共查询到20条相似文献,搜索用时 16 毫秒
1.
V. Totik 《Analysis Mathematica》1981,7(1):81-84
В РАБОтЕ ДАЕтсь ОтВЕт НА ОДИН ВОпРОс, пОстАВ лЕННыИ В. г. кРОтОВыМ. УстАНОВлЕН О, ЧтО ЕслИ Ф(х) — МОНОтОННО ВО жРАстАУЩАь ФУНкцИь,Ф (0)=0, Ф(2х)≦кФ(х), х[0, ∞), тО $$\left\{ {f:\left\| {\sum\limits_{k = 1}^\infty {\mu _k \Phi (\lambda _k \left| {S_k - f} \right|)} } \right\|_c< \infty } \right\} \subseteqq C \Leftrightarrow \sum\limits_{k = 1}^\infty {\mu _k } \Phi (\lambda _k ) = \infty $$ Дль пРОИжВОльНых НЕО тРИцАтЕльНых ЧИслОВ ых пОслЕДОВАтЕльНОстЕ И {Μk} И {λk}. (жДЕсьS k ОБОжНАЧАЕт ЧАстНУУ с УММУ пОРьДкАk РьДА ФУ РьЕ ФУНкцИИf). УстАНОВлЕН О тАкжЕ, ЧтО ВО МНОгИх слУЧАьх $$\left\{ {f:\left\| {\sum\limits_{k = 1}^\infty {\mu _k \Phi (\lambda _k \left| {\tilde S_k - \tilde f} \right|)} } \right\|_c< \infty } \right\} \subseteqq C \Leftrightarrow \sum\limits_{k = 1}^\infty {\frac{1}{{k\lambda _k }}} \Phi ^{ - 1} \left( {\frac{1}{{k\mu _k }}} \right)< \infty .$$ 相似文献
2.
Felipe J. Zó 《Analysis Mathematica》1978,4(2):153-158
Пустьl 1 иl 2 — неотрицательные убывающие функции на (0, ∞). Допустим, что $$\int\limits_0^\infty {S^{n_i - 1} l_i (S)\left( {1 + \log + \frac{1}{{S^{n_i } l_i (S)}}} \right)dS}< \infty ,$$ , гдеn 1 иn 2 — натуральные числа. Тогда для каждой функции \(f \in L^1 (R^{n_1 + n_2 } )\) при почти всех (x0, у0) мы имеем $$\mathop {\lim }\limits_{\lambda \to \infty } \lambda ^{n_1 + n_2 } \int\limits_{R^{n_1 } } {\int\limits_{R^{n_2 } } {l_1 } } (\lambda |x|)l_2 (\lambda |y|)f(x_0 - x,y_0 - y)dx dy = f(x_0 ,y_0 )\int\limits_{R^{n_1 } } {\int\limits_{R^{n_2 } } {l_i (|x|)l_2 } } (|y|)dx dy.$$ 相似文献
3.
A. Yu. Veretennikov 《Mathematical Notes》1977,22(4):804-808
Existence, uniqueness, and ergodicity are proved for a stationary distribution for a service system having countably many servomechanisms with input flow rate μk depending on the number k of servomechanisms occupied, and with arbitrary (identical) distribution of the service time with finite mean μ, under the condition \(\mu \mathop {\overline {\lim } }\limits_{k \to \infty } \frac{{\lambda _k }}{{k + 1}}< 1\) . For this system we have, in particular, Erlang's formula $$p_k (t)\mathop \to \limits_{k + \infty } p_k = \frac{{\lambda _0 ...\lambda _{k - 1} \mu ^k }}{{k!}}p_0 ,k = 0,1,...,p_0^{ - 1} = \sum\nolimits_{k = 0}^\infty {\frac{{\lambda _0 ...\lambda _{k - 1} \mu ^k }}{{k!}}} ,\lambda _{ - 1} = 1.$$ 相似文献
4.
