Discusses the complexities and paradoxes of love as represented in the history of Western philosophy and Christianity. In The Asymptote of Love, James Kellenberger develops a theory of religious love that resists essentialist definitions of the term and brings into conversation historical debates on love in Western philosophy and Christian theology. He argues that if love can be likened to a mathematical asymptote, which is a straight line that infinitely approaches a curve but never quite reaches it, then the asymptote (...) of love reaches toward the infinite endpoint of love at its uttermost, namely, God’s love. Drawing upon a broad range of thinkers who have put forth classic debates on love—such as St. Augustine of Hippo, Anders Nygren, and St. Thomas Aquinas, as well as some lesser known figures in the debate, such as Leo Tolstoy and Albert Schweitzer—Kellenberger explains the profound connection between human agape and God’s infinite love in its capacity to offer both directive guidance and to exist beyond human conception. James Kellenberger is Emeritus Professor of Philosophy at California State University, Northridge and the author of God’s Goodness and God’s Evil. (shrink)

One method to determine the asymptotics of particular solutions of a difference equation is by solving an associated asymptotic functional equation. Here we study the behaviour of the solutions in an asymptotic neighbourhood of such individual solutions. We identify several types of attraction and repulsion, which range from almost orthogonality to almost parallelness. Necessary and sufficient conditions for these types of behaviour are given.

All the major inter-theoretic relations of fundamental science are asymptotic ones, e.g. quantum theory as Planck's constant h 0, yielding (roughly) Newtonian mechanics. Thus asymptotics ultimately grounds claims about inter-theoretic explanation, reduction and emergence. This paper examines four recent, central claims by Batterman concerning asymptotics and reduction. While these claims are criticised, the discussion is used to develop an enriched, dynamically-based account of reduction and emergence, to show its capacity to illuminate the complex variety of inter-theory relationships in (...) physics, and to provide a principled resolution to such persistent philosophical problems as multiple realisability and the nature of the special sciences. Introduction Exposition Examination I: Claims (1) and (2), asymptotic explanation and reference Examination II: Claim (3), reduction and singular asymptotics Examination III: Claim (4), emergence and multiple realisability Conclusion. (shrink)

Asymptotic models in which singular limits are taken are very common in physics. They are often used to investigate the general behaviour of systems undergoing rapid, discontinuous, changes. The singularities in the mathematics of these systems have no physical counterparts; these models operate by containing non-physically interpretable fictional elements. As such there is an intuition that states that asymptotics only offer descriptions of systems not explanations of them. By contrast, in different areas of science other models containing fictional elements (...) which are physically interpretable are claimed to be explanatory. I argue that both types of fictional model possess modal content, and therefore it is unclear why models that contain unphysical fictions are merely descriptions, but models that contain physical fictions are explanations. One must either reject both as unexplanatory because they use false ontology to explain, or one should accept both types as explanatory. (shrink)

We study connections between classical asymptotic density, computability and computable enumerability. In an earlier paper, the second two authors proved that there is a computably enumerable set A of density 1 with no computable subset of density 1. In the current paper, we extend this result in three different ways: The degrees of such sets A are precisely the nonlow c.e. degrees. There is a c.e. set A of density 1 with no computable subset of nonzero density. There is a (...) c.e. set A of density 1 such that every subset of A of density 1 is of high degree. We also study the extent to which c.e. sets A can be approximated by their computable subsets B in the sense that A\B has small density. There is a very close connection between the computational complexity of a set and the arithmetical complexity of its density and we characterize the lower densities, upper densities and densities of both computable and computably enumerable sets. We also study the notion of "computable at density r" where r is a real in the unit interval. Finally, we study connections between density and classical smallness notions such as immunity, hyperimmunity, and cohesiveness. (shrink)

We classify the asymptotic densities of the sets according to their level in the Ershov hierarchy. In particular, it is shown that for, a real is the density of an n‐c.e. set if and only if it is a difference of left‐ reals. Further, we show that the densities of the ω‐c.e. sets coincide with the densities of the sets, and there are ω‐c.e. sets whose density is not the density of an n‐c.e. set for any.

A traditional view of mathematical modeling holds, roughly, that the more details of the phenomenon being modeled that are represented in the model, the better the model is. This paper argues that often times this ‘details is better’ approach is misguided. One ought, in certain circumstances, to search for an exactly solvable minimal model—one which is, essentially, a caricature of the physics of the phenomenon in question.

Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for first-order sentences. Given first-order sentences φ and θ, we consider the structures with domain {1,..., N} that satisfy θ, and compute the fraction of them in which φ is true. We then consider what happens to this fraction as N gets large. This extends the work on 0-1 laws that considers the limiting probability of first-order sentences, by considering asymptotic conditional (...) probabilities. As shown by Liogon'kii [24], if there is a non-unary predicate symbol in the vocabulary, asymptotic conditional probabilities do not always exist. We extend this result to show that asymptotic conditional probabilities do not always exist for any reasonable notion of limit. Liogon'kii also showed that the problem of deciding whether the limit exists is undecidable. We analyze the complexity of three problems with respect to this limit: deciding whether it is well-defined, whether it exists, and whether it lies in some nontrivial interval. Matching upper and lower bounds are given for all three problems, showing them to be highly undecidable. (shrink)

We show that, when the number of participating agents n tends to infinity, all classical social choice rules are asymptotically strategy -proof with the proportion of manipulable profiles being of order O.

In this paper we consider classes of finite structures where we have good control over the sizes of the definable sets. The motivating example is the class of finite fields: it was shown in [1] that for any formulain the language of rings, there are finitely many pairs (d,μ) ∈ω×Q>0so that in any finite fieldFand for any ā ∈Fmthe size |ø(Fn,ā)| is “approximately”μ|F|d. Essentially this is a generalisation of the classical Lang-Weil estimates from the category of varieties to that of (...) the first-order-definable sets. (shrink)

Robert Batterman and others have argued that certain idealizing explanations have an asymptotic form: they account for a state of affairs or behavior by showing that it emerges “in the limit”. Asymptotic idealizations are interesting in many ways, but is there anything special about them as idealizations? To understand their role in science, must we augment our philosophical theories of idealization? This paper uses simple examples of asymptotic idealization in population genetics to argue for an affirmative answer and proposes a (...) general schema for asymptotic idealization, drawing on insights from Batterman’s treatment and from John Norton’s subsequent critique. (shrink)

We introduce the concept of o-asymptotic classes of finite structures, melding ideas coming from 1-dimensional asymptotic classes and o-minimality. Along with several examples and non-examples of these classes, we present some classification theory results of their infinite ultraproducts: Every infinite ultraproduct of structures in an o-asymptotic class is superrosy of U^þ-rank 1, and NTP2 (in fact, inp-minimal).

When the first r lower extreme order statistics of a sample of large size n, 1 < r < s < n, are observed, asymptotic predictive intervals of the future extreme order statistic with a rank s are constructed. The only assumption that we adopt is that the first failure time is attracted to the Weibull distribution. In addition, we suggest an efficient point estimator of its shape parameter and then a confidence interval is constructed for it. Moreover, new interesting (...) asymptotic properties of the distributions that belong to the minimum domain of attraction of the Weibull distribution are revealed. Furthermore, extensive simulation studies are conducted to demonstrate the efficiency of the proposed methods. Two real datasets are analyzed to illustrate and corroborate the obtained results. The main results concern large samples. (shrink)

When the first r lower extreme order statistics of a sample of large size n, 1 < r < s < n, are observed, asymptotic predictive intervals of the future extreme order statistic with a rank s are constructed. The only assumption that we adopt is that the first failure time is attracted to the Weibull distribution. In addition, we suggest an efficient point estimator of its shape parameter and then a confidence interval is constructed for it. Moreover, new interesting (...) asymptotic properties of the distributions that belong to the minimum domain of attraction of the Weibull distribution are revealed. Furthermore, extensive simulation studies are conducted to demonstrate the efficiency of the proposed methods. Two real datasets are analyzed to illustrate and corroborate the obtained results. The main results concern large samples. (shrink)

A heuristic proof is given that the divergent QCD perturbation series is, asymptotic. By treating it as an asymptotic expansion we show that it makes sense to keep only the first few terms. The example of e+e− annihilation is considered. It is shown that by keeping only the first few terms one can get within a per cent (or smaller) of the complete sum of the series even at very low momenta where the coupling is large. More generally, this affords (...) an explanation of the phenomena of precocious scaling and why keeping only leading order corrections generally works so well. (shrink)

We present a general family of asymptotic solutions to Einstein's equation which are asymptotically flat but do not satisfy the peeling theorem. Near scri, the Weyl tensor obeys a logarithmic asymptotic flatness condition and has a partial peeling property. The physical significance of this asymptotic behavior arises from a quasi-Newtonian treatment of the radiation from a collapsing dust cloud. Practically all the scri formalism carries over intact to this new version of asymptotic flatness.

