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Yuuji Tanaka - One of the best experts on this subject based on the ideXlab platform.
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a weak Compactness Theorem of the donaldson thomas instantons on compact kahler threefolds
Journal of Mathematical Analysis and Applications, 2013Co-Authors: Yuuji TanakaAbstract:In Tanaka [18], we introduced a gauge-theoretic equation on symplectic 6-manifolds, which is a version of the Hermitian–Einstein equation perturbed by Higgs fields, and called it a Donaldson–Thomas equation, to analytically approach the Donaldson–Thomas invariants. In this article, we consider the equation on compact Kahler threefolds, and study some of the analytic properties of solutions to them, using analytic methods in higher-dimensional Yang–Mills theory developed by Nakajima (1987) [14], Nakajima (1988) [15] and Tian (2000) [20] with some additional arguments concerning an extra nonlinear term coming from the Higgs fields. We prove that a sequence of solutions to the Donaldson–Thomas equation of a unitary vector bundle over a compact Kahler threefold has a converging subsequence outside a closed subset whose real two-dimensional Hausdorff measure is finite, provided that the L2-norms of the Higgs fields are uniformly bounded. We also prove an n/2-Compactness Theorem of solutions to the equations on compact Kahler threefolds.
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A weak Compactness Theorem of the Donaldson–Thomas instantons on compact Kähler threefolds
Journal of Mathematical Analysis and Applications, 2013Co-Authors: Yuuji TanakaAbstract:Abstract In Tanaka [18] , we introduced a gauge-theoretic equation on symplectic 6-manifolds, which is a version of the Hermitian–Einstein equation perturbed by Higgs fields, and called it a Donaldson–Thomas equation, to analytically approach the Donaldson–Thomas invariants. In this article, we consider the equation on compact Kahler threefolds, and study some of the analytic properties of solutions to them, using analytic methods in higher-dimensional Yang–Mills theory developed by Nakajima (1987) [14] , Nakajima (1988) [15] and Tian (2000) [20] with some additional arguments concerning an extra nonlinear term coming from the Higgs fields. We prove that a sequence of solutions to the Donaldson–Thomas equation of a unitary vector bundle over a compact Kahler threefold has a converging subsequence outside a closed subset whose real two-dimensional Hausdorff measure is finite, provided that the L 2 -norms of the Higgs fields are uniformly bounded. We also prove an n / 2 -Compactness Theorem of solutions to the equations on compact Kahler threefolds.
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a weak Compactness Theorem of the donaldson thomas instantons on compact k ahler threefolds
arXiv: Differential Geometry, 2008Co-Authors: Yuuji TanakaAbstract:In arXiv:0805.2192, we set up a gauge-theoretic equation on symplectic 6-manifolds, which is a version of the Hermitian-Einstein equation perturbed by Higgs fields, and call Donaldson-Thomas equation, to analytically approach the Donaldson-Thomas invariants. In this article, we consider the equation on compact K\"ahler threefolds, and study some of analytic properties of solutions to them, using analytic methods in higher-dimensional Yang-Mills theory developed by Nakajima and Tian with some additional arguments concerning an extra non-linear term coming from the Higgs fields. We prove that a sequence of solutions to the Donaldson-Thomas equation of a unitary vector bundle over a compact K\"ahler threefold has a converging subsequence outside a closed subset whose real 2-dimensional Hausdorff measure is finite, provided that the L^2-norms of the Higgs fields are uniformly bounded. We also prove an n/2-Compactness Theorem of solutions to the equations on compact K\"ahler threefolds.
Bernardo Cascales - One of the best experts on this subject based on the ideXlab platform.
