A simple proof of the Radó and Král theorems on removability of the zero locus for analytic and harmonic functions

A unified approach to the proof of the Radó and Král theorems on removability of the zero locus for analytic and harmonic functions is proposed.Запропоновано єдиний пiдхiд до доведення теорем Радо та Крала про усувнiсть множини нулiв для аналiтичних та гармонiчних функцiй.Предложен единый подход к доказательству теорем Радо и Крала об устранимости множества нулей для аналитических и гармонических функций

Theory of Two-Dimensional Quantum Heisenberg Antiferromagnets with a Nearly Critical Ground State

We present the general theory of clean, two-dimensional, quantum Heisenberg antiferromagnets which are close to the zero-temperature quantum transition between ground states with and without long-range N\'{e}el order. For N\'{e}el-ordered states, `nearly-critical' means that the ground state spin-stiffness, $\rho_s$, satisfies $\rho_s \ll J$, where $J$ is the nearest-neighbor exchange constant, while `nearly-critical' quantum-disordered ground states have a energy-gap, $\Delta$, towards excitations with spin-1, which satisfies $\Delta \ll J$. Under these circumstances, we show that the wavevector/frequency-dependent uniform and staggered spin susceptibilities, and the specific heat, are completely universal functions of just three thermodynamic parameters. Explicit results for the universal scaling functions are obtained by a $1/N$ expansion on the $O(N)$ quantum non-linear sigma model, and by Monte Carlo simulations. These calculations lead to a variety of testable predictions for neutron scattering, NMR, and magnetization measurements. Our results are in good agreement with a number of numerical simulations and experiments on undoped and lightly-doped $La_{2-\delta} Sr_{\delta}Cu O_4$.Comment: 81 pages, REVTEX 3.0, smaller updated version, YCTP-xxx

A Mathematical Analysis of the Effect of the Variability of Input on the Mean Value of Output for a Class of Technological Processes

A class of random dynamical systems is considered. We discuss a class of problems arising from an example in food production likely to have wider applicability in the content of industrial quality control. Permanent address: Institute for Information Transmission Problems, Russian Academy of Science, 19 Bolshoi Karetny lane, Moscow 101447, Russia. Contents 1 Introduction 3 2 General results 6 2.1 Mathematical setup . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Proof of the theorem . . . . . . . . . . . . . . . . . . . . . . . 10 3 Some particular cases 13 3.1 Scalar systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Coordinate-wise monotone systems . . . . . . . . . . . . . . . 14 3.3 Control theory type systems . . . . . . . . . . . . . . . . . . . 15 A Appendix 17 1 Introduction This is basically a paper in mathematical control theory. In general terms, we are interested in..

Hysteresis in thermal expansion of the quasi 1-dimensional conductor TaS3_3: Coupling of the underlying and the electronic crystals

An interferometer-based setup for measurements of length of needle-like samples is developed, and thermal expansion of o-TaS$_3$ crystals is studied. Below the Peierls transition the temperature hysteresis of length $L$ is observed, the width of the hysteresis loop $\delta L/L$ being up to $5\times 10^{-5}$. Curiously, $L(T)$ changes so that it is in front of its equilibrium value. The hysteresis loop couples with that of conductivity. With lowering $T$ the charge-density waves' (CDW) elastic modulus grows and at 100 K becomes comparable with that of the lattice $Y_l$. The results justify the assumption about the strain dependence of the CDW wave vector and clarify the nature of the anomalies of $Y_l$ which occur on the CDW depinning. In particular, $Y_l$, is expected to show a strong drop in the static regime, if measured at sufficiently small sample elongation $(\delta L/L < 10^{ -5})$

Memory effects in population dynamics: spread of infectious disease as a case study

Modification of behaviour in response to changes in the environment or ambient conditions, based on memory, is typical of the human and, possibly, many animal species.One obvious example of such adaptivity is, for instance, switching to a safer behaviour when in danger, from either a predator or an infectious disease. In human society such switching to safe behaviour is particularly apparent during epidemics. Mathematically, such changes of behaviour in response to changes in the ambient conditions can be described by models involving switching. In most cases, this switching is assumed to depend on the system state, and thus it disregards the history and, therefore, memory. Memory can be introduced into a mathematical model using a phenomenon known as hysteresis. We illustrate this idea using a simple SIR compartmental model that is applicable in epidemiology. Our goal is to show why and how hysteresis can arise in such a model, and how it may be applied to describe a variety of memory effects. Our other objective is to introduce a unified paradigm for mathematical modelling with memory effects in epidemiology and ecology. Our approach treats changing behaviour as an irreversible flow related to large ensembles of elementary exchange operations that recently has been successfully applied in a number of other areas, such as terrestrial hydrology, and macroeconomics. For the purposes of illustrating these ideas in an application to biology, we consider a rather simple case study and develop a model from first principles. We accompany the model with extensive numerical simulations which exhibit interesting qualitative effects

Memory Effects in Population Dynamics : Spread of Infectious Disease as a Case Study

Modification of behaviour in response to changes in the environment or ambient conditions, based on memory, is typical of the human and, possibly, many animal species.One obvious example of such adaptivity is, for instance, switching to a safer behaviour when in danger, from either a predator or an infectious disease. In human society such switching to safe behaviour is particularly apparent during epidemics. Mathematically, such changes of behaviour in response to changes in the ambient conditions can be described by models involving switching. In most cases, this switching is assumed to depend on the system state, and thus it disregards the history and, therefore, memory. Memory can be introduced into a mathematical model using a phenomenon known as hysteresis. We illustrate this idea using a simple SIR compartmental model that is applicable in epidemiology. Our goal is to show why and how hysteresis can arise in such a model, and how it may be applied to describe a variety of memory effects. Our other objective is to introduce a unified paradigm for mathematical modelling with memory effects in epidemiology and ecology. Our approach treats changing behaviour as an irreversible flow related to large ensembles of elementary exchange operations that recently has been successfully applied in a number of other areas, such as terrestrial hydrology, and macroeconomics. For the purposes of illustrating these ideas in an application to biology, we consider a rather simple case study and develop a model from first principles. We accompany the model with extensive numerical simulations which exhibit interesting qualitative effects

Chaos in a seasonally perturbed SIR model: avian influenza in a seabird colony as a paradigm

Seasonality is a complex force in nature that affects multiple processes in wild animal populations. In particular, seasonal variations in demographic processes may considerably affect the persistence of a pathogen in these populations. Furthermore, it has been long observed in computer simulations that under seasonal perturbations, a host–pathogen system can exhibit complex dynamics, including the transition to chaos, as the magnitude of the seasonal perturbation increases. In this paper, we develop a seasonally perturbed Susceptible-Infected-Recovered model of avian influenza in a seabird colony. Numerical simulations of the model give rise to chaotic recurrent epidemics for parameters that reflect the ecology of avian influenza in a seabird population, thereby providing a case study for chaos in a host– pathogen system. We give a computer-assisted exposition of the existence of chaos in the model using methods that are based on the concept of topological hyperbolicity. Our approach elucidates the geometry of the chaos in the phase space of the model, thereby offering a mechanism for the persistence of the infection. Finally, the methods described in this paper may be immediately extended to other infections and hosts, including humans