共查询到20条相似文献,搜索用时 15 毫秒
1.
Let M(σ) = sup{| F(σ + it)|: t ∈ ℝ} and μ(σ) = max {| a
n
|exp(σλ n): n ≥ 0}, σ < 0, for a Dirichlet series {fx995-01} with abscissa of absolute convergence σ a = 0. We prove that the condition ln ln n = o(ln λ n), n → ∞, is necessary and sufficient for the equivalence of the relations {fx995-02}, for each series of this type.
Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 851–856, June, 2008. 相似文献
2.
In this paper, we prove the algebraic independence of the reciprocal sums of odd terms in Fibonacci numbers ∑
n=1∞
F
2n−1−1, ∑
n=1∞
F
2n−1−2, ∑
n=1∞
F
2n−1−3 and write each ∑
n=1∞
F
2n−1−s
( s≥4) as an explicit rational function of these three numbers over ℚ. Similar results are obtained for various series including
the reciprocal sums of odd terms in Lucas numbers.
相似文献
3.
For the Dirichlet series F(s) = ?n = 1¥ anexp{ sln } F(s) = \sum\nolimits_{n = 1}^\infty {{a_n}\exp \left\{ {s{\lambda_n}} \right\}} with abscissa of absolute convergence σ
a
=0, we establish conditions for (λ
n
) and (a
n
) under which lnM( s, F ) = TR( 1 + o(1) )exp{ rR | / |
| s| } \ln M\left( {\sigma, F} \right) = {T_R}\left( {1 + o(1)} \right)\exp \left\{ {{{{{\varrho_R}}} \left/ {{\left| \sigma \right|}} \right.}} \right\} for σ ↑ 0, where
M( s, F ) = sup{ | F( s+ it ) |:t ? \mathbbR } M\left( {\sigma, F} \right) = \sup \left\{ {\left| {F\left( {\sigma + it} \right)} \right|:t \in \mathbb{R}} \right\} and T
R
and ϱ
R
are positive constants. 相似文献
4.
Let f( z)=∑
n=1∞
λ
f
( n) n
(κ−1)/2
e( nz) be a holomorphic cusp form of weight κ for the full modular group SL
2(ℤ). In this paper we study the cancellation of the coefficients λ
f
( n) over primes in exponential sums. 相似文献
5.
We consider Dirichlet series z g,a( s)=? n=1¥ g( na) e-ln s{\zeta_{g,\alpha}(s)=\sum_{n=1}^\infty g(n\alpha) e^{-\lambda_n s}} for fixed irrational α and periodic functions g. We demonstrate that for Diophantine α and smooth g, the line Re( s) = 0 is a natural boundary in the Taylor series case λ
n
= n, so that the unit circle is the maximal domain of holomorphy for the almost periodic Taylor series ? n=1¥ g( na) zn{\sum_{n=1}^{\infty} g(n\alpha) z^n}. We prove that a Dirichlet series z g,a( s) = ? n=1¥ g( n a)/ ns{\zeta_{g,\alpha}(s) = \sum_{n=1}^{\infty} g(n \alpha)/n^s} has an abscissa of convergence σ
0 = 0 if g is odd and real analytic and α is Diophantine. We show that if g is odd and has bounded variation and α is of bounded Diophantine type r, the abscissa of convergence σ
0 satisfies σ
0 ≤ 1 − 1/ r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and α is Diophantine, then the Dirichlet series ζ
g,α
( s) has an analytic continuation to the entire complex plane. 相似文献
6.
For given analytic functions ϕ( z) = z + Σ
n=2∞ λ
n
z
n
, Ψ( z) = z + Σ
n=2∞ μ with λ
n
≥ 0, μ
n
≥ 0, and λ
n
≥ μ
n
and for α, β (0≤α<1, 0<β≤1), let E(φ,ψ; α, β) be of analytic functions ƒ( z) = z + Σ
n=2∞
a
n
z
n
in U such that f(z)*ψ(z)≠0 and for z∈U; here, * denotes the Hadamard product. Let T be the class of functions ƒ( z) = z - Σ
n=2∞| a
n
| that are analytic and univalent in U, and let E
T
(φ,ψ;α,β)= E(φ,ψ;α,β)∩ T. Coefficient estimates, extreme points, distortion properties, etc. are determined for the class E
T
(φ,ψ;α,β) in the case where the second coefficient is fixed. The results thus obtained, for particular choices of φ( z) and ψ( z), not only generalize various known results but also give rise to several new results.
University of Bahrain, Isa Town, Bahrain. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 9, pp. 1162–1170,
September, 1997. 相似文献
7.
