It is only when mixing two or more pure substances along a reversible path that the entropy of the mixing can be made physically manifest. It is not, in this case, a mere mathematical artifact. This mixing requires a process of successive stages. In any finite number of stages, the external manifestation of the entropy change, as a definite and measurable quantity of heat, isa fully continuous function of the relevant variables. It is only at an infinite and unattainable limit (...) thata non-uniform convergence occurs. And this occurs when considered in terms of the number of stages together with a distinguishability parameter appropriate to the particular device which is used to achieve reversibility. These considerations, which are of technological interest to chemical engineers, resolve a paradox derived in chemical theory called Gibbs'' Paradox. (shrink)

The main purpose of this paper is to investigate constructively the relationship between proximal convergence, uniform sequential convergence and uniform convergence for sequences of mappings between apartness spaces. It is also shown that if the second space satisfies the Efremovic axiom, then proximal convergence preserves strong continuity.

The history of uniform convergence is typically focused on the contributions of Cauchy, Seidel, Stokes, and Björling. While the mathematical contributions of these individuals to the concept of uniform convergence have been much discussed, Weierstrass is considered to be the actual inventor of today’s concept. This view is often based on his well-known article from 1841. However, Weierstrass’s works on a rigorous foundation of analytic and elliptic functions date primarily from his lecture courses at the University (...) of Berlin up to the mid-1880s. For the history of uniform convergence, these lectures open up an independent branch of development that is disconnected from the approaches of the previously mentioned authors; to my knowledge, Weierstraß never explicitly referred to Cauchy’s continuity theorem (1821 or 1853) or to Seidel’s or Stokes’s contributions (1847). In the present article, Weierstrass’s contributions to the development of uniform convergence will be discussed, mainly based on lecture notes made by Weierstrass’s students between 1861 and the mid-1880s. The emphasis is on the notation and the mathematical rigor of the introductions to the concept, leading to the proposal to re-date the famous 1841 article and thus Weierstrass’s first introduction of uniform convergence. (shrink)

The subject of the present article is the following fact. Consider an effective uniform space. A generally constructed metric from the uniformity has the property that a sequence from the space effectively converges with respect to the uniform topology if and only if it does with respect to the induced metric. This can be shown without assuming the computability of the metric.

The paper deals with proximal convergence and Leader's theorem, in the constructive theory of uniform apartness spaces.

This book explores the question of how international law is applied by domestic courts. Through case studies and analysis the contributors consider how traditions and diversity affect the interpretation of international law, from a mixture of doctrinal, practical, and theoretical approaches.

This paper develops a general theory of uniform probability for compact metric spaces. Special cases of uniform probability include Lebesgue measure, the volume element on a Riemannian manifold, Haar measure, and various fractal measures (all suitably normalized). This paper first appeared fall of 1990 in the Journal of Theoretical Probability, vol. 3, no. 4, pp. 611—626. The key words by which this article was indexed were: ε-capacity, weak convergence, uniform probability, Hausdorff dimension, and capacity dimension.

We define two types of convergence for observables on a quantum logic which we call M-weak and uniform M-weak convergence. These convergence modes correspond to weak convergence of probability measures. They are motivated by the idea that two (in general unbounded) observables are “close” if bounded functions of them are “close.” We show that M-weak and uniform M-weak convergence generalize strong resolvent and norm resolvent convergence for self-adjoint operators on a Hilbert space. (...) Also, these types of convergence strengthen the weak operator convergence and operator norm convergence of bounded self-adjoint operators on a Hilbert space. Finally, we consider spectral perturbation by showing that the spectra of approximating observables approach the spectrum of the limit in a certain sense. (shrink)

I propose a new definition of identification in the limit, as a new success criterion that is meant to complement, rather than replacing, the classic definition due to Gold. The new definition is designed to explain how it is possible to have successful learning in a kind of scenario that Gold's classic account ignores---the kind of scenario in which the entire infinite data stream to be presented incrementally to the learner is not presupposed to completely determine the correct learning target. (...) From a purely mathematical point of view, the new definition employs a convergence concept that generalizes net convergence and sits in between pointwise convergence and uniform convergence. Two results are proved to suggest that the new definition provides a success criterion that is by no means weak: Between the new identification in the limit and Gold's classic one, neither implies the other. If a learning method identifies the correct target in the limit in the new sense, any U-shaped learning involved therein has to be redundant and can be removed while maintaining the new kind of identification in the limit. I conclude that we should have two success criteria that correspond to two senses of identification in the limit: the classic one and the one proposed here. They are complementary: meeting any one of the two is good; meeting both at the same time, if possible, is even better. (shrink)

