Abstract: A vector equilibrium problem is defined as follows: given a closed convex subset K of a real topological Hausdorff vector space and a bifunction F(x, y) valued in a real ordered locally convex vect...

Abstract: The ring of invariant forms of the full linear group GL(V ) of a finite dimensional vector space over the finite field Fq was computed early in the 20th century by L. E. Dickson [5], and was found to be a graded polynomial algebra on certain generators {cn,i}. This ring of invariants, for q = p, has found use in algebraic topology in work of Milgram–Man [9], Singer [16,17], Adams–Wilkerson [1], Rector [13], Lam [6], Mui [11], and Smith–Switzer [18]. The aim of this exposition is to give a simple proof of the structure of the ring of invariants, and to compute the action of the Steenrod algebra on the generators of the invariants. The methods used are implicit in Adams–Wilkerson. Dickson’s viewpoint was to vastly generalize the defining equation of Fq,

Abstract: A generalized form of vectorial equilibria is proposed, and, using an abstract monotonicity condition, an existence result is demonstrated.

Abstract: The purpose of this paper is to extend Himmelberg's fixed point theorem replacing the usual convexity in topological vector spaces by an abstract topological notion of convexity which generalizes classical convexity as well as several metric convexity structures found in the literature. We prove the existence, under weak hypothese, of a fixed point for a compact approachable map and we provide sufficient conditions under which this result applies to maps whose values are convex in the abstract sense mentionned above.

Abstract: We consider the problem of minimizing a separable convex objective function over the linear space given by a system Mx=0 with M a totally unimodular matrix. In particular, this generalizes the usual minimum linear cost circulation and cocirculation problems in a network and the problems of determining the Euclidean distance from a point to the perfect bipartite matching polytope and the feasible flows polyhedron. We first show that the idea of minimum mean cycle canceling originally worked out for linear cost circulations by Goldberg and Tarjan [J. Assoc. Comput. Mach., 36 (1989), pp. 873--886.] and extended to some other problems [T. R. Ervolina and S. T. McCormick, Discrete Appl. Math., 46 (1993), pp. 133--165], [A. Frank and A. V. Karzanov, Technical Report RR 895-M, Laboratoire ARTEMIS IMAG, Universite Joseph Fourier, Grenoble, France, 1992], [T. Ibaraki, A. V. Karzanov, and H. Nagamochi, private communication, 1993], [M. Hadjiat, Technical Report, Groupe Intelligence Artificielle, Faculte des Sciences de Luminy, Marseille, France, 1994] can be generalized to give a combinatorial method with geometric convergence for our problem. We also generalize the computationally more efficient cancel-and-tighten method. We then consider objective functions that are piecewise linear, pure and piecewise quadratic, or piecewise mixed linear and quadratic, and we show how both methods can be implemented to find exact solutions in polynomial time (strongly polynomial in the piecewise linear case). These implementations are then further specialized for finding circulations and cocirculations in a network. We finish by showing how to extend our methods to find optimal integer solutions, to linear spaces of larger fractionality, and to the case when the objective functions are given by approximate oracles.

Abstract: LetT be an interval exchange transformation onN intervals whose lengths lie in a quadratic number field. Let {Tn}n=1∞ be any sequence of interval exchange transformations such thatT1 =T andTn is the first return map induced byTn-1 on one of its exchanged intervals In-1. We prove that {Tn}n=1∞ contains finitely many transformations up to rescaling. If the interval In is chosen according to a consistent pattern of induction, e.g., the first interval is chosen, then there existk,n0 ∈ ℤ+, λ ∈R+ such that for alln ≥n0,In = λIn+k andTn,Tn+k are the same up to rescaling. Rephrased arithmetically, this says that a certain family of vectorial division algorithms, applied to quadratic vector spaces, yields sequences of remainders that are eventually periodic. WhenN = 2 the assertion reduces to Lagrange’s classical theorem that the simple continued fraction expansion of a quadratic irrational is eventually periodic. We also discuss the case of periodic induced sequences.

