scholarly journals Entropy and variational principle for one-dimensional lattice systems with a generala prioriprobability: positive and zero temperature

2014 ◽  
Vol 35 (6) ◽  
pp. 1925-1961 ◽  
Author(s):  
A. O. LOPES ◽  
J. K. MENGUE ◽  
J. MOHR ◽  
R. R. SOUZA
Keyword(s):  
Infinite Product ◽  
A Priori ◽  
Transfer Operator ◽  
Thermodynamic Formalism ◽  
General Setting ◽  
Lattice Systems ◽  
Ruelle Operator ◽  
Finite Set ◽  
Zero Entropy ◽  
Infinite Set

We generalize several results of the classical theory of thermodynamic formalism by considering a compact metric space$M$as the state space. We analyze the shift acting on$M^{\mathbb{N}}$and consider a generala prioriprobability for defining the transfer (Ruelle) operator. We study potentials$A$which can depend on the infinite set of coordinates in$M^{\mathbb{N}}$. We define entropy and by its very nature it is always a non-positive number. The concepts of entropy and transfer operator are linked. If$M$is not a finite set there exist Gibbs states with arbitrary negative value of entropy. Invariant probabilities with support in a fixed point will have entropy equal to minus infinity. In the case$M=S^{1}$, and thea priorimeasure is Lebesgue$dx$, the infinite product of$dx$on$(S^{1})^{\mathbb{N}}$will have zero entropy. We analyze the Pressure problem for a Hölder potential$A$and its relation with eigenfunctions and eigenprobabilities of the Ruelle operator. Among other things we analyze the case where temperature goes to zero and we show some selection results. Our general setting can be adapted in order to analyze the thermodynamic formalism for the Bernoulli space with countable infinite symbols. Moreover, the so-called$XY$model also fits under our setting. In this last case$M$is the unitary circle$S^{1}$. We explore the differentiable structure of$(S^{1})^{\mathbb{N}}$by considering a certain class of smooth potentials and we show some properties of the corresponding main eigenfunctions.

scholarly journals Thermodynamic formalism for quantum channels: Entropy, pressure, Gibbs channels and generic properties

2021 ◽  
Author(s):  
Jader E. Brasil ◽  
Josué Knorst ◽  
Artur O. Lopes
Keyword(s):  
Density Operator ◽  
A Priori ◽  
Thermodynamic Formalism ◽  
Quantum Channels ◽  
Generic Properties ◽  
Ruelle Operator ◽  
Density Operators ◽  
Starting Point ◽  
Generic Set ◽  
Related Process

Denote [Formula: see text] the set of complex [Formula: see text] by [Formula: see text] matrices. We will analyze here quantum channels [Formula: see text] of the following kind: given a measurable function [Formula: see text] and the measure [Formula: see text] on [Formula: see text] we define the linear operator [Formula: see text], via the expression [Formula: see text]. A recent paper by T. Benoist, M. Fraas, Y. Pautrat, and C. Pellegrini is our starting point. They considered the case where [Formula: see text] was the identity. Under some mild assumptions on the quantum channel [Formula: see text] we analyze the eigenvalue property for [Formula: see text] and we define entropy for such channel. For a fixed [Formula: see text] (the a priori measure) and for a given a Hamiltonian [Formula: see text] we present a version of the Ruelle Theorem: a variational principle of pressure (associated to such [Formula: see text]) related to an eigenvalue problem for the Ruelle operator. We introduce the concept of Gibbs channel. We also show that for a fixed [Formula: see text] (with more than one point in the support) the set of [Formula: see text] such that it is [Formula: see text]-Erg (also irreducible) for [Formula: see text] is a generic set. We describe a related process [Formula: see text], [Formula: see text], taking values on the projective space [Formula: see text] and analyze the question of the existence of invariant probabilities. We also consider an associated process [Formula: see text], [Formula: see text], with values on [Formula: see text] ([Formula: see text] is the set of density operators). Via the barycenter, we associate the invariant probability mentioned above with the density operator fixed for [Formula: see text].

PRECISE CALCULATION OF HAUSDORFF DIMENSION OF APOLLONIAN GASKET

Fractals ◽  
2018 ◽  
Vol 26 (04) ◽  
pp. 1850050
Author(s):  
ZAI-QIAO BAI ◽  
STEVEN R. FINCH
Keyword(s):  
Hausdorff Dimension ◽  
Transfer Operator ◽  
Operator Method ◽  
Matrix Approximation ◽  
Infinite Dimensional ◽  
Precise Calculation ◽  
Finite Matrix ◽  
Finite Set ◽  
Infinite Set ◽  
Infinite Sum

