Academic literature on the topic '(potential) infinite divisibility'

We determine all 2- and 3-cocycles for Laver tables, an infinite sequence of finite structures obeying the left-self-distributivity law; in particular, we describe simple explicit bases. This provides a number of new positive braid invariants and paves the way for further potential topological applications. An important tool for constructing a combinatorially meaningful basis of 2-cocycles is the right-divisibility relation on Laver tables, which turns out to be a partial ordering.

Selberg and Morris integral probability distributions are long conjectured to be distributions of the total mass of the Bacry–Muzy Gaussian Multiplicative Chaos measures with non-random logarithmic potentials on the unit interval and circle, respectively. The construction and properties of these distributions are reviewed from three perspectives: Analytic based on several representations of the Mellin transform, asymptotic based on low intermittency expansions, and probabilistic based on the theory of Barnes beta probability distributions. In particular, positive and negative integer moments, infinite factorizations and involution invariance of the Mellin transform, analytic and probabilistic proofs of infinite divisibility of the logarithm, factorizations into products of Barnes beta distributions, and Stieltjes moment problems of these distributions are presented in detail. Applications are given in the form of conjectured mod-Gaussian limit theorems, laws of derivative martingales, distribution of extrema of [Formula: see text] noises, and calculations of inverse participation ratios in the Fyodorov–Bouchaud model.

When the slowness surface S of an anisotropic elastic medium consists of three concentric ellipsoids, solutions of the displacement equations of motion can be generated from functions satisfying scalar wave equations and the problem of constructing the fundamental, or Green’s, tensor G for an infinite region becomes tractable. This paper has two aims: first, to find all the conditions on the linear elastic moduli under which S is ellipsoidal (that is the union of concentric ellipsoids), and, second, to determine G for each case in which S simplifies in this way. The two stages of the investigation have a key idea in common. The ellipsoidal form of S requires the eigenvalues of the acoustical tensor Q ( n ) to be quadratic forms in the unit vector argument n : at least two of the associated eigenvectors are either constant or linear in n and the squared moduli of the linear eigenvectors are divisors of eigenvalue differences. These algebraic properties provide a classification of media with ellipsoidal slowness surfaces and aid in characterizing the membership of each class. The first stage culminates in four sets of conditions, labelled A, B, C(i) and C(ii): case C(i) is a restriction of transverse isotropy and the others are specializations of orthorhombic symmetry. At the second stage n is replaced by the gradient ∂ with respect to spatial position and polynomials in n become differential operators. The construction of G involves two canonical problems of classical type, an initial-value problem for a scalar wave equation and a potential problem for a pair of ‘ charged ’ ellipsoids. The divisibility property indicated above implies that the ellipsoids are confocals carrying equal and opposite charges and these characteristics render the fundamental solution causal in the sense that the entire disturbance excited by the point impulse begins with the first and ends with the last of the wavefront arrivals. The structures of the fundamental solutions in cases A, B, C(i) and C(ii) are described and the latter solution is shown to reduce to a standard result of Stokes in the degenerate case of isotropy. Mention is also made of a specialization of case B, appropriate to a transversely isotropic medium which is inextensible in the direction of the symmetry axis.

