Name der Zeitschrift: Semiotica Jahrgang: 2015 Heft: 204 Seiten: 121-143.

We show how the notion of full Frobenius group of finite Morley rank generalizes that of bad group, and how it seems to be more appropriate when we consider the possible existence (still unknown) of nonalgebraic simple groups of finite Morley rank of a certain type, notably with no involution. We also show how these groups appear as a major obstacle in the analysis of FT-groups, if one tries to extend the Feit-Thompson theorem to groups of finite Morley rank.

The Riemann–Roch theorem for algebraic curves is derived from a theorem for Girard quantales. Serre duality is shown to be a quantalic phenomenon. An example provides a Girard quantale satisfying the Riemann–Roch theorem, where the associated curve is non-connected and irreducible.

We prove some results about the model theory of fields with a derivation of the Frobenius map, especially that the model companion of this theory is axiomatizable by axioms used by Wood in the case of the theory $\operatorname {DCF}_p$ and that it eliminates quantifiers after adding the inverse of the Frobenius map to the language. This strengthens the results from [4]. As a by-product, we get a new geometric axiomatization of this model companion. Along the way we (...) also prove a quantifier elimination result, which holds in a much more general context and we suggest a way of giving “one-dimensional” axiomatizations for model companions of some theories of fields with operators. (shrink)

We prove that the theory of fields with a derivation of Frobenius has the model companion which is stable and admits elimination of quantifiers up to the level of the λ-functions. Along the way, we give new geometric axioms of DCFp.

Charles W. Curtis is a prominent mathematician who has made important contributions to the field of representation theory. His textbooks in this field have been classics for a long time. In Pioneers of Representation Theory he has set out to present the historical development of the main ideas of the discipline, from the work of Georg Ferdinand Frobenius in the 1890s up to 1960. In addition to Frobenius, the book focuses mainly on three other “pioneers”: William Burnside, Issai (...) Schur, and Richard Brauer.Curtis states that his interest in the history of representation theory arose gradually, and jointly with that of Reiner, in the process of writing their textbooks. Thus, a main aim of the book is to show how the pioneers reached their important results while using only the mathematics then available to them. In order to do so, Curtis presents successive and thorough accounts of the works in which the new and seminal ideas of the pioneers were introduced and then developed. At the same time, however, he simplifies some of the original arguments by translating them into modern terminology.In an attempt to make the book appealing and accessible to a wider circle of potential readers, some of the necessary mathematics is explained. Thus Chapter 1 presents an introduction to the basic ideas and problems of nineteenth‐century algebra and number theory; later on, as the need arises, additional mathematical sections addressed to nonexperts are included. Of course, even this very clear presentation will be accessible only to readers with at least a graduate‐level knowledge of algebra.Curtis also attempts to provide some historical context. He is particularly concerned to clarify the main problems that the work of each of the mathematicians discussed here was originally meant to resolve. In a book of this kind this is surely one of the major benefits that the reader may expect to obtain.There are biographical sketches of the major figures involved, as well as additional information about other participants. Some illuminating documents are quoted, and Curtis supplies brief accounts of the mathematical traditions within which each of the pioneers was educated and worked. He acknowledges, however, that Pioneers of Representation Theory has only a limited claim as a work of historical research. He thus provides references to more historically oriented work that interested readers can consult. In particular, he mentions several articles by Thomas Hawkins; it should be stressed that much of Hawkins's work has recently been published in book form: Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics, 1869–1926 . Curtis's and Hawkins's books complement each other in many important ways, and each should appeal to readers of the other.Charles Curtis has written an impressive, authoritative, and well‐informed book on a difficult subject. Mathematicians and historians of twentieth‐century mathematics with the relevant background will find it difficult but rewarding reading. Along the way, historiographical questions may arise. To what extent is it possible to translate, as Curtis does, the original proofs of the pioneers into more modern terminology and yet remain faithful to the sources? In most cases, however, it will require the technical expertise of the author himself to answer questions like this one properly. We can only hope that the firm starting point this book provides will lead Curtis to further historical research in this and related fields. (shrink)

We develop an elimination theory for addition and the Frobenius map over rings of polynomials. As a consequence we show that if F is a countable, recursive and perfect field of positive characteristic p, with decidable theory, then the structure of addition, the Frobenius map x→ xp and the property ‘x∈ F', over the ring of polynomials F[T], has a decidable theory.

