Evaluation of Certain Integrals Involving the Product of Classical Hermite's Polynomials Using Laplace Transform Technique and Hypergeometric Approach

In this paper some novel integrals associated with the product of classical Hermite's polynomials ∫_(-∞)~(+∞)(x~2)~mexp(-x~2){Hr (x)}2dx, ∫_0~∞exp(-x~2)H_(2k)(x)H_(2s+1)(x)dx,∫_0~∞exp(-x~2)H_(2k)(x)H_(2s)(x)dx and ∫_0~∞exp(-x~2)H_(2k+1)(x)H_(2s+1)(x)dx,are evaluated using hypergeometric approach and Laplace transform method, which is a different approach from the approaches given by the other authors in the field of special functions. Also the results may be of significant nature, and may yield numerous other interesting integrals involving the product of classical Hermite's polynomials by suitable simplifications of arbitrary parameters.     

凸函数的几个Hadamard型不等式

对于凸函数建立了几个新的 Hadamard型不等式 ,比如f ∑nk=1qkak∫A+yA- yg(x) dx ∫A+yA- yf (x) g(x) dx ∑nk=1qkf (ak) ∫A+yA- yg(x) dx和f (pa +qb) p∫ξaf (x) g(x) dx∫ξag(x) dx+q∫bξf (x) g(x) dx∫bξg(x) dx pf (a) +qf (b)等 ,推广了前人所做的工作 .     

一类乘积核的本征值分布

本文讨论了一类形如 G( x,y) =m1( x) K( x,y) m2 ( y)的乘积核所诱导的积分算子的本征值的分布问题 ,其中 K ( x,y)∈ CΩ×Ω 是正定的 .当 m1( x) ,m2 ( x)∈ CΩ 并且 m1( x) m2 ( x) 0时 ,我们证明了TG∶ L2 ( Ω)→L2 ( Ω)是迹算子 ,其本征值非负并得到了一个迹公式∑n∈ Nλn( TG) =∫Ωm1( x) K ( x,x) m2 ( x) dx.对于 m1( x) ,m2 ( x)∈ L∞( Ω)的情形 ,我们证明了一个稍弱的结果 .∑n∈ N|λn( TG) | ‖ m1. m2 ‖L∞∫ΩK( x,x) dx.     

A method for the integration of oscillatory functions

A method for the numerical evaluation of the integrals $$I_1 (\lambda ) = \int_{ - 1}^1 {f(x)\sin (\lambda x)dx} andI_2 (\lambda ) = \int_{ - 1}^1 {f(x)\cos (\lambda x)dx} $$ is presented. The functionf(x) is approximated by a partial sum of its Legendre series.     

一个局部波动方程组的解的全局存在性与爆破

我们研究初始值问题(e)u1/(e)t2=(e)2u1/(e)x2+‖u2(·,t)‖p, (e)2u2/(e)t2=(e)2u2/(e)x2+‖u1(·,t)‖q,-∞<x<∞,t>0,u1(x,0)=f1(x), (e)u1/(e)t(x,0)=g1(x),u2(x,0)=f2(x), (e)u2/(e)t(x,0)=g2(x),- ∞<x<∞,where‖ui(·,t)‖=∫∞-∞(4)i(x)|ui(x,t)|dx with (4)i(x)≥0 and ∫∞-∞(4)i(x)dx=1,i=1,2.然后建立解的全局存在和爆破的标准,提出爆破增长率.     

Minimal coupled diffusion process

An attempt is made to classify the boundary of the coupled operator and to construct the corresponding minimal semigroup and minimal coupled diffusion process. The sample properties and the conservative conditions of the process are discussed also.     

