The Mathematical Heritage of C.F. Gauss: A Collection of Papers in Memory of C.F. Gauss

George M. Rassias

World Scientific Pub Co Inc | 1991 | 916 páginas | rar - pdf |27,75 Mb

link (password : matav)

A collection of of original and expository papers in the fields of mathematics in which Gauss has made many fundamental discoveries.

Contents
Preface ix
A summary of C. F. Gauss's life and mathematical work 1
G. M. Rassias
On Gauss' differential equation and its twenty-four Rummer's solutions 12
M. A. Al-Bassam
On the Diophantine equation (X + Y)A = ZB 32
L. J. Alex
Partitions and the Gaussian sum 35
G. E. Andrews
The periodicity of an algorithm over the complex number field (ACF) and the solution of the Hermite problem 43
M. Baica
Lorentzian distance and curvature 53
J. K. Beem
The decomposition theory and its applications 66
N. Bokan
Spaces of morphisms between algebraic schemes 100
P. Cherenack, L. Guerra
Uniqueness of solutions of electromagnetic interaction problems associated with scattering by bianisotropic bodies covered with impedance sheets 119
D. K. Cohoon
On the topological structure of curves 133
P. J. Collins
Gauss, Bayes, Kalman: State-space models 137
P. Daffer
Waves, quanta, E = mc2, and perihelion shifts: A new sciencehistorical perspective and maathematical unification 157
K. Demys
Uniformly continuous multi-valued mappings 172
D. Doitchinov
Computers in algebraic topology 179
E. Dominguez and J. Rubio
Factorization in quaternion orders over number fields 195
D. R. Estes
Thue inequalities with a small number of solutions 204
J.-H. Evertse and K. Gyory
A load balanced algorithm for the calculation of the polynomial knot and link invariants 225
B. Ewing and K. C. Millett
Convex polyhedral models for the finite three-dimensional isometry groups 267
L. L. Foster
The Lorentzian modular group and nonlinear lattices 282
G. J. Fox and P. E. Parker
Integral operators for harmonic functions 304
A. Fry ant
Quadrature and harmonic L1 -approximation 321
M. Goldstein
The Theorema Egregium of Gauss from a viewpoint of partial differential equations 326
C. K. Han
A new cosine functional equation 334
H. Haruki
On paralindelof and metalindelof spaces 342
H. Z. Hdeib
The Maxwell condition in Friedmann cosmology 349
C. D. Hill
Space-time compactification and Riemannian submersions 358
S. Ianus and M. Visinescu
Independence and scrambled sets for chaotic mappings 372
A. Iwanik
A recent modification of iterative methods for solving nonlinear problems 379
A. J. Jerri
Expansions of special cases of Gauss' hypergeometric functions using generalized calculus 405
R. N. Kalia
Gauge-natural operators transforming connections to the tangent bundle 416
I. Koldf
Returns under bounded number of iterations 427
Z. S. Kowalski
On a structural scheme of physical theories proposed by E. Tonti 432
E. A. Lacomb and F. Ongay
Archimedes versus Gauss: The construction of a regular heptagon 454
/. F. Lamb, Jr.
Some properties of covariant operators in Gauge theories 458
K. B. Marathe and G. Giachetta
A signal discriminator , 479
F. McNolty, W. Sherwood, and J. Mirra
Powers of 2, continued fractions, and the class number one problem for real quadratic fields Q(Vd), with d = 1 (mod 8) 505
R. A. Mollin and H. C. Williams
The validity of Gaussian electrodynamics 517
P. Moon, D. E. Spencer, S. Y. Uma, and P, J. Mann
Dimensions, fractals, and sphere packing 526
C. Muses
Second cohomology spaces and flexible Lie-admissible algebras 544
H. C. Myung and A. A. Sagle
New Fibonacci and Lucas identities 562
S. A. Obaid
Hyper-Kahler metrics and monopoles 573
H. Pedersen
On certain mathematical problems connected with the use of the complex variable boundary element method to the problems of plane hydrodynamics, Gauss' variant of the procedure 585
T. Peirila
A survey on the Poincare conjecture of the topology of 3-manifolds 605
G. M. Rassias
Adjoint connections on group manifolds and gauge transformations 621
H. Rund
An abstract fixed-point theorem of Vanderbauwhede-VanGils type 645
K. P. Rybakowski
Formulas for higher-order finite expansions of composite maps 652
K. P. Rybakowski
Infinite order differential operators in generalized Fock spaces 670
J. Schmeelk
Gauss and the electrodynamic force 685
D. E. Spencer and S. Y. Uma
Some operational techniques in the theory of generalized Gaussian and Clausenian functions 712
H. M. Srivastava
Gauss' gamma multiplication theorem: Analogues and extensions 733
K. B. Stolarsky
Some properties of hereditarily locally connected continua related to the Hann-Mazurkiewicz theorem 758
L. B. Treybig
Convex nonholonomic hyper surfaces 769
C. Udri§te, 0. Dogaru
The measure of covering the Euclidean space by group-translates of a set 785
B. Uhrin
A non-Archimedean number field and its applications in modern physics 810
S. T. Wang
The differential geometry of two types of almost contact metric submersions 827
B. Watson
Hypersurface with constant mean curvature in # n + 1 862
B. Q. Wu
The Poincare density 872
S. Yamashita
Sophie Germain primes 882
5. Yates
On sampling theorems and the Gauss-Jacobi mechanical quadrature 887
A. I. Zayed
Author index 901

