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# Topic: Elimination theory

###### In the News (Thu 20 Jun 19)

 Elimination theory - Wikipedia, the free encyclopedia In algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating between polynomials of several variables. The historical development of commutative algebra, which was initially called ideal theory, is closely linked to concepts in elimination theory: ideas of Kronecker, who wrote a major paper on the subject, were adapted by Hilbert and effectively 'linearised' while dropping the explicit constructive content. Elimination of quantifiers is a term used in mathematical logic to explain that in some cases - algebraic geometry of projective space over an algebraically closed field being one - existential quantifiers can be removed. www.wikipedia.org /wiki/Elimination_theory   (321 words)

 Proof theory - Wikipedia, the free encyclopedia Proof theory, studied as a branch of mathematical logic, represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof. Structural proof theory is connected to type theory by means of the Curry-Howard correspondence, which observes a structural analogy between the process of normalisation in the natural deduction calculus and beta reduction in the typed lambda calculus. en.wikipedia.org /wiki/Proof_theory   (942 words)

 Proof theory - Open Encyclopedia   (Site not responding. Last check: 2007-10-20) The subject of proof theory has a significant if somewhat opaque prehistory as metamathematics, the proposed theory under development from the start of the twentieth century, for a generation, under the influence of David Hilbert. Hilbert's ideas seem to have been based on an analogy, in fact false, with the elimination theory of algebraic geometry familiar to him from his early work in algebra. Kurt Gödel's seminal work on proof theory first advanced, then refuted this program: his completeness theorem seemed to bring Hilbert's dream of reducing all mathematics to a small, finitarily meaningful core within reach, then his incompleteness theorems showed that the dream was unattainable. open-encyclopedia.com /Proof_theory   (1063 words)

 Refinements of Theory Model Elimination and a Variant without Contrapositives - Baumgartner (ResearchIndex)   (Site not responding. Last check: 2007-10-20) Theory Reasoning means to build-in certain knowledge about a problem domain into a deduction system or calculus, such as model elimination. Linear and Unit-Resulting Refutations for Horn Theories - Baumgartner (1995) 1 A ModelElimination Calculuswith Built-in Theories (context) - Baumgartner - 1992 citeseer.ist.psu.edu /13762.html   (736 words)

 Ockham's Connotation Theory and Ontological Elimination First, this theory implies that every categorematic term in written or spoken language is necessarily equivocal, since a written or spoken categorematic term (e.g., 'cat') is subordinate to one concept (e.g., "subject-cat") when it is in the subject position, and another concept (e.g., "object-cat") when it is in the predicate position. Briefly, the theory of "exponibles" is a theory that reduces an "exponible" proposition - a categorical proposition in its outward form - to a molecular or hypothetical proposition (e.g., 'Socrates is blind' is reduced to 'Socrates exists', 'Socrates should have sight', and 'Socrates does not have sight'). In general, the theory of "exponibles" is a semantical theory of propositions, while the connotation theory a theory of terms. www.fordham.edu /gsas/phil/klima/ZHENG.htm   (4246 words)

 Dictionary of Philosophy of Mind - eliminativism   (Site not responding. Last check: 2007-10-20) Like its predecessor, the mind-brain identity theory, eliminativism claims that it is an empirical fact, rather than a conceptual necessity, that mental states are identical with brain states, and that this fact is justified only by scientific evidence. Instead the old theory is often eliminated, and replaced with a better theory that rejects or ignores the ontological assumptions of the old theory. The obvious objection to causal theories of reference is that surely not all causal relationships between language users and objects are reference relations, and it is not easy to figure out which causal relations are reference relations and which are not. www.artsci.wustl.edu /~philos/MindDict/eliminativism.html   (9937 words)

 Elimination theory   (Site not responding. Last check: 2007-10-20) In algebraic geometry, elimination theoryis the classical name for algorithmic approaches to eliminating between polynomials of several variables. In the same way, computational techniques for elimination can inpractice be based on Grobner basis methods. The historical development of commutative algebra, whichwas initially called ideal theory, is closely linked to concepts in elimination theory: ideas of Kronecker, who wrote amajor paper on the subject, were adapted by Hilbert and effectively'linearised' while dropping the explicit constructive content. www.therfcc.org /elimination-theory-177662.html   (299 words)

 Read about Proof theory at WorldVillage Encyclopedia. Research Proof theory and learn about Proof theory here!   (Site not responding. Last check: 2007-10-20) recursion theory, proof theory is one of the so-called four pillars of the self-verifying theories, systems strong enough to talk about themselves, but too weak to carry out the diagonal argument that is the key to Gödel's unprovability argument. Structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of encyclopedia.worldvillage.com /s/b/Proof_theory   (993 words)