The dominated convergence theorem implies that if is a sequence of functions on a probability space taking values in the interval [0, 1], and converges pointwise a.e., then converges to the integral of the pointwise limit. Tao [26] has proved a quantitative version of this theorem: given a uniform bound on the rates of metastable convergence in the hypothesis, there is a bound on the rate of metastable convergence in the conclusion that is independent of the (...) sequence and the underlying space. We prove a slight strengthening of Tao’s theorem which, moreover, provides an explicit description of the second bound in terms of the first. Specifically, we show that when the first bound is given by a continuous functional, the bound in the conclusion can be computed by a recursion along the tree of unsecured sequences. We also establish a quantitative version of Egorov’s theorem, and introduce a new mode of convergence related to these notions. (shrink)

In [9] we introduced a new framework for asymptotic probabilities, in which a $\sigma-additive$ measure is defined on the sample space of all sequences $A = $ of finite models, where the universe of An is {1, 2, .., n}. In this framework we investigated the strong 0-1 law for sentences, which states that each sentence either holds in An eventually almost surely or fails in An eventually almost surely. In this paper we define the strong convergence law for (...) formulas, which carries over the ideas of the strong 0-1 law to formulas with free variables, and roughly states that for each formula φ(x), the fraction of tuples a in An, which satisfy the formula φ(x), almost surely has a limit as n tends to infinity. We show that the infinitary logic with finitely many variables has the strong convergence law for formulas for the uniform measure, and further characterize the measures on random graphs for which the strong convergence law holds. (shrink)

This paper explores the possibility of a standard legal language (e.g. English) for a principled evolution of law in line with technological development. In doing so, reference is made to blockchain networks and smart contracts to emphasise the discontinuity with the liberal legal tradition when it comes to decentralisation and binary code language. Methodologically, the argument is built on the underlying relation between law, semiotics and new forms of media adding to natural language; namely: code and symbols. In what follows, (...) I will concentrate on the study of factors that explain why such approach can be fruitful for the future of law and innovation. I have three reasons for selecting this topic. The first is a more pragmatic reason, based on my current research of law as a linguistic phenomenon. Secondly, the topic does also touch the matter on binary code language, rivalrous to legal alphabetic language. Lastly, the study aims at emphasizing the pivotal role of the jurist as an interpreter in a changing society accommodating diverging realms of reality. The study is structured as follows. Firstly, a quick exam of the traits of blockchain networks would provide the contextual link to establish the arguments in support of the need of a standard legal language. Secondly, a comparison between liberal legal institutions and theory of semiotics is set to perceive their functioning and ascertain their limits in the light of todays unprecedented changes. Thirdly, a summary on blockchain networks’ legal features would constitute the thrust behind the idea of a uniform legal language. Methodologically, the argument does also establish some relations with classical laws of physics and philosophy of media. Its aim is to demonstrate how the suggested legal interpretation and semiotic-based approach can contribute to overcome existing stumbling blocks including, but not limited to, the lack of cooperation at the international level as well as the gap in State norms when it comes to innovation. In this sense, the proposed strategies do not intend to replace current advances in the legal thought. In contrast, it seeks to harmonise their results providing a methodical approach that can concur to inform a new technique to address new controversial issues. In practice, the proposed method regarding the adoption of a uniform legal language would lower transactive costs in terms of normative coordination in the matter of international cooperation and in the definition of applicable law among different legal systems. Alternatively, it might contribute to the convergence of legal systems and/or their underlying concepts. Differently put, article’s contribution can be envisaged in fostering juridical consistency with regards to different forms of languages’ coexistence. (shrink)

We expand our effective framework for weak convergence of measures on the real line by showing that effective convergence in the Prokhorov metric is equivalent to effective weak convergence. In addition, we establish a framework for the study of the effective theory of vague convergence of measures. We introduce a uniform notion and a non-uniform notion of vague convergence, and we show that both these notions are equivalent. However, limits under effective vague (...) class='Hi'>convergence may not be computable even when they are finite. We give an example of a finite incomputable effective vague limit measure, and we provide a necessary and sufficient condition so that effective vague convergence produces a computable limit. Finally, we determine a sufficient condition for which effective weak and vague convergence of measures coincide. As a corollary, we obtain an effective version of the equivalence between classical weak and vague convergence of sequences of probability measures. (shrink)

We examined social facilitation and emotional convergence in amusement, sadness, and fear in dynamic interactions. Dyads of friends or strangers jointly watched emotion-eliciting films while they either could or could not communicate nonverbally. We assessed three components of each emotion (expressions, appraisals, and feelings), as well as attention to and social motives toward the co-participant. In Study 1, participants interacted through a mute videoconference. In Study 2, they sat next to each other and either were or were not separated (...) by a partition. Results revealed that facilitation and convergence are not uniform across different emotions and emotion components. Particularly strong supporting patterns emerged for the facilitation of and convergence in smiling. When direct interaction was possible (Study 2), friends showed a general tendency for strong convergence, with the exception of fear-related appraisals. This suggests that underlying processes of emotional contagion and social appraisal are differentially relevant for different emotions. (shrink)