Classics in the History of Greek Mathematics

Jean Christianidis

Springer | 2004 | 463 páginas | pdf | 4 Mb

link

This volume includes a selection of 19 classic papers on the history of Greek mathematics that were published during the 20th century and affected significantly the state of the art of this field. It is divided into six thematic sections and covers all the major issues of the Greek mathematical production. First, the inclusion in one volume of a considerable number of papers that had been published for the first time in old, and in certain cases hard to find, scientific journals representing turning-points in the history of the field, constitutes a particularly useful aid for all those working on the history of mathematics. Second, by means of the selected papers and the introductory texts of six well-known modern historians of ancient mathematics that accompany them, the reader can follow the ways the historiography of Greek mathematics developed. Finally, the introductory texts that precede each chapter help the reader to approach critically the selected papers and at the same time to get an idea of the issues being further clarified by the new historiographical approaches.

The audience of the book includes scholars from history and philosophy of mathematics and mathematical sciences, scholars from history of science, students in the field of history of mathematics and history of sciences.

TABLE OF CONTENTS

Permissions ix

Preface xi

PART 1. THE BEGINNINGS OF GREEK MATHEMATICS

Texts selected and introduced by Hans-Joachim Waschkies

HANS-JOACHIM WASCHKIES / Introduction 3

JÜRGEN MITTELSTRASS / Die Entdeckung der Möglichkeit von Wissenchaft

Archive for History of Exact Sciences 2 (1962-66), 410-435 19

ÁRPÁD SZABÓ / Wie ist die Mathematik zu einer deduktiven Wissenschaft geworden?

Acta Antiqua Academiae Scientiarum Hungaricae IV (1956), 109-151 45

WILBUR RICHARD KNORR / On the early history of axiomatics.

The interaction of mathematics and philosophy in Greek antiquity.

Theory change, ancient axiomatics, and Galileo’s methodology.

J. Hintikka, D. Gruender, E. Agazzi (Eds), Dordrecht/Boston/London, 1981, 145-186 81

PART 2. STUDIES ON GREEK GEOMETRY

Texts selected and introduced by Reviel Netz

REVIEL NETZ / Introduction 113

WILBUR RICHARD KNORR / Construction as Existence Proof in Ancient Geometry

Ancient Philosophy 3 (1983), 125-148 115

KEN SAITO / Book II of Euclid’s Elements in the Light of the Theory of Conic Sections

Historia Scientiarum 28 (1985), 31-60 139

G.E.R. LLOYD / The Meno and the Mysteries of Mathematics

Phronesis 37 (1992), 166-183 169

PART 3. STUDIES ON PROPORTION THEORY AND INCOMMENSURABILITY

Texts selected and introduced by Ken Saito

KEN SAITO / Introduction 187

Proportionenlehre und ihre Spuren bei Aristoteles und Euklid

Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik B.II (1933), 311-330 191

KURT VON FRITZ / The Discovery of Incommensurability by Hippasus of Metapontum

Annals of Mathematics 46 (1954), 242-264 211

HANS FREUDENTHAL / Y avait-il une crise des fondements des mathématiques dans l’antiquité?

