System Of Recurrence Equations Research Articles

Tropical Modeling of Battery Swapping and Charging Station

We propose and investigate a queueing model of a battery swapping and charging station (BSCS) for electric vehicles (EVs). A new approach to the analysis of the queueing model is developed, which combines the representation of the model as a stochastic dynamic system with the use of the methods and results of tropical algebra, which deals with the theory and applications of algebraic systems with idempotent operations. We describe the dynamics of the queueing model by a system of recurrence equations that involve random variables (RVs) to represent the interarrival time of incoming EVs. A performance measure for the model is defined as the mean operation cycle time of the station. Furthermore, the system of equations is represented in terms of the tropical algebra in vector form as an implicit linear state dynamic equation. The performance measure takes on the meaning of the mean growth rate of the state vector (the Lyapunov exponent) of the dynamic system. By applying a solution technique of vector equations in tropical algebra, the implicit equation is transformed into an explicit one with a state transition matrix with random entries. The evaluation of the Lyapunov exponent reduces to finding the limit of the expected value of norms of tropical matrix products. This limit is then obtained using results from the tropical spectral theory of deterministic and random matrices. With this approach, we derive a new exact formula for the mean cycle time of the BSCS, which is given in terms of the expected value of the RVs involved. We present the results of the Monte Carlo simulation of the BSCS’s operation, which show a good agreement with the exact solution. The application of the obtained solution to evaluate the performance of one BSCS and to find the optimal distribution of battery packs between stations in a network of BSCSs is discussed. The solution may be of interest in the case when the details of the underlying probability distributions are difficult to determine and, thus, serves to complement and supplement other modeling techniques with the need to fix a distribution.

Convergence of solutions of a system of recurrence equations

Convergence of solutions of a system of recurrence equations

Utilizing nested mappings to deconvolute photoluminescence decay

A self-similar system of recurrent equations was used for calculating the dynamics of the emission decays of rare-earth ion-doped photoluminescent nanoparticles. The proposed approach provides a smooth transition between different descriptions of all kinetic photoluminescence (PL) data and does not imply a solution through the convolution integral equation.The suggested method introduces a transition mechanism based on elementary mapping of the proposed hierarchical system. The main equation of the system contains two changeable parameters: one of these can be considered the decay value. The calculated dependence of the PL decay time is presented with a division into regions with different asymptotes.

GAME CONTROL OF A FILTER WITH A RANDOM JUMP STRUCTURE

The game problem of synthesis of information-control algorithms of filtering of two opposing sides is considered. The opponents’ control objects are described by multidimensional linear stochastic recurrent equations. Each side controls the covariance of the output signals of its object and the covariance of additive signals in the channels of the enemy object meters. Covariances of noise signals of an object characterize the power of disturbances, and covariances of noise signals of meters characterize the power of interference.The goal of each player’s management is to reduce the filtering error of their system and at the same time increase the filtering error of the opponent’s system. The criterion of optimal control of one player is the minimax, and the other is the maximin of the average value of the quality indicator for a finite period of time. Quality indicators consist of a linear combination of three particular indicators, the first of which is the probability of one of the two states of the Markov jump structure, and the other two are restrictions on the probabilities of transitions of the Markov chain. Observation systems consist of structure indicators and meters of one-time coordinates. Each information-control algorithm is a set of five interconnected blocks: a structure classifier, a structure indicator, a regulator, a dispersiometer and a filter. The obtained information-control algorithms are described by a closed system of recurrent equations. The algorithms are based on the methods of the theory of multistep antagonistic games, the theory of systems with a random jump structure, stochastic dynamic programming and Bayesian filtering. An example of the construction of game information-control algorithms of two unmanned aerial vehicles of the opposing sides, aimed at the objects of the opponents, is considered.

FILTERING OF SIGNALS WITH UNKNOWN INTERFERENCE POWER AND RANDOM JUMPING PERTURBATION

A linear stochastic dynamic system under the influence of random disturbances and interference is considered. The perturbation is a sequence of uncorrelated random variables with a distribution in the range [–1, 1]. This sequence is modeled by a random binary signal with values 1 and 0 and is described by a Markov chain with known probabilities of transitions from one state to another. The modulated signal is fed to the input of the linear control unit. The output signal of the control object is measured with an error, which is a sequence of uncorrelated random variables with an unknown distribution in the range [–1, 1].The problem under consideration differs from the optimal linear filtration problems based on the application of the Kalman filter and its modifications. Its novelty consists in the following: 1) the input signal is a random jumping process – uncorrelated noise modulated by a random binary signal; 2) the variances of random processes – the input signal and the interference characterizing their power are unknown. А posteriori mathematical expectation and the a posteriori variance of the filtering error are determined by the methods of Bayesian estimation and the theory of systems with a random jump structure. The optimal estimation algorithm is described by a system of recurrent equations. It consists of five interconnected blocks: 1) a meter of mathematical expectation and dispersion of an additive mixture of an output signal with interference; 2) an indicator of a random structure; 3) a classifier of a random structure; 4) a dispersiometer; 5) a filter.