La thèse porte sur l'un des sujets les moins étudiés de la philosophie de la nature aristotélicienne latine médiévale (ca. 1250-ca. 1350), à savoir le soi-disant sujet des "minima sensibilia". Si, comme il est affirmé notamment dans "Physique" VI, les grandeurs sont infiniment divisibles en puissance, un dilemme se pose quant aux limites de divisibilité des qualités sensibles à travers la division de la matière (considérée comme une grandeur étendue) à laquelle elles sont unies. Soit les qualités sensibles sont aussi infiniment divisibles en puissance (mais cela implique que les sens doivent avoir un pouvoir infini pour les percevoir, contrairement à un présupposé aristotélicien fondamental concernant les limites de tout pouvoir existant dans la nature), soit elles ne sont pas infiniment divisibles en puissance (dans ce cas, cependant, il y aurait des portions de matière qui ne peuvent être connues ni par les sens ni, évidemment, par l'intellect, et, ce qui est pire, les entités sensibles seraient finalement composées par elles, ce qui est tout à fait inacceptable dans la vision du monde aristotélicienne). Pour résoudre le dilemme, Aristote, au chapitre 6 du "De sensu et sensato" (445b3-446a20), fait usage de la distinction entre acte et puissance, affirmant que les qualités sensibles sont infiniment divisibles en puissance en tant que parties du tout auquel elles appartiennent, mais qu'il y a des quantités minimales de matière qui peuvent exister en acte par elles-mêmes douées de leurs qualités sensibles. La thèse examine la réflexion menée par les commentateurs latins médiévaux au "De sensu et sensato" (toujours lus en relation avec leurs sources grecques et islamiques) sur le sujet des "minima sensibilia", en l'utilisant comme une perspective privilégiée pour étudier à partir d'un point de vue nouveau et original la conception latine médiévale de l'ontologie et de l'épistémologie des qualités sensibles. En effet, à travers un examen attentif du débat (qui s'accompagne d'une reconstruction approfondie de la tradition manuscrite des commentaires latins médiévaux au "De sensu", qui ont jusqu'à présent été largement négligés par les chercheurs), il est démontré que les commentateurs latins médiévaux développèrent progressivement une conception selon laquelle les qualités sensibles peuvent exister par elles-mêmes dans le monde naturel sans être perceptibles en acte en raison de la petitesse de la matière à laquelle elles sont unies. De telles qualités sensibles (que l'on appelle parfois "insensibilia propter parvitatem") peuvent néanmoins devenir perceptibles en acte en s'unissant les unes aux autres. Grâce à ce développement fondamental, non seulement les qualités sensibles commencèrent à être comprises dans une large mesure indépendamment de leur rôle dans la perception, mais le monde sensible devint soudainement beaucoup plus étendu que le monde perceptible par les sens, avec pour conséquence que la confiance en la capacité humaine à connaître sa structure ultime a commença à se désintégrer
The thesis focuses on one of the least studied topics in Medieval Latin Aristotelian natural philosophy (ca. 1250-ca. 1350), i.e., the so-called topic of "minima sensibilia". If, as claimed most notably in "Physics" VI, magnitudes are (potentially) infinitely divisible, a dilemma arises with respect to the limits of the divisibility of sensible qualities through the division of the matter (considered as an extended magnitude) with which they are united. Either sensible qualities are also (potentially) infinitely divisible (but this implies that the senses should have an infinite power in order to perceive them, against a fundamental Aristotelian assumption concerning the limits of every power existing in nature), or they are not (potentially) infinitely divisible (in this case, however, there would be portions of matter that can neither be cognised by the senses nor, evidently, by the intellect, and, what is worse, sensible entities would be ultimately composed of them, something entirely unacceptable in the Aristotelian worldview). To solve the dilemma, Aristotle, in Chapter 6 of the "De sensu et sensato" (445b3-446a20), makes use of the distinction between act and potency, affirming that sensible qualities are infinitely divisible in potency as part of the whole to which they belong, but there are minimal quantities of matter that can exist in act on their own endowed with their sensible qualities. The thesis investigates the reflection conducted by Medieval Latin commentators of the "De sensu et sensato" (always read in connection with their Greek and Islamic sources) on the subject of "minima sensibilia", using it as a privileged gateway to study from a new and original point of view the Medieval Latin conception of the ontology and of the epistemology of sensible qualities. Indeed, through a close scrutiny of the debate (which is accompanied by a thorough reconstruction of the complex manuscript tradition of Medieval Latin "De sensu" commentaries, that have hitherto been largely neglected by scholars) it is demonstrated that Medieval Latin commentators progressively developed a conception according to which sensible qualities can exist on their own in the natural world without being perceptible in act due to the smallness of the matter with which they are united. Such sensible qualities (that are sometimes called "insensibilia propter parvitatem") can, nevertheless, become perceptible in act by uniting with each other. Thanks to this fundamental development, not only sensible qualities started to be understood mostly in autonomy from their role in perception, but the sensible world became suddenly much more extended than the world that can be perceived by the senses, with the consequence that the confidence in the human ability to cognise its ultimate structure began to crumble

This paper argues that, for Ockham, the parts of the continuum exist in act in the continuum: they are already there before any division of the continuum. Yet, they are infinitely many in that no division of the continuum will exhaust all the existing parts of the continuum taken conjointly. This reading of Ockham takes into account the crucial place of his new concept of the infinite in his analysis of the infinite divisibility of the continuum. Like many of his fellow anti-atomists, Ockham stresses that the concept of a potential infinite seems to contradict Aristotle’s modal logic, in particular the central assumption that there is no potency that will never be realized. Ockham, like other fourteenth-century anti-atomists, tried not only to refute atomism, but also to propose an analysis of the infinite divisibility of the continuum that is not incompatible with their modal logic.