Showing papers on "Vector space published in 2002"
Book•
8 May 2002
TL;DR: In this article, the authors present a generalization of the Backus-Gilbert method for linear inverse problems in the context of geophysics, which is based on the theory of functions of a complex variable.
Abstract: Preface. I. Introduction to Inversion Theory. 1. Forward and inverse problems in geophysics. 1.1 Formulation of forward and inverse problems for different geophysical fields. 1.2 Existence and uniqueness of the inverse problem solutions. 1.3 Instability of the inverse problem solution. 2. Ill-posed problems and the methods of their solution. 2.1 Sensitivity and resolution of geophysical methods. 2.2 Formulation of well-posed and ill-posed problems. 2.3 Foundations of regularization methods of inverse problem solution. 2.4 Family of stabilizing functionals. 2.5 Definition of the regularization parameter. II. Methods of the Solution of Inverse Problems. 3. Linear discrete inverse problems. 3.1 Linear least-squares inversion. 3.2 Solution of the purely under determined problem. 3.3 Weighted least-squares method. 3.4 Applying the principles of probability theory to a linear inverse problem. 3.5 Regularization methods. 3.6 The Backus-Gilbert method. 4. Iterative solutions of the linear inverse problem. 4.1 Linear operator equations and their solution by iterative methods. 4.2 A generalized minimal residual method. 4.3 The regularization method in a linear inverse problem solution. 5. Nonlinear inversion technique. 5.1 Gradient-type methods. 5.2 Regularized gradient-type methods in the solution of nonlinear inverse problems. 5.3 Regularized solution of a nonlinear discrete inverse problem. 5.4 Conjugate gradient re-weighted optimization. III. Geopotential Field Inversion. 6. Integral representations in forward modeling of gravity and magnetic fields. 6.1 Basic equations for gravity and magnetic fields. 6.2 Integral representations of potential fields based on the theory of functions of a complex variable. 7. Integral representations in inversion of gravity and magnetic data. 7.1 Gradient methods of gravity inversion. 7.2 Gravity field migration. 7.3 Gradient methods of magnetic anomaly inversion. 7.4 Numerical methods in forward and inverse modeling. IV. Electromagnetic Inversion. 8. Foundations of electromagnetic theory. 8.1 Electromagnetic field equations. 8.2 Electromagnetic energy flow. 8.3 Uniqueness of the solution of electromagnetic field equations. 8.4 Electromagnetic Green's tensors. 9. Integral representations in electromagnetic forward modeling. 9.1 Integral equation method. 9.2 Family of linear and nonlinear integral approximations of the electromagnetic field. 9.3 Linear and non-linear approximations of higher orders. 9.4 Integral representations in numerical dressing. 10. Integral representations in electromagnetic inversion. 10.1 Linear inversion methods. 10.2 Nonlinear inversion. 10.3 Quasi-linear inversion. 10.4 Quasi-analytical inversion. 10.5 Magnetotelluric (MT) data inversion. 11. Electromagnetic migration imaging. 11.1 Electromagnetic migration in the frequency domain. 11.2 Electromagnetic migration in the time domain. 12. Differential methods in electromagnetic modeling and inversion. 12.1 Electromagnetic modeling as a boundary-value problem. 12.2 Finite difference approximation of the boundary-value problem. 12.3 Finite element solution of boundary-value problems. 12.4 Inversion based on differential methods. V. Seismic Inversion. 13. Wavefield equations. 13.1 Basic equations of elastic waves. 13.2 Green's functions for wavefield equations. 13.3 Kirchhoff integral formula and its analogs. 13.4 Uniqueness of the solution of the wavefield equations. 14. Integral representations in wavefield theory. 14.1 Integral equation method in acoustic wavefield analysis. 14.2 Integral approximations of the acoustic wavefield. 14.3 Method of integral equations in vector wavefield analysis. 14.4 Integral approximations of the vector wavefield. 15. Integral representations in wavefield inversion. 15.1 Linear inversion methods. 15.2 Quasi-linear inversion. 15.3 Nonlinear inversion. 15.4 Principles of wavefield migration. 15.5 Elastic field inversion. A. Functional spaces of geophysical models and data. A.1 Euclidean space. A.2 Metric space. A.3 Linear vector spaces. A.4 Hilbert spaces. A.5 Complex Euclidean and Hilbert spaces. A.6 Examples of linear vector spaces. B. Operators in the spaces of models and data. B.1 Operators in functional spaces. B.2 Linear operators. B.3 Inverse operators. B.4 Some approximation problems in the Hilbert spaces of geophysical data. B.5 Gram - Schmidt orthogonalization process. C. Functionals in the spaces of geophysical models. C.1 Functionals and their norms. C.2 Riesz representation theorem. C.3 Functional representation of geophysical data and an inverse problem. D. Linear operators and functionals revisited. D.1 Adjoint operators. D.2 Differentiation of operators and functionals. D.3 Concepts for variational calculus. E. Some formulae and rules from matrix algebra. E.1 Some formulae and rules of operation on matrices. E.2 Eigenvalues and eigenvectors. E.3 Spectral decomposition of a symmetric matrix. E.4 Singular value decomposition (SVD). E.5 The spectral Lanczos decomposition method. F. Some formulae and rules from tensor calculus. F.1 Some formulae and rules of operation on tensor functions. F.2 Tensor statements of the Gauss and Green's formulae. F.3 Green's tensor and vector formulae for Lame and Laplace operators. Bibliography. Index.
