The Experts below are selected from a list of 26622 Experts worldwide ranked by ideXlab platform
Hiroshi Sakai - One of the best experts on this subject based on the ideXlab platform.
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The weakly compact Reflection Principle need not imply a high order of weak compactness
Archive for Mathematical Logic, 2020Co-Authors: Brent Cody, Hiroshi SakaiAbstract:The weakly compact Reflection Principle $${\mathrm{Refl}}_{{\mathrm{wc}}}(\kappa )$$ Refl wc ( κ ) states that $$\kappa $$ κ is a weakly compact cardinal and every weakly compact subset of $$\kappa $$ κ has a weakly compact proper initial segment. The weakly compact Reflection Principle at $$\kappa $$ κ implies that $$\kappa $$ κ is an $$\omega $$ ω -weakly compact cardinal. In this article we show that the weakly compact Reflection Principle does not imply that $$\kappa $$ κ is $$(\omega +1)$$ ( ω + 1 ) -weakly compact. Moreover, we show that if the weakly compact Reflection Principle holds at $$\kappa $$ κ then there is a forcing extension preserving this in which $$\kappa $$ κ is the least $$\omega $$ ω -weakly compact cardinal. Along the way we generalize the well-known result which states that if $$\kappa $$ κ is a regular cardinal then in any forcing extension by $$\kappa $$ κ -c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if $$\kappa $$ κ is a weakly compact cardinal then after forcing with a ‘typical’ Easton-support iteration of length $$\kappa $$ κ the weakly compact ideal equals the ideal generated by the ground model weakly compact ideal.
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The weakly compact Reflection Principle need not imply a high order of weak compactness
Archive for Mathematical Logic, 2019Co-Authors: Brent Cody, Hiroshi SakaiAbstract:The weakly compact Reflection Principle\({\mathrm{Refl}}_{{\mathrm{wc}}}(\kappa )\) states that \(\kappa \) is a weakly compact cardinal and every weakly compact subset of \(\kappa \) has a weakly compact proper initial segment. The weakly compact Reflection Principle at \(\kappa \) implies that \(\kappa \) is an \(\omega \)-weakly compact cardinal. In this article we show that the weakly compact Reflection Principle does not imply that \(\kappa \) is \((\omega +1)\)-weakly compact. Moreover, we show that if the weakly compact Reflection Principle holds at \(\kappa \) then there is a forcing extension preserving this in which \(\kappa \) is the least \(\omega \)-weakly compact cardinal. Along the way we generalize the well-known result which states that if \(\kappa \) is a regular cardinal then in any forcing extension by \(\kappa \)-c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if \(\kappa \) is a weakly compact cardinal then after forcing with a ‘typical’ Easton-support iteration of length \(\kappa \) the weakly compact ideal equals the ideal generated by the ground model weakly compact ideal.
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Martin's Maximum and the Diagonal Reflection Principle
arXiv: Logic, 2019Co-Authors: Sean Cox, Hiroshi SakaiAbstract:We prove that Martin's Maximum does not imply the Diagonal Reflection Principle for stationary subsets of $[ \omega_2 ]^\omega$.
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How much weak compactness does the weakly compact Reflection Principle imply
arXiv: Logic, 2017Co-Authors: Brent Cody, Hiroshi SakaiAbstract:The weakly compact Reflection Principle $\text{Refl}_{\text{wc}}(\kappa)$ states that $\kappa$ is a weakly compact cardinal and every weakly compact subset of $\kappa$ has a weakly compact proper initial segment. The weakly compact Reflection Principle at $\kappa$ implies that $\kappa$ is an $\omega$-weakly compact cardinal. In this article we show that the weakly compact Reflection Principle does not imply that $\kappa$ is $(\omega+1)$-weakly compact. Moreover, we show that if the weakly compact Reflection Principle holds at $\kappa$ then there is a forcing extension preserving this in which $\kappa$ is the least $\omega$-weakly compact cardinal. Along the way we generalize the well-known result which states that if $\kappa$ is a regular cardinal then in any forcing extension by $\kappa$-c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if $\kappa$ is a weakly compact cardinal then after forcing with a `typical' Easton-support iteration of length $\kappa$ the weakly compact ideal equals the ideal generated by the ground model weakly compact ideal.
