AMLI preconditioner for the p‐version of the FEM

From the literature it is known that the conjugate gradient method with domain decomposition preconditioners is one of the most efficient methods for solving systems of linear algebraic equations resulting from p‐version finite element discretizations of elliptic boundary value problems. One ingredient of such a preconditioner is a preconditioner related to the Dirichlet problems. In the case of Poisson's equation, we present a preconditioner for the Dirichlet problems which can be interpreted as the stiffness matrix K_h,k resulting from the h‐version finite element discretization of a special degenerated problem. We construct an AMLI preconditioner C_h,k for the matrix K_h,k and show that the condition number of C K_h,k is independent of the discretization parameter. This proof is based on the strengthened Cauchy inequality. The theoretical result is confirmed by numerical examples. Copyright © 2003 John Wiley & Sons, Ltd.     

On the ideal of functions with compact support in pointfree function rings

As usual, let RL\mathcal{R}L denote the ring of real-valued continuous functions on a completely regular frame L. We consider the ideals R_s(L)\mathcal{R}_{s}(L) and R_K(L)\mathcal{R}_{K}(L) consisting, respectively, of functions with small cozero elements and functions with compact support. We show that, as in the classical case of C(X), the first ideal is the intersection of all free maximal ideals, and the second is the intersection of pure parts of all free maximal ideals. A corollary of this latter result is that, in fact, R_K(L)\mathcal{R}_{K}(L) is the intersection of all free ideals. Each of these ideals is pure, free, essential or zero iff the other has the same feature. We observe that these ideals are free iff L is a continuous frame, and essential iff L is almost continuous (meaning that below every nonzero element there is a nonzero element the pseudocomplement of which induces a compact closed quotient). We also show that these ideals are zero iff L is nowhere compact (meaning that non-dense elements induce non-compact closed quotients).     

Hereditarily indecomposable Banach algebras of diagonal operators

We provide a characterization of the Banach spaces X with a Schauder basis (e _n ) _n∈ℕ which have the property that the dual space X* is naturally isomorphic to the space L _diag(X) of diagonal operators with respect to (e _n ) _n∈ℕ. We also construct a Hereditarily Indecomposable Banach space $ \mathfrak{X} $ \mathfrak{X} _D with a Schauder basis (e _n ) _n∈ℕ such that $ \mathfrak{X} $ \mathfrak{X} *_D is isometric to L _diag($ \mathfrak{X} $ \mathfrak{X} _D) with these Banach algebras being Hereditarily Indecomposable. Finally, we show that every T ∈ L _diag($ \mathfrak{X} $ \mathfrak{X} _D) is of the form T = λI + K, where K is a compact operator.     

Inverse product Toeplitz preconditioners for non-Hermitian Toeplitz systems

In this paper, we first propose product Toeplitz preconditioners (in an inverse form) for non-Hermitian Toeplitz matrices generated by functions with zeros. Our inverse product-type preconditioner is of the form T_F T_L^-1 T_U^-1T_F T_L^{-1} T_U^{-1} where T _F , T _L , and T _U are full, band lower triangular, and band upper triangular Toeplitz matrices, respectively. Our basic idea is to decompose the generating function properly such that all factors T _F , T _L , and T _U of the preconditioner are as well-conditioned as possible. We prove that under certain conditions, the preconditioned matrix has eigenvalues and singular values clustered around 1. Then we use a similar idea to modify the preconditioner proposed in Ku and Kuo (SIAM J Sci Stat Comput 13:1470–1487, 1992) to handle the zeros in rational generating functions. Numerical results, including applications to the computation of the stationary probability distribution of Markovian queuing models with batch arrival, are given to illustrate the good performance of the proposed preconditioners.     

On approximation of functions by trigonometric polynomials with incomplete spectrum in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>, 0 < <Emphasis Type="Italic">p</Emphasis> < 1

Let B be a set of integers with certain arithmetic properties. We obtain estimates of the best approximation of functions in the space L _p , 0 < p <1, by trigonometric polynomials that are constructed by the system {e^ikx}_{k ? \mathbbZ\B} \{e^{ikx}\}_{k\in \mathbb{Z}\backslash B} . Bibliography: 13 titles.     

