Maximal Orders of Abelian Subgroups in Finite Chevalley Groups

In the present paper, for any finite group G of Lie type (except for ² F ₄(q)), the order a(G) of its large Abelian subgroup is either found or estimated from above and from below (the latter is done for the groups F ₄ (q), E ₆ (q), E ₇ (q), E ₈ (q), and ² E ₆(q ²)). In the groups for which the number a(G) has been found exactly, any large Abelian subgroup coincides with a large unipotent or a large semisimple Abelian subgroup. For the groups F ₄ (q), E ₆ (q), E ₇ (q), E ₈ (q), and ² E ₆(q ²)), it is shown that if an Abelian subgroup contains a noncentral semisimple element, then its order is less than the order of an Abelian unipotent group. Hence in these groups the large Abelian subgroups are unipotent, and in order to find the value of a(G) for them, it is necessary to find the orders of the large unipotent Abelian subgroups. Thus it is proved that in a finite group of Lie type (except for ² F ₄(q))) any large Abelian subgroup is either a large unipotent or a large semisimple Abelian subgroup.     

The finite simple groups with at most two fusion classes of every order

Elements a,b of a group G are said to be fused if a = b^σ and to be inverse-fused if a =(b^-1)^σ for some σ ? Aut(G). The fusion class of a ? G is the set {a^σ | σ ? Aut(G)}, and it is called a fusion class of order i if a has order iThis paper gives a complete classification of the finite nonabelian simple groups G for which either (i) or (ii) holds, where:

(i) G has at most two fusion classes of order i for every i (23 examples); and

(ii) any two elements of G of the same order are fused or inversenfused.

The examples in case (ii) are: A₅, A₆,L₂(7),L₂(8), L₃(4), Sz(8), M₁₁ and M₂₃An application is given concerning isomorphisms of Cay ley graphs.     

Some arrowing results for trees versus complete graphs

For an arbitrary tree T of order m and an arbitrary positive integer n, Chvátal proved that the Ramsey number r(T, K_n) = 1 + (m ? 1) (n ? 1). for graphs G, G₁, and G₂, we say that G arrows G₁ and G₂, written G → (G₁, G₂), if for every factorization G = R ⊕ B, either G₁ is a subgraph of R or G₂ is a subgraph of B. it is shown that (i) for each l ≥ 2, K₁+ (m?1)(n?1) ?E(K₁) → (T, K_n) for m ≥ 2/ ? 1 and n ≥ 2; (ii) K₁ +,(m ?1)(n ?1) ? E(H) → (T, K_n), where H is any tree of order m ? 1, m ≥ 3 and n ≥ 2. It is further shown that result (i) is sharp with respect to the inequality m ≥ 2/? 1; in particular, examples are given to show that K1 + (2l?3)(n?1) E(K₁) ? (P_21?2, K_n) for all n ≥ 2, where P_21?2 denotes the path of order 21 ? 2. Also result (ii) is sharp with respect to the order of H; examples aregiven to show that K₁ + _(m?1)(n?1)? E(K(1, m ? 1)) ?(T, K_n)for any tree T of order m and any n ≥ 2.     

Statistical Convergence of Order α for Difference Sequences

Abstract

In this paper the concepts of ?^m_v-statistical convergence of order α and strong (p, ?^m)-Ces`aro summability of order α are introduced for sequences of complex (or real) numbers. Some relations between the ?<sup>m</sup><sub>v</sub>-statistical convergence of order α and strong (p, ?<sup>m</sup><sub>v</sub>)-Ces`aro summability of order α are given. Also some relations between the space ω^α_p (?^m_v, f) and S^α (?^m_v) are examined.     

Angle orders,regular n-gon orders and the crossing number

A finite poset P(X,<) on a set X={ x ₁,...,x _m} is an angle order (regular n-gon order) if the elements of P(X,<) can be mapped into a family of angular regions on the plane (a family of regular polygons with n sides and having parallel sides) such that x _ij if and only if the angular region (regular n-gon) for x _i is contained in the region (regular n-gon) for x _j. In this paper we prove that there are partial orders of dimension 6 with 64 elements which are not angle orders. The smallest partial order previously known not to be an angle order has 198 elements and has dimension 7. We also prove that partial orders of dimension 3 are representable using equilateral triangles with the same orientation. This results does not generalizes to higher dimensions. We will prove that there is a partial order of dimension 4 with 14 elements which is not a regular n-gon order regardless of the value of n. Finally, we prove that partial orders of dimension 3 are regular n-gon orders for n3.This research was supported by the Natural Sciences and Engineering Research Council of Canada, grant numbers A0977 and A2415.     

