Solution of the differential equation (∂2∂x∂y + ax∂∂x + by∂∂y + cxy + ∂∂t) P(x,y, t) = 0 and the Bogoliubov transformation

An eigen function expansion for the solution of the Lambropolous partial differential equation is obtained by the use of a transformation similar to the Bugolubov transformation familiar in Bose gas theory. Also the technique of normal ordering of operators is employed. The orthogonality properties of such solutions are also analysed.     

Observations about the Lie algebra g2⊂so(7), associative 3-planes,and so(4) subalgebras

We make several observations relating the Lie algebra ${\mathfrak{g}}_2?\mathfrak{so}\left(7\right)$, associative 3-planes, and $\mathfrak{so}\left(4\right)$ subalgebras. Some are likely well-known but not easy to find in the literature, while other results are new. We show that an element $X\in {\mathfrak{g}}_2$ cannot have rank 2, and if it has rank 4 then its kernel is an associative subspace. We prove a canonical form theorem for elements of ${\mathfrak{g}}_2$. Given an associative 3-plane $P$ in ${\mathbb{R}}^7$, we construct a Lie subalgebra $\Theta \left(P\right)$ of $\mathfrak{so}\left(7\right)={\Lambda }^2\left({\mathbb{R}}^7\right)$ that is isomorphic to $\mathfrak{so}\left(4\right)$. This $\mathfrak{so}\left(4\right)$ subalgebra differs from other known constructions of $\mathfrak{so}\left(4\right)$ subalgebras of $\mathfrak{so}\left(7\right)$ determined by an associative 3-plane. These are results of an NSERC undergraduate research project. The paper is written so as to be accessible to a wide audience.     

Remarks on Nonassociative Structures in Quantum Projective Field Theory: the Central Extension jl(2,C) of the Double sl(2,C)+sl(2,C) of the Simple Lie Algebra sl(2,C) and Related Topics

This paper is a revised and expanded version of two notes devoted to nonassociative structures in quantum projective field theory.     

An Upper and Lower Solution Theory for the Problem （G＇（y））＇ ＋ f（t,y）= 0 on Finite and Infinite Intervals

A new upper and lower solution theory is presented for the second order problem （G＇（y））＇＋ f（t, y） = 0 on finite and infinite intervals. The theory on finite intervals is based on a Leray-Schauder alternative, where as the theory on infinite intervals is based on results on the finite interval and a diagonalization process.     

微分方程组x=a sum from (i j=2) (aijxiyj),y=b sum from (i j=2)(bijxiyj)的极限环存在的分歧值与代数奇异环

In this paper we consider algebraic critical cycles and bifurcation of limit cycles for the system .     

On the Estimates of Widths of the Classes of Functions Defined by the Generalized Moduli of Continuity and Majorants in the Weighted Space L2,x(0, 1)

Ukrainian Mathematical Journal - The upper and lower estimates for the Kolmogorov, linear, Bernstein, Gelfand, projective, and Fourier widths are obtained in the space L2,x(0, 1) for the classes of...     

一类广义积分integral from n=0 to ∞(((sin(θ_x))/x)~ndx)的计算

利用三角函数幂公式、L′Hospital法则、分部积分公式,得到含有三角函数的第一类广义积分integral from n=0 to ∞(((sin(θ_x))/x)~ndx)的计算公式,其中n≥1且θ≠0.     

关于奇异二阶微分方程（Φ（X’））’＋f（t，X）＝0的某些可解准则

本文讨论了一类右端具有奇异性的二阶常微分方程的边值问题，证明了当ｆ（ｔ，ｘ）关于ｔ的奇异程度大于某值时，边值问题无解，而当奇异程度小于某值时，边值问题有解．关于ｕ在ｕ＝０的奇异程度没有限制．     

On the completeness of the functions e−np(x) cos nx,e−np(x) sin nx for n ≥ 0 and p(x) a 2π periodic function

It is shown how a conjecture of G. B. Whitham on the completeness of the functions { ψ _n } = { e^?np(x) cos nx, e^?np(x) sin nx } in the interval [0, 2π] is indeed true provided p has a Hölder continuous first derivative, i.e., p ∈ C^{1, α}[0, 2π]. Also an elementary proof of Whitham's conjecture is given for p ∈ C^{2, α}[0, 2π].     

Solving the equation −uxx − ϵuyy = f(x,y, u) by an O(h4) finite difference method

The semi‐linear equation −u_xx − ϵu_yy = f(x, y, u) with Dirichlet boundary conditions is solved by an O(h⁴) finite difference method, which has local truncation error O(h²) at the mesh points neighboring the boundary and O(h⁴) at most interior mesh points. It is proved that the finite difference method is O(h⁴) uniformly convergent as h → 0. The method is considered in the form of a system of algebraic equations with a nine diagonal sparse matrix. The system of algebraic equations is solved by an implicit iterative method combined with Gauss elimination. A Mathematica module is designed for the purpose of testing and using the method. To illustrate the method, the equation of twisting a springy rod is solved. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 395–407, 2000     

The asymptotic eigenfunctions of the operator ?D(x, y)? corresponding to Liouville metrics and waves on water captured by bottom irregularities

We construct a set of examples of bottom reliefs for which there exist captured waves corresponding to quasimodes of the wave operator ∇D(x, y)∇. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 414–422, September, 1998. The author thanks S. Yu. Dobrokhotov for stating the problem and numerous helpful discussions.