Abstract: 1 Review of algebra.- 1.1 Introduction.- 1.2 Definitions and examples of groups.- 1.3 Subgroups, cosets, and quotients.- 1.4 Ideals.- 1.5 Mappings.- 1.6 Finitely generated abelian groups.- 1.6.1 Cyclic groups.- 1.6.2 Free abelian groups.- References.- 2 Linear algebra and abelian groups.- 2.1 Introduction.- 2.2 Vector space L(A).- 3 Fourier transform over A.- 3.1 Introduction.- 3.2 Character groups.- 3.3 Character formulas.- 3.4 Duality theory.- 3.5 Character group basis.- 3.6 Fourier transform.- 3.7 Shift and multiplication operators.- References.- Problems.- 4 Poisson summation formula.- 4.1 Introduction.- 4.2 Statement and proof.- 4.3 Fourier transform of periodic functions.- 4.4 Periodization-decimation operators.- References.- Problems.- 5 Zak transform.- 5.1 Introduction.- 5.2 Fourier analysis on A x A*.- 5.3 Zak transform.- 5.4 Functional equation.- 5.5 Fourier and Zak transform.- 5.6 Isometry.- 5.7 Algorithm for computing Zak transform.- References.- Problems.- 6 Weyl-Heisenberg systems.- 6.1 Introduction.- 6.2 Translates.- 6.3 W-H systems.- 6.4 Sampling rates.- 6.5 Divide-and-conquer.- References.- Problems.- 7 Zak transform and W-H systems.- 7.1 Introduction.- 7.2 Basic results.- 7.3 Fundamental formulas.- 7.4 Zak space characterization of W-H systems.- 7.4.1 Critical sampling subgroup.- 7.4.2 Integer over-sampling subgroup.- 7.4.3 General sampling subgroup.- 7.5 Zero set characterization.- 7.5.1 Critical sampling subgroup.- 7.5.2 Integer over-sampling subgroup.- Problems.- 8 Algorithms for W-H systems.- 8.1 Introduction.- 8.2 Critical sampling algorithm.- 8.3 Integer over-sampling algorithm.- 8.3.1 Reducing the problem.- 8.4 General over-sampling algorithm.- 8.4.1 Reducing the problem.- References.- Problems.- 9 Orthogonal projection theorem.- 9.1 Introduction.- 9.2 Orthogonal projection algorithm.- 9.3 Iterative W-H coefficient set algorithm.- 10 Cross-ambiguity function.- 10.1 Introduction.- 10.2 Basic results.- 10.3 Direct algorithm.- 10.4 Critical sampling algorithm.- 10.5 Integer over-sampling algorithm.- 10.6 General divide-and-conquer algorithm.- References.- Problems.- 11 Ambiguity surfaces.- 11.1 Introduction.- 11.2 Fourier transform of ambiguity surfaces.- 11.3 Formulas D1 and D2.- References.- 12 Orthonormal W-H systems.- 12.1 Introduction.- 12.2 Orthonormal W-H systems.- 12.2.1 Critical sampling subgroup.- 12.2.2 Integer over-sampling subgroup.- 12.2.3 Over-sampling subgroup ?.- References.- Problems.- 13 Duality.- 13.1 Introduction.- 13.2 Biorthogonal.- 13.3 Algorithms for computing biorthogonals.- 13.3.1 ?-periodization.- 13.3.2 Critical sampling subgroup.- 13.3.3 Integer over-sampling subgroup.- 13.3.4 General over-sampling subgroup.- References.- Problems.- 14 Frames.- 14.1 Introduction.- 14.2 Frame Operator.- 14.2.1 Critical sampling subgroup.- 14.2.2 Integer over-sampling subgroup.- 14.2.3 General over-sampling subgroup.- 14.3 Frames.- 14.4 Frame biorthogonals.- 14.5 Operator interpretation.- 14.6 Tight frames.- References.- 15 Implementation.- 15.1 Introduction.- 15.2 Tensor product.- 15.3 Multidimensional arrays.- 15.3.1 Two-dimensional arrays.- 15.3.2 Multidimensional arrays.- 15.4 Computing the Zak transform.- 15.4.1 1-dimensional Zak transform.- 15.4.2 Two-dimensional Zak transform.- References.- Problems.- 16 Algebra of multirate structures.- 16.1 Introduction.- 16.2 Algebra.- 16.3 Exact sequences.- 16.4 Main theorem.- 16.5 Representatives mapping theorem.- 16.6 Integer matrices.- References.- Problems.- 17 Multirate structures.- 17.1 Introduction.- 17.2 Decimation operator.- 17.2.1 Elementary formulas.- 17.2.2 Shift-invariant operators.- 17.3 Polyphase representation.- 17.3.1 Analysis bands.- 17.3.2 Synthesis bands.- 17.4 Integer rate bands.- 17.5 Reduction theorems.- 17.5.1 Subgroup of a band.- 17.5.2 Nonuniform integer sampling rate filter banks.- 17.5.3 Rational sampling rate filter banks.- 17.6 Decimators and expanders.- References.- 18 A Time-frequency search for stock market anomalies.- 18.1 Introduction.- 18.2 Time-frequency trees.- 18.2.1 Adaptive segmentation via time-frequency trees.- 18.2.2 Recombination to a nondyadic split.- 18.3 Analysis of the log-differenced DJIA and S&P 500 data.- References.

