共查询到20条相似文献,搜索用时 46 毫秒
1.
Tachen Liang 《Journal of Applied Mathematics and Computing》2007,25(1-2):1-15
Consider a Rayleigh distribution withpdfp(x|θ) = 2xθ - 1 exp(- x 2/θ) and mean lifetime μ = √πθ/2. We study the two-action problem of testing the hypothesesH 0: μ≤ μ0 againstH 1: μ > μ0 using a linear error loss of |μ- μ 0 | via the empirical Bayes approach. We construct a monotone empirical Bayes test δ n * and study its associated asymptotic optimality. It is shown that the regret of δ n * converges to zero at a rate $\frac{{\ln ^2 n}}{n}$ , wheren is the number of past data available when the present testing problem is considered. 相似文献
2.
K. L. Olifirov 《Mathematical Notes》1975,18(5):1050-1053
For an arbitrary element x with spectrum sp(x) in a Banach algebra with identity e ≠ 0 we define the upper (lower) spectral abscissa \(\mathop {\sigma + (x)}\limits_{( - )} = \mathop {\max }\limits_{(\min )} \operatorname{Re} \lambda ,\lambda \in sp(x)\) . With the aid of the spectral radius \(\rho (x) = \mathop {\max }\limits_{\lambda \in sp(x)} \left| \lambda \right| = \mathop {\lim }\limits_{n \to + \infty } \parallel x^n {{1 - } \mathord{\left/ {\vphantom {{1 - } n}} \right. \kern-0em} n}\) we prove the following bounds: γ?(x)?σ?(x)?Γ?(x)?+(x)?σ+(x)?γ+(x), Γ(±)(x)=(2δ(±))?1 (ρ δ 2 )(±)?δ (±) 2 ?ρ 0 2 )(δ(±)≠0), γ(±)(x)= (±)ρδ(±)?δ(±), δ+?0, δ??0 ρ (±) δ = ρ(x+eδ(±)). We mention a case where equality is achieved, some corollaries,and discuss the sharpness of the bounds: for every ? > 0 there is a δ: ¦δ¦ ≥ρ 0 2 /2?, such that Δ: = ¦γ(±) x?Γ(±) x¦?ε and conversely, if the bounds are computed for some δ ≠ 0, then △ ≤ρ 0 2 /2 ¦δ¦. An example is considered. 相似文献
3.
We consider the asymptotic behavior of the solutions to the equation ${u_{t}-u_{xx} = \lambda(1 + {\delta}u_{x}^{2})(1 - u)^{-2}}$ , which comes from Micro-Electromechanical Systems (MEMS) devices modeling. It is shown that when the fringing field exists (i.e., δ?> 0), there is a critical value λ δ * > 0 such that if 0 < λ < λ δ * , the equation has a global solution for some initial data; while for λ > λ δ * , all solutions to the equation will quench at finite time. When the quenching happens, u has only finitely many quenching points for particular initial data. A one-side estimate is deduced for the quenching rate of u. 相似文献
4.
Let ?1<α≤0 and let $$L_n^{(\alpha )} (x) = \frac{1}{{n!}}x^{ - \alpha } e^x \frac{{d^n }}{{dx^n }}(x^{\alpha + n} e^{ - x} )$$ be the generalizednth Laguerre polynomial,n=1,2,… Letx 1,x 2,…,x n andx*1,x*2,…,x* n?1 denote the roots ofL n (α) (x) andL n (α)′ (x) respectively and putx*0=0. In this paper we prove the following theorem: Ify 0,y 1,…,y n ?1 andy 1 ′ ,…,y n ′ are two systems of arbitrary real numbers, then there exists a unique polynomialP(x) of degree 2n?1 satisfying the conditions $$\begin{gathered} P\left( {x_k^* } \right) = y_k (k = 0,...,n - 1) \hfill \\ P'\left( {x_k } \right) = y_k^\prime (k = 1,...,n). \hfill \\ \end{gathered} $$ . 相似文献
5.
