The connections between mathematical logic and combinatorics have a rich history. This paper focuses on one aspect of this relationship: understanding the strength, measured using the tools of computability theory and reverse mathematics, of various partition theorems. To set the stage, recall two of the most fundamental combinatorial principles, König's Lemma and Ramsey's Theorem. We denote the set of natural numbers by ω and the set of finite sequences of natural numbers by ω<ω. We also identify each n ∈ ω (...) with its set of predecessors, so n = {0, 1, 2, …, n − 1}. (shrink)

Les musées contemporains sont en pleine mutation, ils passent du statut de « temples de l’art et de la culture » à celui de pourvoyeurs d’activités culturelles et ludiques. Censés instruire, ils s’engagent dans la voie controversée de plaire en vue d’exister, le but ultime étant de drainer le plus de « clients » possible pour assurer la continuation. Et dans la mouvance, ils n’hésitent pas à s’exporter au même titre que les enseignes commerciales vers des cités-nations à la recherche (...) d’une « présence » sur la mappemonde globalisée. Le Louvre-Abou Dabi est un exemple saillant de cette tendance mondialisée, tout comme Guggenheim, il devient un jalon mobile de l’art et de la culture. Dans cette phénoménologie de « l’existence par l’architecture », les œuvres architecturales deviennent des objets-outils de notoriété et de communication, plus inhérents au marketing qu’à la culture. Mais quelle part d’intérêt va pour les collections dans ce nouveau statut du musée ou le bâtiment est à lui seul un gage d’existence et de contemporanéité ? Et dans quelle mesure, la part de responsabilité publique qui incombe aux musées dans le développement culturel de la collectivité, risque-t-elle d’être compromise par les lois du marché?Contemporary museums are rapidly evolving from “temples of art and culture” to providers of cultural and recreational activities. Although supposed to be instructive, they are embarking on a controversial policy of pleasing the public in order to exist, the ultimate aim being to draw as many “customers” as possible to secure their future. And in pursuing these aims, they have been exporting themselves, just like any commercial brand, to nation-cities seeking a “place” on the globalised map. The Abu Dhabi Louvre is a striking example of this global trend, as is the Guggenheim, which has now become a movable landmark in the arts and culture landscape. This phenomenon of “existence through architecture” is turning architectural works into objects as well as tools to enhance communication and notoriety, a development that has more to do with marketing than with culture. But given this new status, where the building itself guarantees the museum’s existence and contemporaneity, how much interest is there in its collections? And to what extent is every museum’s share of public responsibility for collective cultural development likely to compromised by “market forces”? (shrink)

When, why, and how does interpersonal forgiveness occur? These questions guided recent research that compared the relative abilities of empathy versus motivated reasoning models to account for the influence of relationship closeness on interpersonal forgiveness. Consistent support was provided for the Model of Motivated Interpersonal Forgiveness. This model hypothesizes that following relationship transgressions, relationship closeness leads to a desire to maintain a relationship. Desire to maintain a relationship leads to motivated reasoning. And motivated reasoning fosters interpersonal forgiveness. The goal of (...) the present research was to examine two concerns that emerged from the initial support for the Model of Motivated Interpersonal Forgiveness. First, were the measures of motivated reasoning and interpersonal forgiveness conflated, thus reducing the potential for empathy to account for interpersonal forgiveness? Second, did the analytic estimation used reduce the power to detect the mediational role of empathy? The present research examined these questions. When motivated reasoning was measured by thought listings (in addition to the original questionnaire items) and when the analytic estimation provided greater power, the Model of Motivated Interpersonal Forgiveness was replicated. (shrink)

Background. As the development and use of genetic tests have increased, so have concerns regarding the uses of genetic information. Genetic discrimination, the differential treatment of individuals based on real or perceived differences in their genomes, is a recently described form of discrimination. The range and significance of experiences associated with this form of discrimination are not yet well known and are investigated in this study. Methods. Individuals at-risk to develop a genetic condition and parents of children with specific genetic (...) conditions were surveyed by questionnaire for reports of genetic discrimination. A total of 27,790 questionnaires were sent out by mail. Of 917 responses received, 206 were followed up with telephone interviews. The responses were analyzed regarding circumstances of the alleged discrimination, the institutions involved, issues relating to the redress of grievances, and strategies to avoid discrimination. (shrink)

The objective of this study is to incorporate topological space into the realm of n-Cylindrical Fuzzy Neutrosophic Sets (n-CyFNS), which are the most novel type of fuzzy neutrosophic sets. In this paper, we introduce n-Cylindrical Fuzzy Neutrosophic Topological Spaces (n-CyFNTS), n-Cylindrical Fuzzy Neutrosophic (n-CyFN) open sets, and n-CyFN closed sets. We also defined the n-CyFN base, n-CyFN subbase, and some related theorems here.

