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1.
For r(0,1), let Z
r
={xR
2
|dist(x,Z
2)>r/2} and define τ
r
(x,v)=inf{t>0|x+tv∂Z
r
}. Let Φ
r
(t) be the probability that τ
r
(x,v)≥t for x and v uniformly distributed in Z
r
and §1 respectively. We prove in this paper that
as t→+∞. This result improves upon the bounds on Φ
r
in Bourgain-Golse-Wennberg [Commun. Math. Phys. 190, 491–508 (1998)]. We also discuss the applications of this result in the context of kinetic theory.
Received: 2 August 2002 / Accepted: 27 November 2002
Published online: 14 April 2003
Communicated by G. Gallavotti 相似文献
2.
For aL
∞
(ℝ+)∩L
1
(ℝ+) the truncated Bessel operator B
τ
(a) is the integral operator acting on L
2
[0,τ] with the kernel
where J
ν
stands for the Bessel function with ν>−1. In this paper we determine the asymptotics of the determinant det(I+B
τ
(a)) as τ→∞ for sufficiently smooth functions a for which a(x)≠1 for all x[0,∞). The asymptotic formula is of the form det(I+B
τ
(a))∼G
τ
E with certain constants G and E, and thus similar to the well-known Szeg?-Akhiezer-Kac formula for truncated Wiener-Hopf determinants.
Received: 23 April 2002 / Accepted: 25 September 2002 Published online: 24 January 2003
RID="*"
ID="*" Supported in part by NSF Grant DMS-9970879.
Communicated by J.L. Lebowitz 相似文献
3.
Antonio Ambrosetti Andrea Malchiodi Wei-Ming Ni 《Communications in Mathematical Physics》2003,235(3):427-466
We deal with the existence of positive radial solutions concentrating on spheres to a class of singularly perturbed elliptic
problems like −ɛ2Δu+V(|x|)u=u
p
,uH
1
(ℝ
n
). Under suitable assumptions on the auxiliary potential M(r)=r
n−1
V
θ
(r), θ(p+1)/(p−1)−1/2, we provide necessary and sufficient conditions for concentration as well as the bifurcation of non-radial solutions. 相似文献
4.
Remco van der Hofstad Frank den Hollander Gordon Slade 《Communications in Mathematical Physics》2002,231(3):435-461
We construct the incipient infinite cluster measure (IIC) for sufficiently spread-out oriented percolation on ℤ
d
× ℤ+, for d +1 > 4+1. We consider two different constructions. For the first construction, we define ℙ
n
(E) by taking the probability of the intersection of an event E with the event that the origin is connected to (x,n) ℤ
d
× ℤ+, summing this probability over x ℤ
d
, and normalising the sum to get a probability measure. We let n → ∞ and prove existence of a limiting measure ℙ∞, the IIC. For the second construction, we condition the connected cluster of the origin in critical oriented percolation
to survive to time n, and let n → ∞. Under the assumption that the critical survival probability is asymptotic to a multiple of n
−1, we prove existence of a limiting measure ℚ∞, with ℚ∞ = ℙ∞. In addition, we study the asymptotic behaviour of the size of the level set of the cluster of the origin, and the dimension
of the cluster of the origin, under ℙ∞. Our methods involve minor extensions of the lace expansion methods used in a previous paper to relate critical oriented
percolation to super-Brownian motion, for d+1 > 4+1.
Received: 13 December 2001 / Accepted: 11 July 2002 Published online: 29 October 2002
RID="*"
ID="*" Present address: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513,
5600 MB Eindhoven, The Netherlands. E-mail: rhofstad@win.tue.nl 相似文献
5.
