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GEOMETRIC AND APPROXIMATION PROPERTIES OF GENERALIZED SINGULAR INTEGRALS IN THE UNIT DISK
Anastassiou George A.,Gal Sorin G. Korean Mathematical Society 2006 대한수학회지 Vol.43 No.2
The aim of this paper is to obtain several results in approximation by Jackson-type generalizations of complex Picard, Poisson-Cauchy and Gauss-Weierstrass singular integrals in terms of higher order moduli of smoothness. In addition, these generalized integrals preserve some sufficient conditions for starlikeness and univalence of analytic functions. Also approximation results for vector-valued functions defined on the unit disk are given.
UNIVARIATE LEFT FRACTIONAL POLYNOMIAL HIGH ORDER MONOTONE APPROXIMATION
Anastassiou, George A. Korean Mathematical Society 2015 대한수학회보 Vol.52 No.2
Let $f{\in}C^r$ ([-1,1]), $r{\geq}0$ and let $L^*$ be a linear left fractional differential operator such that $L^*$ $(f){\geq}0$ throughout [0, 1]. We can find a sequence of polynomials $Q_n$ of degree ${\leq}n$ such that $L^*$ $(Q_n){\geq}0$ over [0, 1], furthermore f is approximated left fractionally and simulta-neously by $Q_n$ on [-1, 1]. The degree of these restricted approximations is given via inequalities using a higher order modulus of smoothness for $f^{(r)}$.
Anastassiou, George A. Korean Mathematical Society 2019 대한수학회보 Vol.56 No.6
Here we present the right and left Riemann-Liouville fractional fundamental theorems of fractional calculus without any initial conditions for the first time. Then we establish a Riemann-Liouville fractional Polya type integral inequality with the help of generalised right and left Riemann-Liouville fractional derivatives. The amazing fact here is that we do not need any boundary conditions as the classical Polya integral inequality requires. We extend our Polya inequality to Choquet integral setting.
GENERALIZED SYMMETRICAL SIGMOID FUNCTION ACTIVATED NEURAL NETWORK MULTIVARIATE APPROXIMATION
ANASTASSIOU, GEORGE A. The Korean Society for Computational and Applied M 2022 Journal of applied and pure mathematics Vol.4 No.3/4
Here we exhibit multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or ℝ<sup>N</sup>, N ∈ ℕ, by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last four types. These approximations are achieved by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Fréchet derivatives. Our multivariate operators are defined by using a multidimensional density function induced by the generalized symmetrical sigmoid function. The approximations are point-wise and uniform. The related feed-forward neural network is with one hidden layer.
Common Best Proximity Points For Proximally Commuting Mappings In Non-Archimedean PM-Spaces
Anastassiou, George A.,Cho, Yeol Je,Saadati, Reza,Yang, Young-Oh Eudoxus Press LLC 2016 Journal of computational analysis and applications Vol.20 No.6
<P>In this paper, we prove new common best proximity point theorems for proximally commuting mappings in complete non-Archimedean PM-spaces. Our results generalized the recent results of S. Basha [Common best proximity points: global minimization of multi-objective functions, J. Global Optim. 49(2011), 15-21] and C. Mongkolkeha, P. Kumam [Some common best proximity points for proximity commuting mappings, Optim. Lett. 7 (2013), 1825-1836].</P>
MULTIVARIATE RIGHT FRACTIONAL OSTROWSKI INEQUALITIES
Anastassiou, George A. The Korean Society for Computational and Applied M 2012 Journal of applied mathematics & informatics Vol.30 No.3
Very general multivariate right Caputo fractional Ostrowski inequalities are presented. Some of them are proved to be sharp and attained. Estimates are with respect to ${\parallel}{\cdot}{\parallel}_{\infty}$.
Ephaptic coupling of cortical neurons
Anastassiou, Costas A,Perin, Rodrigo,Markram, Henry,Koch, Christof Nature Publishing Group, a division of Macmillan P 2011 NATURE NEUROSCIENCE Vol.14 No.2
The electrochemical processes that underlie neural function manifest themselves in ceaseless spatiotemporal field fluctuations. However, extracellular fields feed back onto the electric potential across the neuronal membrane via ephaptic coupling, independent of synapses. The extent to which such ephaptic coupling alters the functioning of neurons under physiological conditions remains unclear. To address this question, we stimulated and recorded from rat cortical pyramidal neurons in slices with a 12-electrode setup. We found that extracellular fields induced ephaptically mediated changes in the somatic membrane potential that were less than 0.5 mV under subthreshold conditions. Despite their small size, these fields could strongly entrain action potentials, particularly for slow (<8 Hz) fluctuations of the extracellular field. Finally, we simultaneously measured from up to four patched neurons located proximally to each other. Our findings indicate that endogenous brain activity can causally affect neural function through field effects under physiological conditions.
George A. Anastassiou 한국전산응용수학회 2021 Journal of Applied and Pure Mathematics Vol.3 No.5
Here we extended our earlier high order simultaneous fractional polynomial spline monotone approximation theory to abstract high order simultaneous fractional polynomial spline monotone approximation, with applications to Prabhakar fractional calculus and non-singular kernel fractional calculi. We cover both the left and right sides of this constrained approximation. Let f\in C^{s}\left( \left[ -1,1\right] \right), s\in \mathbb{N} and L^{\ast } be a linear left or right side fractional differential operator such that L^{\ast }\left( f\right) \geq 0 over \left[ 0,1\right] or \left[ -1,0\right] , respectively. Then there exists a sequence Q_{n}, n\in \mathbb{N} of polynomial splines with equally spaced knots of given fixed order such that L^{\ast }\left(Q_{n}\right) \geq 0 on \left[ 0,1\right] or \left[ -1,0\right], respectively. Furthermore f is approximated with rates fractionally and simultaneously by Q_{n} in the uniform norm. This constrained fractional approximation on \left[ -1,1\right] is given via inequalities involving a higher modulus of smoothness of f^{\left( s\right) }.
Generalized symmetrical sigmoid function activated neural network multivariate approximation
George A. Anastassiou 한국전산응용수학회 2022 Journal of Applied and Pure Mathematics Vol.4 No.3
Here we exhibit multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or \mathbb{R}^{N}, N\in \mathbb{N}, by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last four types. These approximations are achieved by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Fr\'{e}chet derivatives. Our multivariate operators are defined by using a multidimensional density function induced by the generalized symmetrical sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.
George A. Anastassiou 한국전산응용수학회 2022 Journal of Applied and Pure Mathematics Vol.4 No.5
In this article we exhibit univariate and multivariate quantitative approximation by Kantorovich-Choquet type quasi-interpolation neural network operators with respect to supremum norm. This is done with rates using the first univariate and multivariate moduli of continuity. We approximate continuous and bounded functions on $\mathbb{R}^{N},$ $N\in \mathbb{N}$. When they are also uniformly continuous we have pointwise and uniform convergences. Our activation functions are induced by the arctangent, algebraic, Gudermannian and generalized symmetrical sigmoid functions.