The Experts below are selected from a list of 33081 Experts worldwide ranked by ideXlab platform
Gunther Uhlmann - One of the best experts on this subject based on the ideXlab platform.
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inverse problems with partial data for a magnetic schrodinger operator in an infinite slab and on a bounded domain
Communications in Mathematical Physics, 2012Co-Authors: Katsiaryna Krupchyk, Matti Lassas, Gunther UhlmannAbstract:In this paper we study inverse boundary value problems with partial data for the magnetic Schrodinger operator. In the case of an infinite slab in \({\mathbb{R}^n}\) , n ≥ 3, we establish that the magnetic field and the electric potential can be determined uniquely, when the Dirichlet and Neumann data are given either on the different boundary Hyperplanes of the slab or on the same Hyperplane. This is a generalization of the results of Li and Uhlmann (Inverse Probl Imaging 4(3):449–462, 2010), obtained for the Schrodinger operator without magnetic potentials.
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inverse problems with partial data for a magnetic schr odinger operator in an infinite slab and on a bounded domain
arXiv: Analysis of PDEs, 2011Co-Authors: Katsiaryna Krupchyk, Matti Lassas, Gunther UhlmannAbstract:In this paper we study inverse boundary value problems with partial data for the magnetic Schr\"odinger operator. In the case of an infinite slab in $R^n$, $n\ge 3$, we establish that the magnetic field and the electric potential can be determined uniquely, when the Dirichlet and Neumann data are given either on the different boundary Hyperplanes of the slab or on the same Hyperplane. This is a generalization of the results of [41], obtained for the Schr\"odinger operator without magnetic potentials. In the case of a bounded domain in $R^n$, $n\ge 3$, extending the results of [2], we show the unique determination of the magnetic field and electric potential from the Dirichlet and Neumann data, given on two arbitrary open subsets of the boundary, provided that the magnetic and electric potentials are known in a neighborhood of the boundary. Generalizing the results of [31], we also obtain uniqueness results for the magnetic Schr\"odinger operator, when the Dirichlet and Neumann data are known on the same part of the boundary, assuming that the inaccessible part of the boundary is a part of a Hyperplane.
Nir Gadish - One of the best experts on this subject based on the ideXlab platform.
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representation stability for families of linear subspace arrangements
Advances in Mathematics, 2017Co-Authors: Nir GadishAbstract:Abstract Church–Ellenberg–Farb [8] used the language of FI-CHA to identify certain sequences of Hyperplane arrangements with S n -actions that satisfy cohomological representation stability. Here we vastly extend their results, and define when a collection of arrangements is “finitely generated”. Using this notion we get stability results to: • General linear subspace arrangements, not necessarily of Hyperplanes. • A wide class of group actions, replacing FI by a general category C. We show that the cohomology of such collections of arrangements satisfies a strong form of representation stability, with many concrete applications. For example, this implies that their Betti numbers are always given by certain polynomials. For this purpose we use the theory of representation stability for quite general classes of groups, developed in a [17] . We apply this theory to get classical cohomological stability of quotients of linear subspace arrangements with coefficients in certain constructible sheaves.
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representation stability for families of linear subspace arrangements
arXiv: Geometric Topology, 2016Co-Authors: Nir GadishAbstract:Church-Ellenberg-Farb used the language of FI-modules to prove that the cohomology of certain sequences of Hyperplane arrangements with S_n-actions satisfies representation stability. Here we lift their results to the level of the arrangements themselves, and define when a collection of arrangements is "finitely generated". Using this notion we greatly widen the stability results to: 1) General linear subspace arrangements, not necessarily of Hyperplanes. 2) A wide class of group actions, replacing FI by a general category C. We show that the cohomology of such collections of arrangements satisfies a strong form of representation stability, with many concrete applications. For this purpose we develop a theory of representation stability and generalized character polynomials for wide classes of groups. We apply this theory to get classical cohomological stability of quotients of linear subspace arrangements with coefficients in certain constructible sheaves.
Anita Schöbel - One of the best experts on this subject based on the ideXlab platform.
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Anchored Hyperplane Location Problems
Discrete and Computational Geometry, 2003Co-Authors: Anita SchöbelAbstract:Abstract. The anchored Hyperplane location problem is to locate a Hyperplane passing through some given points P \subseteq R n and minimizing either the sum of weighted distances (median problem ), or the maximum weighted distance (center problem ) to some other points Q \subseteq R n . This problem of computational geometry is analyzed by using nonlinear programming techniques. If the distances are measured by a norm, it will be shown that in the median case there exists an optimal Hyperplane that passes through at least n - k affinely independent points of Q , if k is the maximum number of affinely independent points of P . In the center case, there exists an optimal Hyperplane which is at maximum distance to at least n- k +1 affinely independent points of Q . Furthermore, if the norm is a smooth norm, all optimal Hyperplanes satisfy these criteria. These results generalize known results about unrestricted Hyperplane location problems.
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Median and center Hyperplanes in Minkowski spaces — a unified approach
Discrete Mathematics, 2001Co-Authors: Horst Martini, Anita SchöbelAbstract:Abstract In this paper we will extend two known location problems from Euclidean n-space to all n-dimensional normed spaces, n ⩾2. Let X be a finite set of weighted points whose affine hull is n-dimensional. Our first objective is to find a Hyperplane minimizing (among all Hyperplanes) the sum of weighted distances with respect to X . Such a Hyperplane is called a median Hyperplane with respect to X , and we will show that for all distance measures d derived from norms one of the median Hyperplanes is the affine hull of n of the demand points. (This approach was already presented in the recent survey (Discrete Appl. Math. 89 (1998) 181), but without proofs. Here we give the complete proofs to all necessary lemmas.) On the other hand, we will prove that one of the Hyperplanes minimizing (among all Hyperplanes) the maximum weighted distance to some point from X has the same maximal distance to least n +1 affinely independent demand points (such a Hyperplane is said to be a center Hyperplane of X ). Both these results allow polynomially bounded algorithmical approaches to median and center Hyperplanes and the respective distance sums or maximal distances for any fixed dimension n ⩾2, and in particular we discuss the algorithms for both the problems in the case of polyhedral norms. Also two independence of norm results for optimal Hyperplanes with fixed slope will be derived, and finally the considerations are even extended to gauges which are no longer combined with a norm.