M. H. Шеремета 《Analysis Mathematica》1991,17(1):47-54
Пусть Λ=(λn) — возрастаю щая к+∞ последователь ность неотрицательных чис ел, λ0=0, а S+(Λ) — класс абсолют но сходящихся в С рядо в Дирихле вида $$F\left( z \right) = \mathop \sum \limits_{k = 0}^\infty a_k \exp \left\{ {z\lambda _k } \right\},$$ где a0=1 и ak>0 (k∈N). Положим $$\begin{gathered} S_n \left( z \right) = \mathop \sum \limits_{k = 1}^\infty a_k \exp \left\{ {z\lambda _k } \right\}, \hfill \\ \sigma _n \left( F \right) = \max \left\{ {\frac{1}{{S_n \left( x \right)}} - \frac{1}{{F\left( x \right)}}:x \in R} \right\}. \hfill \\ \end{gathered} $$ Доказано, что для того, чтобы для любой функц ии F∈S+(Λ) выполнялось равенст во $$\mathop {\lim \sup }\limits_{n \to \infty } \frac{1}{{\ln n}}\ln \frac{1}{{\sigma _n \left( F \right)}} = + \infty ,$$ необходимо и достато чно, чтобы $$\mathop \sum \limits_{n = 1}^\infty \frac{1}{{n\lambda _n }}< + \infty .$$ Аналогичные результ ы получены для различ ных подклассов классаS + (Λ), определяемых условиями на убывани е коэффициентова n. 相似文献
5.
V. Totik 《Analysis Mathematica》1979,5(4):287-299
Пустьf 2π-периодическ ая суммируемая функц ия, as k (x) еë сумма Фурье порядк аk. В связи с известным ре зультатом Зигмунда о сильной суммируемости мы уст анавливаем, что если λn→∞, то сущес твует такая функцияf, что почти всюду $$\mathop {\lim \sup }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = n + 1}^{2n} |s_k (x) - f(x)|^{\lambda _{2n} } } \right\}^{1/\lambda _{2n} } = \infty .$$ Отсюда, в частности, вы текает, что если λn?∞, т о существует такая фун кцияf, что почти всюду $$\mathop {\lim \sup }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = 0}^n |s_k (x) - f(x)|^{\lambda _k } } \right\}^{1/\lambda _n } = \infty .$$ Пусть, далее, ω-модуль н епрерывности и $$H^\omega = \{ f:\parallel f(x + h) - f(x)\parallel _c \leqq K_f \omega (h)\} .$$ . Мы доказываем, что есл и λ n ?∞, то необходимым и достаточным условие м для того, чтобы для всехf∈H ω выполнялос ь соотношение $$\mathop {\lim }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = n + 1}^{2n} |s_k (x) - f(x)|^{\lambda _n } } \right\}^{1/\lambda _n } = 0(x \in [0;2\pi ])$$ является условие $$\omega \left( {\frac{1}{n}} \right) = o\left( {\frac{1}{{\log n}} + \frac{1}{{\lambda _n }}} \right).$$ Это же условие необхо димо и достаточно для того, чтобы выполнялось соотнош ение $$\mathop {\lim }\limits_{n \to \infty } \frac{1}{{n + 1}}\mathop \sum \limits_{k = 0}^n |s_k (x) - f(x)|^{\lambda _k } = 0(f \in H^\omega ,x \in [0;2\pi ]).$$ 相似文献
6.
Prof. Dr. Werner Raab 《Monatshefte für Mathematik》1984,98(4):311-322
The inequalities $$P_{k,l} = \frac{1}{{B(l + 1,k)}}\int\limits_0^{l/(k + l)} {x^l (1 - x)^{k - 1} dx = I_{l/(k + l)} (l + 1,k)< \frac{1}{2}}$$ and $$\Phi _{k,l,\mu } = \frac{1}{{B(k,k\mu + 1)}}\int\limits_0^1 {x^{k - 1} (1 - x)^{k\mu } I_x (l + 1,l\mu )dx< \frac{1}{2}}$$ are shown to be valid for any positive real numbersk, l, μ. 相似文献
7.