Bulletin de la Société mathématique de Belgique 18 (1966), 43-55 233

WILBUR RICHARD KNORR / The Impact of Modern Mathematics on Ancient Mathematics

Revue d’histoire des mathématiques 7 (2001), 121-135 243

PART 4. STUDIES ON GREEK ALGEBRA

Texts selected and introduced by Jacques Sesiano

JACQUES SESIANO / Introduction 257

KURT VOGEL / Zur Berechnung der quadratischen 

Unterrichtsblätter für Mathematik und Naturwissenschaften 39 (1933), 76-81 265

G.J. TOOMER / Lost Greek mathematical works in Arabic translation

Mathematical Intelligencer 6.2 (1984), 32-38 275

THOMAS L. HEATH / Diophantus’ methods of solution

Fourth chapter (pp. 54-98) of Heath’s book Diophantus of Alexandria. A study in the history of Greek algebra. New York, 1964 285

PART 5. DID THE GREEKS HAVE THE NOTION OF COMMON FRACTION? DID THEY USE IT?

Texts selected and introduced by Jean Christianidis

JEAN CHRISTIANIDIS / Introduction 331

WILBUR RICHARD KNORR / Techniques of fractions in Ancient Egypt and Greece.

Historia Mathematica 9 (1982), 133-171 337

DAVID H. FOWLER / Logistic and fractions in early Greek mathematics: a new interpretation.

Histoire des fractions, fraction d’histoire. P. Benoit, K. Chemla, J. Ritter (Eds), Basel/Boston/Berlin, 1992, 133-147 367

PART 6. METHODOLOGICAL ISSUES IN THE HISTORIOGRAPHY OF GREEK MATHEMATICS

Texts selected and introduced by Sabetai Unguru

SABETAI UNGURU / Introduction 383

SABETAI UNGURU / On the Need to Rewrite the History of Greek Mathematics.

Archive for History of Exact Sciences 15 (1975), 67-114 385

B.L. VAN DER WAERDEN / Defence of a ‘Shocking’ Point of View.

Archive for History of Exact Sciences 15 (1976), 199-210 433

ANDRÉ WEIL / Who Betrayed Euclid? (Extract from a letter to the Editor)

Archive for History of Exact Sciences 19 (1978), 91-93 447

SABETAI UNGURU / History of Ancient Mathematics: Some Reflections on the State of the Art

Isis 70 (1979), 555-565 451

Exploring Advanced Euclidean Geometry with GeoGebra

(Classroom Resource Materials)

Gerard A. Venema

Mathematical Association of America | 2013 | 146 páginas | rar - pdf | 820 kb

link (password: matav)

This book provides an inquiry-based introduction to advanced Euclidean geometry. It utilizes dynamic geometry software, specifically GeoGebra, to explore the statements and proofs of many of the most interesting theorems in the subject. Topics covered include triangle centers, inscribed, circumscribed, and escribed circles, medial and orthic triangles, the nine-point circle, duality, and the theorems of Ceva and Menelaus, as well as numerous applications of those theorems. The final chapter explores constructions in the Poincaré disk model for hyperbolic geometry.

The book can be used either as a computer laboratory manual to supplement an undergraduate course in geometry or as a stand-alone introduction to advanced topics in Euclidean geometry. The text consists almost entirely of exercises (with hints) that guide students as they discover the geometric relationships for themselves. First the ideas are explored at the computer and then those ideas are assembled into a proof of the result under investigation. The goals are for the reader to experience the joy of discovering geometric relationships, to develop a deeper understanding of geometry, and to encourage an appreciation for the beauty of Euclidean geometry.