Graph solution of a system of recurrence equations

Graph solution of a system of recurrence equations

Using nested mappings for heat capacity data processing

In the article discrete approach based on self-similar hierarchical system of recurrent equations is proposed. The basic element of this system is linear stationary recurrent equation with constant coefficients. In general, such system has an endless nested structure, limited by the level, where its coefficients become constants.For processing heat capacity data, the proposed system was reduced to equation with uniformly varying coefficients, which is able to successfully describe a wide range of dependencies and make reliable predictions. The flexibility and convenience of this approach were shown by the example of various descriptions of heat capacity data of materials in different states and temperature ranges.

Differential Equations with Variable Coefficients in the Mechanics of Inhomogeneous Bodies

The paper considers differential equations in partial derivatives of an elliptic type with variables, piecewise-smooth coefficients, depending on the coordinates (initial equations). It is shown that the solution of the original equation can be represented as an integral equation through the solution of the accompanying equation with constant coefficients of the same type. This representation includes a fundamental solution to the original equation. Under the assumption that the accompanying solution is smooth, the integral equation implies the representation of the original solution in the form of a series with respect to all possible derivatives of the related solution. The coefficients of the series are called structural functions, since they are determined by the functional dependence of the coefficients of the initial equations either on coordinates, or on time, or on coordinates and time. Structural functions are identically equal to zero in the case when the initial coefficients coincide with the corresponding constant coefficients of the accompanying equation. For structural functions, systems of recurrence equations are obtained. It is shown that in the case of a plate with non-uniform thickness, the structural functions depend only on the coordinate in the thickness of the plate, and structural equations become ordinary differential equations that integrate in a general way. A scheme for solving the plate problem is considered.

About one Approach to a Solution of Linear Differential Equations with Variable Coefficients

Using the uniform approach, linear differential equations of elliptic, hyperbolic and parabolic types with variable coefficients depending on coordinates and time are considered. It is shown that the solution of the initial equation can be expressed by means of an integral formula through the solution of the accompanying equation of the same type, but with constant coefficients. It is considered that the solution of the accompanying equation is known. From the integral formula, assuming smoothness of the accompanying solution, an equivalent representation of the solution of the initial equation is obtained in the form of series in all possible derivatives of the solution of the accompanying equation. For coefficients at derivatives a system of recurrent equations is obtained, which can be solved analytically at some cases.

On the complexity of monitoring Orchids signatures, and recurrence equations

Modern monitoring tools such as our intrusion detection tool Orchids work by firing new monitor instances dynamically. Given an Orchids signature (a.k.a. a rule, a specification), what is the complexity of checking that specification, that signature? In other words, let f(n) be the maximum number of monitor instances that can be fired on a sequence of n events: we design an algorithm that decides whether f(n) is asymptotically exponential or polynomial, and in the latter case returns an exponent d such that $$f(n) = \varTheta (n^d)$$ . Ultimately, the problem reduces to the following mathematical question, which may have other uses in other domains: given a system of recurrence equations described using the operators $$+$$ and $$\max $$ , and defining integer sequences $$u_n$$ , what is the asymptotic behavior of $$u_n$$ as n tends to infinity? We show that, under simple assumptions, $$u_n$$ is either exponential or polynomial, and that this can be decided, and the exponent computed, using a simple modification of Tarjan’s strongly connected components algorithm, in linear time.

Computation of disordered system from the first principles of classical mechanics and ℕℙ hard problem

We study the classical 1D Heisenberg spin glasses in the framework of nearest-neighboring model. Based on the Hamilton equations we obtained the system of recurrence equations which allows to perform node-by-node calculations of a spin-chain. It is shown that calculations from the first principles of classical mechanics lead to ℕℙ hard problem, that however in the limit of the statistical equilibrium can be calculated by ℙ algorithm. For the partition function of the ensemble a new representation is offered in the form of one-dimensional integral of spin-chains’ energy distribution.