 Selected Publications These restart model elimination calculi are proven sound and complete and their implementation by PTTP is depicted. The transformed theory can be used in combination with linear calculi such as e.g.\ (theory) model elimination to yield sound, complete and efficient calculi for full first order clause logic over the given Horn theory. Theory reasoning means to relieve a calculus from explicitly drawing inferences in a given theory by special purpose inference rules (e.g. www.uni-koblenz-landau.de /ag-ki/TheTP/publications.html   (2111 words)

 Elimination Elimination theory plays an important role when working with ideals of multivariate polynomial rings. If the elimination ideals I_k are to be computed for several different k, it is recommended first that a Gröbner basis with respect to lexicographical order for I first be computed as then the elimination ideals can be determined trivially. 4.3.8], the elimination ideal E_y = I intersect (Z[Sqrt(- 5))[y] is shown to be generated by f_5 = (5 + Sqrt(- 5))y^2 - 15y and f_7 = - 2Sqrt(- 5)y^2 + 5(1 + Sqrt(- 5))y. www.math.niu.edu /help/math/magmahelp/text620.html   (1104 words)

 ACA 2003 Session T8: Elimination Theory Elimination Theory is the study of eliminating unknowns from equations (polynomial or differential). Elimination theory is classical topic that has been studied for hundreds of years. Elimination theory of non-commutative algebras of differential invariants math.unm.edu /ACA/2003/Sessions/T8.html   (1408 words)

 Simultaneous equations - Wikipedia, the free encyclopedia Systems of simultaneous linear equations are studied in linear algebra and can always be solved; one uses Gauss-Jordan elimination or the Cholesky decomposition. Algebraic geometry is essentially the theory of simultaneous polynomial equations. The question of effective computation with such equations belongs to elimination theory. en.wikipedia.org /wiki/Simultaneous_equations   (363 words)

 Noncommutative elimination theory   (Site not responding. Last check: 2007-10-20) If G is a Gröbner Basis with respect to an elimination order (see Definition 11.2) and I is the ideal generated by G, then there exist a nested sequence of ideals This result is crucial to assuring that the Gröbner Basis algorithm puts the collection of polynomial equations into a triangular form. Pure lex and, more generally, multigraded lex are examples of such elimination orders. math.ucsd.edu /~ncalg/StrategyPaper/node16.html   (107 words)

 Ph.D. Thesis   (Site not responding. Last check: 2007-10-20) This thesis is on extensions of calculi for automated theorem proving towards theory reasoning. For TME the emphasis lies on the combination with theory reasoners to be obtained by new compilation approach. A theory reasoning system is only operable in combination with an (efficient) background reasoner for the theories of interest. www.uni-koblenz.de /~peter/thesis.html   (292 words)

 ThugLifeArmy.com Keeping The Legacy Alive ‘Elimination Theory,’ written by former FBI and CIA spy TJ Byron, alleges his story of being involved as a double agent and informant in the mid-eighties for the FBI, CIA, and the Republic of South African Intelligence. Project Coast was a program that had the purpose of protecting the white community in South Africa by eliminating most of its fl population, and containing its opponents at its borders. Byron did as he was asked for a number of years, but was continually shocked when his reports about the development and transfer of these agents were not stopped by the U.S. Government, even though it was aware of what was happening based on his reports to the FBI and CIA. www.thuglifearmy.com /news?id=1955   (514 words)

 Math-Angers : Prépublication 191   (Site not responding. Last check: 2007-10-20) The theory of boolean algebras is well known to admit as a model-completion the theory of dense boolean algebras. There is a wide class of lattices in which the Krull-dimension of the spectrum (that is the Stone space of prime filters) is definable in a natural way, we call it the class of scaled lattices. We prove that the theory of scaled lattices of arbitrary dimension N admits a model-completion and give an explicit axiomatization of it, which boils down to the usual one (density) in the zero dimensionnal case. math.univ-angers.fr /preprint/191.html   (195 words)

 CH4 Loss from (CH)4N+revisited: how does this high energy elimination compete with ·CH3 loss?   (Site not responding. Last check: 2007-10-20) However, according to results obtained and presented by RRKM theory, methane elimination through this transition state would be too slow to compete with methyl loss. It is unlikely that CH is actually lost by a concerted elimination because only a small fraction of the reverse activation energy becomes translational energy, whereas concerted eliminations usually convert substantial fractions of their reverse activation energies into translational energy. Thus it is concluded that CH elimination from (CH occurs by loss of ·CH followed by loss of H· at all energies, even though that is a higher energy process than methane elimination. www.impub.co.uk /abs/EMS10_0767.html   (355 words)

