A General Integral

We define an integral, the distributional integral of functions of one real variable, that is more general than the Lebesgue and the Denjoy-Perron-Henstock-Kurzweil integrals, and which allows the integration of functions with distributional values everywhere or nearly everywhere. Our integral has the property that if $f$ is locally distributionally integrable over the real line and $\psi\in\mathcal{D}(\mathbb{R}$ is a test function, then $f\psi$ is distributionally integrable, and the formula% [ =(\mathfrak{dist}) \int_{-\infty}^{\infty}f(x) \psi(x) \,\mathrm{d}% x\,,] defines a distribution $\mathsf{f}\in\mathcal{D}^{\prime}(\mathbb{R})$ that has distributional point values almost everywhere and actually $\mathsf{f}(x) =f(x)$ almost everywhere. The indefinite distributional integral $F(x) =(\mathfrak{dist}) \int_{a}^{x}f(t) \,\mathrm{d}t$ corresponds to a distribution with point values everywhere and whose distributional derivative has point values almost everywhere equal to $f(x).$ The distributional integral is more general than the standard integrals, but it still has many of the useful properties of those standard ones, including integration by parts formulas, substitution formulas, even for infinite intervals --in the Ces\`{a}ro sense--, mean value theorems, and convergence theorems. The distributional integral satisfies a version of Hake's theorem. Unlike general distributions, locally distributionally integrable functions can be restricted to closed sets and can be multiplied by power functions with real positive exponents.Comment: 59 pages, to appear in Dissertationes Mathematica

In-situ production of electrically conductive polyaniline fibres from polymer blends

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Polymers and polymer-based composite materials with electro-conductive properties, respectively, are materials with several potential applications. New materials are being offered in every area and novel products are constantly being introduced. Among these new materials, composites made of electro-conductive monofilaments and insulating polymers are nowadays being used as antistatic materials in the carpets and textiles industries. One promising approach for the manufacture of this kind of material is to generate the electrically conductive fibres in-situ, that is, during the actual forming process of the component. The main objective of this project was to establish the feasibility of producing electrically conductive polyaniline (PANI) fibres within a suitable polymer matrix by means of the development of a suitable processing strategy, which allows the fabrication of an anisotropically conducting composite. It is remarkable, however, that layered structures of the conducting filler were also formed within the matrix material. The latter morphology, particularly observed in compression moulded specimens of a specific polymer system, was also in good agreement with that inferred by means of a mathematical model. Experimentation was carried out with three different PANI conductive complexes (PANIPOLTM). They were initially characterised, which assisted in the identification of the most suitable material to be deformed into fibres. Preliminary processing was carried out with the selected PANIPOLTM complex, which was blended with polystyrene-polybutadiene-polystyrene (SBS), low density polyethylene (LDPE) and polypropylene (PP), respectively. The resultant blends were formed by ram extrusion, using a capillary die, to induce the deformation of the conducting phase into fibres. The morphological analysis performed on the extrudates suggested that the most suitable polymer matrix was SBS. Further experimentation was carried out with the polymer system selected. The relationships between the content of conductive complex in the composites and their electrical conductivity and microstructure were established. The blends were compression moulded and they displayed a morphology of layered domains of the conducting phase within the SBS matrix. The behaviour of the conductivity with respect to the PANIPOLTM complex in the compression moulded blends was found to be characteristic of a percolating system with a threshold as low as 5 weight percent of the conducting filler in the blends. The morphological analysis performed on the extruded blends suggested that the conducting phase was deformed into elongated domains, aligned parallel to the extrusion direction, which in some cases displayed a considerable degree of continuity and uniformity. The level of electrical conductivity in the extrudates was considerably lower than that of their corresponding non-extruded blends. This was attributed to a lack of continuity in the conducting elongated domains produced in-situ within the SBS matrix. Percolation theory and a generalisation of effective media theories were used to model the behaviour of the conductivity with respect to the content of PANIPOLTM in the compression moulded blends. Both approaches yielded similar values for the critical parameters, which were also in good agreement with the percolation threshold experimentally observed. The results of the modelling suggested that, at the percolation threshold, the morphology of the composite may consists of aggregates of flattened polyaniline particles forming very long layered structures within the SBS matrix, which is in agreement with the results of the morphological analysis

