Nonstandard Analysis Research Articles

Ergodicity of Markov Processes via Nonstandard Analysis

The Markov chain ergodic theorem is well-understood if either the time-line or the state space is discrete. However, there does not exist a very clear result for general state space continuous-time Markov processes. Using methods from mathematical logic and nonstandard analysis, we introduce a class of hyperfinite Markov processes-namely, general Markov processes which behave like finite state space discrete-time Markov processes. We show that, under moderate conditions, the transition probability of hyperfinite Markov processes align with the transition probability of standard Markov processes. The Markov chain ergodic theorem for hyperfinite Markov processes will then imply the Markov chain ergodic theorem for general state space continuous-time Markov processes.

Mixing and hitting times for Gibbs samplers and other non-Feller processes

The hitting and mixing times are two often-studied quantities associated with Markov chains. Yuval Peres, Perla Sousi and Roberto Oliveira showed that the mixing times and “worst-case” hitting times of reversible Markov chains on finite state spaces are “equivalent”—that is, equal up to some universal multiplicative constant. We have extended this strong connection between mixing and hitting times to Markov chains satisfying the strong Feller property in an earlier work. In the present paper, we further extend the results to include Metropolis–Hastings chains, the popular Gibbs sampler (from statistics), and Glauber dynamics (from statistical physics), which make “one-dimensional” updates and thus do not satisfy the strong Feller property. We also apply this result to obtain decomposition bounds for such Markov chains. Our main tools come from nonstandard analysis.

Categorical Nonstandard Analysis

In the present paper, we propose a new axiomatic approach to nonstandard analysis and its application to the general theory of spatial structures in terms of category theory. Our framework is based on the idea of internal set theory, while we make use of an endofunctor U on a topos of sets S together with a natural transformation υ, instead of the terms as “standard”, “internal”, or “external”. Moreover, we propose a general notion of a space called U-space, and the category USpace whose objects are U-spaces and morphisms are functions called U-spatial morphisms. The category USpace, which is shown to be Cartesian closed, gives a unified viewpoint toward topological and coarse geometric structure. It will also be useful to further study symmetries/asymmetries of the systems with infinite degrees of freedom, such as quantum fields.

Applications of Non-Standard analysis in Topoi to Mathematical Neurosciences and Artificial Intelligence: Infons, Energons, Receptons (I)

The purpose of this paper is to promote new methods in mathematical modeling inspired by neuroscience—that is consciousness and subconsciousness—with an eye toward artificial intelligence as parts of the global brain. As a mathematical model, we propose topoi and their non-standard enlargements as models, due to the fact that their logic corresponds well to human thinking. For this reason, we built non-standard analysis in a special class of topoi; before now, this existed only in the topos of sets (A. Robinson). Then, we arrive at the pseudo-particles from the title and to a new axiomatics denoted by Intuitionistic Internal Set Theory (IIST); a class of models for it is provided, namely, non-standard enlargements of the previous topoi. We also consider the genetic–epigenetic interplay with a mathematical introduction consisting of a study of the Yang–Baxter equations with new mathematical results.

Some aspects of non-standard multivariate analysis

A broad review is given of some areas of multivariate analysis that are not frequently emphasized. We start with situations in which underlying distributions are far from multivariate normal form, so that standard methods of multivariate analysis based on covariances are likely to be unsatisfactory. We emphasize the important distinction between internal and external analyses associated with multiple outcomes. A second broad theme relates to multiple outcomes generated by time or spatial series in which long-range dependence operates. Some implications are summarized.

NONSTANDARD COMPLETION OF NON-COMPLETE METRIC SPACE

Our aim in this study is to establishing nonstandard foundations, definitions and theorems for completion a noncomplete metric spaces. We have a lot of space or sets X which agree with all usual properties of complete, except at a small size subset of it. In this paper, by using nonstandard analysis tools founded by A. Robinson and axiomatized by E. Nelson, we try to introduce a new notion of completion corresponding to nonstandard modified metric , and to use it to reprove the classical (standard) completion theorem and to overcome the incompetence of those spaces which has deprivation at a small size subset.

Solution of Inhomogeneous Fractional Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard Analysis

Discussions are presented by Morita and Sato in Mathematics 2017; 5, 62: 1–24, on the problem of obtaining the particular solution of an inhomogeneous ordinary differential equation with polynomial coefficients in terms of the Green’s function, in the framework of distribution theory. In the present paper, a compact recipe in nonstandard analysis is presented, which is applicable to an inhomogeneous ordinary and also fractional differential equation with polynomial coefficients. The recipe consists of three theorems, each of which provides the particular solution of a differential equation for an inhomogeneous term, satisfying one of three conditions. The detailed derivation of the applications of these theorems is given for a simple fractional differential equation and an ordinary differential equation.