Xiaoli Han Jiayu Li Liuqing Yang 《Calculus of Variations and Partial Differential Equations》2013,48(1-2):111-129
Let Σ be an immersed symplectic surface in CP 2 with constant holomorphic sectional curvature k > 0. Suppose Σ evolves along the mean curvature flow in CP 2. In this paper, we show that the symplectic mean curvature flow exists for long time and converges to a holomorphic curve if the initial surface satisfies ${|A|^2 \leq \lambda|H|^2 + \frac{2\lambda-1}{\lambda}k}$ and ${\cos\alpha\geq\sqrt{\frac{7\lambda-3}{3\lambda}}\left(\frac{1}{2} < \lambda\leq\frac{2}{3}\right) {\rm or} |A|^2\leq \frac{2}{3}|H|^2+\frac{4}{5}k\cos\alpha\, {\rm and} \cos\alpha\geq 1-\varepsilon}$ , for some ${\varepsilon}$ . 相似文献
8.
V. I. Frolov 《Mathematical Notes》1976,19(4):369-371
Simple estimates are obtained for the spectrum of the operator bundle \(R(\lambda ) = \sum\nolimits_{i = 0}^n {A_{n - i} \lambda ^i }\) in terms of estimates of the maximum and minimum eigenvalues of the operators \(\frac{1}{2}(A_{n - i} - A_{n - i}^* )(i = 0,1,2, \ldots n)\) and the norms of the operators \(\frac{1}{2}(A_{n - i} - A_{n - i}^* )(i = 0,1,2, \ldots n)\) We formulate a criterion of the asymptotic stability of the differential equations $$\sum\nolimits_{i = 1}^n {A_{n - i} } \frac{{d^{(i)} x}}{{dt^i }} = 0.$$ We present examples of the stability conditions for equations with n=2 and n=3. 相似文献
9.
The symmetric group $\operatorname{Sym}(d)$ acts on the Cartesian product (S 2) d by coordinate permutation, and the quotient space $(S^{2})^{d}/\operatorname{Sym}(d)$ is homeomorphic to the complex projective space ?P d . We used the case d=2 of this fact to construct a 10-vertex triangulation of ?P 2 earlier. In this paper, we have constructed a 124-vertex simplicial subdivision $(S^{2})^{3}_{124}$ of the 64-vertex standard cellulation $(S^{2}_{4})^{3}$ of (S 2)3, such that the $\operatorname{Sym}(3)$ -action on this cellulation naturally extends to an action on $(S^{2})^{3}_{124}$ . Further, the $\operatorname{Sym}(3)$ -action on $(S^{2})^{3}_{124}$ is ??good??, so that the quotient simplicial complex $(S^{2})^{3}_{124}/\operatorname{Sym}(3)$ is a 30-vertex triangulation $\mathbb{C}P^{3}_{30}$ of ?P 3. In other words, we have constructed a simplicial realization $(S^{2})^{3}_{124} \to\mathbb{C} P^{3}_{30}$ of the branched covering (S 2)3???P 3. 相似文献
10.
E. M. E. Zayed 《Journal of Applied Mathematics and Computing》2003,12(1-2):81-105
This paper deals with the very interesting problem about the influence of piecewise smooth boundary conditions on the distribution of the eigenvalues of the negative Laplacian inR 3. The asymptotic expansion of the trace of the wave operator $\widehat\mu (t) = \sum\limits_{\upsilon = 1}^\infty {\exp \left( { - it\mu _\upsilon ^{1/2} } \right)} $ for small ?t? and $i = \sqrt { - 1} $ , where $\{ \mu _\nu \} _{\nu = 1}^\infty $ are the eigenvalues of the negative Laplacian $ - \nabla ^2 = - \sum\limits_{k = 1}^3 {\left( {\frac{\partial }{{\partial x^k }}} \right)} ^2 $ in the (x 1,x 2,x 3), is studied for an annular vibrating membrane Ω inR 3 together with its smooth inner boundary surfaceS 1 and its smooth outer boundary surfaceS 2. In the present paper, a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth componentsS * i(i=1, …,m) ofS 1 and on the piecewise smooth componentsS * i(i=m+1, …,n) ofS 2 such that $S_1 = \bigcup\limits_{i = 1}^m {S_i^* } $ and $S_2 = \bigcup\limits_{i = m + 1}^n {S_i^* } $ are considered. The basic problem is to extract information on the geometry of the annular vibrating membrane ω from complete knowledge of its eigenvalues by analyzing the asymptotic expansions of the spectral function $\widehat\mu (t)$ for small ?t?. 相似文献
11.
