relative to a hereditary torsion theory $tau$ we introduce a dimension for a module $m$, called {em $tau$-rank of} $m$, which coincides with the reduced rank of $m$ whenever $tau$ is the goldie torsion theory. it is shown that the $tau$-rank of $m$ is measured by the length of certain decompositions of the $tau$-injective hull of $m$. moreover, some relations between the $tau$-rank of $m$ and c...

In this paper we address the questions of the convergence rate for approximate solutions to conservation laws with piecewise smooth solutions in a weighted W 1,1 space. Convergence rate for the derivative of the approximate solutions is established under the assumption that a weak pointwise-error estimate is given. In other words, we are able to convert weak pointwise-error estimates to optimal...

In this paper, we introduce a notion of weak pointwise Hölder regularity, starting from the definition of the pointwise anti-Hölder irregularity. Using this concept, a weak spectrum of singularities can be defined as for the usual pointwise Hölder regularity. We build a class of wavelet series satisfying the multifractal formalism and thus show the optimality of the upper bound. We also show th...

We now discuss convergence of the Fourier series on compact intervals I. ‘Convergence’ depends on the notion of convergence we use, such as (i) L: uj → u in L if ‖uj − u‖L2 → 0 as j →∞. (ii) uniform, or C: uj → u uniformly if ‖uj−u‖C0 = supx∈I |uj(x)−u(x)| → 0. (iii) uniform with all derivatives, or C∞: uj → u in C∞ if for all non-negative integers k, supx∈I |∂uj(x)− ∂u(x)| → 0. (iv) pointwise:...

A popular simultaneous confidence band for survival functions is the equal precision band of Nair [1984. Confidence bands for survival functions with censored data: a comparative study. Technometrics 26, 265–275] This band is found by adjusting the level of Wald-type pointwise confidence intervals centered on the Kaplan–Meier estimator. The present paper develops a complementary method of adjus...

We now discuss convergence of the Fourier series on compact intervals I. ‘Convergence’ depends on the notion of convergence we use, such as (i) L: uj → u in L if ‖uj − u‖L2 → 0 as j →∞. (ii) uniform, or C: uj → u uniformly if ‖uj−u‖C0 = supx∈I |uj(x)−u(x)| → 0. (iii) uniform with all derivatives, or C∞: uj → u in C∞ if for all non-negative integers k, supx∈I |∂uj(x)− ∂u(x)| → 0. (iv) pointwise:...

Four recent papers 1–4 present simple and elegant proofs of fixed point results for pointwise contractions, asymptotic pointwise contractions, and asymptotic nonexpansive mappings. Kirk and Xu 1 study these mappings in the context of weakly compact convex subsets of Banach spaces, respectively, in uniformly convex Banach spaces. Hussain and Khamsi 2 consider these problems in the framework of m...

There are two kinds of estimates of the degree of approximation of continuous functions on [−1, 1] by algebraic polynomials, Nikolskii-type pointwise estimates and Jackson-type uniform estimates, involving either ordinary moduli of smoothness, or the DitzianTotik (DT) ones, or the recent estimates involving the weighted DTmoduli of smoothness. The purpose of this paper is to complete the table ...

Let D be a dendrite and f : D → D a continuous map. Denote by E(D) and B(D) the sets of endpoints and branch points of D respectively. We show that if E(D) is countable (resp. B(D) is discrete) then f is pointwise-recurrent if and only if f is pointwise periodic homeomorphism (resp. every point in D\E(D) is periodic).