We prove a nonstandard density result. It asserts that if a particular formula is true for functions in a set K of linear continuous functions between Banach spaces E and D, then it remains valid for functions that are limits, in the uniform convergence topology on a given class ℳ of subsets of E, of nets of vectors in K. We then apply this result to various class ℳ and setsK in the context of E-valued Bochner integrable functions (...) defined on a finite measure space. (shrink)

There is a “Pragmatist turn” visible in the field of organization science today, resulting from a renewed interest in the work of Pragmatist philosophers like Dewey, Mead, Peirce, James and others, and in its implications for the study of organizations. Following Wicks and Freeman, in the past decade Pragmatism has also entered the field of business ethics, which, however, has not been uniformly applauded in that field. Some scholars fear that Pragmatism may enhance already existing positivist and managerialist tendencies in (...) current business ethics, while others see more emancipatory potential in Pragmatism, arguing that it complements and supports stakeholder theory. In this paper, a comparison of the philosophical underpinnings of Pragmatist and Critical conceptions of business ethics is offered, concentrating on the Pragmatism of John Dewey and the Critical theory of the Frankfurt School, in particular of Axel Honneth. It is argued that these two developed along two converging lines. Along the first line, Dewey was far more skeptical and critical of capitalism than is often thought. Along the second line, the reactions to Pragmatism of Frankfurt School Critical theorists developed over time from generally hostile, to partially inclusive, to more fully integrative. At the crossroads of these converging lines a Pragmatist Critical perspective is developed and exemplified, and its implications for business ethics are outlined. (shrink)

A nonlinear coupled wing model subject to unknown external disturbances is proposed in this paper. Since the model is modeled by partial differential equations, the traditional control design scheme based on the ordinary differential equation model is not applicable, and the control law design becomes very complex. In this paper, a new antidisturbance boundary control scheme based on a finite time convergent disturbance observer is proposed. The control laws are designed based on the new disturbance observers to make the external (...) disturbance errors converge to zero in a finite time and ensure the uniformly bounded stability of the controlled system. Finally, the effectiveness of the controllers and the finite-time convergence of disturbance errors are verified by the simulation and comparison. (shrink)

Dini's theorem says that compactness of the domain, a metric space, ensures the uniform convergence of every simply convergent monotone sequence of real-valued continuous functions whose limit is continuous. By showing that Dini's theorem is equivalent to Brouwer's fan theorem for detachable bars, we provide Dini's theorem with a classification in the recently established constructive reverse mathematics propagated by Ishihara. As a complement, Dini's theorem is proved to be equivalent to the analogue of the fan theorem, weak König's (...) lemma, in the original classical setting of reverse mathematics started by Friedman and Simpson. (shrink)

In this paper we develop a method for finding, under general conditions, explicit and highly uniform rates of convergence for the Picard iteration sequences for selfmaps on bounded metric spaces from ineffective proofs of convergence to a unique fixed point. We are able to extract full rates of convergence by extending the use of a logical metatheorem recently proved by Kohlenbach. In recent case studies we were able to find such explicit rates of convergence in (...) two concrete cases. Our novel method now provides an explanation in logical terms for these findings. This amounts, loosely speaking, to general conditions under which we in this specific setting can transform a ∀∃∀-sentence into a ∀∃-sentence via an argument involving product spaces. This reduction in logical complexity allows us to use the existing machinery to extract quantitative bounds of the sort we need. (shrink)

We prove a nonstandard density result. It asserts that if a particular formula is true for functions in a set K of linear continuous functions between Banach spaces E and D, then it remains valid for functions that are limits, in the uniform convergence topology on a given class ℳ︁ of subsets of E, of nets of vectors in K. We then apply this result to various class ℳ︁ and setsK in the context of E‐valued Bochner integrable functions (...) defined on a finite measure space. (shrink)

Cauchy’s contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the epsilon-delta doctrine in the context of an Archimedean continuum. What does one see if one refrains from viewing Cauchy as if he had read Weierstrass already? One sees, with Felix Klein, a parallel thread for the development of analysis, in the context of an infinitesimal-enriched continuum. One sees, with Emile Borel, (...) the seeds of the theory of rates of growth of functions as developed by Paul du Bois-Reymond. One sees, with E. G. Björling, an infinitesimal definition of the criterion of uniform convergence. Cauchy’s foundational stance is hereby reconsidered. (shrink)

Let X be a non-reflexive Banach space such that X ∗ is separable. Let N be the set of all equivalent norms on X , equipped with the topology of uniform convergence on bounded subsets of X . We show that the subset Z of N consisting of Fréchet-differentiable norms whose dual norm is not strictly convex reduces any difference of analytic sets. It follows that Z is exactly a difference of analytic sets when N is equipped with (...) the standard Effros–Borel structure. Our main lemma elucidates the topological structure of the norm-attaining linear forms when the norm of X is locally uniformly rotund. (shrink)

We investigate the classes of ideals for which the Egoroff’s theorem or the generalized Egoroff’s theorem holds between ideal versions of pointwise and uniform convergences. The paper is motivated by considerations of Korch :269–282, 2017).

