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ON THE EQUIVALENCE OF PRIMAL AND DUAL SUBSTRUCTURING Preconditioners
, 2008
"... After a short historical review, we present four popular substructuring methods: FETI1, BDD, FETIDP, BDDC, and derive the primal versions to the two FETI methods, called PFETI1 and PFETIDP, as proposed by Fragakis and Papadrakakis. The formulation of the BDDC method shows that it is the same ..."
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Cited by 4 (1 self)
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After a short historical review, we present four popular substructuring methods: FETI1, BDD, FETIDP, BDDC, and derive the primal versions to the two FETI methods, called PFETI1 and PFETIDP, as proposed by Fragakis and Papadrakakis. The formulation of the BDDC method shows
Primality Testing
, 1992
"... Contents 1 Pseudo primes, probable primes 2 1.1 The Fermat test : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.2 The Euler test : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.3 The test of SolovayStrassen : : : : : : : : : : : : : : : : : : : : : : : : : ..."
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Strassen : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.4 The test of Miller : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2 Elementary primality proofs 7 2.1 Pocklington's theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 2.2 Primality proofs for numbers of special form : : : : : : : : : : : : : : : : 8 3
Primality Testing
"... INTRODUCTION Primality Testing is a fundamental problem of Number Theory, for which despite centuries of study no provably efficient algorithms have been devised. Further it has several applications especially in Cryptography. In this treatise we shall survey this beautiful and interesting area. 2. ..."
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INTRODUCTION Primality Testing is a fundamental problem of Number Theory, for which despite centuries of study no provably efficient algorithms have been devised. Further it has several applications especially in Cryptography. In this treatise we shall survey this beautiful and interesting area. 2
Primality of Trees
"... A graph of order n is prime if one can bijectively label its vertices with integers 1,..., n so that any two adjacent vertices get coprime labels. We prove that all bipartite ddegenerate graphs with separators of size at most n 1−Od(1 / ln ln n) are prime. It immediately follows that all large tree ..."
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Cited by 1 (0 self)
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A graph of order n is prime if one can bijectively label its vertices with integers 1,..., n so that any two adjacent vertices get coprime labels. We prove that all bipartite ddegenerate graphs with separators of size at most n 1−Od(1 / ln ln n) are prime. It immediately follows that all large trees are prime, confirming an old conjecture of Entringer and Tout from around 1980. Also, our method allows us to determine the smallest size of a nonprime connected ordern graph for all large n, proving a conjecture of Rao [R. C. Bose Centenary
A trust region method based on interior point techniques for nonlinear programming
 Mathematical Programming
, 1996
"... Jorge Nocedal z An algorithm for minimizing a nonlinear function subject to nonlinear inequality constraints is described. It applies sequential quadratic programming techniques to a sequence of barrier problems, and uses trust regions to ensure the robustness of the iteration and to allow the direc ..."
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Cited by 156 (19 self)
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the direct use of second order derivatives. This framework permits primal and primaldual steps, but the paper focuses on the primal version of the new algorithm. An analysis of the convergence properties of this method is presented. Key words: constrained optimization, interior point method, large
ON THE EQUIVALENCE OF PRIMAL AND DUAL SUBSTRUCTURING PRECONDITIONERS ∗
, 802
"... Abstract. After a short historical review, we present four popular substructuring methods: FETI1, BDD, FETIDP, BDDC, and derive the primal versions to the two FETI methods, called PFETI1 and PFETIDP, as proposed by Fragakis and Papadrakakis. The formulation of the BDDC method shows that it is ..."
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Abstract. After a short historical review, we present four popular substructuring methods: FETI1, BDD, FETIDP, BDDC, and derive the primal versions to the two FETI methods, called PFETI1 and PFETIDP, as proposed by Fragakis and Papadrakakis. The formulation of the BDDC method shows
Primaldual interior methods for nonconvex nonlinear programming
 SIAM Journal on Optimization
, 1998
"... Abstract. This paper concerns largescale general (nonconvex) nonlinear programming when first and second derivatives of the objective and constraint functions are available. A method is proposed that is based on finding an approximate solution of a sequence of unconstrained subproblems parameterize ..."
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Cited by 80 (8 self)
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parameterized by a scalar parameter. The objective function of each unconstrained subproblem is an augmented penaltybarrier function that involves both primal and dual variables. Each subproblem is solved with a modified Newton method that generates search directions from a primaldual system similar
Elliptic periods and primality proving
, 2009
"... We define the ring of elliptic periods modulo an integer n and give an elliptic version of the AKS primality criterion. ..."
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We define the ring of elliptic periods modulo an integer n and give an elliptic version of the AKS primality criterion.
Some Primality Testing Algorithms
 Notices of the AMS
, 1993
"... We describe the primality testing algorithms in use in some popular computer algebra systems, and give some examples where they break down in practice. 1 Introduction In recent years, fast primality testing algorithms have been a popular subject of research and some of the modern methods are now i ..."
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Cited by 3 (0 self)
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were Mathematica 2.1 for Sparc, copyright dates 19881992; Maple V Release 2, copyright dates 19811993; Axiom Release 1.2 (version of February 18, 1993); Pari/GP 1.37.3 (Sparc version, dated November 23, 1992). The tests were performed on Sparc workstations. Primality testing is a large and growing
ELLIPTIC PERIODS AND PRIMALITY PROVING (EXTENTED VERSION)
, 2009
"... We construct extension rings with fast arithmetic using isogenies between elliptic curves. As an application, we give an elliptic version of the AKS primality criterion. ..."
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We construct extension rings with fast arithmetic using isogenies between elliptic curves. As an application, we give an elliptic version of the AKS primality criterion.
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