The Experts below are selected from a list of 141 Experts worldwide ranked by ideXlab platform
Naoki Saito - One of the best experts on this subject based on the ideXlab platform.
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On Rayleigh-type formulas for a non-local boundary value problem associated with an integral operator commuting with the Laplacian
Applied and Computational Harmonic Analysis, 2018Co-Authors: Lotfi Hermi, Naoki SaitoAbstract:Abstract In this article we prove the existence, uniqueness, and simplicity of a Negative Eigenvalue for a class of integral operators whose kernel is of the form | x − y | ρ , 0 ρ ≤ 1 , x , y ∈ [ − a , a ] . We also provide two different ways of producing recursive formulas for the Rayleigh functions (i.e., recursion formulas for power sums) of the Eigenvalues of this integral operator when ρ = 1 , providing means of approximating this Negative Eigenvalue. These methods offer recursive procedures for dealing with the Eigenvalues of a one-dimensional Laplacian with non-local boundary conditions which commutes with an integral operator having a harmonic kernel. The problem emerged in recent work by one of the authors [48] . We also discuss extensions in higher dimensions and links with distance matrices.
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On Rayleigh-Type Formulas for a Non-local Boundary Value Problem Associated with an Integral Operator Commuting with the Laplacian
arXiv: Spectral Theory, 2010Co-Authors: Lotfi Hermi, Naoki SaitoAbstract:In this article we prove the existence, uniqueness, and simplicity of a Negative Eigenvalue for a class of integral operators whose kernel is of the form $|x-y|^\rho$, $0 < \rho \leq 1$, $x, y \in [-a, a]$. We also provide two different ways of producing recursive formulas for the Rayleigh functions (i.e., recursion formulas for power sums) of the Eigenvalues of this integral operator when $\rho=1$, providing means of approximating this Negative Eigenvalue. These methods offer recursive procedures for dealing with the Eigenvalues of a one-dimensional Laplacian with non-local boundary conditions which commutes with an integral operator having a harmonic kernel. The problem emerged in recent work by one of the authors [45]. We also discuss extensions in higher dimensions and links with distance matrices.
Lotfi Hermi - One of the best experts on this subject based on the ideXlab platform.
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On Rayleigh-type formulas for a non-local boundary value problem associated with an integral operator commuting with the Laplacian
Applied and Computational Harmonic Analysis, 2018Co-Authors: Lotfi Hermi, Naoki SaitoAbstract:Abstract In this article we prove the existence, uniqueness, and simplicity of a Negative Eigenvalue for a class of integral operators whose kernel is of the form | x − y | ρ , 0 ρ ≤ 1 , x , y ∈ [ − a , a ] . We also provide two different ways of producing recursive formulas for the Rayleigh functions (i.e., recursion formulas for power sums) of the Eigenvalues of this integral operator when ρ = 1 , providing means of approximating this Negative Eigenvalue. These methods offer recursive procedures for dealing with the Eigenvalues of a one-dimensional Laplacian with non-local boundary conditions which commutes with an integral operator having a harmonic kernel. The problem emerged in recent work by one of the authors [48] . We also discuss extensions in higher dimensions and links with distance matrices.
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On Rayleigh-Type Formulas for a Non-local Boundary Value Problem Associated with an Integral Operator Commuting with the Laplacian
arXiv: Spectral Theory, 2010Co-Authors: Lotfi Hermi, Naoki SaitoAbstract:In this article we prove the existence, uniqueness, and simplicity of a Negative Eigenvalue for a class of integral operators whose kernel is of the form $|x-y|^\rho$, $0 < \rho \leq 1$, $x, y \in [-a, a]$. We also provide two different ways of producing recursive formulas for the Rayleigh functions (i.e., recursion formulas for power sums) of the Eigenvalues of this integral operator when $\rho=1$, providing means of approximating this Negative Eigenvalue. These methods offer recursive procedures for dealing with the Eigenvalues of a one-dimensional Laplacian with non-local boundary conditions which commutes with an integral operator having a harmonic kernel. The problem emerged in recent work by one of the authors [45]. We also discuss extensions in higher dimensions and links with distance matrices.