一类非线性系统解的有界性

该文研究了非线性微分系统(dx/dt)=h(y)-φ(x), (dy/dt)=-h(y)f(x)-g(x)k(y)解的有界性。获得该系统的所有解有界的充分条件。应用此结果于Liénard方程 d^2x/dt^2+f^*(x)(dx/dt)+g)^*(x)=0,改进和推广了文［1-6］中的相应结果。      

The Existence of Almost periodic Solutions of Singularly Perturbed Systems

First the author considers the system (1)$\frac{dx}{dt}=f(t,x,y,\varepsilon),\varepsilon\frac{dy}{dt}=g(t,x,y,\varepsilon)$ and its degenerate system (2)$\frac{dx}{dt}=f(t,x, y, 0), g(f, x, y, 0) =0$. In both noncritical and critical cases, sufficient conditions are established for the existence of almost periodic solutions of system (1) near the given solutions of system (2). The main method of proof is that, by performing suitable transformation, the author establishes exponential dichotomies, and then applies the theory of integral manifolds. Secondly, for the autonomous system (3) $\frac{dx}{dt}=f(x,y,\varepsilon),\varepsilon\frac{dy}{dt}=g(x,y,\varepsilon)$, analogous results are obtained by performing the generalized normal coordinate transformation.     

Regularity Results for Minimizers of Certain Functional Having Nonquadratic Growth with Obstacles

We prove partial regularity for minimizers of degenerate variational integrals ∫_Ω F(x, u, Du)dx with obstacles of either the form (i) μ_f = {u ∈ H^{1,m} (Ω,\mathbb{R}^N)|u^N ≥ f_1(u¹, ... ,u^{N-1}) + f_2(x) a.e.} or (ii) μ_N = {u ∈ H^{1,m}(Ω,\mathbb{R}^N)|u^i(x) ≥ h^i(x), a.e.; i=1, ... ,N} The typical mode of variational integrals is given by ∫_Ω [a^{αβ}(x, u)b_{ij}(x, u)D_αu^i D_βu^i]^{\frac{m}{2}}dx, m ≥ 2     

Almost Periodic Solutions of Equation $[\dot x = {x^3} + \lambda g(t)x + \mu f(t)\]$ and Their Stability

By using the Liapunov function and the contraction mapping principle, the author investigates the existence and stability of almost periodic solutions of the first order nonlinear equations $\frac{dx}{dt}=-h_1(x)+h_2(x)g(t)+f(t)$ and $\frac{dx}{dt}=r(t)x^n+\lambdag(t)x+\muf(t)$, where r(t), g(t), f(t) are given almost periodic functions, n(\geq 2) integer, and \lambda,\mu real parameters.     

Γ-Convergence of some super quadratic functionals with singular weights

We study the Γ-convergence of the following functional (p > 2)

The evaluation and application of some modified moments

Recurrence formulas for the calculation of the modified moments $$\int\limits_{ - 1}^{ + 1} {(1 - x)^\alpha (1 + x)^\beta T_n (x)dx} $$ and $$\int\limits_{ - 1}^{ + 1} {(1 - x)^\alpha (1 + x)^\beta \ln \left( {\frac{{1 + x}}{2}} \right)T_n (x)dx} $$ are presented. Some applications of these modified moments are discussed, such as the numerical calculation of integrals of functions having branch points, the computation of Chebyshev series coefficients and the construction of Gaussian quadrature formulas for integrals with logarithmic singularity.     

常微分方程的區域分析理論(Ⅲ)

<正> 本文是繼續[Ⅰ],[Ⅱ]的發展,並初步總結其中若干部分,定名為分區線性方程.總結其特性及簡易的定性處理法,使這類的非線性方程的規律性非常容易掌握;正如我們掌握常係數線性方程一樣.     

