共查询到20条相似文献,搜索用时 31 毫秒
1.
Jihua Yang 《Journal of Nonlinear Modeling and Analysis》2020,2(3):431-445
This paper is devoted to study the following complete hyper-elliptic integral of the first kind
$$J(h)=\oint\limits_{\Gamma_h}\frac{\alpha_0+\alpha_1x+\alpha_2x^2+\alpha_3x^3}{y}dx,$$
where $\alpha_i\in\mathbb{R}$, $\Gamma_h$ is an oval contained in the level set $\{H(x,y)=h, h\in(-\frac{5}{36},0)\}$ and $H(x,y)=\frac{1}{2}y^2-\frac{1}{4}x^4+\frac{1}{9}x^9$. We show that the 3-dimensional real vector spaces of these integrals are Chebyshev for $\alpha_0=0$ and Chebyshev with accuracy one for $\alpha_i=0\ (i=1,2,3)$. 相似文献
2.
3.
Let ∈ :N → R be a parameter function satisfying the condition ∈(k) + k + 1 > 0and let T∈ :(0,1] →(0,1] be a transformation defined by T∈(x) =-1 +(k + 1)x1 + k-k∈x for x ∈(1k + 1,1k].Under the algorithm T∈,every x ∈(0,1] is attached an expansion,called generalized continued fraction(GCF∈) expansion with parameters by Schweiger.Define the sequence {kn(x)}n≥1of the partial quotients of x by k1(x) = ∈1/x∈ and kn(x) = k1(Tn-1∈(x)) for every n ≥ 2.Under the restriction-k-1 < ∈(k) <-k,define the set of non-recurring GCF∈expansions as F∈= {x ∈(0,1] :kn+1(x) > kn(x) for infinitely many n}.It has been proved by Schweiger that F∈has Lebesgue measure 0.In the present paper,we strengthen this result by showing that{dim H F∈≥12,when ∈(k) =-k-1 + ρ for a constant 0 < ρ < 1;1s+2≤ dimHF∈≤1s,when ∈(k) =-k-1 +1ksfor any s ≥ 1where dim H denotes the Hausdorff dimension. 相似文献
4.
设$W_{\beta}(x)=\exp(-\frac{1}{2}|x|^{\beta})~(\beta > 7/6)$ 为Freud权, Freud正交多项式定义为满足下式$\int_{- \infty}^{\infty}p_{n}(x)p_{m}(x)W_{\beta}^{2}(x)\rd x=\left \{ \begin{array}{ll} 0 & \hspace{3mm} n \neq m , \\ 1 & \hspace{3mm}n = m \end{array} \right.$的 相似文献
5.
Weiyang Chen & Xiaoli Chen 《数学研究》2014,47(2):208-220
In this paper, we are concerned with the properties of positive solutions of the following nonlinear integral systems on the Heisenberg group $\mathbb{H}^n$, \begin{equation} \left\{\begin{array}{ll} u(x)=\int_{\mathbb{H}^n}\frac{v^{q}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ v(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ w(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)v^{q}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ \end{array}\right.\end{equation} for $x\in \mathbb{H}^n$, where $0<\alpha
1$ satisfying $\frac{1}{p+1} $+ $\frac{1}{q+1} + \frac{1}{r+1} = \frac{Q+α+β}{Q}.$ We show that positive solution triples $(u,v,w)\in L^{p+1}(\mathbb{H}^n)\times L^{q+1}(\mathbb{H}^n)\times L^{r+1}(\mathbb{H}^n)$ are bounded and they converge to zero when $|x|→∞.$ 相似文献
6.
本文探索了一种能多变量综合优化的方法,即对喷管进行参数化设计后,用均匀试验设计(UED)将试验样本均匀散布在设计区间内,求出各性能参数后,利用径向基神经网络(RBF)对试验样本进行拟合,再用粒子群算法(PSO)对训练好的神经网络进行寻优,找出了更好的双喉道气动矢量喷管设计参数组合。数值模拟结果显示,优化后的双喉道气动矢量喷管的矢量角有了明显提高。试验表明这种优化方法具有很好的优化能力,可以用来对喷管几何外形进行参数优化。 相似文献
7.
An integral boundary value problem of conformable integro-differential equations with a parameter
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In this article, we consider some properties of positive solutions for a new conformable integro-differential equation with integral boundary conditions and a parameter
$$
\left\{ \begin{array}{l} T_{\alpha}u(t)+\lambda f(t,u(t),I_{\alpha}u(t))=0,t\in[0,1],\u(0)=0,u(1)=\beta\int_{0}^{1}u(t)dt ,\beta\in[\frac 32,2), \ \end{array}\right.\nonumber
$$
where $\alpha\in(1,2]$, $\lambda$ is a positive parameter, $T_{\alpha}$ is the usual conformable derivative and $I_{\alpha}$ is the conformable integral, $f:[0,1]\times\mathbf{R^{+}}\times\mathbf{R^{+}}\rightarrow \mathbf{R^{+}} $ is a continuous function, where $\mathbf{R^{+}}=[0,+\infty)$.
