VOLE POPULATION DYNAMICS UNDER THE INFLUENCE OF SPECIALIST AND GENERALIST PREDATION

Abstract To understand the impact of predation by different types of predators on the vole population dynamics, we formulate a three differential equation model describing the population dynamics of voles, the “specialist predator” and the “generalist predator.” First we perform a local stability study of the different steady states of the basic model and deduce that the predation rates of the “specialist” as well as the “generalist” predator are the main parameters controlling the existence/extinction criteria of the concerned populations. Next we analyze the model from a thermodynamic perspective and study the thermodynamic stability of the different equilibria. Finally using stochastic driving forces, we incorporate the exogenous factor of environmental forcing and investigate the stochastic stability of the system. We compare the stability criteria of the different steady states under deterministic, thermodynamic and stochastic situations. The analysis reveals that when the “specialist” and the “generalist” predator are modeled separately, the system exhibits rich dynamics and the predation rates of both types of predators play a major role in controlling vole oscillation and/or stability. These findings are also seen to resemble closely with the observed behavior of voles in the natural setting. Numerical simulations are carried out to illustrate analytical findings.     

Statistical theory of rarified gases in the coulomb model of substance: Adiabatic approximation and initial atoms

In the framework of the adiabatic approximation for a subsystem of nuclei with the average distance between them significantly exceeding the dimensions of the initial atom, we consider a nonrelativistic Coulomb system consisting of electrons and nuclei of one type for the temperature range where we can restrict ourself to using the ground state to describe the electron subsystem. We show that the equilibrium properties of such a system are equivalent to the thermodynamic properties of the one-component system of initial atoms interacting between themselves via a short-range potential that is the effective potential of the nucleus-nucleus interaction. In the framework of the applicability of Boltzmann statistics, we present quantum group expansions for the thermodynamic properties of a chemically reacting rarified gas that correspond to the method of initial atoms.     

Inhomogeneous Electron Density in the Static Field of Nuclei

Theoretical and Mathematical Physics - We consider the thermodynamic potential of a Coulomb system of electrons and nuclei in the adiabatic approximation for the nuclei subsystem. In this...     

Controlling Bivariate Categorical Processes using Scan Rules

In this paper, we introduce a methodology for efficiently monitoring a health process that classify the intervention outcome, in two dependent characteristics, as “absolutely successful”, “with minor but acceptable complications” and “unsuccessful due to severe complications”. The monitoring procedure is based on appropriate 2-dimensional scan rules. The run length distribution is acquired by studying the waiting time distribution for the first occurrence of a 2-dimensional scan in a bivariate sequence of trinomial trials. The waiting time distribution is derived through a Markov chain embedding technique. The proposed procedure is applied on two simulated cases while it is tested against a competing method showing an excellent performance.     

On Tight‐Binding Approximations in Optical Lattices

The validity of using the tight‐binding approximation for the nonlinear Schrödinger equations with a two‐dimensional optical lattice is considered. This work provides a rigorous foundation for a technique based on “orbital” functions that is central to solid‐state physics and nonlinear optics. Simple and honeycomb lattices are addressed, and it is therefore shown that the use of tight‐binding approximations is justified in complicated situations.     

Perturbation theory for the periodic Anderson model

We investigate the system of conductivity electrons and f-localized electrons described by the periodic Anderson model. Single-site hybridization of the state of two constituent subsystems of electrons is treated as a perturbation. We develop a new diagram technique based on the use of multiparticle one-site irreducible Green’s functions for the f-electrons and the standard Wick theorem for the subsystem of conductivity electrons. We derive the Dyson equations for the one-particle Green’s functions and find the relation between these functions. These results are exact and can be used as a starting point for self-consistent approximations. In the Hubbard-I approximation, we analyze the spectrum of one-particle perturbations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 2, pp. 308–322, February, 1997.     