V. K. Dzjadyk 《Analysis Mathematica》1977,3(3):171-175
п. л. ЧЕБышЕВыМ БылА пОс тАВлЕНА И РЕшЕНА жАДА ЧА: пРИ пРОИжВОльНО жАДАННО М НА [?1,1] пОлОжИтЕльНОМ МНОг ОЧлЕНЕP l (x) жАДАННОИ ст ЕпЕНИl НАИтИ пРИ кАжДОМn≧1 МНОгОЧлЕНР n * (x) стЕпЕН Ип с кОЁФФИцИЕНтОМ п РИx n , РАВНыМ 1, кОтОРыИ ьВл ьлсь Бы МНОгО-ЧлЕНОМ, НАИМЕНЕЕ УклОНьУЩИМ сь От НУль с ВЕсОМP l ?1 (x) В МЕтРИкЕC[? 1,1]. А. А. МАРкОВ ОБОБЩИл ЁтО т РЕжУльтАт И пРИ тЕх ж Е УслОВИьх НАP l (x) пОстРО Ил Дль≧1/2 МНОгОЧлЕНыP n * (x) стЕп ЕНИ п, НАИМЕНЕЕ УклОНьУЩИЕсь От НУль с ВЕсОМP l -1/2 (x). В ДАННОИ стАтьЕ УкАжы ВАЕтсь БОлЕЕ пРОстОИ, ЧЕМ В [3], спОсОБ пОстРОЕНИь МН ОгОЧлЕНОВP n * (x), ДАУЩИх РЕшЕНИЕ жАД АЧИ МАРкОВА, И пРИ ЁтОМ, ВО-пЕРВых, УстАНОВлЕН О, ЧтО ВськИИ тАкОИ МНОгОЧлЕН МОжН О пРЕДстАВИть В ВИДЕ л ИНЕИНОИ кОМБИНАцИИ НЕ БОлЕЕ Ч ЕМ Ижl+1 МНОгОЧлЕНОВ ЧЕБышЕВ АT j (x)=cos (jarc cosx): $$P_n^* (x) = \mathop \Sigma \limits_{k = 0}^l \gamma _k T_{|n - l + k|} (x)$$ , В кОтОРОИ кОЁФФИцИЕН тыγ k НЕ жАВИсьт Отп И, ВО-ВтОРых, УкАжАН спОс ОБ ЁФФЕктИВНОгО РАжы с-кАНИь кОЁФФИцИЕНтОВγ k пО М НОгОЧлЕНУР l (х). 相似文献
6.
I. Joó 《Analysis Mathematica》1975,1(4):273-281
В этой работе мы даем о бобщение понятия нор мальной системы точек, введен ного Фейером [3]. Наше определ ение включает и случа й бесконечного интерв ала (0, ∞). Доказано, в частности, что систе ма точек 0<x 1 (n) /(n)<... n (n) <∞ является нормальной в смысле нашего определения тогда и т олько тогда, когда вып олняются оценки — фиксированное чис ло, 0≦?<1. Мы доказываем, что есл и точкиx k (n) /(n) являются ну лями многочлена ЛагерраL n (α) (x), то они образуют норма льную систему в том и т олько том случае, когда ?1<α≦0. Мы получаем, таким обр азом, положительный интерполяционный пр оцесс для каждой нормальной системы т очек и устанавливаем теорему сходимости для того с лучая, когда эти точки являются ну лямиL n (α) (x) при — 1相似文献
7.
A. Nissenzweig 《Israel Journal of Mathematics》1975,22(3-4):266-272
LetX be an infinite dimensional Banach space, andX* its dual space. Sequences {χ n * } n=1 ∞ ?X* which arew* converging to 0 while inf n ‖x* n ‖>0, are constructed. 相似文献
8.
