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## SERRE DUALITY AND APPLICATIONS Modern Geometry_ Methods and Applications Part II The. a geometry is not de ned by the objects it represents but by their trans- Computer Vision: A Modern Approach, Prentice Hall (2003) Grenoble Universities 4. Master MOSIG Introduction to Projective Geometry (Duality) The set of hyperplanes of a projective space Pn is a projective space of dimension n. Any de nition, property or theorem, Buy Modern Geometry_ Methods and Applications: Part II: The Geometry and Topology of Manifolds (Graduate Texts in Mathematics) (Part 2) on Amazon.com FREE SHIPPING on qualified orders.

### Duality (projective geometry) The Full Wiki

Lecture4_Wave particle duality.pdf Modern Physics. From Wikipedia, the free encyclopedia. In the geometry of the projective plane, duality refers to geometric transformations that replace points by lines and lines by points while preserving incidence properties among the transformed objects. The existence of such transformations leads to a general principle, that any theorem about incidences between points and lines in the projective plane may, In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and duality is the formalization of this concept. There are two approaches to the subject of duality, one through language (В§ Principle of duality) and the other a more functional approach through special mappings..

Generalized complex geometry and T-duality Gil R. Cavalcantiв€— and Marco Gualtieri вЂ  arXiv:1106.1747v1 [math.DG] 9 Jun 2011 Abstract We describe how generalized complex geometry, which interpolates between complex and symplectic geometry, is compatible with T-duality, a relation between quantum field theories discovered by physicists. Generalized complex geometry and T-duality Gil R. Cavalcantiв€— and Marco Gualtieri вЂ  arXiv:1106.1747v1 [math.DG] 9 Jun 2011 Abstract We describe how generalized complex geometry, which interpolates between complex and symplectic geometry, is compatible with T-duality, a relation between quantum field theories discovered by physicists.

Bundles & Gauges, a Math-Physics Duality - the case of Gravity - David Mendes A modern and straight forward summary of the necessary tools and concepts needed to understand and work with gauge theory in a fibre bundle formalism. Due to the aim of being a quick but thorough introduction full derivations are rarely included, but 4.3 Duality in Projective Geometry Printout I think and think for months and years, ninety-nine times, the conclusion is false. The hundredth time I am right. вЂ” Albert Einstein (1879вЂ“1955) Remember that the dual of a statement was defined in the section on axiomatic systems in the first chapter.

understanding of duality to illustrate several examples of dual pairs of curves, including a self-dual cubic curve. 2 Background As a starting point for understanding duality in projective geometry, we rst recall the axioms de ning a general projective plane. De nition 1. A projective plane is a pair (P;L) where Pis a nonempty set of points and Lis PDF The Duality of Time Theory is the result of more than two decades of ceaseless investigation and searching through ancient manuscripts of concealed philosophies and mystical traditions

Projective geometry is an elementary non-metrical form of geometry, meaning that it is not based on a concept of distance.In two dimensions it begins with the study of configurations of points and lines.That there is indeed some geometric interest in this sparse setting was first established by Desargues and others in their exploration of the principles of perspective art. 3 2.1 Projective geometry In ordinary plane geometry, the statement "two points determine a line" is an axiom, and, aside from the case of parallel lines, this statement closely resembles the fact that "two lines determine a point" (i.e., any two non-parallel lines intersect вЂ¦

understanding of duality to illustrate several examples of dual pairs of curves, including a self-dual cubic curve. 2 Background As a starting point for understanding duality in projective geometry, we rst recall the axioms de ning a general projective plane. De nition 1. A projective plane is a pair (P;L) where Pis a nonempty set of points and Lis Topological Gauge Theory, Cartan Geometry, and Gravity by Derek Keith Wise Doctor of Philosophy in Mathematics University of California, Riverside Dr. John C. Baez, Chair We investigate the geometry of general relativity, and of related topological gauge theories, using Cartan geometry. Cartan geometry|an вЂin nitesimalвЂ™ version of KleinвЂ™s