Contents

Preface vii

0 A Quick Review of Elementary Euclidean Geometry 1

0.1 Measurement and congruence. . 1

0.2 Angle addition  . . 2

0.3 Triangles and triangle congruence conditions . . 3

0.4 Separation and continuity. . 4

0.5 The exterior angle theorem . . 5

0.6 Perpendicular lines and parallel lines . . 5

0.7 The Pythagorean theorem. . 7

0.8 Similar triangles . . . 8

0.9 Quadrilaterals  . . 9

0.10 Circles and inscribed angles . . 10

0.11 Area . . 11

1 The Elements of GeoGebra 13

1.1 Getting started: the GeoGebra toolbar. . 13

1.2 Simple constructions and the drag test  . . 16

1.3 Measurement and calculation . . 18

1.4 Enhancing the sketch .. . 20

2 The Classical Triangle Centers 23

2.1 Concurrent lines . . 23

2.2 Medians and the centroid . . 24

2.3 Altitudes and the orthocenter .. 25

2.4 Perpendicular bisectors and the circumcenter . . 26

2.5 The Euler line . . 27

3 Advanced Techniques in GeoGebra 31

3.1 User-defined tools . . 31

3.2 Check boxes . . 33

3.3 The Pythagorean theorem revisited  . . 34

4 Circumscribed, Inscribed, and Escribed Circles 39

4.1 The circumscribed circle and the circumcenter  . . 39

4.2 The inscribed circle and the incenter . . 41

4.3 The escribed circles and the excenters . . 42

4.4 The Gergonne point and the Nagel point . . 43

4.5 Heron’s formula . . 44

5 The Medial and Orthic Triangles 47

5.1 The medial triangle  . . 47

5.2 The orthic triangle . . 48

5.3 Cevian triangles  . . 50

5.4 Pedal triangles . . 51

6 Quadrilaterals 53

6.1 Basic definitions . . 53

6.2 Convex and crossed quadrilaterals. . 54

6.3 Cyclic quadrilaterals . . 55

6.4 Diagonals . . 56

7 The Nine-Point Circle 57

7.1 The nine-point circle . . . 57

7.2 The nine-point center . . 59

7.3 Feuerbach’s theorem  . . 60

8 Ceva’s Theorem 63

8.1 Exploring Ceva’s theorem . . 63

8.2 Sensed ratios and ideal points  . . 65

8.3 The standard form of Ceva’s theorem  . . 68

8.4 The trigonometric form of Ceva’s theorem  . . 71

8.5 The concurrence theorems  . . 72

8.6 Isotomic and isogonal conjugates and the symmedian point  . . 73

9 The Theorem of Menelaus 77

9.1 Duality  . . 77

9.2 The theorem of Menelaus . . 78

10 Circles and Lines 81

10.1 The power of a point  . . 81

10.2 The radical axis . . 83

10.3 The radical center  . . 84

11 Applications of the Theorem of Menelaus 85

11.1 Tangent lines and angle bisectors . . . 85

11.2 Desargues’ theorem  . . 86

11.3 Pascal’s mystic hexagram . . 88

11.4 Brianchon’s theorem  . . 90

11.5 Pappus’s theorem  . . 91

11.6 Simson’s theorem. . 93

11.7 Ptolemy’s theorem . . 96

11.8 The butterfly theorem  . . 97

12 Additional Topics in Triangle Geometry 99

12.1 Napoleon’s theorem and the Napoleon point  . . 99

12.2 The Torricelli point . . 100

12.3 van Aubel’s theorem. . 100

12.4 Miquel’s theorem and Miquel points . . 101

12.5 The Fermat point  . . 101

12.6 Morley’s theorem . . 102

13 Inversions in Circles 105

13.1 Inverting points .  . 105

13.2 Inverting circles and lines  . . 107

13.3 Othogonality  . . 108

13.4 Angles and distances. . 110

14 The Poincar´e Disk 111

14.1 The Poincar´e disk model for hyperbolic geometry  . . 111

14.2 The hyperbolic straightedge . . 113

14.3 Common perpendiculars . . 114

14.4 The hyperbolic compass. . 116

14.5 Other hyperbolic tools . . 117

14.6 Triangle centers in hyperbolic geometry . . 118

References 121

Index 123

Opening the Cage: Critique and Politics of Mathematics Education

Ole Skovsmose e Brian Greer 

Sense Publishers | 2012 | 387 páginas | rar - pdf | 1,6 Mb

link (password: matav)