We give a new account of Björling’s contribution to uniform convergence in connection with Cauchy’s theorem on the continuity of an infinite series. Moreover, we give a complete translation from Swedish into English of Björling’s 1846 proof of the theorem. Our intention is also to discuss Björling’s convergence conditions in view of Grattan-Guinness’ distinction between history and heritage. In connection to Björling’s convergence theory we discuss the interpretation of Cauchy’s infinitesimals.

Because physicians use scientific inference for the generalizations of individual observations and the application of general knowledge to particular situations, the Bayesian probability solution to the problem of induction has been proposed and frequently utilized. Several problems with the Bayesian approach are introduced and discussed. These include: subjectivity, the favoring of a weak hypothesis, the problem of the false hypothesis, the old evidence/new theory problem and the observation that physicians are not currently Bayesians. To the complaint that the prior probability (...) is subjective, Bayesians reply that there will be ultimate convergence, but the rebuttal to this is that there will not be uniform convergence. Secondly, since the Bayesian scheme favors a weak hypothesis, theories turn out to be a gratuitous risk. The problem with the false hypothesis comes out in the denominator of the theorem, revealing that a factor which is not a theory at all is being considered in the reasoning. On the old evidence/new theory problem old evidence cannot confirm a new theory so that the posterior probability will equal the prior probability. Finally, empiric studies have shown that current physicians are not Bayesians. But on consideration of Bayesian inference as a system of inference, it can be reasoned that physicians should be Bayesians. However, the problem of physicians' and patients' own subjectivity continue to plague this system of medical decision making. (shrink)

We study the relations between two consequences of the Continuum Hypothesis discovered by Wacław Sierpiński, concerning uniform continuity of continuous functions and uniform convergence of sequences of real-valued functions, defined on subsets of the real line of cardinality continuum.

Many prominent conceptions of moral progress implicitly assume that progress must lead to convergence in the moral domain. However, given the actual plurality of ways of life and attendant moral outlooks, there is no reason to assume improvement must lead to uniformity. Moreover, as the entanglement of the Enlightenment discourse of progress with colonialism makes evident, the assumption that progress must lead to convergence can license problematic practical conclusions. Drawing on insights from postcolonialist critique, I argue in favor (...) of functionalist conceptions of moral progress. Functionalist conceptions of moral progress do not assume that progress must lead to convergence. By contrast, they make it possible to understand progress in a pluralistic way. Functionalist conceptions of moral progress thus offer one way to develop a conception of moral progress, which can offer practical guidance, while taking into account one important line of critique of the discourse of progress. (shrink)

A key motivation for Universal Grammar (UG) is developmental: UG can help children acquire the linguistic knowledge that they do as quickly as they do from the data that's available to them. Some of the most fruitful recent work in language acquisition has combined ideas about different hypothesis space building blocks with domain‐general statistical learning. Statistical learning can then provide a way to help navigate the hypothesis space in order to converge on the correct hypothesis. Reinforcement learning is a principled (...) way to update the probability of a categorical option that's in competition with other categorical options. There are two prominent options for linking theory theoretical representations: the Uniformity of Theta Assignment Hypothesis (UTAH) and the relativized form of that theory, relativized UTAH. The learning strategy of Pearl and Sprouse shares the intuition with subjacency that there's a local structural anomaly when syntactic islands occur. (shrink)

Given a convergence theorem in analysis, under very general conditions a model-theoretic compactness argument implies that there is a uniform bound on the rate of metastability. We illustrate with three examples from ergodic theory.

