
Professor Gh. G. Hamedani Marquette University
Aug. 19  Nov 19, 2002
School of Mathematics
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.
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