Searcharxiv⌕ Search

arXiv subjects

Armen Bagdasaryan

Publications and source records attributed to Armen Bagdasaryan.

At least 19 recordsLinked to original sources
arXiv2024

The Price of Cognition and Replicator Equations in Parallel Neural Networks

In this paper, we are aiming to propose a novel mathematical model that studies the dynamics of synaptic damage in terms of concentrations of toxic neuropeptides/neurotransmitters during neurotransmission processes. Our primary objective is to employ Wardrop's first and second principles within a neural network of the brain. In order to comprehensively incorporate Wardrop's first and second principles into the neural network of the brain, we introduce two novel concepts: \textit{neuropeptide's (neurotransmitter's) equilibrium} and \textit{synapses optimum}. The \textit{neuropeptide/neurotransmitter equilibrium} refers to \textit{a distribution of toxic neuropeptides/neurotransmitters that leads to uniform damage across all synaptic links}. Meanwhile, \textit{synapses optimum} is \textit{the most desirable distribution of toxic neuropeptides/neurotransmitters that minimizes the cumulative damage experienced by all synapses}. In the context of a neural network within the brain, an analogue of the price of anarchy is \textit{the price of cognition} which is \textit{the most unfavorable ratio between the overall impairment caused by toxic neuropeptide's (neurotransmitter's) equilibrium in comparison to the optimal state of synapses (synapses optimum)}. To put it differently, \textit{the price of cognition} measures \textit{the loss of cognitive ability resulting from increased concentrations of toxic neuropeptides/neurotransmitters}. Additionally, a replicator equation is proposed within this framework that leads to the establishment of the synapses optimum during the neurotransmission process.

q-bio.NC↗
arXiv2014

On The Properties Of $q$-Bernstein-Type Polynomials

The aim of this paper is to give a new approach to modified $q$-Bernstein polynomials for functions of several variables. By using these polynomials, the recurrence formulas and some new interesting identities related to the second Stirling numbers and generalized Bernoulli polynomials are derived. Moreover, the generating function, interpolation function of these polynomials of several variables and also the derivatives of these polynomials and their generating function are given. Finally, we get new interesting identities of modified $q$-Bernoulli numbers and $q$-Euler numbers applying $p$-adic $q$-integral representation on $\mathbb {Z}_p$ and $p$-adic fermionic $q$-invariant integral on $\mathbb {Z}_p$, respectively, to the inverse of $q$-Bernstein polynomials.

math.NT↗
arXiv2013

Systems Theoretic Techniques for Modeling, Control, and Decision Support in Complex Dynamic Systems

We discuss the problems of modeling, control, and decision support in complex dynamic systems from a general system theoretic point of view. The main characteristics of complex systems and of system approach to complex system study are considered. We provide an overview and analysis of known existing paradigms and methods of mathematical modeling and simulation of complex systems, which support the processes of control and decision making. Then we continue with the general dynamic modeling and simulation technique for complex hierarchical systems functioning in control loop. Architectural and structural models of computer information system intended for simulation and decision support in complex systems are presented.

eess.SY↗
arXiv2013

Some New Symmetric Identities for the q-Zeta Type Functions

The main object of this paper is to obtain several symmetric properties of the q-Zeta type functions. As applications of these properties, we give some new interesting identities for the modified q-Genocchi polynomials. Finally, our applications are shown to lead to a number of interesting results which we state in the present paper.

math.NT↗
arXiv2013

On the partition of R^n by hyperplanes

The partitioning of space by hyperplanes in the context of discrete classification problem is considered. We obtain some relations for the number of partitions and establish a recurrence relation for the maximal number of partitions of R^n by m hyperplanes. We rederive an explicit formula for the number of components into which the space can be partitioned by m hyperplanes.

cs.DM↗
arXiv2013

Identities involving the $\left(h,q\right)$-Genocchi polynomials and $\left(h,q\right)$-Zeta-type function