In recent years, the scientific study of meditation and psychedelic drugs has seen remarkable developments. The increased focus on meditation in cognitive neuroscience has led to a cross-cultural classification of standard meditation styles validated by functional and structural neuroanatomical data. Meanwhile, the renaissance of psychedelic research has shed light on the neurophysiology of altered states of consciousness induced by classical psychedelics, such as psilocybin and LSD, whose effects are mainly mediated by agonism of serotonin receptors. Few attempts have been made (...) at bridging these two domains of inquiry, despite intriguing evidence of overlap between the phenomenology and neurophysiology of meditation practice and psychedelic states. In particular, many contemplative traditions explicitly aim at dissolving the sense of self by eliciting altered states of consciousness through meditation, while classical psychedelics are known to produce significant disruptions of self-consciousness, a phenomenon known as drug-induced ego dissolution. In this article, we discuss available evidence regarding convergences and differences between phenomenological and neurophysiological data on meditation practice and psychedelic drug-induced states, with a particular emphasis on alterations of self-experience. While both meditation and psychedelics may disrupt self-consciousness and underlying neural processes, we emphasize that neither meditation nor psychedelic states can be conceived as simple, uniform categories. Moreover, we suggest that there are important phenomenological differences even between conscious states described as experiences of self-loss. As a result, we propose that self-consciousness may be best construed as a multidimensional construct, and that “self-loss,” far from being an unequivocal phenomenon, can take several forms. Indeed, various aspects of self-consciousness, including narrative aspects linked to autobiographical memory, self-related thoughts and mental time travel, and embodied aspects rooted in multisensory processes, may be differently affected by psychedelics and meditation practices. Finally, we consider long-term outcomes of experiences of self-loss induced by meditation and psychedelics on individual traits and prosocial behavior. We call for caution regarding the problematic conflation of temporary states of self-loss with “selflessness” as a behavioral or social trait, although there is preliminary evidence that correlations between short-term experiences of self-loss and long-term trait alterations may exist. (shrink)

Preface. I. BASICS OF LOGIC. Introduction. The Structure of Simple Statements. The Structure of Complex Statements. Simple and Complex Properties. Validity. 2. PROBABILITY AND INDUCTIVE LOGIC. Introduction. Arguments. Logic. Inductive versus Deductive Logic. Epistemic Probability. Probability and the Problems of Inductive Logic. 3. THE TRADITIONAL PROBLEM OF INDUCTION. Introduction. Hume’s Argument. The Inductive Justification of Induction. The Pragmatic Justification of Induction. Summary. IV. THE GOODMAN PARADOX AND THE NEW RIDDLE OF INDUCTION. Introduction. Regularities and Projection. The Goodman Paradox. The Goodman (...) Paradox, Regularity, and the Principle of the Uniformity of Nature. Summary. 5. MILL’S METHODS OF EXPERIMENTAL INQUIRY AND THE NATURE OF CAUSALITY. Introduction. Causality and Necessary and Sufficient Conditions. Mill’s Methods. The Direct Method of Agreement. The Inverse Method of Agreement. The Method of Difference. The Combined Methods. The Application of Mill’s Methods. Sufficient Conditions and Functional Relationships. Lawlike and Accidental Conditions. 6. THE PROBABILITY CALCULUS. Introduction. Probability, Arguments, Statements, and Properties. Disjunction and Negation Rules. Conjunction Rules and Conditional Probability. Expected Value of a Gamble. Bayes’ Theorem. Probability and Causality. 7. KINDS OF PROBABILITY. Introduction. Rational Degree of Belief. Utility. Ramsey. Relative Frequency. Chance. 8. PROBABILITY AND SCIENTIFIC INDUCTIVE LOGIC. Introduction. Hypothesis and Deduction. Quantity and Variety of Evidence. Total Evidence. Convergence to the Truth. ANSWERS TO SELECTED EXERCISES. INDEX. (shrink)

Geach and Thomson have argued that nothing is just plain good, because ‘good’ is, logically, an attributive adjective. The upshot, according to Geach and Thomson, is that consequentialism is unacceptable, since its very formulation requires a predicative use of ‘good’. Reactions to the argument have, for the most part, been uniform. Authors have converged on two challenging objections . First, although the logical tests that Geach and Thomson invoke clearly illustrate that ‘good’, as commonly used, is an attributive, they (...) don’t show that ‘good’ lacks an intelligible predicative interpretation. Second, even if the English word ‘good’ fails to express the property of goodness, we can just stipulate that ‘good*’ expresses goodness and thus formulate consequentialism accordingly. The second objection is one way of voicing skepticism about the method of drawing substantive philosophical conclusions from considerations about ordinary language. In this essay, we present an argument, inspired by Geach and Thomson, which isn’t susceptible to the same objections but which supports the same conclusion. The significance of our argument for ethics is obvious; it challenges the intelligibility of standard consequentialism, and even certain forms of non-consequentialism. One might be inclined to think that a more sophisticated consequentialism, which relies on ‘good {possible world/state of affairs/outcome}’ instead of just ‘good’, evades the criticism. But we explain why the criticism can’t be so easily evaded. (shrink)

