Spaces with Almost Strictly Totally Positive Bases

In spline spaces there are often totally positive bases possessing a strong property called almost strictly total positivity. In this paper, it is proved that, for totally positive bases of continuous functions B, the following concepts are equivalent: (i) B is almost strictly totally positive, (ii) B satifies a Schoenberg-Whitney Theorem, (iii) The functions in B are locally linearly independent. Some classical examples of almost strictly totally positive bases are given, completing the knowledge of their properties known in the mathematical literature. Some criteria to know the existence of almost strictly totally positive bases are also derived.     

What is a multi-parameter renewal process?

The concept of the renewal property is extended to processes indexed by a multidimensional time parameter. The definition given includes not only partial sum processes, but also Poisson processes and many other point processes whose jump points are not totally ordered. A new version of the waiting time paradox is proven for multidimensional Poisson processes, and is shown to imply the renewal property. Finally, martingale properties of renewal processes are studied.     

Stable Functions and Extension of Vietoris’ Theorem

In this work, using subordination results for the Cesáro sum of certain analytic functions, a concept called Cesáro stable is given and two conjectures in this direction are also proposed. Using an extension of Vietoris’ theorem on positivity of cosine sums, some results on positivity of cosine and sine sums are obtained. These results are useful in discussing some particular cases of the proposed conjectures.     

Positive Refinable Operators

Recently, linear positive operators of Bernstein–Schoenberg type, relative to B-splines bases, have been considered. The properties of these operators are derived mainly from the total positivity of normalized B-spline bases. In this paper we shall construct a generalization of the operator considered in [15] by means of normalized totally positive bases generated by a particular class of totally positive scaling functions. Next, we shall study its approximation properties. Our results can be established also for more general sequences of normalized totally positive bases.     

Dividierte Differenzen und Monotonie von Quadraturformeln

Summary As a generalisation of divided differences we consider linear functionals vanishing for polynomials of given degree and with discrete support. It is shown that functionals of that type may be uniquely represented by a linear combination of divided differences. On the basis of this representation theorem we introduce the concept of positivity and definiteness of functions and linear functionals. Next we show that in many cases positivity follows from the number of sign changes of the coefficients of the given linear functional. These results may be applied to the problems of nonexistence of Newton-Côtes and Gegenbauer quadrature formulas with positive weights and to the monotony problem of Gauss and Newton Côtes quadrature.

     

On the characterization of almost strictly totally positive matrices

A nonsingular matrix is called almost strictly totally positive when all its minors are nonnegative and, furthermore, these minors are strictly positive if and only if their diagonal entries are strictly positive. Almost strictly totally positive matrices are useful in Approximation Theory and Computer Aided Geometric Design to generate bases of functions with good shape preserving properties. In this paper we give an algorithmic characterization of these matrices. Moreover, we provide a determinantal characterization of them in terms of the positivity of a very reduced number of their minors and also in terms of their factorizations. Both authors were partially supported by the DGICYT Spain Research Grant PB93-0310     

Totally interpolating biorthogonal multiwavelet systems with approximation order

In this article, compactly supported totally interpolating biorthogonal multiwavelet systems are studied. Necessary and sufficient conditions for such systems to have given approximation orders are stated in simple equations. It is shown that the shorter nontrivial filter component that has the minimum possible length for a given approximation order is uniquely determined up to a discrete parameter. Among systems with such property, we provide all totally interpolating biorthogonal stable multiwavelet systems of approximation orders 2 and 3 with minimal total length whose scaling vectors have minimal lengths as well.     

On the characterization of almost strictly totally positive matrices

A nonsingular matrix is called almost strictly totally positive when all its minors are nonnegative and, furthermore, these minors are strictly positive if and only if their diagonal entries are strictly positive. Almost strictly totally positive matrices are useful in Approximation Theory and Computer Aided Geometric Design to generate bases of functions with good shape preserving properties. In this paper we give an algorithmic characterization of these matrices. Moreover, we provide a determinantal characterization of them in terms of the positivity of a very reduced number of their minors and also in terms of their factorizations.     

A Bijection for Eulerian-equivalence Classes of Totally Cyclic Orientations

Gioan showed that the number of cycle reversing classes of totally cyclic orientations of a given graph can be calculated as an evaluation of the corresponding Tutte polynomial. We note that the concept of cycle reversing classes of orientations coincides with that of Eulerian-equivalence classes considered by Chen and Stanley, and Kochol. Based on this coincidence, we give a bijective proof of Gioan’s result. Precisely, the main result of the paper is an algorithmic bijection between the set of Eulerian-equivalence classes of totally cyclic orientations and the set of spanning trees without internally active edges.      

Generalized even and odd totally positive matrices

A generalization of the definition of an oscillatory matrix based on the theory of cones is given in this paper. The positivity and simplicity of all the eigenvalues of a generalized oscillatory matrix are proved. Classes of generalized even and odd oscillatory matrices are introduced. Spectral properties of the obtained matrices are studied. Criteria of generalized even and odd oscillation are given. Examples of generalized even and odd oscillatory matrices are presented.     

