61. Algebra The Pythagorean theorem. In a right triangle, the longest side is called thehypotenuse. The hypotenuse is always the side opposite the right angle. http://www.dial-a-teacher.com/algebra/page10.html

The Pythagorean Theorem In a right triangle, the longest side is called the hypotenuse. The hypotenuse is always the side opposite the right angle. The other two sides are called the legs of the triangle. We usually use the variables a and b to identify the legs and c for the hypotenuse. They are related as follows. In any right triangle, if a and b are the lengths of the legs and c is the length of the hypotenuse, then a + b = c Commutative Properties Identity Properties Associative Properties Additive Inverses ... Exponents More with Exponents Polynomials Slopes The Pythagorean Theorem Complex Numbers

62. Parikh's Theorem In Commutative Kleene Algebra We prove the following general theorem of commutative Kleene algebra, of which Parikh'stheorem is a special case Every system of polynomial inequalities fi(x1 http://www.computer.org/conferen/proceed/lics/0158/01580394abs.htm

14th Symposium on Logic in Computer Science July 02 - 05, 1999 Trento, Italy p. 394 Parikh's Theorem in Commutative Kleene Algebra Mark W. Hopkins, Adaptive Micro Systems, Inc. Dexter C. Kozen Cornell University ... Kleene algebra, formal methods, universal algebra, logics of programs The full text of lics is available to members of the IEEE Computer Society who have an online subscription and an web account

63. A General Conservative Extension Theorem In Process Algebra A general conservative extension theorem in process algebra. C. VerhoefDepartment of Mathematics and Computing Science, Eindhoven http://adam.wins.uva.nl/~x/ece/finalifip.html

A general conservative extension theorem in process algebra C. Verhoef Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, NL-5600 MB Eindhoven, The Netherlands email chrisv@win.tue.nl NOTE: Since this document contains loads of formulas that cannot be processed with a tex to html facility please find a PS file of this document on my SOS page Abstract: Keyword Codes: F.1.2, F.3.2, F.4.3. Keywords: Concurrency; Operational Semantics; Algebraic language theory. 1 INTRODUCTION In the past few years people working in the area of process algebra have started to extend process theories such as CCS, CSP, and ACP with, for instance, real-time or probabilistics. A natural question that arises is whether or not such an extension is somehow related with its subtheory, for instance, whether or not the extension is conservative in some sense. If we add new operators and/or rules to a particular transition system it would be nice to know whether or not provable transitions of a term in the original system are the same as those in the extended system for that term; we will call this property operational conservativity (cf. [

64. GeoSci 236: The Fundamental Theorem Of Linear Algebra GeoSci 236 The Fundamental theorem of Linear algebra. Gidon Eshel 491 Hinds Dept. Figure1 The forward problem (the fundamental theorem of linear algebra). http://geosci.uchicago.edu/~gidon/geosci236/fundam/

GeoSci 236: The Fundamental Theorem of Linear Algebra Gidon Eshel 491 Hinds Dept. of the Geophysical Sciences, 5734 S. Ellis Ave., The Univ. of Chicago, Chicago, IL 60637 geshel@midway.uchicago.edu Figure 1: The forward problem (the fundamental theorem of linear algebra). A 's domain is the upper-left space, while its range is the lower-right one. In the domain, A 's row-space is shown in red , while its nullspace in blue . A generic vector comprising both a row-space and a nullspace components is the vector on which A operates, mapping it onto the adjoint space (lower-right). In the latter space, the shown b comprises components from A 's range (column-space) and left nullspace Figure 1 represents the operation of a matrix on a vector (the upper-left space). That is, it shows schematically what happens when an arbitrary vector from 's domain (the space corresponding dimensionally to 's row dimension N ) is mapped by onto the range space (the space corresponding dimensionally to 's column dimension M ). Hence the schematic shows what happens to from the upper-left space as transforms it to the range, the lower-right space. Put differently, this schematic represents the

65. Wilson Stothers' Cabri Pages - Algebra these topics. The proofs may be obtained by clicking on the link belowthe statement of each theorem. A at U . Proof of theorem 1. We http://www.maths.gla.ac.uk/~wws/cabripages/algebra.html

