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1.
A. É. Ruuge 《Mathematical Notes》2000,67(2):207-217
We consider the semiclassical asymptotics of eigenfunctions for the Hamiltonian of a quantum-mechanical system ofN identical fermions withd degrees of freedom without spin interaction. In the one-dimensional case (d=1), examples are known in which the ground antisymmetric state in the semiclassical limit is the product ofN(N−1)/2 two-particle wave functions. We construct a nontrivial generalization of this property ford>1.
Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 257–269, February, 2000. 相似文献
2.
In this paper, we consider a periodic problem for the n-dimensional complex Landau--Ginzburg equation. It is shown that in the case of small initial data there exists a unique classical solution of this problem, and an asymptotics of this solution uniform in the space variable is given. The leading term of the asymptotics is exponentially decreasing in time. 相似文献
3.
S. Poghosyan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2010,45(1):43-51
The paper considers interacting Bose gas in a polygonal domain and studies the asymptotics of the log-partition function in
its Feynman-Kac representation as the domain is delated to infinity. It is proved that for repulsive interaction with power
decay at infinity the asymptotics of the log-partition function determines the area of the domain, the length of its boundary
and the constant term defined by the angles of the polygon. This is a natural generalization of the Kac’s famous problem on
computing the asymptotics of the partition function Tre
βδ, where δ is the Dirichlet Laplacian for a polygonal domain. 相似文献
4.
V. P. Maslov 《Mathematical Notes》1995,58(5):1166-1177
We construct a representation in which the asymptotics of the solution to the Kolmogorov-Feller equation in the Fock space
Γ(L
1(ℝ
n
)) is of a form similar to the WKB asymptotic expansion; namely, the Boltzmann equation inL
1(ℝ
n
) plays the role of the Hamilton equations, the linearized Boltzmann equation extended to Γ(L
1(ℝ
n
)) plays the role of the transport equation, and the Hamilton-Jacobi equation follows from the conservation of the total probability
for the solutions of the Boltzmann equation. We also construct the asymptotics of the solution to the Boltzmann equation with
small transfer of momentum; this asymptotics is given by the tunnel canonical operator corresponding to the self-consistent
characteristic equation.
Translated fromMatematicheskie Zametki, Vol. 58, No. 5, pp. 694–709, November, 1995.
The author is deeply grateful to Prof. A. M. Chebotarev, whose assistance has made the writing of this paper possible.
This work was financially supported by the International Science Foundation under grants Nos. MFO000 and MFO300. 相似文献
5.
Boris N. Khoromskij 《Constructive Approximation》2011,34(2):257-280
In the present paper, we discuss the novel concept of super-compressed tensor-structured data formats in high-dimensional
applications. We describe the multifolding or quantics-based tensor approximation method of O(dlog N)-complexity (logarithmic scaling in the volume size), applied to the discrete functions over the product index set {1,…,N}⊗d
, or briefly N-d tensors of size N
d
, and to the respective discretized differential-integral operators in ℝ
d
. As the basic approximation result, we prove that a complex exponential sampled on an equispaced grid has quantics rank 1.
Moreover, a Chebyshev polynomial, sampled over a Chebyshev Gauss–Lobatto grid, has separation rank 2 in the quantics tensor
format, while for the polynomial of degree m over a Chebyshev grid the respective quantics rank is at most 2m+1. For N-d tensors generated by certain analytic functions, we give a constructive proof of the O(dlog Nlog ε
−1)-complexity bound for their approximation by low-rank 2-(dlog N) quantics tensors up to the accuracy ε>0. In the case ε=O(N
−α
), α>0, our approach leads to the quantics tensor numerical method in dimension d, with the nearly optimal asymptotic complexity O(d/αlog 2
ε
−1). From numerical examples presented here, we observe that the quantics tensor method has proved its value in application
to various function related tensors/matrices arising in computational quantum chemistry and in the traditional finite element
method/boundary element method (FEM/BEM). The tool apparently works. 相似文献
6.
