On the order of deferred correction

New deferred correction methods for the numerical solution of initial value problems in ordinary differential equations have recently been introduced by Dutt, Greengard and Rokhlin. A convergence proof is presented for these methods, based on the abstract Stetter-Lindberg-Skeel framework and Spijker-type norms. It is shown that p corrections of an order-r one-step solver yield order-r(p+1) accuracy.     

A basic variable neighborhood search heuristic for the uncapacitated multiple allocation <Emphasis Type="Italic">p</Emphasis>-hub center problem

The uncapacitated multiple allocation p-hub center problem (UMApHCP) consists of choosing p hub locations from a set of nodes with pairwise traffic demands in order to route the traffic between the origin-destination pairs such that the maximum cost between origin-destination pairs is minimum. It is assumed that transportation between non-hub nodes is possible only via chosen hub nodes. In this paper we propose a basic variable neighborhood search (VNS) heuristic for solving this NP hard problem. In addition we apply two mathematical formulations of the UMApHCP in order to detect limitations of the current state-of-the-art solver used for this problem. The heuristics are tested on benchmark instances for p-hub problems. The obtained results reveal the superiority of the proposed basic VNS over the state-of-the-art as well as over a multi-start local search heuristic developed by us in this paper.     

Analysis and application of finite volume element methods to a class of partial differential equations

The finite volume element (FVE) methods for a class of partial differential equations are discussed and analyzed in this paper. The new initial values are introduced in the finite volume element schemes, and we obtain optimal error estimates in Lp and W1,p (2?p?∞) as well as some superconvergence estimates in W1,p (2?p?∞). The main results in this paper perfect the theory of the finite volume element methods.     

Classification theorems for sumsets modulo a prime

Let Zp be the finite field of prime order p and A be a subsequence of Zp. We prove several classification results about the following questions:(1) When can one represent zero as a sum of some elements of A?(2) When can one represent every element of Zp as a sum of some elements of A?(3) When can one represent every element of Zp as a sum of l elements of A?     

Groups with the same prime graph as the orthogonal group B n (3)

Let G be a finite group. The prime graph of G is denoted by Γ(G). It is proved in [1] that if G is a finite group such that Γ(G) = Γ(B _p (3)), where p > 3 is an odd prime, then G ? B _p (3) or C _p (3). In this paper we prove the main result that if G is a finite group such that Γ(G) = Γ(B _n (3)), where n ≥ 6, then G has a unique nonabelian composition factor isomorphic to B _n (3) or C _n (3). Also if Γ(G) = Γ(B ₄(3)), then G has a unique nonabelian composition factor isomorphic to B ₄(3), C ₄(3), or ² D ₄(3). It is proved in [2] that if p is an odd prime, then B _p (3) is recognizable by element orders. We give a corollary of our result, generalize the result of [2], and prove that B _2k+1(3) is recognizable by the set of element orders. Also the quasirecognition of B _2k (3) by the set of element orders is obtained.     

Superconvergence of the continuous Galerkin finite element method for delay-differential equation with several terms

The continuous Galerkin finite element method for linear delay-differential equation with several terms is studied. Adding some lower terms in the remainder of orthogonal expansion in an element so that the remainder satisfies more orthogonal condition in the element, and obtain a desired superclose function to finite element solution, thus the superconvergence of p  -degree finite element approximate solution on (p+1)$(p+1)$-order Lobatto points is derived.     

On the equivariant Tamagawa number conjecture in tame CM-extensions

Let L/K be a finite Galois CM-extension with Galois group G. The Equivariant Tamagawa Number Conjecture (ETNC) for the pair ${(h^0({\rm Spec} (L))(0), {\mathbb Z}G)}$ naturally decomposes into p-parts, where p runs over all rational primes. If p is odd, these p-parts in turn decompose into a plus and a minus part. Let L/K be tame above p. We show that a certain ray class group of L defines an element in ${K_0({\mathbb Z}{_p}G_-, \mathbb Q_p)}$ which is determined by a corresponding Stickelberger element if and only if the minus part of the ETNC at p holds. For this we use the Lifted Root Number Conjecture for small sets of places which is equivalent to the ETNC in the number field case. For abelian G, we show that the minus part of the ETNC at p implies the Strong Brumer?CStark Conjecture at p. We prove the minus part of the ETNC at p for almost all primes p.     

A three-stage, VSVO, Hermite-Birkhoff-Taylor, ODE solver

The new variable-step, variable-order, ODE solver, HBT(p) of order p, presented in this paper, combines a three-stage Runge-Kutta method of order 3 with a Taylor series method of order p-2 to solve initial value problems , where y:R→R^d and f:R×R^d→R^d. The order conditions satisfied by HBT(p) are formulated and they lead to Vandermonde-type linear algebraic systems whose solutions are the coefficients in the formulae for HBT(p). A detailed formulation of variable-step HBT(p) in both fixed-order and variable-order modes is presented. The new method and the Taylor series method have similar regions of absolute stability. To obtain high-accuracy results at high order, this method has been implemented in multiple precision.     

On the structure of the Wishart distribution

In this paper it is shown that every nonnegative definite symmetric random matrix with independent diagonal elements and at least one nondegenerate nondiagonal element has a noninfinitely divisible distribution. Using this result it is established that every Wishart distribution W_p(k, Σ, M) with both p and rank (Σ) ≥ 2 is noninfinitely divisible. The paper also establishes that any Wishart matrix having distribution W_p(k, Σ, 0) has the joint distribution of its elements in the rth row and rth column to be infinitely divisible for every r = 1,2,…,p.     