Recent EEG studies on the early postmortem interval that suggest the persistence of electrophysiological coherence and connectivity in the brain of animals and humans reinforce the need for further investigation of the relationship between the brain’s activity and the dying process. Neuroscience is now in a position to empirically evaluate the extended process of dying and, more specifically, to investigate the possibility of brain activity following the cessation of cardiac and respiratory function. Under the direction of the Center for Healthy (...) Minds at the University of Wisconsin-Madison, research was conducted in India on a postmortem meditative state cultivated by some Tibetan Buddhist practitioners in which decomposition is putatively delayed. For all healthy baseline and postmortem subjects presented here, we collected resting state electroencephalographic data, mismatch negativity, and auditory brainstem response. In this study, we present HB data to demonstrate the feasibility of a sparse electrode EEG configuration to capture well-defined ERP waveforms from living subjects under very challenging field conditions. While living subjects displayed well-defined MMN and ABR responses, no recognizable EEG waveforms were discernable in any of the tukdam cases. (shrink)

A detailed outline is presented of several convergent points of view connecting the self-dual and anti-self-dual fields with their free data. This is done for the Maxwell and for linearized gravity as exemplifying the approaches. The Sparling equation provides one tool of great power and characterizes one approach. The twistor theory of Penrose yields another equally powerful point of view. The links between these two basic approaches given in this paper provide a unification that allows workers and others with (...) interest in this area to proceed more readily toward the goal of understanding the full nonlinear Einstein equations. (shrink)

BackgroundThere is increasing need for peer reviewers as the scientific literature grows. Formal education in biostatistics and research methodology during residency training is lacking. In this pilot study, we addressed these issues by evaluating a novel method of teaching residents about biostatistics and research methodology using peer review of standardized manuscripts. We hypothesized that mentored peer review would improve resident knowledge and perception of these concepts more than non-mentored peer review, while improving review quality.MethodsA partially blinded, randomized, controlled multi-center study (...) was performed. Seventy-eight neurology residents from nine US neurology programs were randomized to receive mentoring from a local faculty member or not. Within a year, residents reviewed a baseline manuscript and four subsequent manuscripts, all with introduced errors designed to teach fundamental review concepts. In the mentored group, mentors discussed completed reviews with residents. Primary outcome measure was change in knowledge score between pre- and post-tests, measuring epidemiology and biostatistics knowledge. Secondary outcome measures included level of confidence in the use and interpretation of statistical concepts before and after intervention, and RQI score for baseline and final manuscripts.ResultsSixty-four residents (82%) completed initial review with gradual decline in completion on subsequent reviews. Change in primary outcome, the difference between pre- and post-test knowledge scores, did not differ between mentored (−8.5%) and non-mentored (−13.9%) residents (p = 0.48). Significant differences in secondary outcomes (using 5-point Likert scale, 5 = strongly agree) included mentored residents reporting enhanced understanding of research methodology (3.69 vs 2.61; p = 0.001), understanding of manuscripts (3.73 vs 2.87; p = 0.006), and application of study results to clinical practice (3.65 vs 2.78; p = 0.005) compared to non-mentored residents. There was no difference between groups in level of interest in peer review (3.00 vs 3.09; p = 0.72) or the quality of manuscript review assessed by the Review Quality Instrument (RQI) (3.25 vs 3.06; p = 0.50).ConclusionsWe used mentored peer review of standardized manuscripts to teach biostatistics and research methodology and introduce the peer review process to residents. Though knowledge level did not change, mentored residents had enhanced perception in their abilities to understand research methodology and manuscripts and apply study results to clinical practice. (shrink)

This collection of essays, offered in honor of the distinguished career of prominent political philosophy professor Clifford Orwin, brings together internationally renowned scholars to provide a wide context and discuss various aspects of the virtue of “humanity” through the history of political philosophy.