Abstract: A vector space is used to represent periodic voltage and current signals in a single port electrical network. This representation is convenient for expressing time average power quantities, such as average power, apparent power, and certain definitions of reactive power. The vector representation for reactive power provides insight into the difficulty of finding a consistent definition for a scalar measure of reactive power when harmonics are present. Instead, a reactive power vector is defined and shown to obey conservation according to Tellegen's theorem. Projections of the reactive power vector are shown to obey conservation and can be used as signed, scalar measures of reactive power.

Abstract: In this paper, we will generalize the vector space construction due to Brickell. This generalization, introduced by Bertilsson, leads to secret sharing schemes with rational information rates in which the secret can be computed efficiently by each qualified group. A one to one correspondence between the generalized construction and linear block codes is stated, and a matrix characterization of the generalized construction is presented. It turns out that the approach of minimal codewords by Massey is a special case of this construction. For general access structures we present an outline of an algorithm for determining whether a rational number can be realized as information rate by means of the generalized vector space construction. If so, the algorithm produces a secret sharing scheme with this information rate.

Abstract: In this paper, we determine the rational orbit decomposition for two prehomogeneous vector spaces associated with the simple group of type G_2.

Abstract: The interpretation of propositions in Lukasiewicz's infinite-valued calculus as answers in Ulam's game with lies--the Boolean case corresponding to the traditional Twenty Questions game--gives added interest to the completeness theorem. The literature contains several different proofs, but they invariably require technical prerequisites from such areas as model-theory, algebraic geometry, or the theory of ordered groups. The aim of this paper is to provide a self-contained proof, only requiring the rudiments of algebra and convexity in finite-dimensional vector spaces.

Abstract: The algorithm for finding a consistent approximation to an inconsistent pairwise comparisons matrix is based on a logarithmic transformation of a pairwise comparisons matrix into a vector space with the Euclidean metric. An orthogonal basis is introduced in the vector space formed by the images of consistent matrices. The required consistent approximation is the orthogonal projection of the transformed matrix onto this space.

Abstract: One of the best-known results of extremal combinatorics is Sperner’s theorem, which asserts that the maximum size of an antichain of subsets of an -element set equals the binomial coefficient , that is, the maximum of the binomial coefficients. In the last twenty years, Sperner’s theorem has been generalized to wide classes of partially ordered sets. It is the purpose of the present paper to propose yet another generalization that strikes in a different direction. We consider the lattice of linear subspaces (through the origin) of the vector space . Because this lattice is infinite, the usual methods of extremal set theory do not apply to it. It turns out, however, that the set of elements of rank of the lattice , that is, the set of all subspaces of dimension of , or Grassmannian, possesses an invariant measure that is unique up to a multiplicative constant. Can this multiplicative constant be chosen in such a way that an analogue of Sperner’s theorem holds for , with measures on Grassmannians replacing binomial coefficients? We show that there is a way of choosing such constants for each level of the lattice that is natural and unique in the sense defined below and for which an analogue of Sperner’s theorem can be proven. The methods of the present note indicate that other results of extremal set theory may be generalized to the lattice by similar reasoning. c 1997 John Wiley & Sons, Inc.