On extended admissible procedures and their nonstandard Bayes risk

For finite parameter spaces, among decision procedures with finite risk functions, a decision procedure is extended admissible if and only if it is Bayes. Various relaxations of this classical equivalence have been established for infinite parameter spaces, but these extensions are each subject to technical conditions that limit their applicability, especially to modern (semi and nonparametric) statistical problems. Using results in mathematical logic and nonstandard analysis, we extend this equivalence to arbitrary statistical decision problems: informally, we show that, among decision procedures with finite risk functions, a decision procedure is extended admissible if and only if it has infinitesimal excess Bayes risk. In contrast to existing results, our equivalence holds in complete generality, that is, without regularity conditions or restrictions on the model or loss function. We also derive a nonstandard analogue of Blyth’s method that yields sufficient conditions for admissibility, and apply the nonstandard theory to derive a purely standard theorem: when risk functions are continuous on a compact Hausdorff parameter space, a procedure is extended admissible if and only if it is Bayes.

Describing limits of integrable functions as grid functions of nonstandard analysis

In functional analysis, there are different notions of limit for a bounded sequence of L^1 functions. Besides the pointwise limit, that does not always exist, the behaviour of a bounded sequence of L^1 functions can be described in terms of its weak-star limit or by introducing a measure-valued notion of limit in the sense of Young measures. Working in Robinson’s nonstandard analysis, we show that for every bounded sequence {z_n}_{n in mathbb {N}} of L^1 functions there exists a function of a hyperfinite domain (i.e. a grid function) that represents both the weak-star and the Young measure limits of the sequence. This result has relevant applications to the study of nonlinear PDEs. We discuss the example of an ill-posed forward–backward parabolic equation.

Non-standard computational analysis of the stochastic COVID-19 pandemic model: An application of computational biology

The present study is conducted to analyse the computational dynamical analysis of the stochastic susceptible-infected-recovered pandemic model of the novel coronavirus. We adopted two ways for stochastic modelling like as transition probabilities and parametric perturbation techniques. We applied different and well-known computational methods like Euler Maruyama, stochastic Euler, and stochastic Runge Kutta to study the dynamics of the model mentioned above. Unfortunately, these computational methods do not restore the dynamical properties of the model like positivity, boundedness, consistency, and stability in the sense of biological reasoning, as desired. Then, for the given stochastic model, we developed a stochastic non-standard finite difference method. Following that, several theorems are presented to support the proposed method, which is shown to satisfy all of the model's dynamical properties. To that end, several simulations are presented to compare the proposed method's efficiency to that of existing stochastic methods.

Qualitative versus quantitative representation: a non-standard analysis of the sorites paradox

Qualitative versus quantitative representation: a non-standard analysis of the sorites paradox

Gordon's conjectures 1 and 2: Pontryagin-van Kampen duality in the hyperfinite setting

Using the ideas of E. I. Gordon we present and farther advancean approach, based on nonstandard analysis, to simultaneousapproximations of locally compact abelian groups and their dualsby (hyper)finite abelian groups, as well as to approximations ofvarious types of Fourier transforms on them by the discrete Fouriertransform. Combining some methods of nonstandard analysis andadditive combinatorics we prove the three Gordon's Conjectureswhich were open since 1991 and are crucial both in the formulationsand proofs of the LCA groups and Fourier transform approximationtheorems

Hausdorff compactifications

Previously, the authors used the insights of Robinson's nonstandard analysis as a powerful tool to extend and simplify the construction of some compactifications of regular spaces. They now show that any Hausdorff compactification is obtainable with their method.

Mathematical modeling and nonstandard finite difference scheme analysis for the environmental and spillover transmissions of Avian Influenza A model

ABSTRACT This work models, analyzes and assesses the impacts of environmental and spillover transmissions on Avian Influenza Virus (AIV) type A infection formulated in terms of nonlinear ordinary differential system that takes into account five spreading pathways: poultry-to-poultry; environment-to-poultry; poultry-to-human (spillover event); environment-to-human and poultry-to-environment. An in-depth theoretical and numerical analysis of the model is performed as follows. The basic reproduction number is computed and shown to be a sharp threshold for the global asymptotic dynamics of the submodel without recruitment of infected poultry. These results are obtained through the construction of suitable Lyapunov functions and the application of Poincaré-Bendixson combined with Lyapunov-LaSalle techniques. When the infected poultry is brought into the population, the model exhibits only a unique endemic equilibrium whose global asymptotic stability is established using the same techniques mentioned earlier. Further, the model is shown to exhibit a transcritical bifurcation with the value one of the basic reproduction number being the bifurcation parameter threshold. We further prove that during avian influenza outbreaks, the recruitment of infected poultry increases the disease endemic level. We show that the classical Runge-Kutta numerical method fails to preserve the positivity of solutions and alternatively design a nonstandard finite difference scheme (NSFD), which preserves the essential properties of the continuous system. Numerical simulations are implemented to illustrate the theoretical results and assess the role of the environmental and spillover transmissions on the disease.