The painting on the front of this book is an illustration for Totakahini: The tale of the parrot, by Rabindranath Tagore, in which he satirized education as a magnificent golden cage. Opening the cage addresses mathematics education as a complex socio-political phenomenon, exploring the vast terrain that spans critique and politics. Opening the cage includes contributions from educators writing critically about mathematics education in diverse contexts. They demonstrate that mathematics education is politics, they investigate borderland positions, they address the nexus of mathematics, education, and power, and they explore educational possibilities. Mathematics education is not a free enterprise. It is carried on behind bars created by economic, political, and social demands. This cage might not be as magnificent as that in Tagore's fable. But it is strong. Opening the cage is a critical and political challenge, and we may be surprised to see what emerges.

Contents
Preface vii
Introduction: Seeing the Cage? The Emergence of Critical Mathematics Education 1
Brian Greer and Ole Skovsmose
Part I: Mathematics Education is Politics
Chapter 1: Mathematics as a Weapon in the Struggle 23
Eric (Rico) Gutstein
Chapter 2: A Critical Approach to Equity 49
Alexandre Pais
Chapter 3: The Role of Mathematics in the Destruction of Communities, and What We can do to Reverse this Process, including Using Mathematics 93
Munir Jamil Fasheh
Chapter 4: The USA Mathematics Advisory Panel: A Case Study 107
Brian Greer
Part II: Borderland Positions
Chapter 5: Mathematics Teaching and Learning of Immigrant Students: An Overview of the Research Field Across Multiple Settings 127
Marta Civil
Chapter 6: Learning of Mathematics Among Pakistani Immigrant Children in Barcelona: A Sociocultural Perspective 143
Sikunder Ali Baber
Chapter 7: Mathematics Education Across Two Language Contexts: A Political Perspective 167
Mamokgethi Setati and Núria Planas
Chapter 8: Genealogy of Mathematics Education in Two Brazilian Rural Forms of Life 187
Gelsa Knijnik and Fernanda Wanderer
Chapter 9: On Becoming and Being a Critical Black Scholar in Mathematics Education: The Politics of Race and Identity 203
Danny Bernard Martin and Maisie Gholson
Intermezzo: Totakahini (The Tale of the Parrot) 223
Rabindranath Tagore (Translated by Swapna Mukhopadhyay)
Part III: Mathematics and Power
Chapter 10: The Hegemony of Mathematics 229
Brian Greer and Swapna Mukhopadhyay
Chapter 11: Bringing Critical Mathematics to Work: But can Numbers Mobilise? 249
Keiko Yasukawa and Tony Brown
Chapter 12: Shaping and being Shaped by Mathematics: Examining a Technology of Rationality 265
Keiko Yasukawa, Ole Skovsmose, and Ole Ravn
Part IV: Searching for Possibilities
Chapter 13: Potentials, Pitfalls, and Discriminations: Curriculum Conceptions Revisited 287
Eva Jablonka and Uwe Gellert
Chapter 14: A Philosophical Perspective on Contextualisations
in Mathematics Education 309
Annica Andersson and Ole Ravn
Chapter 15: Mathematics Education and Democratic Participation Between the Critical and the Ethical: A Socially Response-able Approach 325
Bill Atweh
Chapter 16: Towards a Critical Mathematics Education Research Programme? 343
Ole Skovsmose
Chapter 17: Opening the Cage? Critical Agency in the Face of Uncertainty 369
Ole Skovsmose and Brian Greer

Contributors 387

Our Mathematical Universe: My Quest for the Ultimate Nature of Reality

Max Tegmark

Knopf  | 2014 | 432 paginas | epub | 25 Mb

link

mobi - 8 Mb - link

Max Tegmark leads us on an astonishing journey through past, present and future, and through the physics, astronomy and mathematics that are the foundation of his work, most particularly his hypothesis that our physical reality is a mathematical structure and his theory of the ultimate multiverse. In a dazzling combination of both popular and groundbreaking science, he not only helps us grasp his often mind-boggling theories, but he also shares with us some of the often surprising triumphs and disappointments that have shaped his life as a scientist. Fascinating from first to last—this is a book that has already prompted the attention and admiration of some of the most prominent scientists and mathematicians.