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the weakly compact Reflection Principle need not imply a high order of weak compactness
arXiv e-prints, 2017Co-Authors: Brent Cody, Hiroshi SakaiAbstract:The weakly compact Reflection Principle $\text{Refl}_{\text{wc}}(\kappa)$ states that $\kappa$ is a weakly compact cardinal and every weakly compact subset of $\kappa$ has a weakly compact proper initial segment. The weakly compact Reflection Principle at $\kappa$ implies that $\kappa$ is an $\omega$-weakly compact cardinal. In this article we show that the weakly compact Reflection Principle does not imply that $\kappa$ is $(\omega+1)$-weakly compact. Moreover, we show that if the weakly compact Reflection Principle holds at $\kappa$ then there is a forcing extension preserving this in which $\kappa$ is the least $\omega$-weakly compact cardinal. Along the way we generalize the well-known result which states that if $\kappa$ is a regular cardinal then in any forcing extension by $\kappa$-c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if $\kappa$ is a weakly compact cardinal then after forcing with a `typical' Easton-support iteration of length $\kappa$ the weakly compact ideal equals the ideal generated by the ground model weakly compact ideal.
Eric Toubiana - One of the best experts on this subject based on the ideXlab platform.
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Classical Schwarz Reflection Principle for Jenkins–Serrin type minimal surfaces
Annals of Global Analysis and Geometry, 2020Co-Authors: Ricardo Sa Earp, Eric ToubianaAbstract:We give a proof of the classical Schwarz Reflection Principle for Jenkins–Serrin type minimal surfaces in the homogeneous three manifolds $$\mathbb E(\kappa ,\tau )$$ for $$\kappa \leqslant 0$$ and $$\tau \geqslant 0$$. In our previous paper, we proved a Reflection Principle in Riemannian manifolds. The statements and techniques in the two papers are distinct.
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Classical Schwarz Reflection Principle for Jenkins-Serrin Type Minimal Surfaces
arXiv: Differential Geometry, 2018Co-Authors: Ricardo Sa Earp, Eric ToubianaAbstract:We give a proof of the classical Schwarz Reflection Principle for Jenkins-Serrin type minimal surfaces in the homogeneous three manifolds $E(\kappa, \tau)$ for $\kappa \leqslant 0$ and $\tau \geqslant 0$. In our previous paper we proved a Reflection Principle in Riemannian manifolds. The statements and techniques in the two papers are distinct.
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CLASSICAL SCHWARZ Reflection Principle FOR JENKINS-SERRIN TYPE MINIMAL SURFACES
2018Co-Authors: Ricardo Sa Earp, Eric ToubianaAbstract:We give a proof of the classical Schwarz Reflection Principle for Jenkins-Serrin type minimal surfaces in the homoge- neous three manifolds $E(\kappa, \tau)$ for $\kappa < 0$ and $\tau \geqslant 0$. In our previous paper we proved a Reflection Principle in Riemannian manifolds. The statements and techniques in the two papers are distinct.
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A Reflection Principle for minimal surfaces in smooth three manifolds
arXiv: Differential Geometry, 2017Co-Authors: Ricardo Sa Earp, Eric ToubianaAbstract:We prove a Reflection Principle for minimal surfaces in smooth (non necessarily analytic) three manifolds and we give an explicit application when the ambient space is just a smooth manifold.
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Variants on Alexandrov Reflection Principle and other applications of maximum Principle
2001Co-Authors: Ricardo Sa Earp, Eric ToubianaAbstract:In this article we discuss several derivations based on Alexandrov Reflection Principle and Maximum Principle. Particularly, we give some applications for surfaces of constant mean curvature in Euclidean and hyperbolic space. We also discuss the Perron Process for minimal vertical graphs in hyperbolic space. We infer some new related results in hyperbolic space. Namely, we infer symmetry and half-space results for properly embedded mean curvature one surfaces. Furthermore, we carry out a Molzon-Serrin type theorem for a classical overdetermined elliptic equation in hyperbolic space.
