functions on separable Banach spaces) proved by D. Preiss and the author is stronger than that proved by D. Preiss and J. Lindenstrauss. Further, we show that the set of non-differentiability points of a convex function on n is σ-strongly lower porous. A discussion concerning sets of Fréchet non-differentiability points of continuous convex functions on a separable Hilbert space is also presented.
Steinhaus [9] prove that if a set $A$ has a positive Lebesgue measure in the line then its distance set contains an interval. He obtained even stronger forms of this result in [9], which are concerned with mutual distances between points in an infinite sequence of sets. Similar theorems in the case we replace distance by mutual ratio were established by Bose-Majumdar [1]. In the present paper, we endeavour to obtain some results related to sets with Baire property in locally compact topological spaces, particular cases of which yield the Baire category analogues of the above results of Steinhaus [9] and their corresponding form for ratios by Bose-Majumdar [1].