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1.
We consider a family of Gagliardo–Nirenberg–Sobolev interpolation inequalities which interpolate between Sobolev?s inequality and the logarithmic Sobolev inequality, with optimal constants. The difference of the two terms in the interpolation inequalities (written with optimal constant) measures a distance to the manifold of the optimal functions. We give an explicit estimate of the remainder term and establish an improved inequality, with explicit norms and fully detailed constants. Our approach is based on nonlinear evolution equations and improved entropy–entropy production estimates along the associated flow. Optimizing a relative entropy functional with respect to a scaling parameter, or handling properly second moment estimates, turns out to be the central technical issue. This is a new method in the theory of nonlinear evolution equations, which can be interpreted as the best fit of the solution in the asymptotic regime among all asymptotic profiles. 相似文献
2.
V. Caselles 《Journal of Differential Equations》2011,250(8):3311-3348
In this paper we give a characterization of the notion of entropy solutions of some flux limited diffusion equations for which we can prove that the solution is a function of bounded variation in space and time. This includes the case of the so-called relativistic heat equation and some generalizations. For them we prove that the jump set consists of fronts that propagate at the speed given by Rankine-Hugoniot condition and we give on it a geometric characterization of the entropy conditions. Since entropy solutions are functions of bounded variation in space once the initial condition is, to complete the program we study the time regularity of solutions of the relativistic heat equation under some conditions on the initial datum. An analogous result holds for some other related equations without additional assumptions on the initial condition. 相似文献
3.
We describe and survey in this paper iterative algorithms for solving the discrete maximum entropy problem with linear equality constraints. This problem has applications e.g. in image reconstruction from projections, transportation planning, and matrix scaling. In particular we study local convergence and asymptotic rate of convergence as a function of the iteration parameter. For the trip distribution problem in transportation planning and the equivalent problem of scaling a positive matrix to achieve a priori given row and column sums, it is shown how the iteration parameters can be chosen in an optimal way. We also consider the related problem of finding a matrix X, diagonally similar to a given matrix, such that corresponding row and column norms in X are all equal. Reports of some numerical tests are given. 相似文献
4.
In the first part of this paper, we prove the sharp global Li‐Yau type gradient estimates for positive solutions to doubly nonlinear diffusion equation(DNDE) on complete Riemannian manifolds with nonnegative Ricci curvature. As an application, one can obtain a parabolic Harnack inequality. In the second part, we obtain a Perelman‐type entropy monotonicity formula for DNDE on compact Riemannian manifolds with nonnegative Ricci curvature. These results generalize some works of Ni (JGA 2004), Lu–Ni–Vázquez–Villani (JMPA 2009) and Kotschwar–Ni (Annales Scientifiques de l'École Normale Supérieure 2009). Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
5.
A. A. Selivanov 《Vestnik St. Petersburg University: Mathematics》2011,44(4):301-308
The problem of constructing models for the statistical dynamics of finite-level quantum mechanical systems is considered.
The maximum entropy principle formulated by E.T. Jaynes in 1957 and asserting that the entropy of any physical system increases until it attains its maximum value under constraints imposed by other physical
laws is applied. In accordance with this principle, the von Neumann entropy is taken for the objective function; a dynamical equation
describing the evolution of the density operator in finite-level systems is derived by using the speed gradient principle.
In this case, physical constraints are the mass conservation law and the energy conservation law. The stability of the equilibrium
points of the system thus obtained is investigated. By using LaSalle’s theorem, it is shown that the density function tends
to a Gibbs distribution, under which the entropy attains its maximum. The method is exemplified by analyzing a finite system
of identical particles distributed between cells. Results of numerical simulation are presented. 相似文献
6.
XING Xiusan 《中国科学A辑(英文版)》2001,44(10):1331-1339
Inspired by the evolution equation of nonequilibrium statistical physics entropy and the concise statistical formula of the
entropy production rate, we develop a theory of the dynamic information entropy and build a nonlinear evolution equation of
the information entropy density changing in time and state variable space. Its mathematical form and physical meaning are
similar to the evolution equation of the physical entropy: The time rate of change of information entropy density originates
together from drift, diffusion and production. The concise statistical formula of information entropy production rate is similar
to that of physical entropy also. Furthermore, we study the similarity and difference between physical entropy and information
entropy and the possible unification of the two statistical entropies, and discuss the relationship among the principle of
entropy increase, the principle of equilibrium maximum entropy and the principle of maximum information entropy as well as
the connection between them and the entropy evolution equation. 相似文献
7.
