We show that polynomial-time randomness (p-randomness) is preserved under a variety of familiar operations, including addition and multiplication by a nonzero polynomial-time computable real number. These results follow from a general theorem: If I ⊆ R is an open interval, f : I → R is a function, and r ∈ I is p-random, then f(r) is p-random provided 1. f is p-computable on the dyadic rational ...

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Using the variational characterization of principal (i.e., smallest) eigenvalue below essential spectrum a lower semibounded self-adjoint operator, we prove strict domain monotonicity (with respect to changing finite interval length) Friedrichs extension $$ T_F$$ minimal operator for regular four-coefficient Sturm–Liouville differential expressions. In more general singular context, these expre...

Abstract. Let 1 ≤ p < ∞, f ∈ Lp(R) and Λ ⊆ R. We consider the closed subspace of Lp(R), Xp(f,Λ), generated by the set of translations f(λ) of f by λ ∈ Λ. If p = 1 and {f(λ) : λ ∈ Λ} is a bounded minimal system in L1(R), we prove that X1(f,Λ) embeds almost isometrically into `1. If {f(λ) : λ ∈ Λ} is an unconditional basic sequence in Lp(R), then {f(λ) : λ ∈ Λ} is equivalent to the unit vector ba...

Given a binary relation R, we look for partitions of its row set and its column set, respectively, that behave well with respect to selected algebraic properties, i.e., correspond to congruences related to R. Permutations are derived from these congruences that allow to rearrange R visualizing the decomposition.

V r /2 t,p/2 is a scaled version of Gaussian measure with r p if 1 < p < 2 and r p ′ if 2 < p < ∞ with 1/p 1/p′ 1. Conversely if F is a holomorphic function which is in a “slightly” smaller space, namely L C, V r/2 t,p/2 , then it is shown that there is a function f ∈ L R, ρ such that e−tHf F. However, a single necessary and sufficient condition is obtained for the image of L2 R, ρp/2 under e−t...

and Applied Analysis 3 2. Proof of Theorem 1 We need the following lemmas for the proof of Theorem 1. The difference analogue of the logarithmic derivative lemma was given by Halburd-Korhonen [8, Corollary 2.2] and Chiang-Feng [7, Corollary 2.6], independently. The following Lemma 8 is a variant of [8, Corollary 2.2]. Lemma 8. Let f(z) be a nonconstant meromorphic function of finite order, and ...

We investigate the problem of approximating function f in power-type weighted variable exponent Sobolev space Wr,p(.) ?(.) (0,1), (r = 1, 2, ...), by Hardy averaging operator A (f) (x) 1/x ?x0 f(t)dt. If lies ?(.)(0, 1), it is shown that A||(f)?f|| p(.),?(.)?rp(.) ? C ||f(r) p(.),?(.) , where a positive constant. Moreover, we consider boundedness grand Lebesgue spaces Lp(.),? ?(.)(0,1). The suf...