V. R. Fatalov 《Mathematical Notes》1999,65(3):358-364
In this paper we calculate the exact asymptotics of the probability P{‖w(t)+uct‖
p
>u},u→∞, wherew(t) is the standard Wiener process and ‖x‖
p
is the ordinary norm in the spaceL
p[0,1],p≥2. The result is obtained on the basis of a general theorem due to the author on the asymptotics of the Gaussian measureP(uD),u→∞, for a Borel setD belonging to a separable Banach space.
Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 429–436, March, 1999. 相似文献
7.
We consider a quantum system constituted by N identical particles interacting by means of a mean-field Hamiltonian. It is well known that, in the limit N → ∞, the one-particle state obeys to the Hartree equation. Moreover, propagation of chaos holds. In this paper, we take care
of the dependence by considering the semiclassical expansion of the N-particle system. We prove that each term of the expansion agrees, in the limit N → ∞, with the corresponding one associated with the Hartree equation. We work in the classical phase space by using the Wigner
formalism, which seems to be the most appropriate for the present problem.
Submitted: October 2, 2008., Accepted: December 4, 2008. 相似文献
8.
Summary. Let X,X
1,X
2,… be a sequence of i.i.d. random vectors taking values in a d-dimensional real linear space ℝ
d
. Assume that E
X=0 and that X is not concentrated in a proper subspace of ℝ
d
. Let G denote a mean zero Gaussian random vector with the same covariance operator as that of X. We investigate the distributions of non-degenerate quadratic forms ℚ[S
N
] of the normalized sums S
N
=N
−1/2(X
1+⋯+X
N
) and show that
provided that d≥9 and the fourth moment of X exists. The bound ?(N
−1) is optimal and improves, e.g., the well-known bound ?(N
−
d
/(
d
+1)) due to Esseen (1945). The result extends to the case of random vectors taking values in a Hilbert space. Furthermore, we
provide explicit bounds for Δ
N
and for the concentration function of the random variable ℚ[S
N
].
Received: 9 January 1997 / In revised form: 15 May 1997 相似文献
9.
We obtain exact solutions U(x, y, z, t) of the three-dimensional sine-Gordon equation in a form that Lamb previously proposed
for integrating the two-dimensional sine-Gordon equation. The three-dimensional solutions depend on arbitrary functions F(α)
and ϕ(α,β), whose arguments are some functions α(x, y, z, t) and β(x, y, z, t). The ansatzes must satisfy certain equations. These
are an algebraic system of equations in the case of one ansatz. In the case of two ansatzes, the system of algebraic equations
is supplemented by first-order ordinary differential equations. The resulting solutions U(x, y, z, t) have an important property,
namely, the superposition principle holds for the function tan(U/4). The suggested approach can be used to solve the generalized sine-Gordon equation, which, in contrast to the classical equation,
additionally involves first-order partial derivatives with respect to the variables x, y, z, and t, and also to integrate
the sinh-Gordon equation. This approach admits a natural generalization to the case of integration of the abovementioned types
of equations in a space with any number of dimensions.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 370–377, March, 2009. 相似文献
10.
A continuous change-point problem is studied in which N independent diffusion processes X
j
are observed. Each process X
j
is associated with a “channel”, each has an unknown piecewise constant drift and the unit diffusion coefficient. All the
channels are connected only by a common change-point of drift. As the result, a change-point problem is defined in which the
unknown and unidentifiable drift forms a 2N-dimensional nuisance parameter. The asymptotics of the minimax rate in estimating the change-point is studied as N → ∞. This rate is compared with the case of the known drift. This problem is a special case of an open change-point detection
problem in the high-dimensional diffusion with nonparametric drift.
相似文献
11.
