Asymptotics of Eigenvalues of Differential Operator With Alternating Weight Function

We study a differential operator of the sixth order with an alternating weight function. The potential of the operator has a first-order discontinuity at some point of the segment, where the operator is being considered. The boundary conditions are separated. We study the asymptotics of solutions to the corresponding differential equations and the asymptotics of eigenvalues of the considered differential operator.     

On the Asymptotics of Eigenvalues of a Fourth-Order Differential Operator with Matrix Coefficients

We study a fourth-order differential operator with matrix coefficients whose domain is determined by the Dirichlet boundary conditions. An asymptotics of the weighted average of the eigenvalues of this operator is obtained in the general case. As a consequence of this result, we present the asymptotics of the eigenvalues in several special cases. The obtained results significantly improve the already known asymptotic formulas for the eigenvalues of a one-dimensional fourth-order differential operator.     

Accretivity of the General Second Order Linear Differential Operator

For the general second order linear differential operator ■ with complex-valued distributional coefficients a_(jk), b_j, and c in an open set Ω ? R~n(n ≥ 1), we present conditions which ensure that-L~0 is accretive, i.e., Re-L_0φ, φ ≥ 0 for all φ∈ C_0~∞(Ω).     

A Second Order Differential Operator with Eigenvalue Degeneracy

The second order differential operator Lu = ?u″ + (M ? |x|)u defined on ( ?∞, ∞) is analyzed. It is shown that the pair of successive eigenvalues λ_2n, λ_2n+1 is asymptotically degenerate as M becomes large, for all n.     

Some Approximations of the Characteristics of Second Order Hyperbolic Partial Differential Equations

The characteristic curves of the quasi-linear hyperbolic partialdifferential equation of the second order, having unequallyspaced, discreate initial conditions, are approximated by L₂parabolas, L₁ parabolas and cubic splines.     

一类高阶微分方程第二特征值的上界

本文考虑形如（－１）ｔＤｔ（ｐ（ｘ）Ｄｔｙ）＝λ（－Ｄ２）ｒｙ，ｘ∈（ａ，ｂ），Ｄｋｙ（ａ）＝Ｄｋｙ（ｂ）＝０，ｋ＝０，１，２，…，ｔ－１｛的第二特征值λ２的上界问题，得到了定理１和定理２，其中定理１的估计系数与［ａ，ｂ］无关，定理２的结果在一定条件下比定理１的好．     

一类高阶微分方程第二特征值的上界

本文研究形如（１．１）的第二特征上界问题,得到了定理１ 和定理２,其中定理１ 的估计系数与［ａ,ｂ］无关,定理２ 的结果在一定条件下比定理１ 好．     

The Minimal Process Generated by a Differential Operator of the Second Order in R^n

For a differential operator $\Omega u=\sum\limits_i,j=1^n \frac{\partial}{\partial x_i}(a_ij(x)\frac{\partial u}{\partial x_j})+\sum\limits_{i=1}^n b_i(x)\frac{\partial u}{\partial x_i}+c(x)u$ with unbounded coefficients in R^n, a standard continuous paths process with infinitesimal operator \Omega has been constructed in this paper, and the invariance of such process under a transformation group of phase space has been discussed.     

二阶中立型双曲偏泛函微分方程解的振动性质

研究一类二阶中立型双曲偏泛函微分方程解的振动性质，获得了在齐次Dirichlet，Neumann和Robin边界条件下，解振动或非振动的充分条件．     

Asymptotic Solution of a Non-linear Second Order Hyperbolic Differential Equation with Large Time Delay

The solution is given for a certain hyperbolic partial differentialequation with small non-linearity and large time delay includedinto both unperturbed and perturbed parts of the equation.     

On the Multiplicity of Eigenvalues of a Fourth-Order Differential Operator on a Graph

Differential Equations - We study the properties of eigenvalues of a boundary value problem for a fourth-order differential equation on a geometric graph modeling the elastic deformations of a...     

Eigenvalues of a Differential Operator and Zeros of the Riemann $zeta$-Function

The eigenvalues of a differential operator on a Hilbert-Pόlya space are determined. It is shown that these eigenvalues are exactly the nontrivial zeros of the Riemann $zeta$-function. Moreover, their corresponding multiplicities are the same.     

Two-Dimensional Trace-Normed Canonical Systems of Differential Equations and Selfadjoint Interface Conditions

The class of two-dimensional trace-normed canonical systems of differential equations on is considered with selfadjoint interface conditions at 0. If one or both of the intervals around 0 are H-indivisible the interface conditions which give rise to selfadjoint relations (multi-valued operators) are determined. It is shown that the corresponding generalized Fourier transforms are partially isometric. Compression to the halfline (0, ) results in boundary conditions which depend on the eigenvalue parameter involving a fractional linear transform of the Titchmarsh-Weyl coefficient of the halfline (–, 0). The corresponding generalized Fourier transforms are isometric except possibly on a one-dimensional subspace where they are contractive.     

Second Order Asymptotics of Aggregated Log-Elliptical Risk

In this paper we establish the error rate of first order asymptotic approximation for the tail probability of sums of log-elliptical risks. Our approach is motivated by extreme value theory which allows us to impose only some weak asymptotic conditions satisfied in particular by log-normal risks. Given the wide range of applications of the log-normal model in finance and insurance our result is of interest for both rare-event simulations and numerical calculations. We present numerical examples which illustrate that the second order approximation derived in this paper significantly improves over the first order approximation.     

Positive Solutions to Second Order Singular Differential Equations Involving the One-Dimensional M-Laplace Operator

We consider a class of second order quasilinear differential equations with singular ninlinearities. Our main purpose is to investigate in detail the asymptotic behavior of their solutions defined on a positive half-line. The set of all possible positive solutions is classified into five types according to their asymptotic behavior near infinity, and sharp conditions are established for the existence of solutions belonging to each of the classified types.     

Second Order Regularity for the A-Laplace Operator

In this paper we establish second order regularity for the quasilinear elliptic equation Δ _A u =  f, where Δ _A is the so called A-Laplace operator.     

边界条件含特征参数的奇异不连续Sturm-Liouville算子的渐近特征

研究有限区间内一类边界条件含特征参数的不连续奇异Sturm-Liouville问题.利用函数论和算子理论的方法,证明该问题的自伴性,得到其特征值的相关性质,基本解及其特征值的渐近公式.     

四阶不定微分算子非实特征值的界

给出了四阶正则不定微分算子仅在可积条件下的非实特征值上界和下界的估计.更一般地,非实特征值的下界可以利用Krein空间的自共轭算子得到.