共查询到20条相似文献,搜索用时 673 毫秒
1.
F. E. Lomovtsev 《Differential Equations》2008,44(6):866-871
We prove the well-posed solvability in the strong sense of the boundary value Problems where the unbounded operators A s (t), s > 0, in a Hilbert space H have domains D(A s (t)) depending on t, are subordinate to the powers A 1?(s?1)/2m (t) of some self-adjoint operators A(t) ≥ 0 in H, are [(s+1)/2] times differentiable with respect to t, and satisfy some inequalities. In the space H, the maximally accretive operators A 0(t) and the symmetric operators A s (t), s > 0, are approximated by smooth maximally dissipative operators B(t) in such a way that , where the smoothing operators are defined by .
相似文献
$$\begin{gathered} ( - 1)\frac{{_m d^{2m + 1} u}}{{dt^{2m + 1} }} + \sum\limits_{k = 0}^{m - 1} {\frac{{d^{k + 1} }}{{dt^{k + 1} }}} A_{2k + 1} (t)\frac{{d^k u}}{{dt^k }} + \sum\limits_{k = 1}^m {\frac{{d^k }}{{dt^k }}} A_{2k} (t)\frac{{d^k u}}{{dt^k }} + \lambda _m A_0 (t)u = f, \hfill \\ t \in ]0,t[,\lambda _m \geqslant 1, \hfill \\ {{d^i u} \mathord{\left/ {\vphantom {{d^i u} {dt^i }}} \right. \kern-\nulldelimiterspace} {dt^i }}|_{t = 0} = {{d^j u} \mathord{\left/ {\vphantom {{d^j u} {dt^j }}} \right. \kern-\nulldelimiterspace} {dt^j }}|_{t = T} = 0,i = 0,...,m,j = 0,...,m - 1,m = 0,1,..., \hfill \\ \end{gathered} $$
$$\begin{gathered} \mathop {lim}\limits_{\varepsilon \to 0} Re(A_0 (t)B_\varepsilon ^{ - 1} (t)(B_\varepsilon ^{ - 1} (t))^ * u,u)_H = Re(A_0 (t)u,u)_H \geqslant c(A(t)u,u)_H \hfill \\ \forall u \in D(A_0 (t)),c > 0, \hfill \\ \end{gathered} $$
$$B_\varepsilon ^{ - 1} (t) = (I - \varepsilon B(t))^{ - 1} ,(B_\varepsilon ^{ - 1} (t)) * = (I - \varepsilon B^ * (t))^{ - 1} ,\varepsilon > 0.$$
2.
Danilo Bazzanella 《Archiv der Mathematik》2011,97(5):453-458
Let d(n) denote the number of positive divisors of the natural number n. The aim of this paper is to investigate the validity of the asymptotic formulafor \({x \to + \infty,}\) assuming a hypothetical estimate on the meanwhich is a weakened form of a conjecture of M. Jutila.
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$\begin{array}{lll}\sum \limits_{x < n \leq x+h(x)}d(n)\sim h(x)\log x\end{array}$
$\begin{array}{lll} \int \limits_X^{X+Y}(\Delta(x+h(x))-\Delta (x))^2\,{d}x, \end{array}$
3.
M. M. Kabardov 《Vestnik St. Petersburg University: Mathematics》2009,42(2):90-94
In this paper, we consider a method for inverting the Laplace transform F(s) = \(\int\limits_0^\infty {e^{ - st} f(t)dt} \), which consists in representing the original function by the Laguerre series
$f(t) = \sum\limits_{k = 0}^\infty {a_k L_k (bt).} $
(1)
First, we perform a conformal mapping of the plane (s), which depends on parameter ξ. The value of the parameter is determined by the location of the singular points of the given representation. Under this mapping, series (1) takes the form
$f(t) = \frac{{b - \xi }}{b}\exp (\xi t)\sum\limits_{k = 0}^\infty {c_k L_k ((b - \xi )t).} $
It is demonstrated that such inverting scheme is equivalent to applying the Picone-Tricomi method with further acceleration of the rate of convergence of series (1) using the Euler-Knopp nonlinear procedure
$\sum\limits_{k = 0}^\infty {a_k z^k } = \sum\limits_{k = 0}^\infty {A_k (p)\frac{{z^k }}{{(1 - pz)^{k + 1} }},} A_k (p) = \sum\limits_{j = 0}^k {\left( \begin{gathered} k \hfill \\ j \hfill \\ \end{gathered} \right)( - p)^{k - j} a_j } .$
Under this approach, the original function is represented by the series where parameters ξ and p are related by the formula p = x/(ξ ? b). Unlike many other methods for summation of series, in the scheme suggested, there is no need to investigate the regularity conditions.