Swapan Rana - One of the best experts on this subject based on the ideXlab platform.
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Negative Eigenvalues of partial transposition of arbitrary bipartite states
Physical Review A, 2013Co-Authors: Swapan RanaAbstract:The partial transposition of a two-qubit state has at most one Negative Eigenvalue and all the Eigenvalues lie in [-1/2,1]. In this Brief Report, we extend this result by Sanpera et al. [A. Sanpera, R. Tarrach and G. Vidal, Phys. Rev. A 58, 826 (1998)] to arbitrary bipartite states. We show that partial transposition of an $m\otimes n$ state can not have more than (m-1)(n-1) number of Negative Eigenvalues. Low-dimensional states have been studied to show the tightness of this result and explicit examples have been provided for $mn\le 9$. It is also shown that all the Eigenvalues of partial transposition lie within [-1/2,1]. Some possible applications are also discussed.
Jiming Peng - One of the best experts on this subject based on the ideXlab platform.
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New global algorithms for quadratic programming with a few Negative Eigenvalues based on alternative direction method and convex relaxation
Mathematical Programming Computation, 2018Co-Authors: Hezhi Luo, Xiaodi Bai, Gino J. Lim, Jiming PengAbstract:We consider a quadratic program with a few Negative Eigenvalues (QP-r-NE) subject to linear and convex quadratic constraints that covers many applications and is known to be NP-hard even with one Negative Eigenvalue (QP1NE). In this paper, we first introduce a new global algorithm (ADMBB), which integrates several simple optimization techniques such as alternative direction method, and branch-and-bound, to find a globally optimal solution to the underlying QP within a pre-specified $$\epsilon $$ -tolerance. We establish the convergence of the ADMBB algorithm and estimate its complexity. Second, we develop a global search algorithm (GSA) for QP1NE that can locate an optimal solution to QP1NE within $$\epsilon $$ -tolerance and estimate the worst-case complexity bound of the GSA. Preliminary numerical results demonstrate that the ADMBB algorithm can effectively find a global optimal solution to large-scale QP-r-NE instances when $$r\le 10$$ , and the GSA outperforms the ADMBB for most of the tested QP1NE instances. The software reviewed as part of this submission was given the DOI (digital object identifier) https://doi.org/10.5281/zenodo.1344739 .
Hezhi Luo - One of the best experts on this subject based on the ideXlab platform.
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New global algorithms for quadratic programming with a few Negative Eigenvalues based on alternative direction method and convex relaxation
Mathematical Programming Computation, 2018Co-Authors: Hezhi Luo, Xiaodi Bai, Gino J. Lim, Jiming PengAbstract:We consider a quadratic program with a few Negative Eigenvalues (QP-r-NE) subject to linear and convex quadratic constraints that covers many applications and is known to be NP-hard even with one Negative Eigenvalue (QP1NE). In this paper, we first introduce a new global algorithm (ADMBB), which integrates several simple optimization techniques such as alternative direction method, and branch-and-bound, to find a globally optimal solution to the underlying QP within a pre-specified $$\epsilon $$ -tolerance. We establish the convergence of the ADMBB algorithm and estimate its complexity. Second, we develop a global search algorithm (GSA) for QP1NE that can locate an optimal solution to QP1NE within $$\epsilon $$ -tolerance and estimate the worst-case complexity bound of the GSA. Preliminary numerical results demonstrate that the ADMBB algorithm can effectively find a global optimal solution to large-scale QP-r-NE instances when $$r\le 10$$ , and the GSA outperforms the ADMBB for most of the tested QP1NE instances. The software reviewed as part of this submission was given the DOI (digital object identifier) https://doi.org/10.5281/zenodo.1344739 .
