The Experts below are selected from a list of 222 Experts worldwide ranked by ideXlab platform
Ivan Netuka - One of the best experts on this subject based on the ideXlab platform.
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semipolar sets and intrinsic hausdorff measure
Potential Analysis, 2019Co-Authors: Wolfhard Hansen, Ivan NetukaAbstract:Given a “Green function” G on a locally compact space X with Countable Base, a Borel set A in X is called G-semipolar, if there is no measure ν ≠ 0 supported by A such that \(G\nu :=\int G(\cdot ,y)\,d\nu (y)\) is a continuous real function on X. Introducing an intrinsic Hausdorff measuremG using G-balls B(x, ρ) := {y ∈ X : G(x, y) > 1/ρ}, it is shown that every set A in X with \(m_{G}(A)<\infty \) is contained in a G-semipolar Borel set. This is of interest, since G-semipolar sets are semipolar in the potential-theoretic sense (Countable unions of totally thin sets, hit by a corresponding process at most countably many times), if G is a genuine Green function. The result has immediate consequences for classical potential theory, Riesz potentials and the heat equation (where it solves an open problem). More generally, it is applied to metric measure spaces (X, d, μ), where a continuous heat kernel with upper and lower bounds of the form t−α/βΦj(d(x,y)t− 1/β), j = 1, 2, is given. Then the intrinsic Hausdorff measure on X is equivalent to an ordinary Hausdorff measure mα−β. For the corresponding space-time structure on X × ℝ, the intrinsic Hausdorff measure turns out to be equivalent to an anisotropic Hausdorff measure mα,β.
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Semipolar Sets and Intrinsic Hausdorff Measure
Potential Analysis, 2018Co-Authors: Wolfhard Hansen, Ivan NetukaAbstract:Given a “Green function” G on a locally compact space X with Countable Base, a Borel set A in X is called G-semipolar, if there is no measure ν ≠ 0 supported by A such that \(G\nu :=\int G(\cdot ,y)\,d\nu (y)\) is a continuous real function on X. Introducing an intrinsic Hausdorff measuremG using G-balls B(x, ρ) := {y ∈ X : G(x, y) > 1/ρ}, it is shown that every set A in X with \(m_{G}(A)
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Semipolar sets and intrinsic Hausdorff measure
arXiv: Analysis of PDEs, 2017Co-Authors: Wolfhard Hansen, Ivan NetukaAbstract:Given a "Green function" $G$ on a locally compact space $X$ with Countable Base, a Borel set $A$ in $X$ is called $G$-semipolar, if there is no measure $\nu\ne 0$ supported by $A$ such that $G\nu:=\int G(\cdot,y)\,d\nu(y)$ is a continuous real function on $X$. Introducing an intrinsic Hausdorff measure $m_G$ using $G$-balls $B(x,\rho):=\{y\in X\colon G(x,y)>1/\rho\}$, it is shown that every set $A$ in $X$ with $m_G(A)
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hunt s hypothesis h and triangle property of the green function
Expositiones Mathematicae, 2016Co-Authors: Wolfhard Hansen, Ivan NetukaAbstract:Abstract Let X be a locally compact abelian group with Countable Base and let W be a convex cone of positive numerical functions on X which is invariant under the group action and such that ( X , W ) is a balayage space or (equivalently, if 1 ∈ W ) such that W is the set of excessive functions of a Hunt process on X , W separates points, every function in W is the supremum of its continuous minorants in W , and there exist strictly positive continuous u , v ∈ W such that u / v → 0 at infinity. Assuming that there is a Green function G > 0 for X which locally satisfies the triangle inequality G ( x , z ) ∧ G ( y , z ) ≤ C G ( x , y ) (true for many Levy processes), it is shown that Hunt’s hypothesis (H) holds, that is, every semipolar set is polar.
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hunt s hypothesis h and the triangle property of the green function
arXiv: Analysis of PDEs, 2014Co-Authors: Wolfhard Hansen, Ivan NetukaAbstract:Let $X$ be a locally compact abelian group with Countable Base and let $\mathcal W$ be a convex cone of positive numerical functions on $X$ which is invariant under the group action and such that $(X,\mathcal W)$ is a balayage space or (equivalently, if $1\in \mathcal W$) such that $\mathcal W$ is the set of excessive functions of a Hunt process on $X$, $\mathcal W$ separates points, every function in $\mathcal W$ is the supremum of its continuous minorants in $\mathcal W$, and there exist strictly positive continuous $u,v\in \mathcal W$ such that $u/v\to 0$ at infinity. Assuming that there is a Green function $G>0$ for $X$ which locally satisfies the triangle inequality $G(x,z)\wedge G(y,z)\le C G(x,y)$ (true for many L\'evy processes), it is shown that Hunt's hypothesis (H) holds, that is, every semipolar set is polar.
Wolfhard Hansen - One of the best experts on this subject based on the ideXlab platform.