Jan 04, 2015В В· Happy New Year everyone, and I wish you all the best for 2015! In this video we introduce some basic orientation to the problem of how we represent, and вЂ¦ Bundles & Gauges, a Math-Physics Duality - the case of Gravity - David Mendes A modern and straight forward summary of the necessary tools and concepts needed to understand and work with gauge theory in a fibre bundle formalism. Due to the aim of being a quick but thorough introduction full derivations are rarely included, but

Noncommutative Geometry and String Duality F. Lizzia and R.J. Szabob the continuity criterion. Thus, in general, given a topological space one may naturally associate to it an abelian Cв€—-algebra. That the converse is also true is known as the GelвЂ™fand-Naimark theorem . Namely, there is an isomorphism between the category of Hausdorп¬Ђ TEXTBOOK. UniT OBJEcTiVEs вЂў Geometry is the mathematical study of space. вЂў EuclidвЂ™s postulates form the basis of the geometry we learn in high school. вЂў EuclidвЂ™s fifth postulate, also known as the parallel postulate, stood for over . two thousand years before it was shown to be unnecessary in creating a self-consistent geometry.

understanding of duality to illustrate several examples of dual pairs of curves, including a self-dual cubic curve. 2 Background As a starting point for understanding duality in projective geometry, we rst recall the axioms de ning a general projective plane. De nition 1. A projective plane is a pair (P;L) where Pis a nonempty set of points and Lis View Lecture4_Wave particle duality.pdf from AA 1Modern Physics Particle model of light 1. Blackbody Radiation 2. Photoelectric Effect 3. Emission of Light by Atoms in Gas Discharge Tubes 4. BohrвЂ™s

Department of Mathematics at Columbia University New York. This course is taken in sequence, part 1 in the fall, and part 2 in the spring. The terms duality and dualism were originally associated with (1) scientific investigations of Huygens and Newton into the properties of visible light, (2) philosophical ideas of Descartes and especially the mind-body problem, and (3) development of projective geometry by Descargues in the study of projective art. The investigation into the

SummaryClassical Enumerative GeometryModern ApproachOutline of ProofOn Thms A,B,C Enumerative Geometry: from Classical to Modern Aleksey Zinger Stony Brook University Duality and Geometry in SVM Classi ers Kristin P. Bennett bennek@rpi.edu Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180 USA Erin J. Bredensteiner eb6@evansville.edu Department of Mathematics, University of Evansville, Evansville, IN 47722 USA Abstract We develop an intuitive geometric interpre-

### Free Algebraic Geometry Books Download Ebooks Online Topological Gauge Theory and Gravity Derek Keith Wise. Nov 30, 2018В В· Over the last three decades, string theory has had a profound impact in pure mathematics connected to string theory, including generalized geometry, vertex algebras, topological T-duality and related topics. In the past decade, there has been huge surge of interest in topological aspects of condensed matter physics. More recently, it has started to transpire that the mathematics underlying, Generalized complex geometry and T-duality Gil R. Cavalcantiв€— and Marco Gualtieri вЂ  arXiv:1106.1747v1 [math.DG] 9 Jun 2011 Abstract We describe how generalized complex geometry, which interpolates between complex and symplectic geometry, is compatible with T-duality, a relation between quantum field theories discovered by physicists..

### Enumerative Geometry from Classical to Modern Projectively Dual Varieties arXiv. Yet, it seems that the role of the concept of duality in modern mathematics has been the subject of very few philosophical studies. 2 One such study by Ernest Nagel concerns projective duality [Nagel 1939]. In Euclidean geometry, two points determine a line, and two non-parallel lines determine a point. Duality and Geometry in SVM Classi ers Kristin P. Bennett bennek@rpi.edu Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180 USA Erin J. Bredensteiner eb6@evansville.edu Department of Mathematics, University of Evansville, Evansville, IN 47722 USA Abstract We develop an intuitive geometric interpre-. In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and duality is the formalization of this concept. There are two approaches to the subject of duality, one through language (В§ Principle of duality) and the other a more functional approach through special mappings. Nov 30, 2018В В· Over the last three decades, string theory has had a profound impact in pure mathematics connected to string theory, including generalized geometry, vertex algebras, topological T-duality and related topics. In the past decade, there has been huge surge of interest in topological aspects of condensed matter physics. More recently, it has started to transpire that the mathematics underlying