§1. Introduction. Asymptotic cones of metric spaces were first invented by Gromov. They are metric spaces which capture the ‘large-scale structure’ of the underlying metric space. Later, van den Dries and Wilkie gave a more general construction of asymptotic cones using ultrapowers. Certain facts about asymptotic cones, like the completeness of the metric space, now follow rather easily from saturation properties of ultrapowers, and in this survey, we want to present two applications of the van den Dries-Wilkie approach. Using ultrapowers (...) we obtain an explicit description of the asymptotic cone of a semisimple Lie group. From this description, using semi-algebraic groups and non-standard methods, we can give a short proof of the Margulis Conjecture. In a second application, we use set theory to answer a question of Gromov.§2. Definitions. The intuitive idea behind Gromov's concept of an asymptotic cone was to look at a given metric space from an ‘infinite distance’, so that large-scale patterns should become visible. In his original definition this was done by gradually scaling down the metric by factors 1/nfornϵ ℕ. In the approach by van den Dries and Wilkie, this idea was captured by ultrapowers. Their construction is more general in the sense that the asymptotic cone exists for any metric space, whereas in Gromov's original definition, the asymptotic cone existed only for a rather restricted class of spaces. (shrink)

A recent proposal by Norton (2003) to show that a simple Newtonian system can exhibit stochastic acausal behavior by giving rise to spontaneous movements of a mass on the dome of a certain shape is examined. We discuss the physical significance of an often overlooked and yet important Lipschitz condition the violation of which leads to the existence of anomalous nontrivial solutions in this and similar cases. We show that the Lipschitz condition is closely linked with the time reversibility of (...) certain solutions in Newtonian mechanics and the failure to incorporate this condition within Newtonian mechanics may unsurprisingly lead to physically impossible solutions that have no serious metaphysical implications. ‡I thank Steven Savitt of the Philosophy Department at the University of British Columbia for drawing my attention to the Lipschitz condition, and Alexei Cheviakov of the Mathematics Department at the University of British Columbia for useful discussions. †To contact the author, please write to: Department of Philosophy, University of British Columbia, Vancouver, BC, V6T 1Z1, Canada; e-mail: [email protected]. (shrink)

This paper presents a systematic approach for obtaining results from the area of quantitative investigations in logic and type theory. We investigate the proportion between tautologies (inhabited types) of a given length n against the number of all formulas (types) of length n. We investigate an asymptotic behavior of this fraction. Furthermore, we characterize the relation between number of premises of implicational formula (type) and the asymptotic probability of finding such formula among the all ones. We also deal with a (...) distribution of these asymptotic probabilities. Using the same approach we also prove that the probability that randomly chosen fourth order type (or type of the order not greater than 4), which admits decidable lambda definability problem, is zero. (shrink)

In this work, we study a several species aerobic chemostat model with constant recycle sludge concentration in continuous culture. We reduce the number of parameters by considering a dimensionless model. First, the existence of a global positive uniform attractor for the model with different removal rates is proved using the theory of dissipative dynamical systems. Hence, we investigate the asymptotic behavior of the model under small perturbations using methods of singular perturbation theory and we prove that, in the case of (...) two species in competition, the unique equilibrium which is positive is globally asymptotically stable. Finally, we establish the link between the open problem of the chemostat with different removal rates and monotone functional responses, and our model when two species compete on the same nutrient. We give some numerical simulations to illustrate the results. (shrink)

In this paper, we study the asymptotic behavior of solutions to the Kirchhoff type stochastic plate equation driven by additive noise defined on unbounded domains. We first prove the uniform estimates of solutions and then establish the existence and upper semicontinuity of random attractors.

Matrix models are widely used in biology to predict the temporal evolution of stage-structured populations. One issue related to matrix models that is often disregarded is the sampling variability. As the sample used to estimate the vital rates of the models are of finite size, a sampling error is attached to parameter estimation, which has in turn repercussions on all the predictions of the model. In this study, we address the question of building confidence bounds around the predictions of matrix (...) models due to sampling variability. We focus on a density-dependent Usher model, the maximum likelihood estimator of parameters, and the predicted stationary stage vector. The asymptotic distribution of the stationary stage vector is specified, assuming that the parameters of the model remain in a set of the parameter space where the model admits one unique equilibrium point. Tests for density-dependence are also incidentally provided. The model is applied to a tropical rain forest in French Guiana. (shrink)

Skolem studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant $1$, the identity function ${\mathbf {x}}$, and such that whenever f and g are in the set, $f+g,fg$ and $f^g$ are in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz computed the order type of the fragment below $2^{2^{\mathbf {x}}}$. Here we prove that (...) the set of asymptotic classes within any Archimedean class of Skolem functions has order type $\omega $. As a consequence we obtain, for each positive integer n, an upper bound for the fragment below $2^{n^{\mathbf {x}}}$. We deduce an epsilon-zero upper bound for the fragment below $2^{{\mathbf {x}}^{\mathbf {x}}}$, improving the previous epsilon-omega bound by Levitz. A novel feature of our approach is the use of Conway’s surreal number for asymptotic calculations. (shrink)

