摘要 :
An n-ary Steiner law f(x(sub 1),x(sub 2),(hor ellipsis),x(sub n)) on a projective curve (Gamma) over an algebraically closed field k is a totally symmetric n-ary morphism f from (Gamma)(sup n) to (Gamma) satisfying the universal i...
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An n-ary Steiner law f(x(sub 1),x(sub 2),(hor ellipsis),x(sub n)) on a projective curve (Gamma) over an algebraically closed field k is a totally symmetric n-ary morphism f from (Gamma)(sup n) to (Gamma) satisfying the universal identity f(x(sub 1),x(sub 2),(hor ellipsis),x(sub n-1), f(x(sub 1),x(sub 2),(hor ellipsis),x(sub n))) = x(sub n). An element e in (Gamma) is called an idempotent for f if f(e,e,(hor ellipsis),e) = e. The binary morphism x * y of the classical chord-tangent construction on a nonsingular cubic curve is an example of a binary Steiner law on the curve, and the idempotents of * are precisely the inflection points of the curve. In this paper, the authors prove that if f and g are two 5-ary Steiner laws on an elliptic curve (Gamma) sharing a common idempotent, then f = g. They use a new rule of inference rule =(gL)(implies), extracted from a powerful local-to-global principal in algebraic geometry. This rule is implemented in the theorem-proving program OTTER. Then they use OTTER to automatically prove the uniqueness of the 5-ary Steiner law on an elliptic curve. Very much like the binary case, this theorem provides an algebraic characterization of a geometric construction process involving conics and cubics. The well-known theorem of the uniqueness of the group law on such a curve is shown to be a consequence of this result.
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摘要 :
An n-ary Steiner law f(x(sub 1),x(sub 2),(hor ellipsis),x(sub n)) on a projective curve (Gamma) over an algebraically closed field k is a totally symmetric n-ary morphism f from (Gamma)(sup n) to (Gamma) satisfying the universal i...
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An n-ary Steiner law f(x(sub 1),x(sub 2),(hor ellipsis),x(sub n)) on a projective curve (Gamma) over an algebraically closed field k is a totally symmetric n-ary morphism f from (Gamma)(sup n) to (Gamma) satisfying the universal identity f(x(sub 1),x(sub 2),(hor ellipsis),x(sub n-1), f(x(sub 1),x(sub 2),(hor ellipsis),x(sub n))) = x(sub n). An element e in (Gamma) is called an idempotent for f if f(e,e,(hor ellipsis),e) = e. The binary morphism x * y of the classical chord-tangent construction on a nonsingular cubic curve is an example of a binary Steiner law on the curve, and the idempotents of * are precisely the inflection points of the curve. In this paper, the authors prove that if f and g are two 5-ary Steiner laws on an elliptic curve (Gamma) sharing a common idempotent, then f = g. They use a new rule of inference rule =(gL)(implies), extracted from a powerful local-to-global principal in algebraic geometry. This rule is implemented in the theorem-proving program OTTER. Then they use OTTER to automatically prove the uniqueness of the 5-ary Steiner law on an elliptic curve. Very much like the binary case, this theorem provides an algebraic characterization of a geometric construction process involving conics and cubics. The well-known theorem of the uniqueness of the group law on such a curve is shown to be a consequence of this result.
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A special class of sequences having no adjacent repeated subsequences is discussed. It is shown that repeat-free sequences of three symbols having arbitrary length exist. Moreover, the exponential growth of the number of such sequ...
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A special class of sequences having no adjacent repeated subsequences is discussed. It is shown that repeat-free sequences of three symbols having arbitrary length exist. Moreover, the exponential growth of the number of such sequences with their length is established. (ERA citation 08:057915)
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Our purpose is to introduce a process generating the mass production of certain binary operations by means of a list of mathematical objects, given in advance, and a definition procedure consisting of a finite number of steps. (ER...