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semipolar sets and intrinsic hausdorff measure
Potential Analysis, 2019Co-Authors: Wolfhard Hansen, Ivan NetukaAbstract:Given a “Green function” G on a locally compact space X with Countable Base, a Borel set A in X is called G-semipolar, if there is no measure ν ≠ 0 supported by A such that \(G\nu :=\int G(\cdot ,y)\,d\nu (y)\) is a continuous real function on X. Introducing an intrinsic Hausdorff measuremG using G-balls B(x, ρ) := {y ∈ X : G(x, y) > 1/ρ}, it is shown that every set A in X with \(m_{G}(A)<\infty \) is contained in a G-semipolar Borel set. This is of interest, since G-semipolar sets are semipolar in the potential-theoretic sense (Countable unions of totally thin sets, hit by a corresponding process at most countably many times), if G is a genuine Green function. The result has immediate consequences for classical potential theory, Riesz potentials and the heat equation (where it solves an open problem). More generally, it is applied to metric measure spaces (X, d, μ), where a continuous heat kernel with upper and lower bounds of the form t−α/βΦj(d(x,y)t− 1/β), j = 1, 2, is given. Then the intrinsic Hausdorff measure on X is equivalent to an ordinary Hausdorff measure mα−β. For the corresponding space-time structure on X × ℝ, the intrinsic Hausdorff measure turns out to be equivalent to an anisotropic Hausdorff measure mα,β.
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Semipolar Sets and Intrinsic Hausdorff Measure
Potential Analysis, 2018Co-Authors: Wolfhard Hansen, Ivan NetukaAbstract:Given a “Green function” G on a locally compact space X with Countable Base, a Borel set A in X is called G-semipolar, if there is no measure ν ≠ 0 supported by A such that \(G\nu :=\int G(\cdot ,y)\,d\nu (y)\) is a continuous real function on X. Introducing an intrinsic Hausdorff measuremG using G-balls B(x, ρ) := {y ∈ X : G(x, y) > 1/ρ}, it is shown that every set A in X with \(m_{G}(A)
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Semipolar sets and intrinsic Hausdorff measure
arXiv: Analysis of PDEs, 2017Co-Authors: Wolfhard Hansen, Ivan NetukaAbstract:Given a "Green function" $G$ on a locally compact space $X$ with Countable Base, a Borel set $A$ in $X$ is called $G$-semipolar, if there is no measure $\nu\ne 0$ supported by $A$ such that $G\nu:=\int G(\cdot,y)\,d\nu(y)$ is a continuous real function on $X$. Introducing an intrinsic Hausdorff measure $m_G$ using $G$-balls $B(x,\rho):=\{y\in X\colon G(x,y)>1/\rho\}$, it is shown that every set $A$ in $X$ with $m_G(A)
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Hunt’s hypothesis (H) and triangle property of the Green function
2016Co-Authors: Wolfhard HansenAbstract:Netuka Let X be a locally compact abelian group with Countable Base and let W be a convex cone of positive numerical functions on X which is invariant under the group action and such that (X,W) is a balayage space or (equivalently, if 1 ∈ W) such that W is the set of excessive functions of a Hunt process on X, W separates points, every function in W is the supremum of its continuous minorants in W, and there exist strictly positive continuous u, v ∈ W such that u/v → 0 at infinity. Assuming that there is a Green function G> 0 for X which locally satisfies the triangle inequality G(x, z)∧G(y, z) ≤ CG(x, y) (true for many Lévy pro-cesses), it is shown that Hunt’s hypothesis (H) holds, that is, every semipola
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hunt s hypothesis h and triangle property of the green function
Expositiones Mathematicae, 2016Co-Authors: Wolfhard Hansen, Ivan NetukaAbstract:Abstract Let X be a locally compact abelian group with Countable Base and let W be a convex cone of positive numerical functions on X which is invariant under the group action and such that ( X , W ) is a balayage space or (equivalently, if 1 ∈ W ) such that W is the set of excessive functions of a Hunt process on X , W separates points, every function in W is the supremum of its continuous minorants in W , and there exist strictly positive continuous u , v ∈ W such that u / v → 0 at infinity. Assuming that there is a Green function G > 0 for X which locally satisfies the triangle inequality G ( x , z ) ∧ G ( y , z ) ≤ C G ( x , y ) (true for many Levy processes), it is shown that Hunt’s hypothesis (H) holds, that is, every semipolar set is polar.
Paul J. Szeptycki - One of the best experts on this subject based on the ideXlab platform.
