We present two new data structures for computing values of an n-variate polynomial P of degree at most d over a finite field of q elements. Assuming that d divides q − 1, our first data structure relies on (d + 1)n+2 tabulated values of P to produce the value of P at any of the q points using O(nqd2) arithmetic operations in the finite field. Assuming that s divides d and d/s divides q − 1, our...

An explicit character formula is established for any strongly generic nite-dimensional irreducible g-module, g being an arbitrary nite-dimensional complex Lie superalgebra. This character formula had been conjectured earlier by Vera Serganova and the author for any generic irreducible nite-dimensional g-module, i.e. such that its highest weight is far enough from the walls of the Weyl chambers....

In this paper, we obtain the Rényi entropy rate for irreducible-aperiodic Markov chains with countable state space, using the theory of countable nonnegative matrices. We also obtain the bound for the rate of Rényi entropy of an irreducible Markov chain. Finally, we show that the bound for the Rényi entropy rate is the Shannon entropy rate.

We study a dependence relation between the join-irreducible elements of a finite lattice that generalizes the classical perspectiuity relation between the atoms of a geometric lattice. This relation is useful in the axiomatic approach of latticial consensus problems, since if it is strongly connected one can get arrowian theorems. First we show that the dependence relation is linked with the ar...

A simple object (one point in m-dimensional space) is the resultant of the evolving matrix polynomial of walks in the irreducible aperiodic network structure of the first order DeGroot (weighted averaging) state-space process. This paper draws on a second order generalization the DeGroot model that allows complex object resultants, i.e, multiple points with distinct coordinates, in the convex h...

For an irreducible stochastic matrix T , the Kemeny constant K(T ) measures the expected time to mixing of the Markov chain corresponding to T . Given a strongly connected directed graph D, we consider the set ΣD of stochastic matrices whose directed graph is subordinate to D, and compute the minimum value of K, taken over the set ΣD. The matrices attaining that minimum are also characterised, ...

For every fixed set F ⊆ {0, 1}n the following problems are strongly polynomial time equivalent: given a feasible point x ∈ F and a linear objective function c ∈ ZZn, • find a feasible point x∗ ∈ F that maximizes cx (Optimization), • find a feasible point x ∈ F with cx > cx (Augmentation), and • find a feasible point x ∈ F with cx > cx such that x − x is “irreducible” (Irreducible Augmentation)....

Fix a prime p > 2. Let ρ : Gal(Q/Q) → GL2(I) be the Galois representation coming from a non-CM irreducible component I of Hida’s p-ordinary Hecke algebra. Assume the residual representation ρ̄ is absolutely irreducible. Under a minor technical condition we identify a subring I0 of I containing Zp[[T ]] such that the image of ρ is large with respect to I0. That is, Im ρ contains ker(SL2(I0) → SL2...

Remark 0.1 (Notation). |G| denotes the order of a finite group G. [E : F ] denotes the degree of a field extension E/F. We write H ≤ G to mean that H is a subgroup of G, and N G to mean that N is a normal subgroup of G. If E/F and K/F are two field extensions, then when we say that K/F is contained in E/F , we mean via a homomorphism that fixes F. We assume the following basic facts in this set...

The main result is that, for any projective compact analytic subset Y of dimension q > 0 in a reduced complex space X , there is a neighborhood Ω of Y such that, for any covering space Υ: X̂ → X in which Ŷ ≡ Υ(Y ) has no noncompact connected analytic subsets of pure dimension q with only compact irreducible components, there exists a C∞ exhaustion function φ on X̂ which is strongly q-convex on Ω̂ ...