Large‐deviations/thermodynamic approach to percolation on the complete graph

We present a large‐deviations/thermodynamic approach to the classic problem of percolation on the complete graph. Specifically, we determine the large‐deviation rate function for the probability that the giant component occupies a fixed fraction of the graph while all other components are “small.” One consequence is an immediate derivation of the “cavity” formula for the fraction of vertices in the giant component. As a byproduct of our analysis we compute the large‐deviation rate functions for the probability of the event that the random graph is connected, the event that it contains no cycles and the event that it contains only small components. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Quantum Transport in Degenerate Systems

Transport in nonequilibrium degenerate quantum systems is investigated. The transfer rate depends on the parameters of the system. In this paper we investigate the dependence of the flow (transfer rate) on the angle between “bright” vectors (which define the interaction of the system with the environment). We show that in some approximation for the system under investigation the flow is proportional to the cosine squared of the angle between the “bright” vectors. Earlier the author has shown that in this degenerate quantum system excitation of nondecaying quantum “dark” states is possible; moreover, the effectiveness of this process is proportional to the sine squared of the angle between the “bright” vectors (this phenomenon was discussed as a possible model of excitation of quantum coherence in quantum photosynthesis). Thus quantum transport and excitation of dark states are competing processes; “dark” states can be considered as a result of leakage of quantum states in a quantum thermodynamic machine which performs the quantum transport.     

Smoothing Minimally Supported Frequency Wavelets: Part II

The main purpose of this paper is to give a procedure to "mollify" the low-pass filters of a large number of Minimally Supported Frequency (MSF) wavelets so that the smoother functions obtained in this way are also low-pass filters for an MRA. Hence, we are able to approximate (in the L²-norm) MSF wavelets by wavelets with any desired degree of smoothness on the Fourier transform side. Although the MSF wavelets we consider are bandlimited, this may not be true for their smooth approximations. This phenomena is related to the invariant cycles under the transformation $x\mapsto 2x (\mbox{mod}2\pi).The main purpose of this paper is to give a procedure to “mollify” the low-pass filters of a large number ofMinimally Supported Frequency (MSF) wavelets so that the smoother functions obtained in this way are also low-pass filters for an MRA. Hence, we are able to approximate (in the L ² -norm) MSF wavelets by wavelets with any desired degree of smoothness on the Fourier transform side. Although the MSF wavelets we consider are bandlimited, this may not be true for their smooth approximations. This phenomena is related to the invariant cycles under the transformation x ?2x (mod2π). We also give a characterization of all low-pass filters for MSF wavelets. Throughout the paper new and interesting examples of wavelets are described.     

Application of best rational function approximation for Laplace transform inversion

The application of a method of least squares Laplace transform inversion due to the author is discussed, and examples are given, which also serve to give us best trigonometric and exponential function approximations to some known functions. In particular the unit step function H(t-1) is considered, and the “best” approximations obtained for it would seem to have application in electrical network theory, including the design of delay lines.     

On some estimates for best approximations of bivariate functions by Fourier–Jacobi sums in the mean

Some problems in computational mathematics and mathematical physics lead to Fourier series expansions of functions (solutions) in terms of special functions, i.e., to approximate representations of functions (solutions) by partial sums of corresponding expansions. However, the errors of these approximations are rarely estimated or minimized in certain classes of functions. In this paper, the convergence rate (of best approximations) of a Fourier series in terms of Jacobi polynomials is estimated in classes of bivariate functions characterized by a generalized modulus of continuity. An approximation method based on “spherical” partial sums of series is substantiated, and the introduction of a corresponding class of functions is justified. A two-sided estimate of the Kolmogorov N-width for bivariate functions is given.     

The behavior of plasma with an arbitrary degree of degeneracy of electron gas in the conductive layer

We obtain an analytic solution of the boundary problem for the behavior (fluctuations) of an electron plasma with an arbitrary degree of degeneracy of the electron gas in the conductive layer in an external electric field. We use the kinetic Vlasov–Boltzmann equation with the Bhatnagar–Gross–Krook collision integral and the Maxwell equation for the electric field. We use the mirror boundary conditions for the reflections of electrons from the layer boundary. The boundary problem reduces to a one-dimensional problem with a single velocity. For this, we use the method of consecutive approximations, linearization of the equations with respect to the absolute distribution of the Fermi–Dirac electrons, and the conservation law for the number of particles. Separation of variables then helps reduce the problem equations to a characteristic system of equations. In the space of generalized functions, we find the eigensolutions of the initial system, which correspond to the continuous spectrum (Van Kampen mode). Solving the dispersion equation, we then find the eigensolutions corresponding to the adjoint and discrete spectra (Drude and Debye modes). We then construct the general solution of the boundary problem by decomposing it into the eigensolutions. The coefficients of the decomposition are given by the boundary conditions. This allows obtaining the decompositions of the distribution function and the electric field in explicit form.     

