共查询到20条相似文献,搜索用时 31 毫秒
1.
Erwin Lutwak Deane Yang Gaoyong Zhang 《Transactions of the American Mathematical Society》2001,353(5):1767-1779
New affine invariant functionals for convex polytopes are introduced. Some sharp affine isoperimetric inequalities are established for the new functionals. These new inequalities lead to fairly strong volume estimates for projection bodies. Two of the new affine isoperimetric inequalities are extensions of Ball's reverse isoperimetric inequalities.
2.
We define a classical probability analogue of Voiculescu's free entropy dimension that we shall call the classical probability entropy dimension of a probability measure on Rn. We show that the classical probability entropy dimension of a measure is related with diverse other notions of dimension. First, it can be viewed as a kind of fractal dimension. Second, if one extends Bochner's inequalities to a measure by requiring that microstates around this measure asymptotically satisfy the classical Bochner's inequalities, then we show that the classical probability entropy dimension controls the rate of increase of optimal constants in Bochner's inequality for a measure regularized by convolution with the Gaussian law as the regularization is removed. We introduce a free analogue of the Bochner inequality and study the related free entropy dimension quantity. We show that it is greater or equal to the non-microstates free entropy dimension. 相似文献
3.
The aim of this pedagogical paper is to show how some renowned inequalities may be obtained via a simple argument: entropy projection from the path space onto finite-dimensional coordinates spaces. Some applications are given: ergodic behaviour, perturbation. 相似文献
4.
By using some lattice-like operations which constitute extensions of ones introduced by M.S. Gowda, R. Sznajder and J. Tao for self-dual cones, a new perspective is gained on the subject of isotonicity of the metric projection onto the closed convex sets. The results of this paper are wide range generalizations of some results of the authors obtained for self-dual cones. The aim of the subsequent investigations is to put into evidence some closed convex sets for which the metric projection is isotonic with respect to the order relation which give rise to the above mentioned lattice-like operations. The topic is related to variational inequalities where the isotonicity of the metric projection is an important technical tool. For Euclidean sublattices this approach was considered by G. Isac and respectively by H. Nishimura and E.A. Ok. 相似文献
5.
The article builds on several recent advances in the Monge-
Kantorovich theory of mass transport which have, among other things, led
to new and quite natural proofs for a wide range of geometric inequalities
such as the ones formulated by Brunn-Minkowski, Sobolev, Gagliardo-
Nirenberg, Beckner, Gross, Talagrand, Otto-Villani and their extensions
by many others. While this paper continues in this spirit, we however propose
here a basic framework to which all of these inequalities belong, and
a general unifying principle from which many of them follow. This basic
inequality relates the relative total energy - internal, potential and interactive
- of two arbitrary probability densities, their Wasserstein distance,
their barycentres and their entropy production functional. The framework
is remarkably encompassing as it implies many old geometric - Gaussian
and Euclidean - inequalities as well as new ones, while allowing a direct
and unified way for computing best constants and extremals. As expected,
such inequalities also lead to exponential rates of convergence to equilibria
for solutions of Fokker-Planck and McKean-Vlasov type equations. The
principle also leads to a remarkable correspondence between ground state
solutions of certain quasilinear - or semilinear - equations and stationary
solutions of nonlinear Fokker-Planck type equations. 相似文献
6.
A quantitative version of Minkowski sum, extending the definition of θ-convolution of convex bodies, is studied to obtain extensions of the Brunn–Minkowski and Zhang inequalities, as well as, other interesting properties on Convex Geometry involving convolution bodies or polar projection bodies. The extension of this new version to more than two sets is also given. 相似文献
7.
利用极大熵函数方法将不等式组及变分不等式的求解问题转化为近似可微优化问题,给出了不等式组及变分不等式问题近似解的可微优化方法,得到了不等式组和变分不等式问题的解集合的示性函数. 相似文献
8.
傅白白 《数学的实践与认识》2005,35(8):84-88
利用极大熵函数方法将不等式组及变分不等式的求解问题转化为近似可微优化问题,给出了不等式组及变分不等式问题近似解的可微优化方法,得到了不等式组和变分不等式问题的解集合的示性函数. 相似文献
9.
Yuichi Kanjin 《Monatshefte für Mathematik》1999,127(4):311-319
For the real Hardy spaces , we shall show the Hardy type integral inequalities, and applying the inequalities we shall establish the Hardy’s inequalities
with respect to Hankel transforms.
