Let G be a real form of a complex reductive group. Suppose that we are given involutions σ and θ of G. Let H = G denote the fixed group of σ and let K = G denote the fixed group of θ. We are interested in calculating the double coset space H\G/K. We use moment map and invariant theoretic techniques to calculate the double cosets, especially the ones that are closed. One salient point of our res...

Let G be a real form of a complex reductive group. Suppose that we are given involutions σ and θ of G. Let H = G denote the fixed group of σ and let K = G denote the fixed group of θ. We are interested in calculating the double coset space H\G/K. We use moment map and invariant theoretic techniques to calculate the double cosets, especially the ones that are closed. One salient point of our res...

Let ai, bi, i = 0, 1, 2, . . . be drawn uniformly and independently from the unit interval, and let t be a fixed real number. Let a site (i, j) ∈ N be open if ai + bj ≤ t, and closed otherwise. We obtain a simple, exact expression for the probability Θ(t) that there is an infinite path (oriented or not) of open sites, containing the origin. Θ(t) is continuous and has continuous first derivative...

Answer Consider the case where the initial value is H0 and M ′ = M1‖M2‖ . . . ‖MN . Let the notation H1(M1, I) denote the first iteration of the hash function using message block M1 and initial value I. H1(M1, H0) = EM1(H0)⊕H0 H1(M1, H0) = EM1(H0)⊕H0 by complementary Property of DES H1(M1, H0) = EM1(H0)⊕H0 = H1(M1,H0) A⊕B = A⊕B ∀A,B Thus H1(H0,M1) yields the same result as H1(H0,M1). In additio...

Let X = (X1, ..., X ˆ n) be a random sample from distribution fθ. Let θ = δ(X) be an estimator of θ. Let T (X) be a su cient statistic for θ. As we have seen already, MSE provides one way to compare the quality of di erent estimators. In particular, estimators with smaller MSE are said to be more e cient. On the other hand, once we know T (X), we can discard X. How do these concepts relate to e...

To obtain a contradiction, suppose that there is a cycle; this implies that so that there are q ≥ 2 different governments H1, . . . ,Hq such that φ (Hj) = Hj+1 for all 1 ≤ j < q, and φ (Hq) = H1. Without loss of generality, let H1 be the least competent of these governments. Take H = φ (H2) (if q > 2 then H = H3 and if q = 2 then H = H1). As φ is a political equilibrium, Vi (H) > Vi (H2) holds ...

Let h1, · · · , hn be positive integers. We study new sums m(h1, · · · , hn) = h1−1 ∑ r1=0 · · · hn−1 ∑ rn=0 min { r1 h1 , · · · , rn hn } and M(h1, · · · , hn) = h1−1 ∑ r1=0 · · · hn−1 ∑ rn=0 max { r1 h1 , · · · , rn hn } , the first of which times h1 · · ·hn is the number of lattice points in a pyramid of dimension n + 1. We show that m(h1, · · · , hn) (h1 − 1) · · · (hn − 1) = 1 + ∑ ∅6 =I⊆{1...

Let N be a large parameter, C a constant, and let ΓN (C) denote the C-neighborhood of the cone segment {ξ : 2−CN ≤ |ξ| ≤ 2N}. For fixed N , we take a partition of unity subordinate to a covering of Sd−1 by caps Θ of diameter about N− 1 2 , and use this to form a (smooth) partition of unity yΘ on ΓN(C) in the natural way. We will write ΓN,Θ(C) = supp yΘ. Let ΞΘ be a function whose Fourier transf...

where (hαβ) is the positive definite n × n matrix on M , which is uniquely determined by the Levi form Lθ(u, v) = −idθ(u, v) for u, v ∈ H(M). Let ∆sb be the sub-Laplacian with respect to θ (see [8], [22] and [7] for references) and let μ1 be the first positive eigenvalue of ∆sb on (M, θ). Let Rαβ be the Webster pseudo Ricci curvature, R be the pseudo scalar curvature and Tor be the pseudo torsi...

Let the underlying process Y be a semimartingale taking values in an interval I. Let φ : I → R be a difference of convex functions, and let X := φ(Y ). A typical application takes Y to be a positive price process and φ(y) = log y for y ∈ I = (0,∞). Then [the floating leg of] a forward-starting weighted variance swap or generalized variance swap on φ(Y ) (shortened to “on Y ” if the φ is underst...