A non-uniform basis order for the discontinuous Galerkin method of the acoustic and elastic wave equations

Here, we solve the time-dependent acoustic and elastic wave equations using the discontinuous Galerkin method for spatial discretization and the low-storage Runge-Kutta and Crank-Nicolson methods for time integration. The aim of the present paper is to study how to choose the order of polynomial basis functions for each element in the computational mesh to obtain a predetermined relative error. In this work, the formula 2p+1≈κhk, which connects the polynomial basis order p, mesh parameter h, wave number k, and free parameter κ, is studied. The aim is to obtain a simple selection method for the order of the basis functions so that a relatively constant error level of the solution can be achieved. The method is examined using numerical experiments. The results of the experiments indicate that this method is a promising approach for approximating the degree of the basis functions for an arbitrarily sized element. However, in certain model problems we show the failure of the proposed selection scheme. In such a case, the method provides an initial basis for a more general p-adaptive discontinuous Galerkin method.     

Stable hp mixed finite elements based on the Hellinger–Reissner principle

In the Hellinger–Reissner formulation for linear elasticity, both the displacement u and the stress σ are taken as unknowns, giving rise to a saddle point problem. We present new pairings of quadrilateral ‘trunk’ finite element spaces for this method and prove stability (and optimality) in terms of both h and p. The effect of mesh shape regularity on the stability constant is explicitly tracked. Our results provide a theoretical basis for recent numerical experiments (in the context of a mixed p formulation for viscoelasticity) that showed these spaces worked well computationally.     

The prescribed p-mean curvature equation of low regularity in the Heisenberg group

This work reports on the author’s recent study about regularity and the singular set of a C ¹ smooth surface with prescribed p (or H)-mean curvature in the 3-dimensional Heisenberg group. As a differential equation, this is a degenerate hyperbolic and elliptic PDE of second order, arising from the study of CR geometry. Assuming only the p-mean curvature H ∈ C ⁰, it is shown that any characteristic curve is C ² smooth and its (line) curvature equals ?H. By introducing special coordinates and invoking the jump formulas along characteristic curves, it is proved that the Legendrian (horizontal) normal gains one more derivative. Therefore the seed curves are C ² smooth. This work also obtains the uniqueness of characteristic and seed curves passing through a common point under some mild conditions, respectively. In an on-going project, it is shown that the p-area element is in fact C ² smooth along any characteristic curve and satisfies a certain ordinary differential equation of second order. Moreover, this ODE is analyzed to study the singular set.     

Compact elements of the algebras PM p (G)and PF p (G) of a locally compact group

Compact and weakly compact elements of the group algebra L ¹ (G) of a locally compact group G, have been considered by a number of authors. In these investigations it has been shown that, if G is non-compact, then the only weakly compact element of L ¹ (G ) is zero. Conversely, if G is compact, then every element of L ¹ (G) is compact. For 1<p<∞, let PM _p (G)and PF _p (G) denote the closure of L ¹ (G), considered as an algebra of convolution operators on L ^p (G), with respect to the weak operator topology and the norm topology, respectively, in B(L ^p (G), b), the bounded linear operators on L ¹ (G). We study the question of characterizing compact and weakly compact elements of the algebras PM _p (G)and PF _p (G).     

Recognition of simple groups B p (3) by the set of element orders

Let G be a finite group and let ω(G) be the set of its element orders. We prove that if ω(G) = ω(B _p (3)) where p is an odd prime, then G ? B ₃(3) or D ₄(3) for p = 3 and G ? B _p (3) for p > 3.     

Multiplicative Schwarz algorithms for the p-version Galerkin boundary element method in 3D

We study a 2-level multiplicative Schwarz method for the p version Galerkin boundary element method for a weakly singular integral equation of the first kind in 3D. We prove that the rate of convergence of the multiplicative Schwarz operator for the p version grows only logarithmically in p and is independent of h.     

Implicit peer methods for large stiff ODE systems

Implicit two-step peer methods are introduced for the solution of large stiff systems. Although these methods compute s-stage approximations in each time step one-by-one like diagonally-implicit Runge-Kutta methods the order of all stages is the same due to the two-step structure. The nonlinear stage equations are solved by an inexact Newton method using the Krylov solver FOM (Arnoldi??s method). The methods are zero-stable for arbitrary step size sequences. We construct different methods having order p=s in the multi-implicit case and order p=s?1 in the singly-implicit case with arbitrary step sizes and s??5. Numerical tests in Matlab for several semi-discretized partial differential equations show the efficiency of the methods compared to other Krylov codes.     

A Nontrivial Homotopy Element of Order p2 Detected by the Classical Adams Spectral Sequence

Let p be an odd prime. The authors detect a nontrivial element ã _p of order p² in the stable homotopy groups of spheres by the classical Adams spectral sequence. It is represented by \(a_0^{p - 2} h_1 \in Ext_A^{p - 1,pq + p - 2} (\mathbb{Z}/p,\mathbb{Z}/p)\) in the E₂-term of the ASS and meanwhile p · ã ^p is the first periodic element α ^p .     

Reflexive ideals in Iwasawa algebras

Let G be a torsionfree compact p-adic analytic group. We give sufficient conditions on p and G which ensure that the Iwasawa algebra ΩG of G has no non-trivial two-sided reflexive ideals. Consequently, these conditions imply that every non-zero normal element in ΩG is a unit. We show that these conditions hold in the case when G is an open subgroup of SL₂(Zp) and p is arbitrary. Using a previous result of the first author, we show that there are only two prime ideals in ΩG when G is a congruence subgroup of SL₂(Zp): the zero ideal and the unique maximal ideal. These statements partially answer some questions asked by the first author and Brown.     

Orthomodular semilattices

We present a simple equational characterization of (meet) semilattices with 0 where for each element p the interval [0,p] is an orthomodular lattice or an ortholattice possibly satisfying the compatibility condition.