Received 21 October 1997; in revised form 19 October 1998 相似文献
10.
11.
We shall present several Hanner type inequalities with a weight constant and characterize 2-uniformly smooth and 2-uniformly convex Banach spaces with these inequalities. p-Uniformly smooth and q-uniformly convex Banach spaces will be also characterized with another Hanner type inequalities with a weight in the other side term. The best value of the weight in these inequalities will be determined for Lp spaces. Also we shall present a duality theorem between these inequalities in a generalized form. 相似文献
12.
Bernd Carl 《Journal of Functional Analysis》1981,41(3):290-306
We establish inequalities between entropy numbers and approximation numbers for operators acting between Banach spaces. Furthermore we derive inequalities between eigenvalues and entropy numbers for operators acting on a Banach space. The results are compared with the classical inequalities of Bernstein and Jackson. 相似文献
13.
We prove a general duality result showing that a Brascamp–Lieb type inequality is equivalent to an inequality expressing subadditivity of the entropy, with a complete correspondence of best constants and cases of equality. This opens a new approach to the proof of Brascamp–Lieb type inequalities, via subadditivity of the entropy. We illustrate the utility of this approach by proving a general inequality expressing the subadditivity property of the entropy on ${\mathbb {R}^n}We prove a general duality result showing that a Brascamp–Lieb type inequality is equivalent to an inequality expressing subadditivity
of the entropy, with a complete correspondence of best constants and cases of equality. This opens a new approach to the proof
of Brascamp–Lieb type inequalities, via subadditivity of the entropy. We illustrate the utility of this approach by proving
a general inequality expressing the subadditivity property of the entropy on
\mathbb Rn{\mathbb {R}^n}, and fully determining the cases of equality. As a consequence of the duality mentioned above, we obtain a simple new proof
of the classical Brascamp–Lieb inequality, and also a fully explicit determination of all of the cases of equality. We also
deduce several other consequences of the general subadditivity inequality, including a generalization of Hadamard’s inequality
for determinants. Finally, we also prove a second duality theorem relating superadditivity of the Fisher information and a
sharp convolution type inequality for the fundamental eigenvalues of Schr?dinger operators. Though we focus mainly on the
case of random variables in
\mathbb Rn{\mathbb {R}^n} in this paper, we discuss extensions to other settings as well. 相似文献
14.
František Matúš 《Discrete Mathematics》2007,307(21):2464-2477
Two polymatroids are adhesive if a polymatroid extends both in such a way that two ground sets become a modular pair. Motivated by entropy functions, the class of polymatroids with adhesive restrictions and a class of selfadhesive polymatroids are introduced and studied. Adhesivity is described by polyhedral cones of rank functions and defining inequalities of the cones are identified, among them known and new non-Shannon type information inequalities for entropy functions. The selfadhesive polymatroids on a four-element set are characterized by Zhang-Yeung inequalities. 相似文献
15.
In this paper,we shall be concerned with what happens of Gorenstein homological dimensions when certain modifications are made to a ring.The five structural operations addressed later are the formation of excellent extensions,localizations,Morita equivalences,polynomial extensions and power series extensions. 相似文献
16.
First we present various scalar inequalities that extends the classical Cauchy–Schwarz and Kantorovich inequalities. Some of these extensions are based on the moment problem and the Hölder and Minkowski inequalities. These results are then extended to the matrix case. Many well-known inequalities are recovered ans new ones are obtained. 相似文献
17.
对一些基本的不等式进行了推广,给出了含有n个无关变元的非线性的离散不等式.所得结果推广了已有的一些结论. 相似文献
18.
G-radicalG-Kneser and G-Cogalois extensions were introduced in [2]. In this paper we shall study some properties of these extensions by using techniques of graded rings theory. 相似文献
19.
In this paper,firstty we shall show some equivalent conditions of A>B>0;secondly by using the results of ours we shall show some characterizations of the chaotic order(i.e.,log A≥log B)by norm inequalities. 相似文献
20.
Dao-Jun Wen 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(7):2292-2297
In this paper, we introduce and consider a new generalized system of nonconvex variational inequalities with different nonlinear operators. We establish the equivalence between the generalized system of nonconvex variational inequalities and the fixed point problems using the projection technique. This equivalent alternative formulation is used to suggest and analyze a general explicit projection method for solving the generalized system of nonconvex variational inequalities. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities. 相似文献