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Xingru Zhang - One of the best experts on this subject based on the ideXlab platform.
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On Culler-Shalen Seminorms and Dehn filling
arXiv: Geometric Topology, 1998Co-Authors: Steven Boyer, Xingru ZhangAbstract:Consider the exterior M of a hyperbolic knot lying in a closed, connected, orientable 3-manifold. Culler and Shalen defined norm on H_1(dM;R) using the SL(2,C) character variety of pi_1(M). The Culler-Shalen norm encodes many topological properties of M; in particular it provides information about Dehn fillings of M. Their construction may be applied to arbitrary curves in the SL_2(C)-character variety of a connected, compact, orientable, irreducible 3-manifold whose boundary is a torus, though in this generality one can only guarantee that it will define a Seminorm. The first half of this paper is devoted to the development of the general theory of Culler-Shalen Seminorms defined for curves of PSL_2(C)-characters. By working over PSL_2(C) we obtain a theory that is more generally applicable than its SL_2(C) counterpart, while being only mildly more difficult to set up. In the second half of this paper we apply the theory of Culler-Shalen Seminorms to study the Dehn filling operation. In particular we examine the relationship between fillings which yield manifolds having a positive dimensional PSL_2(C)-character variety with those that yield manifolds having a finite or cyclic fundamental group. In one interesting application of this work we show that manifolds resulting from a nonintegral surgery on a knot in the 3-sphere tend to have a zero-dimensional PSL_2(C)-character variety. As a consequence we obtain an infinite family of closed, orientable, hyperbolic Haken manifolds which have zero-dimensional PSL_2(C)-character varieties.
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On Culler-Shalen Seminorms and Dehn filling
The Annals of Mathematics, 1998Co-Authors: Steven Boyer, Xingru ZhangAbstract:If F is a finitely generated discrete group and G a complex algebraic Lie group, the G-character variety of r is an affine algebraic variety whose points correspond to characters of representations of r with values in G. Marc Culler and Peter Shalen developed the theory of SL2(C)-character varieties of finitely generated groups and applied their results to study the topology of 3-dimensional manifolds in the papers [6], [7], [8]. Consider the exterior M of a hyperbolic knot lying in a closed, connected, orientable 3-manifold. The Mostow rigidity theorem implies that the holonomy representation p: iri(M) Isom+(H3) _ PSL2(C) is unique up to conjugation and taking complex conjugates. The orientability of M can be used to show 1 lifts to a representation p E Hom(-ri (M), SL2(C)) whose character determines an essentially unique point of Xp of X(iri(M)), the SL2(C)-character variety of irl(M). Culler and Shalen [8] proved that the component X0 of X(X1ri(M)) which contains X' is a curve. One of their major contributions was to show how X0 determines a norm on H1(WM; R) which encodes many topological properties of M. In particular it provides information on the Dehn fillings of M. Their construction may be applied to arbitrary curves in the SL2(C)-character variety of a connected, compact, orientable, irreducible 3manifold whose boundary is a torus, though in this generality one can only guarantee that it will define a Seminorm. The first half of this paper is devoted to the development of the general theory of -Culler-Shalen Seminorms defined for curves of PSL2(C)-characters. By working over PSL2(C) we obtain a theory that is more generally applicable than its SL2(C) counterpart, while being only mildly more difficult to set up. In the second half of this paper we apply the theory of Culler-Shalen Seminorms to study the Dehn filling operation. In particular we examine the relationship between fillings which yield manifolds having a positive dimen-
Steven Boyer - One of the best experts on this subject based on the ideXlab platform.
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On Culler-Shalen Seminorms and Dehn filling
arXiv: Geometric Topology, 1998Co-Authors: Steven Boyer, Xingru ZhangAbstract:Consider the exterior M of a hyperbolic knot lying in a closed, connected, orientable 3-manifold. Culler and Shalen defined norm on H_1(dM;R) using the SL(2,C) character variety of pi_1(M). The Culler-Shalen norm encodes many topological properties of M; in particular it provides information about Dehn fillings of M. Their construction may be applied to arbitrary curves in the SL_2(C)-character variety of a connected, compact, orientable, irreducible 3-manifold whose boundary is a torus, though in this generality one can only guarantee that it will define a Seminorm. The first half of this paper is devoted to the development of the general theory of Culler-Shalen Seminorms defined for curves of PSL_2(C)-characters. By working over PSL_2(C) we obtain a theory that is more generally applicable than its SL_2(C) counterpart, while being only mildly more difficult to set up. In the second half of this paper we apply the theory of Culler-Shalen Seminorms to study the Dehn filling operation. In particular we examine the relationship between fillings which yield manifolds having a positive dimensional PSL_2(C)-character variety with those that yield manifolds having a finite or cyclic fundamental group. In one interesting application of this work we show that manifolds resulting from a nonintegral surgery on a knot in the 3-sphere tend to have a zero-dimensional PSL_2(C)-character variety. As a consequence we obtain an infinite family of closed, orientable, hyperbolic Haken manifolds which have zero-dimensional PSL_2(C)-character varieties.