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Sharp Bases and weakly uniform Bases versus point-Countable Bases
Topology and its Applications, 2000Co-Authors: Alexander Arhangel’skii, Winfried Just, E.a. Rezniczenko, Paul J. SzeptyckiAbstract:Abstract A Base B for a topological space X is said to be sharp if for every x∈X and every sequence (U n ) n∈ω of pairwise distinct elements of B with x∈U n for all n the set {⋂ i U i : n∈ω} forms a Base at x . Sharp Bases of T 0 -spaces are weakly uniform. We investigate which spaces with sharp Bases or weakly uniform Bases have point-Countable Bases or are metrizable. In particular, Davis, Reed, and Wage had constructed in a 1976 paper a consistent example of a Moore space with weakly uniform Base, but without a point-Countable Base. They asked whether such an example can be constructed in ZFC. We partly answer this question by showing that under CH, every first-Countable space with a weakly uniform Base and at most ℵ ω isolated points has a point-Countable Base.
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Strongly almost disjoint sets and weakly uniform Bases
arXiv: Logic, 1998Co-Authors: Zoltan Balogh, S W Davis, Saharon Shelah, Winfried Just, Paul J. SzeptyckiAbstract:A combinatorial principle CECA is formulated and its equivalence with GCH+ certain weakenings of Box_lambda for singular lambda is proved. CECA is used to show that certain ``almost point- < tau'' families can be refined to point- < tau families by removing a small set from each member of the family. This theorem in turn is used to show the consistency of ``every first Countable T_1-space with a weakly uniform Base has a point-Countable Base.''
Alexander Arhangel’skii - One of the best experts on this subject based on the ideXlab platform.
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Structure theorems for finite unions of subspaces of special kind
Topology and its Applications, 2016Co-Authors: Alexander Arhangel’skiiAbstract:Abstract We study the internal structure of topological spaces X which can be represented as the union of a finite collection of subspaces belonging to some nice class of spaces. Several closely related structure theorems are established. In particular, they concern the finite unions of subspaces with the weight ≤τ, the finite unions of subspaces with a point-Countable Base, and the finite unions of metrizable subspaces. As a corollary, we extend to finite unions the classical Mischenko's Theorem on metrizability of compacta with a point-Countable Base [11] (see Theorem 11 ). A few other applications of the structure theorems are given, in particular, to homogeneous spaces ( Corollary 5 , Corollary 10 ).
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D-spaces and finite unions
Proceedings of the American Mathematical Society, 2004Co-Authors: Alexander Arhangel’skiiAbstract:This article is a continuation of a recent paper by the author and R. Z. Buzyakova. New results are obtained in the direction of the next natural question: how complex can a space be that is the union of two (of a finite family) nice subspaces? Our approach is Based on the notion of a D-space introduced by E. van Douwen and on a generalization of this notion, the notion of aD-space. It is proved that if a space X is the union of a finite family of subparacompact subspaces, then X is an aD-space. Under (CH), it follows that if a separable normal T 1 -space X is the union of a finite number of subparacompact subspaces, then X is Lindelof. It is also established that if a regular space X is the union of a finite family of subspaces with a point-Countable Base, then X is a D-space. Finally, a certain structure theorem for unions of finite families of spaces with a point-Countable Base is established, and numerous corollaries are derived from it. Also, many new open problems are formulated.
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Sharp Bases and weakly uniform Bases versus point-Countable Bases
Topology and its Applications, 2000Co-Authors: Alexander Arhangel’skii, Winfried Just, E.a. Rezniczenko, Paul J. SzeptyckiAbstract:Abstract A Base B for a topological space X is said to be sharp if for every x∈X and every sequence (U n ) n∈ω of pairwise distinct elements of B with x∈U n for all n the set {⋂ i U i : n∈ω} forms a Base at x . Sharp Bases of T 0 -spaces are weakly uniform. We investigate which spaces with sharp Bases or weakly uniform Bases have point-Countable Bases or are metrizable. In particular, Davis, Reed, and Wage had constructed in a 1976 paper a consistent example of a Moore space with weakly uniform Base, but without a point-Countable Base. They asked whether such an example can be constructed in ZFC. We partly answer this question by showing that under CH, every first-Countable space with a weakly uniform Base and at most ℵ ω isolated points has a point-Countable Base.
Luminiţa Viţă - One of the best experts on this subject based on the ideXlab platform.
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strong and uniform continuity the uniform space case
Lms Journal of Computation and Mathematics, 2003Co-Authors: Douglas S. Bridges, Luminiţa ViţăAbstract:It is proved, within the constructive theory of apartness spaces, that a strongly continuous mapping from a totally bounded uniform space with a Countable Base of entourages to a uniform space is uniformly continuous. This lifts a result of Ishihara and Schuster from metric to uniform apartness spaces. The paper is part of a systematic development of computable topology using apartness as the fundamental notion.
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Strong and Uniform Continuity – the Uniform Space Case
LMS Journal of Computation and Mathematics, 2003Co-Authors: Douglas S. Bridges, Luminiţa ViţăAbstract:It is proved, within the constructive theory of apartness spaces, that a strongly continuous mapping from a totally bounded uniform space with a Countable Base of entourages to a uniform space is uniformly continuous. This lifts a result of Ishihara and Schuster from metric to uniform apartness spaces. The paper is part of a systematic development of computable topology using apartness as the fundamental notion.
