Shape-Preserving Widths of Weighted Sobolev-Type Classes of Positive, Monotone, and Convex Functions on a Finite Interval

Abstract. Let I be a finite interval, r∈ N and ρ(t)= dist {t, I} , t∈ I . Denote by Δ ^s ₊ L _q the subset of all functions y∈ L _q such that the s -difference Δ ^s _τ y(t) is nonnegative on I , $$\forall$$ τ>0 . Further, denote by $$\Delta^s_+W^r_{p,\alpha}$$ , 0≤α<∞ , the classes of functions x on I with the seminorm ||x ^(r) ρ ^α ||_ L _p ≤ 1 , such that Δ ^s _τ x≥ 0 , τ>0 . For s=0,1,2 , we obtain two-sided estimates of the shape-preserving widths $$d_n (\Delta _ + ^s W_{p,\alpha ,}^r \Delta _ + ^s L_q )L_q : = \mathop {\inf }\limits_{M^n \in \mathcal{M}^n } \mathop {\sup }\limits_{x \in \Delta _ + ^s W_{p,\alpha }^r } \mathop {\inf }\limits_{y \in M^n \cap \Delta ^s + L_q } \left\| {x - y} \right\|L_q$$ where M ⁿ is the set of all linear manifolds M ⁿ in L _q such that dim M ⁿ ≤ n , and satisfying $$M^n\cap\Delta^s_+L_q\neq\emptyset$$ .     

Degree of Simultaneous Coconvex Polynomial Approximation

$ {\rm Let}\ f\ \epsilon\ {C^{1}}[-1,1] $ change its convexity finitely many times in the interval, say s times, at ${\rm at}\ {Y_{s}}\:\ -1\ <\ y_{s}\ <\ \dots\ < y_{1}\ < 1 $ . We estimate the degree of simultaneous approximation of ? and its derivative by polynomials of degree n, which change convexity exactly at the points Y _s, and their derivatives. We show that provided n is sufficiently large, depending on the location of the points Y _s, the rate of approximation can be estimated by C(s)/n times the second Ditzian-Totik modulus of smoothness of ?′. This should be compared to a recent paper by the authors together with I. A. Shevchuk where ? is merely assumed to be continuous and estimates of coconvex approximation are given by means of the third Ditzian-Totik modulus of smoothness. However, no simultaneous approximation is given there.     

59Co, 23Na NMR and electric field gradient calculations in the layered cobalt oxides NaCoO2 and HCoO2

⁵⁹Co and ²³Na NMR has been applied to the layered cobalt oxides NaCoO₂ and HCoO₂ at three different magnetic field strengths (4.7, 7.1 and 11.7 T). The ⁵⁹Co and ²³Na quadrupole and anisotropic shift tensors have been determined by iterative fitting of the NMR line shapes at the three magnetic field strengths. Due to the large ⁵⁹Co quadrupole interaction in NaCoO₂, a frequency-swept irradiation procedure was used to alleviate the limited bandwidth of the excitation. While the ⁵⁹Co and ²³Na shift and quadrupole coupling tensors in NaCoO₂ are found to be coincident and axially symmetric in agreement with the crystal symmetry requirements, the fits of the ⁵⁹Co NMR spectra clearly show the presence of structural disorder in HCoO₂. The ²³Na chemical shift anisotropy can be reproduced by shift tensor calculations using a point dipole model and considering that the magnetic susceptibility in NaCoO₂ is due to Van Vleck paramagnetism for Co³⁺. Electric field gradient calculations using either the empirical point charge model or the ab initio full potential-linearized augmented plane wave method are compared with the experimental NMR data.     

Partial dynamical symmetry in a fermion system

The relevance of the partial dynamical symmetry concept for an interacting fermion system is demonstrated. Hamiltonians with partial SU(3) symmetry are presented in the framework of the symplectic shell model of nuclei and shown to be closely related to the quadrupole-quadrupole interaction. Implications are discussed for the deformed light nucleus 20Ne.     

Recovered efficiency of filter control in dispersion-managed solitons for optical regeneration applications: analysis and experimental validation

We demonstrate numerically that local conversion of dispersion-managed return-to-zero pulses into nonlinear Schr?dinger solitons fully restores optical filtering control for energy stabilization, thus permitting the association of optical regeneration with dispersion-managed propagation. Experimental validation is achieved by means of 40-Gbit/s regenerated dispersion-managed loop transmission over 10,000 km.     

Absolute tauberian conditions for absolute hausdorff and quasi-hausdorff methods

The purpose of this paper is to prove that for a large set of absolute Hausdorff and quasi-Hausdorff methods the condition $$\sum\limits_{k = 1}^\infty {\left| {\lambda _n a_n - \lambda _{n - 1} a_{n - 1} } \right|< } \infty $$ is a Tauberian condition, i.e., its fulfillment together with the absolute summability of \(\sum\limits_{n = 0}^\infty {a_n } \) tos implies that \(\sum\limits_{n = 0}^\infty {\left| {a_n } \right|}< \infty \) and \(\sum\limits_{n = 0}^\infty {a_n } = s.\) a _n =s.     

Some vector-valued laplace transforms

Representation theorems for vector-valued Laplace transforms are discussed. Necessary and sufficient conditions are obtained in order that a function be the Laplace transform of a general vector measure and of a vector measure of finite variation, finiteq-variation or finiteq-semi-variation for 1<q≦∞.     

First-order quantum phase transitions: Test ground for emergent chaoticity,regularity and persisting symmetries

We present a comprehensive analysis of the emerging order and chaos and enduring symmetries, accompanying a generic (high-barrier) first-order quantum phase transition (QPT). The interacting boson model Hamiltonian employed, describes a QPT between spherical and deformed shapes, associated with its U(5) and SU(3) dynamical symmetry limits. A classical analysis of the intrinsic dynamics reveals a rich but simply-divided phase space structure with a Hénon–Heiles type of chaotic dynamics ascribed to the spherical minimum and a robustly regular dynamics ascribed to the deformed minimum. The simple pattern of mixed but well-separated dynamics persists in the coexistence region and traces the crossing of the two minima in the Landau potential. A quantum analysis discloses a number of regular low-energy U(5)-like multiplets in the spherical region, and regular SU(3)-like rotational bands extending to high energies and angular momenta, in the deformed region. These two kinds of regular subsets of states retain their identity amidst a complicated environment of other states and both occur in the coexistence region. A symmetry analysis of their wave functions shows that they are associated with partial U(5) dynamical symmetry (PDS) and SU(3) quasi-dynamical symmetry (QDS), respectively. The pattern of mixed but well-separated dynamics and the PDS or QDS characterization of the remaining regularity, appear to be robust throughout the QPT. Effects of kinetic collective rotational terms, which may disrupt this simple pattern, are considered.