相似文献
$f(t) = \exp \left( {\frac{{bpt}}{{p - 1}}} \right)\sum\limits_{k = 0}^\infty {\frac{{A_k (p)}}{{(1 - p)^{k + 1} }}L_k } \left( {\frac{{bpt}}{{1 - p}}} \right),$
4.
A. T. Mukanova 《Moscow University Mathematics Bulletin》2008,63(2):82-84
An asymptotic formula for the mean value of the V. I. Arnold function A(n) = \(\tfrac{{\sigma (n)}}{{\tau (n)}}\) is obtained, here σ(n) = \(\mathop \Sigma \limits_{d|n} \) d is the sum of all divisors of the number n, τ (n) = \(\mathop \Sigma \limits_{d|n} \) 1 is their quantity. 相似文献
5.
We study decomposition of functions in the Hardy space \(H^2(\mathbb{D} )\) into linear combinations of the basic functions (modified Blaschke products) in the systemwhere the points a n ’s in the unit disc \(\mathbb{D}\) are adaptively chosen in relation to the function to be decomposed. The chosen points a n ’s do not necessarily satisfy the usually assumed hyperbolic non-separability conditionin the traditional studies of the system. Under the proposed procedure functions are decomposed into their intrinsic components of successively increasing non-negative analytic instantaneous frequencies, whilst fast convergence is resumed. The algorithm is considered as a variation and realization of greedy algorithm.
相似文献
$\label{Walsh like} {B}_n(z)= \frac{\sqrt{1-|a_n|^2}}{1-\overline{a}_{n}z}\prod\limits_{k=1}^{n-1}\frac{z-a_k}{1-\overline{a}_{k}z}, \quad n=1,2,..., $
(1)
$\label{condition} \sum\limits_{k=1}^\infty (1-|a_k|)=\infty $
(2)
6.
B. Simon 《Functional Analysis and Its Applications》2007,41(2):143-153
We show that the parameters a n , b n of a Jacobi matrix have a complete asymptotic expansion , where 1 < |µj| < R for j ? K(R) and all R, if and only if the Jost function, u, written in terms of z (where E = z + z ?1) is an entire meromorphic function. We relate the poles of u to the µj’s.
相似文献
$a_n^2 - 1 = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n} + O(R^{ - 2n} ),} b_n = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n + 1} + O(R^{ - 2n} )} $
7.
In the space L 2(?2), we consider the operator . We study the spectrum of H and, for V ∈ C 0 2 (?2), prove the trace formula where c 0 = π ?1 \(\smallint _{\mathbb{R}^2 } \) V(x) dx and the µ k (i) are the eigenvalues of H.
相似文献
$H = \left( {\frac{1}{i}\frac{\partial }{{\partial x_1 }} - x_2 } \right)^2 + \left( {\frac{1}{i}\frac{\partial }{{\partial x_2 }} + x_1 } \right)^2 + V,V = V(x) \in L_2 (\mathbb{R}^2 ).$
$\sum\limits_{k = 0}^\infty {\left( {\sum\limits_{i = - k}^\infty {(4k + 2 - \mu _k^{(i)} ) + c_0 } } \right)} = \frac{1}{{8\pi }}\int\limits_{\mathbb{R}^2 } {V^2 (x)dx,} $
8.
Alon Nishry 《Israel Journal of Mathematics》2011,186(1):197-220
Consider the random entire function , where the ? n are independent standard complex Gaussian coefficients, and the a n are positive constants, which satisfy .
$f(z) = \sum\limits_{n = 0}^\infty {{\phi _n}{a_n}{z^n}} $
$\mathop {\lim }\limits_{x \to \infty } {{\log {a_n}} \over n} = - \infty $
We study the probability P H (r) that f has no zeroes in the disk{|z| < r} (hole probability). Assuming that the sequence a n is logarithmically concave, we prove that , where , and r tends to ∞ outside a (deterministic) exceptional set of finite logarithmic measure.
相似文献
$\log {P_H}(r) = - S(r) + o(S(r))$
$S(r) = 2 \cdot \sum\limits_{n:{a_n}{r^n} \ge 1} {\log ({a_n}{r^n})} $
9.