This collection of essays, offered in honor of the distinguished career of prominent political philosophy professor Clifford Orwin, brings together internationally renowned scholars to provide a wide context and discuss various aspects of the virtue of “humanity” through the history of political philosophy.

Peter Abelard (1079–1142 ce) was the most wide‐ranging philosopher of the twelfth century. He quickly established himself as a leading teacher of logic in and near Paris shortly after 1100. After his affair with Heloise, and his subsequent castration, Abelard became a monk, but he returned to teaching in the Paris schools until 1140, when his work was condemned by a Church Council at Sens. His logical writings were based around discussion of the “Old Logic”: Porphyry's Isagoge, aristotle'S Categories and (...) On Interpretation and boethius'S textbook on topical inference. They comprise a freestanding Dialectica (“Logic”; probably c.1116), a set of commentaries (known as the Logica [Ingredientibus], c. 1119) and a later (c. 1125) commentary on the Isagoge (Logica Nostrorum Petititoni Sociorum or Glossulae). In a work Abelard called his Theologia, issued in three main versions (between 1120 and c.1134), he attempted a logical analysis of trinitarian relations and explored the philosophical problems surrounding God's claims to omnipotence and omniscience. The Collationes (“Debates,” also known as “Dialogue between a Christian, a Philosopher and a Jew”; probably c.1130) present a rational investigation into the nature of the highest good, in which the Christian and the Philosopher (who seems to be modeled on a philosopher of pagan antiquity) are remarkably in agreement. The unfinished Scito teipsum (“Know thyself,” also known as the “Ethics”; c.1138) analyses moral action. (shrink)

Physical activity, and likely the motivation for it, varies throughout the day. The aim of this investigation was to create a short assessment (CRAVE: Cravings for Rest and Volitional Energy Expenditure) to measure motivation states (wants, desires, urges) for physical activity and sedentary behaviors. Five studies were conducted to develop and evaluate the construct validity and reliability of the scale, with 1,035 participants completing the scale a total of 1,697 times. In Study 1, 402 university students completed a questionnaire inquiring (...) about the want or desire to perform behaviors “at the present moment (right now).” Items related to physical activity (e.g., “move my body”) and sedentary behaviors (e.g., “do nothing active”). An exploratory structural equation model (ESEM) revealed that 10 items should be retained, loading onto two factors (5 each for Move and Rest). In Study 2, an independent sample (n= 444) confirmed these results and found that Move and Rest desires were associated with stage-of-change for exercise behavior. In Study 3, 127 community-residing participants completed the CRAVE at 6-month intervals over two years- two times each session. Across-session interclass correlations (ICC) for Move (ICC = 0.72–0.95) and Rest (ICC = 0.69–0.88) were higher than when they were measured across 24-months (Move: ICC = 0.53; Rest: ICC = 0.49), indicating wants/desires have state-like qualities. In Study 4, a maximal treadmill test was completed by 21 university students. The CRAVE was completed immediately pre and post. Move desires decreased 26% and Rest increased 74%. Changes in Move and Rest desires were moderately associated with changes in perceived physical fatigue and energy. In Study 5, 41 university students sat quietly during a 50-min lecture. They completed the CRAVE at 3 time points. Move increased 19.6% and Rest decreased 16.7%. Small correlations were detected between move and both perceived energy and tiredness, but not calmness or tension. In conclusion, the CRAVE scale has good psychometric properties. These data also support tenets of the WANT model of motivation states for movement and rest (Stults-Kolehmainen et al., 2020a). Future studies need to explore how desires to move/rest relate to dynamic changes in physical activity and sedentarism. (shrink)

A real is called properly n-generic if it is n-generic but not n+1-generic. We show that every 1-generic real computes a properly 1-generic real. On the other hand, if m > n ≥ 2 then an m-generic real cannot compute a properly n-generic real.

Comments on the views on free will offered by B. D. Slife , M. Gergen , R. N. Williams , M. S. Richardson , and G. S. Howard in light of the classical definition of FW as being capable of doing otherwise. It is argued that FW interpretations differ markedly depending on whether they are viewed as due to a process or to contents within some process. 2012 APA, all rights reserved).

A laboratory-testable, solid-state Maxwell demon is proposed that utilizes the electric field energy of an open-gap p-n junction. Numerical results from a commercial semiconductor device simulator (Silvaco International–Atlas) verify primary results from a 1-D analytic model. Present day fabrication techniques appear adequate for laboratory tests of principle.