We use a recent fixed point theorem for monotone operators in ordered Banach spaces, and then establish the existence and uniqueness of positive solutions for the boundary value problem. Further, we give an iterative sequence to approximate the unique positive solution and some good properties of positive solution about the parameter $\lambda$. A concrete example is given to better demonstrate our main result. 相似文献
8.
考虑一类平坦凸曲线的Hilbert变换■.对于平坦凸曲线,给出Hγ是L2有界的一些必要条件. 相似文献
9.
H. W. GOULD 《数学研究及应用》2019,39(6):603-606
The Catalan numbers $1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862,\ldots$ are given by $C(n)=\frac{1}{n+1}\binom{2n}{n}$ for $n\geq 0$. They are named for Eugene Catalan who studied them as early as 1838. They were also found by Leonhard Euler (1758), Nicholas von Fuss (1795), and Andreas von Segner (1758). The Catalan numbers have the binomial generating function $$\mathbf{C}(z) = \sum_{n=0}^{\infty}C(n)z^n = \frac{1 - \sqrt{1-4z}}{2z}$$ It is known that powers of the generating function $\mathbf{C}(z)$ are given by $$\mathbf{C}^a(z) = \sum_{n=0}^{\infty}\frac{a}{a+2n}\binom{a+2n}{n}z^n.$$ The above formula is not as widely known as it should be. We observe that it is an immediate, simple consequence of expansions first studied by J. L. Lagrange. Such series were used later by Heinrich August Rothe in 1793 to find remarkable generalizations of the Vandermonde convolution. For the equation $x^3 - 3x + 1 =0$, the numbers $\frac{1}{2k+1}\binom{3k}{k}$ analogous to Catalan numbers occur of course. Here we discuss the history of these expansions. and formulas due to L. C. Hsu and the author. 相似文献
10.
多尺度分析生成元的刻画 总被引:1,自引:0,他引:1
本文将给出多尺度分析生成元的一种完全刻画.将证明:函数φ∈L~2(R)是二进多尺度分析生成元的充要条件是(1)存在{a_k}∈l~2,φ(x)=∑_(k∈Z)a_kφ(2x-k);(2)存在正数A相似文献
11.
Infinitely many low- and high-energy solutions for a class of elliptic equations with variable exponent
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This paper is concerned with the $p(x)$-Laplacian equation of the form
$$
\left\{\begin{array}{ll}
-\Delta_{p(x)} u=Q(x)|u|^{r(x)-2}u, &\mbox{in}\ \Omega,\u=0, &\mbox{on}\ \partial \Omega,
\end{array}\right. \eqno{0.1}
$$
where $\Omega\subset\R^N$ is a smooth bounded domain, $1
p^+$ and $Q: \overline{\Omega}\to\R$ is a nonnegative continuous function. We prove that (0.1) has infinitely many small solutions and infinitely many large solutions by using the Clark''s theorem and the symmetric mountain pass lemma. 相似文献
12.
A time fractional functional differential equation driven by the fractional Brownian motion
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Let $B^H$ be a fractional Brownian motion with Hurst index $H>\frac12$. In this paper, we prove the global existence and uniqueness of the equation
$$
\begin{cases}
^CD_t^{\gamma}x(t)=f(x_t)+G(x_t)\frac{d}{dt}B^H(t),\ \ \ \ &t\in(0,T], \x(t)=\eta(t), \ \ \ \ \ &t\in[-r,0],
\end{cases}
$$
where $\max\{H,2-2H\}<\gamma<1$, $^CD_t^{\gamma}$ is the Caputo derivative, and $x_t\in \mathcal{C}_r=\mathcal{C}([-r,0],\mathbb{R})$ with $x_t(u)=x(t+u),u\in[-r,0]$. We also study the dependence of the solution on the initial condition. 相似文献
13.
Nakao Hayashi Pavel I. Naumkin Joel A. Rodriguez-Ceballos 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(3):355-369
We study large time asymptotic behavior of solutions to the periodic problem for the nonlinear damped wave equation
$ \left\{ {l} u_{tt}+2\alpha u_{t}-\beta u_{xx}=-\lambda \left| u\right| ^{\sigma}u,\text{ }x\in \Omega ,t >0 , \\ u(0,x)=\phi \left( x\right) ,\text{}u_{t}(0,x)=\psi \left( x\right) ,\text{ }x\in \Omega , \right. $ \left\{ \begin{array}{l} u_{tt}+2\alpha u_{t}-\beta u_{xx}=-\lambda \left| u\right| ^{\sigma}u,\text{ }x\in \Omega ,t >0 , \\ u(0,x)=\phi \left( x\right) ,\text{}u_{t}(0,x)=\psi \left( x\right) ,\text{ }x\in \Omega , \end{array} \right. 相似文献
14.
