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Localized homology
 In Shape Modeling International
, 2007
"... In this paper, we provide the theoretical foundation and an effective algorithm for localizing topological attributes such as tunnels and voids. Unlike previous work that focused on 2manifolds with restricted geometry, our theory is general and localizes arbitrarydimensional attributes in arbitrar ..."
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Cited by 19 (4 self)
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In this paper, we provide the theoretical foundation and an effective algorithm for localizing topological attributes such as tunnels and voids. Unlike previous work that focused on 2manifolds with restricted geometry, our theory is general and localizes arbitrarydimensional attributes
Local Homology and Local Cohomology
, 2004
"... Let (R,m) be a local ring, I a proper ideal of R and M a finitely generated Rmodule of dimension d. We discuss the local homology modules of H d I (M). When M is CohenMacaulay, it is proved that H d m(M) is coCohenMacaulay of N.dimension d and H x d (Hd m(M)) ∼ = ̂M where x = (x1,..., xd) is ..."
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Let (R,m) be a local ring, I a proper ideal of R and M a finitely generated Rmodule of dimension d. We discuss the local homology modules of H d I (M). When M is CohenMacaulay, it is proved that H d m(M) is coCohenMacaulay of N.dimension d and H x d (Hd m(M)) ∼ = ̂M where x = (x1,..., xd
Local homology and cohomology on schemes
, 1997
"... Abstract. We prove a sheaftheoretic derivedcategory generalization of GreenleesMay duality (a farreaching generalization of Grothendieck’s local duality theorem): for a quasicompact separated scheme X and a “proregular ” subscheme Z—for example, any separated noetherian scheme and any closed su ..."
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Cited by 41 (6 self)
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subscheme—there is a sort of sheafified adjointness between local cohomology supported in Z and leftderived completion along Z. In particular, leftderived completion can be identified with local homology, i.e., the homology of RHom • (RΓ Z OX, −). Sheafified generalizations of a number of duality theorems
Measuring and localizing homology classes
 The Computing Research Repository (CoRR
, 2007
"... We develop a method for measuring and localizing homology classes. This involves two problems. First, we define relevant notions of size for both a homology class and a homology group basis, using ideas from relative homology. Second, we propose an algorithm to compute the optimal homology basis, us ..."
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Cited by 4 (0 self)
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We develop a method for measuring and localizing homology classes. This involves two problems. First, we define relevant notions of size for both a homology class and a homology group basis, using ideas from relative homology. Second, we propose an algorithm to compute the optimal homology basis
Dimension Detection with Local Homology
"... Detecting the dimension of a hidden manifold from a point sample has become an important problem in the current datadriven era. Indeed, estimating the shape dimension is often the first step in studying the processes or phenomena associated to the data. Among the many dimension detection algorithm ..."
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dimension by determining local homology. The computation of this topological structure is much less sensitive to the local distribution of points, which leads to the relaxation of the sampling conditions. Furthermore, by leveraging various developments in computational topology, we show that this local
Homological Algebra of Mirror Symmetry
 in Proceedings of the International Congress of Mathematicians
, 1994
"... Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual Ca ..."
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Cited by 529 (3 self)
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Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual CalabiYau manifolds V, W of dimension n (not necessarily equal to 3) one has dim H p (V, Ω q) = dim H n−p (W, Ω q). Physicists conjectured that conformal field theories associated with mirror varieties are equivalent. Mathematically, MS is considered now as a relation between numbers of rational curves on such a manifold and Taylor coefficients of periods of Hodge structures considered as functions on the moduli space of complex structures on a mirror manifold. Recently it has been realized that one can make predictions for numbers of curves of positive genera and also on CalabiYau manifolds of arbitrary dimensions. We will not describe here the complicated history of the subject and will not mention many beautiful contsructions, examples and conjectures motivated
Approximating Local Homology from Samples
"... Abstract. Recently, multiscale notions of local homology (a variant of persistent homology) have been used to study the local structure of spaces around a given point from a point cloud sample. Current reconstruction guarantees rely on constructing embedded complexes which become difficult in high ..."
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Cited by 2 (0 self)
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Abstract. Recently, multiscale notions of local homology (a variant of persistent homology) have been used to study the local structure of spaces around a given point from a point cloud sample. Current reconstruction guarantees rely on constructing embedded complexes which become difficult in high
ON THE FINITENESS OF LOCAL HOMOLOGY MODULES
, 2009
"... Let (R,m) be a commutative Noetherian complete local ring, a an ideal of R, and A an Artinian Rmodule with Ndim A = d. We prove that if d> 0, then Cosupp(Ha d−1 (A)) is finite and if d ≤ 3, then the set Coass(Ha i (A)) is finite for all i. Moreover, if either d ≤ 2 or the cohomological dimensio ..."
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Let (R,m) be a commutative Noetherian complete local ring, a an ideal of R, and A an Artinian Rmodule with Ndim A = d. We prove that if d> 0, then Cosupp(Ha d−1 (A)) is finite and if d ≤ 3, then the set Coass(Ha i (A)) is finite for all i. Moreover, if either d ≤ 2 or the cohomological
SWISSMODEL: an automated protein homologymodeling server
 Nucleic Acids Research
, 2003
"... SWISSMODEL ..."
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