The theory of nonnegative matrices is an example of a theory motivated in its origins and development by purely mathematical concerns that later proved to have a remarkably broad spectrum of applications to such diverse fields as probability theory, numerical analysis, economics, dynamical programming, and demography. At the heart of the theory is what is usually known as the Perron–Frobenius Theorem. It was inspired by a theorem of Oskar Perron on positive matrices, usually called Perron’s Theorem. This paper is (...) primarily concerned with the origins of Perron’s Theorem in his masterful work on ordinary and generalized continued fractions (1907) and its role in inspiring the remarkable work of Frobenius on nonnegative matrices (1912) that produced, inter alia, the Perron–Frobenius Theorem. The paper is not at all intended exclusively for readers with expertise in the theory of nonnegative matrices. Anyone with a basic grounding in linear algebra should be able to read this article and come away with a good understanding of the Perron–Frobenius Theorem as well as its historical origins. The final section of the paper considers the first major application of the Perron–Frobenius Theorem, namely, to the theory of Markov chains. When he introduced the eponymous chains in 1908, Markov adumbrated several key notions and results of the Perron–Frobenius theory albeit within the much simpler context of stochastic matrices; but it was by means of Frobenius’ 1912 paper that the linear algebraic foundations of Markov’s theory for nonpositive stochastic matrices were first established by R. Von Mises and V.I. Romanovsky. (shrink)

We introduce a graphical framework for Bayesian inference that is sufficiently general to accommodate not just the standard case but also recent proposals for a theory of quantum Bayesian inference wherein one considers density operators rather than probability distributions as representative of degrees of belief. The diagrammatic framework is stated in the graphical language of symmetric monoidal categories and of compact structures and Frobenius structures therein, in which Bayesian inversion boils down to transposition with respect to an appropriate compact (...) structure. We characterize classical Bayesian inference in terms of a graphical property and demonstrate that our approach eliminates some purely conventional elements that appear in common representations thereof, such as whether degrees of belief are represented by probabilities or entropie quantities. We also introduce a quantum-like calculus wherein the Frobenius structure is noncommutative and show that it can accommodate Leifer's calculus of 'conditional density operators'. The notion of conditional independence is also generalized to our graphical setting and we make some preliminary connections to the theory of Bayesian networks. Finally, we demonstrate how to construct a graphical Bayesian calculus within any dagger compact category. (shrink)

It is shown that the theory of fields with an automorphism has a decidable model companion. Quantifier-elimination is established in a natural language. The theory is intimately connected to Ax's theory of pseudofinite fields, and analogues are obtained for most of Ax's classical results. Some indication is given of the connection to nonstandard Frobenius maps.