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Median and center Hyperplanes in Minkowski spaces—a unified approach
Discrete Mathematics, 2001Co-Authors: Horst Martini, Anita SchöbelAbstract:AbstractIn this paper we will extend two known location problems from Euclidean n-space to all n-dimensional normed spaces, n⩾2. Let X be a finite set of weighted points whose affine hull is n-dimensional. Our first objective is to find a Hyperplane minimizing (among all Hyperplanes) the sum of weighted distances with respect to X. Such a Hyperplane is called a median Hyperplane with respect to X, and we will show that for all distance measures d derived from norms one of the median Hyperplanes is the affine hull of n of the demand points. (This approach was already presented in the recent survey (Discrete Appl. Math. 89 (1998) 181), but without proofs. Here we give the complete proofs to all necessary lemmas.) On the other hand, we will prove that one of the Hyperplanes minimizing (among all Hyperplanes) the maximum weighted distance to some point from X has the same maximal distance to least n+1 affinely independent demand points (such a Hyperplane is said to be a center Hyperplane of X). Both these results allow polynomially bounded algorithmical approaches to median and center Hyperplanes and the respective distance sums or maximal distances for any fixed dimension n⩾2, and in particular we discuss the algorithms for both the problems in the case of polyhedral norms. Also two independence of norm results for optimal Hyperplanes with fixed slope will be derived, and finally the considerations are even extended to gauges which are no longer combined with a norm
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Median Hyperplanes in normed spaces
1999Co-Authors: Anita Schöbel, Horst MartiniAbstract:In this paper we deal with the location of Hyperplanes in n-dimensional normed spaces. If d is a distance measure, our objective is to find a Hyperplane H which minimizes f(H) = sum_{m=1}^{M} w_{m}d(x_m,H), where w_m ge 0 are non-negative weights, x_m in R^n, m=1, ... ,M demand points and d(x_m,H)=min_{z in H} d(x_m,z) is the distance from x_m to the Hyperplane H. In robust statistics and operations research such an optimal Hyperplane is called a median Hyperplane. We show that for all distance measures d derived from norms, one of the Hyperplanes minimizing f(H) is the affine hull of n of the demand points and, moreover, that each median Hyperplane is (ina certain sense) a halving one with respect to the given point set.
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Median Hyperplanes in normed spaces — a survey
Discrete Applied Mathematics, 1998Co-Authors: Horst Martini, Anita SchöbelAbstract:AbstractIn this survey we deal with the location of Hyperplanes in n-dimensional normed spaces, i.e., we present all known results and a unifying approach to the so-called median Hyperplane problem in Minkowski spaces. We describe how to find a Hyperplane H minimizing the weighted sum f(H) of distances to a given, finite set of demand points. In robust statistics and operations research such an optimal Hyperplane is called a median Hyperplane. After summarizing the known results for the Euclidean and rectangular situation, we show that for all distance measures d derived from norms one of the Hyperplanes minimizing f(H) is the affine hull of n of the demand points and, moreover, that each median Hyperplane is a halving one (in a sense defined below) with respect to the given point set. Also an independence of norm result for finding optimal Hyperplanes with fixed slope will be given. Furthermore, we discuss how these geometric criteria can be used for algorithmical approaches to median Hyperplanes, with an extra discussion for the case of polyhedral norms. And finally a characterization of all smooth norms by a sharpened incidence criterion for median Hyperplanes is mentioned
Eliza O’reilly - One of the best experts on this subject based on the ideXlab platform.
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The stochastic geometry of unconstrained one-bit data compression
Electronic Journal of Probability, 2019Co-Authors: François Baccelli, Eliza O’reillyAbstract:A stationary stochastic geometric model is proposed for analyzing the data compression method used in one-bit compressed sensing. The data set is an unconstrained stationary set, for instance all of R^n or a stationary Poisson point process in R^n. It is compressed using a stationary and isotropic Poisson Hyperplane tessellation, assumed independent of the data. That is, each data point is compressed using one bit with respect to each Hyperplane, which is the side of the Hyperplane it lies on. This model allows one to determine how the intensity of the Hyperplanes must scale with the dimension n to ensure sufficient separation of different data by the Hyperplanes as well as sufficient proximity of the data compressed together. The results have direct implications in compressed sensing and in source coding.
Patrick J. F. Groenen - One of the best experts on this subject based on the ideXlab platform.
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Classification with support Hyperplanes
Lecture Notes in Computer Science, 2006Co-Authors: Georgi Nalbantov, Jail C. Bioch, Patrick J. F. GroenenAbstract:A new classification method is proposed, called Support Hyperplanes (SHs). To solve the binary classification task, SHs consider the set of all Hyperplanes that do not make classification mistakes, referred to as semi-consistent Hyperplanes. A test object is classified using that semi-consistent Hyperplane, which is farthest away from it. In this way, a good balance between goodness-of-fit and model complexity is achieved, where model complexity is proxied by the distance between a test object and a semi-consistent Hyperplane. This idea of complexity resembles the one imputed in the width of the so-called margin between two classes, which arises in the context of Support Vector Machine learning. Class overlap can be handled via the introduction of kernels and/or slack variables. The performance of SHs against standard classifiers is promising on several widely-used empirical data sets.
