A function f 2 L 1 (R) tiles the line with a constant weight w using the discrete tile set A if P a2A f(x ? a) = w almost everywhere. A set A is of bounded density if there is a constant C such that #fa 2 A : n a n + 1g C for all integers n. This paper characterizes compactly supported f 2 L 1 (R) that admit a tiling of R of bounded density. It shows that for such functions all tile sets of bou...

Let A be an associative algebra over a field F of characteristic zero and let L Lie F. If acts on by derivations, then such action determines its universal enveloping U(L) in this case we refer to as with derivations or L-algebra. Here give characterization the ideal differential identities finite dimensional L-algebras corresponding sequence codimensions ${c_{n}^{L}} (A)$ , n ≥ 1, is polynomia...

Abstract. We consider bounded solutions of the semilinear heat equation ut = uxx + f(u) on R, where f is of the unbalanced bistable type. We examine the ω-limit sets of bounded solutions with respect to the locally uniform convergence. Our goal is to show that even for solutions whose initial data vanish at x = ±∞, the ω-limit sets may contain functions which are not steady states. Previously, ...

Some generalized trapezoid type inequalities for the RiemannStieltjes integral of continuous complex valued integrands de…ned on the complex unit circle C (0; 1) and various subclasses of integrators of bounded variation are given. Natural applications for functions of unitary operators in Hilbert spaces are provided. 1. Introduction In [13], in order to approximate the Riemann-Stieltjes integr...

Let X,Y, Z be Banach spaces and let u : X×Y → Z be a bounded bilinear map. Given a locally compact abelian group G , and two functions f ∈ L(G,X) and g ∈ L(G,Y ), we define the u -convolution of f and g as the Z -valued function f ∗u g(t) = ∫ G u(f(t− s), g(s))dμG(s) where dμG stands for the Haar measure on G . We define the concepts of vector-valued approximate identity and summability kernel ...

We define a cohomology theory for topological semigroups, with seminormed cohomology groups. Our theory can be considered as a topological version of bounded cohomology of discrete semigroups. 1. Definition of the cohomology Bounded cohomology was first defined for discrete groups by F. Trauber and then for topological spaces by M. Gromov [6]. In this paper we establish a topological bounded co...

Given two metric spaces S1, S2 and a measurable function f : S1 → S2, sup­ pose S1 is equipped with some probability measure P. This induces a proba­ bility measure on S2 which is denoted by Pf−1 and is defined by Pf−1(A) = P(f−1(A)) for every measurable set A ⊂ S2. Then for any random variable X : S2 → R, its expectation EPf−1 [X] is equal to EP[X(f)]. (Convince your­ self that this is the cas...

For simplicity, we adopt the following rules: n denotes an element of N, V denotes a subset of the carrier of En T , s, s1, s2, t, t1, t2 denote points of E n T , C denotes a simple closed curve, P denotes a subset of the carrier of E T , and a, p, p1, p2, q, q1, q2 denote points of E 2 T . Next we state several propositions: (1) For all real numbers a, b holds (a − b) = (b − a). (2) Let S, T b...

We will prove that under certain conditions on the parameters operators T+f=max(f,0) and Tf=|f| are bounded mappings Triebel–Lizorkin–Morrey Besov–Morrey spaces. Moreover we show some of mentioned before also necessary. Furthermore in many cases spaces do not have Fubini property.

We consider the Monge-Amp\`ere equation $\det(D^2u)=f$ in $\mathbb{R}^n$, where $f$ is a positive bounded periodic function. prove that $u$ must be sum of quadratic polynomial and For $f\equiv 1$, this classic result by J\"orgens, Calabi Pogorelov. $f\in C^\alpha$, was proved Caffarelli first named author.