The asymptotics for the number of representations ofN asN→∞ is expressed as the sum of a number havingk prime divisors and a product of two natural numbers. The asymptotics is found fork≤(2−ε) ln lnN and (2+ε) ln lnN≤k≤b ln lnN, whereε>0. The results obtained are uniform with respect tok.
Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 585–602, April, 1996.
This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00260. 相似文献
12.
We consider the harmonic crystal, or massless free field, , , that is the centered Gaussian field with covariance given by the Green function of the simple random walk on ℤ
d
. Our main aim is to obtain quantitative information on the repulsion phenomenon that arises when we condition to be larger than , is an IID field (which is also independent of ϕ), for every x in a large region , with N a positive integer and D a bounded subset of ℝ
d
. We are mostly motivated by results for given typical realizations of σ (quenched set–up), since the conditioned harmonic crystal may be seen as a model for an equilibrium interface, living in a (d+1)–dimensional space, constrained not to go below an inhomogeneous substrate that acts as a hard wall. We consider various
types of substrate and we observe that the interface is pushed away from the wall much more than in the case of a flat wall as soon as the upward tail of σ
0
is heavier than Gaussian, while essentially no effect is observed if the tail is sub–Gaussian. In the critical case, that
is the one of approximately Gaussian tail, the interplay of the two sources of randomness, ϕ and σ, leads to an enhanced repulsion effect of additive type. This generalizes work done in the case of a flat wall and also in our case the crucial estimates are optimal Large
Deviation type asymptotics as of the probability that ϕ lies above σ in D
N
.
Received: 6 February 2002 / Revised version: 23 May 2002 / Published online: 30 September 2002
Mathematics Subject Classification (2000): 82B24, 60K35, 60G15
Keywords or phrases: Harmonic Crystal – Rough Substrate – Quenched and Annealed Models – Entropic Repulsion – Gaussian fields – Extrema of Random
Fields – Large Deviations – Random Walks 相似文献
13.
Jiří Janáček 《Czechoslovak Mathematical Journal》2008,58(3):751-758
The variance of the number of lattice points inside the dilated bounded set rD with random position in ℝ
d
has asymptotics ∼ r
d−1 if the rotational average of the squared modulus of the Fourier transform of the set is O(ϰ
−d−1). The asymptotics follow from Wiener’s Tauberian theorem. 相似文献
14.
Djemaïa Bensikaddour Sadek Gala Amina Lahmar-Benbernou 《Periodica Mathematica Hungarica》2008,57(1):1-22
The most important result stated in this paper is to show that the solutions of the Poisson equation −Δu = f, where f ∈ (Ḣ1(ℝ
d
) → (Ḣ−1(ℝ
d
)) is a complex-valued distribution on ℝ
d
, satisfy the regularity property D
k
u ∈ (Ḣ1 → Ḣ−1) for all k, |k| = 2. The regularity of this equation is well studied by Maz’ya and Verbitsky [12] in the case where f belongs to the class of positive Borel measures.
相似文献
15.
Stephan Wagner 《The Ramanujan Journal》2009,20(2):189-206
We consider the distribution of the longest run of equal elements in number partitions (equivalently, the distribution of
the largest gap between subsequent elements); in a recent paper, Mutafchiev proved that the distribution of this random variable
(appropriately rescaled) converges weakly. The corresponding distribution function is closely related to the generating function
for number partitions. In this paper, this problem is considered in more detail—we study the behavior at the tails (especially
the case that the longest run is comparatively small) and extend the asymptotics for the distribution function to the entire
interval of possible values. Additionally, we prove a local limit theorem within a suitable region, i.e. when the longest
run attains its typical order n
1/2, and we observe another phase transition that occurs when the largest gap is of order n
1/4: there, the conditional probability that the longest run has length d, given that it is ≤d, jumps from 1 to 0. Asymptotics for the mean and variance follow immediately from our considerations. 相似文献
16.
