Let C be a closed convex subset of a Banach space E. A mapping T of C into itself is called nonexpansive if ‖Tx−Ty‖ ≤ ‖x− y‖ for all x, y ∈ C. We denote by F(T) the set of fixed points of T . Let T1,T2, . . . ,Tr be a finite family of nonexpansive mappings satisfying that the set F =⋂i=1F(Ti) of common fixed points of T1,T2, . . . ,Tr is nonempty. The problem of finding a common fixed point has...

In 1965 F. E. Browder [3] and D. Göhde [6] proved that each nonempty bounded and convex subset of a uniformly convex Banach space has fixed point property for nonexpansive mappings. Also in 1965 W. A. Kirk [8] came to the same conclusion for weakly compact convex subsets of any Banach space under additional assumption that the set has the so-called normal structure. This condition is much weake...

A fuzzy subset of a universe X (a fuzzy set) is a mathematical object A described by its (generalized) characteristic function (membership function) μA : X → [0, 1] Alternative notation: A(x) In this context, “classical” sets are called crisp or sharp. F(X) denotes the set of all fuzzy subsets of a universe X Range (level set): Range(A) = { α ∈ [0, 1] : (∃x ∈ X : μA(x) = α) } = μA(X) Height: h(...

Let C be a subset of a Banach space E. A mapping T : C → E is nonexpansive if ‖Tx − Ty‖ ≤ ‖x − y‖ for all x, y ∈ C. In 1965, it was proved independently by Browder 1 , Göhde 2 , and Kirk 3 that if C is a bounded closed convex subset of a Hilbert space and T : C → C is nonexpansive, then T has a fixed point. Combining the results above, Ray 4 obtained the following interesting result see 5 for a...

A subset of the d-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent d-dimensional closed balls. The spindle convex body is called a “fat” one, if it contains the centers of its generating balls. The main result of this paper is a proof of the Illumination Conjecture for “fat” spindle convex b...

Abstract. This paper presents a new observability estimate for parabolic equations in × (0, T ), where  is a convex domain. The observation region is restricted over a product set of an open nonempty subset of  and a subset of positive measure in (0, T ). This estimate is derived with the aid of a quantitative unique continuation at one point in time. Applications to the bang-bang property f...

Problem B1 (20 pts). A ray , R, of E is any subset of the form R = {a+ tu | t ∈ R, t ≥ 0}, for some point, a ∈ E and some nonzero vector, u ∈ R. A subset, A ⊆ E, is unbounded iff it is not contained in any ball. Prove that every closed and unbounded convex set, A ⊆ E, contains a ray. Hint : Pick some point, c ∈ A, as the origin and consider the sphere, S(c, r) ⊆ E, of center c and radius r > 0....

We study the Ornstein-Uhlenbeck operator and the Ornstein-Uhlenbeck semigroup in an open convex subset of an infinite dimensional separable Banach space X. This is done by finite dimensional approximation. In particular we prove Logarithmic-Sobolev and Poincaré inequalities, and thanks to these inequalities we deduce spectral properties of the OrnsteinUhlenbeck operator. 2010 Mathematics Subjec...

The paper is motivated by the study of graded representations of Takiff algebras, cominuscule parabolics, and their generalizations. We study certain special subsets of the set of weights (and of their convex hull) of the generalized Verma modules (or GVM’s) of a semisimple Lie algebra g. In particular, we extend a result of Vinberg and classify the faces of the convex hull of the weights of a ...

The set of bistochastic or doubly stochastic N × N matrices form a convex set called Birkhoff’s polytope, that we describe in some detail. Our problem is to characterize the set of unistochastic matrices as a subset of Birkhoff’s polytope. For N = 3 we present fairly complete results. For N = 4 partial results are obtained. An interesting difference between the two cases is that there is a ball...