Nonstandard Analysis Research Articles

A Semi-Constructive Approach to the Hyperreal Line

Using an alternative to Tarskian semantics for first-order logic known as possibility semantics, I introduce an approach to nonstandard analysis that remains within the bounds of semiconstructive mathematics, i.e., does not assume any fragment of the Axiom of Choice beyond the Axiom of Dependent Choices. I define the Fr´echet hyperreal line †R as a possibility structure and show that it shares many fundamental properties of the classical hyperreal line, such as a Transfer Principle and a Saturation Principle. I discuss the technical advantages of †R over some other alternative approaches to nonstandard analysis and argue that it is well-suited to address some of the philosophical and methodological concerns that have been raised against the application of nonstandard methods to ordinary mathematics.

Mixing times and hitting times for general Markov processes

The hitting and mixing times are two fundamental quantities associated with Markov chains. In Peres and Sousi [PS15] and Oliveira [Oli12], the authors show that the mixing times and “worst-case” hitting times of reversible Markov chains on finite state spaces are equal up to some universal multiplicative constant. We use tools from nonstandard analysis to extend this result to reversible Markov chains on general state spaces that satisfy the strong Feller property. Finally, we show that this asymptotic equivalence can be used to find bounds on the mixing times of a large class of Markov chains used in MCMC, such as typical Gibbs samplers and Metropolis–Hastings chains, even though they usually do not satisfy the strong Feller property.

Rank Two Integral Aspects of Three Dimensional Lotka–Volterra Equations with Nonstandard Analysis

Rank Two Integral Aspects of Three Dimensional Lotka–Volterra Equations with Nonstandard Analysis

Analytical approach to piezoelectric model synthesis with the use of Cauer’s method for system design

Background: Piezoceramic materials have unique property which enables direct and bilateral conversion between mechanical and electrical energy. This ability facilitates significant miniaturisation of technology and opens many opportunities in design of new actuators and energy harvesters. Mathematical modelling of piezoelectric modules is notoriously hard due to complex constitutive equations defining mechanical and electrical energy conversion. Methods: The article presents research on a new synthesis method based on the Cauer’s method for electrical and mechanical system design. Mechanical damping is introduced with the use of Rayleigh’s approximation. A discrete electromechanical model is formed based on the Mason’s piezoelectric model. As an additional element of the research, a non-standard model analysis method using edge graphs and structural numbers is employed to investigate potential reductions in computational requirements for model analysis. Results: The electromechanical model analysis has shown an error of 5% in relation to the resonance target. The calculated peak in magnitude of displacement at the resonant frequency was well above the capability of a 3.5mm thick piezoelectric plate. The proposed non-standard method of analysis gave identical results in terms of obtained resonance frequencies. The calculated magnitude of vibrations was varying by 50 – 58%. Conclusions: The synthesis method presented in this paper allows an approximation of piezoelectric parameters of a real system based on the created mathematical model. Currently this method is subject to a considerable error in the determined vibration amplitudes of the system, but it allows a coarse approximation of the parameters while having a very limited number of input data. The additional method of analysis based on structural numbers offers a promising alternative to matrix calculations but requires a more thorough investigation of the computational power required to determine whether it can compete with existing algorithms.

Stateful Realizers for Nonstandard Analysis

In this paper we propose a new approach to realizability interpretations for nonstandard arithmetic. We deal with nonstandard analysis in the context of (semi)intuitionistic realizability, focusing on the Lightstone-Robinson construction of a model for nonstandard analysis through an ultrapower. In particular, we consider an extension of the $\lambda$-calculus with a memory cell, that contains an integer (the state), in order to indicate in which slice of the ultrapower $\cal{M}^{\mathbb{N}}$ the computation is being done. We pay attention to the nonstandard principles (and their computational content) obtainable in this setting. In particular, we give non-trivial realizers to Idealization and a non-standard version of the LLPO principle. We then discuss how to quotient this product to mimic the Lightstone-Robinson construction.