934 citations
TL;DR: In this article, a 1-parameter deformation of the Harish-Chandra homomorphism from the Weyl group of a root system to the Calogero-Moser space was constructed.
Abstract: To any finite group Γ⊂Sp(V) of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, Hκ of the algebra ℂ[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of ℙ
r
, where r=number of conjugacy classes of symplectic reflections in $Γ$. The algebra Hκ, called a symplectic reflection algebra, is related to the coordinate ring of a Poisson deformation of the quotient singularity V/Γ. This leads to a symplectic analogue of McKay correspondence, which is most complete in case of wreath-products. If Γ is the Weyl group of a root system in a vector space ? and V=?⊕?*, then the algebras Hκ are certain ‘rational’ degenerations of the double affine Hecke algebra introduced earlier by Cherednik. Let $Γ=S
n
, the Weyl group of ?=??
n
. We construct a 1-parameter deformation of the Harish-Chandra homomorphism from ?(?)?, the algebra of invariant polynomial differential operators on ??
n
, to the algebra of S
n
-invariant differential operators with rational coefficients on the space ℂ
n
of diagonal matrices. The second order Laplacian on ? goes, under the deformed homomorphism, to the Calogero-Moser differential operator on ℂ
n
, with rational potential. Our crucial idea is to reinterpret the deformed Harish-Chandra homomorphism as a homomorphism: ?(?)? ↠ spherical subalgebra in Hκ, where Hκ is the symplectic reflection algebra associated to the group Γ=S
n
. This way, the deformed Harish-Chandra homomorphism becomes nothing but a description of the spherical subalgebra in terms of ‘quantum’ Hamiltonian reduction. In the ‘classical’ limit κ→∞, our construction gives an isomorphism between the spherical subalgebra in H∞ and the coordinate ring of the Calogero-Moser space. We prove that all simple H∞-modules have dimension n!, and are parametrised by points of the Calogero-Moser space. The family of these modules forms a distinguished vector bundle on the Calogero-Moser space, whose fibers carry the regular representation of S
n
. Moreover, we prove that the algebra $H∞ is isomorphic to the endomorphism algebra of that vector bundle.
606 citations
TL;DR: In this paper, a sharp affine Lp Sobolev inequality for functions on Euclidean n-space is established, which is invariant under all affine transformations of ℝn.
Abstract: A sharp affine Lp Sobolev inequality for functions on Euclidean n-space is established. This new inequality is significantly stronger than (and directly implies) the classical sharp Lp Sobolev inequality of Aubin and Talenti, even though it uses only the vector space structure and standard Lebesgue measure on ℝn. For the new inequality, no inner product, norm, or conformal structure is needed; the inequality is invariant under all affine transformations of ℝn.
364 citations
TL;DR: In this paper, the authors studied the algebra D(G,K) of K-valued locally analytic distributions on G, and applied their results to the locally analytic representation theory of G in vector spaces over K. They showed that the algebra behaves like the ring of functions on a rigid Stein space, and that (at least when G is Qp-analytic) it is a faithfully flat extension of its subring Zp[[G]], where Zp is the completed group ring of G.
