共查询到20条相似文献,搜索用时 31 毫秒
1.
The integral $$\int_0^{{\pi \mathord{\left/ {\vphantom {\pi 4}} \right. \kern-\nulldelimiterspace} 4}} {\ln \left( {\cos ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta \pm \sin ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta } \right)d\theta } $$ where m and n are relatively prime positive integers, is evaluated exactly in terms of elementary functions and the Catalan constant G. 相似文献
2.
D. Suryanarayana 《Periodica Mathematica Hungarica》1983,14(1):69-75
LetL(x) denote the number of square-full integers not exceedingx. It is well-known that $$L\left( x \right) \sim \frac{{\zeta \left( {{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \right)}}{{\zeta \left( 3 \right)}}x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + \frac{{\zeta \left( {{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} \right)}}{{\zeta \left( 2 \right)}}x^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} ,$$ whereζ(s) denotes the Riemann Zeta function, LetΔ(x) denote the error function in the asymptotic formula forL(x). On the assumption of the Riemann hypothesis (R.H.), it is known that $$\Delta x = O\left( {x^{13/81 + 8} } \right)$$ for everyε > 0. In this paper, we prove on the assumption of R.H. that $$\frac{1}{x}\int\limits_x^1 {\left| {\Delta \left( t \right)} \right|dt = O\left( {x^{1/10 + ^8 } } \right)} .$$ In fact, we prove a more general result. We conjecture that $$\Delta x = O\left( {x^{1/10 + ^8 } } \right)$$ under the assumption of the R.H. 相似文献
3.
Ukrainian Mathematical Journal - We study C*-algebras $$ {\mathcal{O}}_{n+m}^{\hat{q}} $$ generated by isometries $$ {\left\{{S}_i\right\}}_{i=1}^n\bigcup {\left\{{t}_j\right\}}_{j=1}^m, $$ where... 相似文献
4.
5.
Jorge Antezana Gustavo Corach Demetrio Stojanoff 《Integral Equations and Operator Theory》2006,55(2):169-188
If
$$\mathcal{H}$$ is a Hilbert space,
$$\mathcal{S}$$ is a closed subspace of
$$\mathcal{H},$$ and A is a positive bounded linear operator on
$$\mathcal{H},$$ the spectral shorted operator
$$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ is defined as the infimum of the sequence
$$\sum (\mathcal{S},A^n )^{1/n} ,$$ where denotes
$$\sum \left( {\mathcal{S},B} \right)$$ the shorted operator of B to
$$\mathcal{S}.$$ We characterize the left spectral resolution of
$$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ and show several properties of this operator, particularly in the case that
dim
$${\mathcal{S} = 1.}$$ We use these results to generalize the concept of Kolmogorov complexity for the infinite dimensional
case and for non invertible operators. 相似文献
6.
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8.
Mathematical Notes - For the classes of functions $${W^r}\left( {{\omega _{\cal M}},\;{\rm{\Phi }}} \right)\;: = \left\{ {f\; \in \;L_2^r\left(\mathbb{R}\right)\;:\;{\omega _{\cal M}}\left(... 相似文献
9.
Kenji Nishihara 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2006,41(6):604-614
We consider the Cauchy problem for the nonlinear dissipative evolution system with ellipticity on one dimensional space
$ \left\{{{ll} {\psi_t=-\left({1-\alpha}\right)\psi-\theta_x+\alpha\psi_{xx},}&{\left( {t,x} \right) \in \left( {0,\infty } \right) \times {\bf R}}\\ {\theta _t = - \left( {1 - \alpha } \right)\theta + \nu ^2 \psi _x + \alpha \theta _{xx} + 2\psi \theta _x ,} } \right. $ \left\{{\begin{array}{ll} {\psi_t=-\left({1-\alpha}\right)\psi-\theta_x+\alpha\psi_{xx},}&{\left( {t,x} \right) \in \left( {0,\infty } \right) \times {\bf R}}\\ {\theta _t = - \left( {1 - \alpha } \right)\theta + \nu ^2 \psi _x + \alpha \theta _{xx} + 2\psi \theta _x ,} \end{array}} \right. 相似文献
10.
