Let G be a group with a metric invariant under left and right translations, and let ????r be the ball of radius r around the identity. A (k,r)-metric approximate subgroup is a symmetric subset X of G...

We prove that each ∀1 free amalgamation class K over a finite relational language L admits a countable generic structure M isometrically embedding all countable structuresin K relative to a fixed metric. We expand L by infinitely many binary predicates expressingdistance, and prove that the resulting expansion of K has a model companion axiomatizedby the first-order theory of M. The model companion is non-finitely axiomatizable, evenover a strong form of the axiom scheme of infinity.

Let G be a group with a metric invariant under left and right translations, and let ????r be the ball of radius r around the identity. A (k,r)-metric approximate subgroup is a symmetric subset X of G...

Citation metrics are statistical measures of scientific output that draw on citation indexes. They purport to capture the impact of scientific articles and the journals in which they appear. As evaluative tools, they are mostly used in the natural sciences, but they are also acquiring an important role in the humanities. While the strengths and weaknesses of citation metrics are extensively debated in a variety of fields, they have only recently started attracting attention in the philosophy of science literature. This (...) paper takes a further step in this direction and presents an analysis of citation metrics from the perspective of a Kuhnian model for the development of science. Starting from Gillies’ argument against the use of citation metrics for scientific research (2008), this paper shows that citation metrics interfere with the development of normal science in certain fields or subfields. The main issue is that citation metrics do not take field-specific differences into account, thereby selectively favoring some fields and arbitrarily hindering the development of others. In other words, this paper shows that current citation metrics fail to “carve science at its joints”. In light of this, the paper cautions against their use for evaluative purposes. (shrink)

It is shown that for compact metric spaces the following statements are pairwise equivalent: “X is Loeb”, “X is separable”, “X has a we ordered dense subset”, “X is second countable”, and “X has a dense set G = ∪{Gn : n ∈ ω}, ∣Gn∣ < ω, with limn→∞ diam = 0”. Further, it is shown that the statement: “Compact metric spaces are weakly Loeb” is not provable in ZF0 , the Zermelo-Fraenkel set theory without the axiom of regularity, and (...) that the countable axiom of choice for families of finite sets CACfin does not imply the statement “Compact metric spaces are separable”. (shrink)

Because ports are considered to be the heart of the maritime transportation system, thereby assessing port performance is necessary for a nation’s development and economic success. This study proposes a novel metric, namely, “_port performance index (PPI)_”, to determine the overall performance and utilization of inland waterway ports based on six criteria,_ port facility, port availability, port economics, port service, port connectivity, and port environment_. Unlike existing literature, which mainly ranks ports based on quantitative factors, this study utilizes a Bayesian (...) Network (BN) model that focuses on both quantitative and qualitative factors to rank a port. The assessment of inland waterway port performance is further analyzed based on different advanced techniques such as sensitivity analysis and belief propagation. Insights drawn from the study show that all the six criteria are necessary to predict PPI. The study also showed that port service has the highest impact while port economics has the lowest impact among the six criteria on PPI for inland waterway ports. (shrink)

Metrics shape our social worlds in many and more ways. Everyday quantifications of our preferences, our behaviors and our relationships, alter us and the institutions that we constitute. This essay takes a brief look at the metrics of business ethics through two analytic devices. Representation explains the notion that metrics can capture or demonstrate ethics and performativity explains the notion that metrics can shape or constitute ethics. The analytic distinction between representation and performativity is obscured in practice when metrics become (...) targets, indeed much of the social power of metrics comes from their use as targets. Hence, we should pay attention to the world of practice in which measuring and doing are entangled. However, we should not lose sight of the limitations of measurement and the possibility that there are areas of ethical life best left unmeasured. (shrink)

£ The existence of a model for a logic program is generally established by lattice-theoretic arguments. We present three examples to show that metric methods can often be used instead, generally in a direct, straightforward way. One example is a game program, which is not stratified or locally stratified, but which has a unique supported model whose existence is easily established using metric methods. The second example is a program without a unique supported model, but having a part that is (...) ‘well-behaved.’ The third example is a program in which one part depends on another, illustrating how modularity might be treated metrically. Finally we use ideas from this third example to prove a general result from [3]. The intention in presenting these examples and the theorem is to stimulate interest in.. (shrink)

