Existence of solutions to time-dependent PDEs in an algebra of generalized functions, in preparation.
Abstract:
We study a nonstandard formulation for a class of time-dependent partial differential equations with distributional or measure-valued initial data. Equations of this form include linear and nonlinear diffusion and systems of conservation laws. Some of these problems, such as certain forward-backward parabolic equations, present many similarities, but are studied with different techniques and have different notions of solutions. Working in the setting of Robinson's analysis with infinitesimals, we discretize the differential operators in space by means of finite differences of an infinitesimal step. The resulting hyperfinite system of ODEs is formally equivalent to the original problem. Under the hypothesis that f is Lipschitz continuous, this system has a unique solution that can be used to define a real solution to the original problem. For a class of forward-backward heat equations, this real solution coincides with the one obtained with a vanishing viscosity approach; moreover, the non-Archimedean model allows to characterize its asymptotic behaviour. We suggest that this nonstandard formulation could be successfully used to highlight the similarities between other classes of problems and to obtain novel qualitative information about their solutions.
Two-track approaches to uniform processes, with Mikhail G. Katz, submitted.
Abstract:
We apply Cartwright’s analysis of scientific modeling to evaluate rival approaches to modeling a notion of chance for an ideal uniform physical process that can be described as a fair spinner. According to Cartwright, it is possible to distinguish (a) pre-mathematical concepts best described as physical and/or naive-probabilistic phenomena; (b) a basic mathematical model thereof; (c) advanced mathematical tools brought to bear on the analysis of the basic mathematical model. A pre-mathematical phenomenon typically admits multiple basic mathematical models, that in turn can be analyzed by means of distinct advanced tools. In the case of the fair spinner, we argue that some authors conflate the pre-mathematical concepts with the basic mathematical model and the basic mathematical model with some advanced mathematical tools. As a consequence, they claim that the fair spinner should satisfy hypotheses such as invariance with respect to rotations by an arbitrary real angle, uniformity and assigning a chance to every point. Moreover, they argue that the only optimal mathematical tool in this context is the Lebesgue measure. Other authors argue that invariance with respect to every real rotation is not an essential feature of the underlying physical process, so it could be relaxed in favour of regularity. We show that, working in ZFC, no subset of these hypotheses determines a unique model. Thus we argue that physically based intuitions do not enable one to pin down a unique mathematical model.
Integration with filters, with Monroe Eskew, Journal of Logic and Analysis.
Abstract:
We introduce a notion of integration defined from filters over families of finite sets. This procedure corresponds to determining the average value of functions whose range lies in any algebraic structure in which finite averages make sense. The average values so determined lie in a proper extension of the range of the original functions. The most relevant scenario involves algebraic structures that extend the field of rational numbers; hence, it is possible to associate to the filter integral an upper and lower standard part. These numbers can be interpreted as upper and lower bounds on the average value of the function that one expects to observe empirically. We discuss the main properties of the filter integral and we show that it is expressive enough to represent every real integral. As an application, we define a geometric measure on an infinite-dimensional vector space that overcomes some of the known limitations valid for real-valued measures. We also discuss how the filter integral can be applied to the problem of non-Archimedean integration, and we develop the iteration theory for these integrals.
An incompatibility result on non-Archimedean integration, p-Adic Numbers, Ultrametric Analysis, and Applications.
Abstract:
We prove that a Riemann-like integral on non-Archimedean extensions of R cannot assign an integral to every function whose standard part is measurable and simultaneously satisfy the fundamental theorem of calculus. We also discuss how existing theories of non-Archimedean integration deal with the incompatibility of these conditions.
Examples and counterexamples regarding hyperfinite representations of real distributions, Examples and Counterexamples.
Abstract:
We provide some examples and counterexamples regarding hyperfinite representations of real distributions. We examine some hyperfinite representatives of the null distribution and of the Dirac distribution, we discuss hypernfiite representations of an infinite Lp norm, and we study the failure of an energy inequality for the discrete Laplacian in dimension 1. These examples are relevant for the study of partial differential equations with hyperfinite techniques.
Describing limits of integrable functions as grid functions of nonstandard analysis, SN Partial Differential Equations and Applications 2:51 (2021).
Abstract:
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 framework of analysis with infinitesimals, we show that for every bounded sequence 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.
A real-valued measure on non-Archimedean field extensions of the field of real numbers, p-Adic Numbers, Ultrametric Analysis, and Applications.
Abstract:
We introduce a real-valued measure m_L on non-Archimedean ordered fields (F;<) that extend the field of real numbers (R;<). The definition of m_L is inspired by the Loeb measures of hyperreal fields in the context of Robinson’s analysis with infinitesimals. The real-valued measure m_L turns out to be general enough to obtain a canonical measurable representative in F for every Lebesgue measurable subset of R, moreover the measure of the two sets is equal. We focus on the properties of the real-valued measure in the case where F is the Levi-Civita field. In particular, we compare m_L with the uniform non-Archimedean measure over R developed by Shamseddine and Berz, and we prove that the first is infinitesimally close to the second, whenever the latter is defined. We also define a real-valued integral for functions on the Levi-Civita field, and we prove that every real continuous function has an integrable representative in the Levi-Civita field. Recall that this result is false for the non-Archimedean integration of Shamseddine and Berz. The paper concludes with some applications to the representation of distributions and parametrized measures by pointwise functions on non-Archimedean domains.