Relative entropy between two quantum states, which quantifies to what extent the quantum states can be distinguished via whatever methods allowed by quantum mechanics, is a central and fundamental quantity in quantum information theory. However, in both theoretical analysis (such as selective measurements) and practical situations (such as random experiments), one is often encountered with quantum ensembles, which are families of quantum states with certain prior probability distributions. How can we quantify the quantumness and distinguishability of quantum ensembles? In this paper, by use of a probabilistic coupling technique, we propose a notion of relative entropy between quantum ensembles, which is a natural generalization of the relative entropy between quantum states. This generalization enjoys most of the basic and important properties of the original relative entropy. As an application, we use the notion of relative entropy between quantum ensembles to define a measure for quantumness of quantum ensembles. This quantity may be useful in quantum cryptography since in certain circumstances it is desirable to encode messages in quantum ensembles which are the most quantum, thus the most sensitive to eavesdropping. By use of this measure of quantumness, we demonstrate that a set consisting of two pure states is the most quantum when the states are 45° apart. 相似文献
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10.
We give a definition of relative entropy with respect to a sublinear expectation and establish large deviation principle for the empirical measures for independent random variables under the sublinear expectation. 相似文献
11.
We show that the ‘directed diffusion equation’ with periodic boundary conditions has a unique weak solution u whenever b is measurable and bounded above and below by positive constants. Also, limt→∞u(t,x) in Lp, 1?p≤∞. 相似文献
12.
We study the utility maximization problem, the problem of minimization of the hedging error and the corresponding dual problems
using dynamic programming approach. We consider an incomplete financial market model, where the dynamics of asset prices are
described by an ℝd-valued continuous semimartingale. Under some regularity assumptions, we derive the backward stochastic PDEs for the value
functions related to these problems, and for the primal problem, we show that the strategy is optimal if and only if the corresponding
wealth process satisfies a certain forward SDE. As examples we consider the mean-variance hedging problem and the cases of
power, exponential, logarithmic utilities, and corresponding dual problems.
__________
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 45, Martingale
Theory and Its Application, 2007. 相似文献
13.
A. F. Ramírez 《Probability Theory and Related Fields》1998,110(3):369-395
Summary. Let η be a diffusion process taking values on the infinite dimensional space T
Z
, where T is the circle, and with components satisfying the equations dη
i
=σ
i
(η) dW
i
+b
i
(η) dt for some coefficients σ
i
and b
i
, i∈Z. Suppose we have an initial distribution μ and a sequence of times t
n
→∞ such that lim
n
→∞μS
tn
=ν exists, where S
t
is the semi-group of the process. We prove that if σ
i
and b
i
are bounded, of finite range, have uniformly bounded second order partial derivatives, and inf
i
,ησ
i
(η)>0, then ν is invariant.
Received: 12 September 1996 / In revised form: 10 November 1997 相似文献
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15.
Sujin Shin 《Transactions of the American Mathematical Society》2006,358(5):2205-2216
Let and be topological dynamical systems and a factor map. A function is a compensation function for if for all . For a factor code between subshifts of finite type, we analyze the associated relative entropy function and give a necessary condition for the existence of saturated compensation functions. Necessary and sufficient conditions for a map to be a saturated compensation function will be provided.
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We consider several structured population models (age structured, size structured, maturity structured) and the general scattering equation. These models are not conservation laws, nevertheless, we show that they admit a common relative entropy structure which uses the first eigenelements of the problem. In case of scattering, it is more general than the usual ‘detailed balance principle’. Three types of consequences are deduced from this entropy structure: a priori bounds, large time convergence to the steady state and in some cases, exponential rates of convergence. To cite this article: P. Michel et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004). 相似文献