We show that the set of points of an overt closed subspace of a metric completion of a Bishop-locally compact metric space is located. Consequently, if the subspace is, moreover, compact, then its collection of points is Bishop-compact. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Info-metrics is a framework for rational inference based on insufficient information. The complete info-metric framework, accompanied with many interdisciplinary examples and case studies, as well as graphical representations of the theory appear in the new book “Foundations of Info-Metrics: Modeling, Inference and Imperfect Information,” Oxford University Press, 2018. In this commentary, I describe that framework in general terms, demonstrate some of the ideas via simple examples, and provide arguments for using it to transform information into useful knowledge.

We introduce a new Turing machine based concept of time complexity for functions on computable metric spaces. It generalizes the ordinary complexity of word functions and the complexity of real functions studied by Ko [19] et al. Although this definition of TIME as the maximum of a generally infinite family of numbers looks straightforward, at first glance, examples for which this maximum exists seem to be very rare. It is the main purpose of this paper to prove that, nevertheless, the (...) definition has a large number of important applications. Using the framework of TTE [40], we introduce computable metric spaces and computability on the compact subsets. We prove that every computable metric space has a c-proper c-admissible representation. We prove that Turing machine time complexity of a function computable relative to c-admissible c-proper representations has a computable bound on every computable compact subset. We prove that computably compact computable metric spaces have concise c-proper c-admissible representations and show by examples that many canonical representations are of this kind. Finally, we compare our definition with a similar one by Labhalla et al. [22]. Several examples illustrate the concepts. By these results natural and realistic definitions of computational complexity are now available for a variety of numerical problems such as image processing, integration, continuous Fourier transform or wave propagation. (shrink)

This work concerns constructive aspects of measure theory. By considering metric completions of Boolean algebras – an approach first suggested by Kolmogorov – one can give a very simple construction of e.g. the Lebesgue measure on the unit interval. The integration spaces of Bishop and Cheng turn out to give examples of such Boolean algebras. We analyse next the notion of Borel subsets. We show that the algebra of such subsets can be characterised in a pointfree and constructive way by (...) an initiality condition. We then use our work to define in a purely inductive way the measure of Borel subsets. (shrink)

Alternative models of similarity judgments that do not rest on metric space assumptions are known to be better descriptions of actual human behaviour but are ignored by Edelman. The internal spaces he postulates are a convenient fiction for artificial intelligence, but not compatible with what is now known about psychophysics at both behavioural and neurological levels of perceptual processing.

De Marchi and Lorenzetti :253-261, 2016) have recently argued that in fields where the journal impact factor is not calculated, such as in the humanities, it is key to find other indicators that would allow the relevant community to assess the quality of scholarly journals and the research outputs that are published in them. The authors' suggestion is that information concerning the journal's rejection rate and the number of subscriptions sold is important and should be used for such assessment. The (...) question addressed by the authors is very important, yet their proposed solutions are problematic. Here I point to some of these problems and illustrate them by considering as a case in point the field of philosophy. Specifically, here I argue for four main claims. First, even assuming that IF provides a reliable indicator of the quality of journals for the assessment of research outputs, De Marchi and Lorenzetti have failed to validate their suggested indicators and proxies. Second, it has not been clarified why, in absence of IF, other journal-based metrics that are currently available should not be used. Third, the relationship between IF and rejection rate is more complex than the authors suggest. Fourth, accepting the number of sold subscriptions as a proxy would result in discrimination against open access journals. The upshot of my analysis is that the question of how to assess journals and research outputs in the humanities is still far from resolved. (shrink)

Previous research on reactivity—defined as changing organizational behaviour to better conform to the criteria of measurement in response to being measured—has found significant variation in company responses toward sustainability metrics. We propose that reactivity is driven by dialogue, motivation, and capacity in a configurational way. Empirically, we use fuzzy set qualitative comparative analysis to analyze company responses to the sustainability index FTSE4Good. We find evidence of complementary and substitute effects between motivation and capacity. Based on these effects, we develop a (...) typology of reactivity to sustainability metrics, which also theorizes the use of metrics as tools for performance feedback and the building of calculative capacity. We show that when reactivity is studied configurationally, we can identify previously underacknowledged types of responses. We discuss the theoretical and practical implications for studying and using sustainability metrics as governance tools for responsible behaviour. (shrink)

