81. The Geometry Of Separation Processes: The Horse-Carrot Theorem For Steady Flow S I38.48 The geometry of Separation Processes The HorseCarrot Theoremfor Steady Flow Systems. Peter Salamon (SDSU), James Nulton (SDCC). http://www.aps.org/BAPSMAR98/abs/S1875048.html

Previous abstract Graphical version Next abstract Session I38 - General Poster Session I. POSTER session, Tuesday morning, March 17 Exhibit Hall, Los Angeles Convention Center The Geometry of Separation Processes: The Horse-Carrot Theorem for Steady Flow Systems Peter Salamon (SDSU), James Nulton (SDCC) The horse-carrot theorem bounding the entropy production in processes with a fixed number of relaxations is extended to steady flow processes. The dissipation turns out to be related to a path of flows rather than states. The example of fractional distillation is presented and shows how null directions for the geometry turn out to be useful in the analysis. The implied distillation column design offers potentially significant energy savings. The distinguishability geometry of thermodynamic state space is reviewed. Part I of program listing

82. The Geometry Of Separation Processes: The Horse-Carrot Theorem For Steady Flow S I38.53 The geometry of Separation Processes The HorseCarrot Theoremfor Steady Flow Systems. Peter Salamon (SDSU), James Nulton (SDCC). http://www.aps.org/BAPSMAR98/abs/S1885053.html

Previous abstract Graphical version Next abstract Session I38 - General Poster Session I. MIXED session, Tuesday morning, March 17 Exhibit Hall, The Geometry of Separation Processes: The Horse-Carrot Theorem for Steady Flow Systems Peter Salamon (SDSU), James Nulton (SDCC) The horse-carrot theorem bounding the entropy production in processes with a fixed number of relaxations is extended to steady flow processes. The dissipation turns out to be related to a path of flows rather than states. The example of fractional distillation is presented and shows how null directions for the geometry turn out to be useful in the analysis. The implied distillation column design offers potentially significant energy savings. The distinguishability geometry of thermodynamic state space is reviewed. Part I of program listing

83. ThinkQuest Library Of Entries geometry Postulate, theorem, and Corollary. Part 9. Corollary 4, Theopposite angles of an inscribed quadrilateral are supplementary. http://library.advanced.org/16284/reference_gc_9.htm

Welcome to the ThinkQuest Internet Challenge of Entries The web site you have requested, Math Planet , is one of over 4000 student created entries in our Library. Before using our Library, please be sure that you have read and agreed to our To learn more about ThinkQuest. You can browse other ThinkQuest Library Entries To proceed to Math Planet click here Back to the Previous Page The Site you have Requested ... Math Planet click here to view this site A ThinkQuest Internet Challenge 1998 Entry Click image for the Site Languages : Site Desciption Math Planet is web site dedicated to the advancement of mathematics. That's good, because today's demanding world requires young people to be equipped with a solid math foundation. Targeted towards high school students, there are many different categories, including Basic Algebra and Geometry, advanced Algebra and Trigonometry, SAT and ACT math preparation courses, and even a chat room for team problem solving. Students Wen Vestavia Hills High School AL, United States Parakash Vestavia Hills High School AL, United States

84. 13.ai Automated geometrytheorem Proving. Phil Liao. In this project, we compareand contrast five automated geometry-theorem-proving techniques. http://www.eecs.berkeley.edu/Research/Areas/Summaries/1995/95.13.ai.html

Chapter 13. Artificial Intelligence Chapter 13. Introduction Artificial intelligence is one of the fastest growing areas of research in computer science and is currently expanding at Berkeley. Well-established projects include Professor Robert Wilensky's research on cognitive approaches to natural language processing, planning, and information retrieval, and Professor Lotfi Zadeh's work on fuzzy logic and its applications. A major new project under the direction of Professor Wilensky addresses the concept of design of digital libraries. The new Berkeley Initiative on Soft Computing (BISC), under the direction of Professor Zadeh, is expanding our frontiers still further. Automated Geometry-Theorem Proving Phil Liao (Professor R. J. Fateman) (NSF) CCR-92-14963 Automated theorem proving has a variety of applications, including program verification and robot motion planning. In this project, we compare and contrast five automated geometry-theorem-proving techniques. They are: (1) Wu's method, (2) Gršbner basis method, (3) Tarski's decision procedure and cylindrical algebraic decomposition, (4) Rege's resultant method, and (5) Chou's point-elimination method. All five methods can be applied to geometries associated with the space of complex coordinates. The third and fourth can be restricted to geometries of the real space. In particular, the third methods are complete decision procedures that solve membership problems in semi-algebraic sets over the reals. Our investigations include the process of translating geometry statements into algebraic equations and inequalities, the cost of reducing the equations to triangular form, which is required by Wu's and Rege's methods, and the formulation of nondegenerate conditions.