On some exotic finite subgroups of E 8 and Springer’s regular elements of the Weyl group

A proof (by Serre and by Cohen, Griess and Lisser) verified, in the special case of E ₈, a conjecture of mine, that the finite projective group L ₂(61) embeds in E₈( \mathbbC ) {E_8}\left( \mathbb{C} \right) . Subsequently, Griess and Ryba have shown (using computers) that L ₂(49) and, in addition, (established by Serre without computers) L ₂(41) also embed in E₈( \mathbbC ) {E_8}\left( \mathbb{C} \right) . That is, if K = 30, 24, 20 and k ∈ K then L ₂(2k + 1) embeds in E₈( \mathbbC ) {E_8}\left( \mathbb{C} \right) . In this paper we show that the “Borel” subgroup B(k) of L ₂(2k + 1), k ∈ K, has a uniform construction. The theorem uses a result of T. Springer on the existence in E₈( \mathbbC ) {E_8}\left( \mathbb{C} \right) of three regular elements of the Weyl group, having orders k ∈ K, and associated to the regular, subregular and subsubregular nilpotent elements. Springer’s result generalizes (in the E ₈ case) a 1959 general result of mine relating the principal nilpotent element with the Coxeter element.     

Coxeter group actions on <Subscript>4</Subscript><Emphasis Type="Italic">F</Emphasis><Subscript>3</Subscript>(1) hypergeometric series

In this paper we investigate a certain linear combination K([(x)\vec])=K(a;b,c,d;e,f,g)K(\vec{x})=K(a;b,c,d;e,f,g) of two Saalschutzian hypergeometric series of type ₄ F ₃(1). We first show that K([(x)\vec])K(\vec{x}) is invariant under the action of a certain matrix group G _K , isomorphic to the symmetric group S ₆, acting on the affine hyperplane V={(a,b,c,d,e,f,g)∈ℂ⁷:e+f+g−a−b−c−d=1}. We further develop an algebra of three-term relations for K(a;b,c,d;e,f,g). We show that, for any three elements μ ₁,μ ₂,μ ₃ of a certain matrix group M _K , isomorphic to the Coxeter group W(D ₆) (of order 23040) and containing the above group G _K , there is a relation among K(m₁[(x)\vec])K(\mu_{1}\vec{x}), K(m₂[(x)\vec])K(\mu_{2}\vec{x}), and K(m₃[(x)\vec])K(\mu_{3}\vec{x}), provided that no two of the μ _j ’s are in the same right coset of G _K in M _K . The coefficients in these three-term relations are seen to be rational combinations of gamma and sine functions in a,b,c,d,e,f,g.     

Finite-Dimensional Subalgebras In Polynomial Lie Algebras Of Rank One

Let W _n ( \mathbb K {\mathbb K} ) be the Lie algebra of derivations of the polynomial algebra \mathbb K {\mathbb K} [X] := \mathbb K {\mathbb K} [x ₁,…,x _n ]over an algebraically closed field \mathbb K {\mathbb K} of characteristic zero. A subalgebra L í W_n(\mathbbK) L \subseteq {W_n}(\mathbb{K}) is called polynomial if it is a submodule of the \mathbb K {\mathbb K} [X]-module W _n ( \mathbb K {\mathbb K} ). We prove that the centralizer of every nonzero element in L is abelian, provided that L is of rank one. This fact allows one to classify finite-dimensional subalgebras in polynomial Lie algebras of rank one.     

On Approximate Spectral Factorization of Matrix Functions

It is proved that if positive definite matrix functions (i.e. matrix spectral densities) S _n , n=1,2,… , are convergent in the L ₁-norm, ||S_n-S||_L1? 0\|S_{n}-S\|_{L_{1}}\to 0, and ò₀^2plogdetS_n(e^iq) dq?ò₀^2plogdetS(e^iq) dq\int_{0}^{2\pi}\log \mathop{\mathrm{det}}S_{n}(e^{i\theta})\,d\theta\to\int_{0}^{2\pi}\log \mathop{\mathrm{det}}S(e^{i\theta})\,d\theta, then the corresponding (canonical) spectral factors are convergent in L ₂, ||S⁺_n-S⁺||_L2? 0\|S^{+}_{n}-S^{+}\|_{L_{2}}\to 0. The formulated logarithmic condition is easily seen to be necessary for the latter convergence to take place.     