Basic Modern Algebraic Geometry. This note covers the following topics: Functors, Isomorphic and equivalent categories, Representable functors, Some constructions in the light of representable functors, Schemes: Definition and basic properties, Properties of morphisms of вЂ¦ Duality and Geometry in SVM Classi ers Kristin P. Bennett bennek@rpi.edu Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180 USA Erin J. Bredensteiner eb6@evansville.edu Department of Mathematics, University of Evansville, Evansville, IN 47722 USA Abstract We develop an intuitive geometric interpre-

3 2.1 Projective geometry In ordinary plane geometry, the statement "two points determine a line" is an axiom, and, aside from the case of parallel lines, this statement closely resembles the fact that "two lines determine a point" (i.e., any two non-parallel lines intersect вЂ¦ 5. Area in Hyperbolic Geometry 6. Showing Consistency - A Model for Hyperbolic Geometry 7. Classifying Theorems 2. Elliptical Geometry 1. A Geometry with no Parallels 2. Two Models 3. Some Results in Elliptic Geometry 3. Geometry in the Real World D. Projective Geometry 1. Introduction 2. The Real Projective Plane 3. Duality 4. Perspectivity 5.

Modern Geometry by Wayne Aitken. This is a course note on Euclidean and non-Euclidean geometries with emphasis on (i) the contrast between the traditional and modern approaches to geometry, and (ii) the history and role of the parallel postulate. Topological Gauge Theory, Cartan Geometry, and Gravity by Derek Keith Wise Doctor of Philosophy in Mathematics University of California, Riverside Dr. John C. Baez, Chair We investigate the geometry of general relativity, and of related topological gauge theories, using Cartan geometry. Cartan geometry|an вЂin nitesimalвЂ™ version of KleinвЂ™s

The terms duality and dualism were originally associated with (1) scientific investigations of Huygens and Newton into the properties of visible light, (2) philosophical ideas of Descartes and especially the mind-body problem, and (3) development of projective geometry by Descargues in the study of projective art. The investigation into the PDF The Duality of Time Theory is the result of more than two decades of ceaseless investigation and searching through ancient manuscripts of concealed philosophies and mystical traditions

diп¬Ђerent aspects of projective duality and points of view on it. We hope, that the exposition is quite informal and requires only a standard knowledge of algebraic geometry and algebraic (or Lie) groups theory. Some chapters are, however, more diп¬ѓcult and use вЂ¦ Geometry. These are the books for those you who looking for to read the Geometry, try to read or download Pdf/ePub books and some of authors may have disable the live reading.Check the book if it available for your country and user who already subscribe вЂ¦

Department of Mathematics at Columbia University New York. This course is taken in sequence, part 1 in the fall, and part 2 in the spring. 5. Area in Hyperbolic Geometry 6. Showing Consistency - A Model for Hyperbolic Geometry 7. Classifying Theorems 2. Elliptical Geometry 1. A Geometry with no Parallels 2. Two Models 3. Some Results in Elliptic Geometry 3. Geometry in the Real World D. Projective Geometry 1. Introduction 2. The Real Projective Plane 3. Duality 4. Perspectivity 5.

This comprehensive, best-selling text focuses on the study of many different geometries -- rather than a single geometry -- and is thoroughly modern in its approach. Each chapter is essentially a short course on one aspect of modern geometry, including finite geometries, the geometry of transformations, convexity, advanced Euclidian geometry, inversion, projective geometry, geometric aspects Geometry, Symmetry, and Physics 8 4.1 Duality Symmetries and BPS states 9 4.1.1 BPS states 9 divide between Mathematics and Physics began to open up in the 19th century. For example in volume 2 of Nature, from 1870, we read of the following challenge from the actually the modern physical developments have required a mathematics that

TEXTBOOK. UniT OBJEcTiVEs вЂў Geometry is the mathematical study of space. вЂў EuclidвЂ™s postulates form the basis of the geometry we learn in high school. вЂў EuclidвЂ™s fifth postulate, also known as the parallel postulate, stood for over . two thousand years before it was shown to be unnecessary in creating a self-consistent geometry. Yet, it seems that the role of the concept of duality in modern mathematics has been the subject of very few philosophical studies. 2 One such study by Ernest Nagel concerns projective duality [Nagel 1939]. In Euclidean geometry, two points determine a line, and two non-parallel lines determine a point.