Estimating the derivative of the Legendre polynomial

For the derivative of the Legendre polynomial, the estimates

On the error term for the counting functions of finite Abelian groups

Leta(n) denote the number of non-isomorphic Abelian groups withn elements, and Δ(x) (resp. Δ _x ) appropriate error terms in the asymptotic formulas for the counting function \(\sum\nolimits_{n \leqslant x} {a(n)} (resp. \sum\nolimits_{m n \leqslant x} {a(m)} a(n))\) . Sharp bounds for $$\int\limits_1^X {\Delta (x) dx} , \int\limits_1^X {\Delta _{ 1} (x) dx} ,\int\limits_1^X {\Delta _1^2 (x) dx} $$ are given by using results on power moments of the Riemann zeta-function.     

A variational difference scheme for the one-dimensional diffusion equation

In this paper we construct a homogeneous variational difference scheme for the diffusion equation assuming its coefficients to be bounded and measurable; the order of convergence of the scheme is O(h²). We consider the boundary value problem (1) $$\frac{d}{{dx}}\left( {K(x)\frac{{du}}{{dx}}} \right) - g(x)u = - \frac{{dF}}{{dx}},0< x< X$$ subject to the boundary conditions (2) $$u(0) = a,u(X) = b$$ .     

How to use the Fundamental Theorem of Calculus?

Now we can put the two part of the Fundamental Theorem of Calculus together named the Fundamental Theorem of Calculus:Suppose f（x）is continuous on[a,b].1.If F（x）=∫₀^xf（t）dt,then F’（x）=f（x）.2.∫_α^bf（x）dx=F（b）-F（a）,where     

Boundedness Characterization of Maximal Commutators on Orlicz Spaces in the Dunkl Setting

On the real line, the Dunkl operators$$D_{\nu}(f)(x):=\frac{d f(x)}{dx} + (2\nu+1) \frac{f(x) - f(-x)}{2x}, ~~ \quad\forall \, x \in \mathbb{R}, ~ \forall \, \nu \ge -\tfrac{1}{2}$$are differential-difference operators associated with the reflection group $\mathbb{Z}_2$ on $\mathbb{R}$, and on the $\mathbb{R}^d$ the Dunkl operators $\big\{D_{k,j}\big\}_{j=1}^{d}$ are the differential-difference operators associated with the reflection group $\mathbb{Z}_2^d$ on $\mathbb{R}^{d}$.In this paper, in the setting $\mathbb{R}$ we show that $b \in BMO(\mathbb{R},dm_{\nu})$ if and only if the maximal commutator $M_{b,\nu}$ is bounded on Orlicz spaces $L_{\Phi}(\mathbb{R},dm_{\nu})$. Also in the setting $\mathbb{R}^{d}$ we show that $b \in BMO(\mathbb{R}^{d},h_{k}^{2}(x) dx)$ if and only if the maximal commutator $M_{b,k}$ is bounded on Orlicz spaces $L_{\Phi}(\mathbb{R}^{d},h_{k}^{2}(x) dx)$.     

Multiplicity of Weak Solutions for a $(p(x), q(x))$-Kirchhoff Equation with Neumann Boundary Conditions

The aim of this study is to investigate the existence of infinitely many weak solutions for the $(p(x), q(x))$-Kirchhoff Neumann problem described by the following equation : \begin{equation*} \left\{\begin{array}{ll} -\left(a_{1}+a_{2}\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\right)\Delta_{p(\cdot)}u-\left(b_{1}+b_{2}\int_{\Omega}\frac{1}{q(x)}|\nabla u|^{q(x)}dx\right)\Delta_{q(\cdot)}u\+\lambda(x)\Big(|u|^{p(x)-2} u+|u|^{q(x)-2} u\Big)= f_1(x,u)+f_2(x,u) &\mbox{ in } \Omega, \\frac{\partial u}{\partial \nu} =0 \quad &\mbox{on} \quad \partial\Omega.\end{array}\right. \end{equation*} By employing a critical point theorem proposed by B. Ricceri, which stems from a more comprehensive variational principle, we have successfully established the existence of infinitely many weak solutions for the aforementioned problem.     

一类亚纯函数的广义积分及其Cauchy主值

利用函数zp(-1相似文献