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178
On Random Walks on Wreath Products
 Ann. Probab
, 2001
"... Wreath products are a type of semidirect products. They play an important role in group theory. This paper studies the basic behavior of simple random walks on such groups and shows that these walks have interesting, somewhat exotic behaviors. The crucial fact is that the probability of return to th ..."
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Cited by 23 (1 self)
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Wreath products are a type of semidirect products. They play an important role in group theory. This paper studies the basic behavior of simple random walks on such groups and shows that these walks have interesting, somewhat exotic behaviors. The crucial fact is that the probability of return to the starting point of certain walks on wreath products is closely related to some functionals of the local times of a walk taking place on a simpler factor group.
Topology on the spaces of orderings of groups
 Bull. London Math. Soc
"... Abstract. A natural topology on the space of left orderings of an arbitrary semigroup is introduced. It is proved that this space is compact and that for free abelian groups it is homeomorphic to the Cantor set. An application of this result is a new proof of the existence of universal Gröbner base ..."
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Cited by 20 (0 self)
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Abstract. A natural topology on the space of left orderings of an arbitrary semigroup is introduced. It is proved that this space is compact and that for free abelian groups it is homeomorphic to the Cantor set. An application of this result is a new proof of the existence of universal Gröbner bases. 1. Orderings for semigroups Given a semigroup G (ie. a set with an associative binary operation), a linear order, <, on G is a left order if a < b implies ca < cb, for any c. Similarly, a linear order,<, is a right order if a < b implies ac < bc, for any c ∈ G. The sets of all left and right orderings of G are denoted by LO(G) and RO(G) respectively. If G is a group then there is a 11 correspondence between these two sets which associates with any left ordering, <l, a right ordering, <r, such that a <r b if and only if b −1 <l a −1. For more about ordering of groups see [4, 7, 8]. Let Ua,b ⊂ LO(G) denote the set of all left orderings, <, for which a < b. We can put a topology on LO(G) in one of the following two ways. Definition 1.1. LO(G) has the smallest topology for which all the sets Ua,b are open. Any open set in this topology is a union of sets of the form
Smarandache MultiSpace Theory
, 2011
"... Our WORLD is a multiple one both shown by the natural world and human beings. For example, the observation enables one knowing that there are infinite planets in the universe. Each of them revolves on its own axis and has its own seasons. In the human society, these rich or poor, big or small countr ..."
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Cited by 13 (5 self)
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Our WORLD is a multiple one both shown by the natural world and human beings. For example, the observation enables one knowing that there are infinite planets in the universe. Each of them revolves on its own axis and has its own seasons. In the human society, these rich or poor, big or small countries appear and each of them has its own system. All of these show that our WORLD is not in homogenous but in multiple. Besides, all things that one can acknowledge is determined by his eyes, or ears, or nose, or tongue, or body or passions, i.e., these six organs, which means the WORLD consists of have and not have parts for human beings. For thousands years, human being has never stopped his steps for exploring its behaviors of all kinds. We are used to the idea that our space has three dimensions: length, breadth and height with time providing the fourth dimension of spacetime by Einstein. In the string or superstring theories, we encounter 10 dimensions. However, we do not even know what the right degree of freedom is, as Witten said. Today, we have known two heartening notions for sciences. One is the Smarandache multispace came into being by purely logic.
Clifford code constructions of operator quantum errorcorrecting codes
, 2006
"... Recently, operator quantum errorcorrecting codes have been proposed to unify and generalize decoherence free subspaces, noiseless subsystems, and quantum errorcorrecting codes. This note introduces a natural construction of such codes in terms of Clifford codes, an elegant generalization of stabil ..."
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Cited by 12 (3 self)
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Recently, operator quantum errorcorrecting codes have been proposed to unify and generalize decoherence free subspaces, noiseless subsystems, and quantum errorcorrecting codes. This note introduces a natural construction of such codes in terms of Clifford codes, an elegant generalization of stabilizer codes due to Knill. Charactertheoretic methods are used to derive a simple method to construct operator quantum errorcorrecting codes from any classical additive code over a finite field, which obviates the need for selforthogonal codes. Introduction. One of the main challenges in quantum information processing is the protection of the quantum information against various sources of errors. A possible remedy is given by encoding the quantum information in a subspace C of the state space H of the quantum system. If such a quantum errorcorrecting code C is wellchosen, then many errors can be
Describing groups
 Bull. Symb. Logic
"... Abstract. Two ways of describing a group are considered. 1. A group is finiteautomaton presentable if its elements can be represented by strings over a finite alphabet, in such a way that the set of representing strings and the group operation can be recognized by finite automata. 2. An infinite f.g ..."