Hermite-Fejer type interpolation of higher order

The paper introduces Hermite-Fejér type (Hermite type) interpolation of higher order denoted by S _mn(f)(S* _mm(f)), and gives some basic properties including expression formulas, convergence relationship between S _mn(f) and H _mn(f) (Hermite-Fejér interpolation of higher order), and the saturation of S _mn(f). Supported by the Science Foundation of Shanxi Province for Returned Scholars.     

Maximum chromatic polynomials of 2-connected graphs

In this paper we obtain chromatic polynomials P(G; λ) of 2-connected graphs of order n that are maximum for positive integer-valued arguments λ ≧ 3. The extremal graphs are cycles C_n and these graphs are unique for every λ ≧ 3 and n ≠ 5. We also determine max{P(G; λ): G is 2-connected of order n and G ≠ C_n} and all extremal graphs relative to this property, with some consequences on the maximum number of 3-colorings in the class of 2-connected graphs of order n having _X(G) = 2 and _X(G) = 3, respectively. For every n ≧ 5 and λ ≧ 4, the first three maximum chromatic polynomials of 2-connected graphs are determined.     

Relative length of longest paths and cycles in 3‐connected graphs

For a graph G, let p(G) denote the order of a longest path in G and c(G) the order of a longest cycle in G, respectively. We show that if G is a 3‐connected graph of order n such that for every independent set {x₁, x₂, x₃, x₄}, then G satisfies c(G) ≥ p(G) ? 1. Using this result, we give several lower bounds to the circumference of a 3‐connected graph. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 137–156, 2001     

Directional approximation of the jacobians in ROW-methods

In this paper a new technique for avoiding exact Jacobians in ROW methods is proposed. The Jacobiansf' _n are substituted by matricesA _n satisfying a directional consistency conditionA _n f _n =f' _n f _n +O(h). In contrast to generalW-methods this enables us to reduce the number of order conditions and we construct a 2-stage method of order 3 and families of imbedded 4-stage methods of order 4(3). The directional approximation of the Jacobians has been realized via rank-1 updating as known from quasi-Newton methods.     

Local polynomial fitting under association

We consider the estimation of multivariate regression functions r(x₁,…,x_d) and their partial derivatives up to a total order p1 using high-order local polynomial fitting. The processes {Y_i,X_i} are assumed to be (jointly) associated. Joint asymptotic normality is established for the estimates of the regression function r and all its partial derivatives up to the total order p. Expressions for the bias and variance/covariance matrix (of the asymptotic distribution) are given.     

Generalized Hadamard Matrices

The paper studies a generalized Hadamard matrix H = (g _{i j}) of order n with entries gi j from a group G of order n. We assume that H satisfies: (i) For m k, G = {g _{m i} g k i ^-1 i = 1,...., n} (ii) g _1i = g _i1 = 1 for each i; (iii) g _ij ^-1 = g _ji for all i, j. Conditions (i) and (ii) occur whenever G is a(P, L) -transitivity for a projective plane of order n. Condition (iii) holds in the case that H affords a symmetric incidence matrix for the plane. The paper proves that G must be a 2-group and extends previous work to the case that n is a square.     

Efficient Simple Groups

Abstract

We prove that the simple group L ₃(5) which has order 372000 is efficient by providing an efficient presentation for it. This leaves one simple group with order less than one million, S ₄(4) which has order 979200, whose efficiency or otherwise remains to be determined.     

On cycle—Complete graph ramsey numbers

A new upper bound is given for the cycle-complete graph Ramsey number r(C_m, K_n), the smallest order for a graph which forces it to contain either a cycle of order m or a set of n independent vertices. Then, another cycle-complete graph Ramsey number is studied, namely r(?C_m, K_n) the smallest order for a graph which forces it to contain either a cycle of order / for some / satisfying 3?/?m or a set of n independent vertices. We obtain the exact value of r(?C_m K_n) for all m > n and an upper bound which applies when m is large in comparison with log n.     

