A. Greenhalgh. Random walks on groups with subgroup invariance properties. PhD thesis, Department of Mathematics, Stanford University, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....S. Questions to consider include how long it takes to get close to uniformly distributed on G for most choices of S. This problem is noted for the integers mod n in Dia1 conis [3] A lower bound for all S where the size k is a constant (in n) and G is the integers mod n appears in Greenhalgh [10]. Further results for specific abelian groups appear in Hildebrand [12] Dou [4] Greenhalgh [11] and Wilson [16] Results for arbitrary groups appear in Dou and Hildebrand [5] and Roichman [14] Also studying arbitrary groups, Pak [13] gets some results for lazy random walks where at each step ....

Greenhalgh, A. Random walks on groups with subgroup invariance properties. Ph.D. thesis, Department of Mathematics, Stanford University, 1989.

....such that for m = bfln 2= k Gamma1) c we have E(kP (m) ea Gamma Uk) where the expectation is taken over a uniform choice of all possible ea such that a 1 ; a k 2 [j1; n[j and such that all values of a 1 ; a k are pairwise distinct. It is worth noting that Greenhalgh [3] has shown the following lower bound, which nicely complements Theorem 3.1. Theorem 3.2. Let p j (j = 1; k) ea and P ea be as above. Then there exists a value fi = fi(p 1 ; p k ) 0 and n 0 = n 0 (p 1 ; n k ) such that for all choices of ea, m = bfin 2= k Gamma1) c and ....

A. Greenhalgh. Random walks on groups with subgroup invariance properties. PhD thesis, Department of Mathematics, Stanford University, 1989.

....of S. Questions to consider include 1 how long does it take to get close to uniformly distributed on G for most choices of S. This problem is noted for the integers mod n in Diaconis [3] A lower bound for all S where the size k is a constant and G is the integers mod n appears in Greenhalgh [10]. Further results for specific abelian group appear in Hildebrand [12] Dou [4] Greenhalgh [11] and Wilson [16] Results for arbitrary groups appear in Dou and Hildebrand [5] and Roichman [14] Also studying arbitrary groups, Pak [13] gets some results for lazy random walks where at each step ....

Greenhalgh, A. Random walks on groups with subgroup invariance properties. Ph.D. thesis, Department of Mathematics, Stanford University, 1989. 17

....that for m = bfln 2= k Gamma1) c we have E(kP (m) a Gamma Uk) where the expectation is taken over a uniform choice of all possible a such that a 1 ; a k 2 [j1; n[j and such that all values of a 1 ; a k are pairwise distinct. It is worth noting that Greenhalgh [3] has shown the following lower bound, which nicely complements Theorem 2. Theorem 3. Let p j (j = 1; k) a and P a be as above. Then there exists a value fi = fi(p 1 ; p k ) 0 and n 0 = n 0 (p 1 ; n k ) such that for all choices of a, m = bfin 2= k Gamma1) c and ....

A. Greenhalgh. Random walks on groups with subgroup invariance properties. PhD thesis, Department of Mathematics, Stanford University, 1989.

....of S. Questions to consider include how long does it take to get close to uniformly distributed on G for most choices of S. This problem is noted for the integers mod n in Diaconis [3] A lower bound for all S where the size k is a constant and G is the integers mod n appears in Greenhalgh [10]. Further results for specific abelian group appear in Hildebrand [12] Dou [4] Greenhalgh [11] and Wilson [16] Results for arbitrary groups appear in Dou and Hildebrand [5] and Roichman [14] Also studying arbitrary groups, Pak [13] gets some results for lazy random walks where at each step ....

Greenhalgh, A. Random walks on groups with subgroup invariance properties. Ph.D. thesis, Department of Mathematics, Stanford University, 1989.

....then by Theorem 2.8 the Fourier analysis would simplify considerably. The following lemma shows that for random walks on groups, averaging the generating measure Q to make it bi invariant will affect the rate of convergence by at most one step. This lemma is the analogue of a lemma of Greenhalgh [6], who obtained a similar result for total variation. Lemma 3.1. Let Q denote any left N invariant probability measure on a group G, let U denote the uniform distribution on G, and let UN denote the uniform distribution on N . Suppose Q = Q UN . Then Q is bi invariant and D( Q k ; U) ....

Greenhalgh, A. Random walks on groups with subgroup invariance properties. Ph.D. Thesis, Dept. of Mathematics, Stanford Univ., 1987.

....with cosets in G=B, where G = GL(n; F q ) and B = b i;j ) ae GL(n; F q ) such that b i;j = 0 for all 1 j k i n. 1. 6 Remark WHEN AND HOW n CHOOSE k 7 As the reader may have noticed, everything in this chapter is very well known and can be generalized in many directions (see e.g. [GR, GJ, H1, H2, NW, P, S1, St]) For example, one may look at other root systems, such as Cn . The analogs of Sn for other root systems are called Weyl group, which in the case of Cn is the group of symmetries of the n dimensional cube. Also, in this case the analog of GL(n) is the symplectic group Sp(2n) see [H1, St, W] ....

....case of Cn is the group of symmetries of the n dimensional cube. Also, in this case the analog of GL(n) is the symplectic group Sp(2n) see [H1, St, W] Note also that if we look at incomplete flags with a fixed sequence of dimensions we get q multinomial coefficients as their cardinalities (see [GR, GJ, S1]) 2. How do you generate them (First part) The mathematical formulation of the question is the following: how do you choose an element of the classical set with equal probability 2.1 Permutations 2.1.1 Here is a simple algorithm. Take an element 1 and put it on a line. Take 2. We have two ....

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A. Greenhalgh, Random walks on groups with subgroup invariance properties, Ph. D. dissertation, Stanford U., 1987.

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A. Greenhalgh. Random walks on groups with subgroup invariance properties. PhD thesis, Department of Mathematics, Stanford University, 1989.

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