The Experts below are selected from a list of 6366 Experts worldwide ranked by ideXlab platform
K. C. Chang - One of the best experts on this subject based on the ideXlab platform.
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Nonlinear extensions of the Perron–Frobenius Theorem and the Krein–Rutman Theorem
Journal of Fixed Point Theory and Applications, 2014Co-Authors: K. C. ChangAbstract:A unification version of the Perron–Frobenius Theorem and the Krein–Rutman Theorem for increasing, positively 1-homogeneous, compact mappings is given on ordered Banach spaces without monotonic norm. A Collatz-type minimax characterization of the positive eigenvalue with positive eigenvector is obtained. The power method in computing the largest eigenpair is also extended.
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nonlinear extensions of the perron Frobenius Theorem and the krein rutman Theorem
Journal of Fixed Point Theory and Applications, 2014Co-Authors: K. C. ChangAbstract:A unification version of the Perron–Frobenius Theorem and the Krein–Rutman Theorem for increasing, positively 1-homogeneous, compact mappings is given on ordered Banach spaces without monotonic norm. A Collatz-type minimax characterization of the positive eigenvalue with positive eigenvector is obtained. The power method in computing the largest eigenpair is also extended.
Akihisa Yamada - One of the best experts on this subject based on the ideXlab platform.
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efficient certification of complexity proofs formalizing the perron Frobenius Theorem invited talk paper
Certified Programs and Proofs, 2018Co-Authors: Jose Divason, Sebastiaan J C Joosten, Ondřej Kuncar, Rene Thiemann, Akihisa YamadaAbstract:Matrix interpretations are widely used in automated complexity analysis. Certifying such analyses boils down to determining the growth rate of An for a fixed non-negative rational matrix A. A direct solution for this task involves the computation of all eigenvalues of A, which often leads to expensive algebraic number computations. In this work we formalize the Perron–Frobenius Theorem. We utilize the Theorem to avoid most of the algebraic numbers needed for certifying complexity analysis, so that our new algorithm only needs the rational arithmetic when certifying complexity proofs that existing tools can find. To cover the Theorem in its full extent, we establish a connection between two different Isabelle/HOL libraries on matrices, enabling an easy exchange of Theorems between both libraries. This connection crucially relies on the transfer mechanism in combination with local type definitions, being a non-trivial case study for these Isabelle tools.
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CPP - Efficient certification of complexity proofs: formalizing the Perron–Frobenius Theorem (invited talk paper)
Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs, 2018Co-Authors: Jose Divason, Sebastiaan J C Joosten, Ondřej Kuncar, Rene Thiemann, Akihisa YamadaAbstract:Matrix interpretations are widely used in automated complexity analysis. Certifying such analyses boils down to determining the growth rate of An for a fixed non-negative rational matrix A. A direct solution for this task involves the computation of all eigenvalues of A, which often leads to expensive algebraic number computations. In this work we formalize the Perron–Frobenius Theorem. We utilize the Theorem to avoid most of the algebraic numbers needed for certifying complexity analysis, so that our new algorithm only needs the rational arithmetic when certifying complexity proofs that existing tools can find. To cover the Theorem in its full extent, we establish a connection between two different Isabelle/HOL libraries on matrices, enabling an easy exchange of Theorems between both libraries. This connection crucially relies on the transfer mechanism in combination with local type definitions, being a non-trivial case study for these Isabelle tools.
Matthias Hein - One of the best experts on this subject based on the ideXlab platform.
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A unifying Perron-Frobenius Theorem for nonnegative tensors via multi-homogeneous maps
arXiv: Spectral Theory, 2018Co-Authors: Antoine Gautier, Francesco Tudisco, Matthias HeinAbstract:Inspired by the definition of symmetric decomposition, we introduce the concept of shape partition of a tensor and formulate a general tensor spectral problem that includes all the relevant spectral problems as special cases. We formulate irreducibility and symmetry properties of a nonnegative tensor $T$ in terms of the associated shape partition. We recast the spectral problem for $T$ as a fixed point problem on a suitable product of projective spaces. This allows us to use the theory of multi-homogeneous order-preserving maps to derive a general and unifying Perron-Frobenius Theorem for nonnegative tensors that either implies previous results of this kind or improves them by weakening the assumptions there considered. We introduce a general power method for the computation of the dominant tensor eigenpair, and provide a detailed convergence analysis.
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The Perron-Frobenius Theorem for Multi-homogeneous Maps
arXiv: Spectral Theory, 2017Co-Authors: Antoine Gautier, Francesco Tudisco, Matthias HeinAbstract:We introduce the notion of order-preserving multi-homogeneous mapping which allows to study Perron-Frobenius type Theorems and nonnegative tensors in unified fashion. We prove a weak and strong Perron-Frobenius Theorem for these maps and provide a Collatz-Wielandt principle for the maximal eigenvalue. Additionally, we propose a generalization of the power method for the computation of the maximal eigenvector and analyse its convergence. We show that the general theory provides new results and strengthens existing results for various spectral problems for nonnegative tensors.
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tensor norm and maximal singular vectors of nonnegative tensors a perron Frobenius Theorem a collatz wielandt characterization and a generalized power method
Linear Algebra and its Applications, 2016Co-Authors: Antoine Gautier, Matthias HeinAbstract:Abstract We study the l p 1 , … , p m -singular value problem for nonnegative tensors. We prove a general Perron–Frobenius Theorem for weakly irreducible and irreducible nonnegative tensors and provide a Collatz–Wielandt characterization of the maximal singular value. Additionally, we propose a higher order power method for the computation of the maximal singular vectors and show that it has an asymptotic linear convergence rate.
