A reduction formula for Waring numbers through generalized Paley graphs

Journal of Algebraic Combinatorics - We give a reduction formula for the Waring number g(k, q) over a finite field $$\mathbb {F}_q$$ . By exploiting the relation between g(k, q)...     

A generalized Ihara zeta function formula for simple graphs with bounded degree

We establish a generalized Ihara zeta function formula for simple graphs with bounded degree. This is a generalization of the formula obtained by G. Chinta, J. Jorgenson and A. Karlsson from vertex-transitive graphs.     

The generalized tilt formula

A convex hull construction in Minkowski space defines a canonical cell decomposition for a cusped hyperbolicn-manifold. An algorithm to compute the canonical cell decomposition uses the concept of the tilt of ann-simplex relative to each of its (n–1)-dimensional faces. An essential tool for computing tilts is the tilt theorem. The tilt theorem was previously known only in dimensionsn3, and the proof was needlessly complicated. Here we offer a new, simplified proof which applies in all dimensions. We also offer a second geometric interpretation of the tilt.     

Ramanujan formula for the generalized Stirling approximation

The aim of this paper is to improve Ramanujan’s formula for approximation of the factorial function, starting from Burnside’s formula in contradistinction with the classical formula that starts from Stirling’s formula.     

The generalized hu-meyer formula for random kernels

A theory of Hilbert-space-valued traces and multiple integration is developed for kernels inL ²([0, 1]^p × Θ). The multiple Ogawa and the multiple Stratonovich integrals for such kernels are introduced and sufficient conditions for their existence are obtained. The derivation of the Hu-Meyer formula connecting the multiple Ogawa and the multiple Stratonovich integrals requires the introduction of traces of random kernels. Such a derivation is obtained under appropriate conditions. This research was supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. F49620 92 J 0154 and the Army Research Office Grant No. DAAL03-92-G-0008.     

Integral representation formula for generalized normal derivations

For generalized normal derivations, acting on the space of all bounded Hilbert space operators, the following integral representation formulas hold:

 

and

 

whenever is a Hilbert-Schmidt class operator and is a Lipschitz class function on Applying this formula, we extend the Fuglede-Putnam theorem concerning commutativity modulo Hilbert-Schmidt class, as well as trace inequalities for covariance matrices of Muir and Wong. Some new monotone matrix functions and norm inequalities are also derived.

     

Product formula for the generalized q-Bessel function

The aim of this paper is to establish a product formula for the generalized q-Bessel function which is a generalization of the known q-Bessel functions of kind 1,2,3, the modified q-Bessel functions of kind 1,2,3, and the new q-analogy of the modified Bessel function presented and studied by Mansour and Al-Shomarani. As an application of the product formula we derive Turán-type inequality for the modified q-Bessel function of third kind.     

Infinite generalized friendship graphs

We give necessary and sufficient conditions for the existence of infinite generalized friendship graphs and show that there are 2° non-isomorphic ones of each admissible order c and chromatic number. Further we prove that such graphs and their complements are almost always regular of degree equal to the order and that various generalizations of the Friendship Theorem do not hold for infinite generalized friendship graphs.     

Skewness of generalized Petersen graphs and related graphs

The skewness of a graph G is the minimum number of edges in G whose removal results in a planar graph. In this paper, we determine the skewness of the generalized Petersen graph P(4k, k) and hence a lower bound for the crossing number of P(4k, k). In addition, an upper bound for the crossing number of P(4k, k) is also given.     

On extremal problems of graphs and generalized graphs

Anr-graph is a graph whose basic elements are its vertices and r-tuples. It is proved that to everyl andr there is anε(l, r) so that forn>n ₀ everyr-graph ofn vertices andn ^{r−ε(l, r)} r-tuples containsr. l verticesx ^(j), 1≦j≦r, 1≦i≦l, so that all ther-tuples occur in ther-graph.     

Pieri’s formula for generalized Schur polynomials

Young's lattice, the lattice of all Young diagrams, has the Robinson-Schensted-Knuth correspondence, the correspondence between certain matrices and pairs of semi-standard Young tableaux with the same shape. Fomin introduced generalized Schur operators to generalize the Robinson-Schensted-Knuth correspondence. In this sense, generalized Schur operators are generalizations of semi-standard Young tableaux. We define a generalization of Schur polynomials as expansion coefficients of generalized Schur operators. We show that the commutation relation of generalized Schur operators implies Pieri's formula for generalized Schur polynomials.     

A generalized Birkhoff-Young-Chebyshev quadrature formula for analytic functions

A generalized N-point Birkhoff-Young quadrature of interpolatory type, with the Chebyshev weight, for numerical integration of analytic functions is considered. The nodes of such a quadrature are characterized by an orthogonality relation. Some special cases of this quadrature formula are derived.