Abstract: We investigate a new denotational model of linear logic based on the purely relational model. In this semantics, webs are equipped with a notion of ‘finitary’ subsets satisfying a closure condition and proofs are interpreted as finitary sets. In spite of a formal similarity, this model is quite different from the usual models of linear logic (coherence semantics, hypercoherence semantics, the various existing game semantics…). In particular, the standard fix-point operators used for defining the general recursive functions are not finitary, although the primitive recursion operators are. This model can be considered as a discrete analogue of the Kothe space semantics introduced in a previous paper: we show how, given a field, each finiteness space gives rise to a vector space endowed with a linear topology, a notion introduced by Lefschetz in 1942, and we study the corresponding model where morphisms are linear continuous maps (a version of Girard's quantitative semantics with coefficients in the field). In this way we obtain a new model of the recently introduced differential lambda-calculus.

Abstract: Classical Abstract Spaces in Functional Analysis.- Linear Functionals and Linear Operators.- Common Function Spaces in Applications.- Differential Calculus in Normed Vector Spaces.- Minimization of Functionals.- Convex Functionals.- Lower Semicontinuous Functionals.

Abstract: We discuss the linear independence of systems ofmvectors in n-dimensional complex vector spaces where the m vectors are time-frequency shifts of one generating vector. Such systems are called Gabor systems. When n is prime, we show that there exists an open, dense subset with full-measure of such generating vectors with the property that any subset of n vectors in the corresponding full Gabor system of n2 vectors is linearly independent. We derive consequences relevant to coding, operator identification and time-frequency analysis in general.

Abstract: General Fierz-type identities are examined and their well-known connection with completeness relations in matrix vector spaces is shown. In particular, I derive the chiral Fierz identities in a simple and systematic way by using a chiral basis for the complex 4×4 matrices. Other completeness relations for the fundamental representations of SU(N) algebras can be extracted using the same reasoning.

Abstract: This is the second in a series math.AG/0312190, math.AG/0410267, math.AG/0410268 on configurations in an abelian category A. Given a finite partially ordered set (I, \sigma(K) in A satisfying some axioms, where J,K are subsets of I. Configurations describe how an object X in A decomposes into subobjects. The first paper math.AG/0312190 defined configurations and studied moduli spaces of (I,<)-configurations in A, using the theory of Artin stacks. It proved well-behaved moduli stacks Obj_A, M(I,<)_A of objects and configurations in A exist when A is the abelian category coh(P) of coherent sheaves on a projective K-scheme P, or mod-KQ of representations of a quiver Q. Write CF(Obj_A) for the vector space of constructible functions on Obj_A. Motivated by Ringel-Hall algebras, we define an associative multiplication * on CF(Obj_A) using pushforwords and pullbacks along 1-morphisms between the M(I,<)_A, making CF(Obj_A) into an algebra. We also study representations of CF(Obj_A), the Lie subalgebra CF^ind(Obj_A) of functions supported on indecomposables, and other algebraic structures on CF(Obj_A). Then we generalize these ideas to stack functions SF(Obj_A), a universal generalization of constructible functions on stacks introduced in math.AG/0509722, containing more information. Under extra conditions on A we can define (Lie) algebra morphisms from SF(Obj_A) to some explicit (Lie) algebras, which will be important in the sequels on invariants counting t-(semi)stable objects in A.

Abstract: We describe a formalization of the elementary algebra, topology and analysis of finite-dimensional Euclidean space in the HOL Light theorem prover (Euclidean space is R N with the usual notion of distance) A notable feature is that the HOL type system is used to encode the dimension N in a simple and useful way, even though HOL does not permit dependent types In the resulting theory the HOL type system, far from getting in the way, naturally imposes the correct dimensional constraints, eg checking compatibility in matrix multiplication Among the interesting later developments of the theory are a partial decision procedure for the theory of vector spaces (based on a more general algorithm due to Solovay) and a formal proof of various classic theorems of topology and analysis for arbitrary N-dimensional Euclidean space, eg Brouwer's fixpoint theorem and the differentiability of inverse functions

Abstract: A type of tridiagonal pair is considered, said to be mild of q-Serre type.It is shown that these tridiagonal pairs induce the structure of a quantum affine algebra Uq( sl2)-module on their underlying vector space.This is done by presenting an explicit basis for the underlying vector space and describing the Uq( sl2)-action on that basis.