In a number of publications A.N. Prior considered the use of what he called ‘metric tense logic’. This is a tense logic in which the past and future operators P and F have an index representing a temporal distance, so that Pnα means that α was true n -much ago, and Fn α means that α will be true n -much hence. The paper investigates the use of metric predicate tense logic in formalising phenomena ormally treated by such devices as (...) multiple indexing or quantification over times. (shrink)

Using the Weyl-type canonical coordinates, an integration of Einstein's field equations in the cylindrosymmetric case considered by Kurşunoğlu is reexamined. It is made clear that the resulting metric is not describing the spacetime in a rotating frame, but in astatic cylindrical elastic medium. The conclusion of Kurşunoğlu that “for an observer on a rotating disk there is no way of escape from a curved spacetime” is therefore not valid. The metric in an empty rotating frame is found as a solution (...) of Einstein's field equations, and is not orthogonal. It is shown that the corresponding orthogonal solution represents spacetime in an inertial frame expressed in cylindrical coordinates. Introducing a noncoordinate basis, the metric in a rotating frame is given the static form of Kurşunoğlu's solution. The essential role played by the nonvanishing structure coefficients in this case is made clear. (shrink)

The paper introduces a broad family of metrics applicable to finite and countably infinite strings, or, by extension, to formal structures serving as semantics for countable languages. The main focus is on applications to sets of pointed Kripke models, a semantics for modal logics. For the resulting metric spaces, the paper classifies topological properties including which metrics are topologically equivalent, providing sufficient conditions for compactness, characterizing clopen sets and isolated points, and characterizing the metrical topologies by a concept of logical (...) convergence. We then apply the approach to maps from dynamic epistemic logic, showing that product updates with action models yield continuous maps, hence allowing for an interpretation of the iterated updates as discrete time dynamical systems. (shrink)

What kind of Theory of Music and Theory of Metric was taught to the young Pindar or the young Sophocles? So far are we from an answer to this question that we do not even know how far extra study was necessary, or usual, for the professional poet as compared with the ordinary educated Greek citizen. The interdependence of music and metric in lyric poetry gave complexity to the word-rhythms but kept the study of music, the subordinate partner, theoretically simple. (...) Doubtless by the time the young poet had learnt by heart the words of past masters of lyric verse, sung or accompanied them on the lyre, and danced them in choir or κμος, he had absorbed by practice and somewhat rule-of-thumb methods of training a great deal of the τχενη which his calling would require. He would have to learn, in common with the χοροδιδσκαλος, how to read and write a simple score, and so he must know the symbols of pitch in such scales as were then in use. The notation of arsis and thesis may have been required too, and if we knew that this was so, and how he was taught to apply it, we might be in a much better position to assess the kind of theory he learnt. But of one thing we may be sure: genius apart, it was his practical training, as performer and spectator, rather than theoretical teaching which gave his ear its cunning and technical skill to compose new rhythms. (shrink)

This paper examines one aspect of the legacy of the Value-Free Ideal in conservation science: the view that measurements and metrics are value-free epistemic tools detached from ideological, ethical, social, and, generally, non-epistemic considerations. Contrary to this view, I will argue that traditional measurement practices entrenched in conservation are in fact permeated with non-epistemic values. I challenge the received view by revealing three non-epistemic assumptions underlying traditional metrics: (1) a human-environment demarcation, (2) the desirability of a people-free landscape, and (3) (...) the exclusion of cultural diversity from biodiversity. I also draw a connection between arguments for retaining traditional metrics to “scientific colonialism,” exemplified by a fortress conservation model. I conclude by advocating for abandoning the myth of the intrinsic value-freedom of measurement practices and embracing metrics aligned with societal and scientific goals. (shrink)

Text, interpretation and metre present a tangled problem in this threnody, and the solutions of editors differ widely. The chief function of detailed metrical study in such corrupt passages of lyric is to weight the scales in favour of—or more often against—certain methods of handling the text. The positive results of this present attempt to apply metrical criteria are necessarily modest and tentative; negatively they are, I think, sometimes decisive.