La cuestión de si los lenguajes intensionalesson más expresivos que los lenguajes nointensionalessurge en el marco de unaperspect i va semánt i ca de l as t eorí as.Desde esta perspectiva, la cuestión esesta. ¿Hay clases modelo que se puedencaracterizar mediante teorías que usanconceptos intensionales que no se puedencaracterizar mediante teorías que no usanconceptos intensionales? Se sugiere unaformulación precisa de esta cuestión, perono se ofrece una respuesta.Para aproxi marse a est a cuest i ón, seresume la teoría de modelos de (...) primerorden [II] y se revisa el enfoque semánticode las teorías que emplea incrementost eóri cos, no i nt ensi onal es, de pri merorden [III].Los i nc r e me nt os t e ór i c os de pr i me rorden se bosquejan pero no se definenr i g u r o s a me n t e [ I V] . E s t e l e n g u a j ei n t e n s i o n a l pr o po r c i o n a e l a p a r a t op a r a a t r i b u i r u s o d e l l e n g u a j e yact i t udes i nt ens i onal es a i ndi vi duoscuyo comportami ento es el obj eto deinvestigación. También proporciona elaparato para habl ar sobre traducci óndel lenguaj e atribuido al lenguaj e delinvestigador.La cuestión inicial se convierte entoncesen si hay clases modelo que se puedanc a r a c t e r i z a r me d i a n t e i n c r e me n t o sintencionales de lógica de primer ordenq ue n o pue de n s e r c a pt ur a do s po rincrementos teóricos no-intensionales [V].The ques t i on of whet her i nt ens i onallanguages are more expressive than nonintensional languages is raised within the framework of a semantic view of theories. Fromthis perspective, the question isthis. Are there model classes that can becharacterized by theories using intensionalconcepts that cannot be characterizedby theories that do not use intensionalconcepts? A precise formulation of thisquestion is suggested, but no answer isgiven.To approach this question, model theory offirst order theories is summarized [II] andthe semantic approach to theories usingnon-intensional, theoretical augmentationsof first order theories is reviewed [III].I n t e n s i o n a l a u g me n t a t i o n s o f f i r s torder theories are sketched [IV] but notr i gor ous l y de f i ne d. Thi s i nt e ns i onall anguage provi des t he apparat us f orattributing language use and intensionalattitudes to individuals whose behavioris the object investigation. It also providesapparatus for talking about translationf r om t he at t r i but e d l anguage t o t heinvestigator’s language.T h e i n i t i a l q u e s t i o n t h e n b e c o me swhet her t here are model cl asses t hatc a n be c ha r a c t e r i z e d by i nt e ns i ona laugmentations of first order logic thatcannot be captured by non-intensionaltheoretical augmentations [V]. (shrink)

BackgroundThe increased risk of developing attention-deficit hyperactivity disorder (ADHD) in extremely preterm infants is well-documented. Better understanding of perinatal risk factors, particularly those that are modifiable, can inform prevention efforts.MethodsWe examined data from the Extremely Low Gestational Age Newborns (ELGAN) Study. Participants were screened for ADHD at age 10 with the Child Symptom Inventory-4 (N = 734) and assessed at age 15 with a structured diagnostic interview (MINI-KID) to evaluate for the diagnosis of ADHD (N = 575). We studied associations (...) of pre-pregnancy maternal body mass index (BMI), pregestational and/or gestational diabetes, maternal smoking during pregnancy (MSDP), and hypertensive disorders of pregnancy (HDP) with 10-year and 15-year ADHD outcomes. Relative risks were calculated using Poisson regression models with robust error variance, adjusted for maternal age, maternal educational status, use of food stamps, public insurance status, marital status at birth, and family history of ADHD. We defined ADHD as a positive screen on the CSI-4 at age 10 and/or meeting DSM-5 criteria at age 15 on the MINI-KID. We evaluated the robustness of the associations to broadening or restricting the definition of ADHD. We limited the analysis to individuals with IQ ≥ 70 to decrease confounding by cognitive functioning. We evaluated interactions between maternal BMI and diabetes status. We assessed for mediation of risk increase by alterations in inflammatory or neurotrophic protein levels in the first week of life.ResultsElevated maternal BMI and maternal diabetes were each associated with a 55–65% increase in risk of ADHD, with evidence of both additive and multiplicative interactions between the two exposures. MSDP and HDP were not associated with the risk of ADHD outcomes. There was some evidence for association of ADHD outcomes with high levels of inflammatory proteins or moderate levels of neurotrophic proteins, but there was no evidence that these mediated the risk associated with maternal BMI or diabetes.ConclusionContrary to previous population-based studies, MSDP and HDP did not predict ADHD outcomes in this extremely preterm cohort, but elevated maternal pre-pregnancy BMI, maternal diabetes, and perinatal inflammatory markers were associated with increased risk of ADHD at age 10 and/or 15, with positive interaction between pre-pregnancy BMI and maternal diabetes. (shrink)