Abstract: We study the Poincar\'e series of the quantum spaces associated to a Hecke operator, i.e., a Yang-Baxter operator satisfying the equation $(x+1)(x-q)=0$. The Poincar\'e series of the corresponding matrix bialgebra is also considered. Using an old result on Poly\'a frequency sequence, we show that the Poincar\'e series of quantum spaces are always rational functions having negative roots and positive poles. In particular, we show that the rank of an even Hecke operator should be rational functions having negative roots and positive poles. In particular, we show that the rank of an even Hecke operator should be greater than the dimension of the vector space it is acting on.

Abstract: In what follows all vector spaces are over the complex field ℂ, H will be a Hilbert space, and elements of H will usually be denoted in lower case Greek letters: ζ, η, …. We shall also write K. for the compact operators on l2. By a concrete operator algebra we mean a subalgebra A of B(H). We shall assume A is norm closed (although this is not usually necessary), but we shall not assume A is selfadjoint (that is, a C*-algebra). In most of the later sections we shall assume the operator algebras have identity of norm 1 or a contractive approximate identity (c.a.i.). This article is a very brief survey of some of our efforts to study the class of all operator algebras1. In other words, what is the “general theory of operator algebras”? There does not appear to be a text in existence which addresses this topic. A few related questions come to mind: What properties does an operator algebra have? What are the good examples of operator algebras? (Good examples might include those of classical interest, or those which illustrate typical behaviour, or which are a good source of counterexamples). Given an algebra, when is it an operator algebra? What are the ‘basic constructions’ with operator algebras? (Such as direct sums, tensor produsts, …) What are interesting classes of modules over operator algebras? For instance what should a “projective module” be? It is fairly clear from talk at this conference that there are several quite diffrent notions of “projective”modules over operator algebras, depending on our context and needs. What is the good notion/notions of isomorphism of operator algebras?

Abstract: Chapter 1: Linear Vector Spaces Chapter 2: Linear Transformations Chapter 3: Finite Dimensional Euclidean Spaces and Cartesian Tensors Chapter 4: Four-Tensors Chapter 5: Applications Appendix 1: Background Appendix 2: Solutions for Selected Problems

Abstract: Functions of a Single Variable: Differentiation Functions of a Single Variable: Integration Series and Limits Functions Defined as Integrals Complex Numbers Differential Equations Power Series Solutions of Differential Equations Orthogonal Polynomials Fourier Series Fourier Transforms Operators Functions of Several Variables Vectors Coordinate Systems The Classical Wave Equation The Schrodinger Equation Determinants Matrices Matrix Eigenvalue Problems Vector Spaces Probability Statistics Numerical Methods Index

Abstract: Chapter 1. New Numbers A Planeful of Integers, Z[i] Circular Numbers, Zn More Integers on the Number Line, Z [v2] Notes Chapter 2. The Division Algorithm Rational Integers Norms Gaussian Numbers Q (v2) Polynomials Notes Chapter 3. The Euclidean Algorithm Bezout's Equation Relatively Prime Numbers Gaussian Integers Notes Chapter 4. Units Elementary Properties Bezout's Equations Wilson's Theorem Orders of Elements: Fermat and Euler Quadratic Residues Z [v2] Notes Chapter 5. Primes Prime Numbers Gaussian Primes Z [v2] Unique Factorization into Primes Zn Notes Chapter 6. Symmetries Symmetries of Figures in the Plane Groups The Cycle Structure of a Permutation Cyclic Groups The Alternating Groups Notes Chapter 7. Matrices Symmetries and Coordinates Two-by-Two Matrices The Ring of Matrices M2(R) Units Complex Numbers and Quaternions Notes Chapter 8. Groups Abstract Groups Subgroups and Cosets Isomorphism The Group of Units of a Finite Field Products of Groups The Euclidean Groups E (1), E (2), and E (3) Notes Chapter 9. Wallpaper Patterns One-Dimensional Patterns Plane Lattices Frieze Patterns Space Groups The 17 Plane Groups Notes Chapter 10. Fields Polynomials Over a Field Kronecker's Construction of Simple Field Extensions Finite Fields Notes Chapter 11. Linear Algebra Vector Spaces Matrices Row Space and Echelon Form Inverses and Elementary Matrices Determinants Notes Chapter 12. Error-Correcting Codes Coding for Redundancy Linear Codes Parity-Check Matrices Cyclic Codes BCH Codes CDs Notes Chapter 13. Appendix: Induction Formulating the n-th Statement The Domino Theory: Iteration Formulating the Induction Statement Squares Templates Recursion Notes Chapter 14. Appendix: The Usual Rules Rings Notes Index

