On efficient computation and asymptotic sharpness of Kalantari's bounds for zeros of polynomials

We study an infinite family of lower and upper bounds on the modulus of zeros of complex polynomials derived by Kalantari. We first give a simple characterization of these bounds which leads to an efficient algorithm for their computation. For a polynomial of degree our algorithm computes the first bounds in Kalantari's family in operations. We further prove that for every complex polynomial these lower and upper bounds converge to the tightest annulus containing the roots, and thus settle a problem raised in Kalantari's paper.

     

Analytical and numerical aspects of a generalization of the complementary error function

In this paper we discuss analytical and numerical properties of the function , with α,β,Rz>0, which can be viewed as a generalization of the complementary error function, and in fact also as a generalization of the Kummer U-function. The function V_ν,μ(α, β, z) is used for certain values of the parameters as an approximate in a singular perturbation problem. We consider the relation with other special functions and give asymptotic expansions as well as recurrence relations. Several methods for its numerical evaluation and examples are given.     

Szeg? type polynomials and para-orthogonal polynomials

Szeg? type polynomials with respect to a linear functional M for which the moments M[tn]=μ−n are all complex, μ−n=μn and Dn≠0 for n?0, are considered. Here, Dn are the associated Toeplitz determinants. Para-orthogonal polynomials are also studied without relying on any integral representation. Relation between the Toeplitz determinants of two different types of moment functionals are given. Starting from the existence of polynomials similar to para-orthogonal polynomials, sufficient conditions for the existence of Szeg? type polynomials are also given. Examples are provided to justify the results.     

Studies on the Cure Reaction and Relationship between Network Structure/Thermal Properties of Styrene Copolymers Based on Adypic/Sebacic Acid Modified Unsaturated (Epoxy) Polyesters

The studies on the relationship between network structure/thermal properties of styrene copolymers based on adypic/sebacic acid modified unsaturated (epoxy) polyesters cured using different hardeners as well as the course of the cure reaction of polyesters with styrene have been presented. The adypic/sebacic acid modified unsaturated polyesters (UP) prepared from 4-cyclohexene-1,2-dicarboxylic anhydride (THPA), maleic anhydride (MA), adypic acid (AA) or sebacic acid (SA) and ethylene glycol (EG) and their epoxy derivatives: adypic/sebacic acid modified unsaturated epoxy polyesters (UEP) were subjected to the cure process with styrene using diacyl peroxide: benzoyl peroxide (BPO) or the mixture of BPO/suitable acid anhydride: 4-cyclohexene-1,2-dicarboxylic anhydride (THPA) or glutaric anhydride (GA). Thermal properties were evaluated by means of DSC, TG and DMA analyses. It was proved that studied properties were significantly depended on polyester's structure and the type of applied curing system. Generally, higher values of E'_20°C, tgδ_max, E”, ν_e, IDT, T_k for styrene copolymers based on UEP were obtained. It was connected with more cross-linked polymer network structure due to the possible copolymerization reaction of carbon-carbon double bonds of polyester with styrene and additional polyaddition of epoxy to anhydride groups or thermal curing of epoxy groups. The additional connections between polyester's chains led to obtain more stiff and thermal stable polymeric materials. Moreover, the increase of saturated aliphatic acid's chain length in polyester backbone caused the decrease of E'_20°C, tgδ_max, E”, ν_e, IDT, T_k values of styrene copolymers. It suggested that copolymers based on polyesters prepared from acid containing more methylene groups in their structure were characterized by more flexible polymer network due to the “laxity” effect of aliphatic chains.     

Hypergeometric series solutions of linear operator equations

Let K be a field and L:K[x]→K[x] be a linear operator acting on the ring of polynomials in x over the field K. We provide a method to find a suitable basis {b_k(x)} of K[x] and a hypergeometric term c_k such that is a formal series solution to the equation L(y(x))=0. This method is applied to construct hypergeometric representations of orthogonal polynomials from the differential/difference equations or recurrence relations they satisfied. Both the ordinary cases and the q-cases are considered.     

Learning intransitive reciprocal relations with kernel methods

In different fields like decision making, psychology, game theory and biology, it has been observed that paired-comparison data like preference relations defined by humans and animals can be intransitive. Intransitive relations cannot be modeled with existing machine learning methods like ranking models, because these models exhibit strong transitivity properties. More specifically, in a stochastic context, where often the reciprocity property characterizes probabilistic relations such as choice probabilities, it has been formally shown that ranking models always satisfy the well-known strong stochastic transitivity property. Given this limitation of ranking models, we present a new kernel function that together with the regularized least-squares algorithm is capable of inferring intransitive reciprocal relations in problems where transitivity violations cannot be considered as noise. In this approach it is the kernel function that defines the transition from learning transitive to learning intransitive relations, and the Kronecker-product is introduced for representing the latter type of relations. In addition, we empirically demonstrate on two benchmark problems, one in game theory and one in theoretical biology, that our algorithm outperforms methods not capable of learning intransitive reciprocal relations.     

Activating the Violation of Steering Inequality for Two‐Qubit X States

The phenomenon of superactivation of quantum steering, that is, the fact that by combining several copies of the unsteerable quantum state one can get a steerable quantum state, has recently attracted much attention. This property is currently only known for isotropic states, and whether there exists an example other than isotropic states is so far an open question. It is of great importance to learn the activation property of non‐isotropic states. In this paper, it is shown that the violation of the local uncertainty relations (LUR) steering inequality can be activated for two‐qubit X type entangled mixed states. Although the activation phenomenon proven in this paper cannot guarantee the superactivation of steering for two‐qubit X states, it does provide clues for this goal. In addition, previous results indicate that two copies are enough for the superactivation of steering, while the price to pay is a high local dimension, and it is unclear whether superactivation is possible with few copies in low‐dimensional systems such as two‐qubit systems. By providing examples of activation of LUR steering inequality with two copies in a two‐qubit system, the work may open a potential way to the solution of this question.     

A Study on the Correlation Between Poles and Cuts in ππ Scattering

In this paper we propose a dispersive method to describe two-body scattering with unitarity imposed. This approach is applied to elastic ππ scattering. The amplitudes keep single-channel unitarity and describe the experimental data well, and the low-energy amplitudes are consistent with that of chiral perturbation theory. The pole locations of the σ, f0(980), ρ(770) and f2(1270) and their couplings to ππ are obtained. A virtual state appearing in the isospin-two S-wave is confirmed. The correlations between the left(and right) hand cut and the poles are discussed. Our results show that the poles are more sensitive to the right hand cut rather than the left hand cut. The proposed method could be used to study other two-body scattering processes.     

Seiberg-Witten/Whitham Equations and Instanton Corrections in N=2 Supersymmetric Yang-Mills Theory

We obtain the instanton correction recursion relations for the low energy effective prepotential in pure N=2 SU(n) supersymmetric Yang-Mills gauge theory from Whitham hierarchy and Seiberg-Witten/Whitham equations. These formulae provide us a powerful tool to calculate arbitrary order instanton corrections coefficients from the perturbative contributions of the effective prepotential in Seiberg-Witten gauge theory. We apply this idea to evaluate one- and two- order instanton corrections coefficients explicitly in SU(n) case in detail through the dynamical scale parameter expressed in terms of Riemann's theta-function.