一类变号(k,n-k)共轭边值问题正解的存在性

通过构造一个特殊的锥,利用范数形式的锥拉伸锥压缩不动点定理,在允许非线性项变号无下界且没有任何单调性假设的条件下,得出了一类高阶(k,n-k)共轭两点边值问题方程组正解的存在性结论.     

Simple Equations Method (SEsM): Algorithm,Connection with Hirota Method,Inverse Scattering Transform Method,and Several Other Methods

The goal of this article is to discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations and to show that several well-known methods for obtaining exact solutions of such equations are connected to SEsM. In more detail, we show that the Hirota method is connected to a particular case of SEsM for a specific form of the function from Step 2 of SEsM and for simple equations of the kinds of differential equations for exponential functions. We illustrate this particular case of SEsM by obtaining the three- soliton solution of the Korteweg-de Vries equation, two-soliton solution of the nonlinear Schrödinger equation, and the soliton solution of the Ishimori equation for the spin dynamics of ferromagnetic materials. Then we show that a particular case of SEsM can be used in order to reproduce the methodology of the inverse scattering transform method for the case of the Burgers equation and Korteweg-de Vries equation. This particular case is connected to use of a specific case of Step 2 of SEsM. This step is connected to: (i) representation of the solution of the solved nonlinear partial differential equation as expansion as power series containing powers of a “small” parameter ϵ; (ii) solving the differential equations arising from this representation by means of Fourier series, and (iii) transition from the obtained solution for small values of ϵ to solution for arbitrary finite values of ϵ. Finally, we show that the much-used homogeneous balance method, extended homogeneous balance method, auxiliary equation method, Jacobi elliptic function expansion method, F-expansion method, modified simple equation method, trial function method and first integral method are connected to particular cases of SEsM.     

Existence of multi-bump solutions for a system with critical exponent in RN

We consider the following system with critical exponent in ${\mathbb{R}}^N$:$\{\begin{array}{c}?\mathrm{\Delta }u=K_1(y)u^{2^{?}?1}+\frac{p}{2^{?}}V(y)u^{p?1}{v}^q\text{in}{\mathbb{R}}^N,\hfill \\ ?\mathrm{\Delta }v=K_2(y)v^{2^{?}?1}+\frac{q}{2^{?}}V(y)u^p{v}^{q?1}\text{in}{\mathbb{R}}^N,\hfill \\ u,v>0,y\in {\mathbb{R}}^N,\hfill \end{array}$ where $N\ge 5$, $p,q>1$ and $p+q=2^{?}=\frac{2N}{N?2}$. Using finite dimensional reduction method, we prove the existence of multi-bump solutions. Their bumps can be placed on arbitrarily many or even infinitely many lattice points in ${\mathbb{R}}^N$. Since $p<2$ or $q<2$, we introduce two new norms to avoid singularity.     

具有四个素因子的奇亏完全数

设n为自然数,σ(n)表示n的所有正因子和函数.令d是n的真因子,若n满足σ(n)=2n-d,则称n为亏因子为d的亏完全数.本文给出了具有四个素因子的奇亏完全数的一些性质的刻画.     

New traveling wave solutions of Drinefel’d–Sokolov–Wilson Equation using Tanh and Extended Tanh methods

Traveling wave solutions are obtained by using a relatively new technique which is called Tanh and extended Tanh method for Drinefel’d–Sokolov–Wilson Equations. Solution procedure and obtained results re-confirm the efficiency of the proposed scheme.     

The Synthesis of B2(SIDip)2 and its Reactivity Between the Diboracumulenic and Diborynic Extremes

A new compound with the formula L‐B₂‐L wherein the stabilizing ligand (L) is 1,3‐bis[diisopropylphenyl]‐4,5‐dihydroimidazol‐2‐ylidene (SIDip) has been synthesized, isolated, and characterized. The π‐acidity of the SIDip ligand, intermediate between the relatively non‐acidic IDip (1,3‐bis[diisopropylphenyl]imidazol‐2‐ylidene) ligand and the much more highly acidic CAAC (1‐[2,6‐diisopropylphenyl]‐3,3,5,5‐tetramethylpyrrolidin‐2‐ylidene) ligand, gives rise to a compound with spectroscopic, electrochemical, and structural properties between those of L‐B₂‐L compounds stabilized by CAAC and IDip. Reactions of all three L‐B₂‐L compounds with CO demonstrate the differences caused by their respective ligands, as the π‐acidities of the CAAC and SIDip carbenes enabled the isolation of bis(boraketene) compounds (L(OC)B‐B(CO)L), which could not be isolated from reactions with B₂(IDip)₂. However, only B₂(IDip)₂ and B₂(SIDip)₂ could be converted into bicyclic bis(boralactone) compounds.     

Rethinking pedagogy for second-order differential equations: a simplified approach to understanding well-posed problems

Knowing an equation has a unique solution is important from both a modelling and theoretical point of view. For over 70 years, the approach to learning and teaching ‘well posedness’ of initial value problems (IVPs) for second- and higher-order ordinary differential equations has involved transforming the problem and its analysis to a first-order system of equations. We show that this excursion is unnecessary and present a direct approach regarding second- and higher-order problems that does not require an understanding of systems.     

Efficient remainder rule

Understanding the solution of a problem may require the reader to have background knowledge on the subject. For instance, finding an integer which, when divided by a nonzero integer leaves a remainder; but when divided by another nonzero integer may leave a different remainder. To find a smallest positive integer or a set of integers following the given conditions, one may need to understand the concept of modulo arithmetic in number theory. The Chinese Remainder Theorem is a known method to solve these types of problems using modulo arithmetic. In this paper, an efficient remainder rule has been proposed based on basic mathematical concepts. These core concepts are as follows: basic remainder rules of divisions, linear equation in slope intercept form, arithmetic progression and the use of a graphing calculator. These are easily understood by students who have taken prealgebra or intermediate algebra.