Skolem studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant $1$, the identity function ${\mathbf {x}}$, and such that whenever f and g are in the set, $f+g,fg$ and $f^g$ are in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz computed the order type of the fragment below $2^{2^{\mathbf {x}}}$. Here we prove that (...) the set of asymptotic classes within any Archimedean class of Skolem functions has order type $\omega $. As a consequence we obtain, for each positive integer n, an upper bound for the fragment below $2^{n^{\mathbf {x}}}$. We deduce an epsilon-zero upper bound for the fragment below $2^{{\mathbf {x}}^{\mathbf {x}}}$, improving the previous epsilon-omega bound by Levitz. A novel feature of our approach is the use of Conway’s surreal number for asymptotic calculations. (shrink)

We show that the 0-1 law does not hold for the class Σ 1 1 (∀∃∀ without =) by finding a sentence in this class which almost surely expresses parity. We also show that every recursive real in the unit interval is the asymptotic probability of a sentence in this class. This expands a result by Lidia Tendera, who in 1994 proved that every rational number in the unit interval is the asymptotic probability of a sentence in the class Σ (...) 1 1 ∀∃∀ with equality. (shrink)

We show that the theory Tlog of the asymptotic couple of the field of logarithmic transseries is distal. As distal theories are NIP, this provides a new proof that Tlog is NIP.

We consider the reduct to the module language of certain theories of fields with a non surjective endomorphism. We show in some cases the existence of a model companion. We apply our results for axiomatizing the reduct to the theory of modules of non principal ultraproducts of separably closed fields of fixed but non zero imperfection degree.

The asymptotic safety program strives for a consistent description of gravity as a non-perturbatively renormalizable quantum field theory. In this framework the gravitational interactions are encoded in a renormalization group flow connecting the quantum gravity regime at trans-Planckian scales to observable low-energy physics. Our proceedings reviews the key elements underlying the predictive power of the construction and summarizes the state-of-the-art in determining its free parameters. The explicit construction of a realistic renormalization group trajectory describing our world shows that the flow (...) possesses two characteristic scales. The Planck scale where Newton’s coupling G becomes constant is generated dynamically. The freeze-out of the cosmological constant \ occurs at a terrestrial scale fixed by the observed value of the dimensionless product \. We also review the perspectives of determining the free parameters of the theory through cosmologicalobservations. (shrink)

In the asymptotic safety paradigm, a quantum field theory reaches a regime with quantum scale invariance in the ultraviolet, which is described by an interacting fixed point of the Renormalization Group. Compelling hints for the viability of asymptotic safety in quantum gravity exist, mainly obtained from applications of the functional Renormalization Group. The impact of asymptotically safe quantum fluctuations of gravity at and beyond the Planck scale could at the same time induce an ultraviolet completion for the Standard Model of (...) particle physics with high predictive power. (shrink)

This paper addresses a relatively common scientific (as opposed to philosophical) conception of intertheoretic reduction between physical theories. This is the sense of reduction in which one (typically newer and more refined) theory is said to reduce to another (typically older and coarser) theory in the limit as some small parameter tends to zero. Three examples of such reductions are discussed: First, the reduction of Special Relativity (SR) to Newtonian Mechanics (NM) as (v/c)20; second, the reduction of wave optics to (...) geometrical optics as 0; and third, the reduction of Quantum Mechanics (QM) to Classical Mechanics (CM) as0. I argue for the following two claims. First, the case of SR reducing to NM is an instance of a genuine reductive relationship while the latter two cases are not. The reason for this concerns the nature of the limiting relationships between the theory pairs. In the SR/NM case, it is possible to consider SR as a regular perturbation of NM; whereas in the cases of wave and geometrical optics and QM/CM, the perturbation problem is singular. The second claim I wish to support is that as a result of the singular nature of the limits between these theory pairs, it is reasonable to maintain that third theories exist describing the asymptotic limiting domains. In the optics case, such a theory has been called catastrophe optics. In the QM/CM case, it is semiclassical mechanics. Aspects of both theories are discussed in some detail. (shrink)

The derivation on the differential-valued field Tlogof logarithmic transseries induces on its value group${{\rm{\Gamma }}_{{\rm{log}}}}$a certain mapψ. The structure${\rm{\Gamma }} = \left$is a divisible asymptotic couple. In [7] we began a study of the first-order theory of$\left$where, among other things, we proved that the theory$T_{{\rm{log}}} = Th\left$has a universal axiomatization, is model complete and admits elimination of quantifiers in a natural first-order language. In that paper we posed the question whetherTloghas NIP. In this paper, we answer that question in the (...) affirmative:Tlogdoes have NIP. Our method of proof relies on a complete survey of the 1-types ofTlog, which, in the presence of QE, is equivalent to a characterization of all simple extensions${\rm{\Gamma }}\left\langle \alpha \right\rangle$of${\rm{\Gamma }}$. We also show thatTlogdoes not have the Steinitz exchange property and we weigh in on the relationship between models ofTlogand the so-calledprecontraction groupsof [9]. (shrink)