The traditional, dogmatic educational sys tem was reinforced by the addition of science instruction to its curriculum. Three errors are reinforced by this move and the subsequent split of the system into streams. a) Pressure is confused with coercion, b) Interactive study is confused with assigned e x e r c i s e s a n d w i t h s e l f- instruction, and c) Aptitude (disposition) is confused with talent (ability). Reform must begin in the (...) public educational system, at least until experimental schooling becomes the norm; yet research must now repudiate traditional v iews and develop the theory of, and the tools for, free interactive study, aiming at imparting universal literacy, including science literacy. (shrink)

This passage has occasioned, if not proelia, at least divergent interpretations, not to mention instances of tergiversation. In 1910 Cyril Bailey modelled his first rendering of the lines closely on that of H. A. J. Munro: ‘But unless the breast is cleared, what battles and dangers must enter into us in our own despite’; ‘but unless the heart is cleansed, what battles and dangers must then find their way into us in our own despite’. But a reprint of Bailey's translation (...) in 1921 produced the following change: ‘…what battles and perils must we then enter into despite our will’. What changed Bailey's mind? The explanation appeared years later in Bailey's great edition and commentary of 1947. In his note on the passage , he cites objections to Munro's interpretation that had been raised by C. N. Cole and W. R. Hardie. In this paper I will examine these objections and present my own interpretation of the passage, which amounts to a qualified vindication of Munro. (shrink)

The article considers the nature of legalistic, or formal, conceptions of the rule of law, focusing particularly on the work of Joseph Raz and Albert Venn Dicey. It asks how such apparently narrow conceptions are generated, and how far they can resist including broader social claims. It concludes that the rationale behind legalistic conceptions compels them to address issues of poverty and the literacy of the law's subjects. However, legalistic conceptions of the rule of law can still avoid sliding (...) into the wider non‐legalistic models advanced by writers such as T. R. S. Allan. (shrink)

An infinite binary sequence X is Kolmogorov–Loveland random if there is no computable non-monotonic betting strategy that succeeds on X in the sense of having an unbounded gain in the limit while betting successively on bits of X. A sequence X is KL-stochastic if there is no computable non-monotonic selection rule that selects from X an infinite, biased sequence.One of the major open problems in the field of effective randomness is whether Martin-Löf randomness is the same as KL-randomness. Our first (...) main result states that KL-random sequences are close to Martin-Löf random sequences in so far as every KL-random sequence has arbitrarily dense subsequences that are Martin-Löf random. A key lemma in the proof of this result is that for every effective split of a KL-random sequence at least one of the halves is Martin-Löf random. However, this splitting property does not characterize KL-randomness; we construct a sequence that is not even computably random such that every effective split yields two subsequences that are 2-random. Furthermore, we show for any KL-random sequence A that is computable in the halting problem that, first, for any effective split of A both halves are Martin-Löf random and, second, for any computable, nondecreasing, and unbounded function g and almost all n, the prefix of A of length n has prefix-free Kolmogorov complexity at least n−g. Again, the latter property does not characterize KL-randomness, even when restricted to left-r.e. sequences; we construct a left-r.e. sequence that has this property but is not KL-stochastic and, in fact, is not even Mises–Wald–Church stochastic.Turning our attention to KL-stochasticity, we construct a non-empty class of KL-stochastic sequences that are not weakly 1-random; by the usual basis theorems we obtain such sequences that in addition are left-r.e., are low, or are of hyperimmune-free degree.Our second main result asserts that every KL-stochastic sequence has effective dimension 1, or equivalently, a sequence cannot be KL-stochastic if it has infinitely many prefixes that can be compressed by a factor of α<1. This improves on a result by Muchnik, who has shown that were they to exist, such compressible prefixes could not be found effectively. (shrink)

