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1.
Yaochen Zhu 《数学学报(英文版)》2000,16(3):395-398
Let f (x) be a continued fraction with elements a
n
x, where coefficients a
n
are positive algebraic numbers. Using the criterion of [l] for any nonzero real algebraic numbers α1,...,αs with distinct absolute values the algebraic independence of the values f(α1), ..., f(αs) is proved under certain assumption concerning only with a
n
. For some transcendental numbers ξ the algebraic independence of values f(ξj)(j∈ℤ) is also established.
Received March 27, 1998, Accepted September 28, 1998 相似文献
2.
Age-dependent branching processes in random environments 总被引:4,自引:0,他引:4
We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ0,ξ1,...) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξn) on R , and reproduce independently new particles according to a probability law p(ξn) on N. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean EξZ(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments. 相似文献
3.
We say that n independent trajectories ξ1(t),…,ξ
n
(t) of a stochastic process ξ(t)on a metric space are asymptotically separated if, for some ɛ > 0, the distance between ξ
i
(t
i
) and ξ
j
(t
j
) is at least ɛ, for some indices i, j and for all large enough t
1,…,t
n
, with probability 1. We prove sufficient conitions for asymptotic separationin terms of the Green function and the transition
function, for a wide class of Markov processes. In particular,if ξ is the diffusion on a Riemannian manifold generated by
the Laplace operator Δ, and the heat kernel p(t, x, y) satisfies the inequality p(t, x, x) ≤ Ct
−ν/2 then n trajectories of ξ are asymptotically separated provided . Moreover, if for some α∈(0, 2)then n trajectories of ξ(α) are asymptotically separated, where ξ(α) is the α-process generated by −(−Δ)α/2.
Received: 10 June 1999 / Revised version: 20 April 2000 / Published online: 14 December 2000
RID="*"
ID="*" Supported by the EPSRC Research Fellowship B/94/AF/1782
RID="**"
ID="**" Partially supported by the EPSRC Visiting Fellowship GR/M61573 相似文献
4.
A. Yu. Zaįtsev 《Probability Theory and Related Fields》1988,79(2):175-200
Summary Let ξ1, ξ2,... be i.i.d random vectors in ℝ
k
with a common distribution ℒ(ξi),... = F, i = 1, 2,.... Let S
n
= ξ1+...+ξ
n
. We investigate how small is the difference between ℒ(S
n
) and ℒ(S
n+ m
) in the case when ξ
i
have symmetric distributions. 相似文献
5.
Let ξ be a real-valued random variable of mean zero and variance 1. Let M
n
(ξ) denote the n × n random matrix whose entries are iid copies of ξ and σ
n
(M
n
(ξ)) denote the least singular value of M
n
(ξ). The quantity σ
n
(M
n
(ξ))2 is thus the least eigenvalue of the Wishart matrix MnMn*{M_nM_n^\ast}. 相似文献
6.
V. P. Kurenok 《Journal of Theoretical Probability》2007,20(4):859-869
The stochastic equation dX
t
=dS
t
+a(t,X
t
)dt, t≥0, is considered where S is a one-dimensional Levy process with the characteristic exponent ψ(ξ),ξ∈ℝ. We prove the existence of (weak) solutions for a bounded, measurable coefficient a and any initial value X
0=x
0∈ℝ when (ℛe
ψ(ξ))−1=o(|ξ|−1) as |ξ|→∞. These conditions coincide with those found by Tanaka, Tsuchiya and Watanabe (J. Math. Kyoto Univ. 14(1), 73–92, 1974) in the case of a(t,x)=a(x). Our approach is based on Krylov’s estimates for Levy processes with time-dependent drift. Some variants of those estimates
are derived in this note. 相似文献
7.
Let ξ, ξ1, ξ2, ... be independent identically distributed random variables, and S
n
:=Σ
j=1
n
,ξ
j
, $
\bar S
$
\bar S
:= sup
n≥0
S
n
. If Eξ = −a < 0 then we call transient those phenomena that happen to the distribution $
\bar S
$
\bar S
as a → 0 and $
\bar S
$
\bar S
tends to infinity in probability. We consider the case when Eξ fails to exist and study transient phenomena as a → 0 for the following two random walk models:
We obtain some results for identically and differently distributed ξ
j
. 相似文献
1. | The first model assumes that ξ j can be represented as ξ j = ζ j + αη j , where ζ1, ζ 2 , ... and η 1, η 2, ... are two independent sequences of independent random variables, identically distributed in each sequence, such that supn≥0Σ j=1 n ζ j = ∞, sup n≥0Σ j=1 n η j < ∞, and $ \bar S $ \bar S < ∞ almost surely. |
2. | In the second model we consider a triangular array scheme with parameter a and assume that the right tail distribution P(ξ j ≥ t) ∼ V (t) as t→∞ depends weakly on a, while the left tail distribution is P(ξ j < −t) = W(t/a), where V and W are regularly varying functions and $ \bar S $ \bar S < ∞ almost surely for every fixed α > 0. |
8.
