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1.
In certain circumstances, the uncertainty, ΔS[φ], of a quantum observable, S, can be bounded from below by a finite overall constant ΔS>0, i.e., ΔS[φ]≥ΔS, for all physical states φ. For example, a finite lower bound to the resolution of distances has been used to model a natural ultraviolet cutoff at
the Planck or string scale. In general, the minimum uncertainty of an observable can depend on the expectation value, t=〈φ,S
φ〉, through a function ΔS
t
of t, i.e., ΔS[φ]≥ΔS
t
, for all physical states φ with 〈φ,S
φ〉=t. An observable whose uncertainty is finitely bounded from below is necessarily described by an operator that is merely symmetric
rather than self-adjoint on the physical domain. Nevertheless, on larger domains, the operator possesses a family of self-adjoint
extensions. Here, we prove results on the relationship between the spacing of the eigenvalues of these self-adjoint extensions
and the function ΔS
t
. We also discuss potential applications in quantum and classical information theory.
相似文献
2.
Summary. We study the 2D Ising model in a rectangular box Λ
L
of linear size O(L). We determine the exact asymptotic behaviour of the large deviations of the magnetization ∑
t∈ΛL
σ(t) when L→∞ for values of the parameters of the model corresponding to the phase coexistence region, where the order parameter m
* is strictly positive. We study in particular boundary effects due to an arbitrary real-valued boundary magnetic field. Using
the self-duality of the model a large part of the analysis consists in deriving properties of the covariance function <σ(0)σ(t)>, as |t|→∞, at dual values of the parameters of the model. To do this analysis we establish new results about the high-temperature
representation of the model. These results are valid for dimensions D≥2 and up to the critical temperature. They give a complete non-perturbative exposition of the high-temperature representation.
We then study the Gibbs measure conditioned by {|∑
t∈ΛL
σ(t) −m|Λ
L
||≤|Λ
L
|L
−
c
}, with 0<c<1/4 and −m
*<m<m
*. We construct the continuum limit of the model and describe the limit by the solutions of a variational problem of isoperimetric
type.
Received: 17 October 1996 / In revised form: 7 March 1997 相似文献
3.
Gerald Kuba 《Mathematica Slovaca》2009,59(3):349-356
Let ℛ
n
(t) denote the set of all reducible polynomials p(X) over ℤ with degree n ≥ 2 and height ≤ t. We determine the true order of magnitude of the cardinality |ℛ
n
(t)| of the set ℛ
n
(t) by showing that, as t → ∞, t
2 log t ≪ |ℛ2(t)| ≪ t
2 log t and t
n
≪ |ℛ
n
(t)| ≪ t
n
for every fixed n ≥ 3. Further, for 1 < n/2 < k < n fixed let ℛ
k,n
(t) ⊂ ℛ
n
(t) such that p(X) ∈ ℛ
k,n
(t) if and only if p(X) has an irreducible factor in ℤ[X] of degree k. Then, as t → ∞, we always have t
k+1 ≪ |ℛ
k,n
(t)| ≪ t
k+1 and hence |ℛ
n−1,n
(t)| ≫ |ℛ
n
(t)| so that ℛ
n−1,n
(t) is the dominating subclass of ℛ
n
(t) since we can show that |ℛ
n
(t)∖ℛ
n−1,n
(t)| ≪ t
n−1(log t)2.On the contrary, if R
n
s
(t) is the total number of all polynomials in ℛ
n
(t) which split completely into linear factors over ℤ, then t
2(log t)
n−1 ≪ R
n
s
(t) ≪ t
2 (log t)
n−1 (t → ∞) for every fixed n ≥ 2.
相似文献
4.
In this article we study the exponential behavior of the continuous stochastic Anderson model, i.e. the solution of the stochastic
partial differential equation u(t,x)=1+∫0tκΔxu (s,x) ds+∫0t W(ds,x) u (s,x), when the spatial parameter x is continuous, specifically x∈R, and W is a Gaussian field on R+×R that is Brownian in time, but whose spatial distribution is widely unrestricted. We give a partial existence result of the
Lyapunov exponent defined as limt→∞t−1 log u(t,x). Furthermore, we find upper and lower bounds for lim supt→∞t−1 log u(t,x) and lim inft→∞t−1 log u(t,x) respectively, as functions of the diffusion constant κ which depend on the regularity of W in x. Our bounds are sharper, work for a wider range of regularity scales, and are significantly easier to prove than all previously
known results. When the uniform modulus of continuity of the process W is in the logarithmic scale, our bounds are optimal.