A transfer operator method is proposed to calculate [Formula: see text], the Hausdorff dimension of the Apollonian gasket. Compared with previous operator-based methods, we make two improvements in this paper. We adopt an infinite set of contractive Möbius transformations (rather than a finite set of parabolic ones) to generate the Apollonian gasket. We also apply an efficient finite matrix approximation of an infinite sum of infinite-dimensional operators. By using this method, a high precision estimate of [Formula: see text] is obtained: [Formula: see text]

Thermodynamic formalism for a modified shift map

2014 ◽  
Vol 14 (02) ◽  
pp. 1350020 ◽  
Author(s):  
Stephen Muir ◽  
Mariusz Urbański
Keyword(s):  
A Priori ◽  
Transfer Operator ◽  
Thermodynamic Formalism ◽  
Lattice Gases ◽  
Hölder Continuous ◽  
Full Generality ◽  
Counting Measure ◽  
Shift Map ◽  
Continuous Potential ◽  
Shift Action

We introduce a transfer operator and use it to prove some theorems of a classical flavor from thermodynamic formalism (including existence and uniqueness of appropriately defined Gibbs states and equilibrium states for potential functions satisfying Dini's condition and stochastic laws for Hölder continuous potential and observable functions) in a novel setting: the "alphabet" E is a compact metric space equipped with an a priori probability measure ν and an endomorphism T. The "modified shift map" S is defined on the product space Eℕ by the rule (x1x2x3…) ↦ (T(x2)x3…). The greatest novelty is found in the variational principle, where a term must be added to the entropy to reflect the transformation of the first coordinate by T after shifting. Our motivation is that this system, in its full generality, cannot be treated by the existing methods of either rigorous statistical mechanics of lattice gases (where only the true shift action is used) or dynamical systems theory (where the a priori measure is always implicitly taken to be the counting measure).

scholarly journals Thermodynamic formalism for Haar systems in noncommutative integration: transverse functions and entropy of transverse measures

2020 ◽  
pp. 1-29
Author(s):  
ARTUR O. LOPES ◽  
JAIRO K. MENGUE
Keyword(s):  
Equivalence Relation ◽  
General Class ◽  
A Priori ◽  
Transfer Operator ◽  
Thermodynamic Formalism ◽  
Modular Functions ◽  
Noncommutative Integration

We consider here a certain class of groupoids obtained via an equivalence relation (the so-called subgroupoids of pair groupoids). We generalize to Haar systems in these groupoids some results related to entropy and pressure which are well known in thermodynamic formalism. We introduce a transfer operator, where the equivalence relation (which defines the groupoid) plays the role of the dynamics and the corresponding transverse function plays the role of the a priori probability. We also introduce the concept of invariant transverse probability and of entropy for an invariant transverse probability, as well as of pressure for transverse functions. Moreover, we explore the relation between quasi-invariant probabilities and transverse measures. Some of the general results presented here are not for continuous modular functions but for the more general class of measurable modular functions.

scholarly journals FEATURES OF SECTIONING THE TURNS OF THE TRANSFORMING ELEMENT OF THE TRANSFORMER-KEY ACTUATING STRUCTURE IN THE BOOST CHANNEL OF THE DC POWER SYSTEM

2020 ◽  
Vol 2020 (6) ◽  
pp. 25-31
Author(s):  
K.O. Lypkivskyi ◽  
◽  
A.G. Mozharovskyi ◽  
Keyword(s):  
High Efficiency ◽  
A Priori ◽  
Current Source ◽  
Voltage Regulation ◽  
Specific Class ◽  
Voltage Supply ◽  
Emergency Situations ◽  
Time Control ◽  
Finite Set ◽  
In Series

One of the effective ways to ensure the normalized operation of the electricity consumer with an unstable primary power source is the organization of the corresponding voltage supply channel. In a system with a direct current source, the voltage supply is implemented by introducing a rectifier semiconductor bridge in series with the load, into the diagonal of which AC energy is supplied, the voltage level of which is purposefully changed by a corresponding converter with a transformer- and-switches executive structure (TSES). To achieve high efficiency of the use of key elements of TSVS, it is proposed to assign the functions of rectification and voltage regulation to a specific class of TSES – a multilevel rectifier consisting of a transformer and a finite set of parallel connected pairs of serially connected thyristors, the common points of which are connected to the corresponding taps of the sectioned secondary turns of the transformer. By discrete-time control of thyristors, it is necessary to regulate voltage levels, it is attached. The linearity of the scale of these levels is ensured by the proposed transformer sectioning law. This power supply system is characterized by small energy losses in semiconductor elements (only two thyristors work at a time), and the a priori impossibility of emergency situations during transitions from one level to another. References 14, figures 3, tables 3.

scholarly journals Categorical Functional Data Analysis. The cfda R Package

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3074
Author(s):  
Cristian Preda ◽  
Quentin Grimonprez ◽  
Vincent Vandewalle
Keyword(s):  
Functional Data ◽  
Multiple Correspondence Analysis ◽  
Real Data ◽  
Jump Process ◽  
R Package ◽  
Finite Basis ◽  
Data Set ◽  
Stochastic Jump ◽  
Finite Set ◽  
Infinite Set

Categorical functional data represented by paths of a stochastic jump process with continuous time and a finite set of states are considered. As an extension of the multiple correspondence analysis to an infinite set of variables, optimal encodings of states over time are approximated using an arbitrary finite basis of functions. This allows dimension reduction, optimal representation, and visualisation of data in lower dimensional spaces. The methodology is implemented in the cfda R package and is illustrated using a real data set in the clustering framework.