Contents
Preface
  1 What Is Reality?
Not What It Seems • What’s the Ultimate Question? • The Journey Begins
Part One: Zooming Out
  2 Our Place in Space
Cosmic Questions • How Big Is Space? • The Size of Earth • Distance to the Moon • Distance to the Sun and the Planets • Distance to the Stars • Distance to the Galaxies • What Is Space?
  3 Our Place in Time
Where Did Our Solar System Come From? • Where Did the Galaxies Come From? • Where Did the Mysterious Microwaves Come From? • Where Did the Atoms Come From?
  4 Our Universe by Numbers
Wanted: Precision Cosmology • Precision Microwave-Background Fluctuations • Precision Galaxy Clustering • The Ultimate Map of Our Universe • Where Did Our Big Bang Come From?
  5 Our Cosmic Origins
What’s Wrong with Our Big Bang? • How Inflation Works • The Gift That Keeps on Giving • Eternal Inflation
  6 Welcome to the Multiverse
The Level I Multiverse • The Level II Multiverse • Multiverse Halftime Roundup
Part Two: Zooming In
  7 Cosmic Legos
Atomic Legos • Nuclear Legos • Particle-Physics Legos • Mathematical Legos • Photon Legos • Above the Law? • Quanta and Rainbows • Making Waves • Quantum Weirdness • The Collapse of Consensus • The Weirdness Can’t Be Confined • Quantum Confusion
  8 The Level III Multiverse
The Level III Multiverse • The Illusion of Randomness • Quantum Censorship • The Joys of Getting Scooped • Why Your Brain Isn’t a Quantum Computer • Subject, Object and Environment • Quantum Suicide • Quantum Immortality? • Multiverses Unified • Shifting Views: Many Worlds or Many Words?
Part Three: Stepping Back
  9 Internal Reality, External Reality and Consensus Reality
External Reality and Internal Reality • The Truth, the Whole Truth and Nothing but the Truth • Consensus Reality • Physics: Linking External to Consensus Reality
10 Physical Reality and Mathematical Reality
Math, Math Everywhere! • The Mathematical Universe Hypothesis • What Is a Mathematical Structure?
11 Is Time an Illusion?
How Can Physical Reality Be Mathematical? • What Are You? • Where Are You? (And What Do You Perceive?) • When Are You?
12 The Level IV Multiverse
Why I Believe in the Level IV Multiverse • Exploring the Level IV Multiverse: What’s Out There? • Implications of the Level IV Multiverse • Are We Living in a Simulation? • Relation Between the MUH, the Level IV Multiverse and Other Hypotheses • Testing the Level IV Multiverse
13 Life, Our Universe and Everything
How Big Is Our Physical Reality? • The Future of Physics • The Future of Our Universe—How Will It End? • The Future of Life • The Future of You—Are You Insignificant?
Acknowledgments
Suggestions for Further Reading
Index
A Note About the Author

Mathematical Models and Methods for Planet Earth

Alessandra Celletti, Ugo Locatelli, Tommaso Ruggeri e Elisabetta Strickland 

Springer | 2014 | 177 páginas | rar - pdf | 2,6 Mb

link (password: matav)

In 2013 several scientific activities have been devoted to mathematical researches for the study of planet Earth. The current volume presents a selection of the highly topical issues presented at the workshop “Mathematical Models and Methods for Planet Earth”, held in Roma (Italy), in May 2013. The fields of interest span from impacts of dangerous asteroids to the safeguard from space debris, from climatic changes to monitoring geological events, from the study of tumor growth to sociological problems. In all these fields the mathematical studies play a relevant role as a tool for the analysis of specific topics and as an ingredient of multidisciplinary problems. To investigate these problems we will see many different mathematical tools at work: just to mention some, stochastic processes, PDE, normal forms, chaos theory.