David Kalaj 《Annali di Matematica Pura ed Applicata》2011,190(2):355-369
Let \({n\in\mathbb{N}}\). For \({k\in\{1,\dots,n\}}\) let \({\Omega_k\subset \mathbb{C}}\) be a simply connected domain with a rectifiable boundary. Let \({\Omega^n=\prod_{k=1}^n\Omega_k\subset \mathbb{C}^n}\) be a generalized polydisk with distinguished boundary \({\partial\Omega^n=\prod_{k=1}^n\partial\Omega_k}\). Let E r (Ω n ) be the holomorphic Smirnov class on Ω n with index r. We show that the generalized isoperimetric inequalityholds for arbitrary \({f_1\in E^p(\Omega^n)}\) and \({f_2\in E^q(\Omega^n)}\), where 0 < p, q < ∞. We also determine necessary and sufficient conditions for equality.
相似文献
$ \int\limits_{\Omega^n} |f_1|^p|f_2|^qdV\le \frac{1}{(4\pi)^n}\int\limits_{\partial \Omega^n}|f_1|^pdS \int\limits_{\partial \Omega^n} |f_2|^qdS, $
10.
Avram Sidi 《Numerische Mathematik》2011,119(4):725-764
The Shanks transformation is a powerful nonlinear extrapolation method that is used to accelerate the convergence of slowly converging, and even diverging, sequences {A n }. It generates a two-dimensional array of approximations \({A^{(j)}_n}\) to the limit or anti-limit of {A n } defined as solutions of the linear systemswhere \({\bar{\beta}_{k}}\) are additional unknowns. In this work, we study the convergence and stability properties of \({A^{(j)}_n}\) , as j → ∞ with n fixed, derived from general linear sequences {A n }, where \({{A_n \sim A+\sum^{m}_{k=1}\zeta_k^n\sum^\infty_{i=0} \beta_{ki}n^{\gamma_k-i}}}\) as n → ∞, where ζ k ≠ 1 are distinct and |ζ 1| = ... = |ζ m | = θ, and γ k ≠ 0, 1, 2, . . .. Here A is the limit or the anti-limit of {A n }. Such sequences arise, for example, as partial sums of Fourier series of functions that have finite jump discontinuities and/or algebraic branch singularities. We show that definitive results are obtained with those values of n for which the integer programming problemshave unique (integer) solutions for s 1, . . . , s m . A special case of our convergence result concerns the situation in which \({{\Re\gamma_1=\cdots=\Re\gamma_m=\alpha}}\) and n = mν with ν = 1, 2, . . . , for which the integer programming problems above have unique solutions, and it reads \({A^{(j)}_n-A=O(\theta^j\,j^{\alpha-2\nu})}\) as j → ∞. When compared with A j ? A = O(θ j j α ) as j → ∞, this result shows that the Shanks transformation is a true convergence acceleration method for the sequences considered. In addition, we show that it is stable for the case being studied, and we also quantify its stability properties. The results of this work are the first ones pertaining to the Shanks transformation on general linear sequences with m > 1.
相似文献
$A_l=A^{(j)}_n +\sum^{n}_{k=1}\bar{\beta}_k(\Delta A_{l+k-1}),\ \ j\leq l\leq j+n,$
$\begin{array}{ll}{\quad\quad\quad\quad\max\limits_{s_1,\ldots,s_m}\sum\limits_{k=1}^{m}\left[(\Re\gamma_k)s_k-s_k(s_k-1)\right],}\\ {{\rm subject\,to}\,\, s_1\geq0,\ldots,s_m\geq0\quad{\rm and}\quad \sum\limits_{k=1}^{m} s_k = n,}\end{array}$
11.
Let f and g be multiplicative functions of modulus 1. Assume that \( {\lim_{x \to \infty }}\frac{1}{x}\left| {\sum\nolimits_{n \leqslant x} {f(n)} } \right| = A > 0 \) and \( {\lim_{x \to \infty }}\frac{1}{x}\left| {\sum\nolimits_{n \leqslant x} {g(n)} } \right| = 0 \). We prove that, under these conditions,Concerning the Liouville function λ, we find an upper estimate for \( \frac{1}{x}\left| {\sum\limits_{n \leqslant x} {\lambda (n)\lambda (n + 1)} } \right| \) under the unproved hypothesis that L(s, χ) have Siegel zeros for an infinite sequence of L-functions.