understanding of duality to illustrate several examples of dual pairs of curves, including a self-dual cubic curve. 2 Background As a starting point for understanding duality in projective geometry, we rst recall the axioms de ning a general projective plane. De nition 1. A projective plane is a pair (P;L) where Pis a nonempty set of points and Lis understanding of duality to illustrate several examples of dual pairs of curves, including a self-dual cubic curve. 2 Background As a starting point for understanding duality in projective geometry, we rst recall the axioms de ning a general projective plane. De nition 1. A projective plane is a pair (P;L) where Pis a nonempty set of points and Lis

Noncommutative Geometry and String Duality. nov 30, 2018в в· over the last three decades, string theory has had a profound impact in pure mathematics connected to string theory, including generalized geometry, vertex algebras, topological t-duality and related topics. in the past decade, there has been huge surge of interest in topological aspects of condensed matter physics. more recently, it has started to transpire that the mathematics underlying, atiyah then discusses some dualities that occur in differential geometry which will be used to model physics. this section is hard-going, but if you persist through, it eventually becomes more informal and comprehensible once again around p10 of the pdf, which can give us some idea of how these dualities relate to modern physics.).

Yet, it seems that the role of the concept of duality in modern mathematics has been the subject of very few philosophical studies. 2 One such study by Ernest Nagel concerns projective duality [Nagel 1939]. In Euclidean geometry, two points determine a line, and two non-parallel lines determine a point. TEXTBOOK. UniT OBJEcTiVEs вЂў Geometry is the mathematical study of space. вЂў EuclidвЂ™s postulates form the basis of the geometry we learn in high school. вЂў EuclidвЂ™s fifth postulate, also known as the parallel postulate, stood for over . two thousand years before it was shown to be unnecessary in creating a self-consistent geometry.

The terms duality and dualism were originally associated with (1) scientific investigations of Huygens and Newton into the properties of visible light, (2) philosophical ideas of Descartes and especially the mind-body problem, and (3) development of projective geometry by Descargues in the study of projective art. The investigation into the understanding of duality to illustrate several examples of dual pairs of curves, including a self-dual cubic curve. 2 Background As a starting point for understanding duality in projective geometry, we rst recall the axioms de ning a general projective plane. De nition 1. A projective plane is a pair (P;L) where Pis a nonempty set of points and Lis

5. Area in Hyperbolic Geometry 6. Showing Consistency - A Model for Hyperbolic Geometry 7. Classifying Theorems 2. Elliptical Geometry 1. A Geometry with no Parallels 2. Two Models 3. Some Results in Elliptic Geometry 3. Geometry in the Real World D. Projective Geometry 1. Introduction 2. The Real Projective Plane 3. Duality 4. Perspectivity 5. of classical synthetic geometry; it is here where one encounters many of the challenging Olympiad problems which helped inspire this book. The third part, вЂњThe roads to modern geometryвЂќ, consists of two4 chapters which treat slightly more advanced topics (inversive and projective geometry).

On four years (1989, 1993, 1997, 1998), the author offered a one-semester course on algebraic curves in the classical complex projective plane. The well-known duality between points and lines in projective geometry can be approached either axiomatically (by giving a symmetric formulation to axioms and thus to proofs) or transformationally (by invoking correlations, often affected by conic 4.3 Duality in Projective Geometry Printout I think and think for months and years, ninety-nine times, the conclusion is false. The hundredth time I am right. вЂ” Albert Einstein (1879вЂ“1955) Remember that the dual of a statement was defined in the section on axiomatic systems in the first chapter.

Generalized complex geometry and T-duality Gil R. Cavalcantiв€— and Marco Gualtieri вЂ  arXiv:1106.1747v1 [math.DG] 9 Jun 2011 Abstract We describe how generalized complex geometry, which interpolates between complex and symplectic geometry, is compatible with T-duality, a relation between quantum field theories discovered by physicists. Yet, it seems that the role of the concept of duality in modern mathematics has been the subject of very few philosophical studies. 2 One such study by Ernest Nagel concerns projective duality [Nagel 1939]. In Euclidean geometry, two points determine a line, and two non-parallel lines determine a point.

