We survey properties of the set of possible exponents of subsets of Zn (equivalently, exponents of primitive circulant digraphs on n vertices). Let En denote this exponent set. We point out that En contains the positive integers up to √ n, the ‘large’ exponents ⌊ 3 ⌋+1, ⌊ 2 ⌋, n−1, and for even n ≥ 4, the additional value n 2 −1. It is easy to see that no exponent in [n 2 +1, n−2] is possible, ...

A two-loop renormalization group analysis of the critical behaviour at an isotropic Lifshitz point is presented. Using dimensional regularization and minimal subtraction of poles, we obtain the expansions of the critical exponents ν and η, the crossover exponent ϕ, as well as the (related) wave-vector exponent βq, and the correction-to-scaling exponent ω to second order in ǫ 8 = 8 − d. These ar...

In this paper, a recursive algorithm is presented to generate some exponent matrices which correspond to Tanner graphs with girth at least 6. For a J × L exponent matrix E, the lower bound Q(E) is obtained explicitly such that (J, L) QC LDPC codes with girth at least 6 exist for any circulant permutation matrix (CPM) size m ≥ Q(E). The results show that the exponent matrices constructed with ou...

Improved cost estimates are given for the problem of computing the inverse of an n×n matrix of univariate polynomials over a field. A deterministic algorithm is demonstrated that has worst case complexity (n3s) field operations, where s ≥ 1 is an upper bound for the average column degree of the input matrix. Here, the “+o(1)” in the exponent indicates a missing factor c1(log ns)2 for positive r...

In this work, we give a rigorous explicit formula for the Lyapunov exponent for some binary infinite products of random 2 × 2 real matrices. All these products are constructed using only two types of matrices, A and B, which are chosen according to a stochastic process. The matrix A is singular, namely its determinant is zero. This formula is derived by using a particular decomposition for the ...

We study stability of linear time-varying differential-algebraic equations (DAEs). The Bohl exponent is introduced and finiteness of the Bohl exponent is characterized, the equivalence of exponential stability and a negative Bohl exponent is shown and shift properties are derived. We also show that the Bohl exponent is invariant under the set of Bohl transformations. For the class of DAEs which...

We study stability of linear time-varying differential-algebraic equations (DAEs). The Bohl exponent is introduced and finiteness of the Bohl exponent is characterized, the equivalence of exponential stability and a negative Bohl exponent is shown and shift properties are derived. We also show that the Bohl exponent is invariant under the set of Bohl transformations. For the class of DAEs which...

Exponent matrices appeared in the study of tiled orders over discrete valuation rings. Many properties of such orders are formulated using this notion. We think that such matrices are of interest in them own right, in particular, it is convenient to write finite partially ordered sets (posets) and finite metric spaces as special exponent matrices. Note that when we defined a quiver Q(E) of a re...