Positive Quadratic Forms on Intersections of Quadrics

The positive definiteness of a quadratic form on the cone defined as the set of points at which given quadratic forms vanish or are nonnegative is studied. Necessary and sufficient conditions for the positivity of a quadratic form on this cone are obtained. The results are applied to study second-order sufficient conditions for abnormal extremal problems.     

Gaussian OS-positive processes

Summary Gaussian processes satisfying Osterwalder-Schrader positivity are studied. A representation of the (generalized) covariance function of an OS-positive process as the Laplace transform of an operator-valued probability measure is given. It is shown that every Gaussian OS-positive process has a unique Gaussian canonical Markov extension. An explicit application is made to the generalized free Euclidean fields.Partially supported by the National Science Foundation under grant MCS-76 06332     

Efficient Recognition of Totally Nonnegative Matrix Cells

The space of m×p totally nonnegative real matrices has a stratification into totally nonnegative cells. The largest such cell is the space of totally positive matrices. There is a well-known criterion due to Gasca and Peña for testing a real matrix for total positivity. This criterion involves testing mp minors. In contrast, there is no known small set of minors for testing for total nonnegativity. In this paper, we show that for each of the totally nonnegative cells there is a test for membership which only involves mp minors, thus extending the Gasca and Peña result to all totally nonnegative cells.     

On Positivity Conditions for the Cauchy Function of Functional-Differential Equations

We study how the statements on estimates of solutions to linear functional-differential equations, analogous to the Chaplygin differential inequality theorem, are connected with the positivity of the Cauchy function and the fundamental solution. We prove a comparison theorem for the Cauchy functions and the fundamental solutions to two functional-differential equations. In the theorem, it is assumed that the difference of the operators corresponding to the equations (and acting from the space of absolutely continuous functions to the space of summable ones) is a monotone totally continuous Volterra operator. We also obtain the positivity conditions for the Cauchy function and the fundamental solution to some equations with delay as long as those of neutral type.     

Locatedness and overt sublocales

Locatedness is one of the fundamental notions in constructive mathematics. The existence of a positivity predicate on a locale, i.e. the locale being overt, or open, has proved to be fundamental in constructive locale theory. We show that the two notions are intimately connected.Bishop defines a metric space to be compact if it is complete and totally bounded. A subset of a totally bounded set is again totally bounded iff it is located. So a closed subset of a Bishop compact set is Bishop compact iff it is located. We translate this result to formal topology. ‘Bishop compact’ is translated as compact and overt. We propose a definition of locatedness on subspaces of a formal topology, and prove that a closed subspace of a compact regular formal space is located iff it is overt. Moreover, a Bishop-closed subset of a complete metric space is Bishop compact — that is, totally bounded and complete — iff its localic completion is compact overt.Finally, we show by elementary methods that the points of the Vietoris locale of a compact regular locale are precisely its compact overt sublocales.We work constructively, predicatively and avoid the use of the axiom of countable choice.     

The Isometry of Riemannian Manifold to a Sphere

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Linear and nonlinear degenerate boundary value problems in Besov spaces

The boundary value problems for linear and nonlinear degenerate differential-operator equations in Banach-valued Besov spaces are studied. Several conditions for the separability of linear elliptic problems are given. Moreover, the positivity and the analytic semigroup properties of associated differential operators are obtained. By using these results, the maximal regularity of degenerate boundary value problems for nonlinear differential-operator equations is derived. As applications, boundary value problems for infinite systems of degenerate equations in Besov spaces are studied.     

Locatedness and overt sublocales

Locatedness is one of the fundamental notions in constructive mathematics. The existence of a positivity predicate on a locale, i.e. the locale being overt, or open, has proved to be fundamental in constructive locale theory. We show that the two notions are intimately connected.Bishop defines a metric space to be compact if it is complete and totally bounded. A subset of a totally bounded set is again totally bounded iff it is located. So a closed subset of a Bishop compact set is Bishop compact iff it is located. We translate this result to formal topology. ‘Bishop compact’ is translated as compact and overt. We propose a definition of locatedness on subspaces of a formal topology, and prove that a closed subspace of a compact regular formal space is located iff it is overt. Moreover, a Bishop-closed subset of a complete metric space is Bishop compact — that is, totally bounded and complete — iff its localic completion is compact overt.Finally, we show by elementary methods that the points of the Vietoris locale of a compact regular locale are precisely its compact overt sublocales.We work constructively, predicatively and avoid the use of the axiom of countable choice.     

A Sufficient condition for strict total positivity of a matrix

We establish a sufficient condition for strict total positivity of a matrix In particular, we show that if the (positive) elements of a square matrix grow sufficiently fast as their distance from the diagonal of the matrix increases, then the matrix is strictly totally positive.     

Analytic expansion of the lyapunov exponent associated to the SchrÖDinger operator with random potential

The invariant measure and Lyapunov exponent associated to the one–dimensional Schrödinger operator with a random potential (or, in other words, to the damped linear oscillator with random restoring force) are studied for small real noise (diffusions). Analytic expression are given via perturbation expansion. As a by-product, the well-known positivity of the Lyapunov exponent (in the undamped case) is reproved