New aspects of spectral fluctuations of (quantum) chaotic and diffusive systems are considered, namely autocorrelations of the spacing between consecutive levels or spacing autocovariances. They can be viewed as a discretized two point correlation function. Their behavior results from two different contributions. One corresponds to (universal) random matrix eigenvalue fluctuations, the other to diffusive or chaotic characteristics of the corresponding classical motion. A closed formula expressing spacing autocovariances in terms of classical dynamical zeta functions, including the Perron–Frobenius operator, is (...) derived. It leads to a simple interpretation in terms of classical resonances. The theory is applied to zeros of the Riemann zeta function. A striking correspondence between the associated classical dynamical zeta functions and the Riemann zeta itself is found. This induces a resurgence phenomenon where the lowest Riemann zeros appear replicated an infinite number of times as resonances and sub-resonances in the spacing autocovariances. The theoretical results are confirmed by existing “data.” The present work further extends the already well known semiclassical interpretation of properties of Riemann zeros. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 42.1 (2004) 100-101 [Access article in PDF] Kathy Eden. Friends Hold All Things in Common: Tradition, Intellectual Property, and the Adages of Erasmus. New Haven, CT: Yale University Press, 2001. Pp. ix + 194. Cloth, $35.00. When Erasmus returned from England to the continent in 1500 almost all his money was confiscated before he embarked, although his patron, Lord Mountjoy, had assured him (...) that customs regulations applied only to English currency. Upon his arrival in Paris—poor as a church mouse and feeling rather unwell—he immediately set out writing a new gift for his patron, lest Mountjoy would assume that the humanist was angry with him. A few days later Erasmus had gathered a modest collection of 818 proverbs or adagia, a "treasury or storehouse of the accumulated wealth of classical antiquity collected for common use and thus one of his principal contributions to a cultural program for cooperation," as Kathy Eden defines them (4); not a material but an intellectual wealth, a wisdom transmitted throughout generations and civilizations in the form of proverbs.This Adagiorum Collectanea became a bestseller because it provided both humanists and careerists with numerous bon mots to embellish their (Latin) style and to enhance their arguments. During his first visit to Italy Erasmus spent a few months in the household of the famous humanist and printer Aldus Manutius in Venice. This resulted in a reissue in [End Page 100] 1508 of what was now entitled Adagiorum chiliades, since the collection had been expounded to 3260 proverbs. It opened with 'Friends hold all things in common', an ancient wisdom acknowledging the commonality between friendship and property. Eight more official editions came from the press with an ever-increasing number of adagia, up to 4151 in the final edition, published as volume two of the collected works (Basle: Froben, 1536). Erasmus kept also enlarging the entries themselves, some of them becoming small essays—for instance, the famous Festina lente (Adag. II i 1) or the equally well-known Dulce bellum inexpertis (Adag. IV i 1)—which were even republished separately.In her introduction to Friends Hold All Things in Common Kathy Eden gives a quick survey of the genesis of the Adagia referring her readers for further information to more specific studies on the topic. In keeping with the opening adage she focuses on the ancient philosophical tradition that Erasmus claims as his intellectual inheritance from the past, a tradition rooted in the legal transfer of property. The humanist inherits his understanding of both friendship and property from an ancient philosophical tradition going back to Pythagoras and Plato, as well as Aristotle and Cicero. This pagan tradition is confronted and linked with the concepts of early-Christian writers, such as Tertullian, Origen, Jerome, Basil and Augustine. Not only is there a similarity between some ideas of pagan philosophers and Christian thinkers; they can even be a source of inspiration to pious living. The publication of the Adages also reflected changing attitudes and practices concerning property: thanks to the press of Manutius in Venice (and later of Frobenius in Basle), an increasing number of people was able to appropriate the heritage of classical wisdom, which had hitherto been the intellectual property of only very few. It was precisely in Venice, around the transition from the fifteenth to the sixteenth century, that the first laws regarding intellectual property—be it copyright of the author or protection of the printer's efforts—began to make an occurrence.The author proves to be well-read in both classical, biblical and patrologic sources—I particularly appreciated her combinations and confrontations of pagan and Christian sources. Of course, she is thoroughly familiar with Erasmus's Adagia, too, with an occasional excursion towards his other pedagogical treatises, such as the Antibarbarus, the Enchiridion or the Paraclesis, an adhortation to study Christian philosophy.Friends Hold All Things in Common is a carefully edited publication with hardly any noticeable language or printing errors. As a classicist I feel slightly disappointed... (shrink)

“From Paideuma to Shari’a”. Paideuma, concept developed by Leo Frobenius at the beginning of the XXth century, seems to regain its proper actuality when we try to understand the Islamic civilization. Among the roots which define a paideuma, in the particular case of Islam we identified the shari’a as being the most significant one. Shari`a also brings with it a new dimension: the continuous extension of the paideuma, which overpasses frontiers and national identities. The famous sartrian expression “hell is (...) the other” may occur only if we don`t know the Other enough. (shrink)