This article points to the highly centralized nature of the British tax system. A first section shows how all tax law derives from Parliament, the “onlie begetter” of legally enforceable instruments. It is suggested that this system is not democratically accountable at sub-national levels of government. Reforms of the Thatcher era have resulted in the privatization of many public services, leading to the stabilization of State expenditure as a proportion of GDP. However, at the same time, both tax revenue and (...) expenditure have become increasingly centralized. Public service tasks are determined and financed by central government, and local government is for the most part merely the agent through which government policy is realized. The creation of the Scottish and Welsh Assemblies does not alter this overall assessment. Given the high level of centralization, taxes are uniform throughout the country and there is no scope for meaningful regional fiscal competition. Since regional economic development in Britain is notoriously uneven, the author notes that fiscal uniformity does not lead to economic convergence. A final section is devoted to showing why this is not surprising and why fiscal harmonisation at a European level would be a source of growing, not reduced, disparities.Cet article met l’accent sur la nature hautement centralisée du système fiscal britannique. Une première section montre comment toutes les lois fiscales proviennent du Parlement, la “source unique” des instruments légaux exécutoires. Nous considérons que ce système n’est pas démocrat- iquement defendable aux niveaux subnationaux de gouvernement. Les réformes de l’ère Thatcher ont conduit à une privatisation de beaucoup de services publics, entrainant une stabilisation des dépenses de l’Etat en pourcentage du PIB. Cependant, dans le meme temps, à la fois les recettes fiscales et les dépenses sont devenues de plus en plus centralisées. Les taches du service public sont définies et financées par le gouvernement central et le gouvernement local est dans la majeure partie des cas l’agent à travers lequel la politique du gouvernement s’exerce. La creation des Assemblées Ecossaises et galloises ne modifie pas cette evaluation d’ensemble. Etant donné le niveau élevé de centralisation, les impôts sont uniformes à travers le pays et il n’y a pas de possibilité pour une concurrence fiscale régionale significative. Alors que le développement économique regional en Grande Bretagne est notoirement inégal, l’auteur remarque que l’uniformité fiscale ne conduit pas à uneconvergence économique. Une section finale est consacrée à montrer pourquoi cela n’est en rien surprenant et pourquoi l’harmonisation fiscale à un niveau européen serait une source de disparités croissantes, et non décroissantes. (shrink)

We call an $\alpha \in \mathbb {R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha $ with $\alpha - a_n n}$ for infinitely many n. Similarly, there exist regainingly approximable sets whose initial segment complexity infinitely often reaches the maximum possible for c.e. sets. Finally, there is a uniform algorithm splitting regular real numbers into two regainingly approximable numbers that are still regular.

Planar cell polarity (PCP) refers to the polarization of a field of cells within the plane of a cell sheet. This form of polarization is required for diverse cellular processes in vertebrates, including convergent extension (CE), the establishment of PCP in epithelial tissues and ciliogenesis. Perhaps the most distinct example of vertebrate PCP is the uniform orientation of stereociliary bundles at the apices of sensory hair cells in the mammalian auditory sensory organ. The establishment of PCP in the mammalian (...) cochlea occurs concurrently with CE in this ciliated epithelium, therefore linking three cellular processes regulated by the vertebrate PCP pathway in the same tissue and emerging as a model system for dissecting PCP signaling. This review summarizes the morphogenesis of this model system to assist the interpretation of the emerging data and proposes molecular mechanisms underlying PCP signaling in vertebrates. BioEssays 29: 120–132, 2007. © 2007 Wiley Periodicals, Inc. (shrink)

A new foundation for constructive nonstandard analysis is presented. It is based on an extension of a sheaf-theoretic model of nonstandard arithmetic due to I. Moerdijk. The model consists of representable sheaves over a site of filter bases. Nonstandard characterisations of various notions from analysis are obtained: modes of convergence, uniform continuity and differentiability, and some topological notions. We also obtain some additional results about the model. As in the classical case, the order type of the nonstandard natural (...) numbers is a dense set of copies of the integers. Every standard set has a hyperfinite enumeration of its standard elements in the model. All arguments are carried out within a constructive and predicative metatheory: Martin-Löf's type theory. (shrink)

The inverse of the distance between two structures $\mathscr{A} \not\equiv \mathscr{B}$ of finite type τ is naturally measured by the smallest integer q such that a sentence of quantifier rank q - 1 is satisfied by A but not by B. In this way the space $\operatorname{Str}^\tau$ of structures of type τ is equipped with a pseudometric. The induced topology coincides with the elementary topology of $\operatorname{Str}^\tau$ . Using the rudiments of the theory of uniform spaces, in this elementary (...) note we prove the convergence of every Cauchy net of structures, for any type τ. (shrink)