Let Y
s,n
denote the number of part sizes ≧ s in a random and uniform partition of the positive integer n that are counted without multiplicity. For s = λ(6 n) 1/2/π + o( n
1/4), 0 ≦ λ < ∞, as n → ∞, we establish the weak convergence of Y
s,n
to a Gaussian distribution in the form of a central limit theorem. The mean and the standard deviation are also asymptotically
determined.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
8.
We establish conditions under which, for a Dirichlet series F( z) = Σ
n = 1
∞
d
n
exp(λ
n
z), the inequality ⋎F(x)⋎≤y(x),x≥x
o, implies the relation Σ
n = 1
∞ | d
n
exp(λ
n
z)| ⪯ γ((1 + o(1)) x) as x→+∞, where γ is a nondecreasing function on (−∞,+∞).
Franko Drohobych State Pedagogic Institute, Drohobych. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12,
pp. 1610–1616. December, 1997 相似文献
9.
In this paper a variant of Lusternik-Schnirelmann category is presented which is denoted by Qcat( X). It is obtained by applying a base-point free version of Q=Ω ∞∑ ∞ fibrewise to the Ganea fibrations. We prove cat( X)≥ Qcat( X)≥ σcat( X) where σcat( X) denotes Y. Rudyak’s strict category weight. However, Qcat( X) approximates cat( X) better, because, e.g., in the case of a rational space Qcat( X)= cat( X) and σcat( X) equals the Toomer invariant.
We show that Qcat( X× Y)≤ Qcat( X)+ Qcat( Y). The invariant Qcat is designed to measure the failure of the formula cat( X× S
r
)= cat( X)+1. In fact for 2-cell complexes Qcat( X)< cat( X)⇔ cat( X× S
r
)= cat( X) for some r≥1.
We note that the paper is written in the more general context of a functor λ from the category of spaces to itself satisfying
certain conditions; λ= Q, Ω
n
Σ
n
, Sp
∞
or L
f
are just particular cases. 相似文献
10.
Let A⊆N={0,1,2,...} and β be an n-ary Boolean function. We call A a β-implicatively selector (β-IS) set if there exists an
n-ary selector general recursive function f such that (∀x 1,...,x n)(β(χ(x 1),...,χ(x n))=1⟹f(x 1,...,x n)∈A), where χ is the characteristic function of A. Let F (m), m≥1, be the family of all d
m+1
*
-IS sets, where
, F (0)=N, and F (∞) is the class of all subsets in N. The basic result of the article says that the family of all β-IS sets coincides with one
of F (m), m≥0, or F (∞), and, moreover, the inclusions F (0)⊂F (1)⊂...⊂F (∞) hold.
Translated from Algebra i Logika, Vol. 35, No. 2, pp. 145–153, March–April, 1996. 相似文献
11.
The nonlinear two-parameter Sturm-Liouville problem u
"+μ g( u)=λ f( u) is studied for μ, λ>0. By using Ljusternik-Schnirelman theory on the general level set developed by Zeidler, we shall show
the existence of an n-th variational eigenvalue λ=λ n(μ). Furthermore, for special f and g, the asymptotic formula of λ 1(μ)) as μ→∞ is established. 相似文献
12.
Here proposed are certain asympotic expansion formulas for L
n
(∞-1)
(λz) and C
n
(∞)
(λz) in which 0<w=0(λ) and C n/( w)(λz), z being a complex number. Also presented are certain estimates for the remainders (error bounds) of the asymptotic
expansions within the regions D 1(-∞<R ez<=1/2(ω/λ) and D 2(1/2(ω/λ)<=Re z<∞), respectively.
Supported by NSERC (Canada) and also by the National Natural Science Foundation of China. 相似文献
13.
Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. random m×m matrices ( X
i)
i
0/∞
over ℂ. Assume that the X
i's are chosen from a finite set { D
0, D
1..., D
t-1(ℂ), with P(X
i=D j)>0, and that the monoid generated by D
0, D 1,…, D q−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case
where the X
i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22].
Our results on λ enable us to provide an approximation for the number N
≠0( F( x) n, r) of nonzero coefficients in F(x)
n.(mod r), where F( x) ∈ ℤ[ x] and r≥2. We prove the existence of and supply a formula for a constant α (<1) such that N
≠0( F( x) n, r) ≈ n
α for “almost” every n.
Supported in part by FWF Project P16004-N05 相似文献
14.
Let F( z)=∑
n=1∞
A( n) q
n
denote the unique weight 6 normalized cuspidal eigenform on Γ 0(4). We prove that A( p)≡0,2,−1(mod 11) when p≠11 is a prime. We then use this congruence to give an application to the number of representations of an integer by quadratic
form of level 4.
