On A‐stability for some two‐step W‐methods

In this paper we prove the A‐stability for some sequential and parallel two‐step W‐methods (TSW‐methods and PTSW‐methods) which have been constructed recently for the numerical solution of large stiff ODE systems. To show that the spectral radius of the amplification matrix of a fixed method is less than one we apply Schur's criterion recursively to the characteristic polynomial of the amplification matrix. The computations can be done automatically using computer algebra tools, e.g. Maple 7.     

Computing the norm ∥A∥∞,1 is NP-hard

It is proved that computing the subordinate matrix norm ∥A∥∞1 is NP-hard, Even more, existence of a polynomial-time algorithm for computing this norm with relative accuracy less than 1/(4n²), where n is matrix size, implies P = NP.     

Locally boundedly A‐p‐convex algebras

We consider locally p‐convex algebras the bornologies of which coincide with the ones of p‐normed or multiplicatively p‐convex topologies. Two new classes of these algebras are introduced here. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the invertibility of the operator A–XB

For a given pair of (A, B) and an arbitrary operator X, expressions for the inverse, the Moore–Penrose inverse and the generalized Drazin inverse of the operator A–XB are derived under some conditions. Copyright © 2009 John Wiley & Sons, Ltd.     

Estimation of Dirichlet kernel difference in the norm of L

This paper is related to the problem of estimation of the norm of trigonometrical polynomials through their coefficients in L. It is proved that the norm of the difference of Dirichlet’s kernels in L has the precise order ln(n − m) and the lower estimate is also valid with the coefficient 4/π ². A theorem and two lemmas are presented showing that the coefficient c at ln(n − m) in an asymptotic estimate uniform with respect to m and n may be greater than 4/π ² and its value in examples depends on arithmetic properties of n and m.     

Error estimate in operator norm for Trotter-Kato product formula

In the present paper we study the error estimate in the operator norm for the Trotter-Kato product formula, having an intention to apply the obtained result to Schrödinger semigroups with singular potentials.     

On the independence number of the Erdős‐Rényi and projective norm graphs and a related hypergraph

The Erd?s‐Rényi and Projective Norm graphs are algebraically defined graphs that have proved useful in supplying constructions in extremal graph theory and Ramsey theory. Their eigenvalues have been computed and this yields an upper bound on their independence number. Here we show that in many cases, this upper bound is sharp in the order of magnitude. Our result for the Erd?s‐Rényi graph has the following reformulation: the maximum size of a family of mutually non‐orthogonal lines in a vector space of dimension three over the finite field of order q is of order q^3/2. We also prove that every subset of vertices of size greater than q^2/2 + q^3/2 + O(q) in the Erd?s‐Rényi graph contains a triangle. This shows that an old construction of Parsons is asymptotically sharp. Several related results and open problems are provided. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 113–127, 2007     

Weighted mixed‐norm inequalities through extrapolation

We prove that operators satistying weighted inequalities with radial weights are bounded in mixed‐norm spaces of radial‐angular type, even with a weight in the radial part. This is achieved by using the usual extrapolation methods, adapted to the radial setting. All the versions of the extrapolation theorem can be adapted to this setting, and in particular we get results in variable Lebesgue spaces and also for multilinear operators. Furthermore, quantitative estimates are obtained with this approach, but their sharpness remains an open question.     

On a mixed Kernel Hilbert‐type integral inequality and its operator expressions with norm

By using some real analysis techniques, we study the structural characteristics of a multi‐parameter Hilbert‐type integral inequality with the hybrid kernel and obtain some equivalent conditions for this inequality. We also consider the operator expression of the equivalent inequalities. The conclusions not only integrate some results of references but also find some new Hilbert‐type integral inequalities with simple form by choosing suitable parameter values.     

Error norm estimation and stopping criteria in preconditioned conjugate gradient iterations

Some techniques suitable for the control of the solution error in the preconditioned conjugate gradient method are considered and compared. The estimation can be performed both in the course of the iterations and after their termination.The importance of such techniques follows from the non‐existence of some reasonable a priori error estimate for very ill‐conditioned linear systems when sufficient information about the right‐hand side vector is lacking. Hence, some a posteriori estimates are required, which make it possible to verify the quality of the solution obtained for a prescribed right‐hand side. The performance of the considered error control procedures is demonstrated using real‐world large‐scale linear systems arising in computational mechanics. Copyright © 2001 John Wiley & Sons, Ltd.     

I‐convexity and Q‐convexity in Orlicz–Bochner function spaces endowed with the Orlicz norm

Some characterizations of I‐convexity and Q‐convexity of Banach space are obtained. Moreover, the criteria is shown for Orlicz–Bochner function spaces $L_M(\mu ,X)$ endowed with the Orlicz norm being I‐convex as well as being Q‐convex.     

On the norm of random matrices

Translated from Matematicheskie Zametki, Vol. 57, No. 5, pp. 688–698, May, 1995.     

条件概率密度函数核估计的误差分析及其最优带宽的选择

本文在α-混合严平稳过程的假设下，研究了条件概率密度核估计的偏和均方误差．在此基础上给出了核估计的渐近最优带宽，并以S＆P500指数为例展示了本文的结果．     

On a minimal complex norm that extends the real euclidean norm

A minimal extension of the real euclidean norm to ⁿ is introduced and some complex analytic properties of its unit ball are studied.Research supported partially by Fulbright Grant (1986–1987) at Universität Osnabrück-Abteilung Vechta.     

Negative norm least‐squares methods for the velocity‐vorticity‐pressure Navier–Stokes equations

We develop and analyze a least‐squares finite element method for the steady state, incompressible Navier–Stokes equations, written as a first‐order system involving vorticity as new dependent variable. In contrast to standard L² least‐squares methods for this system, our approach utilizes discrete negative norms in the least‐squares functional. This allows us to devise efficient preconditioners for the discrete equations, and to establish optimal error estimates under relaxed regularity assumptions. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 237–256, 1999     

On the spectral abscissa and the logarithmic norm

Mathematical Notes - In this paper, both well-known and new properties of the spectral abscissa and the logarithmic norm are described. In addition to well-known formulas for the norm of a matrix...     

On the norm of the Lp-Dunkl transform

In this work, we establish a Babenko–Beckner-type inequality for the Dunkl transform of a radial function and Dunkl transform associated to a finite reflection group generated by the sign changes. As applications, we establish a new version of Young’s-type inequality and uncertainty relation for Renyi entropy.