85. Geometry Site Map Part B Cutting Up. Part C The Midline theorem. The Midline Cut Some GeometryFacts Proving the Midline theorem. Homework. Notes Part B Notes Part C Notes. http://www.learner.org/channel/courses/learningmath/geometry/sitemap.html

Site Map for This Course Geometry home page Overview Who's Who ... What Is Geometry? Part A: Quick Images Part B: Building from Directions Part C: Folding Paper Constructions Constructing Triangles Concurrencies in Triangles ... More Constructions Part D: Basic Objects Homework Notes: Part A Notes ... Triangles and Quadrilaterals Part A: Different Triangles Drawing Triangles Classifying Triangles Part B: Linkage-Strip Constructions Constructing Triangles Constructing Quadrilaterals Properties of Triangles Part C: Building Towers Homework Notes: Part A Notes ... Polygons Part A: Hidden Polygons Identifying Polygons Finding Polygons Part B: Classifying Polygons Properties of Polygons Grouping Polygons Polygon-Classification Game ... More Venn Diagrams Part C: Definitions and Proof Definitions Understanding Definitions Dividing Polygons into Triangles ... Parallel Lines and Circles Part A: Introduction to Geometer's Sketchpad Drawing with Geometer's Sketchpad Tutorial Constructing with Geometer's Sketchpad Part B: Parallel Lines Properties of Angles Reasoning About Properties of Angles Part C: Circles Inscribed Angles More Circle Constructions Part D: Thinking with Technology Homework Notes: Part B Notes ... Dissections and Proof Part A: Tangrams Exploring Tangrams Moving Tangrams Part B: Cutting Up Part C: The Midline Theorem The Midline Cut Some Geometry Facts Proving the Midline Theorem ... The Pythagorean Theorem Part A: The Pythagorean Theorem Calculating Area Squares Around a Right Triangle The Theorem Part B: Proving the Pythagorean Theorem What Is a Theorem?

86. Perseus Update In Progress Equating the sides of a right triangle has had a dramatic effect on the study ofgeometry even up to the present. However, the Pythagorean theorem did lead to http://www.perseus.tufts.edu/GreekScience/Students/Mike/geometry.html

The Perseus Digital Library is Being Updated Notice The main Perseus web site (at Tufts) is unavailable from 5:00 to 6:00, US Eastern time, in order to rebuild its databases with new or changed meta-data. We apologize for this inconvenience.

87. ThinkQuest Library Of Entries geometry Postulate, theorem, and Corollary. Part 6. theorem 7.10, A parallelogramwith a diagonal that bisects opposite angles is a rhombus. http://library.thinkquest.org/16284/reference_gc_6.htm

Welcome to the ThinkQuest Internet Challenge of Entries The web site you have requested, Math Planet , is one of over 4000 student created entries in our Library. Before using our Library, please be sure that you have read and agreed to our To learn more about ThinkQuest. You can browse other ThinkQuest Library Entries To proceed to Math Planet click here Back to the Previous Page The Site you have Requested ... Math Planet click here to view this site A ThinkQuest Internet Challenge 1998 Entry Click image for the Site Languages : Site Desciption Math Planet is web site dedicated to the advancement of mathematics. That's good, because today's demanding world requires young people to be equipped with a solid math foundation. Targeted towards high school students, there are many different categories, including Basic Algebra and Geometry, advanced Algebra and Trigonometry, SAT and ACT math preparation courses, and even a chat room for team problem solving. Students Wen Vestavia Hills High School AL, United States Parakash Vestavia Hills High School AL, United States

88. Luigi Bianchi A mathematician who developed many theorems regarding Riemannian geometry http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Bianchi.html