Basic Modern Algebraic Geometry. This note covers the following topics: Functors, Isomorphic and equivalent categories, Representable functors, Some constructions in the light of representable functors, Schemes: Definition and basic properties, Properties of morphisms of вЂ¦ Generalized complex geometry and T-duality Gil R. Cavalcantiв€— and Marco Gualtieri вЂ  arXiv:1106.1747v1 [math.DG] 9 Jun 2011 Abstract We describe how generalized complex geometry, which interpolates between complex and symplectic geometry, is compatible with T-duality, a relation between quantum field theories discovered by physicists. Noncommutative Geometry and String Duality

(PDF) Duality Principles and Modern Hylomorphism Kevin. 5. area in hyperbolic geometry 6. showing consistency - a model for hyperbolic geometry 7. classifying theorems 2. elliptical geometry 1. a geometry with no parallels 2. two models 3. some results in elliptic geometry 3. geometry in the real world d. projective geometry 1. introduction 2. the real projective plane 3. duality 4. perspectivity 5., 4.3 duality in projective geometry printout i think and think for months and years, ninety-nine times, the conclusion is false. the hundredth time i am right. вђ” albert einstein (1879вђ“1955) remember that the dual of a statement was defined in the section on axiomatic systems in the first chapter.); generalized complex geometry and t-duality gil r. cavalcantiв€— and marco gualtieri вђ  arxiv:1106.1747v1 [math.dg] 9 jun 2011 abstract we describe how generalized complex geometry, which interpolates between complex and symplectic geometry, is compatible with t-duality, a relation between quantum field theories discovered by physicists., on four years (1989, 1993, 1997, 1998), the author offered a one-semester course on algebraic curves in the classical complex projective plane. the well-known duality between points and lines in projective geometry can be approached either axiomatically (by giving a symmetric formulation to axioms and thus to proofs) or transformationally (by invoking correlations, often affected by conic.

Embedding R2 Duality in Projective Geometry

Modern Geometries 5th edition (9780534351885) Textbooks.com. duality and geometry in svm classi ers kristin p. bennett bennek@rpi.edu department of mathematical sciences, rensselaer polytechnic institute, troy, ny 12180 usa erin j. bredensteiner eb6@evansville.edu department of mathematics, university of evansville, evansville, in 47722 usa abstract we develop an intuitive geometric interpre-, noncommutative geometry and string duality f. lizzia and r.j. szabob the continuity criterion. thus, in general, given a topological space one may naturally associate to it an abelian cв€—-algebra. that the converse is also true is known as the gelвђ™fand-naimark theorem . namely, there is an isomorphism between the category of hausdorп¬ђ). SERRE DUALITY AND APPLICATIONS

Embedding R2 Duality in Projective Geometry. in geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and duality is the formalization of this concept. there are two approaches to the subject of duality, one through language (в§ principle of duality) and the other a more functional approach through special mappings., generalized complex geometry and t-duality gil r. cavalcantiв€— and marco gualtieri вђ  arxiv:1106.1747v1 [math.dg] 9 jun 2011 abstract we describe how generalized complex geometry, which interpolates between complex and symplectic geometry, is compatible with t-duality, a relation between quantum field theories discovered by physicists.). Modern Geometries James R. Smart - Google Books

Modern Geometries 5th edition (9780534351885) Textbooks.com. nov 30, 2018в в· over the last three decades, string theory has had a profound impact in pure mathematics connected to string theory, including generalized geometry, vertex algebras, topological t-duality and related topics. in the past decade, there has been huge surge of interest in topological aspects of condensed matter physics. more recently, it has started to transpire that the mathematics underlying, noncommutative geometry and string duality f. lizzia and r.j. szabob the continuity criterion. thus, in general, given a topological space one may naturally associate to it an abelian cв€—-algebra. that the converse is also true is known as the gelвђ™fand-naimark theorem . namely, there is an isomorphism between the category of hausdorп¬ђ). Duality (projective geometry) Wikipedia

Bundles & Gauges a Math-Physics Duality. in geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and duality is the formalization of this concept. there are two approaches to the subject of duality, one through language (в§ principle of duality) and the other a more functional approach through special mappings., pdf the duality of time theory is the result of more than two decades of ceaseless investigation and searching through ancient manuscripts of concealed philosophies and mystical traditions).

