We construct a (Lipschitz) differentiability space which has at generic points a disconnected tangent and thus does not contain positive measure subsets isometric to positive measure subsets of spaces admitting a Poincaré inequality. We also prove that l-valued Lipschitz maps are differentiable a.e., but there are also Lipschitz maps taking values in some other Banach spaces having the Radon-Ni...

This paper presents some new Lie groups preserving fixed subspaces of geometric algebras (or Clifford algebras) under the twisted adjoint representation. We consider cases grades and determined by grade involution reversion. Some considered can be interpreted as generalizations Lipschitz spin groups. The are subgroups these coincide with them in small dimensions. study corresponding algebras.

In this paper, we consider inner automorphisms that leave invariant fixed subspaces of real and complex Clifford algebras—subspaces grades determined by the reversion grade involution. We present groups elements define such study their properties. Some these Lie can be interpreted as generalizations Clifford, Lipschitz, spin groups. corresponding algebras. results reformulated for case more gen...

‎Given a compact space $X$ and a commutative Banach algebra‎ ‎$A$‎, ‎the character spaces of $A$-valued function algebras on $X$ are‎ ‎investigated‎. ‎The class of natural $A$-valued function algebras‎, ‎those whose characters can be described by means of characters of $A$ and‎ ‎point evaluation homomorphisms‎, ‎is introduced and studied‎. ‎For an‎ ‎admissible Banach $A$-valued function algebra...

In this work, we use a notion of convexificator [25] together with the support function [3, 4, 15, 16, 41] to establish necessary optimality conditions for set valued bilevel optimization problems. Fortunately, the Lipschitz property of a set-valued mapping is conserved for its support function. An intermediate set-valued optimization problem is introduced to help us in our investigation.

This paper is an attempt at developing a theory of algebraic systems that would correspond in a natural fashion to the N0-valued prepositional calculus^). For want of a better name, we shall call these algebraic systems MV-algebras where MV is supposed to suggest many-valued logics. It is known that the classical two-valued logic gives rise to the study of Boolean algebras and, as can be expect...

We propose a new concept of generalized di erentiation of setvalued maps that captures rst order information. This concept encompasses the standard notions of Fréchet di erentiability, strict di erentiability, calmness and Lipschitz continuity in single-valued maps, and the Aubin property and Lipschitz continuity in set-valued maps. We present calculus rules, sharpen the relationship between th...

Let A(D) be the disc algebra of all continuous complex-valued functions on the unit disc D holomorphic in its interior. Functions from A(D) act on the set of all contraction operators (‖A‖ 1) on Hilbert spaces. It is proved that the following classes of functions from A(D) coincide: (1) the class of operator Lipschitz functions on the unit circle T; (2) the class of operator Lipschitz functions...