Method for Constructing Piecewise Quadratic Value Functions in a Control Problem for a Switched System

The problem of constructing internal approximations to solvability sets and the control synthesis problem for a piecewise linear system with control parameters and disturbances (uncertainties) are solved. The solution is based on the comparison principle and piecewise quadratic value functions of a special form. Relations defining such functions and, in particular, “continuous binding conditions” for the functions and their first derivatives are obtained. The results are used to construct numerical methods for solving the control synthesis problem for the class of switched systems under study. An example of approximate solution of the control synthesis problem in a target control problem for a nonlinear mathematical model of a pendulum with a flywheel is considered.     

An approximation method using approximate approximations

The aim of this article is to extend the method of approximate approximations to boundary value problems. This method was introduced by V. Maz'ya in 1991 and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present article we develop an approximation procedure for the solution of the interior Dirichlet problem for the Laplace equation in two dimensions using approximate approximations. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In the first step, the unknown source density in the potential representation of the solution is replaced by approximate approximations. In the second, the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in the third, Nyström's method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given.     

The thermodynamic limit of quantum Coulomb systems Part II. Applications

In a previous paper [C. Hainzl, M. Lewin, J.P. Solovej, The thermodynamic limit of quantum Coulomb systems. Part I. General theory, Adv. Math. (2009), doi:10.1016/j.aim.2008.12.010 (this issue)], we have developed a general theory of thermodynamic limits. We apply it here to three different Coulomb quantum systems, for which we prove the convergence of the free energy per unit volume.The first system is the crystal for which the nuclei are classical particles arranged periodically in space and only the electrons are quantum particles. We recover and generalize a previous result of Fefferman. In the second example, both the nuclei and the electrons are quantum particles, submitted to a periodic magnetic field. We thereby extend a seminal result of Lieb and Lebowitz. Finally, in our last example we take again classical nuclei but optimize their position. To our knowledge such a system was never treated before.The verification of the assumptions introduced in [C. Hainzl, M. Lewin, J.P. Solovej, The thermodynamic limit of quantum Coulomb systems. Part I. General theory, Adv. Math. (2009), doi:10.1016/j.aim.2008.12.010 (this issue)] uses several tools which have been introduced before in the study of large quantum systems. In particular, an electrostatic inequality of Graf and Schenker is one main ingredient of our new approach.     

Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration

Error estimates with explicit constants are given for approximations of functions, definite integrals and indefinite integrals by means of the Sinc approximation. Although in the literature various error estimates have already been given for these approximations, those estimates were basically for examining the rates of convergence, and several constants were left unevaluated. Giving more explicit estimates, i.e., evaluating these constants, is of great practical importance, since by this means we can reinforce the useful formulas with the concept of “verified numerical computations.” In this paper we reveal the explicit form of all constants in a computable form under the same assumptions of the existing theorems: the function to be approximated is analytic in a suitable region. We also improve some formulas themselves to decrease their computational costs. Numerical examples that confirm the theory are also given.     

Multiparticle Direct Correlations

We study the properties of multiparticle direct correlations in detail and construct a system of successive approximations of the generating functional of direct correlation functions. Moreover, in contrast to the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy, this approach does not require approximating higher correlations. Our approximations determine the corresponding approximations of the thermodynamic potential. The irreducible parts of multiparticle direct correlations are constructed explicitly.     

The fraction of large random trees representing a given Boolean function in implicational logic

We consider the logical system of Boolean expressions built on the single connector of implication and on positive literals. Assuming all expressions of a given size to be equally likely, we prove that we can define a probability distribution on the set of Boolean functions expressible in this system. Then we show how to approximate the probability of a function f when the number of variables grows to infinity, and that this asymptotic probability has a simple expression in terms of the complexity of f. We also prove that most expressions computing any given function in this system are “simple”, in a sense that we make precise. The probability of all read‐once functions of a given complexity is also evaluated in this model. At last, using the same techniques, the relation between the probability of a function and its complexity is also obtained when random expressions are drawn according to a critical branching process. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Effective numerical evaluation of the double Hilbert transform

In this paper, we propose two methods to compute the double Hilbert transform of periodic functions. First, we establish the quadratic formula of trigonometric interpolation type for double Hilbert transform and obtain an estimation of the remainder. We call this method 2D mechanical quadrature method (2D-MQM). Numerical experiments show that 2D-MQM outperforms the library function “hilbert” in Matlab when the values of the functions being handled are very large or approach to infinity. Second, we propose a complex analytic method to calculate the double Hilbert transform, which is based on the 2D adaptive Fourier decomposition, and the method is called as 2D-HAFD. In contrast to the pointwise approximation, 2D-HAFD provides explicit rational functional approximations and is valid for all signals of finite energy.