Department of Mathematics at Columbia University New York. This course is taken in sequence, part 1 in the fall, and part 2 in the spring. 5. Area in Hyperbolic Geometry 6. Showing Consistency - A Model for Hyperbolic Geometry 7. Classifying Theorems 2. Elliptical Geometry 1. A Geometry with no Parallels 2. Two Models 3. Some Results in Elliptic Geometry 3. Geometry in the Real World D. Projective Geometry 1. Introduction 2. The Real Projective Plane 3. Duality 4. Perspectivity 5.

PDF The Duality of Time Theory is the result of more than two decades of ceaseless investigation and searching through ancient manuscripts of concealed philosophies and mystical traditions Duality and Geometry in SVM Classi ers Kristin P. Bennett bennek@rpi.edu Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180 USA Erin J. Bredensteiner eb6@evansville.edu Department of Mathematics, University of Evansville, Evansville, IN 47722 USA Abstract We develop an intuitive geometric interpre-

Generalized complex geometry and T-duality Gil R. Cavalcantiв€— and Marco Gualtieri вЂ  arXiv:1106.1747v1 [math.DG] 9 Jun 2011 Abstract We describe how generalized complex geometry, which interpolates between complex and symplectic geometry, is compatible with T-duality, a relation between quantum field theories discovered by physicists. On four years (1989, 1993, 1997, 1998), the author offered a one-semester course on algebraic curves in the classical complex projective plane. The well-known duality between points and lines in projective geometry can be approached either axiomatically (by giving a symmetric formulation to axioms and thus to proofs) or transformationally (by invoking correlations, often affected by conic

5. Area in Hyperbolic Geometry 6. Showing Consistency - A Model for Hyperbolic Geometry 7. Classifying Theorems 2. Elliptical Geometry 1. A Geometry with no Parallels 2. Two Models 3. Some Results in Elliptic Geometry 3. Geometry in the Real World D. Projective Geometry 1. Introduction 2. The Real Projective Plane 3. Duality 4. Perspectivity 5. Topological Gauge Theory, Cartan Geometry, and Gravity by Derek Keith Wise Doctor of Philosophy in Mathematics University of California, Riverside Dr. John C. Baez, Chair We investigate the geometry of general relativity, and of related topological gauge theories, using Cartan geometry. Cartan geometry|an вЂin nitesimalвЂ™ version of KleinвЂ™s

On four years (1989, 1993, 1997, 1998), the author offered a one-semester course on algebraic curves in the classical complex projective plane. The well-known duality between points and lines in projective geometry can be approached either axiomatically (by giving a symmetric formulation to axioms and thus to proofs) or transformationally (by invoking correlations, often affected by conic In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and duality is the formalization of this concept. There are two approaches to the subject of duality, one through language (В§ Principle of duality) and the other a more functional approach through special mappings.

In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and duality is the formalization of this concept. There are two approaches to the subject of duality, one through language (В§ Principle of duality) and the other a more functional approach through special mappings. Generalized complex geometry and T-duality Gil R. Cavalcantiв€— and Marco Gualtieri вЂ  arXiv:1106.1747v1 [math.DG] 9 Jun 2011 Abstract We describe how generalized complex geometry, which interpolates between complex and symplectic geometry, is compatible with T-duality, a relation between quantum field theories discovered by physicists.

Geometry, Symmetry, and Physics 8 4.1 Duality Symmetries and BPS states 9 4.1.1 BPS states 9 divide between Mathematics and Physics began to open up in the 19th century. For example in volume 2 of Nature, from 1870, we read of the following challenge from the actually the modern physical developments have required a mathematics that Duality and Geometry in SVM Classi ers Kristin P. Bennett bennek@rpi.edu Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180 USA Erin J. Bredensteiner eb6@evansville.edu Department of Mathematics, University of Evansville, Evansville, IN 47722 USA Abstract We develop an intuitive geometric interpre- (PDF) DUALITY OF TIME Complex-Time Geometry and Perpetual

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