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On Culler-Shalen Seminorms and Dehn filling
The Annals of Mathematics, 1998Co-Authors: Steven Boyer, Xingru ZhangAbstract:If F is a finitely generated discrete group and G a complex algebraic Lie group, the G-character variety of r is an affine algebraic variety whose points correspond to characters of representations of r with values in G. Marc Culler and Peter Shalen developed the theory of SL2(C)-character varieties of finitely generated groups and applied their results to study the topology of 3-dimensional manifolds in the papers [6], [7], [8]. Consider the exterior M of a hyperbolic knot lying in a closed, connected, orientable 3-manifold. The Mostow rigidity theorem implies that the holonomy representation p: iri(M) Isom+(H3) _ PSL2(C) is unique up to conjugation and taking complex conjugates. The orientability of M can be used to show 1 lifts to a representation p E Hom(-ri (M), SL2(C)) whose character determines an essentially unique point of Xp of X(iri(M)), the SL2(C)-character variety of irl(M). Culler and Shalen [8] proved that the component X0 of X(X1ri(M)) which contains X' is a curve. One of their major contributions was to show how X0 determines a norm on H1(WM; R) which encodes many topological properties of M. In particular it provides information on the Dehn fillings of M. Their construction may be applied to arbitrary curves in the SL2(C)-character variety of a connected, compact, orientable, irreducible 3manifold whose boundary is a torus, though in this generality one can only guarantee that it will define a Seminorm. The first half of this paper is devoted to the development of the general theory of -Culler-Shalen Seminorms defined for curves of PSL2(C)-characters. By working over PSL2(C) we obtain a theory that is more generally applicable than its SL2(C) counterpart, while being only mildly more difficult to set up. In the second half of this paper we apply the theory of Culler-Shalen Seminorms to study the Dehn filling operation. In particular we examine the relationship between fillings which yield manifolds having a positive dimen-
In Ah Hwang - One of the best experts on this subject based on the ideXlab platform.
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On the fuzzy Hahn—Banach Theorem—an analytic form
Fuzzy Sets and Systems, 1999Co-Authors: Gil Seob Rhie, In Ah HwangAbstract:Abstract The main goal of this paper is to investigate the relation between fuzzy Seminorms and crisp Seminorms on a linear space X, and to fuzzify the analytic form of the Hahn–Banach Theorem which is one of the most important theorems of functional analysis.
Gil Seob Rhie - One of the best experts on this subject based on the ideXlab platform.
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On the fuzzy Hahn—Banach Theorem—an analytic form
Fuzzy Sets and Systems, 1999Co-Authors: Gil Seob Rhie, In Ah HwangAbstract:Abstract The main goal of this paper is to investigate the relation between fuzzy Seminorms and crisp Seminorms on a linear space X, and to fuzzify the analytic form of the Hahn–Banach Theorem which is one of the most important theorems of functional analysis.
Sebastien Martin - One of the best experts on this subject based on the ideXlab platform.
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Local Error Estimates of the Finite Element Method for an Elliptic Problem with a Dirac Source Term
Numerical Methods for Partial Differential Equations, 2017Co-Authors: Silvia Bertoluzza, Loïc Lacouture, Astrid Decoene, Sebastien MartinAbstract:The solutions of elliptic problems with a Dirac measure in right-hand side are not H1 and therefore the convergence of the finite element solutions is suboptimal. Graded meshes are standard remedy to recover quasi-optimality, namely optimality up to a log-factor, for low order finite elements in L2-norm. Optimal (or quasi-optimal for the lowest order case) convergence has been shown in L2-Seminorm, where the L2-Seminorm is defined as the L2-norm on a subdomain which excludes the singularity. Here we show a quasi-optimal convergence for the Hs-Seminorm, s > 0, and an optimal convergence in H1-Seminorm for the lowest order case, on a family of quasi- uniform meshes in dimension 2. This question is motivated by the use of the Dirac measure as a reduced model in physical problems, and a high accuracy at the singularity of the finite element method is not required. Our results are obtained using local Nitsche and Schatz-type error estimates, a weak version of Aubin-Nitsche duality lemma and a discrete inf-sup condition. These theoretical results are confirmed by numerical illustrations.
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Local Error Estimates of the Finite Element Method for an Elliptic Problem with a Dirac Source Term
arXiv: Numerical Analysis, 2015Co-Authors: Silvia Bertoluzza, Loïc Lacouture, Astrid Decoene, Sebastien MartinAbstract:The solutions of elliptic problems with a Dirac measure in right-hand side are not H1 and therefore the convergence of the finite element solutions is suboptimal. Graded meshes are standard remedy to recover quasi-optimality, namely optimality up to a log-factor, for low order finite elements in L2-norm. Optimal (or quasi-optimal for the lowest order case) convergence has been shown in L2-Seminorm, where the L2-Seminorm is defined as the L2-norm on a subdomain which excludes the singularity. Here we show a quasi-optimal convergence for the Hs-Seminorm, s \textgreater{} 0, and an optimal convergence in H1-Seminorm for the lowest order case, on a family of quasi- uniform meshes in dimension 2. This question is motivated by the use of the Dirac measure as a reduced model in physical problems, and a high accuracy at the singularity of the finite element method is not required. Our results are obtained using local Nitsche and Schatz-type error estimates, a weak version of Aubin-Nitsche duality lemma and a discrete inf-sup condition. These theoretical results are confirmed by numerical illustrations.