The axiom of multiple choice implies that metric spaces are paracompact but the reverse implication cannot be proved in set theory without the axiom of choice.

A one-to-one correspondence is established between linearized space-time metrics of general relativity and the wave equations of quantum mechanics. Also, the key role of boundary conditions in distinguishing quantum mechanics from classical mechanics, will emerge naturally from the procedure. Finally, we will find that the methodology will enable us to introduce not only test charges but also test masses by means of gauges.

Any theory of health justice requires an account of what areas of social life are important enough to be of public concern. What are the goods that ought to be provided as a matter of justice? This is what I will call the metric problem. The capabilities approach puts forward a particular solution to this problem. In this article I will discuss some issues of such an approach in relation to Sridhar Venkatapuram's well-known theory. Another problem I examine is how (...) to determine a threshold of provision within a theory of justice. What is enough in terms of health justice? I argue that we need such a threshold to avoid healthism, the expansion of the pursuit of health over and above the treatment and prevention of disease. This is an especially pertinent problem in public health, which is also the context of Venkatapuram's theory. (shrink)

Computability theoretic aspects of Polish metric spaces are studied by adapting notions and methods of computable structure theory. In this dissertation, we mainly investigate index sets and classification problems for computably presentable Polish metric spaces. We find the complexity of a number of index sets, isomorphism problems, and embedding problems for computably presentable metric spaces. We also provide several computable structure theory results related to some classical Polish metric spaces such as the Urysohn space $\mathbb {U}$, the Cantor space $2^{\mathbb (...) {N}}$, the Baire space $\mathbb {N}^{\mathbb {N}}$, and spaces of continuous functions.Abstract prepared by Teerawat Thewmorakot.E-mail: [email protected]. (shrink)

§ 1. Introduction. In this communication we present some recent results on the classification of Polish metric spaces up to isometry and on the isometry groups of Polish metric spaces. A Polish metric space is a complete separable metric space.Our first goal is to determine the exact complexity of the classification problem of general Polish metric spaces up to isometry. This work was motivated by a paper of Vershik [1998], where he remarks : “The classification of Polish spaces up to (...) isometry is an enormous task. More precisely, this classification is not ‘smooth’ in the modern terminology.” Our Theorem 2.1 below quantifies precisely the enormity of this task.After doing this, we turn to special classes of Polish metric spaces and investigate the classification problems associated with them. Note that these classification problems are in principle no more complicated than the general one above. However, the determination of their exact complexity is not necessarily easier.The investigation of the classification problems naturally leads to some interesting results on the groups of isometries of Polish metric spaces. We shall also present these results below.The rest of this section is devoted to an introduction of some basic ideas of a theory of complexity for classification problems, which will help to put our results in perspective. Detailed expositions of this general theory can be found, e.g., in Hjorth [2000], Kechris [1999], [2001]. (shrink)

This essay defines Maoism as an experiment in intimate governance and an attempt—albeit a failed one—to dismantle the divide between political leaders and ordinary people. The Communist Party’s claim to intimacy with the people needs to be constantly reenacted in the relationship between party cadres and ordinary citizens––a cadre’s gestures, habits, and attitudes are magnified and scrutinized under the lens of party legitimacy. The special privileges (tequan, 特权) of party leaders are what I call metrics of exceptionality, which separate the (...) party from the people. The promise to govern on the basis of intimacy with the people did not eradicate ruling class entitlements and arrogance but delegitimized them and opened them up to public scrutiny. Although the process of eradicating privileges was always several steps behind their elaboration and intensification, the former was a normative goal that shaped how ordinary citizens perceived and interacted with party cadres. This dual movement in opposing directions still constitutes the fundamental contradiction of China’s political system as seen from within. This essay revisits and examines crucial historical junctures in the People’s Republic of China as framed by popular contestations of special privilege. The fact that in the post-Mao period, party leaders still speak in the language of intimacy, proximity, and distance indicates the aesthetic importance of this relationship in party ideology, despite public complaints regarding its absence. Even when intimacy as a political mode of communication is able to communicate nothing but its own failure, the proximity between the party and the people takes center stage as the aesthetic foundation of party legitimacy. (shrink)

We study versions of limit models adapted to the context of metric abstract elementary classes. Under categoricity and superstability-like assumptions, we generalize some theorems from 7, 15-17. We prove criteria for existence and uniqueness of limit models in the metric context.

