关于矩阵迹的不等式

讨论了满足条件R[A,B]≤1的矩阵和的平方的迹的不等式,并利用经典的H(o|¨)lder不等式和Minkowski不等式,证明了半正定Hermiter矩阵迹的H(o|¨)lder不等式和矩阵迹的Minkowski不等式的推广不等式.     

Bounds for eigenvalues using traces

Several new inequalities are obtained for the modulus, the real part, and the imaginary part of a linear combination of the ordered eigenvalues of a square complex matrix. Included are bounds for the condition number, the spread, and the spectral radius. These inequalities involve the trace of a matrix and the trace of its square. Necessary and sufficient conditions for equality are given for each inequality.     

微分代数系统稳定性及镇定新判据

本文利用矩阵迹不等式讨论了微分代数系统的稳定性,并给出了微分代数系统容许的充分条件.提出了用此方法设计状态反馈控制器使微分代数系统容许的新方法,并设计了矩阵迹不等式组的计算机求解程序,并用例子说明了主要定理的有效性.     

On the existence of an extremal function in critical Sobolev trace embedding theorem

Sufficient conditions for the existence of extremal functions in the trace Sobolev inequality and the trace Sobolev-Poincaré inequality are established. It is shown that some of these conditions are sharp.     

一类算子迹的不等式

将一个算子迹的不等式进行加细,并获得了一个新的算子迹不等式.     

Logarithmic Sobolev trace inequality

A logarithmic Sobolev trace inequality is derived. Bounds on the best constant for this inequality from above and below are investigated using the sharp Sobolev inequality and the sharp logarithmic Sobolev inequality.

     

An inequality for nonnegative matrices and the inverse eigenvalue problem

We present two versions of the same inequality, relating the maximal diagonal entry of a nonnegative matrix to its eigenvalues. We demonstrate a matrix factorization of a companion matrix, which leads to a solution of the nonnegative inverse eigenvalue problem (denoted the nniep) for 4×4 matrices of trace zero, and we give some sufficient conditions for a solution to the nniep for 5×5 matrices of trace zero. We also give a necessary condition on the eigenvalues of a 5×5 trace zero nonnegative matrix in lower Hessenberg form. Finally, we give a brief discussion of the nniep in restricted cases.     

Sharp trace Gagliardo–Nirenberg–Sobolev inequalities for convex cones,and convex domains

We find a new sharp trace Gagliardo–Nirenberg–Sobolev inequality on convex cones, as well as a sharp weighted trace Sobolev inequality on epigraphs of convex functions. This is done by using a generalized Borell–Brascamp–Lieb inequality, coming from the Brunn–Minkowski theory.     

Trace inequalities on a generalized Wigner-Yanase skew information

We introduce a generalized Wigner-Yanase skew information and then derive the trace inequality related to the uncertainty relation. This inequality is a non-trivial generalization of the uncertainty relation derived by S. Luo for the quantum uncertainty quantity excluding the classical mixture. In addition, several trace inequalities on our generalized Wigner-Yanase skew information are argued.     

Balls have the worst best Sobolev inequalities

Using transportation techniques in the spirit of Cordero-Erausquin, Nazaret and Villani [7], we establish an optimal non parametric trace Sobolev inequality, for arbitrary locally Lipschitz domains in ℝⁿ. We deduce a sharp variant of the Brézis-Lieb trace Sobolev inequality [4], containing both the isoperimetric inequality and the sharp Euclidean Sobolev embedding as particular cases. This inequality is optimal for a ball, and can be improved for any other bounded, Lipschitz, connected domain. We also derive a strengthening of the Brézis-Lieb inequality, suggested and left as an open problem in [4]. Many variants will be investigated in a companion article [10].     

New Differential Harnack Inequalities for Nonlinear Heat Equations

This paper deals with constrained trace, matrix and constrained matrix Harnack inequalities for the nonlinear heat equation ωt = ?ω + aωln ω on closed manifolds. A new interpolated Harnack inequality for ωt = ?ω-ωln ω +εRω on closed surfaces under ε-Ricci flow is also derived. Finally, the author proves a new differential Harnack inequality for ωt= ?ω-ωln ω under Ricci flow without any curvature condition. Among these Harnack inequalities, the correction terms are all time-exponential functions,...     

Enhancements to the von Neumann trace inequality

Upper trace bounds for the product of two n × n complex matrices are presented. The real component of the trace inequality is tighter than von Neumann’s inequality, and the imaginary component is new.     

A sharp Sobolev trace inequality for the fractional-order derivatives

We establish a sharp Sobolev trace inequality for the fractional-order derivatives. As a close connection with this best estimate, we show a fractional-order logarithmic Sobolev trace inequality with the asymptotically optimal constant, but also sharpen the Poincaré embedding for the conformal invariant energy and BMO spaces.     

一个算子迹的不等式

本文讨论Bellman不等式的相关问题,利用紧算子的极表示以及陈公宁的一个矩阵迹的不等式,得到算子迹的相应不等式.作为其推论,在无穷维Hilbert空间中给出了Bellman问题的一个肯定回答.     

Hermite正定对称矩阵迹的一些结果(英文)

本文研究了一类Hermite正定矩阵迹的不等式问题.利用文献[2-6]中的结果以及放缩法,获得了Hermite正定矩阵迹的极值定理、杨氏不等式和贝努利不等式,并且将许多初等不等式推广到Hermite正定矩阵迹的情形.     

Interpolating between constrained Li–Yau and Chow–Hamilton Harnack inequalities on a surface

We establish a one-parameter family of Harnack inequalities connecting the constrained trace Li–Yau differential Harnack inequality for the heat equation to the constrained trace Chow–Hamilton Harnack inequality for the Ricci flow on a 2-dimensional closed manifold with positive scalar curvature, and thereby generalize Chow’s interpolated Harnack inequality (J. Partial Diff. Eqs. 11 (1998), 137–140).     

Boundary trace inequalities and rearrangements

A new approach to boundary trace inequalities for Sobolev functions is presented, which reduces any trace inequality involving general rearrangement-invariant norms to an equivalent, considerably simpler, one-dimensional inequality for a Hardy-type operator. In particular, improvements of classical boundary trace embeddings and new optimal trace embeddings are derived. This research was partially supported by the Italian research project “Geometric properties of solutions to variational problems” of GNAMPA (INdAM) 2006, by the research project MSM 0021620839 of the Czech Ministry of Education, by grants 201/03/0935, 201/05/2033 and 201/07/0388 of the Grant Agency of the Czech Republic and by the Nečas Center for Mathematical Modeling project no. LC06052 financed by the Czech Ministry of Education.     

A criterion for straightening of a lipschitz surface in the Lizorkin-Triebel sense. I

We study the conditions when the trace of a Lizorkin-Triebel space on a Lipschitz surface coincides with the trace of this space on a hyperplane. A criterion in terms of a dyadic weighted inequality is found for a wide range of indices.     

A refined Brunn–Minkowski inequality for convex sets

Starting from a mass transportation proof of the Brunn–Minkowski inequality on convex sets, we improve the inequality showing a sharp estimate about the stability property of optimal sets. This is based on a Poincaré-type trace inequality on convex sets that is also proved in sharp form.     

矩阵迹的几个不等式

给出了实对称矩阵的Ｈｏｌｄｅｒ不等式，Ｍｉｎｋｏｗｓｋｉ不等式和算术几何平均值不等式．