Helly-type theorems for pseudoline arrangements in P2

We prove duals of Radon's theorem, Helly's theorem, Carathéodory's theorem, and Kirchberger's theorem for arrangements of pseudolines in the real projective plane, which generalize the original versions of those theorems for plane configurations of points. We also prove a topological generalization of the pseudoline-dual of Helly's theorem.     

Linear equations over cones with interior: a solvability theorem with applications to matrix theory

The solvability of linear equations with solutions in the interior of a closed convex cone is characterized. Corollaries include Lyapunov's theorem characterizing stable matrices and a generalization of Stiemke's theorem of the alternative for complex linear inequalities.     

A short proof of the Lagrange-Good formula

A new proof of Good's generalization to several variables of the Lagrange inversion formula is given, which is mainly based on methods used in Rota's theory of polynomial sequences of binomial type.     

Fixed point theorems for precomplete numberings

In the context of his theory of numberings, Ershov showed that Kleene's recursion theorem holds for any precomplete numbering. We discuss various generalizations of this result. Among other things, we show that Arslanov's completeness criterion also holds for every precomplete numbering, and we discuss the relation with Visser's ADN theorem, as well as the uniformity or nonuniformity of the various fixed point theorems. Finally, we base numberings on partial combinatory algebras and prove a generalization of Ershov's theorem in this context.     

An example from power residues of the critical problem of Crapo and Rota

A natural density arising from the author's recent work on a generalization of Artin's conjecture for primitive roots is shown to be essentially the characteristic polynomial of a geometric lattice, as defined by Crapo and Rota. Necessary and sufficient conditions are obtained for the vanishing of this density.     

On Schensted's construction and the multiplication of schur functions

In this paper, certain aspects of Schensted's construction are examined and a generalization is described. From this generalization, the Littlewood-Richardson rule for the multiplication of Schur functions is derived by purely combinatorial methods.     

Finite rank multivariable Hankel forms

We obtain a several variables generalization of Kronecker's well-known result on finite rank Hankel matrices.     

A simple theorem on 3-connectivity

It is well known that for any element of a connected matroid, either the deletion or the contraction of that element preserves connectivity. We prove a simple and natural generalization to 3-connected matroids. This result is used to prove Seymour's generalization of a theorem of Kelmans.     

Projective diagrams over partially ordered sets are free

A generalization of Kaplansky's theorem on projective modules over local rings is used to show that every projective diagram over a poset is free.     

Incidence matrices of t-designs

We point out a generalization of the matrix equation NN^T=(r? λ)I+λJ to t-designs with t>2 and derive extensions of Fisher's, Connor's, and Mann's inequalities for block designs.     

Santaló's inequality on Cn by complex interpolationInégalité de Santaló sur Cn par interpolation complexe

A new approach to Santaló's inequality on $\text{C}{}^{\mathrm{n}}$ is obtained by combining complex interpolation and Berndtsson's generalization of Prékopa's inequality. To cite this article: D. Cordero-Erausquin, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 767–772.     

Matrix Tree Theorems

A simple proof of a directed graph generalization of the Matrix Tree Theorem, sometimes called Maxwell's rule or Kirchhoff's rule, is given. It is based on the idea A. Rényi used to prove Cayley's tree counting formula. The theorem counts rooted arborescences (analogs of forests) in a directed graph with the determinant of a submatrix in a special adjacency matrix. In the proof we show two n-k degree homogeneous polynomials in n variables are equal by applying induction to those terms lacking one variable. An application to a well-known identity and related theorems are given.     

Cone characterizations of positive semidefinite operators on a Hilbert space

We extend finite dimensional results of Han and Mangasarian characterizing positive semidefinite matrices. We solve a linear complementarity problem for an operator defined on a Hilbert space, and state a generalization of Moreau's theorem.     

A generalization of Sylvester's identity on determinants and some applications

A generalization of Sylvester's identity on determinants is proved by elimination techniques. As an application some inequalities for determinants of totally positive or positive definite matrices are derived.     

Hypergeometric series well-poised in SU(n) and a generalization of Biedenharn's G-functions

We prove that Holman's hypergeometric series well-poised in SU(n) satisfy a general difference equation. We make use of the “path sum” function developed by Biedenharn and this equation to show that a special class of these series, multiplied by simple products, may be regarded as a U(n) generalization of Biedenharn and Louck's G(Δ; X) functions for U(3). The fact that these generalized G-functions are polynomials follows from a detailed study of their symmetries and zeros. As a further application of our general difference equations, we give an elementary proof of Holman's U(n) generalization of the ₅F₄(1) summation theorem.     

New generating functions for Jacobi and related polynomials

In this paper the authors prove a generalization of certain generating functions for Jacobi and related polynomials, given recently by H. M. Srivastava. The method used is due to Pólya and Szegö, and it is based on Rodrigues' formula for the Jacobi polynomials and Lagrange's expansion theorem. A number of special and limiting cases of the main result will give rise to a class of generating functions for ultraspherical, Laguerre and Bessel polynomials.     

Torsion of a composite,viscoelastic prismatic bar of rectangular cross-section

The generalization of Ilyushin's approximation method is used to determine the stresses in the quasistatic problem of torsion of a composite, layered viscoelastic prismatic bar of rectangular cross-section. Numerical computations are carried out for the special case of a step function torsional moment.     

Counting non-isomorphic three-connected planar maps

A counting formula for the number of non-isomorphic planar maps with m edges was obtained by V. A. Liskovets, and non-isomorphic 2-connected planar maps were counted by Liskovets and the author. R. W. Robinson's generalization of Polya's counting theory can be applied to these formulae to count, in polynomial time, non-isomorphic planar maps satisfying various sets of restrictions. Here two sets of planar maps are counted: maps with given 2-connected components, and 3-connected maps.     

Asymptotic structural characteristics of fuzzy measure and their applications

In fuzzy measure theory, as Sugeno's fuzzy measures lose additivity in general, the concept ‘almost’, which is well known in classical measure theory, splits into two different concepts, ‘almost’ and ‘pseudo-almost’. In order to replace the additivity, it is quite necessary to investigate some asymptotic behaviors of a fuzzy measure at sequences of sets which are called ‘waxing’ and ‘waning’, and to introduce some new concepts, such as ‘autocontinuity’, ‘converse-autocontinuity’ and ‘pseudo-autocontinuity’. These concepts describe some asymptotic structural characteristics of a fuzzy measure.In this paper, by means of the asymptotic structural characteristics of fuzzy measure, we also give four forms of generalization for both Egoroff's theorem, Riesz's theorem and Lebesgue's theorem respectively, and prove the almost everywhere (pseudo-almost everywhere) convergence theorem, the convergence in measure (pseudo-in measure) theorem of the sequence of fuzzy integrals. In the last two theorems, the employed conditions are not only sufficient, but also necessary.     

A faster,more stable method for computing the pth roots of positive definite matrices

An accelerated, more stable generalization of Newton's method for finding matrix pth roots is developed in a form suitable for finding the positive definite pth root of a positive definite matrix. Numerical examples are given and compared with the corresponding Newton iterates.