The Emergence of Analysis in the Renaissance and After

Paper by Salomon Bochne

Hypermaps: constructions and operations

It is conjectured that given positive integers l, m, n with l-1 + m-1 + n-1 < 1and an integer g ≥ 0, the triangle group Δ = Δ (l, m, n) = ⟨X,Y,Z|X l = Y m =Z n = X Y Z = 1⟩ contains infinitely many subgroups of finite index and of genusg. This conjecture can be rewritten in another form: given positive integers l,m, n with l¡1 +m¡1 +n¡1 < 1 and an integer g ≥ 0, there are infinitely manynonisomorphic compact orientable hypermaps of type (l, m, n) and genus g.We prove that the conjecture is true, when two of the parameters l, m, n areequal, by showing how to construct those hypermaps, and we extend the resultto nonorientable hypermaps.A classification of all operations of finite order in oriented hypermaps isgiven, and a detailed study of one of these operations (the duality operation)is developed. Adapting the notion of chirality group, the duality group ofH can be defined as the minimal subgroup D(H) ≤¦ M on (H) such thatH = D (H) is a self-dual hypermap. We prove that for any positive integer d,we can find a hypermap of that duality index (the order of D (H) ), even whensome restrictions apply, and also that, for any positive integer k, we can find anon self-dual hypermap such that |Mon (H) | = d = k. We call this k the dualitycoindex of the hypermap. Links between duality index, type and genus of aorientably regular hypermap are explored.Finally, we generalize the duality operation for nonorientable regular hypermaps and we verify if the results about duality index, obtained for orientably regular hypermaps, are still valid

Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations

We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Regarding regularity, we show that the RG map, defined on a suitable space of interactions (= formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the pathological side, we make rigorous some arguments of Griffiths, Pearce and Israel, and prove in several cases that the renormalized measure is not a Gibbs measure for any reasonable interaction. This means that the RG map is ill-defined, and that the conventional RG description of first-order phase transitions is not universally valid. For decimation or Kadanoff transformations applied to the Ising model in dimension $d \ge 3$, these pathologies occur in a full neighborhood $\{ \beta > \beta_0 ,\, |h| < \epsilon(\beta) \}$ of the low-temperature part of the first-order phase-transition surface. For block-averaging transformations applied to the Ising model in dimension $d \ge 2$, the pathologies occur at low temperatures for arbitrary magnetic-field strength. Pathologies may also occur in the critical region for Ising models in dimension $d \ge 4$. We discuss in detail the distinction between Gibbsian and non-Gibbsian measures, and give a rather complete catalogue of the known examples. Finally, we discuss the heuristic and numerical evidence on RG pathologies in the light of our rigorous theorems.Comment: 273 pages including 14 figures, Postscript, See also ftp.scri.fsu.edu:hep-lat/papers/9210/9210032.ps.

Finitism--an essay on Hilbert's programme

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 1991.Includes bibliographical references (p. 213-219).by David Watson Galloway.Ph.D

“TEACHING REAL NUMBERS IN THE HIGH SCHOOL: AN ONTO-SEMIOTIC APPROACH TO THE INVESTIGATION AND EVALUATION OF THE TEACHERS' DECLARED CHOICES”