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One-sided James' Compactness Theorem☆
Journal of Mathematical Analysis and Applications, 2017Co-Authors: Bernardo Cascales, José Orihuela, Antonio PérezAbstract:Abstract We present some extensions of classical results that involve elements of the dual of Banach spaces, such as Bishop–Phelp's Theorem and James' Compactness Theorem, but restricting ourselves to sets of functionals determined by geometrical properties. The main result, which answers a question posed by F. Delbaen, is the following: Let E be a Banach space such that ( B E ⁎ , ω ⁎ ) is convex block compact. Let A and B be bounded, closed and convex sets with distance d ( A , B ) > 0 . If every x ⁎ ∈ E ⁎ with sup ( x ⁎ , B ) inf ( x ⁎ , A ) attains its infimum on A and its supremum on B, then A and B are both weakly compact. We obtain new characterizations of weakly compact sets and reflexive spaces, as well as a result concerning a variational problem in dual Banach spaces.
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One side James' Compactness Theorem
arXiv: Functional Analysis, 2015Co-Authors: Bernardo Cascales, José Orihuela, Antonio PérezAbstract:We present some extensions of classical results that involve elements of the dual of Banach spaces, such as Bishop-Phelp's Theorem and James' Compactness Theorem, but restricting to sets of functionals determined by geometrical properties. The main result, which answers a question posed by F. Delbaen, is the following: Let $E$ be a Banach space such that $(B_{E^\ast}, \omega^\ast)$ is convex block compact. Let $A$ and $B$ be bounded, closed and convex sets with distance $d(A,B) > 0$. If every $x^\ast \in E^\ast$ with \[ \sup(x^\ast,B) < \inf(x^\ast,A) \] attains its infimum on $A$ and its supremum on $B$, then $A$ and $B$ are both weakly compact. We obtain new characterizations of weakly compact sets and reflexive spaces, as well as a result concerning a variational problem in dual Banach spaces.
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A quantitative version of James's Compactness Theorem
Proceedings of the Edinburgh Mathematical Society, 2012Co-Authors: Bernardo Cascales, Ondřej F. K. Kalenda, Jiří SpurnýAbstract:AbstractWe introduce two measures of weak non-Compactness JaE and Ja that quantify, via distances, the idea of boundary that lies behind James's Compactness Theorem. These measures tell us, for a bounded subset C of a Banach space E and for given x* ∈ E*, how far from E or C one needs to go to find x** ∈ $\overline{C}^{w^*}$ ⊂ E** with x**(x*) = sup x*(C). A quantitative version of James's Compactness Theorem is proved using JaE and Ja, and in particular it yields the following result. Let C be a closed convex bounded subset of a Banach space E and r > 0. If there is an element$x_0^{**}$in$\overline{C}^{w^*}$whose distance to C is greater than r, then there is x* ∈ E* such that each x** ∈$\overline{C}^{w^*}$at which sup x*(C) is attained has distance to E greater than ½r. We indeed establish that JaE and Ja are equivalent to other measures of weak non-Compactness studied in the literature. We also collect particular cases and examples showing when the inequalities between the different measures of weak non-Compactness can be equalities and when the inequalities are sharp.
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A quantitative version of James's Compactness Theorem
2012Co-Authors: Bernardo Cascales, Ondřej F. K. Kalenda, Jiří SpurnýAbstract:Y Abstract. We introduce two measures of weak non-Compactness JaE and Ja that quantify, via distances, the idea of boundary behind James' Compactness Theorem. These measures tell us, for a bounded subset C of a Banach space E and for given x � 2 E � , how far from E or C one needs to go to find is attained has distance to E greater than r/2. We indeed establish that JaE and Ja are equivalent to other measures of weak non-Compactness studied in the literature. We also collect particular cases and examples showing when the inequalities between the different measures of weak non-Compactness can be equalities and when the inequalities are sharp.
