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STRUCTURAL THEOREMS FOR SYMBOLIC SUMMATION
"... Starting with Karr’s structural theorem for summation —the discrete version of Liouville’s structural theorem for integration — we work out crucial properties of the underlying difference fields. This leads to new and constructive structural theorems for symbolic summation. E.g., these results can b ..."
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Cited by 9 (7 self)
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Starting with Karr’s structural theorem for summation —the discrete version of Liouville’s structural theorem for integration — we work out crucial properties of the underlying difference fields. This leads to new and constructive structural theorems for symbolic summation. E.g., these results can
Recursive Functions of Symbolic Expressions and Their Computation by Machine, Part I
, 1960
"... this paper in L a T E Xpartly supported by ARPA (ONR) grant N000149410775 to Stanford University where John McCarthy has been since 1962. Copied with minor notational changes from CACM, April 1960. If you want the exact typography, look there. Current address, John McCarthy, Computer Science Depa ..."
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Cited by 457 (3 self)
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and the practical role of an interpreter. Then we describe the representation of Sexpressions in the memory of the IBM 704 by list structures similar to those used by Newell, Shaw and Simon [2], and the representation of Sfunctions by program. Then we mention the main features of the LISP programming system
Greatest Factorial Factorization and Symbolic Summation
 J. SYMBOLIC COMPUT
, 1995
"... This paper is selfcontained, no difference field knowledge but only basic facts from algebra are required. In the following we briefly review its sections. Section 2 presents the basic GFF notions, in particular the Fundamental Lemma and an algorithm for computing the GFFform of a polynomial. In S ..."
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Cited by 66 (7 self)
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in summation. Section 5 presents a new and algebraically motivated approach to Gosper's algorithm; together with the basic notions of GFF and Symbolic Summation 3 Section 2 this section can be read independently from the rest of the paper. In Section 6 we consider the general rational summation problem
Symbolic Summation and Higher Orders in Perturbation Theory ∗
, 2005
"... Higher orders in perturbation theory require the calculation of Feynman integrals at multiple loops. We report on an approach to systematically solve Feynman integrals by means of symbolic summation and discuss the underlying algorithms. Examples such as the nonplanar vertex at two loops, or integr ..."
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Higher orders in perturbation theory require the calculation of Feynman integrals at multiple loops. We report on an approach to systematically solve Feynman integrals by means of symbolic summation and discuss the underlying algorithms. Examples such as the nonplanar vertex at two loops
Algorithms for the indefinite and definite summation
 KONRADZUSEZENTRUM BERLIN (ZIB), PREPRINT SC
, 1994
"... The celebrated Zeilberger algorithm which finds holonomic recurrence equations for definite sums of hypergeometric terms F(n, k) is extended to certain nonhypergeometric terms. An expression F(n, k) is called hypergeometric term if both F(n + 1, k)/F(n, k) and F(n, k + 1)/F(n, k) are rational functi ..."
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Cited by 4 (4 self)
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terms, binomial coefficients, and Pochhammer symbols that are rationallinear with respect to n and k in their arguments, and present an extended version of Zeilberger’s algorithm for this case, using an extended version of Gosper’s algorithm for indefinite summation. In a similar way the Wilf
Using Symbolic Summation and Polynomial Algebra for Imperative Program Verification In Theorema
, 2006
"... An approach utilizing combinatorics, algebraic methods and logic is presented for generating polynomial loop invariants for a family of imperative programs operating on numbers. The approach has been implemented in the Theorema system, which seems ideal for such an integration given that it is built ..."
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that it is built on top of the computer algebra system Mathematica, has a theorem prover for firstorder logic as well as for mechanizing induction. These invariant assertions are then used for generating the necessary verification conditions as firstorder logical formulae, based on Hoare logic and the weakest
On the locality of codeword symbols
 IEEE Trans. Inform. Theory
, 2012
"... Consider a linear [n, k, d]q code C. We say that that ith coordinate of C has locality r, if the value at this coordinate can be recovered from accessing some other r coordinates of C. Data storage applications require codes with small redundancy, low locality for information coordinates, large dis ..."
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Cited by 49 (2 self)
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the bound as optimal. We prove some structure theorems about optimal codes, which are particularly strong for small distances. This gives a fairly complete picture of the tradeoffs between codewords length, worstcase distance and locality of information symbols. We then consider the locality of parity
Summation formulae involving the Laguerre polynomials
 J. Comput. Appl. Math
, 1998
"... Abstract A general result involving the generalized hypergeometric function is deduced by the elementary manipulation of series. Kummer's ÿrst theorem for the con uent hypergeometric function and two summation formulae for the Gauss hypergeometric function are then applied and new summation fo ..."
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Cited by 2 (0 self)
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Abstract A general result involving the generalized hypergeometric function is deduced by the elementary manipulation of series. Kummer's ÿrst theorem for the con uent hypergeometric function and two summation formulae for the Gauss hypergeometric function are then applied and new summation
Symbolic Invariant Verification for Systems with Dynamic Structural Adaptation
, 2006
"... The next generation of networked mechatronic systems will be characterized by complex coordination and structural adaptation at runtime. Crucial safety properties have to be guaranteed for all potential structural configurations. Testing cannot provide safety guarantees, while current model checkin ..."
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Cited by 44 (10 self)
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checking and theorem proving techniques do not scale for such systems. We present a verification technique for arbitrarily large multiagent systems from the mechatronic domain, featuring complex coordination and structural adaptation. We overcome the limitations of existing techniques by exploiting
An essential hybrid reasoning system: knowledge and symbol level accounts of KRYPTON
 In Proceedings of the 9th International Joint Conference on Artificial Intelligence
, 1985
"... Hybrid inference systems are an important way to address the fact that intelligent systems have muiltifaceted representational and reasoning competence. KRYPTON is an experimental prototype that competently handles both terminological and assertional knowledge; these two kinds of information are tig ..."
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Cited by 82 (1 self)
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are tightly linked by having sentences in an assertional component be formed using structured complex predicates denned in a complementary terminological component. KRYPTON is unique in that it combines in a completely integrated fashion a framebased description language and a firstorder resolution theorem
Results 1  10
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441