I. I. Sharapudinov 《Mathematical Notes》2013,94(1-2):281-293
We study new series of the form $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ in which the general term $f_k^{ - 1} \hat P_k^{ - 1} (x)$ , k = 0, 1, …, is obtained by passing to the limit as α→?1 from the general term $\hat f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)$ of the Fourier series $\sum\nolimits_{k = 0}^\infty {f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)} $ in Jacobi ultraspherical polynomials $\hat P_k^{\alpha ,\alpha } (x)$ generating, for α> ?1, an orthonormal system with weight (1 ? x 2)α on [?1, 1]. We study the properties of the partial sums $S_n^{ - 1} (f,x) = \sum\nolimits_{k = 0}^n {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ of the limit ultraspherical series $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ . In particular, it is shown that the operator S n ?1 (f) = S n ?1 (f, x) is the projection onto the subspace of algebraic polynomials p n = p n (x) of degree at most n, i.e., S n (p n ) = p n ; in addition, S n ?1 (f, x) coincides with f(x) at the endpoints ±1, i.e., S n ?1 (f,±1) = f(±1). It is proved that the Lebesgue function Λ n (x) of the partial sums S n ?1 (f, x) is of the order of growth equal to O(ln n), and, more precisely, it is proved that $\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1$ . 相似文献
9.
We study the asymptotic behavior of the eigenvalues the Sturm-Liouville operator Ly = ?y″ + q(x)y with potentials from the Sobolev space W 2 θ?1 , θ ≥ 0, including the nonclassical case θ ∈ [0, 1) in which the potential is a distribution. The results are obtained in new terms. Let s 2k (q) = λ k 1/2 (q) ? k, s 2k?1(q) = μ k 1/2 (q) ? k ? 1/2, where {λ k } 1 ∞ and {μ k } 1 ∞ are the sequences of eigenvalues of the operator L generated by the Dirichlet and Dirichlet-Neumann boundary conditions, respectively,. We construct special Hilbert spaces t 2 θ such that the mapping F:W 2 θ?1 →t 2 θ defined by the equality F(q) = {s n } 1 ∞ is well defined for all θ ≥ 0. The main result is as follows: for θ > 0, the mapping F is weakly nonlinear, i.e., can be expressed as F(q) = Uq + Φ(q), where U is the isomorphism of the spaces W 2 θ?1 and t 2 θ , and Φ(q) is a compact mapping. Moreover, we prove the estimate ∥Ф(q)∥τ ≤ C∥q∥θ?1, where the exact value of τ = τ(θ) > θ ? 1 is given and the constant C depends only on the radius of the ball ∥q∥θ? ≤ R, but is independent of the function q varying in this ball. 相似文献
10.
С. А. ТЕЛЯКОВСКИЙ 《Analysis Mathematica》1982,8(4):305-319
Let σ n 2 (f, x) be the Cesàro means of second order of the Fourier expansion of the function f. Upper bounds of the deviationf(x)-σ n 2 (f, x) are studied in the metricC, while f runs over the class \(\bar W^1 C\) , i. e., of the deviation $$F_n^2 (\bar W^1 ,C) = \mathop {\sup }\limits_{f \in \bar W^1 C} \left\| {f(x) - \sigma _n^2 (f,x)} \right\|_c$$ . It is proved that the function $$g^* (x) = \frac{4}{\pi }\mathop \sum \limits_{v = 0}^\infty ( - 1)^v \frac{{\cos (2v + 1)x}}{{(2v + 1)^2 }}$$ , for whichg *′(x)=sign cosx, satisfies the following asymptotic relation: $$F_n^2 (\bar W^1 ,C) = g^* (0) - \sigma _n^2 (g^* ,0) + O\left( {\frac{1}{{n^4 }}} \right)$$ , i.e.g * is close to the extremal function. This makes it possible to find some of the first terms in the asymptotic formula for \(F_n^2 (\bar W^1 ,C)\) asn → ∞. The corresponding problem for approximation in the metricL is also considered. 相似文献
11.
