We consider the Hardy-type operator (Tf) (x) := v(x) ∫ x a u(t)f(t)dt, x > a, and establish properties of T as a map from L(a, b) into L(a, b) for 1 < p ≤ q ≤ 2, 2 ≤ p ≤ q < ∞ and 1 < p ≤ 2 ≤ q < ∞. The main result is that, with appropriate assumptions on u and v, the approximation numbers an(T ) of T satisfy the inequality c1 ∫ b a |uv|dt ≤ lim inf n→∞ nan(T ) ≤ lim sup n→∞ nan(T ) ≤ c2 ∫ b a ...

An additive and hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let P and Q be two additive and hereditary graph properties and let r, s be integers such that r ≥ s. Then an r s -fractional (P,Q)-total coloring of a finite graphG = (V,E) is a mapping f , which assigns an s-element subset of the set {1, 2, . . . , r} to each vert...

Consider the equation dx dt +Q(t)G(x(t− σ(t))) = f(t) (∗) where f, σ,Q ∈ C([0,∞), [0,∞)), G ∈ C(R,R),G(−x) = −G(x), xG(x) > o for x 6= 0, G is non-decreasing, t > σ(t), σ(t) is decreasing and t − σ(t) → ∞ as t → ∞. Whenf(t) ≡ 0, a sufficient condition in terms of the constants k = lim inf t→∞ ∫ t t−σ(t) Q(s)ds and L = lim sup t→∞ ∫ t

in this note we are going to survey several invariants of finite groups related either to their orders or to generating sets or to lattices of subgroups. some relations among these invariants will be exhibited. special attention will be paid to monotonicity of them.

and Applied Analysis 3 In 2005, C. E. Chidume and C. O. Chidume 3 proved the convergence theorems for fixed points of uniformly continuous generalized Φ-hemicontractive mappings and published in 3 . However, there exists a gap in the proof course of their theorems. The aim of this paper is to show the convergence of the multistep iteration with errors for fixed points of uniformly continuous Φ-...

In this paper we investigate the dynamics of solitons occurring in the nonlinear Schroedinger equation when a parameter h → 0. We prove that under suitable assumptions, the the soliton approximately follows the dynamics of a point particle, namely, the motion of its barycenter q h (t) satisfies the equation ¨ q h (t) + ∇V (q h (t)) = H h (t) where sup t∈R |H h (t)| → 0 as h → 0.

We demonstrate fiber nonlinearity mitigation by using multiple optical phase conjugations (OPCs) in the WDM transmission systems of both 8 × 32-Gbaud PDM QPSK channels and 8 × 32-Gbaud PDM 16-QAM channels, showing improved performance over a single mid-span OPC and no OPC in terms of nonlinear threshold and a best achievable Q2 factor after transmission. In addition, after an even nu...

A new class of Embedded Multiple Description Scalar Quantizers (EMDSQ) is proposed. EMDSQ meet the desired features of high redundancy level, fine grain rate adaptation and progressive transmission of each description. Experimental results show that EMDSQ yield better rate-distortion performance in comparison to Multiple Description Uniform Scalar Quantizers (MDUSQ) previously proposed in the l...

This paper is concerned with the relationship between contexts, closure spaces, and complete lattices. It is shown that, for a unital quantale L, both formal concept lattices and property oriented concept lattices are functorial from the category L-Ctx of L-contexts and infomorphisms to the category L-Sup of complete L-lattices and suprema-preserving maps. Moreover, the formal concept lattice f...

We report direct measurements of the frequency ratio 642 THz ${}^2S_{1/2} (F=0)$--${}^2F_{7/2} (F=3)$ electric octupole transition in ${}^{171}\mathrm{Yb}^+$ and 429 ${}^1S_0$--${}^3P_0$ ${}^{87}\mathrm{Sr}$. A series 107 has been performed at Physikalisch-Technische Bundesanstalt between December 2012 October 2019. Long-term variations are larger than expected from individual measurement uncer...