Poles, polars and duality -the algebraic version In this section, we give an algebraic treatment of these topics. The proofs may be obtained by clicking on the link below the statement of each theorem. A plane conic has an equation of the form ax +bxy +cy +fx+gy+h=0. In terms of homogeneous coordinates , this becomes ax +bxy +cy +fxz+gyz+hz which can be written as x T M x where x =(x,y,z) , and M is a symmetric 3x3 matrix. For a non-degenerate conic, M must be non-singular and have eigenvalues of different sign. Note that, if a conic contains three (distinct) collinear points, then it must be degenerate. Definition If C: x T M x is a non-degenerate conic and U=[ u is any point, then the algebraic polar of U with respect to C is the line u T M x Note that, as M is non-singular, we cannot have u T M= , so that the line always exists. A line L has an equation a T x . Now, u T M x and a T x give the same line if and only if u ]=[M a Thus L is the polar of a unique point U=[ u Definition If C: x T M x is a non-degenerate conic and L is any line, then the algebraic pole of L with respect to C is the point U=[ u such that L has equation u T M x Remark If L has equation a T x , then, as we have seen, the pole of L is U=[M a Theorem 1 If C: x T M x is a non-degenerate conic and U is any point on C then the algebraic polar of U with respect to C is the tangent to C at U Proof of Theorem 1 We now show that the algebraic polar coincides with the geometrical polar.

66. Katholieke Universiteit Nijmegen The Faculty of Mathematics and Computer Science of Eindhoven University of Technologyis strong in computer algebra, theorem proving and applying Web http://mowgli.cs.unibo.it/html_yes_frames/sites/nijmegen.html

MoWGLI: Mathematics on the Web: Get It by Logic and Interfaces Katholieke Universiteit Nijmegen The Netherlands Subfaculteit Informatica, Faculteit Natuurwetenschappen, Wiskunde en Informatica, Katholieke Universiteit Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands Visit the institution home page. Site responsible: Prof. Herman Geuvers Site members: Prof. Herman Geuvers Prof. Arjeh Cohen Prof. Henk Barendregt Dr. Freek Wiedijk ... Dan Synek The Sub-faculty of Computer Science at the University of Nijmegen hosts a broad experience in logic, formal methods and theorem proving. The Faculty of Mathematics and Computer Science of Eindhoven University of Technology is strong in computer algebra, theorem proving and applying Web technology to mathematics. Nijmegen and Eindhoven have a long history in cooperation on topics related to this FET proposal, notably type theory, theorem proving and combining various computer mathematics applications, especially using OpenMath. This cooperation was mainly taking place between the research groups of Geuvers and Barendregt in Nijmegen and the research group of Cohen in Eindhoven. The research group of Geuvers and Barendregt is part of the EC sponsored Thematic Network ``TYPES'' (IST-1999-29001) and of its ancestor, the EC Working Group ``Types for Proofs and Programs'', which testifies there interest in theorem proving, especially using type theory based theorem provers. The FTA project (Fundamental Theorem of Algebra), started in 1999 and to be finished in 2001, has as its main goal to formalize (in Coq) a large body of undergraduate mathematics (algebra and analysis), culminating in a proof of the fundamental theorem of algebra. The formalization of the mathematics is now finished and the next step is to make the formalization accessible and usable by others, preferably through the World Wide Web.

67. Computer Algebra Meets Automated Theorem Proving: Integrating Maple And PVS Computer algebra Meets Automated theorem Proving Integrating Mapleand PVS. Authors. Andrew Adams, Martin Dunstan, Hanne Gottliebsen http://www.csl.sri.com/users/owre/papers/tphols01/

Computer Algebra Meets Automated Theorem Proving: Integrating Maple and PVS Authors Andrew Adams, Martin Dunstan, Hanne Gottliebsen, Tom Kelsey, Ursula Martin, and Sam Owre Abstract We describe an interface between version 6 of the Maple computer algebra system with the PVS automated theorem prover. The interface is designed to allow Maple users access to the robust and checkable proof environment of PVS. We also extend this environment by the provision of a library of proof strategies for use in real analysis. We demonstrate examples using the interface and the real analysis library. These examples provide proofs which are both illustrative and applicable to genuine symbolic computation problems. PDF BibTeX Entry Sam Owre: owre@csl.sri.com

68. Brownian Motion And The Fundamental Theorem Of Algebra proof of this fact and show how it can be applied in some surprising ways We'llsee an elementary proof of the fundamental theorem of algebra, and (with the http://random.gromoll.org/research/talks/eurandom200210