Abstract: CHAPTER 1 Normed Vector Spaces CHAPTER 2 The Lebesgue Integral CHAPTER 3 Hilbert Spaces and Orthonormal Systems CHAPTER 4 Linear Operators on Hilbert Spaces CHAPTER 5 Applications to Integral and Differential Equations CHAPTER 6 Generalized Functions and Partial Differential Equations CHAPTER 7 Mathematical Foundations of Quantum Mechanics CHAPTER 8 Wavelets and Wavelet Transforms CHAPTER 9 Optimization Problems and Other Miscellaneous Applications

Abstract: In this paper, we present a novel framework to carry out computations on tensors, i.e. symmetric positive definite matrices. We endow the space of tensors with an affine-invariant Riemannian metric, which leads to strong theoretical properties: The space of positive definite symmetric matrices is replaced by a regular and geodesically complete manifold without boundaries. Thus, tensors with non-positive eigenvalues are at an infinite distance of any positive definite matrix. Moreover, the tools of differential geometry apply and we generalize to tensors numerous algorithms that were reserved to vector spaces. The application of this framework to the processing of diffusion tensor images shows very promising results. We apply this framework to the processing of structure tensor images and show that it could help to extract low-level features thanks to the affine-invariance of our metric. However, the same affine-invariance causes the whole framework to be noise sensitive and we believe that the choice of a more adapted metric could significantly improve the robustness of the result.

Abstract: We prove the upper semicontinuity (in term of the closedness) of the solution set with respect to parameters of vector quasivariational inequalities involving multifunctions in topological vector spaces under the semicontinuity of the data, avoiding monotonicity assumptions. In particular, a new quasivariational inequality problem is proposed. Applications to quasi-complementarity problems are considered

Abstract: A process for evaluating and geocoding of GIS data elements utilizes a plurality of "locate" tests and a weighting scheme to express the match results as a multidimensional vector. Multiple inputs and data sources, as well as ambiguous and partial input data, are used to generate an output with improved precision by applying a weighting function to each input element and generating a set of test vectors (i.e., the input data element weighted by the known accuracy of the element/source). A sum of a plurality of tests is then generated as the "characteristic vector" of the test set. By using two (or more) different sets of test, two (or more) characteristic vectors are formed. Various well-known algebraic techniques can then be used to evaluate the results of each set of tests and select the "best match" result.

Abstract: Matrix theory has been one of the most utilised concepts in fuzzy models and neutrosophic models. From solving equations to characterising linear transformations or linear operators, matrices are used. Matrices find their applications in several real models. In fact it is not an exaggeration if one says that matrix theory and linear algebra (i.e. vector spaces) form an inseparable component of each other.

Abstract: The vanishing ideal of an arrangement of linear subspaces in a vector space is considered, and the paper investigates when this ideal can be generated by products of linear forms. A combinatorial construction (blocker duality) is introduced which yields such generators in cases with a great deal of combinatorial structure, and examples are presented that inspired the work. A construction is given which produces all elements of this type in the vanishing ideal of the arrangement. This leads to an algorithm for deciding if the ideal is generated by products of linear forms. Generic arrangements of points in and lines in are also considered.

Abstract: The possibility that our space is multi–rather than singly–connected has gained renewed interest after the discovery of the low power for the first multipoles of the CMB by WMAP. To test the possibility that our space is a multi-connected spherical space, it is necessary to know the eigenmodes of such spaces. Except for lens and prism space, and to some extent for dodecahedral space, this remains an open problem. Here we derive the eigenmodes of all spherical spaces. For dodecahedral space, the demonstration is much shorter, and the calculation method much simpler than before. We also apply our method to tetrahedric, octahedric and icosahedric spaces. This completes the knowledge of eigenmodes for spherical spaces, and opens the door to new observational tests of the cosmic topology. The vector space of the eigenfunctions of the Laplacian on the 3-sphere , corresponding to the same eigenvalue λk = −k(k + 2), has dimension (k + 1)2. We show that the Wigner functions provide a basis for such a space. Using the properties of the latter, we express the behaviour of a general function of under an arbitrary rotation G of SO(4). This offers the possibility of selecting those functions of which remain invariant under G. Specifying G to be a generator of the holonomy group of a spherical space X, we give the expression of the vector space of the eigenfunctions of X. We provide a method to calculate the eigenmodes up to an arbitrary order. As an illustration, we give the first modes for the spherical spaces mentioned.