Contents
Mathematics of Planet Earth . 1
Christiane Rousseau
The Role of Boundary Layers in the Large-scale Ocean Circulation . . . 11
Laure Saint-Raymond
Noise-induced Periodicity: Some Stochastic Models for Complex Biological Systems . 25
Paolo Dai Pra, Giambattista Giacomin and Daniele Regoli
Kinetic Equations and Stochastic Game Theory for Social Systems . . . 37
Andrea Tosin
Using Mathematical Modelling as a Virtual Microscope to Support Biomedical Research .59
Chiara Giverso and Luigi Preziosi
Ferromagnetic Models for Cooperative Behavior: Revisiting Universality in Complex Phenomena . 73
Elena Agliari, Adriano Barra, Andrea Galluzzi, Andrea Pizzoferrato and Daniele Tantari
The Near Earth Asteroid Hazard and Mitigation . . . 87
Ettore Perozzi
Mathematical Models of Textual Data: A Short Review .  . 99
Mirko Degli Esposti
Space Debris Long Term Dynamics . .  . 111
Anne Lemaitre and Charles Hubaux
Mathematical Models for Socio-economic Problems . 123
Maria Letizia Bertotti and Giovanni Modanese
Climate as a Complex Dynamical System .. . . 135
Antonello Provenzale
Periodic Orbits of the N-body Problem with the Symmetry of Platonic Polyhedra . 143
Giovanni Federico Gronchi
Superprocesses as Models for Information Dissemination in the Future Internet. . 157
Laura Sacerdote, Michele Garetto, Federico Polito and Matteo Sereno
Appendix: Pictures from INdAMWorkshop .  . 171

Greek Mathematical Thought and the Origin of Algebra

(Dover Books on Mathematics)

Jacob Klein

Dover Publications | 1992 | 384 páginas | epub | 19 Mb

link
link1

Important study focuses on the revival and assimilation of ancient Greek mathematics in the 13th–16th centuries, via Arabic science, and the 16th-century development of symbolic algebra. This brought about the crucial change in the concept of number that made possible modern science — in which the symbolic "form" of a mathematical statement is completely inseparable from its "content" of physical meaning. Includes a translation of Vieta's Introduction to the Analytical Art. 1968 edition

Contents
Author’s note
Translator’s note
Short titles frequently used
PART I
1 Introduction: Purpose and plan of the inquiry.
2 The opposition of logistic and arithmetic in the Neoplatonists.
3 Logistic and arithmetic in Plato.
4 The role of the theory of proportions in Nicomachus, Theon, and Domninus.
5 Theoretical logistic and the problem of fractions.
6 The concept of arithmos.
7 The ontological conception of the arithmoi in Plato.
A. The science of the Pythagoreans.
B. Mathematics in Plato—logistike and dianoia.
C. The arithmos eidetikos.
8 The Aristotelian critique and the possibility of a theoretical logistic.
PART II
9 On the difference between ancient and modern conceptualization.
10 The Arithmetic of Diophantus as theoretical logistic. The concept ofeidos in Diophantus
11 The formalism of Vieta and the transformation of thearithmos concept.
A. The life of Vieta and the general characteristics of his work.
B. Vieta’s point of departure: the concept of synthetic apodeixisin Pappus and in Diophantus.
C. The reinterpretation of the Diophantine procedure by Vieta:
1. The procedure for solutions “in the indeterminate form” as an analogue to geometric analysis.
2. The generalization of the eidos concept and its transformation into the “symbolic” concept of the species.
3. The reinterpretation of the katholou pragmateia as Mathesis Universalis in the sense of ars analytice.
12 The concept of “number.”
A. In Stevin.
B. In Descartes.
C. In Wallis.
NOTES
Part I, Notes 1–125
Part II, Notes 126–348

APPENDIX