This author's research partially supported by NSF grant no. : 0204999 相似文献
5.
Fu Bao XI 《数学学报(英文版)》2005,21(3):457-464
In this paper, we consider the Markov process (X^∈(t), Z^∈(t)) corresponding to a weakly coupled elliptic PDE system with a small parameter ∈ 〉 0. We first prove that (X^∈(t), Z^∈(t)) has the Feller continuity by the coupling method, and then prove that (X^∈(t), Z^∈(t)) has an invariant measure μ^∈(·) by the Foster-Lyapunov inequality. Finally, we establish a large deviations principle for μ^∈(·) as the small parameter e tends to zero. 相似文献
6.
Wenjing ZHAO 《数学年刊B辑(英文版)》2011,32(2):215-222
The Cauchy problem to the Oldroyd-B model is studied. In particular, it is shown that if the smooth solution (u, τ) to this system blows up at a finite time T*, then ∫0
T* ‖▿u(t)‖
L
∞dt = ∞. Furthermore, the global existence of smooth solution to this system is given with small initial data. 相似文献
7.
J. Donald Monk 《Order》2009,26(2):163-175
A tower in a Boolean algebra (BA) is a strictly increasing sequence, of regular order type, of elements of the algebra different
from 1 but with sum 1. A pseudo-tree is a partially ordered set T such that the set T↓t = {s ∈ T:s < t} is linearly ordered for every t ∈ T. If that set is well-ordered, then T is a tree. For any pseudo-tree T, the BA Treealg(T) is the algebra of subsets of T generated by all of the sets T↑t = {s ∈ T:t ≤ s}. The main theorem of this note is a characterization in tree terms of when Treealg(T) has a tower of order type κ (given in advance). 相似文献
8.
Gea Hwa Kwoun Yoshihiro Yajima 《Annals of the Institute of Statistical Mathematics》1986,38(1):297-309
Summary As one of the non-stationary time series model, we consider a firstorder autoregressive model in which the autoregressive
coefficient is assumed to be a function,f
t
(θ), of timet. We establish several assumptions onf
t
(θ), not on the terms in the Taylor expansion of log-likelihood function, and show that the estimators of unknown parameters
involved inf
t
(θ) have strong consistency and asymptotic normality under these assumptions when sample size tends to infinity. 相似文献
9.
We consider the periodic boundary-value problem u
tt
− u
xx
= g(x, t), u(0, t) = u(π, t) = 0, u(x, t + ω) = u(x, t). By representing a solution of this problem in the form u(x, t) = u
0(x, t) + ũ(x, t), where u
0(x, t) is a solution of the corresponding homogeneous problem and ũ(x, t) is the exact solution of the inhomogeneous equation such that ũ(x, t + ω) u x = ũ(x, t), we obtain conditions for the solvability of the inhomogeneous periodic boundary-value problem for certain values of the
period ω. We show that the relation obtained for a solution includes known results established earlier.
__________
Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 912–921, July, 2005. 相似文献
10.
