A. CHARACTERIZATIONS OF UNIVARIATE CONTINUOUS DISTRIBUTIONS.

A BRIEF DESCRIPTION:

    The problem of characterizing a distribution is an important problem which has attracted the attention of many researchers in recent years. Consequently, variuos characterization results have been reported in the literature. These characterizations have been established in many different directions.

    Characterization problems are related to such areas as: ARITHMETIC OF PROBABILITY DISTRIBUTIONS; ESTIMATION THEORY; SUFFICIENCY; RELIABILITY THEORY, just to name a few.

    Characterization results often clarify the role of the assumptions on statistical models. Characterization problems are often mathematically elegant and they lead to new problems in THEORY OF FUNCTIONS, FUNCTIONAL
EQUATIONS, etc.

    Generally speaking, continuous distributions require more elegant mathematical treatment than discrete distributions.

B. OSCILLATORY BEHAVIOR OF THE SOLUTIONS OF FORCED FUNCTIONAL
DIFFERENTIAL EQUATIONS.

A BRIEF DESCRIPTION
 

    Researchers in many different fields have come up with various models of Functional Differential Equations. This, consequently, has motivated research in the qualitative theory of Functional Differential Equations, in particular the Oscillation Theory of Functional Differential Equations.

    A nontrivial solution of a differential equation is called oscillatory if it has arbitrary large zeros. Otherwise, the solution is said to be nonoscillatory, i.e., it is eventually positive or eventually negative.

    A.G. Kartsatos made the following statement: " One of the major, and generally unstudied, problems in the theory of oscillation of nonlinear differential equations, is the problem of maintaining oscillations under the effect of a forcing term." He pointed out that the oscillation of all the solutions of unforced equation :
 

                                                              x^n(t)+ a(t) f(x(t)) = 0, t>to,

    is not generally maintained if one considers the forcing equation, by adding the term e(t) to the right hand side of the above equation. Thus, one must impose more conditions on the function e(t) to ensure oscillation of the equation:

                                                               x^n(t) + a(t) f(x(t)) = e(t), t>to.

Therefore, oscillation criteria of interest are those which can be applied to a class of both forced and unforced equations.