Abstract: Let G be a compact, locally L-analytic group, where L is a finite extension of Qp. Let K be a discretely valued extension field of L. We study the algebra D(G,K) of K-valued locally analytic distributions on G, and apply our results to the locally analytic representation theory of G in vector spaces over K. Our objective is to lay a useful and powerful foundation for the further study of such representations.
We show that the noncommutative, nonnoetherian ring D(G,K) "behaves" like the ring of functions on a rigid Stein space, and that (at least when G is Qp-analytic) it is a faithfully flat extension of its subring K\otimes Zp[[G]], where Zp[[G]] is the completed group ring of G. We use this point of view to describe an abelian subcategory of D(G,K) modules that we call coadmissible.
We say that a locally analytic representation V of G is admissible if its strong dual is coadmissible as D(G,K)-module. For noncompact G, we say V is admissible if its strong dual is coadmissible as D(H,K) module for some compact open subgroup H. In this way we obtain an abelian category of admissible locally analytic representations. These methods allow us to answer a number of questions raised in our earlier papers on p-adic representations; for example we show the existence of analytic vectors in the admissible Banach space representations of G that we studied in "Banach space representations ...", Israel J. Math. 127, 359-380 (2002).
Finally we construct a dimension theory for D(G,K), which behaves for coadmissible modules like a regular ring, and show that smooth admissible representations are zero dimensional.
231 citations
1 Jan 2002
TL;DR: In this article, the notion of locally trivial fibration has been introduced and equivalence classes of such vector bundles over a CW-complex can be used to define groups K*(X) in such a way that K* becomes a cohomology theory.
Abstract: In Chapter 4 we defined the notion of a fibre bundle (a locally trivial fibration); in this chapter we consider an important class of fibre bundles—those for which every fibre has the structure of a vector space in a way which is compatible on neighboring fibres. We show how equivalence classes of such vector bundles over a CW-complex can be used to define groups K*(X) in such a way that K* becomes a cohomology theory.
227 citations
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TL;DR: In this paper, the authors studied the structure of racks, their cohomology groups and the corresponding Nichols algebras of braided vector spaces arising from groups, where the most important class of vector spaces is the class of Braided Vector Spaces (CX, c^q), where C is the field of complex numbers, X is a rack and q is a 2-cocycle on X with values in C^*.
Abstract: A fundamental step in the classification of finite-dimensional complex pointed Hopf algebras is the determination of all finite-dimensional Nichols algebras of braided vector spaces arising from groups. The most important class of braided vector spaces arising from groups is the class of braided vector spaces (CX, c^q), where C is the field of complex numbers, X is a rack and q is a 2-cocycle on X with values in C^*. Racks and cohomology of racks appeared also in the work of topologists. This leads us to the study of the structure of racks, their cohomology groups and the corresponding Nichols algebras. We will show advances in these three directions. We classify simple racks in group-theoretical terms; we describe projections of racks in terms of general cocycles; we introduce a general cohomology theory of racks contaninig properly the existing ones. We introduce a "Fourier transform" on racks of certain type; finally, we compute some new examples of finite-dimensional Nichols algebras.
219 citations
TL;DR: In this paper, an object of the abelian category of mixed Tate motive associated to multiple zeta values was constructed and proved the inequality of the dimension of the vector space generated by multiple Zeta values, which is conjectured by Zagier.
Abstract: In this paper, we construct an object of the abelian category of mixed Tate motive associated to multiple zeta values. as a consequence, we prove the inequality of the dimension of the vector space generated by multiple zeta values, which is conjectured by Zagier. (Some errors was corrected from the first version. Namely, we changed the definition of the boundary $\bold B^0$.)
168 citations
TL;DR: In this article, the growth and pruning operators on the vector space of rooted trees are defined and an inner product with respect to which the two operators are adjoint is defined, and several results about the multiplicities associated with each operator.
Abstract: We begin by considering the graded vector space with a basis consisting of rooted trees, graded by the count of non-root vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the operator that multiplies a rooted tree by its number of vertices. We define an inner product with respect to which the growth and pruning operators are adjoint, and obtain several results about the multiplicities associated with each operator.