Curious Beginnings

2016 ◽  
Author(s):  
Joseph Mazur
Keyword(s):  
Historical Event ◽  
Mathematical Notation ◽  
Human Language ◽  
Mathematical Writing ◽  
Finite Set ◽  
And Algebra ◽  
Simple Equations ◽  
Infinite Set

This chapter traces the beginnings of mathematical notation. For tens of thousands of years, humans had been leaving signification marks in their surroundings, gouges on trees, footprints in hard mud, scratches in skin, and even pigments on rocks. A simple mark can represent a thought, indicate a plan, or record a historical event. Yet the most significant thing about human language and writing is that speakers and writers can produce a virtually infinite set of sounds, declarations, notions, and ideas from a finite set of marks and characters. The chapter discusses the emergence of the alphabet, counting, and mathematical writing. It also considers the discovery of traces of Sumerian number writing on clay tablets in caves from Europe to Asia, the use of Egyptian hieroglyphics, and algebra problems in the Rhind (or Ahmes) papyrus that presented simple equations without any symbols other than those used to indicate numbers.

scholarly journals Generalized r-cohesiveness and the arithmetical hierarchy: a correction to “Generalized cohesiveness”

2002 ◽  
Vol 67 (3) ◽  
pp. 1078-1082
Author(s):  
Carl G. Jockusch ◽  
Tamara J. Lakins
Keyword(s):  
Computable Function ◽  
Arithmetical Hierarchy ◽  
Finite Set ◽  
Infinite Set

AbstractFor X ⊆ ω, let [X]n denote the class of all n-element subsets of X. An infinite set A ⊆ ω is called n-r-cohesive if for each computable function f: [ω]n → {0, 1} there is a finite set F such that f is constant on [A − F]n. We show that for each n > 2 there is no Πn0 set A ⊆ ω which is n-r-cohesive. For n = 2 this refutes a result previously claimed by the authors, and for n ≥ 3 it answers a question raised by the authors.

A generalization of McDiarmid's theorem for mixed Bernoulli percolation

1980 ◽  
Vol 88 (1) ◽  
pp. 167-170 ◽  
Author(s):  
J. M. Hammersley
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Finite Sequence ◽  
Random Set ◽  
Percolation Probability ◽  
Open Path ◽  
Finite Set ◽  
Image Position ◽  
Infinite Set ◽  
Mutually Independent

Let G be an infinite partially directed graph of finite outgoing degree. Thus G consists of an infinite set of vertices, together with a set of edges between certain prescribed pairs of vertices. Each edge may be directed or undirected, and the number of edges from (but not necessarily to) any given vertex is always finite (though possibly unbounded). A path on G from a vertex V1 to a vertex Vn (if such a path exists) is a finite sequence of alternate edges and vertices of the form E12, V2, E23, V3, …, En − 1, n, Vn such that Ei, i + 1 is an edge connecting Vi and Vi + 1 (and in the direction from Vi to Vi + 1 if that edge happens to be directed). In mixed Bernoulli percolation, each vertex Vi carries a random variable di, and each edge Eij carries a random variable dij. All these random variables di and dij are mutually independent, and take only the values 0 or 1; the di take the value 1 with probability p, while the dij take the value 1 with probability p. A path is said to be open if and only if all the random variables carried by all its edges and all its vertices assume the value 1. Let S be a given finite set of vertices, called the source set; and let T be the set of all vertices such that there exists at least one open path from some vertex of S to each vertex of T. (We imagine that fluid, supplied to all the source vertices, can flow along any open path; and thus T is the random set of vertices eventually wetted by the fluid). The percolation probabilityis defined to be the probability that T is an infinite set.

scholarly journals Wilansky's query on outer measures

1985 ◽  
Vol 31 (3) ◽  
pp. 325-328
Author(s):  
Choo-Whan Kim
Keyword(s):  
Outer Measure ◽  
Finite Set ◽  
Infinite Set

On a set X, let μ* be an outer measure and μ the measure induced by μ*. We show that if X is a finite set, then the measure μ is saturated. We give two examples of non-regular outer measures on an infinite set X which induce non-saturated and saturated measures, respectively. These answer a query posed by Wilansky.

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