L-effect algebras are introduced as a class of L-algebras which specialize to all known generalizations of effect algebras with a \-semilattice structure. Moreover, L-effect algebras X arise in connection with quantum sets and Frobenius algebras. The translates of X in the self-similar closure S form a covering, and the structure of X is shown to be equivalent to the compatibility of overlapping translates. A second characterization represents an L-effect algebra in the spirit of closed categories. As an application, it (...) is proved that every lattice effect algebra is an interval in a right \-group, the structure group of the corresponding L-algebra. A block theory for generalized lattice effect algebras, and the existence of a generalized OML as the subalgebra of sharp elements are derived from this description. (shrink)

A split preorder is a preordering relation on the disjoint union of two sets, which function as source and target when one composes split preorders. The paper presents by generators and equations the category SplPre, whose arrows are the split preorders on the disjoint union of two finite ordinals. The same is done for the subcategory Gen of SplPre, whose arrows are equivalence relations, and for the category Rel, whose arrows are the binary relations between finite ordinals, and which has (...) an isomorphic image within SplPre by a map that preserves composition, but not identity arrows. It was shown previously that SplPre and Gen have an isomorphic representation in Rel in the style of Brauer.The syntactical presentation of Gen and Rel in this paper exhibits the particular Frobenius algebra structure of Gen and the particular bialgebraic structure of Rel, the latter structure being built upon the former structure in SplPre. This points towards algebraic modelling of various categories motivated by logic, and related categories, for which one can establish coherence with respect Rel and Gen. It also sheds light on the relationship between the notions of Frobenius algebra and bialgebra. The completeness of the syntactical presentations is proved via normal forms, with the normal form for SplPre and Gen being in some sense orthogonal to the composition-free, i.e. cut-free, normal form for Rel. The paper ends by showing that the assumptions for the algebraic structures of SplPre, Gen and Rel cannot be extended with new equations without falling into triviality. (shrink)

We first show that a theorem by Cartan that generalizes the Frobenius integrability theorem allows us (given certain conditions) to obtain noncurvature solutions for the differential Bianchi conditions and for higher-degree similar relations. We then prove that there is no algorithmic procedure to determine, for a reasonable restricted algebra of functions on spacetime, whether a given connection form satisfies the preceding conditions. A parallel result gives a version of Gödel's first incompleteness theorem within an (axiomatized) theory of gauge fields.

