Summations and transformations for multiple basic and elliptic hypergeometric series by determinant evaluations

Using multiple q-integrals and a determinant evaluation, we establish a multivariable extension of Bailey's nonterminating 1009 transformation. From this result, we deduce new multivariable terminating ₁₀φ₉ transformations, ₈φ₇ summations and other identities. We also use similar methods to derive new multivariable l ₁ψ₁ summations. Some of our results are extended to the case of elliptic hypergeometric series.     

Logic of Determination of Objects (LDO): How to Articulate ��Extension�� with ��Intension�� and ��Objects�� with ��Concepts��

From a logical viewpoint, object is never defined, even by a negative definition. This paper is a theoretical contribution about object using a new constructivist logical approach called Logic of Determination of Objects founded on a basic operation, called determination. This new logic takes into account cognitive problems such as the inheritance of properties by non typical occurrences or by indeterminate atypical objects in opposition to prototypes that are typical completely determinate objects. We show how extensional classes, intensions, more and less determined objects, more or less typical representatives of a concept and prototypes are defined and organized, using a determination operation that constructs a class of indeterminate objects from an object representation of a concept called typical object.     

Implicit multifunction theorems for the sensitivity analysis of variational conditions

We study implicit multifunctions (set-valued mappings) obtained from inclusions of the form 0∈M(p,x), whereM is a multifunction. Our basic implicit multifunction theorem provides an approximation for a generalized derivative of the implicit multifunction in terms of the derivative of the multifunctionM. Our primary focus is on three special cases of inclusions 0∈M(p,x) which represent different kinds of generalized variational inequalities, called “variational conditions”. Appropriate versions of our basic implicit multifunction theorem yield approximations for generalized derivatives of the solutions to each kind of variational condition. We characterize a well-known generalized Lipschitz property in terms of generalized derivatives, and use our implicit multifunction theorems to state sufficient conditions (and necessary in one case) for solutions of variational conditions to possess this Lipschitz, property. We apply our results to a general parameterized nonlinear programming problem, and derive a new second-order condition which guarantees that the stationary points associated with the Karush-Kuhn-Tucker conditions exhibit generalized Lipschitz continuity with respect to the parameter.     

An iterative algorithm based on -proximal mappings for a system of generalized implicit variational inclusions in Banach spaces

In this paper, we give the notion of M-proximal mapping, an extension of P-proximal mapping given in [X.P. Ding, F.Q. Xia, A new class of completely generalized quasi-variational inclusions in Banach spaces, J. Comput. Appl. Math. 147 (2002) 369–383], for a nonconvex, proper, lower semicontinuous and subdifferentiable functional on Banach space and prove its existence and Lipschitz continuity. Further, we consider a system of generalized implicit variational inclusions in Banach spaces and show its equivalence with a system of implicit Wiener–Hopf equations using the concept of M-proximal mappings. Using this equivalence, we propose a new iterative algorithm for the system of generalized implicit variational inclusions. Furthermore, we prove the existence of solution of the system of generalized implicit variational inclusions and discuss the convergence and stability analysis of the iterative algorithm.     

Comments on monoreflections in unital archimedean lattice-ordered groups and rings

In quite general concrete categories, the monoreflections which are H-closed (closed under homomorphic image) are the “naturally occuring” ones, since each is comprised of objects closed under some family of functorial implicit operations. In the specific categories under consideration here, we have earlier shown the class of H-closed monoreflections to be a set, indeed, relatively small. In spite, or because of that very few non-H-closed monoreflections have been exhibited, in fact, none in the categories considered here. We shall exhibit many and assuming no measurable cardinal, a proper class.     

Proof of Andrews' conjecture on a 4φ3 summation

We give a new proof of a ₄φ₃ summation due to G.E. Andrews and confirm another ₄φ₃ summation conjectured by him recently. Some variations of these two ₄φ₃ summations are also given.     

Improved Bounds on the Union Complexity of Fat Objects

We introduce a new class of fat, not necessarily convex or polygonal, objects in the plane, namely locally γ-fat objects. We prove that the union complexity of any set of n such objects is O(λ _s+2(n)log ² n). This improves the best known bound, and extends it to a more general class of objects. The research was supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 639.023.301.     

COMPOSITION SUPERALGEBRAS

ABSTRACT

The super counterparts of both unital composition algebras and of symmetric composition algebras are classified. It turns out that new objects arise only over fields of characteristic 3. The Principle of Local Triality is considered for this new objects. It is related to some order 3 automorphisms of the simple Liesuperalgebras osp(1,2) and osp(4,2) in characteristic 3.     

An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities

Summary It is shown that every proper-hypergeometric multisum/integral identity, orq-identity, with a fixed number of summations and/or integration signs, possesses a short, computer-constructible proof. We give a fast algorithm for finding such proofs. Most of the identities that involve the classical special functions of mathematical physics are readily reducible to the kind of identities treated here. We give many examples of the method, including computer-generated proofs of identities of Mehta-Dyson, Selberg, Hille-Hardy,q-Saalschütz, and others. The prospect of using the method for proving multivariate identities that involve an arbitrary number of summations/integrations is discussed.Supported in part by the U.S. Office of Naval ResearchSupported in part by the U.S. National Science Foundation     

Analogues of Euler and Poisson summation formulae

Euler-Maclaurin and Poisson analogues of the summations ε _{a <n ≤b} χ(n)f(n), have been obtained in a unified manner, where (χ(n)) is a periodic complex sequence;d(n) is the divisor function andf(x) is a sufficiently smooth function on [a, b]. We also state a generalised Abel’s summation formula, generalised Euler’s summation formula and Euler’s summation formula in several variables.     

