The order of a group of even order

We will give an estimation of the order of a group of even order by the order of the centralizer of an involution using the classification of finite simple groups.

     

Mixtures of order matrices and generalized order matrices

This paper deals with matrix representations of linear orders, mixtures of order matrices and the non-integral solutions of the linear systems defining them.     

On the order of a group of even order

We present an upper bound for |G| of a group G of even order possessing a unique conjugacy class of involutions.     

Hadamard matrices of order 28 with automorphisms of order 7

The only primes which can divide the order of the automorphism group of a Hadamard matrix of order 28 are 13, 7, 3, and 2, and there are precisely four inequivalent matrices with automorphisms of order 13 (Tonchev, J. Combin. Theory Ser. A35 (1983), 43–57). In this paper we show that there are exactly twelve inequivalent Hadamard matrices of order 28 with automorphisms of order 7. In particular, there are precisely seven matrices with transitive automorphism groups.     

Sequential order statistics with an order statistics prior

In the model of sequential order statistics, prior distributions are considered for the model parameters, which, for example, describe increasing load put on remaining components. Gamma priors are examined as well as priors out of a class of extended truncated Erlang distributions (ETED), which is introduced along with some properties. The choice of independent priors in both set-ups leads to respective independent, conjugate posterior distributions for the model parameters of sequential order statistics. Since, in practical applications, the model parameters will often be increasingly ordered, a multivariate prior is applied being the joint distribution of common ETED-order statistics. Whatever baseline distribution of the sequential order statistics is chosen, the joint posterior distribution turns out to be a Weinman multivariate exponential distribution. Posterior moments are given explicitly, and HPD credible sets for the model parameters are stated.     

Ergodic properties where order 4 implies infinite order

For a family of dynamical properties, knowing that the condition holds for order 4 implies that it holds for all orders. Here we establish this for the properties minimal self-joinings, simplicity and for cartesiandisjointness. An application of the first yields an analog to Kalikow’s celebrated result that for rank-1 transformations, 2-fold mixing implies 3-fold mixing. Via a joining argument we show that for any rank-1 ℤ ^D -action, 4-fold mixing implies mixing of all orders. Indeed, the rank need only be sufficiently close to 1 for the implication to hold and so this result is new even when the acting group is ℤ. By means of limit-joinings, we settle affirmatively an old open question by establishing, for anyM, thatM-fold Rényi-mixing impliesM-fold mixing. Partially supported by National Science Foundation Postdoctoral Research Fellowship.     

Transformation of high order linear differential-algebraic systems to first order

We study general nonsquare linear systems of differential-algebraic systems of arbitrary order. We analyze the classical procedure of turning the system into a first order system and demonstrate that this approach may lead to different solvability results and smoothness requirements. We present several examples that demonstrate this phenomenon and then derive existence and uniqueness results for differential-algebraic systems of arbitrary order and index. We use these results to identify exactly those variables for which the order reduction to first order does not lead to extra smoothness requirements and demonstrate the effects of this new formulation with a numerical example.Dedicated to Richard S. Varga on the occasion of his 77th birthday.     

On approximating higher order MAPs with MAPs of order two

We show that the autocorrelation sequence of interarrival times for a Markovian arrival process (MAP) of order two is geometric. We determine the set of feasible values for the autocorrelation decay parameter and the first two or three moments of the interarrival time distribution. A method is derived for matching these parameters to a MAP of order two and some numerical examples are included to illustrate approximating higher dimensional MAPs by two dimensional ones. The numerical examples have helped us pose important questions regarding the significance of correlation in a MAP of order two when it is used as input to a queueing model.     

The lower order and linear order of multiple dirichlet series

The article investigates the growth of multiple Dirichlet series.The lower order and the linear order of n-tuple Dirichlet series in C~n are defined and some relations between them and the coefficients and exponents of n-tuple Dirichlet series are obtained.     

Lawless order

R. Baer asked whether the group operation of every (totally) ordered group can be redefined, keeping the same ordered set, so that the resulting structure is an Abelian ordered group. The answer is no. We construct an ordered set (G, ) which carries an ordered group (G, , ) but which islawless in the following sense. If (G, *, ) is an ordered group on the same carrier (G, ), then the group (G, *) satisfies no nontrivial equational law.Research partially supported by NSERC of Canada Grants #A4044 and A3040.Research partially supported by NSERC of Canada Grant #U0075.Research partially supported by a grant from the BSF.     

Hadamard matrices of order 28 with automorphism groups of order two

We give two examples H₁ and H₂ of Hadamard matrices of order 28 with trivial automorphism groups and show that H₁, H₁^T, H₂ and H₂^T are non-equivalent to each other as Hadamard matrices.     

Two-step fourth order methods for linear ODEs of the second order

A family of two step difference schemes of the fourth order has been developed for linear ODEs of the second order. Stability properties for such schemes are discussed and results of numerical tests are given. It is shown how the proposed technique can be extended to non-linear ODEs of second order.     

Classification of Hadamard matrices of order 44 with automorphisms of order 7

The Hadamard matrices of order 44 possessing automorphisms of order 7 are classified. The number of their equivalence classes is 384. The order of their full automorphism group is calculated. These Hadamard matrices yield 1683 nonisomorphic 3-(44,22,10) designs, 57932 nonisomorphic 2-(43,21,10) designs, and two inequivalent extremal binary self-dual doubly even codes of length 88 (one of them being new).     

GB-splines of arbitrary order

Explicit formulae and recurrence relations for the calculation of generalized B-splines (GB-splines) of arbitrary order are given. We derive main properties of GB-splines and their series, i.e. partition of unity, shape-preserving properties, invariance with respect to affine transformations, etc. It is shown that such splines have the variation diminishing property and are Chebyshevian splines.