The symmetric algebra on the vector space of rooted trees (after a degree shift) can be endowed with a coproduct to make a Hopf algebra; this was defined by Kreimer in connection with renormalization. We extend the growth and pruning operators, as well as the inner product mentioned above, to Kreimer's Hopf algebra. On the other hand, the vector space of rooted trees itself can be given a noncommutative multiplication: with an appropriate coproduct, this gives the Hopf algebra of Grossman and Larson. We show the inner product on rooted trees leads to an isomorphism of the Grossman-Larson Hopf algebra with the graded dual of Kreimer's Hopf algebra, correcting an earlier result of Panaite.
104 citations
TL;DR: In this paper, a logarithmic conformal field theory with a chiral algebra C and the corresponding space of states V is constructed via a two-step construction: i) deforming the chiral algebras representation on V\tensor End K[[z, 1/z]], where K is an auxiliary finite-dimensional vector space, and ii) extending C by operators corresponding to the endomorphisms End K.
Abstract: We construct logarithmic conformal field theories starting from an ordinary conformal field theory -- with a chiral algebra C and the corresponding space of states V -- via a two-step construction: i) deforming the chiral algebra representation on V\tensor End K[[z,1/z]], where K is an auxiliary finite-dimensional vector space, and ii) extending C by operators corresponding to the endomorphisms End K. For K=C^2, with End K being the two-dimensional Clifford algebra, our construction results in extending C by an operator that can be thought of as \partial^{-1}E, where \oint E is a fermionic screening. This covers the (2,p) Virasoro minimal models as well as the sl(2) WZW theory.
88 citations
1 Jan 2002
TL;DR: This work applies a general method to estimate vector components of a novel vector, given only a subset of its dimensions, to recover 3D shape of human faces from 2D image positions of a small number of feature points.
Abstract: Based on the assumption that a class of objects or data can be represented as a vector space spanned by a set of examples, we present a general method to estimate vector components of a novel vector, given only a subset of its dimensions. We apply this method to recover 3D shape of human faces from 2D image positions of a small number of feature points. The application demonstrates two aspects of the estimation of novel vector components: (1) From 2D image positions, we estimate 3D coordinates, and (2) from a small set of points, we obtain vertex positions of a high-resolution surface mesh. We provide an evaluation of the technique on laser scans of faces, and present an example of 3D shape reconstruction from a photograph. Our technique involves a tradeoff between reconstruction of the given measurements, and plausibility of the result. This is achieved in a Bayesian approach, and with a statistical analysis of the examples.
83 citations
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TL;DR: Let V be an n-dimensional real vector space endowed with a rank-n lattice Γ and ιΦ(λ) is the number of solutions of the equation ∑N k=1 xkβk = λ in nonnegative integers xk .
Abstract: Given a finite set of vectors spanning a lattice and lying in a halfspace of a real vector space, to each vector $a$ in this vector space one can associate a polytope consisting of nonnegative linear combinations of the vectors in the set which sum up to $a$. This polytope is called the partition polytope of $a$. If $a$ is integral, this polytope contains a finite set of lattice points corresponding to nonnegative integral linear combinations. The partition polytope associated to an integral $a$ is a rational convex polytope, and any rational convex polytope can be realized canonically as a partition polytope. We consider the problem of counting the number of lattice points in partition polytopes, or, more generally, computing sums of values of exponential-polynomial functions on the lattice points in such polytopes. We give explicit formulae for these quantities using a notion of multi-dimensional residue due to Jeffrey-Kirwan. We show, in particular, that the dependence of these quantities on $a$ is exponential-polynomial on "large neighborhoods" of chambers. Our method relies on a theorem of separation of variables for the generating function, or, more generally, for periodic meromorphic functions with poles on an arrangement of affine hyperplanes.
TL;DR: A more general form of vector equilibrium problems with a moving ordering cone and set-valued mappings is introduced, and some existence theorems for generalizedvector equilibrium problems are obtained, which extend and unify some existence results for similar problems.
Abstract: In this paper, we introduce a more general form of vector equilibrium problems with a moving ordering cone and set-valued mappings, and obtain some existence theorems for generalized vector equilibrium problems, which extend and unify some existence results for similar problems.
Book•
26 Sep 2002
TL;DR: In this paper, the authors studied the existence of variational inequalities and their applications to generalized complementary problems, fixed point theory for acylic maps between topological vector spaces having Sufficiently many linear functionals, and generalized contractive maps with closed values between complete metric spaces.