Abstract: The theme of doing quantum mechanics on all abelian groups goes back to Schwinger and Weyl. If the group is a vector space of finite dimension over a non-archimedean locally compact division ring, it is of interest to examine the structure of dynamical systems defined by Hamiltonians analogous to those encountered over the field of real numbers. In this letter a path integral formula for the imaginary time propagators of these Hamiltonians is derived.

Abstract: Dans le present travail, nous etudions une extension du probleme vectoriel generalise d'inequations de type variationnel et nous demontrons l'existence de sa solution dans le contexte des espaces vectoriels topologiques. Divers cas speciaux sont aussi traites.

Abstract: A new orthogonal four-field two-dimensional (2-D) quarter-plane lattice structure with a complete set of reflection coefficients is developed by employing appropriately defined auxiliary prediction errors. This work is the generalization of the three-parameter lattice filter proposed by Parker and Kayran (1984). After the first stage, four auxiliary forward and four auxiliary backward prediction errors are generated in order to obtain a growing number of 2-D reflection coefficients at successive stages. The theory has been proven by using a geometrical formulation based on the mathematical concepts of vector space, orthogonal projection, and subspace decomposition. It is shown that all four quarter-plane filters are orthogonal and thus optimum for all stages. In addition to developing the basic theory, a set of orthogonal backward prediction error fields for successive lattice parameter model stages is presented.

Abstract: Contents Introduction x1 Invariant theory of the space ^ 3 k 8 x2 The xed point set of H 0 x x3 Lie algebra structures on W x4 Intermediate groups x5 An analogue of the Oppenheim conjecture Introduction This is part one of a series of papers. In this series of papers, we consider problems analogous to the Oppenheim conjecture from the viewpoint of prehomogeneous vector spaces. Throughout this paper, k is a eld of characteristic zero. The following theorem, known as the Oppenheim conjecture, was proved by Margulis 19]. Theorem (0.1) (Margulis) Let Q be a real non-degenerate indeenite quadratic form in n 3 variables. Suppose that the corresponding point in P(Sym 2 (R n)) is irrational. Then the set of values of Q at primitive integer points is dense in R. The above theorem for n 5 was conjectured by Oppenheim in 21]. Margulis originally proved that values of Q at integer points can be arbitrarily small (non-trivially of course), which implies that the set of values of Q at integer points is dense in R due to the result of Lewis 18]. The above improved version (the primitive part) is due to Dani{Margulis 8]. A further improvement of this result was obtained by Borel{Prasad 4]. Some partial results were known prior to the work of Margulis. Let f(x) be a degree d form in real n variables x = (x 1 ; ; x n). In the following, we always assume that f is not a multiple of an integral form. Consider the following questions. (1) For any > 0, does there exist x 2 Z n n f0g such that jf(x)j < ? (2) For any > 0, does there exist x 2 Z n n f0g such that 0 < jf(x)j < ? (3) Is the set ff(x) j x 2 Z n g dense in R? For non-degenerate quadratic forms, one typically gets stronger results such as (2) or (3). For forms of higher degree, there is no notion of \ on-degenerate\" forms. However, for (1), this is not necessary. For example, if the form does not depend on the variable x 1 , we may choose x 1 = 1 and x 2 = = x n = 0. For non-degenerate quadratic forms, the result of Lewis 18] implies that (2) implies (3). Oppenheim himself had partial results for quadratic forms in 22], 23]. …

Abstract: It is shown that, for purposes of likelihood calculation, the standard definition of residual in linear models is unsatisfactory. An alternative definition of residual as an element of a quotient space is proposed and exploited. It is shown that the standard REML, or residual, likelihood arises naturally as the normal-theory likelihood based on the ‘observation’ y+X in the quotient space. In addition, for non-normal linear models, the adjusted profile likelihood is shown to be a Laplace approximation to the distribution of the residual in the quotient space.