On the strong maximum principle

This paper presents a necessary and sufficient condition on the convex function in order that continuous solutions to

satisfy a Strong Maximum Principle on any open connected .

     

A one-parameter quadratic-base version of the Baillie-PSW probable prime test

The well-known Baillie-PSW probable prime test is a combination of a Rabin-Miller test and a ``true' (i.e., with The well-known Baillie-PSW probable prime test is a combination of a Rabin-Miller test and a ``true' (i.e., with ) Lucas test. Arnault mentioned in a recent paper that no precise result is known about its probability of error. Grantham recently provided a probable prime test (RQFT) with probability of error less than 1/7710, and pointed out that the lack of counter-examples to the Baillie-PSW test indicates that the true probability of error may be much lower.

In this paper we first define pseudoprimes and strong pseudoprimes to quadratic bases with one parameter: , and define the base-counting functions:

and

Then we give explicit formulas to compute B and SB, and prove that, for odd composites ,

and point out that these are best possible. Finally, based on one-parameter quadratic-base pseudoprimes, we provide a probable prime test, called the One-Parameter Quadratic-Base Test (OPQBT), which passed by all primes and passed by an odd composite odd primes) with probability of error . We give explicit formulas to compute , and prove that

The running time of the OPQBT is asymptotically 4 times that of a Rabin-Miller test for worst cases, but twice that of a Rabin-Miller test for most composites. We point out that the OPQBT has clear finite group (field) structure and nice symmetry, and is indeed a more general and strict version of the Baillie-PSW test. Comparisons with Gantham's RQFT are given.

     

ON THE GENERALIZATIONS OF KALANDIA''''S LEMMA

Based on Bernstein's Theorem, Kalandia's Lemma describes the error estimate and the smoothness of the remainder under the second part of Holder norm when a Holder function is approximated by its best polynomial approximation. In this paper, Kalandia's Lemma is generalized to the cases that the best polynomial is replaced by one of its four kinds of Chebyshev polynomial expansions, the error estimates of the remainder are given out under Holder norm or the weighted Holder norms.     

Super efficiency in convex vector optimization

We establish a Lagrange Multiplier Theorem for super efficiency in convex vector optimization and express super efficient solutions as saddle points of appropriate Lagrangian functions. An example is given to show that the boundedness of the base of the ordering cone is essential for the existence of super efficient points.Research is supported partially by NSERC.Research is supported partially by NSERC and Mount St. Vincent University grant.     

关于图论中棱形式的Menger定理的注记

本注记给出图论中棱形式Menger定理的一个直接而又简单的证明.     

A shortcut to asymptotics for orthogonal polynomials

We consider the asymptotic behavior of the ratios q_n+1(z)/q_n(z) of polynomials orthonormal with respect to some positive measure μ. Let the recurrence coefficients _n and β_n converge to 0 and , respectively. Then, q_n+1(z)/q_n(z) Φ(z),for n→∞ locally uniformly for , where Φ maps conformally onto the exterior of the unit disc (Nevai (1979)). We provide a new and direct proof for this and some related results due to Nevai, and apply it to convergence acceleration of diagonal Padé approximants.     

A Complete Solution of the Wave Equations for Transversely Isotropic Media

A transversely isotropic material in the sense of Green is considered. A complete solution in terms of retarded potential functions for the wave equations in transversely isotropic media is presented. In this paper we reduce the number of potential functions to only one, and we discuss the required conditions. As a special case, the torsionless and rotationally symmetric configuration with respect to the axis of symmetry of the material is discussed. The limiting case of elastostatics is cited, where the solution is reduced to the Lekhnitskii–Hu–Nowacki solution. The solution is simplified for the special case of isotropy. In this way, a new series of potential functions (to the best knowledge of the author) for the elastodynamics problem of isotropic materials is presented This solution is reduced to a special case of the Cauchy–Kovalevski–Somigliana solution, if the displacements satisfy specific conditions. Finally, Boggio's Theorem is generalized for transversely isotropic media which may be of interest to the reader beyond the present application. Dedicated to Morton E. Gurtin     

Some Oscillation Theorems for Second Order Differential Equations

In this paper we establish some oscillation or nonoscillation criteria for the second order half-linear differential equation

Nonlinear discrete Sturm-Liouville problems

In this paper we study nonlinear boundary value problems of the form