Analytic combinatorics of connected graphs

We enumerate the connected graphs that contain a number of edges growing linearly with respect to the number of vertices. So far, only the first term of the asymptotics and a bound on the error were known. Using analytic combinatorics, that is, generating function manipulations, we derive a formula for the coefficients of the complete asymptotic expansion. The same result is derived for connected multigraphs.     

Pseudo almost periodic in distribution solutions to impulsive partial stochastic functional differential equations

In this paper, we study the piecewise pseudo almost periodicity in distribution for a stochastic process. Using the analytic semigroup theory and fixed point strategy with stochastic analysis theory, we obtain the existence and the exponential stability of piecewise pseudo almost periodic in distribution mild solutions for impulsive partial neutral stochastic functional differential equations under non-Lipschitz conditions. Moreover, an example is given to illustrate the general theorems.     

Integrable symplectic maps associated with discrete Korteweg‐de Vries‐type equations

In this paper, we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable partial difference equations which are the discrete counterparts of integrable partial differential equations of Korteweg‐de Vries‐type (KdV‐type). As a consequence it is demonstrated that several distinct Hamiltonian systems lead to one and the same difference equation by means of the Liouville integrability framework. Thus, these integrable symplectic maps may provide an efficient tool for characterizing, and determining the integrability of, partial difference equations.     

Backward transfer influences from quadratic functions instruction on students’ prior ways of covariational reasoning about linear functions

The study reported in this article examined the ways in which new mathematics learning influences students’ prior ways of reasoning. We conceptualize this kind of influence as a form of transfer of learning called backward transfer. The focus of our study was on students’ covariational reasoning about linear functions before and after they participated in a multi-lesson instructional unit on quadratic functions. The subjects were 57 students from two authentic algebra classrooms at two local high schools. Qualitative analysis suggested that quadratic functions instruction did influence students’ covariational reasoning in terms of the number of quantities and the level of covariational reasoning they reasoned with. These results further the field’s understanding of backward transfer and could inform how to better support students’ abilities to engage in covariational reasoning.     

A new result on irreducible NFSRs with respect to cascade connection

Nonlinear feedback shift registers (NFSRs) are widely used in stream cipher design as building blocks. The cascade connection of NFSRs, known as an important architecture, has been adopted in Grain family of stream ciphers. In this paper, a new sufficient condition under which an NFSR cannot be decomposed into the cascade connection of two smaller NFSRs is presented, which is easy to be verified from the algebraic normal form (ANF) of the characteristic function. In fact, our results are also applicable to nonsingular Boolean functions, which actually improve a previous research of Rhodes [6] where the characteristic functions of NFSRs cannot be contained.     

Le Système d'Euler de Kato en Famillie(Ⅱ)

This article is the second article on the generalization of Kato's Euler system. The main subject of this article is to construct a family of Kato's Euler systems over the cuspidal eigencurve, which interpolate the Kato's Euler systems associated to the modular forms parametrized by the cuspidal eigencurve. We also explain how to use this family of Kato's Euler system to construct a family of distributions on Z_p over the cuspidal eigencurve; this distribution gives us a two-variable p-adic L function which interpolate the p-adic L function of modular forms.     

Application of h-adaptive,high order finite element method to solve radial Schrödinger equation

The h-adaptive, high order finite element method is applied to solve a second order one dimension eigenvalue problem. The finite element formulation for the Lobatto basis is given, for which basis functions of arbitrary order can be constructed. The adaptive algorithm is simple, yet very efficient and straightforward to implement. The algorithm is based on the observation that the expansion coefficients of Lobatto basis functions decay rapidly. It allows evaluating the smallest eigenvalues simultaneously with the comparable accuracy for all eigenvalues. The presented algorithm is applied to solve the radial Schrödinger equation with the Coulomb and the Woods–Saxon potentials. For both potentials the convergence rate is presented. After seven adaptive iterations nine-digit accuracy was obtained.     

Fortran codes for the correlation functions of hard sphere fluids

Our Fortran codes for hard sphere fluids and their mixtures for the correlation functions that arise from the Percus–Yevick theory and the Verlet–Weis semi-empirical correction have proven useful during a period of nearly four decades and continue to be useful. In order to make these codes even more widely available, a brief summary is presented here and listings of these codes are given in the electronically accessible Supplementary Material to this paper.     

Thermodynamics of substituted coumarin VII: Potentiometric and spectrophotometric studies of 1-acetyl-2-(coumarin-iminecarboxamide-3-yl) hydrazine transition metal complexes

The dissociation constant of l-acetyl-2-(coumariniminecarboxamide-3-yl) hydrazine (ACCH) has been determined potentiometrically in 0.1 M KC1 and 40% (v/v) ethanol-water mixture. The stepwise stability constants of the formed complexes of Mn²⁺, Co²⁺, Ni²⁺, Cu²⁺, Zn²⁺ and UO₂ ²⁺ with ACCH have been determined. The stability constants were found to UO₂ ²⁺ > Cu²⁺> Zn²⁺ > Ni²⁺ < CO²⁺ < Mn²⁺. The effect of temperature on the dissociation of ACCH and the stability of its formed complexes were studied. The corresponding thermodynamic functions were also determined and discussed. The spectral data of ACCH were investigated in pure organic solvents as well as in Britton Robinson buffer solutions of varying pH values. The dissociation constant pK^H ₁ of ACCH and the overall stability constants logβ of their complexes were determined in 20% (v/v) ethanol-water mixture at 298K.