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Cited by 10 (3 self)
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Abstract. Two ways of describing a group are considered. 1. A group is finiteautomaton presentable if its elements can be represented by strings over a finite alphabet, in such a way that the set of representing strings and the group operation can be recognized by finite automata. 2. An infinite f.g. group is quasifinitely axiomatizable if there is a description consisting of a single firstorder sentence, together with the information that the group is finitely generated. In the first part of the paper we survey examples of FApresentable groups, but also discuss theorems restricting this class. In the second part, we give examples of quasifinitely axiomatizable groups, consider the algebraic content of the notion, and compare it to the notion of a group which is a prime model. We also show that if a structure is biinterpretable in parameters with the ring of integers, then it is prime and
Twisted Alexander polynomials and symplectic structures
, 2006
"... Abstract. Let N be a closed, oriented 3–manifold. A folklore conjecture states that S 1 ×N admits a symplectic structure only if N admits a fibration over the circle. The purpose of this paper is to provide evidence to this conjecture studying suitable twisted Alexander polynomials of N, and showing ..."
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Cited by 9 (8 self)
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Abstract. Let N be a closed, oriented 3–manifold. A folklore conjecture states that S 1 ×N admits a symplectic structure only if N admits a fibration over the circle. The purpose of this paper is to provide evidence to this conjecture studying suitable twisted Alexander polynomials of N, and showing that their behavior is the same as of those of fibered 3–manifolds. In particular, we will obtain new obstructions to the existence of symplectic structures and to the existence of symplectic forms representing certain cohomology classes of S 1 ×N. As an application of these results we will show that S 1 ×N(P) does not admit a symplectic structure, where N(P) is the 0–surgery along the pretzel knot P = (5, −3, 5), answering a question of Peter Kronheimer. 1.
The local linearization problem for smooth SL(n)actions
 Enseign. Math
, 1997
"... Abstract. This paper considers SL(n, R)actions on Euclidean space fixing the origin. We show that all C 1actions on R n are linearizable. We give C ∞actions of SL(2, R) on R 3 and of SL(3, R) on R 8 which are not linearizable. We classify the C 0actions of SL(n, R) on R n. Finally, the paper con ..."
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Cited by 9 (0 self)
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Abstract. This paper considers SL(n, R)actions on Euclidean space fixing the origin. We show that all C 1actions on R n are linearizable. We give C ∞actions of SL(2, R) on R 3 and of SL(3, R) on R 8 which are not linearizable. We classify the C 0actions of SL(n, R) on R n. Finally, the paper concludes with a study of the linearizability of SL(n, Z)actions. RÉSUMÉ. Dans cet article, on considère les actions de SL(n, R) sur l’espace euclidien qui fixent l’origine. On montre que les actions C1 sur Rn sont linéarisables. On donne des actions C ∞ de SL(2, R) sur R3 et de SL(3, R) sur R8 qui ne sont pas linéarisables. On classifie les actions C0 de SL(n, R) sur Rn. L’article s’achève par une étude de la linéarisabilité des actions de SL(n, Z). 1.
Imprimitive permutation groups and trapdoors in iterated block ciphers
 6th International Workshop, FSE’99
, 1999
"... Abstract. An iterated block cipher can be regarded as a means of producing a set of permutations of a message space. Some properties of the group generated by the round functions of such a cipher are known to be of cryptanalytic interest. It is shown here that if this group acts imprimitively on the ..."
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Cited by 8 (0 self)
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Abstract. An iterated block cipher can be regarded as a means of producing a set of permutations of a message space. Some properties of the group generated by the round functions of such a cipher are known to be of cryptanalytic interest. It is shown here that if this group acts imprimitively on the message space then there is an exploitable weakness in the cipher. It is demonstrated that a weakness of this type can be used to construct a trapdoor that may be difficult to detect. An example of a DESlike cipher, resistant to both linear and differential cryptanalysis that generates an imprimitive group and is easily broken, is given. Some implications for block cipher design are noted. 1