Hertz, J. H. Saadya gaon.--Altmann, A. Saadya's theory of revelation.--Herzog, D. The polemic treatise against Saadya.--Krauss, S. Saadya's Tafsir of the seventy hapax legomena explained and continued.--Leveen, J. Saadya's lost commentary on Leviticus.--Markon, I. explained by Saadya and his successors.--Marmorstein, A. The doctrine of redemption in Saadya's theological system.--Mittwoch, E. An unknown fragment by Gaon Saadya.--Rabin, C. Saadya gaon's Hebrew prose style.--Rawidowicz, S. Saadya's purification of the idea of God.-- Robertson, E. The relationship of the Arabic translation of the Samaritan (...) Pentateuch to that of Saadya.--Rosenthal, E. I. J. Saadya's exegesis of the Book of Job.--Stein, S. Saadya's Piyyuṭ of the alphabet.--Weis, P. R. The anti-Ḳaraite tendency of R. Saadya gaon's Arabic version of the Pentateuch.--Wieder, N. Fourteen new Genizah-fragments of Saadya's Siddur together with a reproduction of a missing part. (shrink)

Erasmus: rechtzinnig of vrijzinnig theoloog, door J. Coppens.--Erasmus: wereldburger en pacifist, door A. Gerlo.--Het humanisme van Erasmus, door J. N. Bakhuizen van den Brink.--Erasmus en de muziek, door R. B. Lenaerts.

The tower number ${\mathfrak t}$ and the ultrafilter number $\mathfrak {u}$ are cardinal characteristics from set theory. They are based on combinatorial properties of classes of subsets of $\omega $ and the almost inclusion relation $\subseteq ^*$ between such subsets. We consider analogs of these cardinal characteristics in computability theory.We say that a sequence $(G_n)_{n \in {\mathbb N}}$ of computable sets is a tower if $G_0 = {\mathbb N}$, $G_{n+1} \subseteq ^* G_n$, and $G_n\smallsetminus G_{n+1}$ is infinite for each n. (...) A tower is maximal if there is no infinite computable set contained in all $G_n$. A tower ${\left \langle {G_n}\right \rangle }_{n\in \omega }$ is an ultrafilter base if for each computable R, there is n such that $G_n \subseteq ^* R$ or $G_n \subseteq ^* \overline R$ ; this property implies maximality of the tower. A sequence $(G_n)_{n \in {\mathbb N}}$ of sets can be encoded as the “columns” of a set $G\subseteq \mathbb N$. Our analogs of ${\mathfrak t}$ and ${\mathfrak u}$ are the mass problems of sets encoding maximal towers, and of sets encoding towers that are ultrafilter bases, respectively. The relative position of a cardinal characteristic broadly corresponds to the relative computational complexity of the mass problem. We use Medvedev reducibility to formalize relative computational complexity, and thus to compare such mass problems to known ones.We show that the mass problem of ultrafilter bases is equivalent to the mass problem of computing a function that dominates all computable functions, and hence, by Martin’s characterization, it captures highness. On the other hand, the mass problem for maximal towers is below the mass problem of computing a non-low set. We also show that some, but not all, noncomputable low sets compute maximal towers: Every noncomputable (low) c.e. set computes a maximal tower but no 1-generic $\Delta ^0_2$ -set does so.We finally consider the mass problems of maximal almost disjoint, and of maximal independent families. We show that they are Medvedev equivalent to maximal towers, and to ultrafilter bases, respectively. (shrink)