LetX be a Banach space and letA be the infinitesimal generator of a differentiable semigroup {T(t) |t ≥ 0}, i.e. aC
0-semigroup such thatt ↦T(t)x is differentiable on (0, ∞) for everyx εX. LetB be a bounded linear operator onX and let {S(t) |t ≥ 0} be the semigroup generated byA +B. Renardy recently gave an example which shows that {S(t) |t ≥ 0} need not be differentiable. In this paper we give a condition on the growth of ‖T′(t)‖ ast ↓ 0 which is sufficient to ensure that {S(t) |t ≥ 0} is differentiable. Moreover, we use Renardy’s example to study the optimality of our growth condition. Our results can
be summarized roughly as follows:
We also show that if lim sup
t→0+t
p ‖T′(t)‖<∞ for a givenp ε [1, ∞), then lim sup
t→0+t
p‖S′(t)‖<∞; it was known previously that if limsup
t→0+t
p‖T′(t)‖<∞, then {S(t) |t ≥ 0} is differentiable and limsup
t→0+t
2p–1‖S′(t)‖<∞. 相似文献
(i) | If lim sup t→0+t log‖T′(t)‖/log(1/2) = 0 then {S(t) |t ≥ 0} is differentiable. |
(ii) | If 0<L=lim sup t→0+t log‖T′(t)‖/log(1/2)<∞ thent ↦S(t ) is differentiable on (L, ∞) in the uniform operator topology, but need not be differentiable near zero |
(iii) | For each function α: (0, 1) → (0, ∞) with α(t)/log(1/t) → ∞ ast ↓ 0, Renardy’s example can be adjusted so that limsup t→0+t log‖T′(t)‖/α(t) = 0 andt →S(t) is nowhere differentiable on (0, ∞). |
11.
Vincent Grandjean 《Bulletin of the Brazilian Mathematical Society》2008,39(4):515-535
Given a definable function f: ℝ
n
↦ ℝ, enough differentiable, we study the continuity of the total curvature function t → K(t), total curvature of the level f
−1(t), and the total absolute curvature function t → |K|(t), total absolute curvature of the level f
−1(t). We show they admits at most finitely many discontinuities.
Partially supported by the European research network IHP-RAAG contract number HPRN-CT-2001-00271 and partially supported by
Deutsche Forschungs-Gemeinschaft in the Priority Program Global Differential Geometry. 相似文献
12.
We consider the M/G/1 queue with an arrival rate λ that depends weakly upon time, as λ = λ(εt) where ε is a small parameter. In the asymptotic limit ε → 0, we construct approximations to the probability p
n(t)that η customers are present at time t. We show that the asymptotics are different for several ranges of the (slow) time scale Τ= εt. We employ singular perturbation techniques and relate the various time scales by asymptotic matching.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
13.
Tomasz Komorowski 《Probability Theory and Related Fields》2001,121(4):525-550
We consider the asymptotic behavior of the solutions ofscaled convection-diffusion equations ∂
t
u
ɛ
(t, x) = κΔ
x
(t, x) + 1/ɛV(t/ɛ2,xɛ) ·∇
x
u
ɛ
(t, x) with the initial condition u
ɛ(0,x) = u
0(x) as the parameter ɛ↓ 0. Under the assumptions that κ > 0 and V(t, x), (t, x) ∈R
d
is a d-dimensional,stationary, zero mean, incompressible, Gaussian random field, Markovian and mixing in t we show that the laws of u
ɛ(t,·), t≥ 0 in an appropriate functional space converge weakly, as ɛ↓ 0, to a δ-type measureconcentrated on a solution of a certain
constant coefficient heat equation.
Received: 23 March 2000 / Revised version: 5 March 2001 / Published online: 9 October 2001 相似文献
14.
Mario Abundo 《Methodology and Computing in Applied Probability》2010,12(3):473-490
It is studied the first-passage time (FPT) of a time homogeneous one-dimensional diffusion, driven by the stochastic differential
equation dX(t) = μ(X(t))dt + σ(X(t)) dB
t
, X(0) = x
0, through b + Y(t), where b > x
0 and Y(t) is a compound Poisson process with rate λ > 0 starting at 0, which is independent of the Brownian motion B
t
. In particular, the FPT density is investigated, generalizing a previous result, already known in the case when X(t) = μt + B
t
, for which the FPT density is the solution of a certain integral equation. A numerical method is shown to calculate approximately
the FPT density; some examples and numerical results are also reported. 相似文献
15.
We consider the M(t)/M(t)/m/m queue, where the arrival rate λ(t) and service rate μ(t) are arbitrary (smooth) functions of time. Letting pn(t) be the probability that n servers are occupied at time t (0≤ n≤ m, t > 0), we study this distribution asymptotically, for m→∞ with a comparably large arrival rate λ(t) = O(m) (with μ(t) = O(1)). We use singular perturbation techniques to solve the forward equation for pn(t) asymptotically. Particular attention is paid to computing the mean number of occupied servers and the blocking probability
pm(t). The analysis involves several different space-time ranges, as well as different initial conditions (we assume that at t = 0 exactly n0 servers are occupied, 0≤ n0≤ m). Numerical studies back up the asymptotic analysis.