Abstract: A Laurent-Ore algebra L over a field F is a mathematical abstraction of common properties of linear partial differential and difference operators. A linear (partial) functional system is of the form A(z) = 0 where A is a matrix over L and z is a vector of unknowns. Typically, it is a system consisting of linear partial differential, shift and q-shift operators, or any mixture thereof. We associate to a linear functional system A(z) = 0 an L-module MA, which is called the module of formal solutions. For our purpose, the dimension of an L-module is defined to be the dimension of the module as a vector space over F . A system A(z) = 0 is said to be ∂-finite if MA has finite dimension. A Picard-Vessiot extension for a ∂-finite system A(z) = 0 is a ring containing “all” solutions of A(z) = 0. We prove the existence of Picard-Vessiot extensions for all ∂-finite linear functional systems and show that the dimension of the solution space of a ∂-finite system equals the dimension of its module of formal solutions. The Grobner basis techniques for left ideals in Ore algebras are extended to left submodules over Laurent-Ore algebras. This extension enables us to determine whether a linear functional system is ∂-finite. We present an algorithm for finding all submodules of an L-module with finite dimension. This algorithm allows us to find all “subsystems” whose solution spaces are contained in that of a given ∂-finite linear functional system.

Abstract: In this paper, we present a new learning framework for image style transforms. Considering that the images in different style representations constitute different vector spaces, we propose a novel framework called coupled space learning to learn the relations between different spaces and use them to infer the images from one style to another style. Observing that for each style, only the components correlated to the space of the target style are useful for inference, we first develop the correlative component analysis to pursue the embedded hidden subspaces that best preserve the inter-space correlation information. Then we develop the coupled bidirectional transform algorithm to estimate the transforms between the two embedded spaces, where the coupling between the forward transform and the backward transform is explicitly taken into account. To enhance the capability of modelling complex data, we further develop the coupled Gaussian mixture model to generalize our framework to a mixture-model architecture. The effectiveness of the framework is demonstrated in the applications including face super-resolution and bidirectional portrait style transforms

Abstract: We propose an approach to embed time series data in a vector space based on the distances obtained from Dynamic Time Warping (DTW), and to classify them in the embedded space. Under the problem setting in which both labeled data and unlabeled data are given beforehand, we consider three embeddings, embedding in a Euclidean space by MDS, embedding in a Pseudo-Euclidean space, and embedding in a Euclidean space by the Laplacian eigenmap technique. We have found through analysis and experiment that the embedding by the Laplacian eigenmap method leads to the best classification result. Furthermore, the proposed approach with Laplacian eigenmap embedding shows better performance than k-nearest neighbor method.

Abstract: The notion of comparative similarity ‘X is more similar or closer to Y than to Z’ has been investigated in both foundational and applied areas of knowledge representation and reasoning, e.g., in concept formation, similarity-based reasoning and areas of bioinformatics such as protein sequence alignment. In this paper we analyse the computational behaviour of the ‘propositional’ logic with the binary operator ‘closer to a set τ1 than to a set τ2’ and nominals interpreted over various classes of distance (or similarity) spaces. In particular, using a reduction to the emptiness problem for certain tree automata, we show that the satisfiability problem for this logic is ExpTime-complete for the classes of all finite symmetric and all finite (possibly non-symmetric) distance spaces. For finite subspaces of the real line (and higher dimensional Euclidean spaces) we prove the undecidability of satisfiability by a reduction of the solvability problem for Diophantine equations. As our ‘closer’ operator has the same expressive power as the standard operator > of conditional logic, these results may have interesting implications for conditional logic as well.

Abstract: Data structures for similarity search are commonly evaluated on data in vector spaces, but distance-based data structures are also applicable to non-vector spaces with no natural concept of dimensionality. The intrinsic dimensionality statistic of Chavez and Navarro provides a way to compare the performance of similarity indexing and search algorithms across different spaces, and predict the performance of index data structures on non-vector spaces by relating them to equivalent vector spaces. We characterise its asymptotic behaviour, and give experimental results to calibrate these comparisons.