相似文献
$ \mathop {\lim }\limits_{x \to \infty } \frac{1}{x}\sum\limits_{n \leqslant x} {f(n)g(n + 1) = 0.}$
12.
Expansions for the distribution and the maximum from distributions with an asymptotically gamma tail when a trend is present
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We give expansions about the Gumbel distribution in inverse powers of n and log n for Mn, the maximum of a sample size n or n + 1 when the j-th observation is μ(j/n) + ej, μ is any smooth trend function and the residuals {ej } are independent and identically distributed with P(e r) ≈ exp(-δx)xd0∑∞k=1ckx-kβ as x →∞. We illustrate practical value of the expansions using simulated data sets. 相似文献
13.
We consider some class of non-linear systems of the form where A(·) ∈ ? n × n , A i (·) ∈ ? n × n , b(·) ∈ ? n , whose coefficients are arbitrary uniformly bounded functionals.
$\dot x = A( \cdot )x + \sum\limits_{i = 1}^l {A_i ( \cdot )x(t - \tau _i (t)) + b( \cdot )u} ,$
A special type of the Lyapunov-Krasovskii functional is used to synthesize dynamic control described by the equation where ρ(·) ∈ ?1, m(·) ∈ ? n , which makes the system globally asymptotically stable. Also, the situation is considered where the control u enters into the system not directly but through a pulse element performing an amplitude-frequency modulation.
相似文献
$\dot u = \rho ( \cdot )u + (m( \cdot ),x),$
14.
Huixue Lao 《Acta Appl Math》2010,110(3):1127-1136
Let L(sym j f,s) be the jth symmetric power L-function attached to a holomorphic Hecke eigencuspform f(z) for the full modular group, and \(\lambda_{\mathrm{sym}^{j}f}(n)\) denote its nth coefficient. In this paper we are able to prove that and
相似文献
$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{3}f}(n)\bigg|^{2}dy=O\bigl(x^{2}\bigr),$
$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{4}f}(n)\bigg|^{2}dy=O\bigl(x^{\frac{11}{5}}\log x\bigr).$
15.
16.
Avram Sidi 《Numerical Algorithms》2017,74(3):937-949
Let {x m } be a vector sequence that satisfies s being the limit or antilimit of {x m } and \(\{\boldsymbol {g}_{i}(m)\}^{\infty }_{i=1}\) being an asymptotic scale as m → ∞, in the sense that The vector sequences \(\{\boldsymbol {g}_{i}(m)\}^{\infty }_{m=0}\), i = 1, 2,…, are known, as well as {x m }. In this work, we analyze the convergence and convergence acceleration properties of a vectorized version of the generalized Richardson extrapolation process that is defined via the equations s n, k being the approximation to s. Here, y is some nonzero vector, 〈? ,?〉 is an inner product, such that \(\langle \alpha \boldsymbol {a},\beta \boldsymbol {b}\rangle =\overline {\alpha }\beta \langle \boldsymbol {a},\boldsymbol {b}\rangle \), and Δx m = x m + 1? x m and Δg i (m) = g i (m + 1)?g i (m). By imposing a minimal number of reasonable additional conditions on the g i (m), we show that the error s n, k ? s has a full asymptotic expansion as n→∞. We also show that actual convergence acceleration takes place, and we provide a complete classification of it.
相似文献
$$\boldsymbol{x}_{m}\sim \boldsymbol{s}+\sum\limits^{\infty}_{i=1}\alpha_{i} \boldsymbol{g}_{i}(m)\quad\text{as \(m\to\infty\)}, $$
$$\lim\limits_{m\to\infty}\frac{\|\boldsymbol{g}_{i+1}(m)\|}{\|\boldsymbol{g}_{i}(m)\|}=0,\quad i=1,2,\ldots. $$
$$\sum\limits^{k}_{i=1}\langle\boldsymbol{y},{\Delta}\boldsymbol{g}_{i}(m)\rangle\widetilde{\alpha}_{i}=\langle\boldsymbol{y},{\Delta}\boldsymbol{x}_{m}\rangle,\quad n\leq m\leq n+k-1;\quad \boldsymbol{s}_{n,k}=\boldsymbol{x}_{n}+\sum\limits^{k}_{i=1}\widetilde{\alpha}_{i}\boldsymbol{g}_{i}(n), $$
17.