Eva Špániková 《Czechoslovak Mathematical Journal》2000,50(4):879-887
The purpose of this paper is to obtain oscillation criteria for the differential system
11.
D. Suryanarayana 《Periodica Mathematica Hungarica》1979,10(4):261-271
LetL(x) denote the number of square full integers ≤x. By a square-full integer, we mean a positive integer all of whose prime factors have multiplicity at least two. It is well known that $$\left. {L(x)} \right| \sim \frac{{\zeta ({3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2})}}{{\zeta (3)}}x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + \frac{{\zeta ({2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3})}}{{\zeta (2)}}x^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} ,$$ where ζ(s) denotes the Riemann Zeta function. Let Δ(x) denote the error function in the asymptotic formula forL(x). On the basis of the Riemann hypothesis (R.H.), it is known that \(\Delta (x) = O(x^{\tfrac{{13}}{{81}} + \varepsilon } )\) for every ε>0. In this paper, we prove the following results on the assumption of R.H.:
12.
V. Totik 《Analysis Mathematica》1980,6(2):165-184
стАтьь ьВльЕтсь пРОД ОлжЕНИЕМ пРЕДыДУЩЕИ ОДНОИМЕННОИ РАБОты АВтОРА, гДЕ ИжУ ЧАлсь пОРьДОк ВЕлИЧИН пРИ УслОВИьх, ЧтО α>-1/2, Рα >- 1 И ЧтО МАтРИцАt nk УДОВлЕтВОРьЕт НЕкОт ОРОМУ УслОВИУ РЕгУльРНОстИ. жДЕсь ДОкАжыВАЕтсь, Ч тО ЕслИf∈H Ω, тО ВыпОлНь Етсь ОцЕНкА $$\left\{ {\frac{1}{{\lambda _n }}\mathop \Sigma \limits_{k = n - \lambda _n + 1}^n \left| {\sigma _k^\alpha \left( x \right) - f\left( x \right)} \right|^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} = O\left( {\left\{ {\frac{1}{{\lambda _n }}\mathop \Sigma \limits_{k = n - \lambda _n + 1}^n \left( {\frac{1}{k}\mathop \smallint \limits_{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}}^{2\pi } \frac{{\omega \left( t \right)}}{{t^2 }}dt} \right)^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} + \left( {\frac{{\lambda _n }}{n}} \right)^\alpha \omega \left( {\frac{1}{n}} \right)} \right)$$ (λ1=1, λn+1-λn≦1), А тАкжЕ ЧтО Ёт А ОцЕНкА ОкОНЧАтЕльН А В сВОИх тЕРМИНАх; пОДОБ НыИ РЕжУль-тАт спРАВЕДлИВ тАкжЕ И Дль сОпРьжЕННОИ ФУНкцИИ . ДОкАжыВАЕтсь, ЧтО Усл ОВИьα>?1/2 Иpα>?1, кОтОРыЕ Б ылИ НАлОжЕНы В УпОМьНУтО И ВышЕ ЧАстИ I, сУЩЕстВЕН Ны. 相似文献
13.
Czechoslovak Mathematical Journal - Let χ be a nonprincipal Dirichlet character modulo a prime number p ? 3 and let $${\mathfrak{a}_{\cal X}}: = {1 \over 2}\left( {1{ - _{\cal X}}\left(... 相似文献
14.
In this paper, the following sharp estimate is proved: $$\int_{0}^{2{\pi }} {\left| {F\prime \left( {e^{i\theta } } \right)} \right|^p d\theta \leqslant \sqrt {\pi } 2^{1 + p} \frac{{\gamma \left( {{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} + {p \mathord{\left/ {\vphantom {p 2}} \right. \kern-\nulldelimiterspace} 2}} \right)}} {{\gamma \left( {1 + {p \mathord{\left/ {\vphantom {p 2}} \right. \kern-\nulldelimiterspace} 2}} \right)}}} ,\quad p > - 1,$$ where F is the conformal mapping of the domain $D^ - = \left\{ {\zeta :\left| \zeta \right| > 1} \right\}$ onto the exterior of a convex curve, with $F\prime \left( \infty \right) = 1$ . For p=1, this result is due to Pólya and Shiffer. We also obtain several generalizations of this estimate under other geometric assumptions about the structure of the domain F(D -). 相似文献
15.
