We study continuity and equicontinuity of semigroups on norming dual pairs with respect to topologies defined in terms of the duality. In particular, we address the question whether continuity of a semigroup already implies (local/quasi) equicontinuity. We apply our results to transition semigroups and show that, under suitable hypothesis on E, every transition semigroup on Cb(E) which is conti...

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...

Today, the importance of having 100% uptime for businesses and industries is clear: financial reasons and often strict government regulations for certain industries require 100% business continuity. The concept of business continuity (BC), as Microsoft defines it: the ability of an organization to continue to function even after a disastrous event, accomplished through the deployment of redunda...

Given a compact Riemannian manifold, we study the regularity of the optimal transport map between two probability measures with cost given by the squared Riemannian distance. Our strategy is to define a new form of the so-called Ma-Trudinger-Wang condition and to show that this condition, together with the strict convexity on the nonfocal domains, implies the continuity of the optimal transport...

2 Finite-Dimensional Optimization 5 2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.3 Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

A type-2 computable real function is necessarily continuous; and this remains true for computations relative to any oracle. Conversely, by the Weierstrass Approximation Theorem, every continuous f : [0; 1]→ R is computable relative to some oracle. In their search for a similar topological characterization of relatively computable multi-valued functions f : [0; 1] ⇒ R (also known as multi-functi...

We present the basic analytical and algebraic properties of triangular norms. We discuss continuity as well as the important classes of Archimedean, strict and nilpotent t-norms. Triangular conorms and De Morgan triples are also mentioned. Finally, a brief historical survey on triangular norms is given. Submitted for publication Johannes Kepler Universität Linz Fuzzy Logic Laboratorium Linz-Hag...

In this paper we study the initial-boundary value problems for nonlinear parabolic equations without Bernstein-Nagumo condition. Sufficient conditions guaranteeing the nonexistence of gradient blow-up are formulated. In particular, we show that for a wide class of nonlinearities the Lipschitz continuity in the space variable together with the strict monotonicity with respect to the solution gua...

An inequality preorder is defined as a complete preorder on a simplex which satisfies the properties of continuity and strict Schur-convexity (the mathematical equivalent of Dalton's "principle of transfers"). The paper shows that it is possible to aggregate individual inequality preorders into a collective one if we are interested in continuous anonymous aggregation rules that respect unanimit...

This paper characterizes lexicographic preferences over alternatives that are identified by a finite number of attributes. Our characterization is based on two key concepts: weaker notion continuity called “mild continuity” (strict preference order between any different with respect to every attribute preserved around their small neighborhoods) and an “unhappy set” (any alternative outside such...