Adel Mahmoud Gomaa 《Czechoslovak Mathematical Journal》2017,67(2):339-365
We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation where {?(t): t ∈ R} is a family of linear operators from a Banach space E into itself and f: R × E → E. By L(E) we denote the space of linear operators from E into itself. Furthermore, for a < b and d > 0, we let C([?d, 0],E) be the Banach space of continuous functions from [?d, 0] into E and f d : [a, b] × C([?d, 0],E) → E. Let \(\hat {\mathcal{L}}:[a,b] \to L(E)\) be a strongly measurable and Bochner integrable operator on [a, b] and for t ∈ [a, b] define τ t x(s) = x(t + s) for each s ∈ [?d, 0]. We prove that, under certain conditions, the differential equation with delay has at least one weak solution and, under suitable assumptions, the differential equation (Q) has a solution. Next, under a generalization of the compactness assumptions, we show that the problem (Q) has a solution too.
相似文献
$$\dot x\left( t \right) = \mathcal{L}\left( t \right)x\left( t \right) + f\left( {t,x\left( t \right)} \right),t \in \mathbb{R}$$
(P)
$$\dot x\left( t \right) = \hat {\mathcal{L}}\left( t \right)x\left( t \right) + {f^d}\left( {t,{\tau _t}x} \right),ift \in \left[ {a,b} \right],$$
(Q)
18.
S. Gulzar 《Analysis Mathematica》2016,42(4):339-352
For a polynomial P(z) of degree n having no zeros in |z| < 1, it was recently proved in [9] that for every β ∈ C with |β| ≤ 1, 1 ≤ s ≤ n and |z| = 1. In this paper, we obtain the L p mean extension of the above and other related results for the sth derivative of polynomials.
相似文献
$$\left| {{z^s}{P^{\left( s \right)}}\left( z \right) + \beta \frac{{n\left( {n - 1} \right)...\left( {n - s + 1} \right)}}{{{2^s}}}P\left( z \right)} \right| \leqslant \frac{{n\left( {n - 1} \right)...\left( {n - s + 1} \right)}}{2}\left( {\left| {1 + \frac{\beta }{{{2^s}}}} \right| + \left| {\frac{\beta }{{{2^s}}}} \right|} \right)\mathop {\max }\limits_{\left| z \right| = 1} \left| {P\left( z \right)} \right|$$
19.
We give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in R n , n ≥ 1, so that the following inequalities hold for all \({u \in C_{0}^{\infty}(B)}\) : This characterization makes a very useful connection between Hardy-type inequalities and the oscillatory behaviour of certain ordinary differential equations, and helps in the identification of a large number of such couples (V, W)—that we call Bessel pairs—as well as the best constants in the corresponding inequalities. This allows us to improve, extend, and unify many results—old and new—about Hardy and Hardy–Rellich type inequalities, such as those obtained by Caffarelli et al. (Compos Math 53:259–275, 1984), Brezis and Vázquez (Revista Mat. Univ. Complutense Madrid 10:443–469, 1997), Wang and Willem (J Funct Anal 203:550–568, 2003), Adimurthi et al. (Proc Am Math Soc 130:489–505, 2002), and many others.
相似文献
$\label{one} \int\limits_{B}V(x)|\nabla u |^{2}dx \geq \int\limits_{B} W(x)u^2dx,$
$\label{two} \int\limits_{B}V(x)|\Delta u |^{2}dx \geq\int\limits_{B} W(x)|\nabla u|^{2}dx+(n-1)\int\limits_{B}\left(\frac{V(x)}{|x|^2}-\frac{V_r(|x|)}{|x|}\right)|\nabla u|^2dx.$
20.
S. V. Astashkin 《Functional Analysis and Its Applications》2008,42(2):144-147
Let X be a separable or maximal rearrangement invariant space on [0, 1]. Necessary and sufficient conditions are found under which the generalized Khintchine inequality holds for an arbitrary sequence {?k} k=1 ∞ ? X of mean zero independent variables. Moreover, the subspace spanned in a rearrangement invariant space by the Rademacher system with independent vector coefficients is studied.
相似文献
$\left\| {\sum\limits_{k = 1}^\infty {f_k } } \right\|_X \leqslant C\left\| {\left( {\sum\limits_{k = 1}^\infty {f_k^2 } } \right)^{1/2} } \right\|_X $