Beyond being the first substantial publication on Fourier, this work contains the text of Fourier's seminal paper of 1807 on the propagation of heat, marking the first time it has ever appeared in print. This paper incorporates many of the mathematical creations on which Fourier's fame rests, including derivation of the diffusion equation, the separation of the treatment of surface phenomena from internal phenomena, the use of boundary values and initial conditions, and the development of "Fourier series" and the so-called (...) "Bessel functions."When submitted to the examiners of the Institut de France, the originality of the paper and the surprising nature of some of its mathematical revelations caused great controversy, and it was denied publication both in 1807 and in later years. Fourier had the support, among the examiners, of Laplace and Monge, but Lagrange was adamantly in opposition, so that Fourier's work did not appear in print until 1822, reworked into book form.Fourier's mathematical discoveries are intimately related to his interest in the solution of physical problems and their experimental verification. The mathematical methods he developed in connection with heat diffusion apply to physical situations far beyond the boundaries of this area. Generally, Fourier may be credited with one of the first major extensions of mathematical physics beyond the applications of Newton's laws of motion and universal gravitation.The opening biographical chapter of this book follows Fourier's career up to the submission of the 1807 paper, and the two closing chapters take up his life and work from that point on. Fourier had strong political motivations and spent much of his life in the public service. These chapters trace his political difficulties, both before and after 1807, when he was the prefect of a department of France and was subjected to the dislocations of Napoleon's ups and downs. These chapters also describe aspects of the turbulent but productive development of French science from the Revolution to 1830.The core of the book presents the paper of 1807 in its original French and with the original notation. Grattan-Guinness has divided the paper into sections by the sequence of the problems taken up, and he introduces and, where necessary, closes each section with commentary relevant to Fourier's later work in these areas. The paper itself (cllows the chronology of Fourier's discoveries, and among the topics treated are, in this order: heat diffusion between disjoint bodies and in continuous bodies; the appearance of partial differential equations; the special solution for the lamina; sine and cosine series for an arbitrary function; reflections on the vibrating string problem; solution for the annulus; the full Fourier series for an arbitrary function; reflections on n-body analysis; solution for the sphere; solution for the cylinder; steady-state diffusion in the rectangular prism; time-dependent diffusion in the cube; and Fourier's experimental work. (shrink)

This is a collection of thirteen original essays, of uniformly high quality, on various aspects of Hegel’s political thought. The contributors include younger as well as established Hegel authorities. The opening and concluding pieces are by Z. Pelczynski, who also invited and selected the contributions and served as general editor. The other contributors are John Plamenatz, J.-F. Suter, Judith Shklar, D.-H. Ilting, G. Heiman, Manfred Riedel, D. Cooper, D. P. Verene, W. H. Walsh, R. N. Berki, and Eugène Fleischmann. Their (...) collective effort contributes much to clarifying the wealth of historical, sociological, psychological and philosophical matter that is packed into Hegel’s often obscure theoretical texts on society and polity. Attention is also paid to his less theoretical writings such as The German Constitution and ‘The English Reform Bill’, which helps to clarify the relationship between these and the major theoretical treatises, part III of the Encyclopedia and, especially, the Philosophy of Right. (shrink)

We study the sets that are computable from both halves of some (Martin–Löf) random sequence, which we call 1/2-bases. We show that the collection of such sets forms an ideal in the Turing degrees that is generated by its c.e. elements. It is a proper subideal of the K-trivial sets. We characterize 1/2-bases as the sets computable from both halves of Chaitin’s Ω, and as the sets that obey the cost function c(x,s)=Ωs−Ωx−−−−−−−√. Generalizing these results yields a dense hierarchy of (...) subideals in the K-trivial degrees: For k<n, let Bk/n be the collection of sets that are below any k out of n columns of some random sequence. As before, this is an ideal generated by its c.e. elements and the random sequence in the definition can always be taken to be Ω. Furthermore, the corresponding cost function characterization reveals that Bk/n is independent of the particular representation of the rational k/n, and that Bp is properly contained in Bq for rational numbers p<q. These results are proved using a generalization of the Loomis–Whitney inequality, which bounds the measure of an open set in terms of the measures of its projections. The generality allows us to analyze arbitrary families of orthogonal projections. As it turns out, these do not give us new subideals of the K-trivial sets; we can calculate from the family which Bp it characterizes. We finish by studying the union of Bp for p<1; we prove that this ideal consists of the sets that are robustly computable from some random sequence. This class was previously studied by Hirschfeldt [D. R. Hirschfeldt, C. G. Jockusch, R. Kuyper and P. E. Schupp, Coarse reducibility and algorithmic randomness, J. Symbolic Logic81(3) (2016) 1028–1046], who showed that it is a proper subclass of the K-trivial sets. We prove that all such sets are robustly computable from Ω, and that they form a proper subideal of the sets computable from every (weakly) LR-hard random sequence. We also show that the ideal cannot be characterized by a cost function, giving the first such example of a Σ03 subideal of the K-trivial sets. (shrink)

