We derive necessary and sufficient conditions for the existence of a Hermitian nonnegative-definite solution to the matrix equation AXB = C. Moreover, we derive a representation of a general Hermitian nonnegative-defi...We derive necessary and sufficient conditions for the existence of a Hermitian nonnegative-definite solution to the matrix equation AXB = C. Moreover, we derive a representation of a general Hermitian nonnegative-definite solution. We then apply our solution to two examples, including a comparison of our solution to a proposed solution by Zhang in [1] using an example problem given from [1]. Our solution demonstrates that the proposed general solution from Zhang in [1] is incorrect. We also give a second example in which we derive the general covariance structure so that two matrix quadratic forms are independent.展开更多
We generalize the decomposition method of the finite Markov chains for Poincare inequality in Jerrum et al.（Ann.Appl.Probab.,14,1741-1765（2004）） to the reversible continuous-time Markov chains.And inductively,we g...We generalize the decomposition method of the finite Markov chains for Poincare inequality in Jerrum et al.（Ann.Appl.Probab.,14,1741-1765（2004）） to the reversible continuous-time Markov chains.And inductively,we give the lower bound of spectral gap for the ergodic open Jackson network by the decomposition method and the symmetrization procedure.The upper bound of the spectral gap is also presented.展开更多
This paper develops further the theory of symmetrization of fractional Laplacian operators contained in recent works of two of the authors. This theory leads to optimal estimates in the form of concentration compariso...This paper develops further the theory of symmetrization of fractional Laplacian operators contained in recent works of two of the authors. This theory leads to optimal estimates in the form of concentration comparison inequalities for both elliptic and parabolic equations. The authors extend the theory for the so-called restricted fractional Laplacian defined on a bounded domain Ω of R^N with zero Dirichlet conditions outside of Ω. As an application, an original proof of the corresponding fractional Faber-Krahn inequality is derived. A more classical variational proof of the inequality is also provided.展开更多
In this paper we extend the notion of rearrangement of nonnegative functions to the setting of Carnot groups. We define rearrangement with respect to a given family of anisotropic balls Br , or equivalently with respe...In this paper we extend the notion of rearrangement of nonnegative functions to the setting of Carnot groups. We define rearrangement with respect to a given family of anisotropic balls Br , or equivalently with respect to a gauge‖x‖, and prove basic regularity properties of this construction. If u is a bounded nonnegative real function with compact support, we denote by u*its rearrangement. Then, the radial function u* is of bounded variation. In addition, if u is continuous then u* is continuous, and if u belongs to the horizontal Sobolev space W 1,ph , then Dhu*(x)/Dh( ‖x‖ )| is in Lp. Moreover, we found a generalization of the inequality of P(o)lya and Szeg(o) ∫|Dhu*|p/Dh(‖x‖)|pdx≤C ∫|Dhu|pdx,where p ≥ 1.展开更多
A set E ? R^d whose indicator function 1_E has maximal Gowers norm, among all sets of equal measure, is an ellipsoid up to Lebesgue null sets. If 1_E has nearly maximal Gowers norm then E nearly coincides with some el...A set E ? R^d whose indicator function 1_E has maximal Gowers norm, among all sets of equal measure, is an ellipsoid up to Lebesgue null sets. If 1_E has nearly maximal Gowers norm then E nearly coincides with some ellipsoid, in the sense that their symmetric difference has small Lebesgue measure.展开更多
A preliminary design for a heavy ion driver inertial fusion（HIDIF） target is presented. The effect of target material and dimensions on transfer efficiency and symmetrical irradiation in the hohlraum are investigate...A preliminary design for a heavy ion driver inertial fusion（HIDIF） target is presented. The effect of target material and dimensions on transfer efficiency and symmetrical irradiation in the hohlraum are investigated.The analysis led to the evaluation of optimal target materials and dimensions to achieve a positive power balance of an ICF power plant.The results show that the best choice is a high Z material for cavity wall materials and a low Z material for the capsule ablator.It is concluded that for achieving the highest transfer efficiency and best symmetrization we need an area ratio between 5≤A2/A1≤9.展开更多
The affine Pólya-Szegö inequality on the Steiner rearrangement in any codimension is proved.We not only define the k-Orlicz-Sobolev balls on Sobolev functions and prove the corresponding affine Pólya-Sz...The affine Pólya-Szegö inequality on the Steiner rearrangement in any codimension is proved.We not only define the k-Orlicz-Sobolev balls on Sobolev functions and prove the corresponding affine Pólya-Szegö inequality,but also define the k-Orlicz-Sobolev balls on functions of bounded variation and prove the corresponding affine Pólya-Szegö inequality.展开更多
Lutwak et al.(2010) established the Orlicz centroid inequality for convex bodies and conjectured that their Orlicz centroid inequality could be extended to star bodies. Zhu(2012) confirmed the conjectured Lutwak, Yang...Lutwak et al.(2010) established the Orlicz centroid inequality for convex bodies and conjectured that their Orlicz centroid inequality could be extended to star bodies. Zhu(2012) confirmed the conjectured Lutwak, Yang and Zhang(LYZ) Orlicz centroid inequality and solved the equality condition for the case that φis strictly convex. Without the condition that φ is strictly convex, this paper studies the equality condition of the conjectured LYZ Orlicz centroid inequality for star bodies.展开更多
文摘We derive necessary and sufficient conditions for the existence of a Hermitian nonnegative-definite solution to the matrix equation AXB = C. Moreover, we derive a representation of a general Hermitian nonnegative-definite solution. We then apply our solution to two examples, including a comparison of our solution to a proposed solution by Zhang in [1] using an example problem given from [1]. Our solution demonstrates that the proposed general solution from Zhang in [1] is incorrect. We also give a second example in which we derive the general covariance structure so that two matrix quadratic forms are independent.