Modelling Heavy Tailed Phenomena Using a LogNormal Distribution Having a Numerically Verifiable Infinite Variance

One-sided heavy tailed distributions have been used in many engineering applications, ranging from teletraffic modelling to financial engineering. In practice, the most interesting heavy tailed distributions are those having a finite mean and a diverging variance. The LogNormal distribution is sometimes discarded from modelling heavy tailed phenomena because it has a finite variance, even when it seems the most appropriate one to fit the data. In this work we provide for the first time a LogNormal distribution having a finite mean and a variance which converges to a well-defined infinite value. This is possible thanks to the use of Non-Standard Analysis. In particular, we have been able to obtain a Non-Standard LogNormal distribution, for which it is possible to numerically and experimentally verify whether the expected mean and variance of a set of generated pseudo-random numbers agree with the theoretical ones. Moreover, such a check would be much more cumbersome (and sometimes even impossible) when considering heavy tailed distributions in the traditional framework of standard analysis.

Nonstandard analysis in electrical engineering. The analysis of the direct current circles with ideal reactive elements.

The article proposes the use of ideas and methods of non-standard analysis in the field of theoretical electronics. The article shows that the analysis of DC circuits, including ideal inductances and capacitances, by standard methods of theoretical electrical engineering is too complicated or almost impossible. To solve this problem, it is proposed to extend the methods of non-standard analysis by the tasks of analyzing electrical circuits with ideal reactive elements. The authors have defined a class of non-standard electrotechnical problems aimed at the analysis of DC electrical circuits, including ideal reactive elements — ideal inductances and capacitances. It is shown that the solution of the selected class of problems by standard methods of theoretical electrical engineering is too difficult or almost impossible. It is proposed to extend the methods of non-standard analysis by the tasks of analyzing electrical circuits with ideal reactive elements. The obtained advantages of this approach are confirmed by examples of calculations of electrical circuits with inductances and capacitances, as well as magnetic circuits.

Limit analysis and design-oriented approach for out-of-plane loaded masonry walls strengthened by grouted anchors

Masonry structures without box-like behaviour are often prone to the complete or partial out-of-plane failure of their walls under horizontal actions, i.e. mainly the seismic ones. Steel grouted anchors or tie-rods have been widely used in the past to improve connections between orthogonal walls and avoid out-of-plane mechanisms. However, reliable analytical approaches to design such strengthening systems are lacking, whereas the evaluation of their effectiveness is still a matter of ongoing research. The paper presents a force-based design approach to assess the performance of three-wall masonry systems strengthened by grouted anchors, using a frictional macro-block model and a non-standard limit analysis, previously developed for unreinforced masonry structures. The extension of this model to strengthened masonry systems, with specific reference to the use of grouted anchors, allows assessing their horizontal capacity in terms of minimum load multiplier and expected failure. The novel aspects introduced in the updated model are: (1) a simplified closed-form formulation for the calculation of the stabilising moments provided by the grouted anchors at the onset of the failure mechanism, (2) the possibility of placing the grouted anchors with variable inclinations, which may improve their stabilising contribution and optimise the intervention. The effectiveness of the simplified formulation is firstly proved by the comparison with literature results; then, several parametric analyses are carried out to highlight the influence of some variables on the capacity of both the unreinforced and the strengthened three-wall system. Finally, a literature example related to a traditional intervention with steel tie-rods is used to apply the proposed procedure and investigate the performance of different design solutions with innovative grouted anchors.

The Existence Theorem for Optimal Control with Non-Standard Cost Functional and Analysis

The Existence Theorem for Optimal Control with Non-Standard Cost Functional and Analysis

Non-standard analysis for fractal calculus

Non-standard analysis for fractal calculus

Asymptotic analysis of stretching modes for a folded plate

<abstract><p>In this paper, we show that the spectral problem associated to stretching modes in a thin folded plate can be derived from the three-dimensional eigenvalue problem of linear elasticity through a rigourous convergence analysis as the thickness of the plate goes to zero. We show, using a nonstandard asymptotic analysis technique, that each stretching frequency of an elastic thin folded plate is the limit of a family of high frequencies of the three-dimensional linearized elasticity system in the folded plate, as the thickness approaches zero.</p></abstract>

Non-standard analysis in electrical engineering. Transient analysis in second-order electrical circuits with violation of switching laws

For the first time, the authors proposed the use of the mathematical apparatus of non-standard analysis to solve certain non-trivial problems of theoretical electrical engineering. It was established that the axiomatics of non-standard analysis allows to simplify the analysis of transient processes in high-order electric circuits when the switching laws are violated. It is shown that the application of non-standard analysis methods in theoretical electrical engineering provides an opportunity to use the traditional classical method of transient processes analysis of circuits with violation of switching laws. Only by using methods of non-standard analysis, it is possible to strictly prove the fulfillment of the law of energy conservation in such schemes. Also, examples of solving such tasks are given. It is recommended to expand the scope of application of non-standard methods of analysis in problems of various branches of science and technology, which use differential calculus and boundary transitions, and the solution of which is limited or impossible by standard approaches.