Abstract: In securities markets, the characterization of the absence of arbitrage by the existence of state price deflators is generally obtained through the use of the Kreps–Yan theorem.This paper deals with the validity of this theorem (see Kreps, D.M., 1981. Arbitrage and equilibrium in economies with infinitely many commodities. Journal of Mathematical Economics 8, 15–35; Yan, J.A., 1980. Caracterisation d'une classe d'ensembles convexes de L1 ou H1. Sem. de Probabilites XIV. Lecture Notes in Mathematics 784, 220–222) in a general framework. More precisely, we say that the Kreps–Yan theorem is valid for a locally convex topological space (X,?), endowed with an order structure, if for each closed convex cone C in X such that CX? and C?X+={0}, there exists a strictly positive continuous linear functional on X, whose restriction to C is non-positive.We first show that the Kreps–Yan theorem is not valid for spaces if fails to be sigma-finite.Then we prove that the Kreps–Yan theorem is valid for topological vector spaces in separating duality X,Y, provided Y satisfies both a “completeness condition” and a “Lindelof-like condition”.We apply this result to the characterization of the no-arbitrage assumption in a general intertemporal framework.

Abstract: Let $K$ denote a field, and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*: V \to V$ that satisfy (i), (ii) below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a {\em Leonard pair} on $V$. In this paper we investigate the commutator $AA^*-A^*A$. Our results are as follows. First assume the dimension of $V$ is even. We show $AA^*-A^*A$ is invertible and display several attractive formulae for the determinant. Next assume the dimension of $V$ is odd. We show that the null space of $AA^*-A^*A$ has dimension 1. We display a nonzero vector in this null space. We express this vector as a sum of eigenvectors for $A$ and as a sum of eigenvectors for $A^*$.

Abstract: The possibility that our space is multi - rather than singly - connected has gained a renewed interest after the discovery of the low power for the first multipoles of the CMB by WMAP. To test the possibility that our space is a multi-connected spherical space, it is necessary to know the eigenmodes of such spaces. Excepted for lens and prism space, and in some extent for dodecahedral space, this remains an open problem. Here we derive the eigenmodes of all spherical spaces. For dodecahedral space, the demonstration is much shorter, and the calculation method much simpler than before. We also apply to tetrahedric, octahedric and icosahedric spaces. This completes the knowledge of eigenmodes for spherical spaces, and opens the door to new observational tests of cosmic topology. The vector space V^k of the eigenfunctions of the Laplacian on the three-sphere S^3, corresponding to the same eigenvalue \lambda_k = -k (k+2), has dimension (k+1)^2. We show that the Wigner functions provide a basis for such space. Using the properties of the latter, we express the behavior of a general function of V^k under an arbitrary rotation G of SO(4). This offers the possibility to select those functions of V^k which remain invariant under G. Specifying G to be a generator of the holonomy group of a spherical space X, we give the expression of the vector space V_X^k of the eigenfunctions of X. We provide a method to calculate the eigenmodes up to arbitrary order. As an illustration, we give the first modes for the spherical spaces mentioned.

Abstract: We study the algebra of complex polynomials which remain invariant under the action of the local Clifford group under conjugation. Within this algebra, we consider the linear spaces of homogeneous polynomials degree by degree and construct bases for these vector spaces for each degree, thereby obtaining a generating set of polynomial invariants. Our approach is based on the description of Clifford operators in terms of linear operations over GF(2). Such a study of polynomial invariants of the local Clifford group is mainly of importance in quantum coding theory, in particular in the classification of binary quantum codes. Some applications in entanglement theory and quantum computing are briefly discussed as well.

Abstract: We calculate the invariant subspaces in the linear representation of the group of algebraic automorphisms of ℂn on the vector space of algebraic vector fields on ℂn and more generally we do this in a setting with parameter As an application to the field of Several Complex Variables we get a new proof of the Andersen–Lempert observation and a parametric version of the Andersen–Lempert theorem Further applications to the question of embeddings of ℂk into ℂn are announced

Abstract: Let $K$ be an infinite field and $R=K[x_1,...,x_n]$ be the polynomial ring. Let $V=V_1, ..., V_m$ be a collection of vector spaces of linear forms. Denote by $A(V)$ the $K$-subalgebra of $R$ generated by the elements of the product $V_1... V_m$. Our goal is to investigate the properties of the algebra $A(V)$ and the relations with two problems in algebraic combinatorics White's and related conjectures on polymatroids and the study of integral posets.