69. Abstract Display For Parikh's Theorem In Commutative Kleene Algebra We prove the following general theorem ofcommutative Kleene algebra, of which Parikh'sand Pilling's theorems arespecial cases Every system of polynomial http://techreports.library.cornell.edu:8081/Dienst/UI/1.0/Summarize/cul.cs/TR99-

70. Parikh's Theorem In Commutative Kleene Algebra http://techreports.library.cornell.edu:8081/Dienst/UI/1.0/Display/cul.cs/TR99-17

71. Combining Algebra And Universal Algebra In First-Order Theorem Proving Combining algebra and Universal algebra in FirstOrder theorem Proving The Caseof Commutative Rings. Leo Bachmair, Harald Ganzinger and Jürgen Stuber (1995). http://www.loria.fr/~stuber/publications/COMPASS94.html

Coordinates Publications Software Research interests ... Other interests Combining Algebra and Universal Algebra in First-Order Theorem Proving: The Case of Commutative Rings Egidio Astesiano, Gianna Reggio and Andrzej Tarlecki (eds), Recent Trends in Data Types Specification, 10th Workshop on Specification of Abstract Data Types joint with the 5th COMPASS Workshop, Santa Margherita Ligure, Italy, 1994. Selected papers. LNCS 906, Springer Verlag. Abstract We present a general approach for integrating certain mathematical structures in first-order equational theorem provers. More specifically, we consider theorem proving problems specified by sets of first-order clauses that contain the axioms of a commutative ring with a unit element. Associative-commutative superposition forms the deductive core of our method, while a convergent rewrite system for commutative rings provides a starting point for more specialized inferences tailored to the given class of formulas. We adopt ideas from the Gröbner basis method to show that many inferences of the superposition calculus are redundant. This result is obtained by the judicious application of the simplification techniques afforded by convergent rewriting and by a process called symmetrization that embeds inferences between single clauses and ring axioms. Bibtex DVI Postscript Jürgen Stuber ... stuber@loria.fr

72. MA 109 College Algebra Notes Equations; Exercises. Chapter 4 The Fundamental theorem of AlgebraThe Overall Strategy for Proving the Fundamental theorem; Continuity; http://www.msc.uky.edu/ken/ma109/notes.htm

College Algebra Table of Contents Chapter 1: Algebra and Geometry Review Algebra Simplifying Expressions Solving Equations ... Exercises The button will return you to class homepage Revised: Aug 21, 2001

73. Theorem Proving And Computer Algebra This talk will describe the possible relationships between computer algebra systemsand Automated theorem Proving, and the benefits offered by each, and then http://www.dcs.gla.ac.uk/provers/stp/0397/steve.html

Theorem Proving and Computer Algebra Steve Linton Computer Algebra systems have a number of features that distinguish them from other software systems of similar size and complexity: on the one hand, there is the natural underlying semantics of pure mathematics, on the other hand, many of the algorithms used rely on sophisticated mathematical results for their correctness and may impose conditions on their inputs that go far beyond the scope of normal type checking. Finally, these systems tend to grow over time, and without a formal framework for the precise communication of the functions and restrictions of existing components, this process is inefficient and unsafe. This talk will describe the possible relationships between computer algebra systems and Automated Theorem Proving, and the benefits offered by each, and then outline the work being done in St Andrews on one of them.

74. [math/0204141] A Quasi-Hopf Algebra Freeness Theorem From Peter Schauenburg schauen@mathematik.unimuenchen.de Date Wed,10 Apr 2002 190250 GMT (9kb) A quasi-Hopf algebra freeness theorem. http://arxiv.org/abs/math/0204141

Mathematics, abstract math.QA/0204141 A quasi-Hopf algebra freeness theorem Authors: Peter Schauenburg Comments: 7 pages Subj-class: Quantum Algebra MSC-class: We prove the quasi-Hopf algebra version of the Nichols-Zoeller theorem: A finite-dimensional quasi-Hopf algebra is free over any quasi-Hopf subalgebra. Full-text: PostScript PDF , or Other formats References and citations for this submission: CiteBase (autonomous citation navigation and analysis) Links to: arXiv math find abs

75. Mega-Theorem Of Death (of Linear Algebra) many other equivalent properties As far as I can tell, this name for this theoremwas coined by Dane Johnson of the famous Linear algebra class of 1990. http://www.willamette.edu/~mjaneba/help/mtd.html

