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Maxim J. Goldberg - One of the best experts on this subject based on the ideXlab platform.
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Some Extensions of E. Stein’s Work on Littlewood–Paley Theory Applied to Symmetric Diffusion Semigroups
The Journal of Geometric Analysis, 2020Co-Authors: Ronald R. Coifman, Maxim J. GoldbergAbstract:Discrete diffusion semigroups have proven to be highly effective tools for machine learning and data analysis due to the interplay between diffusion processes (of inferences), and their naturally associated geometries. Inspired by the harmonic analysis machinery that E. Stein developed for symmetric diffusion semigroups acting on $$L_p$$ L p Spaces, we show that a correspondence between the rate of diffusion approximation and a Besov-type version of smoothness exists in the general continuous case, even without a local kernel representation. Specifically, let $$\left\{ A_t\right\} _{t\ge 0}$$ A t t ≥ 0 be a symmetric diffusion semigroup on $$L_p(X)$$ L p ( X ) , for X a complete positive $$\sigma $$ σ -Finite Measure Space. We first establish that for $$1
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Some Extensions of E. Stein’s Work on Littlewood–Paley Theory Applied to Symmetric Diffusion Semigroups
The Journal of Geometric Analysis, 2020Co-Authors: Ronald R. Coifman, Maxim J. GoldbergAbstract:Discrete diffusion semigroups have proven to be highly effective tools for machine learning and data analysis due to the interplay between diffusion processes (of inferences), and their naturally associated geometries. Inspired by the harmonic analysis machinery that E. Stein developed for symmetric diffusion semigroups acting on $$L_p$$ L p Spaces, we show that a correspondence between the rate of diffusion approximation and a Besov-type version of smoothness exists in the general continuous case, even without a local kernel representation. Specifically, let $$\left\{ A_t\right\} _{t\ge 0}$$ A t t ≥ 0 be a symmetric diffusion semigroup on $$L_p(X)$$ L p ( X ) , for X a complete positive $$\sigma $$ σ -Finite Measure Space. We first establish that for $$1
Ronald R. Coifman - One of the best experts on this subject based on the ideXlab platform.
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Some Extensions of E. Stein’s Work on Littlewood–Paley Theory Applied to Symmetric Diffusion Semigroups
The Journal of Geometric Analysis, 2020Co-Authors: Ronald R. Coifman, Maxim J. GoldbergAbstract:Discrete diffusion semigroups have proven to be highly effective tools for machine learning and data analysis due to the interplay between diffusion processes (of inferences), and their naturally associated geometries. Inspired by the harmonic analysis machinery that E. Stein developed for symmetric diffusion semigroups acting on $$L_p$$ L p Spaces, we show that a correspondence between the rate of diffusion approximation and a Besov-type version of smoothness exists in the general continuous case, even without a local kernel representation. Specifically, let $$\left\{ A_t\right\} _{t\ge 0}$$ A t t ≥ 0 be a symmetric diffusion semigroup on $$L_p(X)$$ L p ( X ) , for X a complete positive $$\sigma $$ σ -Finite Measure Space. We first establish that for $$1
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Some Extensions of E. Stein’s Work on Littlewood–Paley Theory Applied to Symmetric Diffusion Semigroups
The Journal of Geometric Analysis, 2020Co-Authors: Ronald R. Coifman, Maxim J. GoldbergAbstract:Discrete diffusion semigroups have proven to be highly effective tools for machine learning and data analysis due to the interplay between diffusion processes (of inferences), and their naturally associated geometries. Inspired by the harmonic analysis machinery that E. Stein developed for symmetric diffusion semigroups acting on $$L_p$$ L p Spaces, we show that a correspondence between the rate of diffusion approximation and a Besov-type version of smoothness exists in the general continuous case, even without a local kernel representation. Specifically, let $$\left\{ A_t\right\} _{t\ge 0}$$ A t t ≥ 0 be a symmetric diffusion semigroup on $$L_p(X)$$ L p ( X ) , for X a complete positive $$\sigma $$ σ -Finite Measure Space. We first establish that for $$1
Benoit Simon - One of the best experts on this subject based on the ideXlab platform.
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RELATIVE REARRANGEMENTON A Finite Measure Space APPLICATION TO THE REGULARITY OF WEIGHTED MONOTONE
1997Co-Authors: Jean Michel Rakotoson, Benoit SimonAbstract:We extend the notion of relative rearrangement introdu ced by J. Mossino and R. Temam [1] (see also [2], [3], [4]) to any Finite Measure Space. This type of a result finds ap plication in the problems set in weighted Spaces. Webegin by reviewing the classical properties of the monotone rea rrangement and we give an original proof of the directional derivative theorem using convex analysis in the general fra mework of a Finite Measure Space. Then, we focus on more regular Measure Spaces, the weighted Spaces, and we study, as in [5], the regularity of the monotone rearrangement. We give a characterization ensuring the continuity. To obtain dif ferentiability results, we introduce inequualities ofDe Giorgi type thoroughly linked to weighted Sobolev imbeddings.