The inverse of the distance between two structures $\mathscr{A} \not\equiv \mathscr{B}$ of finite type $\tau$ is naturally measured by the smallest integer $q$ such that a sentence of quantifier rank $q - 1$ is satisfied by $\mathscr{A}$ but not by $\mathscr{B}$. In this way the space $\operatorname{Str}^\tau$ of structures of type $\tau$ is equipped with a pseudometric. The induced topology coincides with the elementary topology of $\operatorname{Str}^\tau$. Using the rudiments of the theory of uniform spaces, in this elementary note (...) we prove the convergence of every Cauchy net of structures, for any type $\tau$. (shrink)

Modern longitudinal studies collect feature data at many timepoints, often of the same order of sample size. Such studies are typically affected by dropout and positivity violations. We tackle these problems by generalizing effects of recent incremental interventions to accommodate multiple outcomes and subject dropout. We give an identifying expression for incremental intervention effects when dropout is conditionally ignorable and derive the nonparametric efficiency bound for estimating such effects. Then we present efficient nonparametric estimators, showing that they converge at fast (...) parametric rates and yield uniform inferential guarantees, even when nuisance functions are estimated flexibly at slower rates. We also study the variance ratio of incremental intervention effects relative to more conventional deterministic effects in a novel infinite time horizon setting, where the number of timepoints can grow with sample size and show that incremental intervention effects yield near-exponential gains in statistical precision in this setup. Finally, we conclude with simulations and apply our methods in a study of the effect of low-dose aspirin on pregnancy outcomes. (shrink)

I argue that sentence contents should be assigned truth-values relative to parameters other than a possible world only if those parameters are fixed by the context of assessment rather than the context of use. Standard counterexamples, including tense, de se attitudes, and knowledge ascriptions, all admit of alternative analyses which do not make use of such parameters. Moreover, allowing such indices greatly complicates the task of defining disagreement, and forces an odd separation between what is true, and what someone has (...) truthfully said. If non-world indices are always fixed by the context of assessment, a characterization of semantic theories as ?relativist? in terms of assessment-sensitivity converges with a characterization in terms of sensitivity to non-world indices. More tentatively, I suggest that even a possible world index, when used in the assignment of truth-values to sentence contents, should be fixed by the context of assessment, not the context of use. This eliminates MacFarlane's category of ?non-indexical contextualism?, and results in a system in which parameters fixed by the context of use are used only in the assignment of contents to linguistic expressions, and parameters used in the assignment of truth-values to contents are uniformly fixed by the context of assessment. (shrink)

Drawing on C. S. Peirce's notion of an "unlimited community of investigators," Jürgen Habermas seeks to construct binding moral norms apart from independently existing moral absolutes. But Habermas fatally compromises the usefulness of an unlimited community for his project. Where Peirce relied on the notion of a uniform "outward clash" with external reality or "secondness" to lead all members of his unlimited community of investigators to a "predestinate" point of convergence, Habermas lacks any notion of an independent moral (...) reality to bring about this convergence throughout his unlimited community of constructors. Apart from devising a substitute secondness to replace the clash with external moral reality, Habermas' unlimited community will spin apart into countless divergent moral constructs, Habermas' attempts to devise a substitute secondness fouruier on a lifeworld dilemma he never resolves, Habermas' difficulties are instructive, suggesting a possible way forward. (shrink)

Recently, decomposition-based multiobjective evolutionary algorithms have good performances in the field of multiobjective optimization problems and have been paid attention by many scholars. Generally, a MOP is decomposed into a number of subproblems through a set of weight vectors with good uniformly and aggregate functions. The main role of weight vectors is to ensure the diversity and convergence of obtained solutions. However, these algorithms with uniformity of weight vectors cannot obtain a set of solutions with good diversity on some (...) MOPs with complex Pareto optimal fronts. To deal with this problem, an improved decomposition-based multiobjective evolutionary algorithm with adaptive weight adjustment is proposed. Firstly, a new method based on uniform design and crowding distance is used to generate a set of weight vectors with good uniformly. Secondly, according to the distances of obtained nondominated solutions, an adaptive weight vector adjustment strategy is proposed to redistribute the weight vectors of subobjective spaces. Thirdly, a selection strategy is used to help each subobjective space to obtain a nondominated solution. Comparing with six efficient state-of-the-art algorithms, for example, NSGAII, MOEA/D, MOEA/D-AWA, EMOSA, RVEA, and KnEA on some benchmark functions, the proposed algorithm is able to find a set of solutions with better diversity and convergence. (shrink)