A nonstandard approach to asymptotic fixed point theorems

We use methods from nonstandard analysis to unify and extend results from the theory of asymptotic fixed points on metric spaces. Among the topics we consider are Kirk’s asymptotic fixed point theorem with extensions, Boyd and Wong’s fixed point theorem, and Suzuki’s fixed point theorem for asymptotic contractions of the Meier–Keeler type. We also show how Suzuki’s asymptotic contractions of the final type can be modified to yield uniform convergence on compact and bounded sets.

“NON-STANDARD” FORMALISM OF QUANTUM THEORY I: MASS SPECTRUM

In quantum theory, there are both point and integral objects. In this case, the computational technique is based exclusively on set-theoretic structures associated with the point model of the continuum. The transition to integral dynamic structures makes it possible to develop new methods that allow us to obtain useful results, in particular, to derive the formula for the spectrum of a certain class of particles: m = me ( l + k /2 m )( ĺ + s /2 t ). This formula is a generalization of the formula obtained by V.V. Varlamov in 2017. In general, such an approach is consonant with the concept of the so-called. “non-standard analysis”, in which “number-monads” play a key role.

CONSTRUCTING NONSTANDARD HULLS AND LOEB MEASURES IN INTERNAL SET THEORIES

AbstractCurrently the two popular ways to practice Robinson’s nonstandard analysis are the model-theoretic approach and the axiomatic/syntactic approach. It is sometimes claimed that the internal axiomatic approach is unable to handle constructions relying on external sets. We show that internal frameworks provide successful accounts of nonstandard hulls and Loeb measures. The basic fact this work relies on is that the ultrapower of the standard universe by a standard ultrafilter is naturally isomorphic to a subuniverse of the internal universe.

Likelihood estimation of sparse topic distributions in topic models and its applications to Wasserstein document distance calculations

This paper studies the estimation of high-dimensional, discrete, possibly sparse, mixture models in the context of topic models. The data consists of observed multinomial counts of p words across n independent documents. In topic models, the p×n expected word frequency matrix is assumed to be factorized as a p×K word-topic matrix A and a K×n topic-document matrix T. Since columns of both matrices represent conditional probabilities belonging to probability simplices, columns of A are viewed as p-dimensional mixture components that are common to all documents while columns of T are viewed as the K-dimensional mixture weights that are document specific and are allowed to be sparse. The main interest is to provide sharp, finite sample, ℓ1-norm convergence rates for estimators of the mixture weights T when A is either known or unknown. For known A, we suggest MLE estimation of T. Our nonstandard analysis of the MLE not only establishes its ℓ1 convergence rate, but also reveals a remarkable property: the MLE, with no extra regularization, can be exactly sparse and contain the true zero pattern of T. We further show that the MLE is both minimax optimal and adaptive to the unknown sparsity in a large class of sparse topic distributions. When A is unknown, we estimate T by optimizing the likelihood function corresponding to a plug in, generic, estimator Aˆ of A. For any estimator Aˆ that satisfies carefully detailed conditions for proximity to A, we show that the resulting estimator of T retains the properties established for the MLE. Our theoretical results allow the ambient dimensions K and p to grow with the sample sizes. Our main application is to the estimation of 1-Wasserstein distances between document generating distributions. We propose, estimate and analyze new 1-Wasserstein distances between alternative probabilistic document representations, at the word and topic level, respectively. We derive finite sample bounds on the estimated proposed 1-Wasserstein distances. For word level document-distances, we provide contrast with existing rates on the 1-Wasserstein distance between standard empirical frequency estimates. The effectiveness of the proposed 1-Wasserstein distances is illustrated by an analysis of an IMDB movie reviews data set. Finally, our theoretical results are supported by extensive simulation studies.