An n x n matrix A is nonsingular if and only if... There is a matrix B such that AB=BA=I [this is the definition of having an inverse OR A is row-equivalent to I [i.e. a finite number of elementary row operations will reduce A to I OR The only n x 1 matrix X such that AX=0 is X=0 (i.e. the linear system AX=0 has only the trivial solution). OR The null space of A OR For all n x 1 matrices B, AX=B has at least one solution. OR For all n x 1 matrices B, AX=B has at most one solution. OR det( A ) is nonzero. OR The columns of A span R n OR The columns of A are linearly independent. OR The rows of A are linearly independent. OR A has rank n OR A has nullity 0. OR A has no zero eigenvalue. OR ... many other equivalent properties ... As far as I can tell, this name for this theorem was coined by Dane Johnson of the famous Linear Algebra class of 1990 . You can see Dane in a picture of the class , in the back row looking down at his book. How widespread has this name become? If you have heard it elsewhere, e-mail me A student view of the Mega-theorem A printer-friendly version of this page Last Modified May 9, 1998. Prof. Janeba's Home Page

76. Fundamental Theorem Of Algebra Fundamental theorem of algebra. Institution Nijmegen university. Keywords FTA,Fundamental, theorem, algebra, Constructive, Real, Complex, Polynomial. http://pauillac.inria.fr/cdrom/www/coq/contribs/fta.html

Fundamental Theorem of Algebra A constructive proof of the Fundamental Theorem of Algebra (every non-trivial polynomial equation P(z)=0 always has a solution in the complex plane) Download (archive compatible with Coq V7.3) Authors: Herman Geuvers Freek Wiedijk Jan Zwanenburg Randy Pollack Henk Barendregt Institution: Nijmegen university Keywords: FTA, Fundamental, Theorem, Algebra, Constructive, Real, Complex, Polynomial This page was automatically generated from this description file

77. Pythagorean Theorem Pythagorean theorem a 2 + b 2 = c 2 , where c is the length of the hypotenuseand a and b are the lengths of the legs. Back to algebra Solutions http://www.gomath.com/algebra/pythagorean.asp

78. Boolean Cubes And Uniting Theorem Of Boolean Algebra Slide 2 of 22. http://www.cse.ucsc.edu/classes/cmpe126/Winter03/slides/TwoLevelMin/sld002.htm

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79. An Integrated Framework For Computer Algebra And Computer Theorem Proving An Integrated Framework for Computer algebra and Computer theorem Proving.A MITACS project led by Dr. William M. Farmer. Dr. Farmer's homepage. http://www.mitacs.math.ca/projects/wmfarmer/

An Integrated Framework for Computer Algebra and Computer Theorem Proving A MITACS project led by Dr. William M. Farmer Dr. Farmer's homepage Project Description A mechanized mathematics system (MMS) is a computer environment that is intended to support, improve, and sometimes automate mathematical reasoning and computing. Computer algebra systems and computer theorem proving systems are both examples of MMSs. Each type of MMS has its own strengths and weaknesses. Our objective is develop a new approach to mechanized mathematics that combines the strengths of symbolic algebra systems with those of theorem proving systems to provide a system that is both powerful and sound. Project Participants Martin von Mohrenschildt (CS, McMaster) David Parnas (CS, McMaster) MITACS is a joint initiative of...

80. GraspMath College Algebra Video 6- Polynomial Functions, Rational Root Theorem Video 6 Polynomial Functions, Rational Root theorem. Polynomial Functions. Divisionof Polynomials, the Remainder theorem and the Factor theorem. http://www.graspmath.com/graspmath/college6.html

Video 6- Polynomial Functions, Rational Root Theorem. Polynomial Functions. This segment covers graphing and finding horizontal and vertical intercepts for general polynomial functions. Division of Polynomials, the Remainder Theorem and the Factor Theorem. This segment covers long division for polynomials, synthetic division, the use of the remainder theorem for calculating function values by synthetic division, and the use of the factor theorem for finding roots of polynomials. Rational Root Theorem. This segment covers the use of the rational root theorem for factorization of polynomials. Vertical and Horizontal Asymptotes. This segment covers methods of finding vertical, horizontal, and oblique asymptotic lines to graphs of rational functions. Purchase Our Price: $29.95 To order video products for your school, please click here

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