We show that if Y is a separable subspace of a Banach space X such that both X and the quotient X/Y have Cp-smooth Lipschitz bump functions, and U is a bounded open subset of X, then, for every uniformly continuous function f : Y ∩U → R and every ε > 0, there exists a Cp-smooth Lipschitz function F : X → R such that |F (y)− f(y)| ≤ ε for every y ∈ Y ∩U . If we are given a separable subspace Y o...

We consider the Cauchy problem for Klein–Gordon equation in $${\mathbb{R}}^{d},d\ge 1,$$ with random initial data. introduce a family of measures $${\mu }_{0}^{\varepsilon },\varepsilon >0$$ depending on small parameter ε. The }$$ are assumed to be locally homogeneous or slowly varying under spatial shifts order o(ε−1) and inhomogeneous ε−1. Moreover, correlation functions decrease uniformly ε ...

and Applied Analysis 3 then there exists ε0 > 0 such that for all 0 < ε ≤ ε0 the mapping Θ : Ω ε → R3 is an injective mapping and the three vectors gεi x ε : ∂εiΘ x ε ∂α ∂/∂xα, ∂ ε 3 ∂/∂x ε 3 are linear independent for each x ∈ Ωε. The injectivity of the mapping Θ : Ωε → R3 ensures that the physical problem described below is meaningful. The three vectors gεi x ε form the covariant basis at the...

We consider a bichromatic two-center problem for pairs of points. Given set S n points in the plane, every pair, we want to assign red color one point and blue other, such way that value max⁡{r1,r2} is minimized, where r1 (resp., r2) radius smallest enclosing disk all blue) Previously, an exact algorithm O(n3log2⁡n) time (1+ε)-approximate O(n+(1/ε)6log2⁡(1/ε)) were known. In this paper, propose...

Proof of Observation 1. Before the proof, we remind that what we want to prove is the following statement: (∀ε ∈ R) ε > 0⇒ ((∃N ∈ N)(∀n ∈ N) n ≥ N ⇒ |an − x | < ε). To this end, choose arbitrary positive real ε > 0 and pick a positive rational ε′ with 0 < ε′ < ε by utilizing the Archimedean property. Then there exists N (depending on ε′ and hence on ε) such that j, k ≥ N ⇒ |a j − ak | < ε′. Now...

Consider the singular perturbation problem for εu(t; ε) + u(t; ε) = Au(t; ε) + ∫ t 0 K(t− s)Au(s; ε) ds+ f(t; ε) , where t ≥ 0, u(0; ε) = u0(ε), u (0; ε) = u1(ε), and w(t) = Aw(t) + ∫ t 0 K(t− s)Aw(s)ds+ f(t) , t ≥ 0 , w(0) = w0 , in a Banach space X when ε → 0. Here A is the generator of a strongly continuous cosine family and a strongly continuous semigroup, and K(t) is a bounded linear opera...

In this paper, we study ε-generalized weak s...