My subject on this occasion goes uneasily with my piety. Lordship did not as such much interest my teachers William E. Lunt and Joseph R. Strayer, who were leaders in their turn of the Medieval Academy of America. In their presidential addresses of 1954 and 1968 both scholars dealt magisterially with subjects each had studied for forty years. Lunt spoke on financial relations of the papacy with England, Strayer on the place of Normandy and Languedoc in the building of (...) an administrative monarchy in France. It is easy to see that in a considerable sense both historians were working on medieval government, and it is some measure of their achievement that even today no one wishing to be informed in this grand subject can do much better than read Lunt and Strayer. I wish I could discourse on my own years of research as did those admirable scholars. If I cannot, it is not simply because I lack the accumulated erudition to reflect summarily on my work; it is also because I lack the conceptual serenity of my teachers in our common subject matter. Like Lunt and Strayer I have worked on justice, finance, and the beginnings of parliamentary life in the twelfth and thirteenth centuries; unlike those mentors I became uneasy about the paradigm in which these studies were framed. I came to wonder how Strayer could have held students, myself included, spellbound with details about the deportment of judges or sergeants while saying nothing about the adequacy of concepts like “government” or “administration” to describe such behavior. Believing there was something timeless about the action of people in power, he was gifted at evoking the ways of modern bureaucrats in those “corridors of power” he knew from his own experience in Washington. He assumed with other historians of his day and ours that all societies have governments, so that he could speak of feudalism as essentially a “method of government,” of “political organization … reduced to the simplest possible terms.”. (shrink)

A [1983] review, 'Relativity before Einstein' made no mention of the work of Joseph Larmor, whose early derivation of the Lorentz transformation seems to be less well known than those of Lorentz and Poincare. In 1897, Larmor, starting from a first-order transformation similar to Lorentz's first order version, presented the correct form of what is now known as the Lorentz transformation. In his presentation of the theory in 1900 Larmor saw the time dilation effect as a consequence of (...) class='Hi'>Maxwell's electromagnetic theory. It was Lorentz who, in 1895, introduced the notion of the relativity of simultaneity (local time), without the time dilation effect. Poincare in 1900 discussed how Lorentz's local time would arise from the procedure of synchronizing moving clocks by exchanging light signals assumed to travel at the same speed in either direction. Lorentz presented the correct version of the transformation in 1899, and discussed the variation of mass with velocity arising from it. In 1902 Lorentz was aware of Larmor's 1897 work but apparently missed its significance. Nevertheless, the credit for the first presentation of the Lorentz transformation including the crucial time dilation belongs to Larmor. (shrink)

We consider the question of randomness of the probability ΩU[X] that an optimal Turing machine U halts and outputs a string in a fixed set X. The main results are as follows: ΩU[X] is random whenever X is $\Sigma _{n}^{0}$-complete or $\Pi _{n}^{0}$-complete for some n ≥ 2. However, for n ≥ 2, ΩU[X] is not n-random when X is $\Sigma _{n}^{0}$ or $\Pi _{n}^{0}$ Nevertheless, there exists $\Delta _{n+1}^{0}$ sets such that ΩU[X] is n-random. There are $\Delta _{2}^{0}$ sets (...) X such that ΩU[X] is rational. Also, for every n ≥ 1, there exists a set X which is $\Delta _{n+1}^{0}$ and $\Sigma _{n}^{0}$-hard such that ΩU[X] is not random. We also look at the range of ΩU as an operator. We prove that the set {ΩU[X]: X ⊆ 2<ω} is a finite union of closed intervals. It follows that for any optimal machine U and any sufficiently small real r, there is a set X ⊆ 2<ω recursive in ∅′ ⊕ r, such that ΩU[X] = r. The same questions are also considered in the context of infinite computations, and lead to similar results. (shrink)

8.3 The consistency proof -- 8.4 Applications of the consistency proof -- 8.5 Second-order arithmetic -- Problems -- Chapter 9: Set Theory -- 9.1 Axioms for sets -- 9.2 Development of set theory -- 9.3 Ordinals -- 9.4 Cardinals -- 9.5 Interpretations of set theory -- 9.6 Constructible sets -- 9.7 The axiom of constructibility -- 9.8 Forcing -- 9.9 The independence proofs -- 9.10 Large cardinals -- Problems -- Appendix The Word Problem -- Index.