A generalization of bounded variation

Generalized Wiener classes are considered. For these classes the exact order of Fourier coefficients with respect to the trigonometric system is established and the estimation of ‖S _n(·, f)-f(·)‖_{C [0,2π]} where S _n(·, f) are the Fourier partial sums, is given. In particular, a uniform convergence criterion for the Fourier trigonometric series is obtained. This revised version was published online in August 2006 with corrections to the Cover Date.     

A generalization of the variation diminishing property

For totally positive matrices, a new variation diminishing property on the sign of consecutive minors is obtained. this property is used to show shape preserving properties of curves generated by totally positive bases and, in particular, of B-spline curves. Partially supported by DGICYT PS93-0310.     

A generalization of the variation diminishing property

For totally positive matrices, a new variation diminishing property on the sign of consecutive minors is obtained. this property is used to show shape preserving properties of curves generated by totally positive bases and, in particular, of B-spline curves.     

A two-parameter generalization of the complete elliptic integral of second kind

A two-parameter generalization of the complete elliptic integral of second kind is expressed in terms of the Appell function F ₄. This function is further reduced to a quite simple bilinear form in the complete elliptic integrals K and E. Some physical applications are briefly mentioned.     

On subfields of the second generalization of the GK maximal function field

The second generalized GK function fields $K_n$ are a recently found family of maximal function fields over the finite field with $q^{2n}$ elements, where q is a prime power and $n\ge 1$ an odd integer. In this paper we construct many new maximal function fields by determining various Galois subfields of $K_n$. In case $\gcd (q+1,n)=1$ and either q is even or $q\equiv 1(\mathrm{mod}4)$, we find a complete list of Galois subfields of $K_n$. Our construction adds several previously unknown genera to the genus spectrum of maximal curves.     

A generalization of N-matrices

Ky Fan defines an N-matrix to be a matrix of the form A = tI ? B, B ? 0, λ < t < ?(B), where ?(B) is the spectral radius of B and λ is the maximum of the spectral radii of all principal submatrices of B of order n ? 1. In this paper, we define the closure (N₀-matrices) of N-matrices by letting λ ? t. It is shown that if A ∈ Z and A^-1 < 0, then A ∈ N. Certain inequalities of N-matrices are shown to hold for N₀-matrices, and a method for constructing an N-matrix from an M-matrix is given.     

A generalization ofL-splines

A theory of generalized splines is developed for all regular formally self adjoint differential operatorsL with real coefficients. A special case of such operators are those which may be factored in the formL =L ₁ ^* L ₁, such as those related to the generalized splines of Ahlberg, Nilson, and Walsh [1, 2], and theL-splines of Schultz and Varga [6]. Theorems giving unique interpolation, integral relations, and convergence rates are established. IfL has a certain positivity property, a useful extremal result is proven.This research was supported in part by a NASA Traineeship, at the Georgia Institute of Technology.     

The second variation of nonorientable minimal submanifolds

Suppose is a complete nonorientable minimal submanifold of a Riemannian manifold . We derive a second variation formula for the area of with respect to certain perturbations, giving a sufficient condition for the instability of . Some simple applications are given: we show that the totally geodesic is the only stable surface in , and we show the non-existence of stable nonorientable cones in . We reproduce and marginally extend some known results in the truly non-compact setting.

     

A generalization of finiteness

We explore an analogy between the family ₁, of finite/cofinite languages and the family ₁ of languages whose syntactic monoids are J-trivial. It is shown that (a) J-trivial monoids, (b)L-trivial monoids, (c) R-trivial monoids, and (d) a recently studied family, that we callK, of aperiodic monoids are natural generalization of the families of syntactic semigroups of (a) finite/cofinite languages, (b) definite languages, (c) reverse definite languages, and (d) generalized definite languages, respectively. In the case of alphabets of one and two letters, the languages corresponding to the familyK of monoids are characterized, illustrating the above-mentioned analogy explicitly.This work was done partly at the University of Paris VI and VII under the scientific exchange program between Canada and France, and partly at the Institut für Rechner- und Programmstrukturen, Gesellschaft für Mathematik und Datenverarbeitung mbH. Bonn, Germany.     

On a generalization of the class of functions with bounded Mocanu variation

The object of this paper is to generalise the well-known class of functions analytic in the unit disc having bounded Mocanu variation. Certain properties of this more general class are investigated using convolution techniques.