Geometric Properties of Generalized Hypergeometric Functions

Let denote the generalized hypergeometric function where no denominator parameter can be zero or a negative integer and (a,n) denotes the ascending factorial notation. Ponnusamy and Vuorinen raised the problem of finding conditions on the parameters a_j > 0, b_j > 0 so that the function is univalent in . The main aim of this paper is to discuss this problem in detail for the case q = 2.     

A new bound for an extension of Mason’s theorem for functions of several variables

In this paper, using the generalized Wronskian, we obtain a new sharp bound for the generalized Masons theorem [1] for functions of several variables. We also show that the Diophantine equation (The generalized Fermat-Catalan equation) where , such that k out of the n-polynomials are constant, and under certain conditions for has no non-constant solution. Received: 20 March 2003     

A new limit theorem for Gaussian stochastic processes possessing the Baxter property

Let(t) be a Gaussian stochastic process with zero mean and correlation functionR(t, s), and let . We obtain conditions for the weak convergence of the sequence of stochastic processes ,t [0, 1] and we find the form of the limiting distribution. As a corollary, in the stationary case we find the limiting distribution asn for .Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 90–97, 1987.     

Asymptotic Formulas for the Solutions of Integro-Differential Equations

We study the problem of asymptotic integration of the linear integro-differential equation

Some boundary-value problems for linear multidimensional second-order hyperbolic equations

For the linear hyperbolic equations

Global convergence result for conjugate gradient methods

Conjugate gradient optimization algorithms depend on the search directions,

On a Jacobian Identity Associated with Real Hyperplane Arrangements

For the mapping is onto R. It was shown by G. Boole in the 1850's that We give an n-dimensional analogue of this result. The proof makes use of the Griffiths–Harris residue theorem from algebraic geometry.     

Coloured Permutations Containing and Avoiding Certain Patterns

Let be the set of all coloured permutations on the symbols 1, 2, . . . , n with colours 1, 2, . . . , r, which is the analogous of the symmetric group when r = 1, and the hyperoctahedral group when r = 2. Let be a subset of d colours; we define to be the set of all coloured permutations . We prove that the number of -avoiding coloured permutations in . We then prove that for any , the number of coloured permutations in which avoid all patterns in except for and contain exactly once equals . Finally, for any , this number equals . These results generalize recent results due to Mansour, Mansour and West, and Simion.AMS Subject Classification: 05A05, 05A15.     

On the Representation of Numbers by the Direct Sums of Some Binary Quadratic Forms

The systems of bases are constructed for the spaces of cusp forms and . Formulas are obtained for the number of representations of a positive integer by the sum of k binary quadratic forms of the kind , of the kind and of the kind .     

Monotonicity of Average Power Means

A new numerical inequality for average power means is presented. Let and let be a sequence of positive numbers. Consider the operator . We denote by the superposition of these operators. The following assertion is proved: if . Bibliography: 2 titles.     

Inequalities of Bernstein type for polynomial splines in L2

For 2-periodic polynomial splines of order r, of minimal defect, with nodes at the points k/n, n , there are established the sharp inequalities

Truncation error bounds for modified continued fractions with applications to special functions

Much recent work has been done to investigate convergence of modified continued fractions (MCF's), following the proof by Thron and Waadeland [35] in 1980 that a limit-periodic MCFK(a _n , 1;x ₁), with andnth approximant

Asymptotics of Laurent Polynomials of Odd Degree Orthogonal with Respect to Varying Exponential Weights

Let denote the linear space over spanned by . Define the (real) inner product , where V satisfies: (i) V is real analytic on ; (ii) ; and (iii) . Orthogonalisation of the (ordered) base with respect to yields the even degree and odd degree orthonormal Laurent polynomials , and . Define the even degree and odd degree monic orthogonal Laurent polynomials: and . Asymptotics in the double-scaling limit such that of (in the entire complex plane), , and (in the entire complex plane) are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on , and then extracting the large-n behaviour by applying the non-linear steepest-descent method introduced in [1] and further developed in [2],[3].     

Alternating Sign Matrices and Some Deformations of Weyl's Denominator Formulas

An alternating sign matrix is a square matrix whose entries are 1, 0, or –1, and which satisfies certain conditions. Permutation matrices are alternating sign matrices. In this paper, we use the (generalized) Littlewood's formulas to expand the products and 2 as sums indexed by sets of alternating sign matrices invariant under a 180° rotation. If we put t = 1, these expansion formulas reduce to the Weyl's denominator formulas for the root systems of type B _n and C _n. A similar deformation of the denominator formula for type D _n is also given.     

Oscillation Theorems for Neutral Differential Equations of Higher Order

In this paper we present some new oscillatory criteria for the n-th order neutral differential equations of the form The results obtained extend and improve a number of existing criteria.     

Asymptotic Invertibility and the Collective Asymptotic Spectral Behavior of Generalized One-Dimensional Discrete Convolutions

We study the asymptotic invertibility as of matrices of the form and , where a and b are functions defined on the sets . The joint asymptotic behavior of the spectrum of these matrices is analyzed.     

Modified (0, 2)-Interpolation on the Roots of Jacobi Polynomials. I (Explicit Formulae)

Let the set of knots

Reduced Words and Plane Partitions

Let w ₀ be the element of maximal length in thesymmetric group S _n , and let Red(w ₀) bethe set of all reduced words for w ₀. We prove the identity which generalizes Stanley's [20] formula forthe cardinality of Red(w ₀), and Macdonald's [11] formula .Our approach uses anobservation, based on a result by Wachs [21], that evaluation of certainspecializations of Schubert polynomials is essentially equivalent toenumeration of plane partitions whose parts are bounded from above. Thus,enumerative results for reduced words can be obtained from the correspondingstatements about plane partitions, and vice versa. In particular, identity(*) follows from Proctor's [14] formula for the number of planepartitions of a staircase shape, with bounded largest part.Similar results are obtained for other permutations and shapes;q-analogues are also given.     

On inequalities for norms of intermediate derivatives on a finite interval

For functionsf which have an absolute continuous (n–1)th derivative on the interval [0, 1], it is proved that, in the case ofn>4, the inequality

Strict inequalities for the derivatives of functions satisfying certain boundary conditions

For functions satisfying the boundary conditions