Domains for Computation in Mathematics, Physics and Exact Real Arithmetic

1997 ◽  
Vol 3 (4) ◽  
pp. 401-452 ◽  
Author(s):  
Abbas Edalat
Keyword(s):  
Statistical Physics ◽  
Computational Models ◽  
Metric Spaces ◽  
Probability Distributions ◽  
Period Doubling ◽  
Computer Arithmetic ◽  
Iterated Function Systems ◽  
Integration Theory ◽  
Elementary Functions ◽  
Invariant Distributions

AbstractWe present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chaos; we also show how efficient algorithms have been obtained for computing elementary functions in exact real arithmetic.

The formal ball model for -categories

2010 ◽  
Vol 21 (1) ◽  
pp. 41-64 ◽  
Author(s):  
MATEUSZ KOSTANEK ◽  
PAWEŁ WASZKIEWICZ
Keyword(s):  
Metric Space ◽  
Computational Models ◽  
Metric Spaces ◽  
List Type ◽  
Type Number ◽  
Open Problems

We generalise the construction of the formal ball model for metric spaces due to A. Edalat and R. Heckmann in order to obtain computational models for separated-categories. We fully describe-categories that are(a)Yoneda complete(b)continuous Yoneda completevia their formal ball models. Our results yield solutions to two open problems in the theory of quasi-metric spaces by showing that:(a)a quasi-metric spaceXis Yoneda complete if and only if its formal ball model is a dcpo, and(b)a quasi-metric spaceXis continuous and Yoneda complete if and only if its formal ball modelBXis a domain that admits a simple characterisation of approximation.

Computational Models in Developmental Psychology

2013 ◽  
pp. 476-504
Author(s):  
Thomas R. Shultz
Keyword(s):  
Neural Networks ◽  
Computational Modeling ◽  
Computational Models ◽  
Developmental Psychology ◽  
Probability Distributions ◽  
Network Models ◽  
Developmental Theory ◽  
Negative Evidence ◽  
Developmental Robotics ◽  
Neural Network Models

Computational modeling implements developmental theory in a precise manner, allowing generation, explanation, integration, and prediction. Several modeling techniques are applied to development: symbolic rules, neural networks, dynamic systems, Bayesian processing of probability distributions, developmental robotics, and mathematical analysis. The relative strengths and weaknesses of each approach are identified and examples of each technique are described. Ways in which computational modeling contributes to developmental issues are documented. A probabilistic model of the vocabulary spurt shows that various psychological explanations for it are unnecessary. Constructive neural networks clarify the distinction between learning and development and show how it is possible to escape Fodor’s paradox. Connectionist modeling reveals different versions of innateness and how learning and evolution might interact. Agent-based models analyze the basic principles of evolution in a testable, experimental fashion that generates complete evolutionary records. Challenges posed by stimulus poverty and lack of negative examples are explored in neural-network models that learn morphology or syntax probabilistically from indirect negative evidence.

Notes on statistical ensembles in the Cell Model

2020 ◽  
Vol 29 (08) ◽  
pp. 2050060
Author(s):  
M. Gazdzicki ◽  
M. I. Gorenstein ◽  
O. Savchuk ◽  
L. Tinti
Keyword(s):  
Statistical Physics ◽  
Cell Model ◽  
Probability Distributions ◽  
Joint Probability ◽  
Second Moments ◽  
Statistical Ensembles ◽  
Volume Limit ◽  
Joint Probability Distributions ◽  
Infinite Volume Limit ◽  
Number Of Particles

Properties of basic statistical ensembles in the Cell Model are discussed. The simplest version of the model with a fixed total number of particles is considered. The microcanonical ensembles of distinguishable and indistinguishable particles, with and without a limit on the maximum number of particles in a single cell, are discussed. The joint probability distributions of particle multiplicities in cells for different ensembles are derived, and their second moments are calculated. The results for infinite volume limit are calculated. The obtained results resemble those in the statistical physics of bosons, fermions and boltzmanions.

scholarly journals The q-exponential family in statistical physics

Open Physics ◽  
2009 ◽  
Vol 7 (3) ◽  
Author(s):  
Jan Naudts
Keyword(s):  
Statistical Physics ◽  
Exponential Family ◽  
Canonical Ensemble ◽  
Probability Distributions

AbstractThe notion of a generalized exponential family is considered in the restricted context of non-extensive statistical physics. Examples are given of models belonging to this family. In particular, the q-Gaussians are discussed and it is shown that the configurational probability distributions of the micro-canonical ensemble belong to the q-exponential family.

