Dual operator norms and the spectra of matrices

Let ∥·∥ be an operator norm and ∥·∥^D its dual. Then it is shown that ∥A∥^D? ∑|λ_i(A)|, where λ_i(A) are the eigenvalues of A, holds for all matrices A if and only if ∥·∥ is the operator norm subordinate to a Euclidian vector norm.     

Order of approximation by electrostatic fields due to electrons

LetD be a Jordan domain in the complex plane andA _q (D) the Bers space with norm ∥ ∥ _q . IfD is the unit disk, it is known that ∥S _n 0∥₂≥π/18, whereS _n =∑ _k=1 ⁿ l/(z?z _nk ) withz _nk ∈?D, so that approximation in ∥ ∥ _q ,q<-2, is not possible. In this paper, we give an order of estimate of ∥f?S _n ∥ _q for 2<q<∞ when ?D is a sufficiently smooth Jordan curve, and prove that this order of approximation is in general best possible.     

A note on the evaluation of bounded L2-functionals at B-splines and its application to singular integral equations

An algorithm computing recursively the values of ∫g(t)v(t) dt, whereg is anL ₂-function andv is aB-spline, is presented. For the functionsg _s(t)=log∥s?t∥ the starting values of the recursion formula can be computed analytically. The problem is related to the numerical solution of integral equations with a logarithmic singularity.     

On the parity of the partition function

Let N be the set of all positive integers and D a subset of N. Let p(D,n) be the number of partitions of n with parts in D and let |D(x)| denote the number of elements of D not exceeding x. It is proved that if D is an infinite subset of N such that p(D,n) is even for all n?n₀, then |D(x)|?logx/log2−logn₀/log2. Moreover, if D is an infinite subset of N such that p(D,n) is odd for all n?n₀ and , then |D(x)|?logx/log2−logn₀/log2. These lower bounds are essentially the best possible.     

Inner derivations on ultraprime normed real algebras

We show that the set of all inner derivations of an ultraprime real Banach algebra is closed within all bounded derivations. More concretely, we show that for such an algebra A there exists a positive number γ (depending only on the “constant of ultraprimeness” of A) satisfying γ ∥ a+Z(A) ∥≦∥ D _a ∥ for all a in A, where Z(A) denotes the centre of A and D _a denotes the inner derivation on A induced by a. This result is an extension of the corresponding complex version obtained by the authors in [Proc. Amer. Math. Soc., to appear]. The proof relies on the following theorem: ultraproducts of a family of central ultraprime real Banach algebras with a unit and with constant of ultraprimeness greater than or equal to a fixed positive constant K are central ultraprime Banach algebras with a unit. This fact is obained via a general result for real Banach algebras that reads as follows: If A is a central real Banach algebra with a unit 1, then for every a in A satisfying ∥ 1+a ² ∥<1 we have [1+√1?||1+1a ²||]²≦2(|?l+M _a ||+||D _a ||) where M _a denotes the two-sided multiplication operator by a on A.     

Asymptotics of the complexity of systems of integer linear forms for additive computations

The computational complexity of integer linear forms is studied. By l ₂(A) we denote the minimal number of the additions and subtractions required for computing the system of p linear forms in q variables x ₁, x ₂, …, x _q that are defined by an integer matrix A of size p × q (repeated use of the results of intermediate computation is permitted). We show that l ₂(A) ? log D(A), where D(A) is the maximum of the absolute values of the minors of A over all minors from order 1 to order min (p, q) (Theorem 1). Moreover, for each sequence of matrices A(n) of size p(n) × q(n) satisfying the condition p + q = o ((log log D(A))^1/2) as n → ∞ the bound l ₂(A) ? log D(A) + o(log D(A)) is valid (Theorem 2). Hence, for all fixed (and even weakly increasing) sizes of matrices that determine a system of integer linear forms, the upper bound on the computational complexity of this system is asymptotically equal to the lower bound.     

Pyramid computer solutions of the closest pair problem

Given an N × N array of 0s and 1s, the closest pair problem is to determine the minimum distance between any pair of ones. Let D be this minimum distance (or D = 2N if there are fewer than two 1s). Two solutions to this problem are given, one requiring O(log(N) + D) time and the other O(log(N)). These solutions are for two types of parallel computers arranged in a pyramid fashion with the base of the pyramid containing the matrix. The results improve upon an algorithm of Dyer that requires O(N) time on a more powerful computer.     

The error bound of the perturbation of the Drazin inverse

Let A and E be n×n matrices and B = A + E. Denote the Drazin inverse of A by A^D. In this paper we give an upper bound for the relative error ∥B^D ? A^D∥/∥A^D∥₂ and a lower bound for ∥B^D∥₂ under certain circumstances. The continuity properties and the derivative of the Drazin inverse are also considered.     