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 42.1 (2004) 100-101 [Access article in PDF] Kathy Eden. Friends Hold All Things in Common: Tradition, Intellectual Property, and the Adages of Erasmus. New Haven, CT: Yale University Press, 2001. Pp. ix + 194. Cloth, $35.00. When Erasmus returned from England to the continent in 1500 almost all his money was confiscated before he embarked, although his patron, Lord Mountjoy, had assured him (...) that customs regulations applied only to English currency. Upon his arrival in Paris—poor as a church mouse and feeling rather unwell—he immediately set out writing a new gift for his patron, lest Mountjoy would assume that the humanist was angry with him. A few days later Erasmus had gathered a modest collection of 818 proverbs or adagia, a "treasury or storehouse of the accumulated wealth of classical antiquity collected for common use and thus one of his principal contributions to a cultural program for cooperation," as Kathy Eden defines them (4); not a material but an intellectual wealth, a wisdom transmitted throughout generations and civilizations in the form of proverbs.This Adagiorum Collectanea became a bestseller because it provided both humanists and careerists with numerous bon mots to embellish their (Latin) style and to enhance their arguments. During his first visit to Italy Erasmus spent a few months in the household of the famous humanist and printer Aldus Manutius in Venice. This resulted in a reissue in [End Page 100] 1508 of what was now entitled Adagiorum chiliades, since the collection had been expounded to 3260 proverbs. It opened with 'Friends hold all things in common', an ancient wisdom acknowledging the commonality between friendship and property. Eight more official editions came from the press with an ever-increasing number of adagia, up to 4151 in the final edition, published as volume two of the collected works (Basle: Froben, 1536). Erasmus kept also enlarging the entries themselves, some of them becoming small essays—for instance, the famous Festina lente (Adag. II i 1) or the equally well-known Dulce bellum inexpertis (Adag. IV i 1)—which were even republished separately.In her introduction to Friends Hold All Things in Common Kathy Eden gives a quick survey of the genesis of the Adagia referring her readers for further information to more specific studies on the topic. In keeping with the opening adage she focuses on the ancient philosophical tradition that Erasmus claims as his intellectual inheritance from the past, a tradition rooted in the legal transfer of property. The humanist inherits his understanding of both friendship and property from an ancient philosophical tradition going back to Pythagoras and Plato, as well as Aristotle and Cicero. This pagan tradition is confronted and linked with the concepts of early-Christian writers, such as Tertullian, Origen, Jerome, Basil and Augustine. Not only is there a similarity between some ideas of pagan philosophers and Christian thinkers; they can even be a source of inspiration to pious living. The publication of the Adages also reflected changing attitudes and practices concerning property: thanks to the press of Manutius in Venice (and later of Frobenius in Basle), an increasing number of people was able to appropriate the heritage of classical wisdom, which had hitherto been the intellectual property of only very few. It was precisely in Venice, around the transition from the fifteenth to the sixteenth century, that the first laws regarding intellectual property—be it copyright of the author or protection of the printer's efforts—began to make an occurrence.The author proves to be well-read in both classical, biblical and patrologic sources—I particularly appreciated her combinations and confrontations of pagan and Christian sources. Of course, she is thoroughly familiar with Erasmus's Adagia, too, with an occasional excursion towards his other pedagogical treatises, such as the Antibarbarus, the Enchiridion or the Paraclesis, an adhortation to study Christian philosophy.Friends Hold All Things in Common is a carefully edited publication with hardly any noticeable language or printing errors. As a classicist I feel slightly disappointed... (shrink)

Aus ganz unterschiedlichen Blickwinkeln, sei es aus musikhistorischer oder ethnomusikologischer Sicht, umkreisen zehn Beitrage das Thema aMusik - Zu Begriff und Konzepteno. Unter dieses Thema ein internationales Symposion in Berlin zu stellen, das zum Gedenken an Hans Heinrich Eggebrecht (1919-1999) veranstaltet wurde, erschien umso naheliegender, zumal Eggebrecht die Frage aWas ist Musik'o existenziell beruhrte, sie grundierte und sein Forscherleben als Musikwissenschaftler von den fruhen Veroffentlichungen nach dem Zweiten Weltkrieg bis zu den letzten Arbeiten durchzog. Inhalt Bruno Nettl: Was ist Musik? (...) Ethnomusikologische Perspektive Sabine Ehrmann-Herfort: Die Kantate. Musik zwischen asthetischer Autonomie und Ideologie Christian Berger: aMusiko nach Kant Erich Reimer: aMusikanto und aMusiko. Zur Geschichte einer wechselvollen Beziehung Christian Kaden: Musik bei denen, die keine aMusiko haben Hans-Joachim Hinrichsen: Hugo Riemann und der Musikbegriff der Musikwissenschaft Klaus-Jurgen Sachs: Musik - erfahren und erortert durch Franz Rosenzweig (1886-1929) Wolf Frobenius: Gestische Musik. Zur Fruhgeschichte des Begriffs Albrecht von Massow: Tonkorper - ein eigenstandiger Parameter der Wesensbestimmung und der Analyse von Musik Christian Thorau: Symphony in White - Musik als modus der Referenz. (shrink)

Fields of characteristic zero with several commuting derivations can be treated as fields equipped with a space of derivations that is closed under the Lie bracket. The existentially closed instances of such structures can then be given a coordinate-free characterization in terms of differential forms. The main tool for doing this is a generalization of the Frobenius Theorem of differential geometry.