This work copiles key metrics to measure and evalute the impact of science and technology on academia, industry and government. it covers such topics as ...

The idea that sentences can be closer or further apart in meaning is highly intuitive. Not only that, it is also a pillar of logic, semantic theory and the philosophy of science, and follows from other commitments about similarity. The present paper proposes a novel way of comparing the ‘distance’ between two pairs of propositions. We define ‘\ is closer in meaning to \ than \ is to \’ and thereby give a precise account of comparative propositional similarity facts. Notably, (...) our definition eschews metric assumptions, which are unrealistic in most applications of interest. (shrink)

In tone languages, some case studies showed that the word-level tonal representation was closely related to the underlying metrical pattern. Based on different tonal patterns in prosodic units, the metrical structures could generally be divided into the left- and right-dominant types in Chinese dialects. Yet the cross-dialectal phonetic realizations (e.g., duration and pitch) between or within these two metrical structures were still unrevealed. The current study investigated the duration and pitch realizations of disyllabic prosodic words in Changsha and Chengdu dialects (...) (the left-dominant structure), and in Fuzhou and Xiamen dialects (the right-dominant structure). Results showed that not all the duration patterns across four Chinese dialects were sensitive to different metrical structures, indicating that the duration might not be the universal cue for metrical prominence in Chinese dialects. In terms of pitch realization across all the four Chinese dialects, level tones (sometimes falling tones) generally appeared in the metrically weak unit, while underlying pitch forms appeared in the metrically strong unit. Compared with duration, pitch might be more robust for prosodic realizations of metrical structures in Chinese dialects. Furthermore, there was an interaction between duration and pitch patterns in Chinese dialects, which could shed new light on the phenomenon of “metrical tone sandhi”. Meanwhile, this study also provides some references for the judgment of the metrical stress and prosodic realizations in other Chinese dialects. (shrink)

The appearance of a new and modern Teubner text of Valerius Flaccus, eliminating much consecrated chaff, is greatly to be welcomed. Its editor, E. Courtney of King's College, University of London, kindly lent me his text in typescript several years ago, so that my metrical statistics might be published at about the same time.

What has the dispositional analysis of properties and laws (e.g. Molnar, Powers, Oxford University Press, Oxford, 2003; Mumford, Laws in nature, Routledge London, 2004; Bird, Nature’s metaphysics, Clarendon Press, Oxford, 2007) to offer to the scientific understanding of physical properties?—The article provides an answer to this question for the case of spacetime points and their metrical properties in General Relativity. The analysis shows that metrical properties are not ‘powers’, i.e. they cannot be understood as producing the effects of spacetime on (...) matter with metaphysical necessity. Instead they possess categorical characteristics which, in connection with specific laws, explain those effects. Thus, the properties of spacetime do not favor the metaphysics of powers with respect to properties and laws. (shrink)

We review the relativistic classical and quantum mechanics of Stueckelberg, and introduce the compensation fields necessary for the gauge covariance of the Stueckelbert–Schrödinger equation. To achieve this, one must introduce a fifth, Lorentz scalar, compensation field, in addition to the four vector fields with compensate the action of the space-time derivatives. A generalized Lorentz force can be derived from the classical Hamilton equations associated with this evolution function. We show that the fifth (scalar) field can be eliminated through the introduction (...) of a conformal metric on the spacetime manifold. The geodesic equation associated with this metric coincides with the Lorentz force, and is therefore dynamically equivalent. Since the generalized Maxwell equations for the five dimensional fields provide an equation relating the fifth field with the spacetime density of events, one can derive the spacetime event density associated with the Friedmann–Robertson–Walker solution of the Einstein equations. The resulting density, in the conformal coordinate space, is isotropic and homogeneous, decreasing as the square of the Robertson–Walker scale factor. Using the Einstein equations, one see that both for the static and matter dominated models, the conformal time slice in which the events which generate the world lines are contained becomes progressively thinner as the inverse square of the scale factor, establishing a simple correspondence between the configurations predicted by the underlying Friedmann–Robertson–Walker dynamical model and the configurations in the conformal coordinates. (shrink)

There is a long tradition in economics of evaluating social arrangements by the extent to which individuals' preferences are satisfied. This is the tradition of welfarism, which has developed from nineteenth-century utilitarianism. Increasingly, however, the presumption that preference-satisfaction is the appropriate standard for evaluating social arrangements is being challenged by an alternative view: that we should focus on the set of opportunities open to each individual.