Null Spaces of Radon Transforms

We obtain new descriptions of the null spaces of several projectively equivalent transforms in integral geometry. The paper deals with the hyperplane Radon transform, the totally geodesic transforms on the sphere and the hyperbolic space, the spherical slice transform, and the Cormack-Quinto spherical mean transform for spheres through the origin. The consideration extends to the corresponding dual transforms and the relevant exterior/interior modifications. The method relies on new results for the Gegenbauer-Chebyshev integrals, which generalize Abel type fractional integrals on the positive half-line.Comment: 24 pages. arXiv admin note: text overlap with arXiv:1410.411

On distributional point values and boundary values of analytic functions

We give the following version of Fatou's theorem for distributions that are boundary values of analytic functions. We prove that if $f\in\mathcal{D}^{\prime}(a,b)$ is the distributional limit of the analytic function $F$ defined in a region of the form $(a,b) \times(0,R),$ if the one sided distributional limit exists, $f(x_{0}+0) =\gamma,$ and if $f$ is distributionally bounded at $x=x_{0}$, then the \L ojasiewicz point value exists, $f(x_{0})=\gamma$ distributionally, and in particular $F(z)\to \gamma$ as $z\to x_{0}$ in a non-tangential fashion.Comment: 7 page

General Stieltjes moment problems for rapidly decreasing smooth functions

We give (necessary and sufficient) conditions over a sequence $\left\{ f_{n}\right\} _{n=0}^{\infty}$ of functions under which every generalized Stieltjes moment problem $$\int_{0}^{\infty} f_{n}(x)\phi(x)\mathrm{d} x=a_{n}, \ \ \ n\in\mathbb{N},$$ has solutions $\phi\in\mathcal{S}(\mathbb{R})$ with $\operatorname*{supp} \phi\subseteq[0,\infty)$. Furthermore, we consider more general problems of this kind for measure or distribution sequences $\left\{ f_{n}\right\} _{n=0}^{\infty}$. We also study vector moment problems with values in Frechet spaces and multidimensional moment problems.Comment: 25 page

On Romanovski's lemma

Romanovski introduced a procedure, Romanovski's lemma, to construct the Denjoy integral without the use of transfinite induction. Here we give two versions of Romanovski's lemma which hold in general topological spaces. We provide several applications in various areas of mathematics

A generalization of the Banach-Steinhaus theorem for finite part limits

It is well known, as follows from the Banach-Steinhaus theorem, that if a sequence $\left\{ y_{n}\right\} _{n=1}^{\infty}$ of linear continuous functionals in a Fr\'{e}chet space converges pointwise to a linear functional $Y,$ $Y\left( x\right) =\lim_{n\rightarrow\infty}\left\langle y_{n} ,x\right\rangle$ for all $x,$ then $Y$ is actually continuous. In this article we prove that in a Fr\'{e}chet space\ the continuity of $Y$ still holds if $Y$ is the \emph{finite part} of the limit of $\left\langle y_{n},x\right\rangle$ as $n\rightarrow\infty.$ We also show that the continuity of finite part limits holds for other classes of topological vector spaces, such as \textsl{LF}-spaces, \textsl{DFS}-spaces, and \textsl{DFS} $^{\ast}$-spaces,\ and give examples where it does not hold

The effect of the increasing demand for elite schools on stratification

I use detailed applications data to document a case in which, contrary to prevailing concerns, increasing school stratification by ability co-existed with stable stratification by family income: Mexico City public high schools. To understand this puzzle, I develop a model that shows that the effect of an overall increase in the demand for elite schools on school stratification by family income is a horse race between the correlations of family income and ability, and family income and demand. My empirical analysis reveals an initial (and decreasing) demand gap by family income that explains the observed stability in stratification