P. Contucci M. Degli Esposti C. Giardinà S. Graffi 《Communications in Mathematical Physics》2003,236(1):55-63
Let {E Σ (N)} ΣΣN be a family of |Σ N |=2 N centered unit Gaussian random variables defined by the covariance matrix C N of elements c N (Σ,τ):=Av(E Σ (N)E τ (N)) and the corresponding random Hamiltonian. Then the quenched thermodynamical limit exists if, for every decomposition N=N 1 +N 2 , and all pairs (Σ,τ)Σ N ×Σ N :
where π k (Σ),k=1,2 are the projections of ΣΣ N into Σ Nk . The condition is explicitly verified for the Sherrington-Kirkpatrick, the even p-spin, the Derrida REM and the Derrida-Gardner GREM models. 相似文献
6.
John A. Toth 《Communications in Mathematical Physics》2003,238(1-2):225-256
Let (M,g) be a C
∞
compact Riemann manifold with classical Hamiltonian, HC
∞
(T
*
M). Assume that the corresponding -quantization P
1
:=Op
(H) is quantum completely integrable. We establish an -microlocal Weyl law on short spectral intervals of size 2−ε;∀ε>0 for various families of operators P
1
u
;uI containing P
1
, both in the mean and pointwise a.e. for uI. The -microlocalization refers to a small tubular neighbourhood of a non-degenerate, stable periodic bicharacteristic γ⊂T
*
M−0.
Received: 10 December 2001 / Accepted: 23 January 2003
Published online: 2 April 2003
RID="⋆"
ID="⋆" Supported in part by an Alfred P. Sloan Research Fellowship and NSERC grant OGP01720280
Communicated by P. Sarnak 相似文献
7.
Let Ω be a region in ℝn and letp = Pi )
i
1m
, be a partition ofΩ into a finite number of closed subsets having piecewise C2 boundaries of finite(n - 1 )dimensional measure. Let τ:Ω→Ω be piecewise C2 onP where, τi = τ|pi is aC
2 diffeomorphism onto its image, and expanding in the sense that there exists α > 1 such that for anyi = 1, 2,...,m ‖Dτi
-1 ‖ < α-1, where Dτi
-1 is the derivative matrixτ
i
- 1 and |‖·‖ is the Euclidean matrix norm. By means of an example, we will show that the simple bound of one-dimensional dynamics
cannot be generalized to higher dimensions. In fact, we will construct a piecewise expanding C2 transformation on a fixed partition with a finite number of elements in ℝ2, but which has an arbitrarily large number of ergodic, absolutely continuous invariant measures 相似文献
8.
9.
Viêt Hà Hoàng 《Communications in Mathematical Physics》2000,214(2):411-428
The paper considers the singularly perturbed Dirichlet problem −ɛΔu
ɛ+u
ɛ=f in a randomly perforated domain Ωɛ, which is obtained from a bounded open set Ω in R
N
after removing many holes of size ɛ
q
. The perforated domain is described in terms of an ergodic dynamical system acting on a probability space. Imposing certain
conditions on the domain, the behaviour of u
ɛ when ɛ→ 0 in Lebesgue spaces L
n
(Ω) is studied. Test functions together with the Birkhoff ergodic theorem are the main tools of analysis. The Poisson distribution
of holes of size ɛ
p
with the intensity λɛ−
r
is then considered. The above results apply in some cases; other cases are treated by the Wiener sausage approach.
Received: 15 December 1999 / Accepted: 14 April 2000 相似文献
10.
We study a reaction-diffusion system of N equations with k nonzero and N − k zero diffusion coefficients. More exactly, the first k equations of the system contain the terms a
i
Δu
i
− f
j
(u, v), i = 1, …, k, with the diffusion coefficient a
i
> 0. The right-hand sides of the other N − k equations contain only nonlinear interaction functions −h
j
(u, v), j = k + 1, …, N, with zero diffusion. Here u = (u
1, …, u
k
) and v = (υ
k+1, …, υ
N
) are unknown concentration vectors. Under appropriate assumptions on the interaction functions f(·) and h(·), we construct the trajectory attractor of this reaction-diffusion system. We also find the trajectory attractors , δ = (δ
1, …, δ
k
), for the analogous reaction-diffusion systems having the terms δ
j
Δυ
j
− h
j
(u, v), j = k + 1, …, N, with small diffusion coefficients δ
j
⩾ 0 in the last N − k equations. We prove that the trajectory attractors converge to (in an appropriate topology) as δ → 0+.