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Our purpose is to introduce a process generating the mass production of certain binary operations by means of a list of mathematical objects, given in advance, and a definition procedure consisting of a finite number of steps. (ERA citation 08:019366)
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Banach often remarked Good mathematicians see analogies between theorems and proofs; the very best ones see analogies between analogies. Mark Kac certainly belongs to the latter group. His work on problems in statistical mechanics...
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Banach often remarked Good mathematicians see analogies between theorems and proofs; the very best ones see analogies between analogies. Mark Kac certainly belongs to the latter group. His work on problems in statistical mechanics and in number theory, disciplines so different from each other, exhibits a feeling for the role of the ideas of probability analogous in some way in these two domains which are so far apart. In what follows an elementary approach to the notion of analogy is presented and a few mathematical problems are pointed out that pose themselves once one tries to discuss this notion in a somewhat general way; namely, similarity or proximity of proofs and counting binary and unary operations at each stage with similar trees on the set of axioms. (ERA citation 07:013595)
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We address the issue of endowing a logical framework with a logically justified notion of negation. Logical frameworks with a logic programming interpretation such as hereditary Harrop formulae cannot directly express negative inf...
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We address the issue of endowing a logical framework with a logically justified notion of negation. Logical frameworks with a logic programming interpretation such as hereditary Harrop formulae cannot directly express negative information, although negation is a useful specification tool. Since negation-as-failure does not fit well in a logical framework, especially one endowed with hypothetical and parametric judgments, we adapt the idea of elimination of negation from Horn logic to a fragment of higher-order hereditary Harrop formulae. The idea is to replace occurrences of negative predicates with positive ones which are operationally equivalent. This entails two separate phases. Complementing terms, i.e. in our case higher order patterns. Due the presence of partially applied lambda terms, intuitionistic lambda calculi are not closed under complementation. We thus develop a strict lambda calculus, where we can directly express whether a function depends on its argument. Complementing clauses: this can be seen as a negation normal form procedure which is consistent with intuitionistic provability. It entails finding a middle ground between the Closed World Assumption usually associated with negation and the Open World Assumption typical of logical frameworks. As this is in general not possible, we restrict ourselves to a fragment in which clause complementation is viable and that has proven to be expressive enough for the practice of logical frameworks. The main technical idea is to isolate a set of programs where static and dynamic clauses do not overlap.
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A generalization of the first order predicate logic is defined. On the premises of this logic, a preference relation can be specified. By using this preference relation, it is possible to describe non-monotonic reasoning in a very...
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A generalization of the first order predicate logic is defined. On the premises of this logic, a preference relation can be specified. By using this preference relation, it is possible to describe non-monotonic reasoning in a very natural way. In this non-monotonic logic, default rules can be formulated. It is also possible to derive new default rules. Unlike other forms of non-monotonic reasoning, this logic has an executable deduction process. In this deduction process, truth maintenance is integrated. (Copyright (c) 1988 by Faculty of Technical Mathematics and Informatics, Delft, The Netherlands.)
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The theory of differential and pseudo-differential forms on supermanifolds is constructed. The definition and notations of superanalogy of the Pontryagin and Chern characteristic classes are given. The theory considered is purely ...
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The theory of differential and pseudo-differential forms on supermanifolds is constructed. The definition and notations of superanalogy of the Pontryagin and Chern characteristic classes are given. The theory considered is purely local. The scheme suggested here generalizes the so-called Weil homomorphism for superspace which lies on the basis of the Chern and Potryagin characteristic class theory. The theory can be extended to the global supermanifolds. (Atomindex citation 11:506222)
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We prove that for a bounded domain (Omega) is contained in R(sup n) there exists an extension operator E:W(sup m,(infinity)) ((Omega)) (yields) W(sup m,(infinity)) (R(sup n)) for all m is an element of N if and only if the geodesi...
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We prove that for a bounded domain (Omega) is contained in R(sup n) there exists an extension operator E:W(sup m,(infinity)) ((Omega)) (yields) W(sup m,(infinity)) (R(sup n)) for all m is an element of N if and only if the geodesic distance in (Omega) is equivalent with the Euclidean distance. (author). 6 refs. (Atomindex citation 27:046053)
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