We introduce the counting function π 2,8 * (x) of the primes with difference 8 between consecutive primes ( ****p n,pn+1 =p n + 8) can be approximated by logarithm integralLi 2,8 * . We calculate the values of π 2,8 * (x) and the sumC 2,8(x) of reciprocals of primes with difference 8 between consecutive primes (p n,pn+1 =p n +8)) wherex is counted up to 7 x 1010. From the results of these calculations, we obtain π 2,8 * (7 x 1010) = 133295081 andC 2,8(7 x 1010) = 0.3374 ±2.6 x 10-4. 相似文献
12.
Я. Л. Геронимус 《Analysis Mathematica》1977,3(2):95-108
Рассматривается сис тема ортогональных м ногочленов {P n (z)} 0 ∞ , удовлетворяющ их условиям $$\frac{1}{{2\pi }}\int\limits_0^{2\pi } {P_m (z)\overline {P_n (z)} d\sigma (\theta ) = \left\{ {\begin{array}{*{20}c} {0,m \ne n,P_n (z) = z^n + ...,z = \exp (i\theta ),} \\ {h_n > 0,m = n(n = 0,1,...),} \\ \end{array} } \right.} $$ где σ (θ) — ограниченная неу бывающая на отрезке [0,2π] функция с бесчисленным множе ством точек роста. Вводится последовательность параметров {аn 0 ∞ , независимых дру г от друга и подчиненных единств енному ограничению { ¦аn¦<1} 0 ∞ ; все многочлены {Р n (z)} 0/∞ можно найти по формуле $$P_0 = 1,P_{k + 1(z)} = zP_k (z) - a_k P_k^ * (z),P_k^ * (z) = z^k \bar P_k \left( {\frac{1}{z}} \right)(k = 0,1,...)$$ . Многие свойства и оце нки для {P n (z)} 0 ∞ и (θ) можн о найти в зависимости от этих параметров; например, условие \(\mathop \Sigma \limits_{n = 0}^\infty \left| {a_n } \right|^2< \infty \) , бо лее общее, чем условие Г. Cerë, необходимо и достато чно для справедливости а симптотической форм улы в области ¦z¦>1. Пользуясь этим ме тодом, можно найти также реш ение задачи В. А. Стекло ва. 相似文献
13.
We consider the function system {cos4nθ} n=0 ∞ , {sin(4n ? 1)θ} n=1 ∞ , which arises in the Frankl problem in the theory of elliptic-hyperbolic equations. We show that this system is a Riesz basis in the space L 2(0, π/2) and construct the biorthogonal system. 相似文献
14.
V. A. Rodin 《Analysis Mathematica》1990,16(4):291-302
пУстьS k (f,x) — ЧАстНАь сУ ММА РьДА ФУРьЕ ФУНкцИ Иf пО тРИгОНОМЕтРИЧЕскОИ сИстЕМЕ,s k (f,x) — ЧАстНАь сУММА сО пРьжЕННОгО РьДА. Дль \(\Delta _k^n = \left[ {\frac{n}{n},\frac{{k + 1}}{n}} \right)\) , гДЕk=0, 1, ...,n?1, пОлОжИМ , ЕслИt?δ k n И , ЕслИt?[0, 1)δ k n . пОкАжАНО, ЧтО ОпЕРАтО Ры ИМЕУт слАБыИ тИп (1,1). РАссМОтР ЕН РьД слЕДстВИИ О пОВ ЕДЕНИИ сИльНых сРЕДНИх РьДА ФУРьЕ сУММИРУЕМОИ ФУНкцИИ. 相似文献
15.
Liqing Yan 《Journal of Theoretical Probability》2009,22(4):827-836
Several sharp upper and lower bounds for the ratio of two normal probabilities $\mathbb{P}\Biggl(\,\bigcap_{i=1}^{n}\bigl\{\xi^{(1)}_i\leq \mu_i\bigr\}\Biggr)\Big/\mathbb{P}\Biggl(\,\bigcap_{i=1}^{n}\bigl\{\xi^{(0)}_i\leq \mu_i\bigr\}\Biggr)$ are given in this paper for various cases, where (ξ 1 (0) ,ξ 2 (0) ,…,ξ n (0) ) and (ξ 1 (1) ,ξ 2 (1) , …,ξ n (1) ) are standard normal random variables with covariance matrices R 0=(r ij 0 ) and R 1=(r ij 1 ), respectively. 相似文献
16.