Ricardo Sa Earp - One of the best experts on this subject based on the ideXlab platform.
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Classical Schwarz Reflection Principle for Jenkins–Serrin type minimal surfaces
Annals of Global Analysis and Geometry, 2020Co-Authors: Ricardo Sa Earp, Eric ToubianaAbstract:We give a proof of the classical Schwarz Reflection Principle for Jenkins–Serrin type minimal surfaces in the homogeneous three manifolds $$\mathbb E(\kappa ,\tau )$$ for $$\kappa \leqslant 0$$ and $$\tau \geqslant 0$$. In our previous paper, we proved a Reflection Principle in Riemannian manifolds. The statements and techniques in the two papers are distinct.
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Classical Schwarz Reflection Principle for Jenkins-Serrin Type Minimal Surfaces
arXiv: Differential Geometry, 2018Co-Authors: Ricardo Sa Earp, Eric ToubianaAbstract:We give a proof of the classical Schwarz Reflection Principle for Jenkins-Serrin type minimal surfaces in the homogeneous three manifolds $E(\kappa, \tau)$ for $\kappa \leqslant 0$ and $\tau \geqslant 0$. In our previous paper we proved a Reflection Principle in Riemannian manifolds. The statements and techniques in the two papers are distinct.
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CLASSICAL SCHWARZ Reflection Principle FOR JENKINS-SERRIN TYPE MINIMAL SURFACES
2018Co-Authors: Ricardo Sa Earp, Eric ToubianaAbstract:We give a proof of the classical Schwarz Reflection Principle for Jenkins-Serrin type minimal surfaces in the homoge- neous three manifolds $E(\kappa, \tau)$ for $\kappa < 0$ and $\tau \geqslant 0$. In our previous paper we proved a Reflection Principle in Riemannian manifolds. The statements and techniques in the two papers are distinct.
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A Reflection Principle for minimal surfaces in smooth three manifolds
arXiv: Differential Geometry, 2017Co-Authors: Ricardo Sa Earp, Eric ToubianaAbstract:We prove a Reflection Principle for minimal surfaces in smooth (non necessarily analytic) three manifolds and we give an explicit application when the ambient space is just a smooth manifold.
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Variants on Alexandrov Reflection Principle and other applications of maximum Principle
2001Co-Authors: Ricardo Sa Earp, Eric ToubianaAbstract:In this article we discuss several derivations based on Alexandrov Reflection Principle and Maximum Principle. Particularly, we give some applications for surfaces of constant mean curvature in Euclidean and hyperbolic space. We also discuss the Perron Process for minimal vertical graphs in hyperbolic space. We infer some new related results in hyperbolic space. Namely, we infer symmetry and half-space results for properly embedded mean curvature one surfaces. Furthermore, we carry out a Molzon-Serrin type theorem for a classical overdetermined elliptic equation in hyperbolic space.
Brent Cody - One of the best experts on this subject based on the ideXlab platform.
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The weakly compact Reflection Principle need not imply a high order of weak compactness
Archive for Mathematical Logic, 2020Co-Authors: Brent Cody, Hiroshi SakaiAbstract:The weakly compact Reflection Principle $${\mathrm{Refl}}_{{\mathrm{wc}}}(\kappa )$$ Refl wc ( κ ) states that $$\kappa $$ κ is a weakly compact cardinal and every weakly compact subset of $$\kappa $$ κ has a weakly compact proper initial segment. The weakly compact Reflection Principle at $$\kappa $$ κ implies that $$\kappa $$ κ is an $$\omega $$ ω -weakly compact cardinal. In this article we show that the weakly compact Reflection Principle does not imply that $$\kappa $$ κ is $$(\omega +1)$$ ( ω + 1 ) -weakly compact. Moreover, we show that if the weakly compact Reflection Principle holds at $$\kappa $$ κ then there is a forcing extension preserving this in which $$\kappa $$ κ is the least $$\omega $$ ω -weakly compact cardinal. Along the way we generalize the well-known result which states that if $$\kappa $$ κ is a regular cardinal then in any forcing extension by $$\kappa $$ κ -c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if $$\kappa $$ κ is a weakly compact cardinal then after forcing with a ‘typical’ Easton-support iteration of length $$\kappa $$ κ the weakly compact ideal equals the ideal generated by the ground model weakly compact ideal.
