Numerical bifurcation and stability analysis for steady-states of reaction diffusion equations

Solutions of a semilinear elliptic boundary value problem, (with bounded below) can be put into a one-to-one correspondence with zeros of a function . Often d is small. The function is called the bifurcation function. It can also be shown that the eigenvalues of the matrix characterize the stability properties of the solutions of the elliptic problem as rest points of . A finite element method that can be used for computing B and B _c has recently been proposed. An overview of these results and the finite element method is given. This revised version was published online in June 2006 with corrections to the Cover Date.     

Asymptotic zero distribution of hypergeometric polynomials

We show that the zeros of the hypergeometric polynomials , , cluster on the loop of the lemniscate as . We also state the equations of the curves on which the zeros of , lie asymptotically as . Auxiliary results for the asymptotic zero distribution of other functions related to hypergeometric polynomials are proved, including Jacobi polynomials with varying parameters and associated Legendre functions. Graphical evidence is provided using Mathematica. This revised version was published online in June 2006 with corrections to the Cover Date.     

Construction of conjugate quadrature filters with specified zeros

Let ℂ denote the complex numbers and denote the ring of complex-valued Laurent polynomial functions on ℂ\{0}. Furthermore, we denote by the subsets of Laurent polynomials whose restriction to the unit circle is real, nonnegative, respectively. We prove that for any two Laurent polynomials , which have no common zeros in ℂ\{0} there exists a pair of Laurent polynomials satisfying the equation Q ₁ P ₁ + Q ₂ P ₂ = 1. We provide some information about the minimal length Laurent polynomials Q ₁ and Q ₂ with these properties and describe an algorithm to compute them. We apply this result to design a conjugate quadrature filter whose zeros contain an arbitrary finite subset Λ⊂ℂ\{0} with the property that for every implies and . This revised version was published online in June 2006 with corrections to the Cover Date.     

Accelerating infinite products

Slowly convergent infinite products are considered, where is a sequence of numbers, or a sequence of linear operators. Using an asymptotic expansion for the “remainder” of the infinite product a method for convergence acceleration is suggested. The method is in the spirit of the d-transformation for series. It is very simple and efficient for some classes of sequences . For complicated sequences it involves the solution of some linear systems, but it is still effective. This revised version was published online in June 2006 with corrections to the Cover Date.     

Stable rational modification of a weight

Consider a weight and a rational modification of it is a polynomial. An algorithm is stabilized for the following problem: “Given the coefficients of the three-term recurrence equation satisfied by the orthogonal polynomials relative to , compute the coefficients of the recurrence equation satisfied by the orthogonal polynomials relative to .” This revised version was published online in June 2006 with corrections to the Cover Date.     

Temperature-Dependence of Equilibrium States of a Two-Phase Elastic Medium with the Zero Surface Tension Coefficient

We consider the energy functional of a two-phase elastic medium with quadratic energy densities defined for such that ,where is a measurable characteristic function. Under some natural conditions on the data of the problem, we prove the existence of an interval (t ^-,t ⁺) of the change of temperature such that the energy functional has only a minimizer such that for or such that t^ + $$ " align="middle" border="0"> . The energy functional has no minimizers such that or if . We derive two-sided estimates for the numbers in terms of the characteristics of the two-phase elastic medium and the boundary condition. Bibliography: 3 titles.     

Profinite and Continuous Higher K-Theory of Exact Categories, Orders, and Group Rings

Let be a rational prime, an exact category. In this article, we define and study for all , the profinite higher K-theory of , that is as well as , where is the -dimensional mod- Moore space. We study connections between and prove several -completeness results involving these and associated groups including the cases where is the category of finitely generated (resp. finitely generated projective) modules over orders in semi-simple algebras over number fields and p-adic fields. We also define and study continuous K-theory of orders in p-adic semi-simple algebras and show some connection between the profinite and continuous K-theory of .     

Approximation of (<Emphasis Type="Italic">ψ, β</Emphasis>)-differentiable functions defined on the real axis by Weierstrass operators

We obtain asymptotic equalities for the upper bounds of approximations by Weierstrass operators on the functional classes and in the metrics of the spaces Ĉ and respectively. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 9, pp. 1201–1220, September, 2007.     

The C r-fundamental splines of Clough–Tocher and Powell–Sabin types for Lagrange interpolation on a three direction mesh

Let Δ⁽¹⁾ be the uniform three direction mesh of the plane whose vertices are integer points of .Let (respectively of degree d=3r (respectively d=3r+1 ) for r odd (respectively even) on the triangulation , and of degree d=2r (respectively d=2r+1) for r odd (respectively even) on the triangulation . Using linear combinations of translates of these splines we obtain Lagrange interpolants whose corresponding order of approximation is optimal. This revised version was published online in June 2006 with corrections to the Cover Date.     