Resolving the Singularity by Looking at the Dot and Demonstrating the Undecidability of the Continuum Hypothesis

Einsteinian gravity, of which Newtonian gravity is a part, is fraught with the problem of singularity that has been established as a theorem by Hawking and Penrose. The hypothesis that founds the basis of both Einsteinian and Newtonian theories of gravity is that bodies with unequal magnitudes of masses fall with the same acceleration under the gravity of a source object. Since, the Einstein’s equations is one of the assumptions that underlies the proof of the singularity theorem, therefore, the above hypothesis is implicitly one of the founding pillars of the same. In this work, I demonstrate how one can possibly write a non-singular theory of gravity which manifests that the above mentioned hypothesis is only valid in an approximate sense in the “large distance” scenario. To mention a specific instance, under the gravity of the earth, a 5 kg and a 500 kg fall with accelerations which differ by approximately \(113.148\times 10^{-32}\) meter/sec\(^2\) and the more massive object falls with less acceleration. Further, I demonstrate why the concept of gravitational field is not definable in the “small distance” regime which automatically justifies why the Einstein’s and Newton’s theories fail to provide any “small distance” analysis. In course of writing down this theory, I demonstrate why the continuum hypothesis as spelled out by Goedel, is undecidable. The theory has several aspects which provide the following realizations: (i) Descartes’ self-skepticism concerning exact representation of numbers by drawing lines (ii) Born’s wish of taking into account “natural uncertainty in all observations” while describing “a physical situation” by means of “real numbers” (iii) Klein’s vision of having “a fusion of arithmetic and geometry” where “a point is replaced by a small spot” (iv) Goedel’s assertion about “non-standard analysis, in some version” being “the analysis of the future”.

Existence of matching priors on compact spaces

Summary A matching prior at level $1-\alpha$ is a prior such that an associated $1-\alpha$ credible region is also a $1-\alpha$ confidence set. We study the existence of matching priors for general families of credible regions. Our main result gives topological conditions under which matching priors for specific families of credible regions exist. Informally, we prove that, on compact parameter spaces, a matching prior exists if the so-called rejection-probability function is jointly continuous when we adopt the Wasserstein metric on priors. In light of this general result, we observe that typical families of credible regions, such as credible balls, highest-posterior density regions, quantiles, etc., fail to meet this topological condition. We show how to design approximate posterior credible balls and highest-posterior density regions that meet these topological conditions, yielding matching priors. Finally, we evaluate a numerical scheme for computing approximately matching priors based on discretization and iteration. The proof of our main theorem uses tools from nonstandard analysis and establishes new results about the nonstandard extension of the Wasserstein metric that may be of independent interest.

A Roadmap for Edge Computing Enabled Automated Multidimensional Transmission Electron Microscopy

Abstract: The advent of modern, high-speed electron detectors has made the collection of multidimensional hyperspectral transmission electron microscopy datasets, such as 4D-STEM, a routine. However, many microscopists find such experiments daunting since analysis, collection, long-term storage, and networking of such datasets remain challenging. Some common issues are their large and unwieldy size that often are several gigabytes, non-standardized data analysis routines, and a lack of clarity about the computing and network resources needed to utilize the electron microscope. The existing computing and networking bottlenecks introduce significant penalties in each step of these experiments, and thus, real-time analysis-driven automated experimentation for multidimensional TEM is challenging. One solution is to integrate microscopy with edge computing, where moderately powerful computational hardware performs the preliminary analysis before handing off the heavier computation to high-performance computing (HPC) systems. Here we trace the roots of computation in modern electron microscopy, demonstrate deep learning experiments running on an edge system, and discuss the networking requirements for tying together microscopes, edge computers, and HPC systems.

Non-smooth variational problems and applications.

Mathematical methods based on the variational approach are successfully used in a broad range of applications, especially those fields that are oriented on partial differential equations. Our problem area addresses a wide class of nonlinear variational problems described by all kinds of static and evolution equations, inverse and ill-posed problems, non-smooth and non-convex optimization, and optimal control including shape and topology optimization. Within these directions, we focus but are not limited to singular and unilaterally constrained problems arising in mechanics and physics, which are governed by complex systems of generalized variational equations and inequalities. Whereas classical mathematical tools are not applicable here, we aim at a non-standard well-posedness analysis, numerical methods, asymptotic and approximation techniques including homogenization, which are successful within the primal as well as the dual variational formalism. In a broad scope, the theme issue objectives are directed toward advances that are attained in the mathematical theory of non-smooth variational problems, its physical consistency, numerical simulation and application to engineering sciences.This article is part of the theme issue ‘Non-smooth variational problems and applications’.