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The weakly compact Reflection Principle need not imply a high order of weak compactness
Archive for Mathematical Logic, 2019Co-Authors: Brent Cody, Hiroshi SakaiAbstract:The weakly compact Reflection Principle\({\mathrm{Refl}}_{{\mathrm{wc}}}(\kappa )\) states that \(\kappa \) is a weakly compact cardinal and every weakly compact subset of \(\kappa \) has a weakly compact proper initial segment. The weakly compact Reflection Principle at \(\kappa \) implies that \(\kappa \) is an \(\omega \)-weakly compact cardinal. In this article we show that the weakly compact Reflection Principle does not imply that \(\kappa \) is \((\omega +1)\)-weakly compact. Moreover, we show that if the weakly compact Reflection Principle holds at \(\kappa \) then there is a forcing extension preserving this in which \(\kappa \) is the least \(\omega \)-weakly compact cardinal. Along the way we generalize the well-known result which states that if \(\kappa \) is a regular cardinal then in any forcing extension by \(\kappa \)-c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if \(\kappa \) is a weakly compact cardinal then after forcing with a ‘typical’ Easton-support iteration of length \(\kappa \) the weakly compact ideal equals the ideal generated by the ground model weakly compact ideal.
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How much weak compactness does the weakly compact Reflection Principle imply
arXiv: Logic, 2017Co-Authors: Brent Cody, Hiroshi SakaiAbstract:The weakly compact Reflection Principle $\text{Refl}_{\text{wc}}(\kappa)$ states that $\kappa$ is a weakly compact cardinal and every weakly compact subset of $\kappa$ has a weakly compact proper initial segment. The weakly compact Reflection Principle at $\kappa$ implies that $\kappa$ is an $\omega$-weakly compact cardinal. In this article we show that the weakly compact Reflection Principle does not imply that $\kappa$ is $(\omega+1)$-weakly compact. Moreover, we show that if the weakly compact Reflection Principle holds at $\kappa$ then there is a forcing extension preserving this in which $\kappa$ is the least $\omega$-weakly compact cardinal. Along the way we generalize the well-known result which states that if $\kappa$ is a regular cardinal then in any forcing extension by $\kappa$-c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if $\kappa$ is a weakly compact cardinal then after forcing with a `typical' Easton-support iteration of length $\kappa$ the weakly compact ideal equals the ideal generated by the ground model weakly compact ideal.
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the weakly compact Reflection Principle need not imply a high order of weak compactness
arXiv e-prints, 2017Co-Authors: Brent Cody, Hiroshi SakaiAbstract:The weakly compact Reflection Principle $\text{Refl}_{\text{wc}}(\kappa)$ states that $\kappa$ is a weakly compact cardinal and every weakly compact subset of $\kappa$ has a weakly compact proper initial segment. The weakly compact Reflection Principle at $\kappa$ implies that $\kappa$ is an $\omega$-weakly compact cardinal. In this article we show that the weakly compact Reflection Principle does not imply that $\kappa$ is $(\omega+1)$-weakly compact. Moreover, we show that if the weakly compact Reflection Principle holds at $\kappa$ then there is a forcing extension preserving this in which $\kappa$ is the least $\omega$-weakly compact cardinal. Along the way we generalize the well-known result which states that if $\kappa$ is a regular cardinal then in any forcing extension by $\kappa$-c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if $\kappa$ is a weakly compact cardinal then after forcing with a `typical' Easton-support iteration of length $\kappa$ the weakly compact ideal equals the ideal generated by the ground model weakly compact ideal.
Sergey Nadtochiy - One of the best experts on this subject based on the ideXlab platform.