89. Biography Of Pappus Of Alexandria Wrote treatise, the Mathematical Collection, as a guide to Greek geometry, discusses theorems and constructions of more than thirty different mathematicians of antiquity. http://www.lib.virginia.edu/science/parshall/pappus.html

Biography of Pappus of Alexandria Pappus of Alexandria flourished in the first half of the fourth century. He wrote his treatise, the Mathematical Collection , as a guide to Greek geometry. Here Pappus discusses theorems and constructions of more than thirty different mathematicians of antiquity, including Euclid , Archimedes and Ptolemy. Sometimes, as in the case of the problem of inscribing the five regular solids in a given sphere, Pappus provides alternatives to the proofs given in earlier works. In other cases, he generalizes theorems of earlier writers, as he does with the Pythagorean Theorem found in Euclid's Elements MAIN DOCUMENT CONTENTS FIRST MENTION To return to place in document from which you came, click on your browser's BACK BUTTON. Selected Biographical References Gillispie, Charles C. ed. The Dictionary of Scientific Biography , 16 vols. 2 supps. New York: Charles Scribner's Sons, 1970-1990. S.v. "Pappus of Alexandria" by Ivor Bulmer-Thomas. Heath, Thomas L. A History of Greek Mathematics , 2 vols. Oxford: Oxford University Press, Clarendon Press, 1921. 1:355-439.

90. Pascal's Theorem -- From MathWorld Coxeter, H. S. M. and Greitzer, S. L. Pascal's theorem. §3.8 in GeometryRevisited. Washington, DC Math. Assoc. Amer., pp. 7476, 1967. http://mathworld.wolfram.com/PascalsTheorem.html

Geometry Line Geometry Incidence Geometry ... Barile Pascal's Theorem The dual of Brianchon's theorem (Casey 1888, p. 146), discovered by B. Pascal in 1640 when he was just 16 years old (Leibniz 1640; Wells 1986, p. 69). It states that, given a (not necessarily regular , or even convex hexagon inscribed in a conic section , the three pairs of the continuations of opposite sides meet on a straight line , called the Pascal line -gon inscribed in a conic section are collinear, then the same is true for the remaining point. Braikenridge-Maclaurin Construction Brianchon's Theorem Cayley-Bacharach Theorem Conic Section ... Steiner's Theorem References Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Casey, J. "Pascal's Theorem." §255 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Cayley, A.

91. Publications Of Xiao-Shan Gao SC Chou, XS Gao, and JZ Zhang, A Deductive Database Approach To Automated GeometryTheorem Proving and Discovering, {\it J. Automated Reasoning}, 25(3), 219246 http://www.mmrc.iss.ac.cn/~xgao/publ.html

Publications of Xiao-Shan Gao Preprints X.S. Gao and M. Li, Approximate Implicitization of Planar Parametric Curves using Quadratic Splines, MM-Preprpints, No21, 2002. X.S. Gao and J. Tang, On the Number of Solutions for the P4P Problem, MM-Preprpints, No21, 2002. X.-S. Gao, Implicitization for Differential Rational Parametric Equations, Preprint 2000, submitted to JSC. J. Wang and X.-S. Gao, An Algorithm for Solving Partial Differential Parametric Systems, Books A. Cohen, X.S. Gao, N. Takayama (eds), Mathematical Software , World Scientific Pub., Singapore, 2002. X. S. Gao and D. Wang, Mathematics Mecanization and Applications , Academic Press, London, 2000. X. S. Gao and D. Wang, Computer Mathematics-Proc. of ASCM'2000 , World Scientific, Singapore, 2000. X. S. Gao, D. Wang and L. Yang, Automated Deduction in Geometry, Procs. of ADG98 Springer, Berlin, 1999. X. S. Gao, J. Z. Zhang, and S.C. Chou, Geometry Expert (in Chinese), Nine Chapters Pub., Taiwan, 1998. S.C. Chou, X.S. Gao, and J.Z. Zhang, Machine Proofs in Geometry , World Scientific, Singapore, 1994.

92. Records Of Type Records of Type GEO. Solutions of problems from mechanized geometrytheorem proving. The GEO table contains solutions of problems http://www.symbolicdata.org/SD_HTML/Data/GEO/?fr

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