The thesis addresses the topics of investigating teachers' declared choices of practices concerning real numbers and the continuum in the high school in Italy, evaluating their didactical suitability and the impact of a deep reflexion about some historical and didactical issues on the teachers' decision-making process. Our research hypothesis was that teachers' choices of teaching sequences concerning real numbers, with particular attention to the representations of real numbers, could be very relevant in order to interpret some of the well-known students' difficulties. After a pilot study in form of a teaching experiment and a literature review concerning students' and teachers' difficulties with real numbers and the continuum, we observed that some causes of potential difficulties could be situated indeed in the very beginning of the teaching-learning process, even before entering the classrooms: the phase in which a teacher choose the practices and objects by means of whom introducing and work with real numbers and the continuum. In particular the choice of the objects involved in the practice seemed to be relevant, since every object emerge from previous practices and its meaning is identified by the practices in which it emerged. Thus we got interested in investigating the personal factors that affect the process of selection of practices: personal meaning, goals and orientations, as it was stressed by Schoenfeold in his goal-oriented decision-making approach to the analysis of teachers choices. Furthermore we decided to explore the teachers' choices of sequences of practices and of representation of the mathematical objects and then to evaluate their suitability in relation to the literature review concerning students' difficulties with real numbers and to the complexity of the mathematical object as it emerge from an historical analysis. After having analysed the theoretical frameworks in mathematics education that could permit us to carry out our research, we decided to use the OSA, (onto-semiotioc approach) elaborated by Godino, Batanero & Font, described in their paper in 2007, and some evolutions like the CDM (conoscimiento didactico matematico) model proposed by Godino in 2009. We evaluated also other frameworks, in particular the ATD (Chevallard, 2014), but we found the OSA better for the analysis we would like to carry out. In particular the operationalization of the methodologies of analysis of the teachers' personal meaning of mathematical objects and the construct of didactical suitability were more effective for our porpouses. Our main results are the following: mny teachers' personal meanings of real numbers are far from the epistemic one; many of the teachers who studied real numbers at a formal level at school and at the University and perceived them as difficult and unuseful try to avoid to deepen the issues concerning real numbers with their stundent, thinking they would not understand; in general the experiences as students affect the teachers' choices; the teachers usually refer to real numbers also when the meaning is partial and doesn't coincide with one of the most general epistemic meanings of real numbers; very few teachers are aware of the complexity of the real numbers and are as aware of it to be able to control the relations between their many facets; also the teachers with a PhD in Mathematics operate choices that we can evaluate as unsuitable standing on the literature review and our framework; the teacher consider very hard to work with discrete and dense sets and prefer the intuitive approach to continuous sets rather then deepen the relation between dense and continuous sets, different degrees of infinity and so on; some teachers reasoning during the interviews changed their mind, getting aware of the complexity and admitting that simplifying too much can constitute a further cause of difficulty; the teachers refer to the students difficulties to justify their choice of simplifying, but when they face some crucial issues, often they admit to consider them unuseful or too difficult; nevertheless no teachers declare that would renounce to introduce the field of real numbers, at least intuitively; the most of the teachers declare that nothing more is introduced about real numbers in the last years and that the partial meanings introduced in the first years are used to face the Calculus problems (intuItive approach to the Calculus); all the teachers consider necessary to introduce R or adequate subsets of R as domains of the functions expressed analytically because of their continuous graphic.The thesis addresses the topics of investigating teachers' declared choices of practices concerning real numbers and the continuum in the high school in Italy, evaluating their didactical suitability and the impact of a deep reflexion about some historical and didactical issues on the teachers' decision-making process. Our research hypothesis was that teachers' choices of teaching sequences concerning real numbers, with particular attention to the representations of real numbers, could be very relevant in order to interpret some of the well-known students' difficulties. After a pilot study in form of a teaching experiment and a literature review concerning students' and teachers' difficulties with real numbers and the continuum, we observed that some causes of potential difficulties could be situated indeed in the very beginning of the teaching-learning process, even before entering the classrooms: the phase in which a teacher choose the practices and objects by means of whom introducing and work with real numbers and the continuum. In particular the choice of the objects involved in the practice seemed to be relevant, since every object emerge from previous practices and its meaning is identified by the practices in which it emerged. Thus we got interested in investigating the personal factors that affect the process of selection of practices: personal meaning, goals and orientations, as it was stressed by Schoenfeold in his goal-oriented decision-making approach to the analysis of teachers choices. Furthermore we decided to explore the teachers' choices of sequences of practices and of representation of the mathematical objects and then to evaluate their suitability in relation to the literature review concerning students' difficulties with real numbers and to the complexity of the mathematical object as it emerge from an historical analysis. After having analysed the theoretical frameworks in mathematics education that could permit us to carry out our research, we decided to use the OSA, (onto-semiotioc approach) elaborated by Godino, Batanero & Font, described in their paper in 2007, and some evolutions like the CDM (conoscimiento didactico matematico) model proposed by Godino in 2009. We evaluated also other frameworks, in particular the ATD (Chevallard, 2014), but we found the OSA better for the analysis we would like to carry out. In particular the operationalization of the methodologies of analysis of the teachers' personal meaning of mathematical objects and the construct of didactical suitability were more effective for our porpouses. Our main results are the following: mny teachers' personal meanings of real numbers are far from the epistemic one; many of the teachers who studied real numbers at a formal level at school and at the University and perceived them as difficult and unuseful try to avoid to deepen the issues concerning real numbers with their stundent, thinking they would not understand; in general the experiences as students affect the teachers' choices; the teachers usually refer to real numbers also when the meaning is partial and doesn't coincide with one of the most general epistemic meanings of real numbers; very few teachers are aware of the complexity of the real numbers and are as aware of it to be able to control the relations between their many facets; also the teachers with a PhD in Mathematics operate choices that we can evaluate as unsuitable standing on the literature review and our framework; the teacher consider very hard to work with discrete and dense sets and prefer the intuitive approach to continuous sets rather then deepen the relation between dense and continuous sets, different degrees of infinity and so on; some teachers reasoning during the interviews changed their mind, getting aware of the complexity and admitting that simplifying too much can constitute a further cause of difficulty; the teachers refer to the students difficulties to justify their choice of simplifying, but when they face some crucial issues, often they admit to consider them unuseful or too difficult; nevertheless no teachers declare that would renounce to introduce the field of real numbers, at least intuitively; the most of the teachers declare that nothing more is introduced about real numbers in the last years and that the partial meanings introduced in the first years are used to face the Calculus problems (intutive approach to the Calculus); all the teachers consider necessary to introduce R or adequate subsets of R as domains of the functions expressed analytically because of their continuous graphic

Course Catalogue of the Massachusetts Institute of Technology 1959 - 1960

Massachusetts Institute of Technology Bulletin, General Catalogue Issue 1959-1960. Includes information about the Institute; its government; staff; regulations; requirements for admission; facilities; and courses of instruction for both undergraduate and graduate students. This edition also includes two appendices on student aid and prizes, and student housing; as well as an index of faculty and staff. Digitized from microfiche copies. Digital version may contain microfiche headers and targets

1971-1972 CATALOG ISSUE- BULLETIN

Course catalog for 1971-1972https://digitalrepository.unm.edu/course_catalogs/1079/thumbnail.jp