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Weak Reflection Principle for L\'evy processes
The Annals of Applied Probability, 2015Co-Authors: Erhan Bayraktar, Sergey NadtochiyAbstract:In this paper, we develop a new mathematical technique which allows us to express the joint distribution of a Markov process and its running maximum (or minimum) through the marginal distribution of the process itself. This technique is an extension of the classical Reflection Principle for Brownian motion, and it is obtained by weakening the assumptions of symmetry required for the classical Reflection Principle to work. We call this method a weak Reflection Principle and show that it provides solutions to many problems for which the classical Reflection Principle is typically used. In addition, unlike the classical Reflection Principle, the new method works for a much larger class of stochastic processes which, in particular, do not possess any strong symmetries. Here, we review the existing results which establish the weak Reflection Principle for a large class of time-homogeneous diffusions on a real line and then proceed to extend this method to the L\'{e}vy processes with one-sided jumps (subject to some admissibility conditions). Finally, we demonstrate the applications of the weak Reflection Principle in financial mathematics, computational methods and inverse problems.
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Weak Reflection Principle for spectrally negative L´ evy processes
2013Co-Authors: Erhan Bayraktar, Sergey NadtochiyAbstract:In this paper, we develop a new mathematical technique which can be used to express the joint distribution of a Markov process and its running maximum (or minimum) through the distribution of the process itself. This technique is an extension of the classical Reflection Principle for Brownian motion, and it is obtained by weakening the assumptions of symmetry required for the standard Reflection Principle to work. We call this method a weak Reflection Principle and show that it provides solutions to many problems for which the classical Reflection Principle is typically used. In addition, unlike the standard Reflection Principle, the new method works for a much larger class of stochastic processes which, in particular, do not possess any strong symmetries. Here, we review the existing results which establish the weak Reflection Principle for a large class of time-homogeneous diffusions on a real line and, then, proceed to develop this method for all L´ evy processes with one-sided jumps (subject to some admissibility conditions). Finally, we demonstrate the applications of the weak Reflection Principle in Financial Mathematics, Computational Methods, and Inverse Problems.
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weak Reflection Principle for spectrally negative l evy processes
2013Co-Authors: Erhan Bayraktar, Sergey NadtochiyAbstract:In this paper, we develop a new mathematical technique which can be used to express the joint distribution of a Markov process and its running maximum (or minimum) through the distribution of the process itself. This technique is an extension of the classical Reflection Principle for Brownian motion, and it is obtained by weakening the assumptions of symmetry required for the standard Reflection Principle to work. We call this method a weak Reflection Principle and show that it provides solutions to many problems for which the classical Reflection Principle is typically used. In addition, unlike the standard Reflection Principle, the new method works for a much larger class of stochastic processes which, in particular, do not possess any strong symmetries. Here, we review the existing results which establish the weak Reflection Principle for a large class of time-homogeneous diffusions on a real line and, then, proceed to develop this method for all L´ evy processes with one-sided jumps (subject to some admissibility conditions). Finally, we demonstrate the applications of the weak Reflection Principle in Financial Mathematics, Computational Methods, and Inverse Problems.
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Weak Reflection Principle for Levy Processes
SSRN Electronic Journal, 2013Co-Authors: Erhan Bayraktar, Sergey NadtochiyAbstract:In this paper, we develop a new mathematical technique which allows us to express the joint distribution of a Markov process and its running maximum (or minimum) through the marginal distribution of the process itself. This technique is an extension of the classical Reflection Principle for Brownian motion, and it is obtained by weakening the assumptions of symmetry required for the classical Reflection Principle to work. We call this method a weak Reflection Principle and show that it provides solutions to many problems for which the classical Reflection Principle is typically used. In addition, unlike the classical Reflection Principle, the new method works for a much larger class of stochastic processes which, in particular, do not possess any strong symmetries. Here, we review the existing results which establish the weak Reflection Principle for a large class of time-homogeneous diffusions on a real line and then proceed to extend this method to the L\'{e}vy processes with one-sided jumps (subject to some admissibility conditions). Finally, we demonstrate the applications of the weak Reflection Principle in financial mathematics, computational methods and inverse problems.