Dedicated to the memory of Vladimir Borovikov
Partially supported by the Russian Foundation for Basic Research (projects nos. 08-01-00784 and 07-01-00500). 相似文献
11.
A new analytical potential energy function for diatomic molecular ion XY+ is proposed based on the energy consistent method (ECM). The Coulomb potential included in the new ionic potential contains
multipole corrections, converges quickly and is variationally, changeable. The new potential and the ECM are applied to variationally
studying the potential energies of eight electronic states of several diatomic molecular ions: the A2π state of CO+, the X2∑g+ state of Li2+, the X2∑g+ state of He2+, the 12∏u state of Na2+, the A2∏u state of N2+, the X1∑+ state of KrH+, the X2∑+ state of SiO+ and the A2π state of SO+ ion. The present results agree excellently with the experiment-based Rydberg-Klein-Rees (RKR) potentials, and are superior
to the commonly used Huxley-Murrell-Sorbie (HMS) analytical potentials, and are better in some cases than some quantum mechanicalab initio potentials in the ionic asymptotic and dissociation regions. 相似文献
12.
We use equivariant methods to define and study the orbifold K-theory of an orbifold X. Adapting techniques from equivariant K-theory, we construct a Chern character and exhibit a multiplicative decomposition
for K
*
orb
(X)⊗ℚ, in particular showing that it is additively isomorphic to the orbifold cohomology of X. A number of examples are provided. We then use the theory of projective representations to define the notion of twisted
orbifold K–theory in the presence of discrete torsion. An explicit expression for this is obtained in the case of a global
quotient.
Received: 21 August 2001 / Accepted: 27 January 2003
Published online: 13 May 2003
RID="*"
ID="*" Both authors were partially supported by the NSF
RID="*"
ID="*" Both authors were partially supported by the NSF
Communicated by R.H. Dijkgraaf 相似文献
13.
Indranil Chakrabarty Sovik Roy Nirman Ganguly Binayak S. Choudhury 《International Journal of Theoretical Physics》2009,48(6):1833-1840
In this work we describe a protocol by which two of three parties generate two bipartite entangled state among themselves
without involving third party, from a non maximal W-state or W-type state |X〉=α|001〉123+β|010〉123+γ|100〉123,α
2+β
2+γ
2=1 shared by three distant partners. Also we have considered the case β=γ, to obtain a range for α
2, for which the local output states are separable and non local output states are inseparable. We also find out the dependence
of the mixedness of inseparable states with their amount of inseparability, for that range of α
2. 相似文献
14.
Electrical conductivity and dielectric properties of single-crystal TlGaSe2 have been studied as a function of γ irradiation dose in the 100–280 K range including the existence of an incommensurate phase. Anomalies in the form of maxima
have been observed in the σ=f(T), tan δ=f(T), and ɛ=f(T) curves at the points of transition from the paraphase to incommensurate (IC) phase, T
i, and from the IC to commensurate phase, T
c. The increase in the quantities σ, tan δ, and ɛ observed initially with increasing irradiation dose is followed by their strong decrease and disappearance of the anomalies.
It has been established that γ irradiation does not affect the phase transition temperatures T
i and T
c.
Fiz. Tverd. Tela (St. Petersburg) 40, 1328–1331 (July 1998) 相似文献
15.
Zhang Jian 《Communications in Mathematical Physics》1994,162(2):249-260
Consider the nonlinear Schrödinger equationu
t
–iu=f(u). Forf(u)=±|u|1+p
, ±i|u|1+p
, ±u|u|
p
(p>0), and the Dirichlet boundary or nonlinear boundary (including the Neumann boundary and the Robin boundary) conditions, we establish the local estimates for the timet to the solutions of the initial-boundary value problems. Being based up on these estimates, we investigate the blowing-up properties of the solutions.Research supported in part by the Youth Foundation of Sichuan Education Committee and the Natural Science Foundation of China 相似文献
16.