Abstract: Positive L^TP and Continuous Solutions for Fredholm Integral Inclusions. A Note on the Structure of the Solution Set for the Cauchy Differential Inclusion in Banach Spaces. Fixed Point Theory for Acylic Maps between Topological Vector Spaces having Sufficiently many Linear Functionals, and Generalized Contractive Maps with Closed Values between Complete Metric Spaces. Using the Integral Manifolds to Solvability of Boundary Value Problems. On the Semicontinuity of Nonlinear Spectra. Existence Results for Two-Point Boundary Value Problems. Generalized Strongly Nonlinear Implicit Quasi-Variational Inequalities for Fuzzy Mappings. Vector Variational Inequalities, Multi-Objective Optimizations, Pareto Optimality and Applications. Variational Principle and Fixed Points. On the Baire Category Method in Existence Problems for Ordinary and Partial Differential Inclusions. Maximal Element Principles on Generalized Convex Spaces and their Applications. Fixed Point Results for Multi-valued Contractions on Gauge Spaces. The Study of Variational Inequalities and Applications to Generalized Complementarity Problems, Fixed Point Theorems of Set-Valued Mappings and Minimization Problems. Remarks on the Existence of Maximal Elements with Respect to a Binary Relation in Non-Compact Topological Spaces. Periodic Solutions of a Singularly Perturbed System of Differential Inclusions in Banach Space. Constrained Differential Inclusions. Nonlinear Boundary Value Problems with Multi-valued Terms. Optimal Control of a Class of Nonlinear Parabolic Problems with Multi-valued Terms. Continuation Theory for A-Proper Mappings and their Uniform Limits and Nonlinear Perturbations of Fredholm Mappings. Existence Theorems for Strongly Accretive Operators in Banch Spaces. A Kneser Type Property for the Solution Set of a Semilinear Differential Inclusion with Lower Semicontinuous Nonlinearity. A Nonlinear Multi-valued Problem with Nonlinear Boundary Conditions. Extensions of Monotone Sets. Convergence of Iterates of Nonexpansive Set-Valued Mappings. A Remark on the Intersection of a Lower Semicontinuous Multi-function and Fixed Set. Random Approximations and Random Fixed Point Theorems for Set-Valued Random Maps.Existence Theorems for Two-Variable Functions and Fixed Point Theorems for Set-Valued Mappings. An Extension Theorem and Duals of Gale-Mas-Colell's and Shafer-Sonnenschein's Theorems. Iterative Algorithms for Nonlinear Variational Inequalities Involving Set-Valued H-Cocoercive Mappings.
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TL;DR: In this paper, it was shown that for any field k of characteristic p>0, any separated scheme X of finite type over k, and any overconvergent F-isocrystal E over X, the rigid cohomology H^i(X, E) and rigid co-hocomology with compact supports h^i_c (X,E) are finite dimensional vector spaces.
Abstract: We prove that for any field k of characteristic p>0, any separated scheme X of finite type over k, and any overconvergent F-isocrystal E over X, the rigid cohomology H^i(X, E) and rigid cohomology with compact supports H^i_c(X,E) are finite dimensional vector spaces. We also establish Poincare duality and the Kunneth formula with coefficients. The arguments use a pushforward construction in relative dimension 1, based on a relative version of Crew's conjecture on the quasi-unipotence of certain p-adic differential equations.
TL;DR: In this paper, the authors studied vector semispaces from a realistic way with the intention to define a natural metric, adapted to their peculiar structure, which reside on the essential positive definiteness of their elements.
Abstract: Vector semispaces are studied from a realistic way with the intention to define a natural metric, adapted to their peculiar structure, which reside on the essential positive definiteness of their elements From this point of view, Minkowski norms allow classifying semispaces in shells, that is: subsets where all the vector elements possess the same norm values Shell structure appears to be a possible disjoint partition of any semispace and so shells become equivalence classes Then, the unit shell appears to be the core of the semispace homothetic construction as well as the origin of the semispace metrics Minkowski or root scalar products permit to connect two or more semispace elements and conduct towards generalized definitions of Pth order root distances and cosines Finally, the unit shell of a given semispace, in company of both Boolean tagged sets, inward matrix products and with the aid of the matrix signatures as well, it is seen as the seed to construct any arbitrary element of the semispace connected vector space Finite and infinite dimensional vector spaces application examples are provided along the work discussion
TL;DR: In this paper, the authors introduce the topological pseudomonotonicity to vector valued bifunctions, and derive some existence results for vector equilibrium problems with the corresponding bifunction topologically pseudomonotone.