D. V. Gusak 《Ukrainian Mathematical Journal》2000,52(2):234-248
For a process ξ(t = ξ1(t)+χ(t), t≥0, ξ(0) = 0, inhomogeneous with respect to time, we investigate the ruin problem associated with the corresponding random
walk in a finite interval, (here, ξ1 (t) is a homogeneous Poisson process with positive integer-valued jumps and χ(t) is an inhomogeneous lower-semicontinuous process with integer-valued jumps ξ
n
≥-1). 相似文献
9.
Frank Jonas 《manuscripta mathematica》1990,67(1):35-39
Zusammenfassung LetD:G→GL(n,C) be an irreducible linear representation of a finite groupG with the characterX. IfD is realizible in Q(ξ
m
) and Q(ξ
m′
) we give a condition for then realizability ofD in Q(ξ(m′)). If the degreen is a prime ≠ 2, we show thatD realizible in Q(ξ
f
), wheref is the conductor of the abelian extensionQ(X)/Q. 相似文献
10.
Artūras Dubickas 《Israel Journal of Mathematics》2009,170(1):95-111
Let f
1 and f
2 be two positive numbers of the field , and let f
n+2 = f
n+1 + f
n
for each n ≥ 1. Let us denote by {x} the fractional part of a real number x. We prove that, for each ξ ∉ K, the inequality {ξf
n
} > 2/3 holds for infinitely many positive integers n. On the other hand, we prove a result which implies that there is a transcendental number ξ such that {ξf
n
} < 39/40 for each n ≥ 1. Moreover, it is shown that, for every a > 1, there is an interval of positive numbers that contains uncountably many numbers ξ such that {a
n
} 6 min 2/(a − 1), (34a
2 − 32a + 7)/(9(2a − 1)2) for each n > 1. Here, the minimum is strictly smaller than 1 for each a > 1. In contrast, by an old result of Weyl, for any a > 1, the sequence {ξa
n
}, n = 1, 2, ..., is uniformly distributed in [0, 1] (and so everywhere dense in [0, 1]) for almost all real numbers ξ. 相似文献
11.
本文研究单位圆盘上Bergman型空间到Zygmund型空间上的一类推广的Volterra复合算子.利用符号函数ϕ和g刻画这类算子的有界性、紧性,并计算其本性范数. 相似文献
12.
In the present paper, necessary and sufficient conditions are given for the equality of the power rezidue symbols
( \fracaa )n {\left( {\frac{\alpha }{a}} \right)_n} and
( \fracaa )n {\left( {\frac{\alpha }{a}} \right)_n} in the cyclotomic field ℚ(ζ
n
), 2 ∤ n, for a ∈ ℤ, (a, n) = 1. This result is a generalization of the classical Eisenstein reciprocity law and its continuation in a Hasse’s paper.
Bibliography: 3 titles. 相似文献
13.
V. S. Atabekyan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2010,45(2):112-122
In the present paper for arbitrary automorphism φ of the free Bunside group B(m, n) and for any odd number n ≥ 1003 a sufficient condition for existence of non-φ-admissible normal subgroup of B(m, n) was found. In particular, if automorphism φ is normal, then for any basis {a
1, a
2, …, a
m
} of the group B(m, n) there is an integer k such that for each i the elements a
i
and φ(a
i)
k
are conjugates. 相似文献
14.
A characteristic property of spheres 总被引:1,自引:1,他引:0
A. D. Alexandrov 《Annali di Matematica Pura ed Applicata》1962,58(1):303-315
Summary We prove: Let S be a closed n-dimensional surface in an(n+1)-space of constant curvature (n ≥ 2); k1 ≥ ... ≥ kn denote its principle curvatures. Let φ(ξ1, ..., ξn) be such that
. Then if φ(k1, ..., kn)=const on S and S is subject to some additional general conditions (those(II
0) or(II) no 1), S is a sphere.