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relative rearrangementon a Finite Measure Space application to weighted Spaces and to p d e relative rearrangement weighted sobolev Spaces weighted relative osoperimetric inequality degenerate equations
1997Co-Authors: Jean Michel Rakotoson, Benoit Simon, J Ildefonso, Diaz DiazAbstract:To obtain differentiability results for weighted monotone re arrangement, we_have introduced in [1], the general classes of weights Q and Q. Here, we give a few examples of weights belonging to these classes. In particular, we show that weights behaving like distance functions belong to Q and Q. As an ap plication ofthe regularity results found in [1], we produce sorne continuous imbeddings for weighted Sobolev Spaces, with an explicit estimate of constants. We also see that it is possible to extend the differentiability results to others weights and to de fine a weighted perimeter of De Giorgi having nice properties. FinalIy, we study the regularity of the solutions of a quasili near degenerate problem, using weighted rearrangement.
Jean Michel Rakotoson - One of the best experts on this subject based on the ideXlab platform.
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RELATIVE REARRANGEMENTON A Finite Measure Space APPLICATION TO THE REGULARITY OF WEIGHTED MONOTONE
1997Co-Authors: Jean Michel Rakotoson, Benoit SimonAbstract:We extend the notion of relative rearrangement introdu ced by J. Mossino and R. Temam [1] (see also [2], [3], [4]) to any Finite Measure Space. This type of a result finds ap plication in the problems set in weighted Spaces. Webegin by reviewing the classical properties of the monotone rea rrangement and we give an original proof of the directional derivative theorem using convex analysis in the general fra mework of a Finite Measure Space. Then, we focus on more regular Measure Spaces, the weighted Spaces, and we study, as in [5], the regularity of the monotone rearrangement. We give a characterization ensuring the continuity. To obtain dif ferentiability results, we introduce inequualities ofDe Giorgi type thoroughly linked to weighted Sobolev imbeddings.
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relative rearrangementon a Finite Measure Space application to weighted Spaces and to p d e relative rearrangement weighted sobolev Spaces weighted relative osoperimetric inequality degenerate equations
1997Co-Authors: Jean Michel Rakotoson, Benoit Simon, J Ildefonso, Diaz DiazAbstract:To obtain differentiability results for weighted monotone re arrangement, we_have introduced in [1], the general classes of weights Q and Q. Here, we give a few examples of weights belonging to these classes. In particular, we show that weights behaving like distance functions belong to Q and Q. As an ap plication ofthe regularity results found in [1], we produce sorne continuous imbeddings for weighted Sobolev Spaces, with an explicit estimate of constants. We also see that it is possible to extend the differentiability results to others weights and to de fine a weighted perimeter of De Giorgi having nice properties. FinalIy, we study the regularity of the solutions of a quasili near degenerate problem, using weighted rearrangement.
Nobusumi Sagara - One of the best experts on this subject based on the ideXlab platform.
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Relaxation and Purification for Nonconvex Variational Problems in Dual Banach Spaces: The Minimization Principle in Saturated Measure Spaces
SIAM Journal on Control and Optimization, 2017Co-Authors: Nobusumi SagaraAbstract:We formulate bang-bang, purification, and minimization principles in dual Banach Spaces with Gelfand integrals and provide a complete characterization of the saturation property of Finite Measure Spaces. We also present an application of the relaxation technique to large economies with inFinite-dimensional commodity Spaces, where the Space of agents is modeled as a Finite Measure Space. We propose a “relaxation” of large economies, which is regarded as a reasonable convexification of original economies. Under the saturation hypothesis, the relaxation and purification techniques enable us to prove the existence of Pareto optimal allocations without convexity assumptions.
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Relaxation and Purification for Nonconvex Variational Problems in Dual Banach Spaces: The Minimization Principle in Saturated Measure Spaces
IFAC-PapersOnLine, 2016Co-Authors: Nobusumi SagaraAbstract:Abstract: We formulate bang-bang, purification, minimization principles in dual Banach Spaces with Gelfand integrals and provide a complete characterization of the saturation property of Finite Measure Spaces. We also present a new application of the relaxation technique to large economies with inFinite-dimensional commodity Spaces, where the Space of agents is modeled as a Finite Measure Space. We propose a “relaxation” of large economies, which is regarded as a reasonable convexification of original economies. Under the saturation hypothesis, the relaxation and purification techniques enable us to prove the existence of Pareto optimal allocations without convexity assumptions.