TL;DR: The theory of oriented matroids gives rise to a combinatorial analogue of the algebra of Orlik—Terao, which is the main tool of the proofs.
Abstract: Let V be a vector space of dimension d over a field K and let A be a central arrangement of hyperplanes in V. To answer a question posed by K. Aomoto, P. Orlik and H. Terao construct a commutative K -algebra U(A) in terms of the equations for the hyperplanes of A. In the course of their work the following question naturally occurred:
\circ Is U(A) determined by the intersection lattice L(A) of the hyperplanes of A?
We give a negative answer to this question. The theory of oriented matroids gives rise to a combinatorial analogue of the algebra of Orlik--Terao, which is the main tool of our proofs.
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TL;DR: In this article, the authors consider finite groups and finite-dimensional vector spaces and general underlying fields, including p-adic fields, which are quite interesting for analysis in the context of analysis of topological groups.
Abstract: On the one hand the algebras of linear operators here act on finite-dimensional vector spaces, and on the other hand the point of view is generally an analysts' Also, one might think of algebras as being used to add more data to basic geometry as on a graph, for instance Of course this is a common theme which is considered in numerous settings From an analysts' perspective, compact groups, their representations, and more general topological groups and their representations are basic objects of study Finite groups are like groups which are especially compact, and with some extra structure As long as one considers finite groups and finite-dimensional vector spaces, one might as well consider general underlying fields k too This includes p-adic fields, which are quite interesting for analysis
TL;DR: This paper defines the operations sum, product, tensor product, Hom, and intersection of fuzzy subspaces and in each case they characterise the corresponding flag.
TL;DR: In this article, the authors present a sequence of linear maps of vector spaces with fixed bases, where each term of a sequence is a linear space of differentials of metric values ascribed to the elements of a simplicial complex -a triangulation of a manifold.
Abstract: We write out some sequences of linear maps of vector spaces with fixed bases. Each term of a sequence is a linear space of differentials of metric values ascribed to the elements of a simplicial complex - a triangulation of a manifold. If the sequence turns out to be an acyclic complex then one can construct a manifold invariant out of its torsion. We demonstrate this first for three-dimensional manifolds, and then we conduct the part, related to moves 2 4, of the corresponding work for four-dimensional manifolds.
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TL;DR: In this article it was shown that a strongly rigid complex Lie algebra has to be rigid as a Lie algebra, and that its second scalar cohomology group has to vanish (which excludes nilpotent Lie algebras of dimension greater than two).
Abstract: We call a finite-dimensional complex Lie algebra $\mathfrak{g}$ strongly rigid if its universal enveloping algebra $\Ug$ is rigid as an associative algebra, i.e. every formal associative deformation is equivalent to the trivial deformation. In quantum group theory this phenomenon is well-known to be the case for all complex semisimple Lie algebras.
We show that a strongly rigid Lie algebra has to be rigid as Lie algebra, and that in addition its second scalar cohomology group has to vanish (which excludes nilpotent Lie algebras of dimension greater or equal than two). Moreover, using Kontsevitch's theory of deformation quantization we show that every polynomial deformation of the linear Poisson structure on $\mathfrak{g}^*$ which induces a nonzero cohomology class of $\mathfrak{g}$ leads to a nontrivial deformation of $\mathcal{U}\mathfrak g$. Hence every Poisson structure on a vector space which is zero at some point and whose linear part is a strongly rigid Lie algebra is therefore formally linearizable in the sense of A. Weinstein. Finally we provide examples of rigid Lie algebras which are not strongly rigid, and give a classification of all strongly rigid Lie algebras up to dimension 6.
TL;DR: The aim of the present paper is to give a new kind of point of view in the theory of variational inequalities, which makes possible the study of both scalar and vector Variational inequalities under a great variety of assumptions.
Abstract: The aim of the present paper is to give a new kind of point of view in the theory of variational inequalities. Our approach makes possible the study of both scalar and vector variational inequalities under a great variety of assumptions. One can include here the variational inequalities defined on reflexive or nonreflexive Banach spaces, as well as the vector variational inequalities defined on topological vector spaces.
