• Volume 111, Issue 4

November 2001,   pages  371-514

• Cyclic Codes of Length 2𝑚

In this paper explicit expressions of 𝑚 + 1 idempotents in the ring $R = F_q[X]/\langle X^{2^m}-1\rangle$ are given. Cyclic codes of length 2𝑚 over the finite field 𝐹𝑞, of odd characteristic, are defined in terms of their generator polynomials. The exact minimum distance and the dimension of the codes are obtained.

• Unitary Tridiagonalization in $M(4, \mathbb{C})$

A question of interest in linear algebra is whether all 𝑛 × 𝑛 complex matrices can be unitarily tridiagonalized. The answer for all 𝑛 ≠ 4 (affirmative or negative) has been known for a while, whereas the case 𝑛 = 4 seems to have remained open. In this paper we settle the 𝑛 = 4 case in the affirmative. Some machinery from complex algebraic geometry needs to be used.

• On Ricci Curvature of 𝐶-totally Real Submanifolds in Sasakian Space Forms

Let 𝑀𝑛 be a Riemannian 𝑛-manifold. Denote by $S(p)$ and $\overline{Ric}(p)$ the Ricci tensor and the maximum Ricci curvature on 𝑀𝑛, respectively. In this paper we prove that every 𝐶-totally real submanifold of a Sasakian space form $\overline{M}^{2m + 1}(c)$ satisfies $S≤ \left(\frac{(n - 1)(c + 3)}{4} + \frac{n^2}{4}H^2\right)g$, where $H^2$ and 𝑔 are the square mean curvature function and metric tensor on 𝑀𝑛, respectively. The equality holds identically if and only if either 𝑀𝑛 is totally geodesic submanifold or 𝑛 = 2 and 𝑀𝑛 is totally umbilical submanifold. Also we show that if a 𝐶-totally real submanifold 𝑀𝑛 of $\overline{M}^{2n + 1}(c)$ satisfies $\overline{Ric}=\frac{(n-1)(c+3)}{4} + \frac{n^2}{4}H^2$ identically, then it is minimal.

• A variational Proof for the Existence of a Conformal Metric with Preassigned Negative Gaussian Curvature for Compact Riemann Surfaces of Genus > 1

Given a smooth function 𝐾 < 0 we prove a result by Berger, Kazhdan and others that in every conformal class there exists a metric which attains this function as its Gaussian curvature for a compact Riemann surface of genus 𝑔 > 1. We do so by minimizing an appropriate functional using elementary analysis. In particular for 𝐾 a negative constant, this provides an elementary proof of the uniformization theorem for compact Riemann surfaces of genus 𝑔 > 1.

• Homogeneous Operators and Projective Representations of the Möbius Group: A Survey

This paper surveys the existing literature on homogeneous operators and their relationships with projective representations of $PSL(2, \mathbb{R})$ and other Lie groups. It also includes a list of open problems in this area.

• Multiwavelet Packets and Frame Packets of $L^2(\mathbb{R}^d)$

The orthonormal basis generated by a wavelet of $L^2(\mathbb{R})$ has poor frequency localization. To overcome this disadvantage Coifman, Meyer, and Wickerhauser constructed wavelet packets. We extend this concept to the higher dimensions where we consider arbitrary dilation matrices. The resulting basis of $L^2(\mathbb{R}^d)$ is called the multiwavelet packet basis. The concept of wavelet frame packet is also generalized to this setting. Further, we show how to construct various orthonormal bases of $L^2(\mathbb{R}^d)$ from the multiwavelet packets.

• A Variational Principle for Vector Equilibrium Problems

A variational principle is described and analysed for the solutions of vector equilibrium problems.

• On a Generalized Hankel Type Convolution of Generalized Functions

The classical generalized Hankel type convolution are defined and extended to a class of generalized functions. Algebraic properties of the convolution are explained and the existence and significance of an identity element are discussed.

• Nonlinear Elliptic Differential Equations with Multivalued Nonlinearities

In this paper we study nonlinear elliptic boundary value problems with monotone and nonmonotone multivalued nonlinearities. First we consider the case of monotone nonlinearities. In the first result we assume that the multivalued nonlinearity is defined on all $\mathbb{R}$. Assuming the existence of an upper and of a lower solution, we prove the existence of a solution between them. Also for a special version of the problem, we prove the existence of extremal solutions in the order interval formed by the upper and lower solutions. Then we drop the requirement that the monotone nonlinearity is defined on all of $\mathbb{R}$. This case is important because it covers variational inequalities. Using the theory of operators of monotone type we show that the problem has a solution. Finally in the last part we consider an eigenvalue problem with a nonmonotone multivalued nonlinearity. Using the critical point theory for nonsmooth locally Lipschitz functionals we prove the existence of at least two nontrivial solutions (multiplicity theorem).

• Subject Index

• Author Index

• # Proceedings – Mathematical Sciences

Current Issue
Volume 127 | Issue 5
November 2017