L. G. Pál 《Analysis Mathematica》1975,1(3):197-205
пУсть жАДАНы Ужлы $$ - \infty< x_1< x_2< ...< x_k< x_{k + 1}< ...< x_n< + \infty ,$$ , И пУстьx 1 * <x 2 * <...<x n-1 * — кОРНИ МНОгО ЧлЕНА Ω′(х). гДЕ $$\omega (x) = \prod\limits_{k = 1}^n {(x - x_k ).} $$ В РАБОтЕ ИсслЕДУЕтсь жАДАЧА: кАк ОпРЕДЕлИт ь МНОгОЧлЕНР(х) МИНИМАльНОИ стЕп ЕНИ, Дль кОтОРОгО ВыпОлНь Утсь слЕДУУЩИЕ ИНтЕР пОльцИОННыЕ УслОВИь гДЕ {y k И {y k′}-жАДАННы Е сИстЕМы жНАЧЕНИИ. 相似文献
17.
Diego Marques 《Bulletin of the Brazilian Mathematical Society》2013,44(3):455-468
For k ≥ 2, the k-generalized Fibonacci sequence (F n (k) ) n is defined by the initial values 0, 0, …, 0,1 (k terms) and such that each term afterwards is the sum of the k preceding terms. In 2005, Noe and Post conjectured that the only solutions of Diophantine equation F m (k) = F n (?) , with ? > k > 1, n > ? + 1, m > k + 1 are $(m,n,\ell ,k) = (7,6,3,2)and(12,11,7,3)$ . In this paper, we confirm this conjecture. 相似文献
18.
19.
A. M. Savchuk A. A. Shkalikov 《Proceedings of the Steklov Institute of Mathematics》2013,283(1):181-196
We consider the inverse problem of recovering the potential for the Sturm-Liouville operator Ly = ?y″ + q(x)y on the interval [0, π] from the spectrum of the Dirichlet problem and norming constants (from the spectral function). For a fixed θ ≥ 0, with this problem we associate a map F: W 2 θ → l D θ , F(σ) = {s k } 1 ∞ , where W 2 θ = W 2 θ [0, π] is the Sobolev space, σ = ∫ q is a primitive of the potential q ∈ W 2 θ ? 1 , and l D θ is a specially constructed finite-dimensional extension of the weighted space l 2 θ ; this extension contains the regularized spectral data s = {s k } 1 ∞ for the problem of recovering the potential from the spectral function. The main result consists in proving both lower and upper uniform estimates for the norm of the difference ‖σ ? σ 1‖ θ in terms of the l D θ norm of the difference of the regularized spectral data ‖s ? s1‖ θ . The result is new even for the classical case q ∈ L 2, which corresponds to the case θ = 1. 相似文献
20.
S. G. Pribegin 《Mathematical Notes》2007,82(5-6):643-652
Suppose that H p (E 2n + ) is the Hardy space for the first octant $$E_{2n}^ + = \{ z \in \mathbb{C}^n :\operatorname{Im} z_j > 0, j = 1, \ldots ,n\} $$ and P ? l (f, x), l > 0, is the generalized Abel-Poisson means of a function f ? H p (E 2n + ). In this paper, we prove the inequalities $$C_1 (l,p)\widetilde\omega _l (\varepsilon ,f)_p \leqslant \left\| {f(x) - P_\varepsilon ^l (f,x)} \right\|_p \leqslant C_2 (l,p)\omega _l (\varepsilon ,f)_p ,$$ where $\widetilde\omega _l (\varepsilon ,f)_p $ and ω l (?, f) p are the integral moduli of continuity of lth order. For n = 1 and an integer l, this result was obtained by Soljanik. 相似文献