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A quantitative version of James' Compactness Theorem
arXiv: Functional Analysis, 2010Co-Authors: Bernardo Cascales, Ondřej F. K. Kalenda, Jiří SpurnýAbstract:We introduce two measures of weak non-Compactness $Ja_E$ and $Ja$ that quantify, via distances, the idea of boundary behind James' Compactness Theorem. These measures tell us, for a bounded subset $C$ of a Banach space $E$ and for given $x^*\in E^*$, how far from $E$ or $C$ one needs to go to find $x^{**}\in \bar{C}^{w^*}\subset E^{**}$ with $x^{**}(x^*)=\sup x^* (C)$. A quantitative version of James' Compactness Theorem is proved using $Ja_E$ and $Ja$, and in particular it yields the following result: {\it Let $C$ be a closed convex bounded subset of a Banach space $E$ and $r>0$. If there is an element $x_0^{**}$ in $\bar{C}^{w^*}$ whose distance to $C$ is greater than $r$, then there is $x^*\in E^*$ such that each $x^{**}\in\bar{C}^{w^*}$ at which $\sup x^*(C)$ is attained has distance to $E$ greater than $r/2$.} We indeed establish that $Ja_E$ and $Ja$ are equivalent to other measures of weak non-Compactness studied in the literature. We also collect particular cases and examples showing when the inequalities between the different measures of weak non-Compactness can be equalities and when the inequalities are sharp.
Jiří Spurný - One of the best experts on this subject based on the ideXlab platform.
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A quantitative version of James's Compactness Theorem
Proceedings of the Edinburgh Mathematical Society, 2012Co-Authors: Bernardo Cascales, Ondřej F. K. Kalenda, Jiří SpurnýAbstract:AbstractWe introduce two measures of weak non-Compactness JaE and Ja that quantify, via distances, the idea of boundary that lies behind James's Compactness Theorem. These measures tell us, for a bounded subset C of a Banach space E and for given x* ∈ E*, how far from E or C one needs to go to find x** ∈ $\overline{C}^{w^*}$ ⊂ E** with x**(x*) = sup x*(C). A quantitative version of James's Compactness Theorem is proved using JaE and Ja, and in particular it yields the following result. Let C be a closed convex bounded subset of a Banach space E and r > 0. If there is an element$x_0^{**}$in$\overline{C}^{w^*}$whose distance to C is greater than r, then there is x* ∈ E* such that each x** ∈$\overline{C}^{w^*}$at which sup x*(C) is attained has distance to E greater than ½r. We indeed establish that JaE and Ja are equivalent to other measures of weak non-Compactness studied in the literature. We also collect particular cases and examples showing when the inequalities between the different measures of weak non-Compactness can be equalities and when the inequalities are sharp.
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A quantitative version of James's Compactness Theorem
2012Co-Authors: Bernardo Cascales, Ondřej F. K. Kalenda, Jiří SpurnýAbstract:Y Abstract. We introduce two measures of weak non-Compactness JaE and Ja that quantify, via distances, the idea of boundary behind James' Compactness Theorem. These measures tell us, for a bounded subset C of a Banach space E and for given x � 2 E � , how far from E or C one needs to go to find is attained has distance to E greater than r/2. We indeed establish that JaE and Ja are equivalent to other measures of weak non-Compactness studied in the literature. We also collect particular cases and examples showing when the inequalities between the different measures of weak non-Compactness can be equalities and when the inequalities are sharp.
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A quantitative version of James' Compactness Theorem
arXiv: Functional Analysis, 2010Co-Authors: Bernardo Cascales, Ondřej F. K. Kalenda, Jiří SpurnýAbstract:We introduce two measures of weak non-Compactness $Ja_E$ and $Ja$ that quantify, via distances, the idea of boundary behind James' Compactness Theorem. These measures tell us, for a bounded subset $C$ of a Banach space $E$ and for given $x^*\in E^*$, how far from $E$ or $C$ one needs to go to find $x^{**}\in \bar{C}^{w^*}\subset E^{**}$ with $x^{**}(x^*)=\sup x^* (C)$. A quantitative version of James' Compactness Theorem is proved using $Ja_E$ and $Ja$, and in particular it yields the following result: {\it Let $C$ be a closed convex bounded subset of a Banach space $E$ and $r>0$. If there is an element $x_0^{**}$ in $\bar{C}^{w^*}$ whose distance to $C$ is greater than $r$, then there is $x^*\in E^*$ such that each $x^{**}\in\bar{C}^{w^*}$ at which $\sup x^*(C)$ is attained has distance to $E$ greater than $r/2$.} We indeed establish that $Ja_E$ and $Ja$ are equivalent to other measures of weak non-Compactness studied in the literature. We also collect particular cases and examples showing when the inequalities between the different measures of weak non-Compactness can be equalities and when the inequalities are sharp.
