The goal of this study examines the quantitative implications of the Malmquist index in a standard Real Business Cycle (RBC) model as a measure of technology shock. To achieve this, the paper first investigates the empirical validity of the equivalence proposition on the two technology shock measures: a relatively new Malmquist Index and the predominant Solow residual. On the basis of permutati...

For a random permutation of n objects, as n → ∞, the process giving the proportion of elements in the longest cycle, the second longest cycle, and so on, converges in distribution to the Poisson-Dirichlet process with parameter 1. This was proved in 1977 by Kingman and by Vershik and Schmidt. For soft reasons, this is equivalent to the statement that the random permutations and the Poisson-Diri...

Lauren Williams (joint work with Einar Steingrímsson) We introduce and study a class of tableaux which we call permutation tableaux; these tableaux are naturally in bijection with permutations, and they are a distinguished subset of the " Le-diagrams " of Alex Postnikov. The structure of these tableaux is in some ways more transparent than the structure of permutations; therefore we believe tha...

Dear Charlie, I promised to write to you about the "characteristic-free" extension of Richard Brauer's Lemma: if a row permutation and a column permutation have the same effect on a nonsingular matrix, the two permutations must have the same number of cycles of any given length. Differently put: if two permutation matrices are conjugate in the general linear group, they are already conjugate in...

Answering a question of Bóna, it is shown that for n ≥ 2 the probability that 1 and 2 are in the same cycle of a product of two n-cycles on the set {1, 2, . . . , n} is 1/2 if n is odd and 1 2 − 2 (n−1)(n+2) if n is even. Another result concerns the polynomial Pλ(q) = ∑ w q κ((1,2,...,n)·w), where w ranges over all permutations in the symmetric group Sn of cycle type λ, (1, 2, . . . , n) denote...

We consider the function m[k](q) that counts the number of cycle permutations of a finite field Fq of fixed length k such that their permutation polynomial has the smallest possible degree. We prove the upper–bound m[k](q) ≤ (k−1)!(q(q−1))/k for char(Fq) > e(k−3)/e and the lower–bound m[k](q) ≥ φ(k)(q(q−1))/k for q ≡ 1 (mod k). This is done by establishing a connection with the Fq–solutions of ...

Let H be a subgroup of the multiplicative group of a finite field. In this note we give a method for constructing permutation polynomials over the field using a bijective map from H to a coset of H. A similar, but inequivalent, method for lifting permutation behaviour of a polynomial to an extension field is also given.

Let L be chosen uniformly at random from among the latin squares of order n ≥ 4 and let r, s be arbitrary distinct rows of L. We study the distribution of σr,s, the permutation of the symbols of L which maps r to s. We show that for any constant c > 0, the following events hold with probability 1− o(1) as n → ∞: (i) σr,s has more than (log n)1−c cycles, (ii) σr,s has fewer than 9√n cycles, (iii...

‎In this work‎, ‎a regular polygon with $n$ sides is described by a periodic (circular) sequence with period $n$‎. ‎Each element of the sequence represents a vertex of the polygon‎. ‎Each symmetry of the polygon is the rotation of the polygon around the center-point and/or flipping around a symmetry axis‎. ‎Here each symmetry is considered as a system that takes an input circular sequence and g...

In this text, extending results of [Ni94] and [Ne05], we study the asymptotics of the number of cycles of a given length of a word in several independent random permutations with restricted cycle lengths. Specifically, for A1,. . . , Ak non empty sets of positive integers and for w word in the letters g1, g −1 1 , . . . , gk, g −1 k , we consider, for all n such that it is possible, an independ...