Concentration nonlinearity of a suspension of transparent microspheres under the action of a gradient force in a periodically modulated laser field

Based on a one-dimensional Smoluchowski equation we have developed the theory of concentration nonlinearity of a suspension of transparent microspheres under the action of a gradient force in an interference laser field. The numerical solution of a system of recurrence equations resulting from the Smoluchowski equation after expansion of the microsphere concentration in the harmonic series has allowed us to determine the dependence of the concentration nonlinearity settling time on the intensity of the incident radiation. In the diffusion limit, we have derived the expression for the optical Kerr coefficient, which is found to be for an aqueous suspension of latex microspheres with a radius of and a concentration of . Diffraction of a probe wave on a light-induced concentration grating is considered as a method for studying a nonlinear concentration response of an artificial highly efficient nonlinear medium for laser radiation of long pulse duration.

Model of a laser gyroscope with frequency dithering

The model of a laser gyroscope (LG) with frequency dithering is described by a system of recurrent equations for the electric fields of counterpropagating waves. The phenomenon of frequency locking is taken into account in the form of the wave coupling through backward scattering; the frequency bias factor is the controlled phase nonreciprocity. The character of the output signal is considered, which corresponds to two types of frequency dithering, namely, sinusoidal and in the form of meander that are produced by various methods, including intracavity phase modulation. Results of calculation of a frequency characteristic of the LG are presented as functions of frequency dithering, rotational velocity and LG parameters. It is shown that the method of processing an output signal by measuring the time interval between intensity oscillations has an advantage due to the absence of so-called dynamic lock-in zones in the output characteristic.

A few $c_2$ invariants of circulant graphs

The $c_2$ invariant is an arithmetic graph invariant introduced by Schnetz and developed by Brown and Schnetz in order to better understand Feynman integrals. This document looks at the special case where the graph in question is a 4-regular circulant graph with one vertex removed; call such a graph a decompletion of a circulant graph. The $c_2$ invariant for the prime $2$ is computed in the case of the decompletion of circulant graphs $C_n(1,3)$ and $C_{2k+2}(1,k)$. For any prime $p$ and for the previous two families of circulant graphs along with the further families $C_n(1,4)$, $C_n(1,5)$, $C_n(1,6)$, $C_n(2,3)$, $C_n(2,4)$, $C_n(2,5)$, and $C_n(3,4)$, the same technique gives the $c_2$ invariant of the decompletions as the solution to a finite system of recurrence equations.

First-passage-time statistics of a Brownian particle driven by an arbitrary unidimensional potential with a superimposed exponential time-dependent drift

In one-dimensional systems, the dynamics of a Brownian particle are governed by the force derived from a potential as well as by diffusion properties. In this work, we obtain the first-passage-time statistics of a Brownian particle driven by an arbitrary potential with an exponential temporally decaying superimposed field up to a prescribed threshold. The general system analyzed here describes the sub-threshold signal integration of integrate-and-fire neuron models, of any kind, supplemented by an adaptation-like current, whereas the first-passage-time corresponds to the declaration of a spike. Following our previous studies, we base our analysis on the backward Fokker–Planck equation and study the survival probability and the first-passage-time density function in the space of the initial condition. By proposing a series solution we obtain a system of recurrence equations, which given the specific structure of the exponential time-dependent drift, easily admit a simpler Laplace representation. Naturally, the present general derivation agrees with the explicit solution we found previously for the Wiener process in (2012 J. Phys. A: Math. Theor. 45 185001). However, to demonstrate the generality of the approach, we further explicitly evaluate the first-passage-time statistics of the underlying Ornstein–Uhlenbeck process. To test the validity of the series solution, we extensively compare theoretical expressions with the data obtained from numerical simulations in different regimes. As shown, agreement is precise whenever the series is truncated at an appropriate order. Beyond the fact that both the Wiener and Ornstein–Uhlenbeck processes have a direct interpretation in the context of neuronal models, given their ubiquity in different fields, our present results will be of interest in other settings where an additive state-independent temporal relaxation process is being developed as the particle diffuses.

Implicit co‐simulation methods: Stability and convergence analysis for solver coupling approaches with algebraic constraints

The analysis of the numerical stability of co‐simulation methods with algebraic constraints is subject of this manuscript. Three different implicit coupling schemes are investigated. The first method is based on the well‐known Baumgarte stabilization technique. Basis of the second coupling method is a weighted multiplier approach. Within the third method, a classical projection technique is applied. The three methods are discussed for different approximation orders. Concerning the decomposition of the overall system into subsystems, we consider all three possible approaches, i.e. force/force‐, force/displacement‐ and displacement/displacement‐decomposition. The stability analysis of co‐simulation methods with algebraic constraints is inherently related to the definition of a test model. Bearing in mind the stability definition for numerical time integration schemes, i.e. Dahlquist's stability theory based on the linear single‐mass oscillator, a linear two‐mass oscillator is used here for analyzing the stability of co‐simulation methods. The two‐mass co‐simulation test model may be regarded as two Dahlquist equations, coupled by an algebraic constraint equation. By discretizing the co‐simulation test model with a linear co‐simulation approach, a linear system of recurrence equations is obtained. The stability of the recurrence system, which reflects the stability of the underlying coupling method, can simply be determined by an eigenvalue analysis.