相似文献
15.
We investigate the behaviour of solution u = u( x, t; λ) at λ = λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ut = (un)xx + lf(u)/(ò-11 f(u)dx)2{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s
n-1
f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution.
For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* = u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}. 相似文献
16.
The study of jointly ergodic measure preserving transformations of probability spaces, begun in [1], is continued, and notions
of joint weak and strong mixing are introduced. Various properties of ergodic and mixing transformations are shown to admit
analogues for several transformations. The case of endomorphisms of compact abelian groups is particularly emphasized. The
main result is that, given such commuting endomorphisms σ 1σ 2,...,σ, of G, the sequence ((1/ N)Σ
n=0
N−1
σ
1
n
f
1·σ
2
n
f
2· ··· · σ
s
n
f
sconverges in L
2( G) for every f
1, f
2,…, f
s∈ L
∞( G). If, moreover, the endomorphisms are jointly ergodic, i.e., if the limit of any sequence as above is Π
i=1
s
∫
G
f
1
d
μ, where μ is the Haar measure, then the convergence holds also μ-a.e. 相似文献
17.
Let ( A, D( A)) be the infinitesimal generator of a Feller semigroup such that C
c
∞(ℝ
n
)⊂ D( A) and A| C
c
∞(ℝ
n
) is a pseudo-differential operator with symbol − p( x,ξ) satisfying | p(•,ξ)| ∞≤ c(1+|ξ| 2) and |Im p( x,ξ)|≤ c
0Re p( x,ξ). We show that the associated Feller process { X
t
}
t
≥0 on ℝ
n
is a semimartingale, even a homogeneous diffusion with jumps (in the sense of [21]), and characterize the limiting behaviour
of its trajectories as t→0 and ∞. To this end, we introduce various indices, e.g., β ∞
x
:={λ>0: lim
|ξ|→∞
|
x
−
y
|≤2/|ξ|| p( y,ξ)|/|ξ| λ=0} or δ ∞
x
:={λ>0: liminf
|ξ|→∞
|
x
−
y
|≤2/|ξ|
|ε|≤1| p( y,|ξ|ε)|/|ξ| λ=0}, and obtain a.s. (ℙ
x
) that lim
t
→0
t
−1/λ
s
≤
t
| X
s
− x|=0 or ∞ according to λ>β ∞
x
or λ<δ ∞
x
. Similar statements hold for the limit inferior and superior, and also for t→∞. Our results extend the constant-coefficient (i.e., Lévy) case considered by W. Pruitt [27].
Received: 21 July 1997 / Revised version: 26 January 1998 相似文献
18.
Let 𝔄 denote the C *-algebra of bounded operators on L
2 ℝ generated by: (i) all multiplications a( M) by functions a∈ C[ − ∞, + ∞], (ii) all multiplications by 2π-periodic continuous functions, and (iii) all operator of the form F
−1
b( M) F, where F denotes the Fourier transform and b∈ C[ − ∞, + ∞]. A given A ∈ 𝔄 is a Fredholm operator if and only if σ( A) and γ( A) are invertible, where σ denotes the continuous extension of the usual principal symbol, while γ denotes an operator-valued “boundary principal symbol” (the “boundary” here consists of two copies of the circle, one at
each end of the real line). We give two proofs of the fact that K
0(𝔄) is isomorphic to ℤ and that K
1(𝔄) is isomorphic to ℤ ⊕ ℤ . We do it first by computing the connecting mappings in the six-term exact sequence associated
to σ. For the second proof, we show that the image of γ is isomorphic to the direct sum of two copies of the crossed product
, where α denotes the translation-by-one automorphism. Its K-theory can be computed using the Pimsner–Voiculescu exact sequence,
and that information suffices for the analysis of the standard cyclic exact sequence associated to γ.
Received: February 2006 相似文献
20.
Summary Let S
i have the Wishart distribution W
p(∑ i,n i) for i=1,2. An asymptotic expansion of the distribution of
for large n=n 1+n 2 is derived, when ∑
1∑
2
−1
=I+n −1/2θ, based on an asymptotic solution of the system of partial differential equations for the hypergeometric function 2
F
1, obtained recently by Muirhead [2]. Another asymptotic formula is also applied to the distributions of −2 log λ and −log| S
2( S
1+ S
2) −1| under fixed ∑
1∑
2
−1
, which gives the earlier results by Nagao [4]. Some useful asymptotic formulas for 1
F
1 were investigated by Sugiura [7]. 相似文献
|