Helge Holden - One of the best experts on this subject based on the ideXlab platform.
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An improvement of the Kolmogorov-Riesz Compactness Theorem
Expositiones Mathematicae, 2019Co-Authors: Harald Hanche-olsen, Helge Holden, Eugenia MalinnikovaAbstract:The purpose of this short note is to provide a new and very short proof of a result by Sudakov, offering an important improvement of the classical result by Kolmogorov-Riesz on compact subsets of Lebesgue spaces.
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Addendum to “The Kolmogorov-Riesz Compactness Theorem” [Expo. Math. 28 (2010) 385–394]
Expositiones Mathematicae, 2010Co-Authors: Harald Hanche-olsen, Helge HoldenAbstract:Abstract We show that the Arzela–Ascoli Theorem and Kolmogorov Compactness Theorem both are consequences of a simple lemma on Compactness in metric spaces. Their relation to Helly's Theorem is discussed. The paper contains a detailed discussion on the historical background of the Kolmogorov Compactness Theorem.
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The Kolmogorov–Riesz Compactness Theorem
Expositiones Mathematicae, 2010Co-Authors: Harald Hanche-olsen, Helge HoldenAbstract:AbstractWe show that the Arzelà–Ascoli Theorem and Kolmogorov Compactness Theorem both are consequences of a simple lemma on Compactness in metric spaces. Their relation to Helly's Theorem is discussed. The paper contains a detailed discussion on the historical background of the Kolmogorov Compactness Theorem
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addendum to the kolmogorov riesz Compactness Theorem expo math 28 2010 385 394
Expositiones Mathematicae, 2010Co-Authors: Harald Hancheolsen, Helge HoldenAbstract:Abstract We show that the Arzela–Ascoli Theorem and Kolmogorov Compactness Theorem both are consequences of a simple lemma on Compactness in metric spaces. Their relation to Helly's Theorem is discussed. The paper contains a detailed discussion on the historical background of the Kolmogorov Compactness Theorem.
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the kolmogorov riesz Compactness Theorem
arXiv: Classical Analysis and ODEs, 2009Co-Authors: Harald Hancheolsen, Helge HoldenAbstract:We show that the Arzela-Ascoli Theorem and Kolmogorov Compactness Theorem both are consequences of a simple lemma on Compactness in metric spaces. Their relation to Helly's Theorem is discussed. The paper contains a detailed discussion on the historical background of the Kolmogorov Compactness Theorem.
Yasemin Soylu - One of the best experts on this subject based on the ideXlab platform.
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A Compactness Theorem in Riemannian manifolds
Journal of Geometry, 2018Co-Authors: Yasemin SoyluAbstract:In this paper, we use the m-Bakry–Emery Ricci tensor on a complete n-dimensional Riemannian manifold to obtain a Compactness Theorem including a diameter estimate. The proof is based on the Riccati comparison Theorem.
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A Myers-type Compactness Theorem by the use of Bakry–Emery Ricci tensor
Differential Geometry and Its Applications, 2017Co-Authors: Yasemin SoyluAbstract:Abstract Let ( M , g ) be a complete and connected Riemannian manifold of dimension n ≥ 2 . By using the Bakry–Emery Ricci curvature tensor on M, we prove a Myers-type Compactness Theorem which corresponds to the Compactness Theorem proved by Cheeger–Gromov–Taylor.
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a myers type Compactness Theorem by the use of bakry emery ricci tensor
Differential Geometry and Its Applications, 2017Co-Authors: Yasemin SoyluAbstract:Abstract Let ( M , g ) be a complete and connected Riemannian manifold of dimension n ≥ 2 . By using the Bakry–Emery Ricci curvature tensor on M, we prove a Myers-type Compactness Theorem which corresponds to the Compactness Theorem proved by Cheeger–Gromov–Taylor.