Aldo Procacci Benedetto Scoppola Victor Gerasimov 《Communications in Mathematical Physics》2003,235(2):215-231
Given an infinite graph 𝔾 quasi-transitive and amenable with maximum degree Δ, we show that reduced ground state degeneracy
per site W
r
(𝔾, q) of the q-state antiferromagnetic Potts model at zero temperature on 𝔾 is analytic in the variable 1/q, whenever |2Δe
3
/q|<1. This result proves, in an even stronger formulation, a conjecture originally sketched in [12] and explicitly formulated
in [16 and 19], based on which a sufficient condition for W
r
(𝔾, q) to be analytic at 1/q=0 is that 𝔾 is a regular lattice.
Received: 16 January 2002 / Accepted: 17 October 2002 Published online: 18 February 2003
RID="*"
ID="*" Partially supported by CNPq (Brazil)
RID="**"
ID="**" Partially supported by CNR, G.N.F.M. (Italy)
Communicated by H. Spohn 相似文献
17.
An analytical method has been proposed for calculating the probabilities P
i
(2)(s) of existence of X-X, X−□, and □−□ pair interactions in the nonmetal sublattice of M
2tX2t−1 superstructures formed in strongly nonstoichiometric compounds MX
y
(MX
y
□1−y) and M
2X
y
(MXy/2□1−y/2) with a high content of structural vacancies □. The main characteristics necessary for the quantitative determination of
the probabilities Pi(2)(s) as functions of the composition, degree of long-range order, symmetry, and structure type have been determined for all the
known superstructures M
2tX2t−1. 相似文献
18.
Gastão A. Braga Leandro M. Ciolleti Rémy Sanchis 《Journal of statistical physics》2007,129(3):587-591
In this short note we consider mixed short-long range independent bond percolation models on ℤ
k+d
. Let p
uv
be the probability that the edge (u,v) will be open. Allowing a x,y-dependent length scale and using a multi-scale analysis due to Aizenman and Newman, we show that the long distance behavior
of the connectivity τ
xy
is governed by the probability p
xy
. The result holds up to the critical point. 相似文献
19.
Makoto Matsumoto 《Reports on Mathematical Physics》1979,15(3):375-386
We consider the length of a vector in a Finsler space with the fundamental function L(x,y). The length of a vector X is usually defined as the value L(x,X) of L. On the other hand, we have an essential tensor gij(x,y), called the fundamental tensor, and the concept of relative length |Xy| of X may be introduced by |X|yy = gij(x,y)XiXj with re spect to a supporting element y. The question arises whether is L(x,X) the minimum of |X|y or not? If there exists a supporting element y satisfying |X|y < L(x,X), then a curve x(t) in the Finsler space will be measured shorter than the usual length, by integrating |dx/dt|y with the field of such supporting element y(t) along the curve. 相似文献
20.
Tetsuya Hattori 《Journal of statistical physics》2007,127(3):609-627
Let W(x,y) = ax
3+ bx
4+ f
5
x
5+ f
6
x
6+ (3 ax
2)2
y+ g
5
x
5
y + h
3
x
3
y
2 + h
4
x
4
y
2 + n
3
x
3
y
3+a
24
x
2
y
4+a
05
y
5+a
15
xy
5+a
06
y
6, and X = , , where the coefficients are non-negative constants, with a > 0, such that X
2(x,x
2)−Y(x,x
2) is a polynomial of x with non-negative coefficients.
Examples of the 2 dimensional map Φ: (x,y)↦ (X(x,y),Y(x,y)) satisfying the conditions are the renormalization group (RG) maps (modulo change of variables) for the restricted self-avoiding
paths on the 3 and 4 dimensional pre-gaskets.
We prove that there exists a unique fixed point (x
f
,y
f
) of Φ in the invariant set .
2000 Mathematics Subject Classification Numbers: 82B28; 60G99; 81T17; 82C41. 相似文献