We classify countable metrically homogeneous graphs of diameter 3.

Metrical scholia to the Agamemnon appear in the manuscripts commonly called F and Tr — also in G , but these are the same as some of those in F and are of interest only at one point . In the lyrical passages these scholia are of two types, which I shall call ‘long’ and ‘short’; Professor Eduard Fraenkel, who prints illustrative examples, has pointed out the distinction . ‘Long’ scholia give a comparatively full description of the metre of the (...) passage to which they refer: they state adequately the number and order of the cola, classified according to the metrical system which Demetrius Triclinius learnt from Hephaestion's work. (shrink)

From Einstein's point of view, his General Relativity Theory had strengths as well as failings. For him, its shortcoming mainly was that it did not unify gravitation and electromagnetism and did not provide solutions to field equations which can be interpreted as particle models with discrete mass and charge spectra, As a consequence, General Relativity did not solve the quantum problem, either. Einstein tried to get rid of the shortcomings without losing the achievements of General Relativity Theory. Stimulated by papers (...) of Weyl 465) and Eddington 194), from 1923 onward, he believed that, to reach this goal, one has to transit to space–times which possess more comprehensive geometrical structures than the Riemann space–time. This was the beginning of a decade's lasting search for a unitary field theory. We describe this exciting part of the history of physics, discuss achievements and failures of this development, and ask how these early attempts of a unified theory strike us today. Taking into account the fact that the Equivalence Principle only speaks for a geometrization of gravitation, we consider an alternative way to give those non-Riemannian structures which were introduced by the unitary field approach a physical meaning, namely the meaning of a generalized gravitational field. This is interesting since there are arguments in favor of such a generalization of General Relativity Theory, e.g., the problems the latter theory meets with if one tries to quantize it and to unify gravitation with other interactions. (shrink)

A form of Kripke's schema turns out to be equivalent to each of the following two statements from metric topology: every open subspace of a separable metric space is separable; every open subset of a separable metric space is a countable union of open balls. Thus Kripke's schema serves as a point of reference for classifying theorems of classical mathematics within Bishop-style constructive reverse mathematics.

This paper introduces a concept called task muddiness as a metric for higher intelligence. Task muddiness is meant to be inclusive and expendable in nature. The intelligence required to execute a task is measured by the composite muddiness of the task described by multiple muddiness factors. The composite muddiness explains why many challenging tasks are muddy and why autonomous mental development is necessary for muddy tasks. It facilitates better understanding of intelligence, what the human adult mind can do, and how (...) to build a machine to acquire higher intelligence. The task-muddiness indicates a major reason why a higher biological mind is autonomously developed from autonomous, simple-to-complex experience. The paper also discusses some key concepts that are necessary for understanding the mind and intelligence, such as intelligence metrics, the mode a task is conveyed to the task executor, a human and a machine being a joint task performer in the traditional artificial intelligence (AI), a developmental agent (human or machine) being a sole task performer, and the need for autonomy in task-nonexplicit learning. (shrink)

This paper addresses some critical issues in the applicability of quantitative performance measures (including bibliometric, economic, and co-occurrence measures) to the assessment of basic research. The strengths and weaknesses of metrics applied as research performance measures are examined. It is concluded that metrics have a useful role to play in the evaluation of research. Each metric employed, whether bibliometric, economic, co-occurrence, or others, brings a new dimension of potential insight to the complex problem of research assessment. However, when used in (...) a stand-alone mode, metrics can be easily misinterpreted and result in misleading conclusions. Metrics should be an integral part of a more comprehensive approach to research evaluation, in which the leading role is assumed by expert peer review. Under this structure, the reviewers must be broadly constituted, so that the technical correctness of the research approaches being taken (doing the job right) can be evaluated by technical experts and the correctness of the research target (doing the right job) can be evaluated as well. This comprehensive approach insures quality in the implementation of existing research paradigms and allows the introduction of revolutionary new research paradigms as well. (shrink)