Nonlinear analysis of nonprismatic Timoshenko beam for different geometric nonlinearity models

This paper deals with the statics and stability of a nonprismatic beam described according to Timoshenko theory, with various geometric nonlinearity models taken into account. The investigated models differed in the complexity of the nonlinear part of the strain tensor. In the simplest model only one nonlinear deformation component, i.e. (W′)2/2, was taken into account. Most of the works on geometrically nonlinear beam models, known to the present authors, analyze the simplest model. As demonstrated here this model yields correct results only in for beams with nonslidable supports. An analysis of slidable systems carried out in this paper indicates big differences between the solutions obtained using the different nonlinearity models and shows that in the case of the simplest model the solutions differ considerably from the ones obtained by, e.g., FEM. It also shown that when certain additional strain tensor elements are taken into account, this, although correct from the mathematical point of view, leads to incorrect solutions. One original contribution of this paper is the application of the approximation method to solve the nonlinear problem. The method uses the Chebyshev series whose expansion coefficients are determined from a certain system of recurrence equations. The method enables one to solve equations with variable coefficients. As shown in the previous papers by the author, in the case of solutions to linear problems this method leads to very accurate (also in comparison with analytical solutions) results. The other original contribution is the demonstration of the influence of the particular nonlinear strain tensor components on the solutions to the analyzed problems.

A self-consistent field method applied to the dynamics of viscous suspensions

A method for the approximate solution of the problem of many bodies of spherical form in a viscous fluid is developed in the Stokes approximation. Using a purely hydrodynamic approach, based on the use of the concept of a self-consistent field, the classical boundary value problem is reduced to a formal procedure for solving a linear system of algebraic equations in the tensor coefficients, which occur in the solution obtained for the velocity field and pressure of the liquid. A procedure for the approximate solution of this system of equations is constructed for the case of dilute suspensions, when the ratio of the size of the dispersed particles to the characteristic distance between them is a small parameter. Finally, the initial boundary value problem is reduced to solving a recurrent system of equations, in which each subsequent approximation for all the required quantities depends solely on the previous approximations. The system of recurrent equations obtained can be solved analytically in any specified approximation with respect to a small parameter. It is shown that this system of equations contains in itself all possible physical formulations of the problems, and, within the frameworks of the mathematical procedure constructed, they are distinguished solely by a set of specified and required functions. The practical possibilities of the method are in no way limited by the number of dispersed particles in the fluid.

Joint spectral radius, dilation equations, and asymptotic behavior of radix-rational sequences

Radix-rational sequences are solutions of systems of recurrence equations based on the radix representation of the index. For each radix-rational sequence with complex values we provide an asymptotic expansion, essentially in the scale NαlogℓN. The precision of the asymptotic expansion depends on the joint spectral radius of the linear representation of the sequence of first-order differences. The coefficients are Hölderian functions obtained through some dilation equations, which are usual in the domains of wavelets and refinement schemes. The proofs are ultimately based on elementary linear algebra.

A new parallel algorithm for simulation of spin glasses on scales of space-time periods of external fields with consideration of relaxation effects

We have investigated the statistical properties of an ensemble of disordered 1D spatial spin chains (SSCs) of finite length, placed in an external field, with consideration of relaxation effects. The short-range interaction complex-classical Hamiltonian was first used for solving this problem. A system of recurrent equations is obtained on the nodes of the spin-chain lattice. An efficient mathematical algorithm is developed on the basis of these equations with consideration of advanced Sylvester conditions which allows one to step by step construct a huge number of stable spin chains in parallel. The distribution functions of different parameters of spin glass system are constructed from first principles by analyzing the calculation results of the 1D SSCs ensemble. It is shown that the behaviors of different distributions parameters are quite different even at weak external fields. The ensemble energy and constants of spin-spin interactions are being changed smoothly depending on the external field in the limit of statistical equilibrium, while some of them such as the mean value of polarizations of the ensemble and parameters of its orderings are frustrated. We have also studied some critical properties of the ensemble such as catastrophes in the Clausius-Mossotti equation depending on the value of the external field. We have shown that the generalized complex-classical approach excludes these catastrophes, which allows one to organize continuous parallel computing on the whole region of values of the external field including critical points. A new representation of the partition function is suggested based on these investigations. Being opposite to the usual definition, it is a complex function and its derivatives are everywhere defined, including at critical points.