Some possible q-exponential type probability distribution in the non-extensive statistical physics

2016 ◽  
Vol 30 (22) ◽  
pp. 1650252 ◽  
Author(s):  
Won Sang Chung
Keyword(s):  
Probability Distribution ◽  
Exponential Type ◽  
Statistical Physics ◽  
Probability Distributions ◽  
Type I ◽  
Quantum Harmonic Oscillator ◽  
Text Type ◽  
First Order ◽  
Gibbs Entropy ◽  
Level Problem

In this paper, we present two exponential type probability distributions which are different from Tsallis’s case which we call Type I: one given by [Formula: see text] (Type IIA) and another given by [Formula: see text] (Type IIIA). Starting with the Boltzman–Gibbs entropy, we obtain the different probability distribution by using the Kolmogorov–Nagumo average for the microstate energies. We present the first-order differential equations related to Types I, II and III. For three types of probability distributions, we discuss the quantum harmonic oscillator, two-level problem and the spin-[Formula: see text] paramagnet.

scholarly journals Period Doubling Bifurcations in a Forced Cell-Free Genetic Oscillator

2021 ◽  
Author(s):  
Lukas Aufinger ◽  
Johann Brenner ◽  
Friedrich C Simmel
Keyword(s):  
Gene Networks ◽  
Computational Models ◽  
Expression System ◽  
Period Doubling ◽  
Genetic Oscillator ◽  
A Cell ◽  
Free Gene ◽  
Favorable Conditions ◽  
Period Doubling Bifurcations ◽  
Good Agreement

Complex non-linear dynamics such as period doubling and chaos have been previously found in computational models of the oscillatory gene networks of biological circadian clocks, but their experimental study is difficult. Here, we present experimental evidence of period doubling in a forced synthetic genetic oscillator operated in a cell-free gene expression system. To this end, an oscillatory negative feedback gene circuit is established in a microfluidic reactor, which allows continuous operation of the system over extended periods of time. We first thoroughly characterize the unperturbed oscillator and find good agreement with a four-species ODE model of the system. Guided by simulations, microfluidics is then used to periodically perturb the system by modulating the concentration of one of the oscillator components with a given amplitude and frequency. When the ratio of the external `zeitgeber' period and the intrinisic period is close to 1, we experimentally find period doubling and quadrupling in the oscillator dynamics, whereas for longer zeitgeber periods, we find stable entrainment. Our theoretical model suggests favorable conditions for which the oscillator can be utilized as an externally synchronized clock, but also demonstrates that related systems could, in principle, display chaotic dynamics.

scholarly journals Parallel encoding of information into visual short-term memory

2018 ◽  
Author(s):  
Edwin S. Dalmaijer ◽  
Sanjay G. Manohar ◽  
Masud Husain
Keyword(s):  
Computational Models ◽  
Short Term Memory ◽  
Feature Integration ◽  
Integration Theory ◽  
Healthy Individuals ◽  
Short Term ◽  
Term Memory ◽  
Visual Short Term Memory ◽  
Feature Integration Theory ◽  
Parallel Encoding

AbstractHumans can temporarily retain information in their highly limited short-term memory. Traditionally, objects are thought to be attentionally selected and committed to short-term memory one-by-one. However, few studies directly test this serial encoding assumption. Here, we demonstrate that information from separate objects can be encoded into short-term memory in parallel. We developed models of serial and parallel encoding that describe probabilities of items being present in short-term memory throughout the encoding process, and tested them in a whole-report design. Empirical data from four experiments in healthy individuals were fitted best by the parallel encoding model, even when items were presented unilaterally (processed within one hemisphere). Our results demonstrate that information from several items can be attentionally selected and consequently encoded into short-term memory simultaneously. This suggests the popular feature integration theory needs to be reformulated to account for parallel encoding, and provides important boundaries for computational models of short-term memory.

Fermi energy in the q-deformed quantum mechanics

2020 ◽  
Vol 35 (11) ◽  
pp. 2050074
Author(s):  
Won Sang Chung ◽  
Hassan Hassanabadi
Keyword(s):  
Quantum Mechanics ◽  
Algebraic Structure ◽  
Fermi Energy ◽  
Statistical Physics ◽  
Heisenberg Algebra ◽  
Time Dependent ◽  
Potential Well ◽  
Elementary Functions ◽  
Mathematical Background ◽  
Infinite Potential Well

In this paper, we use the q-derivative emerging in the non-extensive statistical physics to formulate the q-deformed quantum mechanics. We find the algebraic structure related to this deformed theory and investigate some properties of the q-deformed elementary functions. Using this mathematical background, we formulate the q-deformed Heisenberg algebra and q-deformed time-dependent Schrödinger equation. As an example, we deal with the infinite potential well and compute the Fermi energy in the q-deformed theory.

scholarly journals Quantum speedup of Monte Carlo methods

2015 ◽  
Vol 471 (2181) ◽  
pp. 20150301 ◽  
Author(s):  
Ashley Montanaro
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Methods ◽  
Statistical Physics ◽  
Probability Distributions ◽  
General Setting ◽  
Quantum Algorithm ◽  
Quantum Walks ◽  
Partition Functions ◽  
Performance Bounds ◽  
Quantum Speedup

Monte Carlo methods use random sampling to estimate numerical quantities which are hard to compute deterministically. One important example is the use in statistical physics of rapidly mixing Markov chains to approximately compute partition functions. In this work, we describe a quantum algorithm which can accelerate Monte Carlo methods in a very general setting. The algorithm estimates the expected output value of an arbitrary randomized or quantum subroutine with bounded variance, achieving a near-quadratic speedup over the best possible classical algorithm. Combining the algorithm with the use of quantum walks gives a quantum speedup of the fastest known classical algorithms with rigorous performance bounds for computing partition functions, which use multiple-stage Markov chain Monte Carlo techniques. The quantum algorithm can also be used to estimate the total variation distance between probability distributions efficiently.

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