Influence of Ni / NiO Ratio on the Performance of Thermoelectrochemical Waste Heat Harvester Based on Hollow Microspheres

Technologies harvesting and converting the heat of low-temperature sources into electricity are one of the current trends in modern renewable energy. Thermoelectrochemical cells based on a nickel oxide electrode are gaining increasing popularity among researchers due to the high values of the hypothetical Seebeck coefficient. This work is devoted to the study of the reduction modes effect of hollow nickel oxide microspheres on the ratio of oxide and metal nickel forms in their structure. The paper presents the synthesis of microspheres by the pyrolysis of ultrasonic aerosols of an aqueous solution of nickel nitrate. Composition and morphology are studied by scanning electron microscopy and X-ray non-standard quantitative analysis. It was shown that the proportion of metallic nickel increases with an increase in the reduction temperature to about 375°C and then changes slightly. For microspheres, reduced at 350 °C, the main parameters of the thermoelectrochemical cell were determined. It was shown that the hypothetical Seebeck coefficient in the assembled cell system is 3.1 mV/K, and the resistance to charge transfer across the electrode-electrolyte interface was less than 1 Ohm. It confirms the applicability of these materials as electrodes for electrochemical systems and the possibility of obtaining high values of output power.

Non-standard finite difference scheme and analysis of smoking model with reversion class

Smokers are at more risk to COVID-19 as the entertainment of smoking because their fingers are in touch with lips regularly during smoking that increases the probability of transmission of virus from hand to mouth. On other hand the smokers may have lung disease (or reduced lung capacity) which would greatly increase risk of serious illness especially COVID-19. For this esteem, in this research work, we first formulate a mathematical model contains the reversion class. Then, using different techniques for finding the local and global stability of the presented model related to equilibrium points that are free smoking and positive smoking equilibrium points. As the model consisting on the nonlinear equations, so we use the non-standard finite difference (NSFD) scheme, ODE45 and RK4 methods to find the numerical results. Finally, we show the graphs numerically through MATLAB.

Continuous Operations and Series of Indeterminate Terms

It is known that functions involving natural numbers are generalized to the real ones, for instance the gamma function can be viewed as a generalization of the factorial operator. In this paper, we propose to generalize the repetition of an operation over a function (composition, derivatives and integrals) toward the field of reals. It means repeating q times an operation over a function, where q is a real number. As a result, it is explained what functional and analytical dimensional extensions are and it is given a proof to theorems related to the indeterminate terms. The main finding is that every real number is expressible as a bijection of an infinite sum of elements whose coefficients are real numbers and their main values are either an indeterminate value or an infinite value. The concept of series of indeterminate values becomes relevant, as a novelty to operate with infinite, zero and indeterminate terms, which cannot be deductible from the non-standard analysis.

Association between water for human consumption and health of the population of the state of Rio de Janeiro: the case of hepatitis A between 2007 and 2018

This article analyzes the relationship between incidences of hepatitis A — a water-borne disease that can also be transmitted by lack of hygiene — and quality indicators in the provision of water supply services, through secondary data extracted from the National Information Systems for Notifiable Diseases and on Sanitation, in the period between 2007 and 2018, for the municipalities of the State of Rio de Janeiro, Brazil. The indicators were initially submitted to Principal Component Analysis (PCA) to reduce numerous variables that showed autocorrelation with each other. Subsequently, the principal components were submitted to Pearson’s correlation analysis with the incidence of hepatitis A. A correlation coefficient of -0.32 was observed, at the level of significance (p < 0.05) between the cumulative incidences of hepatitis A and the principal component (PC3) formed by a set of quality indicators for the provision of water supply services. The indicators that best described PC3, with factor loadings ranging from -0.88 to 0.70, were those related to: hydrometer, water and revenue losses, water revenue, the participation of residential water savings, micro-measurement related to water consumption, water consumption and average water consumption per economy, average duration of outages, and the incidence of non-standard total coliform analyses. Thus, the observed results point to an association between the quality of water supply services and the incidence of hepatitis A, which may indicate both failures in treatment efficiency and lack of hygiene.

Nonstandard Analysis for Some Convergence Sequences Theories Via a Transfer Principle

The purpose of this article to introduce nonstandard models for a familiar types of convergent sequences theories through a transfer principle. The nonstandard analysis principle discuss, that any statements on areal system can be extended to a similar structure over larger hyperreal system therefore the results that hold true on the original system, remains true in a hyperreal system if and only if its ∗-transform is true. We apply such technique to transfer a classical proof for real sequences theorems, therefore we obtain an equivalents nonstandard proof on hyperreal system which is often clear, shorter and uncoblicated.

Nonstandard analysis of uniform spaces

There are nonstandard characterizations of classical topological concepts like total boundedness, completeness, comapactness etc., in metric spaces. In this article we present nonstandard extensions of these characterizations to uniform spaces. We also furnish completion of uniform spaces. We accomplish this by extending the methods in metric spaces by way of passing from sequences to filters and nets.