基金Supported in part by 985 Project973 Project(Grant No.2011CB808000)+2 种基金NSFC(Grant No.11131003)SRFDP(Grant No.20100003110005)the Fundamental Research Funds for the Central Universities
文摘We generalize the decomposition method of the finite Markov chains for Poincare inequality in Jerrum et al.（Ann.Appl.Probab.,14,1741-1765（2004）） to the reversible continuous-time Markov chains.And inductively,we give the lower bound of spectral gap for the ergodic open Jackson network by the decomposition method and the symmetrization procedure.The upper bound of the spectral gap is also presented.
基金supported by the ANR projects“HAB”and“NONLOCAL”,the Spanish Research Project MTM2011-24696the INDAM-GNAMPA Project 2014“Analisi qualitativa di soluzioni di equazioni ellittiche e di evoluzione”(Italy)
文摘This paper develops further the theory of symmetrization of fractional Laplacian operators contained in recent works of two of the authors. This theory leads to optimal estimates in the form of concentration comparison inequalities for both elliptic and parabolic equations. The authors extend the theory for the so-called restricted fractional Laplacian defined on a bounded domain Ω of R^N with zero Dirichlet conditions outside of Ω. As an application, an original proof of the corresponding fractional Faber-Krahn inequality is derived. A more classical variational proof of the inequality is also provided.
基金supported in part by NSF(Grant No.DMS-9970687)SECTyP-UNCuyo,Argentina(Res.3853/16-R)
文摘In this paper we extend the notion of rearrangement of nonnegative functions to the setting of Carnot groups. We define rearrangement with respect to a given family of anisotropic balls Br , or equivalently with respect to a gauge‖x‖, and prove basic regularity properties of this construction. If u is a bounded nonnegative real function with compact support, we denote by u*its rearrangement. Then, the radial function u* is of bounded variation. In addition, if u is continuous then u* is continuous, and if u belongs to the horizontal Sobolev space W 1,ph , then Dhu*(x)/Dh( ‖x‖ )| is in Lp. Moreover, we found a generalization of the inequality of P(o)lya and Szeg(o) ∫|Dhu*|p/Dh(‖x‖)|pdx≤C ∫|Dhu|pdx,where p ≥ 1.
基金Research supported in part by NSF(Grant DMS-1363324)
文摘A set E ? R^d whose indicator function 1_E has maximal Gowers norm, among all sets of equal measure, is an ellipsoid up to Lebesgue null sets. If 1_E has nearly maximal Gowers norm then E nearly coincides with some ellipsoid, in the sense that their symmetric difference has small Lebesgue measure.
文摘A preliminary design for a heavy ion driver inertial fusion（HIDIF） target is presented. The effect of target material and dimensions on transfer efficiency and symmetrical irradiation in the hohlraum are investigated.The analysis led to the evaluation of optimal target materials and dimensions to achieve a positive power balance of an ICF power plant.The results show that the best choice is a high Z material for cavity wall materials and a low Z material for the capsule ablator.It is concluded that for achieving the highest transfer efficiency and best symmetrization we need an area ratio between 5≤A2/A1≤9.
基金supported by National Natural Science Foundation of China(Grant No.11971080)the Basic and Advanced Research Project of Chongqing,Chongqing Science and Technology Commission(Grant Nos.cstc2015jcyj A00009 and cstc2018jcyj AX0790)Scientific and Technological Research Program of Chongqing Municipal Education Commission(Grant No.KJ1500628)。
文摘The affine Pólya-Szegö inequality on the Steiner rearrangement in any codimension is proved.We not only define the k-Orlicz-Sobolev balls on Sobolev functions and prove the corresponding affine Pólya-Szegö inequality,but also define the k-Orlicz-Sobolev balls on functions of bounded variation and prove the corresponding affine Pólya-Szegö inequality.
基金supported by National Natural Science Foundation of China(Grant No.11671325)the PhD Program of Higher Education Research Fund(Grant No.2012182110020)Fundamental Research Funds for the Central Universities(Grant No.XDJK2016D026)
文摘Lutwak et al.(2010) established the Orlicz centroid inequality for convex bodies and conjectured that their Orlicz centroid inequality could be extended to star bodies. Zhu(2012) confirmed the conjectured Lutwak, Yang and Zhang(LYZ) Orlicz centroid inequality and solved the equality condition for the case that φis strictly convex. Without the condition that φ is strictly convex, this paper studies the equality condition of the conjectured LYZ Orlicz centroid inequality for star bodies.