In this paper, we consider the 2D Skyrme model $$E(u)=\frac{1}{2} \int\limits_{R^2}|du|^2dx+\frac{\lambda}{4} \int\limits_{R^2}|du\wedge du|^2dx+\frac{\mu}{16} \int\limits_{R^2}|u-{\bf{n}}|^4dx,$$ where ?? and??? > 0 are positive coupling constants and n = (0, 0, 1) is the north pole of S 2. We derive a lower bound of 2D Skyrme model. Using this estimate, we prove the existence of 2D Skyrmion for any positive coupling constants ??, ??. 相似文献
12.
Я. С. Бугров 《Analysis Mathematica》1979,5(2):119-133
Получены новые оценк иL-нормы тригонометр ических полиномов $$T_n (t) = \frac{{\lambda _0 }}{2} + \mathop \sum \limits_{k = 1}^n \lambda _k \cos kt$$ в терминах коэффицие нтовλ k и их разностейΔλ k=λ k?λ k?1: (1) $$\mathop \smallint \limits_{ - \pi }^\pi |T_n (t)|dt \leqq \frac{c}{n}\mathop \sum \limits_{k = 0}^n |\lambda _\kappa | + c\left\{ {x(n,\varphi )\mathop \sum \limits_{k = 0}^n \Delta \lambda _\kappa \mathop \sum \limits_{l = 0}^n \Delta \lambda _l \delta _{\kappa ,l} (\varphi )} \right\}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ,$$ где $$\kappa (n,\varphi ) = \mathop \smallint \limits_{1/n}^\pi [t^2 \varphi (t)]^{ - 1} dt, \delta _{k,1} (\varphi ) = \mathop \smallint \limits_0^\infty \varphi (t)\sin \left( {k + \frac{1}{2}} \right)t \sin \left( {l + \frac{1}{2}} \right)t dt,$$ a ?(t) — произвольная фун кция ≧0, для которой опр еделены соответствующие инт егралы. Из (1) следует, что методы $$\tau _n (f;t) = (N + 1)^{ - 1} \mathop \sum \limits_{k = 0}^{\rm N} S_{[2^{k^\varepsilon } ]} (f;t), n = [2^{N\varepsilon } ],$$ являются регулярным и для всех 0<ε≦1/2. ЗдесьS m (f, x) частные суммы ряда Фу рье функцииf(x). В статье исследуется многомерный случай. П оказано, что метод суммирования (о бобщенный метод Рисса) с коэффиц иентами $$\lambda _{\kappa ,l} = (R^v - k^\alpha - l^\beta )^\delta R^{ - v\delta } (0 \leqq k^\alpha + l^\beta \leqq R^v ;\alpha \geqq 1,\beta \geqq 1,v< 0)$$ является регулярным, когда δ > 1. 相似文献
13.
D. G. Sanikidze 《Mathematical Notes》1974,15(4):313-318
We examine a specific approximating process for the singular integral $$S^* (f;x) \equiv \frac{1}{\pi }\int_{ - 1}^{ + 1} {\frac{{f(t)}}{{\sqrt {1 - l^2 } (t - x)}}} dt( - 1< x< 1)$$ taken in the principal value sense. We study the influence of some local properties of the functionf on the convergence of the approximations. Next, assuming that \(S^* (f;c) \equiv \mathop {\lim }\limits_{x \to c} S^* (f;x)\) , where c is an arbitrary one of the endpoints ?1 and 1, we show that the conditions which guarantee the existence of the limiting values S*(f; c) (c=±1) and, moreover, the convergence of the process at an arbitrary point x∈ (?1, 1) are not always sufficient for convergence of the approximations at the endpoints. 相似文献
14.