On an Adaptive Estimator of the Least Contrast in a Model with Nonlinear Functional Relations

We consider an implicit nonlinear functional model with errors in variables. On the basis of the concept of deconvolution, we propose a new adaptive estimator of the least contrast of the regression parameter. We formulate sufficient conditions for the consistency of this estimator. We consider several examples within the framework of the L ₁- and L ₂-approaches.     

An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation

We report a new unconditionally stable implicit alternating direction implicit (ADI) scheme of O(k² + h²) for the difference solution of linear hyperbolic equation u_tt + 2αu_t + β²u = u_xx + u_yy + f(x, y, t), αβ ≥ 0, 0 < x, y < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where α > 0 and β ≥ 0 are real numbers. The resulting system of algebraic equations is solved by split method. Numerical results are provided to demonstrate the efficiency and accuracy of the method. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 684–688, 2001     

Study of a special nonlinear problem arising in convex semi-infinite programming

We consider convex problems of semi-infinite programming (SIP) using an approach based on the implicit optimality criterion. This criterion allows one to replace optimality conditions for a feasible solution x ⁰ of the convex SIP problem by such conditions for x ⁰ in some nonlinear programming (NLP) problem denoted by NLP(I(x ⁰)). This nonlinear problem, constructed on the base of special characteristics of the original SIP problem, so-called immobile indices and their immobility orders, has a special structure and a diversity of important properties. We study these properties and use them to obtain efficient explicit optimality conditions for the problem NLP(I(x ⁰)). Application of these conditions, together with the implicit optimality criterion, gives new efficient optimality conditions for convex SIP problems. Special attention is paid to SIP problems whose constraints do not satisfy the Slater condition and to problems with analytic constraint functions for which we obtain optimality conditions in the form of a criterion. Comparison with some known optimality conditions for convex SIP is provided.     

Cubic algebraic spline curves design

In this paper we propose a construction method of the planar cubic algebraic spline curve with endpoint interpolation conditions and a specific analysis of its properties. The piecewise cubic algebraic curve has G ² continuous contact with the control polygon at two endpoints and is G ² continuous between each segments of itself. The process of this method is simple and clear, and provides a new way of thinking to design implicit curves.     

Efficient higher order implicit one-step methods for integration of stiff differential equations

A new implicit integration method is presented which can efficiently be applied in the solution of (stiff) differential equations. The given formulas are of a modified implicit Runge-Kutta type and areA-stable. They may containA-stable embedded methods for error estimation and step-size control.     

Efficient similarity factorization of rank-1 modifications

For the solution of stiff ODEs by implicit methods one has to solve sequences of algebraic systems, whose Jacobian is the identity minus a multiple of the 'system' Jacobian of the right hand side. The latter can be successively approximated by a new 'two-sided rank-1' formula yielding only a moderate increase in the number of iterations compared to Newton's method. However, so far we have only been able to achieve the reduction in the linear algebra effort per integration step from O (n 3) to O (n 2) as long as the size is kept constant. To obtain an effort of O (n 2) seems to require an update of a similarity reduction of the system Jacobian to Hessenberg form or even a band matrix. As we have found no O (n 2) algorithms for updating such factorizations in literature, we discuss an idea to approximate the update with enough accuracy to ensure rapid convergence of the underlying implicit equation solver. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the modularity of the GL<Subscript>2</Subscript>-twisted spinor L-function

In modern number theory there are famous theorems on the modularity of Dirichlet series attached to geometric or arithmetic objects. There is Hecke’s converse theorem, Wiles proof of the Taniyama-Shimura conjecture, and Fermat’s Last Theorem to name a few. In this article in the spirit of the Langlands philosophy we raise the question on the modularity of the GL₂-twisted spinor L-function Z _{G ⊗ h} (s) related to automorphic forms G,h on the symplectic group GSp₂ and GL₂. This leads to several promising results and finally culminates into a precise very general conjecture. This gives new insights into the Miyawaki conjecture on spinor L-functions of modular forms. We indicate how this topic is related to Ramakrishnan’s work on the modularity of the Rankin-Selberg L-series.     

A Graph b-coloring Framework for Data Clustering

The graph b-coloring is an interesting technique that can be applied to various domains. The proper b-coloring problem is the assignment of colors (classes) to the vertices of one graph so that no two adjacent vertices have the same color, and for each color class there exists at least one dominating vertex which is adjacent (dissimilar) to all other color classes. This paper presents a new graph b-coloring framework for clustering heterogeneous objects into groups. A number of cluster validity indices are also reviewed. Such indices can be used for automatically determining the optimal partition. The proposed approach has interesting properties and gives good results on benchmark data set as well as on real medical data set.