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Tensor norm and maximal singular vectors of nonnegative tensors — A Perron–Frobenius Theorem, a Collatz–Wielandt characterization and a generalized power method
Linear Algebra and its Applications, 2016Co-Authors: Antoine Gautier, Matthias HeinAbstract:Abstract We study the l p 1 , … , p m -singular value problem for nonnegative tensors. We prove a general Perron–Frobenius Theorem for weakly irreducible and irreducible nonnegative tensors and provide a Collatz–Wielandt characterization of the maximal singular value. Additionally, we propose a higher order power method for the computation of the maximal singular vectors and show that it has an asymptotic linear convergence rate.
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Tensor norm and maximal singular vectors of non-negative tensors - a Perron-Frobenius Theorem, a Collatz-Wielandt characterization and a generalized power method
arXiv: Spectral Theory, 2015Co-Authors: Antoine Gautier, Matthias HeinAbstract:We study the l^{p_1,...,p_m} singular value problem for non-negative tensors. We prove a general Perron-Frobenius Theorem for weakly irreducible and irreducible nonnegative tensors and provide a Collatz-Wielandt characterization of the maximal singular value. Additionally, we propose a higher order power method for the computation of the maximal singular vectors and show that it has an asymptotic linear convergence rate.
Jose Divason - One of the best experts on this subject based on the ideXlab platform.
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efficient certification of complexity proofs formalizing the perron Frobenius Theorem invited talk paper
Certified Programs and Proofs, 2018Co-Authors: Jose Divason, Sebastiaan J C Joosten, Ondřej Kuncar, Rene Thiemann, Akihisa YamadaAbstract:Matrix interpretations are widely used in automated complexity analysis. Certifying such analyses boils down to determining the growth rate of An for a fixed non-negative rational matrix A. A direct solution for this task involves the computation of all eigenvalues of A, which often leads to expensive algebraic number computations. In this work we formalize the Perron–Frobenius Theorem. We utilize the Theorem to avoid most of the algebraic numbers needed for certifying complexity analysis, so that our new algorithm only needs the rational arithmetic when certifying complexity proofs that existing tools can find. To cover the Theorem in its full extent, we establish a connection between two different Isabelle/HOL libraries on matrices, enabling an easy exchange of Theorems between both libraries. This connection crucially relies on the transfer mechanism in combination with local type definitions, being a non-trivial case study for these Isabelle tools.
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CPP - Efficient certification of complexity proofs: formalizing the Perron–Frobenius Theorem (invited talk paper)
Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs, 2018Co-Authors: Jose Divason, Sebastiaan J C Joosten, Ondřej Kuncar, Rene Thiemann, Akihisa YamadaAbstract:Matrix interpretations are widely used in automated complexity analysis. Certifying such analyses boils down to determining the growth rate of An for a fixed non-negative rational matrix A. A direct solution for this task involves the computation of all eigenvalues of A, which often leads to expensive algebraic number computations. In this work we formalize the Perron–Frobenius Theorem. We utilize the Theorem to avoid most of the algebraic numbers needed for certifying complexity analysis, so that our new algorithm only needs the rational arithmetic when certifying complexity proofs that existing tools can find. To cover the Theorem in its full extent, we establish a connection between two different Isabelle/HOL libraries on matrices, enabling an easy exchange of Theorems between both libraries. This connection crucially relies on the transfer mechanism in combination with local type definitions, being a non-trivial case study for these Isabelle tools.
Masahito Ueda - One of the best experts on this subject based on the ideXlab platform.
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Perron-Frobenius Theorem on the superfluid transition of an ultracold Fermi gas
arXiv: Quantum Gases, 2012Co-Authors: Naoyuki Sakumichi, Norio Kawakami, Masahito UedaAbstract:The Perron-Frobenius Theorem is applied to identify the superfluid transition of a two-component Fermi gas with a zero-range s-wave interaction. According to the quantum cluster expansion method of Lee and Yang, the grand partition function is expressed by the Lee-Yang contracted 0-graphs. A singularity of an infinite series of ladder-type Lee-Yang contracted 0-graphs is analyzed. We point out that the singularity is governed by the Perron-Frobenius eigenvalue of a certain primitive matrix which is defined in terms of the two-body cluster functions and the Fermi distribution functions. As a consequence, it is found that there exists a unique fugacity at the phase transition point, which implies that there is no fragmentation of Bose-Einstein condensates of dimers and Cooper pairs at the ladder-approximation level of Lee-Yang contracted 0-graphs. An application to a Bose-Einstein condensate of strongly bounded dimers is also made.
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Perron-Frobenius Theorem on the superfluid transition of an ultracold Fermi gas
2012Co-Authors: Naoyuki Sakumichi, Norio Kawakami, Masahito UedaAbstract:ERATO Macroscopic Quantum Project, JST, Tokyo 113-0033, JapanE-mail: sakumichi@scphys.kyoto-u.ac.jpAbstract. The Perron-Frobenius Theorem is applied to identify the superfluidtransition of a two-component Fermi gas with a zero-range s-wave interaction.According to the quantum cluster expansion method of Lee and Yang, the grandpartition function is expressed by the Lee-Yang contracted 0-graphs. A singularity ofan infinite series of ladder-type Lee-Yang contracted 0-graphs is analyzed. We pointout that the singularity is governed by the Perron-Frobenius eigenvalue of a certainprimitive matrix which is defined in terms of the two-body cluster functions and theFermi distribution functions. As a consequence, it is found that there exists a uniquefugacity at the phase transition point, which implies that there is no fragmentationof Bose-Einstein condensates of dimers and Cooper pairs at the ladder-approximationlevel of Lee-Yang contracted 0-graphs. An application to a Bose-Einstein condensateof strongly bounded dimers is also made.PACS numbers: 34.10.+x, 03.75.Ss, 05.30.Fk, 03.75.Hh