Quasi-uniform completions of partially ordered spaces

In this paper we define partially ordered quasi-uniform spaces (X, , ≤) (PO-quasi-uniform spaces) as those space with a biconvex quasi-uniformity on the poset (X, ≤) and give a construction of a (transitive) biconvex compatible quasi-uniformity on a partially ordered topological space when its topology satisfies certain natural conditions. We also show that under certain conditions on the topology of a PO-quasi-uniform space (X, , ≤), the bicompletion of (X, ) is also a PO-quasi-uniform space ( , ⪯) with a partial order ⪯ on that extends ≤ in a natural way.      

Well-posedness for the 2D dissipative quasigeostrophic equations in the Besov space

In this paper, we consider the initial value problem of the 2D dissipative quasi-geostrophic equations. Existence and uniqueness of the solution global in time are proved in the homogenous Besov space Bp,∞ s p with small data when 1 /2<α≤1,2/2α-1< p<∞,sp=2/p-(2α-1). Our proof is based on a new characterization of the homogenous Besov space and Kato's method.     

Optimal constant in a new estimate for the degree

In this paper, we prove the estimate

Cofinality of normal ideals on<Emphasis Type="Italic">P</Emphasis><Subscript>κ</Subscript>(λ) II

For an idealJ on an infinite setX with add(J)=κ, let be the smallest size of any subfamilyY ofJ with the property that any member ofJ can be covered by less than κ members ofY. We study the value of forA in , where denotes the smallest [δ]^<θ ideal onP _κ(λ). We also discuss the problem of whether there exists a setA such that , or even . Some of the material in this paper originally appeared as part of the author's doctoral dissertation completed at the Université de Caen, 1998. Partially supported by the Israel Science Foundation. Publication 813.     

Boundary Estimates for Solutions of the Parabolic Free Boundary Problem

Let u and solve the problem

On the area swept under the occupation process of an M/M/1 queue in a busy period

We compute in this paper the distribution of the area swept under the occupation process of an M/M/1 queue during a busy period. For this purpose, we use the expression of the Laplace transform of the random variable established in earlier studies as a fraction of Bessel functions. To get information on the poles and the residues of , we take benefit of the fact that this function can be represented by a continued fraction. We then show that this continued fraction is the even part of an S fraction and we identify its successive denominators by means of Lommel polynomials. This allows us to numerically evaluate the poles and the residues. Numerical evidence shows that the poles are very close to the numbers as . This motivated us to formulate some conjectures, which lead to the derivation of the asymptotic behaviour of the poles and the residues. This is finally used to derive the asymptotic behaviour of the probability survivor function . The outstanding property of the random variable is that the poles accumulate at 0 and its tail does not exhibit a nice exponential decay but a decay of the form for some positive constants c and , which indicates that the random variable has a Weibull-like tail. This revised version was published online in June 2006 with corrections to the Cover Date.     

Generalized (anti) Yetter-Drinfeld modules as components of a braided T-category

If H is a Hopf algebra with bijective antipode and α, β ∈ Aut _Hopf (H), we introduce a category , generalizing both Yetter-Drinfeld modules and anti-Yetter-Drinfeld modules. We construct a braided T-category having all the categories as components, which, if H is finite dimensional, coincides with the representations of a certain quasitriangular T-coalgebra DT(H) that we construct. We also prove that if (α, β) admits a so-called pair in involution, then is isomorphic to the category of usual Yetter-Drinfeld modules . Research partially supported by the programme CERES of the Romanian Ministry of Education and Research, contract no. 4-147/2004.     

On Transversally Holomorphic Maps of Kählerian Foliations

Any transversally holomorphic foliated map of Kählerianfoliations with harmonic, is shown to be a transversallyharmonic map and an absolute minimum of the energy functional inits foliated homotopy class.     

On the ‘Standard’ Condition for Noncompact Homogeneous Einstein Spaces

A nonflat Einstein solvmanifold ( , g) is said to be of standard type if in the associated metric Lie algebra , the orthogonal complement of the derived algebra is Abelian. It is an open question whether the standard condition is automatically satisfied for all nonflat Einstein solvmanifolds. We derive certain properties of the metric Lie algebra of a nonflat Einstein solvmanifold ( , g) under the assumption . In particular, we obtain some new sufficient conditions which imply standard type.     

The First-Order Nonholonomic Connections with the Galilean Groups of Local Transformations. III

In the canonical smooth fiber bundles endowed with the metric tensor fields of relevant structure, we consider natural representations of the Galilean groups (1, n) and construct (1, n)-invariant generalized differential-geometric connections. In both regular and special cases of the representations of the considered groups (1, n), we find all affine nonholonomic , and _1,2-connections of the first order (see [1]–[3]) possessing the local Lie groups of transformations (1, n) and also describe the corresponding (1, n)invariant planar connections.