The distal order of a minimal flow

The capturing operation, a kind of reverse orbit closure, is defined. In terms of this operation generalized proximal relationsP _γ(X) for a minimal flow (X, T) are defined. The distal order of (X, T) is then defined to be the least ordinal numberγ for which the distal structure relationS _d(X) =P _γ(X). It is shown that the distal order of a minimal flow is either finite orω, the first infinite ordinal. Examples are given which show that minimal flows of any allowable distal order actually occur. An analogous theory is developed for the equicontinuous relationS _eq(X).     

Existence of frame sols of type 2nu1

An SOLS (self-orthogonal latin square) of order n with n₁ missing sub-SOLS (holes) of order h_i (1 ? i ? k), which are disjoint and spanning (i.e., Σ _1?i?kn_ih_i = n), is called a frame SOLS and denoted by FSOLS(h₁ⁿ¹ h₂ⁿ² …h_k^nk). In this article, it is shown that for u ? 2, an FSOLS(2ⁿu¹) exists if and only if n ? 1 + u. © 1995 John Wiley & Sons, Inc.     

On automorphisms of a class of special p-groups

Let G be a finite group of order n, for some n\geqq 1 n\geqq 1 , and p be an odd prime number. In [5] Verardi has constructed a special p-group P_G P_G of exponent p such that |P_G|=p³ⁿ |P_G|=p^{3n} . In this paper, we calculate the order of Aut(P_G) (P_G) and prove that Aut(P_G) (P_G) is the semidirect product of two subgroups.     

A highly accurate numerical solution of a biharmonic equation

The coefficients for a nine–point high–order accuracy discretization scheme for a biharmonic equation ∇ ⁴u = f(x, y) (∇² is the two–dimensional Laplacian operator) are derived. The biharmonic problem is defined on a rectangular domain with two types of boundary conditions: (1) u and ∂²u/∂n² or (2) u and ∂u/part;n (where ∂/part;n is the normal to the boundary derivative) are specified at the boundary. For both considered cases, the truncation error for the suggested scheme is of the sixth-order O(h⁶) on a square mesh (h_x = h_y = h) and of the fourth-order O(h⁴_xh²_xh²_y h⁴_y) on an unequally spaced mesh. The biharmonic equation describes the deflection of loaded plates. The advantage of the suggested scheme is demonstrated for solving problems of the deflection of rectangular plates for cases of different boundary conditions: (1) a simply supported plate and (2) a plate with built-in edges. In order to demonstrate the high–order accuracy of the method, the numerical results are compared with exact solutions. © John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 375–391, 1997     

Upper triangular operators and <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">L</Emphasis>-functions

For a newform f for Γ₀(N) of even weight k supersingular at a prime p ≥ 5, by using infinite dimensional p-adic analysis, we prove that the p-adic L-function L _p (f,α; χ) has finite order of vanishing at any character of the form [(c)\tilde] _s ( x ) = x^s\tilde \chi _s \left( x \right) = x^s. In particular, under the natural embedding of ℤ _p in the group of ℂ* _p -valued continuous characters of ℤ* _p , the order of vanishing at any point is finite.     

Modeling structural variability in reduced order models of machine tool assembly groups via parametric MOR

We present a parametric model order reduction (PMOR) method applied to a parameter depending generalized state-space system, which describes the evolution of the temperature field on a vertical stand of a machine tool assembly group induced by a moving tool slide. The position of this slide parametrizes the input matrix of the associated system. The main idea is to compute projection matrices V_j, W_j in certain parameter sample points μ_j and concatenate them to the projection bases V, W, respectively, as described in [1]. Instead of using the iterative rational Krylov algorithm (IRKA) to produce the projection matrices in each parameter sample point as suggested there, here we use the well known method of balanced truncation (BT). The numerical results show that for the same reduced order r obtained from V, W ∈ ℝ^n×r, BT produces a parametric reduced order model (ROM) of similar accuracy as IRKA in less time. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)