To Enrico Bompiani on his scientific Jubilee 相似文献
15.
Nadine Hilgert J. Adolfo Minjárez-Sosa 《Mathematical Methods of Operations Research》2001,54(3):491-505
We consider a class of time-varying stochastic control systems, with Borel state and action spaces, and possibly unbounded
costs. The processes evolve according to a discrete-time equation x
n + 1=G
n (x
n , a
n , ξn), n=0, 1, … , where the ξn are i.i.d. ℜk-valued random vectors whose common density is unknown, and the G
n are given functions converging, in a restricted way, to some function G
∞ as n→∞. Assuming observability of ξn, we construct an adaptive policy which is asymptotically discounted cost optimal for the limiting control system
x
n+1=G
∞ (x
n , a
n , ξn). 相似文献
16.
Let ξ,ξ
1,ξ
2,… be positive i.i.d. random variables, S=∑
j=1∞
a(j)ξ
j
, where the coefficients a(j)≥0 are such that P(S<∞)=1. We obtain an explicit form of the asymptotics of −ln P(S<x) as x→0 for the following three cases:
The research partially supported by the RFBR grants 05-01-00810 and 06-01-00738, the Russian President’s grant NSh-8980-2006.1,
and the INTAS grant 03-51-5018. The second author also supported by the Lavrentiev SB RAS grant for young scientists. 相似文献
(i) | the sequence {a(j)} is regularly varying with exponent −β<−1, and −ln P(ξ<x)=O(x −γ+δ ) as x→0 for some δ>0, where γ=1/(β−1), |
(ii) | −ln P(ξ<x) is regularly varying with exponent −γ<0 as x→0, and a(j)=O(j −β−δ ) as j→∞ for some δ>0, where γ=1/(β−1), |
(iii) | {a(j)} decreases faster than any power of j, and P(ξ<x) is regularly varying with positive exponent as x→0. |
17.
René L. Schilling 《Probability Theory and Related Fields》1998,112(4):565-611
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 |Imp(x,ξ)|≤c
0Rep(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.
We study the asymptotic behavior of a set of random vectors ξi, i = 1,..., m, whose coordinates are independent and identically distributed in a space of infinitely increasing dimension. We investigate
the asymptotics of the distribution of the random vectors, the consistency of the sets M
m(n) = ξ1,..., ξm and X
nλ = x ∈ X
n: ρ(x) ≤ λn, and the mutual location of pairs of vectors.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1706–1711, December, 1998. 相似文献
19.
Vilius Stakénas 《Lithuanian Mathematical Journal》1999,39(1):86-101
This paper is concerned with the sieve problem for Farey fractions (i.e., rational numbers with denominators less thanx) lying in an interval (λ1, λ2). An asymptotic formula for the sifting function is derived under the assumption that (λ1, λ2)x→∞ asx→∞. Two applications of this result are made. In the first one, the value distribution of the vector η(m/n)=(ξ(m), ξ(n)) is considered; here, fork=p
1
p
2...p
s
,p
1≥p
2>-..., ξk)_is defined by ξ(k)=(logp
1/logk, logp
2/logk,..., logp
s
/logk, 0, ...); allp
i
are prime numbers. It is shown that the limit distribution is π×π, where π is the Poisson-Dirichlet distribution. The asymptotical
behavior of finite-dimensional distributions of ξ(k) for natural numbers was studied by Billingsley, Knuth, Trabb Pardo, Vershik, and others; the result of weak convergence
to the Poisson-Dirichlet distribution appears in Donnelly and Grimmett. The second application is concerned with the density
of sets {m/n: f(m/n)=a}, wheref is a function with the almost squareful kernel.
Supported by the Lithuanian State Science and Studies Foundation.
Vilnius University, Naugarduko 24, 2600, Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 1,
pp. 108–127, January–March, 1999.
Translated by V. Stakénas 相似文献
20.
Hei-Chi Chan 《数学学报(英文版)》2011,27(4):625-634
In this paper, we study a certain partition function a(n) defined by Σ
n≥0
a(n)q
n
:= Π
n=1(1 − q
n
)−1(1 − q
2n
)−1. We prove that given a positive integer j ≥ 1 and a prime m ≥ 5, there are infinitely many congruences of the type a(An + B) ≡ 0 (mod m
j
). This work is inspired by Ono’s ground breaking result in the study of the distribution of the partition function p(n). 相似文献