 Mortgage Elimination Scams At first blush, it would seem there are dozens of court cases in which the judge actually did what they claimed, that is, declare a mortgage void because the lender used borrowed funds for the loan. Debt elimination programs that claim Federal Reserve approval or acquiescence and the satisfaction of legitimate debts through the presentation of suspicious documents are totally bogus. Members of the public are being harmed as borrowers generally pay significant amounts of money without eliminating or reducing their overall debt obligations - which of course is not in fact possible through any of these programs. www.legalwiz.com /articles/mortgage-elimination-scam.htm   (1540 words)

 Refinements of Theory Model Elimination and a Variant without Contrapositives - Baumgartner (ResearchIndex)   (Site not responding. Last check: 2007-10-20) Abstract: Theory Reasoning means to build-in certain knowledge about a problem domain into a deduction system or calculus, which is in our case model elimination. Several versions of theory model elimination (TME) calculi are presented and proven complete: on the one hand we have highly restricted versions of total and partial TME. 6 Horn equational theories and paramodulation (context) - Furbach, olldobler et al. citeseer.ist.psu.edu /70063.html   (963 words)

 Mathematics Colloquium: Coble Memorial Lectures Later, regarding two polynomials in several variables as polynomials in the last variable whose coefficiets were polynomials in the first variables, the same tools were used to systematically ``eliminate'' the last variable, simplifying the system. If R is a commutative ring and I is an ideal in R, the subring R[It] of the polynomial ring R[t] in one variable over R is called the Rees algebra of I. It is closely related to the construction of the blow-up in algebraic geometry. The geometric formulation of elimination, the problem of finding the image of a map of varieties, can be re-interpreted in terms of this algebra. www.math.uiuc.edu /Colloquia/04SP/eisenbud_mar16-04.html   (464 words)

 Pardo   (Site not responding. Last check: 2007-10-20) The main statement says that a particular class of elimination procedures requires simply exponential running time and this lower time bound is unavoidable. This particular class of procedures (universal elimination procedures) is not so particular since it contains all known elimination procedures. Universal elimination is a notion rising naturally from a suitable research context which is also introduced in the talk. www.bath.ac.uk /~masdr/pardo.html   (198 words)

 Application to Elimination Theory   (Site not responding. Last check: 2007-10-20) elimination theory in general is about eliminating a number of unknowns from a system of polynomial equations in one or more variables to get an easier equivalent system. As you can guess, this is not an efficient way especially if the goal is only to show the existence of a solution. To understand the importance of elimination theory, let us start by considering the following “simple” example. aix1.uottawa.ca /~jkhoury/elimination.htm   (555 words)

 Papers relevant to PROTEIN and LC This paper introduces the ancestry variant of restart model elimination; it is the preferred version for computing minimal answers. This is the primary text for the theory extensions of model elimination as to be found in PROTEIN. To demonstrate this, we show how a theorem prover, based on restart model elimination calculus, can be modified for abductive reasoning and thus for minimal model reasoning. www.uni-koblenz.de /ag-ki/Systems/Protein/papers.html   (955 words)

 Further directions in Galois theory Galois theory is an important tool for studying the arithmetic of ``number fields'' (finite extensions of Q) and ``function fields'' (finite extensions of F Other aspects of the theory of F[X] require more interesting modifications in the setting of p-polynomials, since polynomial multiplication is replaced by composition of p-polynomials, which is not commutative! Elimination theory: resultants, etc. How to actually compute things like the minimal polynomial of x+y where f(x)=g(y)=0, or where x,y are distinct roots of f(x)? www.math.harvard.edu /~elkies/M250.01/galois_topix.html   (611 words)

 Author T.J. Byron interviewed at TheBookInsider.com about his book, Elimination Theory: The Secret Covert Networks of ... THEBOOKINSIDER.COM: "Elimination Theory" is an amazing true-life story of your your involvement as an informant and agent for the FBI's South African intelligence during the apartheid era and the CIA. Project Coast was a program that had the purpose of protecting Afrikaanerdom in South Africa, eliminate most of its fl population and contain its opponents at its borders. Elimination Theory is the true story of my involvement as an informant/agent for the FBI, South African Intelligence, and the CIA during Project Coast and its networks in both the USA and South Africa. www.prweb.com /releases/2005/9/prweb284472.htm   (1369 words)

 A K Peters, Ltd. - Regular Sequences and Resultants   (Site not responding. Last check: 2007-10-20) This carefully prepared manuscript presents elimination theory in weighted projective spaces over arbitrary noetherian commutative base rings. Elimination theory is a classical topic in commutative algebra and algebraic geometry, and it has become of renewed importance recently in the context of applied and computational algebra. This monograph provides a valuable complement to sparse elimination theory in that it presents in careful detail the algebraic difficulties from working over general base rings. www.akpeters.com /product.asp?ProdCode=1519   (183 words)

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