Two classes of nonlinear operators generalizing the notion of a local operator between ideal function spaces are introduced. The first class, called atomic, contains in particular all the linear shifts, while the second one, called coatomic, contains all the adjoints to former, and, in particular, the conditional expectations. Both classes include local (in particular, Nemytski\v{\i}) operators and are closed with respect to compositions of operators. Basic properties of operators of introduced classes in the Lebesgue spaces of vector-valued functions are studied. It is shown that both classes inherit from Nemytski\v{\i} operators the properties of non-compactness in measure and weak degeneracy, while having different relationships of acting, continuity and boundedness, as well as different convergence properties. Representation results for the operators of both classes are provided. The definitions of the introduced classes as well as the proofs of their properties are based on a purely measure theoretic notion of memory of an operator, also introduced in this paper.

AMS 2000 Mathematics subject classification: Primary 47B38; 47A67; 34K05

This paper provides mathematical analysis of optical tomography in a situation when the examined object, for example the human brain, is strongly scattering with non-scattering inclusions. Light propagation in biological tissue is often modelled by the diffusion approximation of the radiative transfer equation. To be justified, the diffusion approximation demands that the medium is strongly scattering. Naturally, this is not true for non-scattering inclusions, for which some other model is needed. This is found through geometrical optics. Combination of the two models leads to an elliptic partial differential equation with boundary conditions on the outer boundary as well as on the boundaries of the non-scattering regions. The well-posedness of this forward problem is the main concern of this work.

AMS 2000 Mathematics subject classification: Primary 35Q60; 35R30

The uniform stability of the zero solution and the asymptotic behaviour of all solutions of the neutral delay differential equation

$$ [x(t)-P(t)x(t-\tau)]'+Q(t)x(t-\sigma)=0,\quad t\ge t_0, $$

are investigated, where $\tau,\sigma\in(0,\infty)$, $P\in C([t_0,\infty),\mathbb{R})$, and $Q\in C([t_0,\infty), [0,\infty))$. The obtained sufficient conditions improve the existing results in the literature.

AMS 2000 Mathematics subject classification: Primary 34K20; 34K15; 34K40

We prove that spaces with an uncountable $\omega$-independent family fail the Kunen–Shelah property. Actually, if $\{x_i\}_{i\in I}$ is an uncountable $\omega$-independent family, there exists an uncountable subset $J\subset I$ such that $x_j\notin\overline{\co}(\{x_i\}_{i\in J\setminus\{j\}})$ for every $j\in J$. This improves a previous result due to Sersouri, namely that every uncountable $\omega$-independent family contains a convex right-separated subfamily.

AMS 2000 Mathematics subject classification: Primary 46B20; 46B26

In this paper, we establish the existence of infinitely many solutions to a Neumann problem involving the p-Laplacian and with discontinuous nonlinearities. The technical approach is mainly based on a very recent result on critical points for possibly non-smooth functionals in a Banach space due to Marano and Motreanu, namely Theorem 1.1 in a paper that is to appear in the journal J. Diff. Eqns (see Theorem 2.3 in the body of this paper). Some applications are presented.

AMS 2000 Mathematics subject classification: Primary 35A15; 35J65; 35R05

As a consequence of the main result of the paper we obtain that every 2-local isometry of the $C^*$-algebra $B(H)$ of all bounded linear operators on a separable infinite-dimensional Hilbert space $H$ is an isometry. We have a similar statement concerning the isometries of any extension of the algebra of all compact operators by a separable commutative $C^*$-algebra. Therefore, on those $C^*$-algebras the isometries are completely determined by their local actions on the two-point subsets of the underlying algebras.

AMS 2000 Mathematics subject classification: Primary 47B49

Complete positivity of ‘atomically extensible’ bounded linear operators between $C^*$-algebras is characterized in terms of positivity of a bilinear form on certain finite-rank operators. In the case of an elementary operator on a $C^*$-algebra, the approach leads us to characterize k-positivity of the operator in terms of positivity of a quadratic form on a subset of the dual space of the algebra and in terms of a certain inequality involving factorial states of finite type I.

As an application we characterize those $C^*$-algebras where every k-positive elementary operator on the algebra is completely positive. They are either k-subhomogeneous or k-subhomogeneous by antiliminal. We also give a dual approach to the metric operator space introduced by Arveson.

AMS 2000 Mathematics subject classification: Primary 46L05. Secondary 47B47; 47B65

Under an extra hypothesis satisfied in every known case, we show that the Euler class of an orientable odd-dimensional Poincaré duality group over any ring has order at most two. We construct groups that are of type FL over the complex numbers but are not FL over the rationals. We construct group algebras over fields for which $K_0$ contains torsion, and construct non-free stably free modules for the group algebras of certain virtually free groups.

AMS 2000 Mathematics subject classification: Primary 19A31. Secondary 16S34; 20J05; 55N15; 57P10

A cotorsion theory is defined as a pair of classes Ext-orthogonal to each other. We give a hereditary condition (HC) which is satisfied by the (flat, cotorsion) cotorsion theory and give properties satisfied by arbitrary cotorsion theories with an HC. Given a cotorsion theory with an HC, we consider the class of all modules having a special precover with respect to the first class in the cotorsion theory and show that this class is closed under extensions. We then raise the question of whether this class is resolving or coresolving.

AMS 2000 Mathematics subject classification: Primary 18G15. Secondary 18E40

The purpose of this paper is to derive an integral representation of the Drazin inverse of an element of a Banach algebra in a more general situation than previously obtained by the second author, and to give an application to the Moore–Penrose inverse in a $C^*$-algebra.

AMS 2000 Mathematics subject classification:Primary 46H05; 46L05

The paper shows that specializations of finitely generated graded modules are also graded and that many important invariants of graded modules and ideals are preserved by specializations.

AMS 2000 Mathematics subject classification: Primary 13A02

In this note we show that two conjectures of George Weiss on admissible and weakly admissible observation operators fail to hold in general for the right shift semigroup in the case that the output space is infinite dimensional.

AMS 2000 Mathematics subject classification: Primary 47D06; 32A37; 93B28

The main result establishes that a weak solution of degenerate elliptic equations can be approximated by a sequence of solutions of non-degenerate elliptic equations. To this end we prove an approximation theorem for $A_p$-weights.

AMS 2000 Mathematics subject classification: Primary 35J50. Secondary 35D05