Paper's Title:

Derivation of the Existence Theorem of the Solution of the Stochastic Functional Differential Equation Using Conditions Given Partial Weights

Author(s):

Young-Ho Kim

Department of Mathematics,
Changwon National University
Changwon, Gyeongsangnam 51140,
Korea.
E-mail: yhkim@changwon.ac.kr

Abstract:

The main purpose of this note was to demonstrate the solution existence theorem for stochastic functional differential equations under sufficient conditions. As an alternative to the stochastic process theory of the stochastic functional differential equations, we impose a partial weighting condition and a weakened linear growth condition. We first show that the condition guarantees existence and uniqueness and then show some exponential estimates for the solution.

Paper's Title:

Extreme Curvature of Polynomials and Level Sets

Author(s):

Stephanie P. Edwards, arah J. Jensen, Edward Niedermeyer, and Lindsay Willett

Department of Mathematics,
Hope College,
Holland, MI 49423,
U.S.A.
E-mail: sedwards@hope.edu
E-mail: tarahjaye@gmail.com
E-mail: eddie.niedermeyer@gmail.com
E-mail: willettlm1@gmail.com
WWW: http://math.hope.edu/sedwards/

Abstract:

Let f be a real polynomial of degree n. Determining the maximum number of zeros of kappa, the curvature of f, is an easy problem: since the zeros of kappa are the zeros of f'', the curvature of f is 0 at most n-2 times. A much more intriguing problem is to determine the maximum number of relative extreme values for the function kappa. Since kappa'=0 at each extreme point of kappa, we are interested in the maximum number of zeros of kappa'. In 2004, the first author and R. Gordon showed that if all the zeros of f'' are real, then f has at most n-1 points of extreme curvature. We use level curves and auxiliary functions to study the zeros of the derivatives of these functions. We provide a partial solution to this problem, showing that f has at most n-1 points of extreme curvature, given certain geometrical conditions. The conjecture that f has at most n-1 points of extreme curvature remains open.

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