We consider the regularity of measurable solutions χ to the cohomological equation φ = χ ◦ T − χ, where (T,X, μ) is a dynamical system and φ : X → R is a C valued cocycle in the setting in which T : X → X is a piecewise C Gibbs– Markov map, an affine β-transformation of the unit interval or more generally a piecewise C uniformly expanding map of an interval. We show that under mild assumptions,...

In a supersymmetric theory with a spontaneously broken global symmetry, G, if the scale of supersymmetry breaking, M s is smaller than the scale M G of the global symmetry, the Nambu-Goldstone boson, χ is accompanied by two massive superpartners (a fermionic, Ψ χ and a scalar boson, σ χ) with mass of order M s. Cosmological considerations imply stringent constraints on the couplings of Ψ χ and ...

as χ varies over all multiplicative characters of k. For each χ, S(χ) is real, and (by Weil) has absolute value at most 2. Evans found empirically that, for large q = #k, these q − 1 sums were approximately equidistributed for the “Sato-Tate measure” (1/2π) √ 4− x2dx on the closed interval [−2, 2], and asked if this equidistribution was provably true. Rudnick had done numerics on the sums T (χ)...

In this paper, we give the theoretical analysis of χ attack proposed by Knudsen and Meier on the RC6 block cipher. To this end, we propose the novel method of security evaluation against χ attack precisely including key dependency by introducing a technique “Transition Matrix Computing.” On the other hand, the way of security evaluation against χ attack has not been known except the computer ex...

Let L(G) denote the set of all odd cycle lengths of a graph G. Gyárfás gave an upper bound for χ(G) depending on the size of this set: if |L(G)| = k ≥ 1, then χ(G) ≤ 2k+1 unless some block of G is a K2k+2, in which case χ(G) = 2k+2. This bound is generally tight, but when investigating L(G) of special forms, better results can be obtained. Wang completely analyzed the case L(G) = {3, 5}; Camach...

For a graph G on n vertices with chromatic number χ(G), the Nordhaus–Gaddum inequalities state that d2 √ ne ≤ χ(G) + χ(G) ≤ n + 1, and n ≤ χ(G) · χ(G) ≤ ⌊( n+1 2 )2⌋ . Much analysis has been done to derive similar inequalities for other graph parameters, all of which are integer-valued. We determine here the optimal Nordhaus–Gaddum inequalities for the circular chromatic number and the fraction...

The efforts of Roy-Sridhar-Close-Cho-Wise-Trivedi to resolve the CDF ψ ′ anomaly with cascades from above-threshold χ ′ c states require well defined signatures [a small total width and a large branching fraction for χ ′ cJ → γ+ψ ′ ] for the solution to be viable. Here we estimate the production of such states from BR(B → χ ′ cJ +X)BR(χ ′ cJ → γψ ′) and γγ production of χ ′ c2 at CLEO II, and c...

as χ varies over all multiplicative characters of k. For each χ, S(χ) is real, and (by Weil) has absolute value at most 2. Evans found empirically that, for large q = #k, these q − 1 sums were approximately equidistributed for the “Sato-Tate measure” (1/2π) √ 4− x2dx on the closed interval [−2, 2], and asked if this equidistribution was provably true. Rudnick had done numerics on the sums T (χ)...

We obtain formulae and estimates for character sums of the type S(χ, f, pm) = ∑pm x=1 χ(f(x)), where p m is a prime power with m ≥ 2, χ is a multiplicative character (mod pm), and f = f1/f2 is a rational function over Z. In particular, if p is odd, d = deg(f1) + deg(f2) and d∗ = max(deg(f1), deg(f2)) then we obtain |S(χ, f, pm)| ≤ (d− 1)pm(1− 1 d∗ ) for any non constant f (mod p) and primitive ...

Paul Garrett [email protected] http://www.math.umn.edu/ g̃arrett/ [This document is http://www.math.umn.edu/ ̃garrett/m/mfms/notes 2015-16/06d functional equation.pdf] 1. L(s, χ) for even χ 2. L(s, χ) for odd χ We prove the analytic continuation and function equation of Dirichlet L-functions L(s, χ) imitating the argument Riemann used for proving the analytic continuation of ζ(s) and its funct...