William Burnside was one of the three most important algebraists who were involved in the transformation of group theory from its nineteenth-century origins to a deeper twentieth-century subject. Building on work of earlier mathematicians, they were able to develop sophisticated tools for solving difficult problems. His works are of enormous historical importance; they remain also a source of inspiration and information. The works of his contemporaries, such as Klein, Frobenius, Schur, have been published as also have the works of (...) his immediate successors such as Phillip Hall. All of Burnside's papers are reproduced here, organized chronologically and with a detailed bibliography. Walter Feit has contributed a foreword, and a collection of introductory essays are included to provide a commentary on Burnside's work and set it in perspective along with a modern biography that draws on archive material. This is the first reference volume of Burnside's collected papers, enhanced by a series of critical essays on his work, and is important for group theorists and historians and philosophers of mathematics and fills the gap in this area of literature. (shrink)

The hexagonal structure for ‘the geometry of logical opposition’, as coming from Aristoteles–Apuleius square and Sesmat–Blanché hexagon, is presented here in connection with, on the one hand, geometrical ideas on duality on triangles (construction of ‘companion’), and on the other hand, constructions of tripartitions, emphasizing that these are exactly cases of borromean objects. Then a new case of a logical interest introduced here is the double magic tripartition determining the semi-ring ${\mathcal{B}_3}$ and this is a borromean object again, in the (...) heart of the semi-ring ${{\rm Mat}_{3}(\mathbb{B}_{\rm Alg})}$ . With this example we understand better in which sense the borromean object is a deepening of the hexagon, in a logical vein. Then, and this is our main objective here, the Post-Mal’cev full iterative algebra ${\mathbb{P}_4 = \mathbb{P}(\mathbb{F}_4)}$ of functions of all arities on ${\mathbb{F}_4}$ , is proved to be a borromean object, generated by three copies of ${\mathbb{P}_2}$ in it. This fact is induced by a hexagonal structure of the field ${\mathbb{F}_4}$ . This hexagonal structure is seen as precisely a geometrical addition to standard boolean logic, exhibiting ${\mathbb{F}_4}$ as a ‘boolean manifold’. This structure allows to analyze also ${\mathbb{P}_4}$ as generated by adding to a boolean set of logical functions a very special modality, namely the Frobenius squaring map in ${\mathbb{F}_4}$ . It is related to the splitting of paradoxes, to modified logic, to specular logic. It is a setting for a theory of paradoxical sentences, seen as computations of movements on the bi-hexagonal link among the 12 classical logics on a set of 4 values. (shrink)

We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we establish a partial connection between coarse dimension and transformal transcendence degree in these difference fields.

We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we also discuss the possible connection between coarse dimension and transformal transcendence degree in these difference fields.

We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we also discuss the possible connection between coarse dimension and transformal transcendence degree in these difference fields.

Lue et appréciée par Schelling et Hegel, mais férocement critiquée par Lobeck et les cercles philologiques rationalistes, la Symbolik de Creuzer ne manqua pas d'exercer son influence sur Bachofen et le jeune Nietzsche, jusqu'à Klages, Frobenius, W. Fr. Otto et Kerényi. Point de repère incontournable de toute recherche historique et philologique sensible aux racines mystériques, dionysiaques et orientales de la grécité, la Symbolik a eu le mérite de conjuguer l'érudition typique du XVIII e siècle et une vision métaphysique et (...) religieuse de l'histoire de l'antiquité et de son immense patrimoine symbolique et mythologique. (shrink)

In this paper we study the logic of relational and partial variable sets, seen as a generalization of set-valued presheaves, allowing transition functions to be arbitrary relations or arbitrary partial functions. We find that such a logic is the usual intuitionistic and co-intuitionistic first order logic without Beck and Frobenius conditions relative to quantifiers along arbitrary terms. The important case of partial variable sets is axiomatizable by means of the substitutivity schema for equality. Furthermore, completeness, incompleteness and independence results (...) are obtained for different kinds of Beck and Frobenius conditions. (shrink)

Review of a recently published collection of the complete writings of Suzanne C ésaire, arguing that it is an important moment for the emerging field of Afro-Caribbean philosophy.