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Expander Graphs and their Applications
, 2003
"... Contents 1 The Magical Mystery Tour 7 1.1 Some Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Hardness results for linear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Error Correcting Codes . . . . . . . ..."
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Cited by 186 (5 self)
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Contents 1 The Magical Mystery Tour 7 1.1 Some Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Hardness results for linear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.3 Derandomizing Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Magical Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.1 A Super Concentrator with O(n) edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.2 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.3 Derandomizing Random Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
NP and Mathematics  a computational complexity perspective
 Proc. of the ICM 06
"... “P versus N P – a gift to mathematics from Computer Science” ..."
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Cited by 1 (0 self)
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“P versus N P – a gift to mathematics from Computer Science”
A note on Agrawal conjecture
"... Abstract. We prove that Lenstra proposition suggesting existence of many counterexamples to Agrawal conjecture is true in a more general case. At the same time we obtain a strictly ascending chain of subgroups of the group (Zp[X]/(Cr(X))) * and state the modified conjecture that the set {X1, X+2} ..."
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Abstract. We prove that Lenstra proposition suggesting existence of many counterexamples to Agrawal conjecture is true in a more general case. At the same time we obtain a strictly ascending chain of subgroups of the group (Zp[X]/(Cr(X))) * and state the modified conjecture that the set {X1, X+2} generate big enough subgroup of this group. 1
MATHEMATICAL ENGINEERING TECHNICAL REPORTS
, 2007
"... Generalized zigzag products of regular digraphs and bounds on their spectral expansions ..."
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Generalized zigzag products of regular digraphs and bounds on their spectral expansions
Summary
"... Our main aim in the present paper is the extension of the Encryption/Decryption processes using products of primes. We now show in this paper how to generate a group from any general natural number or a product of such natural numbers. We then show how this group can be used for generation of a simp ..."
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Our main aim in the present paper is the extension of the Encryption/Decryption processes using products of primes. We now show in this paper how to generate a group from any general natural number or a product of such natural numbers. We then show how this group can be used for generation of a simple (yet as secure, as the one that is generated with the help of larger primes) encryption /decryption process. This work is continuation of the work that the first author had undertaken with Dr. H. Chandrashekhar in the 90’s using Farey Fractions summary.
TEST DE PRIMALITÉ AKS ET APPLICATIONS
"... Abstract. Les mathématiques ont tenté jusqu’à ce jour de découvrir une régularité dans la suite des nombres premiers, et nous avons de bonnes raisons de croire qu’il y a là un mystère que l’esprit humain ne pénétrera jamais. Il suffit d’ailleurs, pour s’en convaincre, de jeter un regard sur une tabl ..."
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Abstract. Les mathématiques ont tenté jusqu’à ce jour de découvrir une régularité dans la suite des nombres premiers, et nous avons de bonnes raisons de croire qu’il y a là un mystère que l’esprit humain ne pénétrera jamais. Il suffit d’ailleurs, pour s’en convaincre, de jeter un regard sur une table de nombres premiers (que certains ont pris la peine de calculer jusqu’à plusieurs centaines de milliers); on est alors instantanément convaincu qu’il n’y règne ni loi, ni ordre, ni règle. Leonhard Euler (17071783) 1.
DRAFT DRAFT DRAFT DRAFT DRAFT
, 2006
"... A major consideration we had in writing this survey was to make it accessible to mathematicians as well as computer scientists, since expander graphs, the protagonists of our story come up in numerous and often surprising contexts in both fields. A glossary of some basic terms and facts from compute ..."
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A major consideration we had in writing this survey was to make it accessible to mathematicians as well as computer scientists, since expander graphs, the protagonists of our story come up in numerous and often surprising contexts in both fields. A glossary of some basic terms and facts from computer science can be found at the end of this article. But, perhaps, we should start with a few words about graphs in general. They are, of course, one of the prime objects of study in Discrete Mathematics. However, graphs are among the most ubiquitous models of both natural and humanmade structures. In the natural and social sciences they model relations among species, societies, companies, etc. In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more. In Mathematics, Cayley graphs are useful in Group Theory. Graphs carry a natural metric and are therefore useful in Geometry, and though they are “just ” onedimensional complexes they are useful in certain parts of Topology, e.g. Knot Theory. In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems. The study of these models calls, then, for the comprehension of the significant structural properties of the relevant graphs. But are there nontrivial structural properties which are universally important? The property of being an expander seems significant in many of these mathematical, computational and physical contexts. It is not surprising that expanders are useful in the design and analysis of communication networks. What is less obvious is that expanders