TL;DR: In this paper, a method for the explicit determination of the polar decomposition (and the related problem of finding tensor square roots) when the underlying vector space dimension n is arbitrary (but finite), is proposed.
Abstract: A method for the explicit determination of the polar decomposition (and the related problem of finding tensor square roots) when the underlying vector space dimension n is arbitrary (but finite), is proposed. The method uses the spectral resolution, and avoids the determination of eigenvectors when the tensor is invertible. For any given dimension n, an appropriately constructed van der Monde matrix is shown to play a key role in the construction of each of the component matrices (and their inverses) in the polar decomposition.
TL;DR: In this paper, the superoptimal Frobenius operators in several matrix vector spaces and in particular in the circulant algebra were studied, by emphasizing both the algebraic and geometric properties.
Abstract: We study the superoptimal Frobenius operators in several matrix vector spaces and in particular in the circulant algebra, by emphasizing both the algebraic and geometric properties. More specifically we prove a series of "negative" results that explain why this approximation procedure is not competitive with the optimal Frobenius approximation, although it could be used for regularization purposes.
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TL;DR: In this paper, a relationship of the classical dynamical Yang-Baxter equation with the problem of Clifford algebras has been studied, where the exponential of an element in a vector space under exterior algebra multiplication is related to its exponential under Clifford multiplication.
Abstract: We describe a relationship of the classical dynamical Yang-Baxter equation with the following elementary problem for Clifford algebras: Given a vector space $V$ with quadratic form $Q_V$, how is the exponential of an element in $\wedge^2(V)$ under exterior algebra multiplication related to its exponential under Clifford multiplication?
TL;DR: In this paper, two user-selectable algorithms for Clifford product are implemented: cmulNUM' and cmulRS' based on non-recursive Rota-Stein sausage.
Abstract: CLIFFORD performs various computations in Grassmann and Clifford algebras. It can compute with quaternions, octonions, and matrices with entries in Cl(B) - the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. Two user-selectable algorithms for Clifford product are implemented: 'cmulNUM' - based on Chevalley's recursive formula, and 'cmulRS' - based on non-recursive Rota-Stein sausage. Grassmann and Clifford bases can be used. Properties of reversion in undotted and dotted wedge bases are discussed.
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TL;DR: In this paper, the authors show that the eigenvalues of the Jordan Osserman Jacobi operator are real and that the curvature tensor is diagonalizable if p < q.
Abstract: Let $R$ be an algebraic curvature tensor on a vector space of signature $(p,q)$ defining a spacelike Jordan Osserman Jacobi operator $\JJ_R$. We show that the eigenvalues of $\JJ_R$ are real and that $\JJ_R$ is diagonalizable if $p
TL;DR: In this paper, an example of quadratic Poisson structure which does not arise this way is presented. But it does not admit a deformation quantization stemming from the construction of V.
Abstract: Any classical r-matrix on the Lie algebra of linear operators on a real vector space V gives rise to a quadratic Poisson structure on V which admits a deformation quantization stemming from the construction of V. Drinfel'd [Dr], [Gr]. We exhibit in this article an example of quadratic Poisson structure which does not arise this way.
6 Aug 2002
TL;DR: In this article, a spectral transform interpretation of AND-EXOR representations of switching functions and related decision diagrams in the vector space over GF(2) was given, which was uniformly extended to the Fourier series-like expressions of functions in the complex vector space and the decision diagrams for integer-valued functions.
Abstract: In this paper we give a spectral transform interpretation of AND-EXOR representations of switching functions and related decision diagrams in the vector space over GF(2). The consideration is uniformly extended to the Fourier series-like expressions of functions in the complex vector space and the decision diagrams for integer-valued functions. It is shown that the multi-terminal decision diagrams, MTBDDs, and edge-valued decision diagrams, EVBDDs, for integer-valued functions are derived by using the same sets of basic functions already applied for the decision diagrams attached to some AND-EXOR expressions, but considered over the complex field. The algebraic transform decision diagrams, ATDDs, are considered as the integer counterparts of the functional decision diagrams, FDDs, attached to the algebraic transform in the same way as the FDDs are attached to the Reed-Muller expressions. It is shown that the EVBDDs are the ATDDs in different notation.