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Affine linear sieve, expanders, and sumproduct
"... This paper is concerned with the following general problem. For j = 1, 2,...,k let Aj be invertible integer coefficient polynomial maps of Z n to Z n (here n ≥ 1 and the inverses of Aj’s are assumed to be of the same type). Let Λ be the group generated by A1,...,Ak and ..."
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Cited by 42 (8 self)
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This paper is concerned with the following general problem. For j = 1, 2,...,k let Aj be invertible integer coefficient polynomial maps of Z n to Z n (here n ≥ 1 and the inverses of Aj’s are assumed to be of the same type). Let Λ be the group generated by A1,...,Ak and
WALKS ON GRAPHS AND LATTICES – EFFECTIVE BOUNDS AND APPLICATIONS
, 2008
"... Abstract. We continue the investigations started in [7, 8]. We consider the following situation: G is a finite directed graph, where to each vertex of G is assigned an element of a finite groupΓ. We consider all walks of length N on G, starting from vi and ending at vj. To each such walk w we assign ..."
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Cited by 3 (1 self)
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Abstract. We continue the investigations started in [7, 8]. We consider the following situation: G is a finite directed graph, where to each vertex of G is assigned an element of a finite groupΓ. We consider all walks of length N on G, starting from vi and ending at vj. To each such walk w we assign the element ofΓequal to the product of the elements along the walk. The set of all walks of length N from vi to vj thus induces a probability distribution FN,i,j onΓ. In [7] we give necessary and sufficient conditions for the limit as N goes to infinity of FN,i,j to exist and to be the uniform density onΓ(a detailed argument is presented in [8]). The convergence speed is then exponential in N. In this paper we consider (G,Γ), whereΓis a group possessing Kazhdan’s property T (or, less restrictively, propertyτwith respect to representations with finite image), and a family of homomorphismsψk:Γ→Γk with finite image. Each FN,i,j induces a distribution Fk N,i,j onΓk (by pushforward underψk). Our main result is that, under mild technical assumptions, the exponential rate of convergence of Fk N,i,k to the uniform distribution onΓk does not depend on k. As an application, we prove effective versions of the results of [8] on the probability that a random (in a suitable sence) element of SL(n,Z) or Sp(n,Z) has irreducible characteristic polynomial, generic Galois group, etc.
Rights Creative Commons: Attribution 3.0 Hong Kong License Research Article Protective Effects of Testosterone on Presynaptic Terminals against Oligomeric
"... Protective effects of testosterone on presynaptic terminals against oligomeric betaamyloid peptide in primary culture of hippocampal neurons ..."
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Protective effects of testosterone on presynaptic terminals against oligomeric betaamyloid peptide in primary culture of hippocampal neurons