Fractional Covers for Convolution Products

A non-zero vector-valued sequence u ∈ ?^q(X′) is a cover for a subset M of ?_P(X) if, for some 0 < α ≤ 1, ∥u * h∥ _∞ ≥ α ∥u∥_q ∥h∥_p for all h ∈ M. Covers of ?¹ = ?¹(R) are important in worst case system identification in ?¹ and in the reconstruction of elements in a normed space from corrupted functional values. We investigate the existence of covers for certain naturally occurring subspaces of ?^p(X). We show that there exist finitely supported covers for some subspaces, and obtain lower bounds for their ’lengths’. We also obtain similar results for covers associated with convolution products for spaces of measurable vector-valued functions defined on the positive real axis.     

On operator-valued analytic functions with constant norm

Let X be a complex Banach space and $\text{D}$ a domain in the complex plane. Let f: $\text{D}$ → X be an analytic function such that ∥f(ζ)∥ is constant as ζ ? $\text{D}$. If X is the complex plane, then by the classical maximum modulus theorem f;(ζ) itself is constant on $\text{D}$. This is not the case in general. In the paper we study the norm-constant analytic functions whose values are bounded linear operators over an uniformly convex complex Banach space or, in particular, over a complex Hilbert space.     

Sharp nonlogarithmic Kneser theorems for fourth-order elliptic equations

Let D(?) be the Doob's class containing all functions f(z) analytic in the unit disk Δ such that f(0) = 0 and lim $\text{inf}\text{¦f(z) ¦ ? 1}$ on an arc A of ?Δ with length $\text{¦A ¦? ?}$. It is first proved that if f?D(?) then the spherical norm ∥ f ∥ = sup_z?Δ$\text{(}\text{1 ? ¦z¦}{}^2\text{)¦f′(z)¦}\text{(1 + ¦f(z)¦}{}^2\text{) ? C}{}_1\text{sin}\text{(π ? (}\text{?}\text{2}\text{))/ (π ? (}\text{g9}\text{2}\text{))}$, where C₁ = lim_n→∞$\text{∥ z}{}^{\mathrm{n}}\text{∥}\text{and}\text{1}\text{2}\text{< C}{}_1\text{<}\text{2}\text{e}$. Next, U represents the Seidel's class containing all non-constant functions f(z) bounded analytic in Δ such that $\text{¦tf(e}{}^{\mathrm{i0}}\text{)¦ = 1}$ almost everywhere. It is proved that inf_f?U∥f∥ = 0, and if f has either no singularities or only isolated singularities on ?Δ, then ∥f∥ ? C₁. Finally, it is proved that if f is a function normal in Δ, namely, the norm ∥f∥< ∞, then we have the sharp estimate ∥f^p∥ ? p∥f∥, for any positive integer p.     

Function algebra invariants and the conformal structure of planar domains

Let A(S) be the sup-normed Banach algebra of analytic functions with continuous boundary values on the compact bordered Riemann surface S.For (?) in $\text{A(S)}{}^{\mathrm{?1}}\text{exp}\text{(A(S))}$, the colength of (?) is defined by $\text{∥(?)∥ =}\text{1}\text{2}\text{log inf}\text{{∥ g ∥ ∥ g}{}^{\mathrm{?1}}\text{∥; g ? (?)}}$. Colength is shown to induce a norm on the cohomology group H¹(S,R) dual to the norm induced on the homology group H₁(S,R) by harmonic length, or, equivalently, dual to the norm on Re A(S)^⊥.The existence and uniqueness of extremal functions for the colength functional is demonstrated. The aforementioned norms are shown to determine the conformal structure of S (up to reflection) and to be related to the mapping properties of S.     

The additive and logical complexities of linear and bilinear arithmetic algorithms

Let C_A(±) be the additive complexity of a (bi)linear algorithm A for a given problem; D(A) and $\text{D}\text{(A)}$ are two acyclic diagraphs that represent A, each of them is obtained from another one by reversing directions of all edges; ir(D) and do(D) are two numbers that are introduced to measure the structural deficiencies of an acyclic digraph D. K and Q are the numbers of outputs and input-variables. $\text{do(}\text{D}\text{(A))}$, do(D(A)), and ir(D(A)) characterize the logical complexity of A. It is shown that C_A(±) + do(D(A)) + ir(D(A)) = ω(K + Q)log(K + Q) and $\text{C}{}_{\mathrm{A}}\text{(±) + do(}\text{D}\text{(A)) = ω(K + Q)}\text{log}\text{(K + Q)}$ in the cases of DFT, vector convolution, and matrix multiplication. Also lower bounds on C_A(±) + do(D(A)) and on C_A(±) are expressed in terms of algebraic quantities such as the ranks of matrices and of multidimensional tensors associated with the problems.     