This paper presents a problem of observer-based adaptive fuzzy predefined performance control of a class of nonlinear pure-feedback systems with input delay and unknown control direction. Compared with the existing research, a novel predefined performance controller is proposed, which relaxes the assumption that the initial error is known. In addition, it is difficult to design the controllers due to input delay and nonaffine properties of the pure-feedback systems, which can be simplified by Pade approximation. Moreover, dynamic surface control and Nussbaum (...) functions are applied to overcome the calculation explosion caused by repeated differentiations and unknown control direction, respectively. Based on the above methods, an adaptive fuzzy predefined performance controller is proposed, and it is proved that all the signals of a closed-loop system are semiglobally uniformly ultimately bounded. The tracking errors converge within a predefined range, while the observer estimation errors converge within a small zero region. Finally, the simulation results demonstrate the effectiveness of the proposed control system. (shrink)

Einstein wrote his famous sentence "God does not play dice with the universe" in a letter to Max Born in 1920. All experiments have confirmed that quantum mechanics is neither wrong nor “incomplete”. One can says that God does play dice with the universe. Let quantum mechanics be granted as the rules generalizing all results of playing some imaginary God’s dice. If that is the case, one can ask how God’s dice should look like. God’s dice turns out to be (...) a qubit and thus having the shape of a unit ball. Any item in the universe as well the universe itself is both infinitely many rolls and a single roll of that dice for it has infinitely many “sides”. Thus both the smooth motion of classical physics and the discrete motion introduced in addition by quantum mechanics can be described uniformly correspondingly as an infinite series converges to some limit and as a quantum jump directly into that limit. The second, imaginary dimension of God’s dice corresponds to energy, i.e. to the velocity of information change between two probabilities in both series and jump. (shrink)

The research presented in this paper was motivated by our aim to study a problem due to J. Bourgain [3]. The problem in question concerns the uniform boundedness of the classical separation rank of the elements of a separable compact set of the first Baire class. In the sequel we shall refer to these sets (separable or non-separable) as Rosenthal compacta and we shall denote by ∝(f) the separation rank of a real-valued functionfinB1(X), withXa Polish space. Notice that in (...) [3], Bourgain has provided a positive answer to this problem in the case ofKsatisfyingwithXa compact metric space. The key ingredient in Bourgain's approach is that whenever a sequence of continuous functions pointwise converges to a functionf, then the possible discontinuities of the limit function reflect a local ℓ1-structure to the sequence (fn)n. More precisely the complexity of this ℓ1-structure increases as the complexity of the discontinuities offdoes. This fruitful idea was extensively studied by several authors (c.f. [5], [7], [8]) and for an exposition of the related results we refer to [1]. It is worth mentioning that A.S. Kechris and A. Louveau have invented the rank rND(f) which permits the link between thec0-structure of a sequence (fn)nof uniformly bounded continuous functions and the discontinuities of its pointwise limit. Rosenthal'sc0-theorem [11] and thec0-index theorem [2] are consequences of this interaction.Passing to the case where either (fn)nare not continuous orXis a non-compact Polish space, this nice interaction is completely lost. (shrink)

This paper presents an adaptive neural tracking control approach for a two-joint robotic manipulator with unknown time-varying delays. In order to work out the effect of unknown time-varying delays on the two-joint robotic manipulator, the appropriate Lyapunov–Krasovskii functionals and separation technology are chosen to settle this matter. The neural networks work as an approximator that has the advantage of estimating the unknown function in the system. In this paper, Lyapunov stability analysis can prove that all signals of the closed-loop system (...) are semiglobal uniformly ultimately bounded and the tracking error can converge to a compact neighborhood with respect to zero. The simulation consequences demonstrate the availability of the feedforward control approach. (shrink)

In recent years, proof theoretic transformations that are based on extensions of monotone forms of Gödel’s famous functional interpretation have been used systematically to extract new content from proofs in abstract nonlinear analysis. This content consists both in effective quantitative bounds as well as in qualitative uniformity results. One of the main ineffective tools in abstract functional analysis is the use of sequential forms of weak compactness. As we recently verified, the sequential form of weak compactness for bounded closed and (...) convex subsets of an abstract Hilbert space can be carried out in suitable formal systems that are covered by existing metatheorems developed in the course of the proof mining program. In particular, it follows that the monotone functional interpretation of this weak compactness principle can be realized by a functional Ω∗ definable from bar recursion of lowest type. While a case study on the analysis of strong convergence results that are based on weak compactness indicates that the use of the latter seems to be eliminable, things apparently are different for weak convergence theorems such as the famous Baillon nonlinear ergodic theorem. For this theorem we recently extracted an explicit bound on a metastable version of this theorem that is primitive recursive relative to Ω∗.In the current paper we for the first time give the construction of Ω∗ . As a corollary to the fine analysis of the use of bar recursion in this construction we obtain that Ω∗ elevates arguments in Tn at most to resulting functionals in Tn+2 . In particular, one can conclude from this that our bound on Baillon’s theorem is at least definable in T4. (shrink)