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COMPUTATIONAL DETAILS ON THE DISPROOF OF MODULARITY
, 709
"... Abstract. The purpose of these notes is to provide the details of the Jacobian ring computations carried out in [1], based on the computer algebra system Magma [2]. 1. Finding an appropriate matrix Our first target is to construct the Jacobian ring associated to a CalabiYau ˜ X as described in sect ..."
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: = (aij) with aij = λ i j, where (λ1,..., λ8) denotes a tuple of 8 distinct numbers in C. The Magma program below produces such an admissible matrix. There are three options: (i) Work with a userdefined matrix. In this case one sets randmat:=false and hyperell:=false. The line A:=RMatrixSpace(K,4
Failed attempt to disproof the Riemann Hypothesis
"... In this paper we are going to describe the results of the computer experiment, which in principle can rule out the Riemann Hypothesis. We use the sequence ck appearing in the BáezDuarte criterion for the RH. Namely we calculate c100000 with thousand digits of accuracy using two different formulas ..."
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formulas for ck with the aim to disproof the Riemann Hypothesis in the case these two numbers will differ. We found the discrepancy only on the 996 decimal place (accuracy of 10−996). The reported here experiment can be of interest for developers of Mathematica and PARI/GP. 1
A disproof of a conjecture of Erdős in Ramsey theory
 J. London Math. Soc
, 1989
"... Denote by kt(G) the number of complete subgraphs of order f in the graph G. Let where G denotes the complement of G and \G \ the number of vertices. A wellknown conjecture of Erdos, related to Ramsey's theorem, is that Mmn^K ct(ri) = 2 1 ~ { * ). This latter number is the proportion of mono ..."
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Denote by kt(G) the number of complete subgraphs of order f in the graph G. Let where G denotes the complement of G and \G \ the number of vertices. A wellknown conjecture of Erdos, related to Ramsey's theorem, is that Mmn^K ct(ri) = 2 1 ~ { * ). This latter number is the proportion
Disproof Of Wiles ’ Proof For Fermat’s Last Theorem
"... Dedicated to Prof. R. M. Santilli on the occasion of his 70th Birthday There cannot be number theory of the twentieth century, the ShimuraTaniyamaWeil conjecture (STWC) and the Langlands program (LP) without the Riemann hypothesis (RH). By using RH it is possible to prove five hundred theorems or ..."
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Dedicated to Prof. R. M. Santilli on the occasion of his 70th Birthday There cannot be number theory of the twentieth century, the ShimuraTaniyamaWeil conjecture (STWC) and the Langlands program (LP) without the Riemann hypothesis (RH). By using RH it is possible to prove five hundred theorems
Parallel depth first proof number search
 In Proc. AAAI10
, 2010
"... The depth first proof number search (dfpn) is an effective and popular algorithm for solving andor tree problems by using proof and disproof numbers. This paper presents a simple but effective parallelization of the dfpn search algorithm for a sharedmemory system. In this parallelization, multi ..."
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The depth first proof number search (dfpn) is an effective and popular algorithm for solving andor tree problems by using proof and disproof numbers. This paper presents a simple but effective parallelization of the dfpn search algorithm for a sharedmemory system. In this parallelization
A disproof of Henning’s conjecture on irredundance perfect graphs
, 2001
"... Let ir(G) and γ(G) be the irredundance number and the domination number of a graph G, respectively. A graph G is called irredundance perfect if ir(H) = γ(H), for every induced subgraph H of G. In this paper we disprove the known conjecture of Henning [3, 11] that a graph G is irredundance perfect i ..."
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Let ir(G) and γ(G) be the irredundance number and the domination number of a graph G, respectively. A graph G is called irredundance perfect if ir(H) = γ(H), for every induced subgraph H of G. In this paper we disprove the known conjecture of Henning [3, 11] that a graph G is irredundance perfect
EvaluationFunction Based ProofNumber Search
"... Abstract. This article introduces EvaluationFunction based Proof– Number Search (EFPN) and its secondlevel variant EFPN2. It is a framework for setting the proof and disproof number of a leaf node with a heuristic evaluation function. Experiments in LOA and Surakarta show that compared to PN and ..."
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Abstract. This article introduces EvaluationFunction based Proof– Number Search (EFPN) and its secondlevel variant EFPN2. It is a framework for setting the proof and disproof number of a leaf node with a heuristic evaluation function. Experiments in LOA and Surakarta show that compared to PN
Lambda Depthfirst Proof Number Search and its Application to Go
, 2007
"... Thomsen’s λ search and Nagai’s depthfirst proofnumber (DFPN) search are two powerful but very different AND/OR tree search algorithms. Lambda DepthFirst Proof Number search (LDFPN) is a novel algorithm that combines ideas from both algorithms. λ search can dramatically reduce a search space by fin ..."
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by finding different levels of threat sequences. DFPN employs the notion of proof and disproof numbers to expand nodes expected to be easiest to prove or disprove. The method was shown to be effective for many games. Integrating λ order with proof and disproof numbers enables LDFPN to select moves more
About the Completeness of DepthFirst ProofNumber Search
"... Abstract. Depthfirst proofnumber (dfpn) search is a powerful member of the family of algorithms based on proof and disproof numbers. While dfpn has succeeded in practice, its theoretical properties remain poorly understood. This paper resolves the question of completeness of dfpn: its ability t ..."
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Abstract. Depthfirst proofnumber (dfpn) search is a powerful member of the family of algorithms based on proof and disproof numbers. While dfpn has succeeded in practice, its theoretical properties remain poorly understood. This paper resolves the question of completeness of dfpn: its ability
GameTree Search Using Proof Numbers: THE FIRST TWENTY YEARS
"... Solving games is a challenging and attractive task in the domain of Artificial Intelligence. Despite enormous progress, solving increasingly difficult games or game positions continues to pose hard technical challenges. Over the last twenty years, algorithms based on the concept of proof and disproo ..."
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and disproof numbers have become dominating techniques for game solving. Prominent examples include solving the game of checkers to be a draw, and developing checkmate solvers for shogi, which can find mates that take over a thousand moves. This article provides an overview of the research on ProofNumber
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