On the spaces S p , an upper estimate is found for the norm of the error functional δ N (f) of cubature formulas possessing the Haar d-property in the two-dimensional case. An asymptotic relation is proved for $ \left\| {\delta _N (f)} \right\|_{S_p^* } On the spaces S
p
, an upper estimate is found for the norm of the error functional δ
N
(f) of cubature formulas possessing the Haar d-property in the two-dimensional case. An asymptotic relation is proved for with the number of nodes N ∼ 2
d
, where d → ∞. For N ∼ 2
d
with d → ∞, it is shown that the norm of δ
N
for the formulas under study has the best convergence rate, which is equal to N
−1/p
.
Original Russian Text ? K.A. Kirillov, M.V. Noskov, 2009, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi
Fiziki, 2009, Vol. 49, No. 1, pp. 3–13. 相似文献
17.
In previous papers [MS 1, 2], we considered stationary critical points of solutions of the initial-boundary value problems
for the heat equation on bounded domains in ℝN,N ≧ 2. In [MS 1], we showed that a solutionu has a stationary critical pointO if and only ifu satisfies a certain balance law with respect toO for any time. Furthermore, we proved necessary and sufficient conditions relating the symmetry of the domain to the initial
datau
0; in this way, we gave a characterization of the ball in ℝN([MS 1]) and of centrosymmetric domains ([MS 2]). In the present paper, we consider a rotationA
dby an angle 2π/d,d ≧ 2 for planar domains and give some necessary and some sufficient conditions onu
0 which relate to domains invariant underA
d. We also establish some conjectures.
This research was partially supported by a Grant-in-Aid for Scientific Research (C) (# 10640175) and (B) (# 12440042) of the
Japan Society for the Promotion of Science. The first author was supported also by the Italian MURST. 相似文献
18.
Domain constants are numbers attached to regions in the complex plane ℂ. For a region Ω in ℂ, letd(Ω) denote a generic domain constant. If there is an absolute constantM such thatM
−1≤d(Ω)/d(Δ)≤M whenever Ω and Δ are conformally equivalent, then the domain constant is called quasiinvariant under conformal mappings.
IfM=1, the domain constant is conformally invariant. There are several standard problems to consider for domain constants. One
is to obtain relationships among different domain constants. Another is to determine whether a given domain constant is conformally
invariant or quasi-invariant. In the latter case one would like to determine the best bound for quasi-invariance. We also
consider a third type of result. For certain domain constants we show there is an absolute constantN such that |d(Ω)−d(Δ)|≤N whenever Ω and Δ and conformally equivalent, sometimes determing the best possible constantN. This distortion inequality is often stronger than quasi-invariance. We establish results of this type for six domain constants.
Research partially supported by a National Science Foundation Grant. 相似文献
19.
We study the asymptotic behavior in time of solutions to the initial value problem of the nonlinear Schrödinger equation with a subcritical dissipative nonlinearity λ|u|p−1u, where 1<p<1+2/n, n is the space dimension and λ is a complex constant satisfying Imλ<0. We show the time decay estimates and the large-time asymptotics of the solution, when the space dimension n?3, p is sufficiently close to 1+2/n and the initial data is sufficiently small. 相似文献
20.
A. V. Romanov 《Mathematical Notes》2000,68(3):378-385
An example of a dissipative semilinear parabolic equation in a Hilbert space without smooth inertial manifolds is constructed.
Moreover, the attractor of this equation can be embedded in no finite-dimensionalC
1 invariant submanifold of the phase space. The class of scalar reaction-diffusion equations in bounded domains Ω ⊂ ℝm without inertial manifolds
with the property of absolute normal hyperbolicity on the setE of stationary points of the phase semiflow is described. Such equations may have inertial manifolds with the weaker property
of normal hyperbolicity onE. Three-dimensional reaction-diffusion systems without inertial manifolds normally hyperbolic at stationary points are found.
Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 439–447, September, 2000. 相似文献