Let M be an even dimensional compact smooth manifold admitting an almost complex structure. Let ${{(\lambda, \mu)} \in \mathbb{R}^2 - (0,0)}$ . We discuss the critical points of the functional ${\mathcal {F}_{\lambda, \mu} (J, g) = \int_M (\lambda \tau + \mu \tau^* ) dv_g}$ on the space of all almost Hermitian structures ${\mathcal{AH}(M)}$ on M and its subspace ${{\mathcal{AH}_{c}(M)}}$ with a certain positive constant c, where τ and τ * are the scalar curvature and the *-scalar curvature of (J, g), respectively. Further, we provide some examples illustrating our arguments. 相似文献
15.
G. A. Afrouzi Nguyen Thanh Chung Z. Naghizadeh 《Mediterranean Journal of Mathematics》2014,11(3):891-903
Using variational methods, we study the existence and nonexistence of nontrivial weak solutions for the quasilinear elliptic system $$\left\{\begin{array}{ll}- {\rm div}(h_1(|\nabla u|^2)\nabla u) = \frac{\mu}{|x|^2}u + \lambda F_u(x, u, \upsilon)\quad {\rm in}\,\Omega,\\- {\rm div}(h_2(|\nabla \upsilon|^2)\nabla \upsilon) = \frac{\mu}{|x|^2}\upsilon + \lambda F_\upsilon(x,u,\upsilon)\quad {\rm in}\,\Omega,\\u = \upsilon = 0 \qquad \qquad \qquad \qquad \qquad \qquad {\rm in}\, \partial\Omega, \end{array}\right.$$ where \({\Omega \subset \mathbb{R}^N,N \geq 3}\) , is a bounded domain containing the origin with smooth boundary \({\partial \Omega ; h_i, i = 1, 2}\) , are nonhomogeneous potentials; \({(F_u, F_v) = \nabla F}\) stands for the gradient of a sign-changing C 1-function \({F : \Omega \times \mathbb{R}^2 \to \mathbb{R}}\) in the variable \({{w = (u, v) \in \mathbb{R}^2}}\) ; and λ and μ are parameters. 相似文献
16.
In this article, we prove that the following weighted Hardy inequality $$\begin{array}{ll}\big(\frac{|{d-p}|}{p}\big)^{p}\, \int\limits_{\Omega}\, \frac{|{u}|^{p}}{|{x}|^{p}}\;d\mu \\ \quad \quad \le \int\limits_{\Omega}\,|{\nabla u}|^{p}\;d\mu+ \big(\frac{|{d-p}|}{p}\big)^{p-1}\,\textrm{sgn}(d-p)\,\int\limits_{\Omega}|{u}|^{p}\,\frac{(x^{t}Ax)^{p/2}}{|{x}|^{p}}\; d\mu \quad \quad \quad (1) \end{array}$$ holds with optimal Hardy constant ${\big(\frac{|d-p|}{p}\big)^{p}}$ for all ${u \in W^{1,p}_{\mu,0}(\Omega)}$ if the dimension d ≥ 2, 1 < p < d, and for all ${u \in W^{1,p}_{\mu,0}(\Omega{\setminus}\{0\})}$ if p > d ≥ 1. Here we assume that Ω is an open subset of ${\mathbb{R}^{d}}$ with ${0 \in \Omega}$ , A is a real d × d-symmetric positive definite matrix, c > 0, and $$ d \mu: = \rho(x) \,dx \qquad \textrm{with} \quad \rho(x) = c \cdot \exp(-\frac{1}{p}(x^{t}Ax)^{p/2}), \quad x \in\Omega.\quad \quad (2) $$ If p > d ≥ 1, then we can deduce from (1) a weighted Poincaré inequality on ${W^{1,p}_{\mu,0}(\Omega \setminus\{0\})}$ . Due to the optimality of the Hardy constant in (1), we can establish the nonexistence (locally in time) of positive weak solutions of a p-Kolmogorov parabolic equation perturbed by a singular potential in dimension d = 1, for 1 < p < + ∞, and when Ω = ]0, + ∞[. 相似文献
17.