A grid function formulation of a class of ill-posed parabolic equations, Journal of Differential Equations.
Abstract:
We study a nonstandard formulation of the Neumann initial value problem
u_t(x,t)=Δϕ(u(x,t)), x∈Ω⊆R^k, t∈R
u(x,0)=u_0(x), x∈Ω.
with Neumann boundary conditions. The function ϕ∈C^1(R) is assumed to be decreasing either in a bounded interval (u−,u+), or in an unbounded interval (u−,+∞): under this hypothesis, the aforementioned problem is ill-posed and only allows for measure-valued solutions. Moreover, such solutions are in general not unique. By using nonstandard analysis, we derive from very simple physical principles a continuous-in-time and discrete-in-space model for the ill-posed pde, and we prove that this model is well-posed. We will also prove that the solution of the nonstandard formulation is coherent with the measure-valued solutions and still retains relevant physical properties, chiefly among them an entropy condition that characterizes physically admissible solutions to the original problem. We then study the asymptotic behaviour of the nonstandard solutions. In doing so, we will give a positive answer to a conjecture by Smarrazzo on the coarsening of the solutions to the ill-posed problem under the hypothesis that ϕ is decreasing in the interval (u−,+∞).
Internality, transfer and infinitesimal modeling (with Mikhail G. Katz), Philosophia Mathematica.
Abstract:
A probability model is underdetermined when there is no rational reason to assign a particular infinitesimal value as the probability of single events. Pruss claims that hyperreal probabilities are underdetermined, however this claim is based upon external hyperreal-valued measures. We show that internal hyperfinite measures are not underdetermined. The importance of internality stems from the fact that Robinson’s transfer principle only applies to internal entities. We also evaluate the claim that transferless ordered fields (such as the surreals, Levi-Civita field, Laurent series) may have advantages over hyperreals in probabilistic modeling. We show that probabilities developed over such fields are less expressive than hyperreal probabilities.
Infinite lotteries, spinners, and the applicability of hyperreals (with Mikhail G. Katz), Philosophia Mathematica.
Abstract:
We analyze recent criticisms of the use of hyperreal probabilities as expressed by Pruss, Easwaran, Parker, and Williamson. We show that the alleged arbitrariness of hyperreal fields can be avoided by working in the Kanovei–Shelahmodel or in saturated models. We argue that some of the objections to hyperreal probabilities arise from hidden biases that favor Archimedean models. We discuss the advantage of the hyperreals over transferless fields with infinitesimals. In the companion paper Internality, transfer and infinitesimal modeling we analyze two underdetermination theorems by Pruss and show that they hinge upon parasitic external hyperreal-valued measures, whereas internal hyperfinite measures are not underdetermined.
Spaces of measurable functions on the Levi-Civita field, Indagationes Matematicae.
From the introduction:
In the last twenty years, Shamseddine et al. developed an uniform measure theory on the Levi-Civita field. We aim at establishing the basis of a theory of generalized functions on the Levi-Civita field, in analogy to what has been done in other non-Archimedean extensions of the real numbers, as shown by Colombeau, Todorov et al., Benci et al. and Bottazzi. Crucial to this goal is the extension of measurable real functions to measurable functions in the Levi-Civita field. At first, we introduce the L^p spaces in the Levi-Civita field, that however are not complete with respect to strong convergence. In order to address this issue, we introduce the completions of the L^p spaces with respect to strong convergence. It turns out that these are Banach spaces and that the completion of L^2 can be equipped with an inner product that turns it into a Hilbert space. Despite these positive results, these spaces are still not rich enough to represent every real continuous function. For this reason, we settle upon the representation of measurable real functions as sequences of measurable functions in the Levi-Civita field that weakly converge in the L^p norm. We have adopted this route in order to establish a continuity with the measure theory developed so far; however, we suggest that a radically different notion of measurable sets and functions similar to the Loeb measure of Robinson's framework for mathematics with infinitesimals might allow for a simpler representation of real continuous functions with measurable functions on the Levi-Civita field.
Homomorphisms Between Rings with Infinitesimals and Infinitesimal Comparisons, Mat. Stud, issue 1, vol 52 (2019).
Abstract:
We examine an argument of Reeder suggesting that the nilpotent infinitesimals in Paolo Giordano’s ring extension of the real numbers •R are smaller than any infinitesimal hyperreal number from Abraham Robinson’s nonstandard analysis *R. Our approach consists in the study of two canonical order-preserving homomorphisms taking values in •R and *R , respectively, and whose domain is Henle’s extension of the real numbers in the framework of “non-nonstandard” analysis. In particular, we will show that there exists a nonzero element in Henle’s ring that is “too small” to be registered as nonzero in •R , while it is seen as a nonzero infinitesimal in *R . This result suggests that some infinitesimals in *R are smaller than the infinitesimals in •R. We argue that the apparent contradiction with the conclusions by Reeder is only due to the presence of nilpotent elements in •R .
