Anastassiou, G. A., O. Duman.
Fractional Korovkin Theory Based on Statistical Convergence
(pp. 381-396)
A B S T R A C T S
PROBABILISTIC APPROACH TO THE NEUMANN PROBLEM FOR A SYMMETRIC OPERATOR
Abdelatif Benchérif-Madani
lotfi_madani@yahoo.fr
2000 Mathematics Subject Classification:
Primary 60J45, 60J50, 35Cxx; Secondary 31Cxx.
Key words:
Neumann and Steklov problems, exponential ergodicity, double layer potential, reflecting diffusion, Lipschitz domain.
We give a probabilistic formula for the solution of a non-homogeneous Neumann problem for a symmetric nondegenerate operator of second order in a bounded domain. We begin with a g-Hölder matrix and a C^{1,g} domain, g > 0, and then consider extensions. The solutions are expressed as a double layer potential instead of a single layer potential; in particular a new boundary function is discovered and boundary random walk methods can be used for simulations. We use tools from harmonic analysis and probability theory.
POTAPOV-GINSBURG TRANSFORMATION AND FUNCTIONAL MODELS OF NON-DISSIPATIVE OPERATORS
Vladimir A. Zolotarev
Vladimir.A.Zolotarev@univer.kharkov.ua,
Raéd Hatamleh
raedhat@yahoo.com
2000 Mathematics Subject Classification:
Primary 47A20, 47A45; Secondary 47A48.
Key words:
Colligations, Non-Dissipative
Operator, Functional Model, Resolvent Operator.
A relation between an arbitrary bounded operator A and dissipative
operator A_{+}, built by A in the following way
A_{+} = A+ij^{*}Q_{-}j, where A-A^{*} = ij^{*}Jj, (J = Q_{+}-Q_{-} is involution), is studied.
The characteristic functions of the operators A and A_{+} are expressed
by each other using the known Potapov-Ginsburg linear-fractional
transformations. The explicit form of the resolvent (A-lI)^{-1} is
expressed by (A_{+}-lI)^{-1} and (A_{+}^{*}-lI)^{-1}
in terms of these transformations. Furthermore, the functional model [10,
12] of non-dissipative operator A in terms of a model for A_{+}, which
evolves the results, was obtained by Naboko, S. N. [7].
The main constructive elements of the present construction are shown to be
the elements of the Potapov-Ginsburg transformation for corresponding
characteristic functions.
ESTIMATION OF A REGRESSION FUNCTION ON A POINT PROCESS AND ITS APPLICATION TO FINANCIAL RUIN RISK FORECAST
Galaye Dia
galayedia@hotmail.com,
2000 Mathematics Subject Classification:
Primary 60G55; secondary 60G25.
Key words:
Point process, regressogram, superposition, claim amount, aggregate claim amount, mean inter-arrival claim intensity, mean intensity of the claim process, ruin time.
We estimate a regression function on a point process by the Tukey regressogram method in a general setting and we give an application in the case of a Risk Process. We show among other things that, in classical Poisson model with parameter r, if W is the amount of the claim with finite espectation E(W) = m, S_{n} (resp. R_{n}) the accumulated interval waiting time for successive claims (resp. the aggregate claims amount) up to the nth arrival, the regression curve of R on S predicts ruin arrival time when the premium intensity c is less than rm whatever be the initial reverve.
FRACTIONAL KOROVKIN THEORY BASED ON STATISTICAL CONVERGENCE
George A. Anastassiou
ganastss@memphis.edu,
Oktay Duman
oduman@etu.edu.tr,
2000 Mathematics Subject Classification:
41A25, 41A36, 40G15.
Key words:
The Korovkin theorem, statistical convergence, fractional
calculus, Caputo fractional derivatives.
In this paper, we obtain some statistical Korovkin-type approximation theorems including fractional derivatives of functions. We also show that our new results are more applicable than the classical ones.
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