The fundamental objective of this paper is to obtain some interesting properties for $\left(h,q\right)$-Genocchi numbers and polynomials by using the fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$ and mentioned in the paper $q$-Bernstein polynomials. By considering the $q$-Euler zeta function defined by T. Kim, which can also be obtained by applying the Mellin transformation to the generating function of $\left(h,q\right)$-Genocchi polynomials, we study $\left(h,q\right)$-Zeta-type function. We derive symmetric properties of $\left(h,q\right)$-Zeta function and from these properties we give symmetric property of $\left(h,q\right)$-Genocchi polynomials.

math.NT↗
arXiv2013

On the modified q-Genocchi numbers and polynomials and their applications

The main objective of this paper is to introduce the modified q-Genocchi polynomials and to define their generating function. In the paper, we show new relations, which are explicit formula, derivative formula, multiplication formula, and some others, for mentioned q-Genocchi polynomials. By applying Mellin transformation to the generating function of the modified q-Genocchi polynomials, we define q-Genocchi zeta-type functions which are interpolated by the modified q-Genocchi polynomials at negative integers.

math.NT↗
arXiv2012

Some Euler-type formulas for planar graphs

The aim of this paper is to derive on the basis of the Euler's formula several analytical relations which hold for certain classes of planar graphs and which can be useful in algorithmic graph theory.

cs.DM↗
arXiv2011

On a new general method of summation

A new general and unified method of summation, which is both regular and consistent, is invented. It is based on the idea concerning a way of integers reordering. The resulting theory includes a number of explicit and closed form summation formulas, and assigns limits to certain unbounded or oscillating functions. Some problems and future lines of research are briefly discussed.

math.CA↗
arXiv2010

An elementary and real approach to values of the Riemann zeta function

An elementary approach for computing the values at negative integers of the Riemann zeta function is presented. The approach is based on a new method for ordering the integers and a new method for summation of divergent series. We show that the values of the Riemann zeta function can be computed, without using the theory of analytic continuation and functions of complex variable.

math.NT↗
arXiv2009

A Note on the 2F1 Hypergeometric Function

The special case of the hypergeometric function $_{2}F_{1}$ represents the binomial series $(1+x)^α=\sum_{n=0}^{\infty}(\:αn\:)x^{n}$ that always converges when $|x|<1$. Convergence of the series at the endpoints, $x=\pm 1$, depends on the values of $α$ and needs to be checked in every concrete case. In this note, using new approach, we reprove the convergence of the hypergeometric series $_{2}F_{1}(α,β;β;x)$ for $|x|<1$ and obtain new result on its convergence at point $x=-1$ for every integer $α\neq 0$. The proof is within a new theoretical setting based on the new method for reorganizing the integers and on the regular method for summation of divergent series.

math.CA↗
arXiv2009

On one number-theoretic conception: towards a new theory

In this paper we present a new mathematical conception based on a new method for ordering the integers. The method relies on the assumption that negative numbers are beyond infinity, which goes back to Wallis and Euler. We also present a new axiom system, the model of which is arithmetics. We define regular method for summation of infinite series which allows us to discover general and unified approach to summation of divergent series, and determine the limits of unbounded and oscillating functions. Several properties for divergent series and explicit formulas for sums of some infinite series are established. A number of finite and new recurrence formulas for Bernoulli numbers are obtained. We rederive some known results, but in a simpler and elementary way, and establish new results by means of techniques of the theoretical background developed.

math.GM↗
arXiv2009

System approach to synthesis, modeling and control of complex dynamical systems

We consider the basic features of complex dynamical and control systems. Special attention is paid to the problems of synthesis of dynamical models of complex systems, construction of efficient control models, and to the development of simulation techniques. We propose an approach to the synthesis of dynamic models of complex systems that integrates expert knowledge with the process of modeling. A set-theoretic model of complex system is defined and briefly analyzed. A mathematical model of complex dynamical system with control, based on aggregate description, is also proposed. The structure of the model is described, and architecture of computer simulation system is presented, requirements to and components of computer simulation systems are analyzed.

cs.CE↗