<Emphasis Type="Italic">v</Emphasis>-Adic maximal extensions,spectral norms and absolute Galois groups

Let (K, v) be a perfect rank one valued field and let ([`(K<sub>v</sub>)],[`(v)]){(\overline{K_{v}},\overline{v})} be the canonical valued field obtained from (K, v) by the unique extension of the valuation [(v)\tilde]{\widetilde{v}} of K _v , the completion of K relative to v, to a fixed algebraic closure [`(K<sub>v</sub>)]{\overline{K_{v}}} of K <sub>v</sub> . Let [`(K)]{\overline{K}} be the algebraic closure of K in [`(K<sub>v</sub>)]{\overline {K_{v}}}. An algebraic extension L of K, L ì [`(K)]{L\subset\overline{K}}, is said to be a v-adic maximal extension, if [`(v)] | <sub>L</sub>{\overline{v}\mid_{L}} is the only extension of v to L and L is maximal with this property. In this paper we describe some basic properties of such extensions and we consider them in connection with the v-adic spectral norm on [`(K)]{\overline{K}} and with the absolute Galois groups Gal([`(K)]/K){(\overline{K}/K)} and Gal([`(K_v)] /K_v){(\overline{K_{v}} /K_{v})}. Some other auxiliary results are given, which may be useful for other purposes.     

Automatic continuity for linear surjective mappings decreasing the local spectral radius at some fixed vector

Let X be a complex Banach space and denote by L( X){\mathcal{L}\left( X\right)} the space of bounded linear operators on X. Let e be a nonzero element of X. We prove that if φ is a linear and surjective mapping from L( X){ \mathcal{L}\left( X\right) } into itself which decreases the local spectral radius at e, then φ is automatically continuous.     

A valuation criterion for normal basis generators in local fields of characteristic p

Let K be a complete local field of characteristic p with perfect residue field, and let L/K be a finite, totally ramified, Galois p-extension with G = Gal(L/K). Let v _L be the normalized valuation with ${v_L(L^{\times})=\mathbb{Z} }Let K be a complete local field of characteristic p with perfect residue field, and let L/K be a finite, totally ramified, Galois p-extension with G = Gal(L/K). Let v _L be the normalized valuation with v_L(L^×)=\mathbbZ {v_L(L^{\times})=\mathbb{Z} }. Let p_L ? L{\pi_L\in L} be a prime element, and let p′ (x) be the derivative of the minimal polynomial for π _L over K. We show that any element r ? L{\rho\in L} with v_L(r) o -v_L(p￠(p_L))-1 mod [L:K]{v_L(\rho)\equiv -v_L(p'(\pi_L))-1\bmod[L:K]} generates a normal basis: K[G]ρ = L. This criterion is tight: Given any integer i with i\not o -v_L(p￠(p_L))-1 mod [L:K]{i\not\equiv -v_L(p'(\pi_L))-1\bmod[L:K]}, there is a r_i ? L{\rho_i\in L} with v _L (ρ _i ) = i such that K[G]r_i\subsetneq L{K[G]\rho_i\subsetneq L}.     

Exterior powers of the reflection representation in Springer theory

Let H^*( B_e ) {H^*}\left( {{\mathcal{B}_e}} \right) be the total Springer representation of W for the nilpotent element e in a simple Lie algebra \mathfrakg \mathfrak{g} . Let Λ ⁱ V denote the ith exterior power of the reflection representation V of W. The focus of this paper is on the algebra of W-invariants in

Doubly eulerian trails on rectangular grids

Let K be the 1-skeleton of the regular tessellation of Euclidean n-space by n-cubes, n ≥ 2. We show that K admits a doubly Eulerian trail (simply Eulerian trail), that is, a doubly infinite path π = … e_?1e₀e₁ … where, out of each pair {e, e^?1} of oppositely directed edges, both (exactly one) appear(s) exactly once in π, and where no e_i+1 = e_i^?1 (there are no U-turns).     