设E=■或■,■(x)∈L~2(R~2)且■_(jk)(x)=2■(E~jx-k),其中j∈Z,k∈Z~2.若{■_(jk)|jJ∈Z,k∈Z~2}构成L~2(R~2)的紧框架,则称■(x)为E-紧框架小波.本文给出E-紧框架小波是MRA E-紧框架小波的一个充要条件,即E紧框架小波■来自多尺度分析当且仅当线性空间F_■(ξ)的维数为0或1,其中F_■(ξ)=■(ξ)|j■1},■_j(ξ)={■((E~T)~j(ξ+2kπ))}_(k∈EZ~2,j■1。 相似文献
15.
E. M. E. Zayed 《计算数学(英文版)》1989,7(3):301-312
The spectral function $\hatμ(t)=\sum\limits_{j=1}^\infty e^{-itλ^{\frac{1}{2}}_j}$ where $\{λ_j\}^\infty_{j=1}$ are the eigenvalues of the three-dimensional Laplacian is studied for a variety of domains, where $- \infty<t<\infty$ and $i=\sqrt{-1}$. The dependence of $\hat{\mu}(t)$ on the connectivity of a domain and the impedance boundary condition (Robbin conditions) are analyzed. Particular attention is given to the spherical shell together with Robbin boundary conditions on its surface. 相似文献
16.
17.
Existence and concentration result for Kirchhoff equations with critical exponent and Hartree nonlinearity
![]() This paper is concerned with the following Kirchhoff-type equations
$$
\left\{
\begin{array}{ll}
\displaystyle
-\big(\varepsilon^{2}a+\varepsilon b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\mathrm{d}x\big)\Delta u
+ V(x)u+\mu\phi |u|^{p-2}u=f(x,u), &\quad \mbox{ in }\mathbb{R}^{3},\(-\Delta)^{\frac{\alpha}{2}} \phi=\mu|u|^{p},~u>0, &\quad \mbox{ in }\mathbb{R}^{3},\\end{array}
\right.
$$
where $f(x,u)=\lambda K(x)|u|^{q-2}u+Q(x)|u|^{4}u$, $a>0,~b,~\mu\geq0$ are constants, $\alpha\in(0,3)$, $p\in[2,3),~q\in[2p,6)$ and $\varepsilon,~\lambda>0$ are parameters. Under some mild conditions on $V(x),~K(x)$ and $Q(x)$, we prove that the above system possesses a ground state solution $u_{\varepsilon}$ with exponential decay at infinity for $\lambda>0$ and $\varepsilon$ small enough. Furthermore, $u_{\varepsilon}$ concentrates around a global minimum point of $V(x)$ as $\varepsilon\rightarrow0$. The methods used here are based on minimax theorems and the concentration-compactness principle of Lions. Our results generalize and improve those in Liu and Guo (Z Angew Math Phys 66: 747-769, 2015), Zhao and Zhao (Nonlinear Anal 70: 2150-2164, 2009) and some other related literature. 相似文献
18.
Todd Cochrane Christopher Pinner 《Proceedings of the American Mathematical Society》2005,133(2):313-320
For a sparse polynomial , with and , we show that
thus improving upon a bound of Mordell. Analogous results are obtained for Laurent polynomials and for mixed exponential sums. 19.
设$\omega_1,\omega_2$为正规函数, $\varphi$是$B_n$ 上的全纯自映射,$ g\in H(B_n)$ 满足 $g(0)=0$. 对所有的$0
相似文献 20.
Bang-He Li 《数学研究》2016,49(4):319-324
Let $ζ(s)$ be the Riemann zeta function, $s=\sigma+it$. For $0 < \sigma < 1$, we expand $ζ(s)$ as the following series convergent in the space of slowly increasing distributions
with variable $t$ : $$ζ(\sigma+it)=\sum\limits^∞_{n=0}a_n(\sigma)ψ_n(t),$$ where $$ψ_n(t)=(2^nn!\sqrt{\pi})^{-1 ⁄ 2}e^{\frac{-t^2}{2}}H_n(t),$$ $H_n(t)$ is the Hermite polynomial, and $$a_n(σ)=2\pi(-1)^{n+1}ψ_n(i(1-σ))+(-i)^n\sqrt{2\pi}\sum\limits^∞_{m=1}\frac{1}{m^σ}ψ_n(1nm).$$ This paper is concerned with the convergence of the above series for $σ > 0.$ In the deduction,
it is crucial to regard the zeta function as Fourier transfomations of Schwartz'
distributions. 相似文献
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