AMS subject classification: 60K25,34E10
Supported in part by NSF grants DMS-99-71656 and DMS-02-02815 相似文献
16.
We show that if the pseudodifferential operator −q(x,D) generates a Feller semigroup (Tt)t≥0 then the Feller semigroups (Tt(v))t≥0 generated by the pseudodifferential operators with symbol will converge strongly to (Tt)t≥0 as ν →∞. 相似文献
17.
Alexander Alvarez Fabien Panloup Monique Pontier Nicolas Savy 《Statistical Inference for Stochastic Processes》2012,15(1):27-59
This paper is concerned with the estimation of the volatility process in a stochastic volatility model of the following form:
dX
t
= a
t
dt + σ
t
dW
t
, where X denotes the log-price and σ is a càdlàg semi-martingale. In the spirit of a series of recent works on the estimation of the cumulated volatility, we
here focus on the instantaneous volatility for which we study estimators built as finite differences of the power variations of the log-price. We provide central limit theorems with an optimal rate depending on the local behavior of σ. In particular, these theorems yield some confidence intervals for σ
t
. 相似文献
18.
Gérard Ben Arous Jiří Černý Thomas Mountford 《Probability Theory and Related Fields》2006,134(1):1-43
Let E
x
be a collection of i.i.d. exponential random variables. Symmetric Bouchaud's model on ℤ2 is a Markov chain X(t) whose transition rates are given by w
xy
= ν exp (−βE
x
) if x, y are neighbours in ℤ2. We study the behaviour of two correlation functions: ℙ[X(t
w
+t) = X(t
w
)] and ℙ[X(t') = X(t
w
) ∀ t'∈ [t
w
, t
w
+ t]]. We prove the (sub)aging behaviour of these functions when β > 1. 相似文献
19.
Patrícia A. Filipe Carlos A. Braumann Carlos J. Roquete 《Methodology and Computing in Applied Probability》2012,14(1):49-56
The evolution of the growth of an individual in a random environment can be described through stochastic differential equations
of the form dY
t
= β(α − Y
t
)dt + σdW
t
, where Y
t
= h(X
t
), X
t
is the size of the individual at age t, h is a strictly increasing continuously differentiable function, α = h(A), where A is the average asymptotic size, and β represents the rate of approach to maturity. The parameter σ measures the intensity of the effect of random fluctuations on growth and W
t
is the standard Wiener process. We have previously applied this monophasic model, in which there is only one functional form
describing the average dynamics of the complete growth curve, and studied the estimation issues. Here, we present the generalization
of the above stochastic model to the multiphasic case, in which we consider that the growth coefficient β assumes different values for different phases of the animal’s life. For simplicity, we consider two phases with growth coefficients
β
1 and β
2. Results and methods are illustrated using bovine growth data. 相似文献
20.
Throughout this article we assume that the df H of a random vector (X,Y) is in the max-domain of attraction of an extreme value distribution function (df) G with reverse exponential margins. Therefore, the asymptotic dependence structure of H can be represented by a Pickands dependence function D with D = 1 representing the case of asymptotic independence. One of our aims is to test the null hypothesis of tail-dependence against
the alternative of tail-independence. Thus we want to prove the validity of the model where D = 1. The test is based on the radial component X + Y. Under a certain spectral expansion it is verified that the df of X + Y, conditioned on X + Y > c, converges to F(t) = t, as c ↑0, if D ≠ 1 and, respectively, to F(t) = t
1 + ρ
, if D = 1, where ρ > 0 determines the rate at which independence is attained. Based on the limiting dfs we find a uniformly most powerful test procedure for testing tail-dependence against rates of tail-independence. In addition,
an estimator of the parameter ρ is proposed. The relationship of ρ to another dependence measure, given in the literature, is indicated.
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