Error analysis of explicit TSERKN methods for highly oscillatory systems

In this paper, we are concerned with the error analysis for the two-step extended Runge-Kutta-Nyström-type (TSERKN) methods [Comput. Phys. Comm. 182 (2011) 2486–2507] for multi-frequency and multidimensional oscillatory systems y″(t) + My(t) = f(t, y(t)), where high-frequency oscillations in the solutions are generated by the linear part My(t). TSERKN methods extend the two-step hybrid methods [IMA J. Numer. Anal. 23 (2003) 197–220] by reforming both the internal stages and the updates so that they are adapted to the oscillatory properties of the exact solutions. However, the global error analysis for the TSERKN methods has not been investigated. In this paper we construct a new three-stage explicit TSERKN method of order four and present the global error bound for the new method, which is proved to be independent of ∥M∥ under suitable assumptions. This property of our new method is very important for solving highly oscillatory systems (1), where ∥M∥ may be arbitrarily large. We also analyze the stability and phase properties for the new method. Numerical experiments are included and the numerical results show that the new method is very competitive and promising compared with the well-known high quality methods proposed in the scientific literature.     

Exponential sums over primes and the prime twin problem

The purpose of this paper is to obtain an effective estimate of the exponential sum $\sum_{n\le x}\Lambda(n)e\left(\left(\frac{a}{q}+\beta\right)n\right)$ (where e(??)=e ^{2?? i ??} , ??,?????, (a,q)=1 and ?? is the von Mangoldt function) in the range ${(\log x)}^{1/2+\varepsilon}\le q\le \frac{x}{{(\log\log\log x)}^{1+\varepsilon}}$ and $|\beta|<\frac{1}{q{(\log\log\log x)}^{1+\varepsilon}}$ . It improves Daboussi??s estimate [2, Theorem 1] in the range q??(log?x) ^D and x(log?x)^?D ??q, D>0 and is valid in a wider range for ??.     

On the mean oscillation of the Hessian of solutions to the Monge-Ampère equation

We give interior a priori estimates for the mean oscillation of second derivatives of solutions to the Monge-Ampère equation detD²u=f(x) with zero boundary values, where f(x) is a non-Dini continuous function. If the modulus of continuity of f(x) is φ(r) such that limr→0φ(r)log(1/r)=0, then D²u∈VMO.     

Estimates on the first eigenvalue of an integral equation

We show how inequalities of the type $\text{∥F∥}{}_{\mathrm{p}}\text{? C(p, q) a}{}^{\mathrm{<mtext>1\; +\; (</mtext><mtext>1</mtext><mtext>p</mtext><mtext>)?\; (</mtext><mtext>1</mtext><mtext>q</mtext><mtext>)</mtext>}}\text{∥ F ′ ∥}{}_{\mathrm{q\prime }}$ when F(0) = 0 can be used to find lower bounds of the first eigenvalue of the integral equation F(z) = λ ∝₀^ak(s, z)F(s) ds.     

On the distribution of the roots of certain congruences and a problem for additive arithmetic functions

The asymptotic distribution of the roots of the congruence ax ≡ b (mod D), 1 ≤ x ≤ D, as D varies, is investigated. Quantitative estimates are obtained by means of exponential sums combined with sieve methods. As an application of the results it is shown that if an additive arithmetic function satisfies f(an + b) ? f(cn + d) = O(1) for all positive integers n, ad ≠ bc, then f(n) = O((log n)³) must hold. This result is apparently the first bound of any kind in such a situation.     

The distribution of Brownian motion in Rn at a natural stopping time

For a measure μ on Rⁿ let ((B_t, P^μ) be Brownian motion in Rⁿ with initial distribution μ. Let D be an open subset of Rⁿ with exit time ζ ≡ inf {t > 0: B_t ? D}. In the case where D is a Green region with Green function G and μ is a measure in D such that G_μ is not identically infinite on any component of D, we have given necessary and sufficient conditions for a measure ν in D to be of the form ν(dx) = P^μ(B_T ? dx, T <ζ), where T is some natural stopping time for (B_t), and we have applied this characterization to show that a measure ν in D satisfies G_ν ? G_μ iff ν is of the form ν(dx) = P^α(B_T ? dx, T <ζ) + β(dx), where T is some natural stopping time for (B_t) and α and β are measures in D such that α + β = μ and β lives on a polar set. We have proved analogous results in the case where D = R² and μ is a finite measure on R² such that ∫ log⁺ ∥x∥ du(x) < ∞, and applied this to give a characterization of the stopping times T for Brownian motion in R² such that (log⁺ ∥B_T∧t∥)_0<t<∞ is P^μ-uniformly integrable.