On mathematical realism and the applicability of hyperreals (with Vladimir Kanovei, Mikhail G. Katz, Thomas Mormann and David Sherry), Mat. Stud., issue 2, vol 51 (2019).
From the conclusions:
Easwaran and Towsner argue that the hyperreal number system of Robinson’s infinitesimal analysis is a good instrumental theory, i.e. a theory that is “useful for proving theorems about the real numbers,” but at the same time it is not suitable for the description of physical phenomena, since the infinitesimals of Robinson’s framework are “idealisations that don’t correspond to the world.” However, these claims rest upon an outdated conception of mathematical realism and on the identification of the physical continuum with the Cantor–Dedekind continuum of real numbers as understood by these classical authors. By abandoning the idea that the relation between mathematics and physical reality must be that of an isomorphism, one sees that the arguments proposed by ET against the applicability of the hyperreals [...] are inadequate. [...] In fact, there are many applications of Robinson’s framework to physics, economics and other sciences, that are unfortunately not discussed in any detail by ET, who choose instead to argue that the hyperreals are inapplicable from first principles and with ad-hoc arguments.
A transfer principle for the continuation of real functions to the Levi-Civita field, p-Adic Numbers, Ultrametric Analysis, and Applications, issue 3, vol 10 (2018).
Abstract:
We discuss the properties of the continuations of real functions to the Levi-Civita field. In particular, we show that, whenever a function f is analytic on a bounded interval [a, b] ⊆ R, the function and its canonical continuation satisfy the same properties that can be expressed in the language of real closed ordered fields. If the function is not analytic, then this equivalence does not hold. These results suggest an analogy with the internal and external functions of nonstandard analysis: while the canonical continuations of analytic functions resemble internal functions, the continuations of non-analytic functions behave like external functions. Inspired by this analogy, we suggest some directions for further research.
Grid functions of nonstandard analysis in the theory of distributions and in partial differential equations, Advances in Mathematics, Volume 345, 17 March 2019, Pages 429-482. See also the Corrigendum.
Abstract:
We introduce the space of grid functions, a space of generalized functions of nonstandard analysis that provides a coherent generalization both of the space of distributions and of the space of Young measures. We will show that in the space of grid functions it is possible to formulate problems from many areas of functional analysis in a way that coherently generalizes the standard approaches. As an example, we discuss some applications of grid functions to the calculus of variations and to the nonlinear theory of distributions. Applications to nonlinear partial differential equations will be discussed in a subsequent paper.
Some applications of numerosities in measure theory (with Vieri Benci and Mauro Di Nasso), Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. vol 26 (2015).
From the introduction:
In this paper we will present three applications of numerosity in topics of measure theory. The first one is about the existence of inner measures associated to any given non-atomic pre-measure. The second application is focused on sets of real numbers. We show that elementary numerosities provide a useful tool with really strong compatibility properties with respect to the Lebesgue measure The third application is about non-Archimedean probability. Following ideas from [1], we consider a model for infinite sequences of coin tosses which is coherent with the original view of Laplace. Moreover, the probability of cylindrical sets exactly coincides with the usual Kolmogorov probability.
[1] V. Benci, L. Horsten, S. Wenmackers (2013), Non-Archimedean probability, vol. 81, pp. 121-151.
Elementary Numerosities and Measures (with Vieri Benci and Mauro Di Nasso), Journal of Logic and Analysis, vol 6 (2014).
Abstract:
Generalizing the notion of numerosity, first introduced in [1], we say that a function n defined on the powerset of a given set X is an elementary numerosity if
1. its range is the non-negative part of a non-archimedean field that extends the field of real numbers;
2. it is finitely additive;
3. n({x}) = 1 for every element x of X.
It turns out that the elementary numerosities are quite general: every non-atomic finitely additive or sigma-additive measure can be obtained as the ratio of an elementary numerosity by a fixed element of F. This theorem can be proved via an ultrapower construction and provides a refinement of a theorem of C. W. Henson about nonstandard representation of measures ([2] Theorem 1). Applications of this result about elementary numerosities will be discussed in a subsequent paper.
[1] V. Benci, M. Di Nasso, Numerosities of labelled sets: a new way of counting, Advances in Mathematics, vol. 173 (2003), pp. 50-67.
[2] C.W. Henson, On the nonstandard representation of measures, Transactions of the American Mathematical Society, vol. 172 (1972), pp. 437-446.
Fermat, Leibniz, Euler, and the gang: the true history of the concepts of limit and shadow (with Tiziana Bascelli, Frederik Herzberg, Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz, Tahl Nowik, David Sherry, and Steven Shnider), Notices of the American Mathematical Society, vol 61, number 8 (2014).
Abstract:
Fermat, Leibniz, Euler, and Cauchy all used one or another form of approximate equality, or the idea of discarding "negligible" terms, so as to obtain a correct analytic answer. Their inferential moves find suitable proxies in the context of modern theories of infinitesimals, and specifically the concept of shadow. We give an application to decreasing rearrangements of real functions.