Dong Guangchang 《数学年刊B辑(英文版)》1986,7(3):277-302
In this paper, the author proves the existence and uniqueness of nonnegative solution for the first boundary value problem of uniform degenerated parabolic equation
$$\[\left\{ {\begin{array}{*{20}{c}}
{\frac{{\partial u}}{{\partial t}} = \sum {\frac{\partial }{{\partial {x_i}}}\left( {v(u){A_{ij}}(x,t,u)\frac{{\partial u}}{{\partial {x_j}}}} \right) + \sum {{B_i}(x,t,u)} \frac{{\partial u}}{{\partial {x_i}}}} + C(x,t,u)u\begin{array}{*{20}{c}}
{}&{(x,t) \in [0,T]}
\end{array},}\{u{|_{t = 0}} = {u_0}(x),x \in \Omega ,}\{u{|_{x \in \partial \Omega }} = \psi (s,t),0 \le t \le T}
\end{array}} \right.\]$$
$$\[\left( {\frac{1}{\Lambda }{{\left| \alpha \right|}^2} \le \sum {{A_{ij}}{\alpha _i}{\alpha _j}} \le \Lambda {{\left| \alpha \right|}^2},\forall a \in {R^n},0 < \Lambda < \infty ,v(u) > 0\begin{array}{*{20}{c}}
{and}&{v(u) \to 0\begin{array}{*{20}{c}}
{as}&{u \to 0}
\end{array}}
\end{array}} \right)\]$$
under some very weak restrictions, i.e. $\[{A_{ij}}(x,t,r),{B_i}(x,t,r),C(x,t,r),\sum {\frac{{\partial {A_{ij}}}}{{\partial {x_j}}}} ,\sum {\frac{{\partial {B_i}}}{{\partial {x_i}}} \in \overline \Omega } \times [0,T] \times R,\left| {{B_i}} \right| \le \Lambda ,\left| C \right| \le \Lambda ,\],\[\left| {\sum {\frac{{\partial {B_i}}}{{\partial {x_i}}}} } \right| \le \Lambda ,\partial \Omega \in {C^2},v(r) \in C[0,\infty ).v(0) = 0,1 \le \frac{{rv(r)}}{{\int_0^r {v(s)ds} }} \le m,{u_0}(x) \in {C^2}(\overline \Omega ),\psi (s,t) \in {C^\beta }(\partial \Omega \times [0,T]),0 < \beta < 1\],\[{u_0}(s) = \psi (s,0).\]$ 相似文献
16.
Let H be a Hilbert space and A, B: H ⇉ H two maximal monotone operators. In this paper, we investigate the properties of the following proximal type algorithm:
17.
A Littlewood-Paley type
inequality 总被引:2,自引:0,他引:2
In this note we prove the following theorem: Let u be a
harmonic function in the unit ball
and
. Then there is a
constant C =
C(p,
n) such that
18.
RenZiYANG JunXiangXU 《数学学报(英文版)》2004,20(3):525-532
In this paper we consider the effective reducibility of the following linear differentialequation: x = (A ∈Q(t,∈))x, |∈| ≤ ∈0, where A is a constant matrix, Q(t,e) is quasiperiodic in t, and e is a small perturbation parameter. We prove that if the eigenvalues of A and the basic frequencies of Q satisfy some non-resonant conditions, the linear differential equation can be reduced to y = (A^*(∈) R^*(t, ∈))y, |∈| ≤ ∈o, where R^* is exponentially small in ∈. 相似文献
19.
Kenji Nishihara 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2006,57(4):604-614
We consider the Cauchy problem for the nonlinear dissipative evolution system with ellipticity on one dimensional space
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