Guoqing Zhang Xiaozhi Wang Sanyang Liu 《Calculus of Variations and Partial Differential Equations》2013,46(1-2):97-111
In this paper, firstly, we investigate a class of singular eigenvalue problems with the perturbed Hardy–Sobolev operator, and obtain some properties of the eigenvalues and the eigenfunctions. (i.e. existence, simplicity, isolation and comparison results). Secondly, applying these properties of eigenvalue problem, and the linking theorem for two symmetric cones in Banach space, we discuss the following singular elliptic problem $$\left\{\begin{array}{ll}-\Delta_{p}u-a(x)\frac{|u|^{p-2}u}{|x|^{p}}= \lambda \eta(x)|u|^{p-2}u+ f(x,u) \quad x \in \Omega, \\ u =0 \quad\quad\quad\quad\quad\quad\quad x\in\partial \Omega, \end{array} \right.$$ where ${a(x)=(\frac{n-p}{p})^{p}q(x),}$ if 1 < p < n, ${a(x)=(\frac{n-1}{n})^{n} \frac{q(x)}{({\rm log}\frac{R}{|x|})^{n}},}$ if p = n, and prove the existence of a nontrivial weak solution for any ${\lambda \in \mathbb{R}.}$ 相似文献
18.
Let T be a Hilbert-space operator of class $$\left\| {S^{ - 1} TS} \right\| \leqslant 1\,\,\,\,and\,\,\left\| {S^{ - 1} } \right\| \cdot \left\| S \right\| \leqslant \max (1,\rho )$$ in the sense of Sz.-Nagy and Foia?. Then there exists an invertible operator S such that ∥S?1TS∥≤1 and ∥S?1∥·∥S∥≤max(1,ρ). The following estimate is valid for the bilateral limit: $$\mathop {lim}\limits_{n \to \infty } \{ \left\| {T^n h} \right\|^2 + \left\| {T*^n h} \right\|^2 \} \leqslant max(2,\rho )\left\| h \right\|^2 .$$ Here the constants max(1,ρ) and max(2,ρ) are the best possible in their respective cases. 相似文献
19.
Ai Hua Fan 《Monatshefte für Mathematik》1994,118(1-2):83-89
Consider the Riesz product $\mu _a = \mathop \prod \limits_{n = 1}^\infty (1 + r\cos (q^n t + \varphi _n ))$ . We prove the following approximative formula for the dimension ofμ a. $$\dim \mu _a = 1 - \frac{1}{{\log q}}\int_0^{2\pi } {(1 + r\cos x)\log (1 + r\cos x)\frac{{dx}}{{2\pi }} + 0\left( {\frac{r}{{q^2 \log q}}} \right).}$$ 相似文献
20.
In this paper, we are concerned with the multibump solutions for the following quasilinear Schrödinger system in ${\mathbb{R}^N}$ : $$\left\{\begin{array}{ll}-\Delta{u} + \lambda{a(x)u} - \frac{1}{2}(\Delta|u|^2)u = \frac{2\alpha}{\alpha + \beta}|u|^{\alpha-2}|\upsilon|^\beta u, \\-\Delta{\upsilon} + \lambda{b(x)\upsilon} - \frac{1}{2}(\Delta|\upsilon|^2)\upsilon = \frac{2\beta}{\alpha + \beta}|u|^\alpha|\upsilon|^{\beta-2} \upsilon, \\u(x) \rightarrow 0, \upsilon(x) \rightarrow 0 \quad as|x| \rightarrow \infty,\end{array}\right.$$ where λ > 0 is a parameter, α, β > 2 satisfying α + β < 2 · 2*, here ${2^{*} = \frac{2N}{N-2}}$ is the critical Sobolev exponent for ${N \geq 3}$ and a(x), b(x) are nonnegative potentials. Using variational methods, we prove that if the zero sets of a(x) and b(x) have several common isolated connected components ${\Omega_{1}, . . . ,\Omega_{k}}$ such that the interior of ${\Omega_{i} (i = 1, 2, . . . , k)}$ is not empty and ${\partial\Omega_{i} (i = 1, 2, . . . , k)}$ is smooth, then for λ sufficiently large, the system admits, for any nonempty subset ${J \subset \{1, 2, . . . , k\}}$ , a solution which is trapped in a neighborhood of ${\cup_{j\epsilon{J}} \Omega_{j}}$ . 相似文献