A Polynomial-Time Algorithm to Approximate the Mixed Volume within a Simply Exponential Factor

Let K=(K ₁,…,K _n ) be an n-tuple of convex compact subsets in the Euclidean space R ⁿ , and let V(⋅) be the Euclidean volume in R ⁿ . The Minkowski polynomial V _K is defined as V _K (λ ₁,…,λ _n )=V(λ ₁ K ₁+⋅⋅⋅+λ _n K _n ) and the mixed volume V(K ₁,…,K _n ) as

Modified tangential frequency filtering decomposition and its fourier analysis

In this paper, a modified tangential frequency filtering decomposition (MTFFD) preconditioner is proposed. The optimal order of the modification and the optimal relaxation parameter is determined by Fourier analysis. With the choice of optimal order of modification, the Fourier results show that the condition number of the preconditioned matrix is O(h^-\frac23){{\mathcal O}(h^{-\frac{2}{3}})}, and the spectrum distribution of the preconditioned matrix can be predicted by the Fourier results. The performance of MTFFD preconditioner is compared with tangential frequency filtering (TFFD) preconditioner on a variety of large sparse matrices arising from the discretization of PDEs with discontinuous coefficients. The numerical results show that the MTFFD preconditioner is much more efficient than the TFFD preconditioner.     

Dense analytic subspaces in fractalL 2-spaces

We show that for certain self-similar measures μ with support in the interval 0≤x≤1, the analytic functions {e ^i2πnx :n=0,1,2, …} contain an orthonormal basis inL ² (μ). Moreover, we identify subsetsP ⊂ ℕ₀ = {0,1,2,...} such that the functions {e _n :n ∈ P} form an orthonormal basis forL ² (μ). We also give a higher-dimensional affine construction leading to self-similar measures μ with support in ℝ ^ν , obtained from a given expansivev-by-v matrix and a finite set of translation vectors. We show that the correspondingL ² (μ) has an orthonormal basis of exponentialse ^i2πλ·x , indexed by vectors λ in ℝ ^ν , provided certain geometric conditions (involving the Ruelle transfer operator) hold for the affine system. Work supported by the National Science Foundation.     

Semidiscrete Galerkin Approximation for a Linear Stochastic Parabolic Partial Differential Equation Driven by an Additive Noise

We study the semidiscrete Galerkin approximation of a stochastic parabolic partial differential equation forced by an additive space-time noise. The discretization in space is done by a piecewise linear finite element method. The space-time noise is approximated by using the generalized L₂ projection operator. Optimal strong convergence error estimates in the L₂ and norms with respect to the spatial variable are obtained. The proof is based on appropriate nonsmooth data error estimates for the corresponding deterministic parabolic problem. The error estimates are applicable in the multi-dimensional case. AMS subject classification (2000) 65M, 60H15, 65C30, 65M65.Received April 2004. Revised September 2004. Communicated by Anders Szepessy.     

Compact Hypergroup Characters and Finite Universal Korovkin Sets in L 1-Hypergroup Algebra

Let K be a compact hypergroup.We investigate Trig(K), the linear span of coordinate functions of the irreducible representations of K. Contrary to the group case, Trig(K) endowed with the usual multiplication does not bear an algebra structure, but it has a natural normed algebra structure when it inherits the convolution from , the algebra of all bounded Radon measures. We characterize the center of the algebras , L _p (K) and Trig(K) respectively, and consequently we obtain, for a certain class of hypergroups, the correspondence between the structure space of the center of L ₁(K) and the center of Trig(K). As an application we study the existence of a finite universal Korovkin set w.r.t. positive operators in the center of L ₁(K), in particular in L ₁(K), whenever K is commutative. The author was partially supported by the Romanian Academy under the grant No. 22/2007.     

Additive polylogarithms and their functional equations

Let ${k[\varepsilon]_{2}:=k[\varepsilon]/(\varepsilon^{2})}Let k[e]₂:=k[e]/(e²){k[\varepsilon]_{2}:=k[\varepsilon]/(\varepsilon^{2})} . The single valued real analytic n-polylogarithm L_n: \mathbbC ? \mathbbR{\mathcal{L}_{n}: \mathbb{C} \to \mathbb{R}} is fundamental in the study of weight n motivic cohomology over a field k, of characteristic 0. In this paper, we extend the construction in ünver (Algebra Number Theory 3:1–34, 2009) to define additive n-polylogarithms li_n:k[e]₂? k{li_{n}:k[\varepsilon]_{2}\to k} and prove that they satisfy functional equations analogous to those of L_n{\mathcal{L}_{n}}. Under a mild hypothesis, we show that these functions descend to an analog of the nth Bloch group B_n￠(k[e]₂){B_{n}' (k[\varepsilon]_{2})} defined by Goncharov (Adv Math 114:197–318, 1995). We hope that these functions will be useful in the study of weight n motivic cohomology over k[ε]₂.