The notions of permutable and weak-permutable convergence of a series ∑∞ n=1 an of real numbers are introduced. Classically, these two notions are equivalent, and, by Riemann’s two main theorems on the convergence of series, a convergent series is permutably convergent if and only if it is absolutely convergent. Working within Bishop-style constructive mathematics, we prove that Ishihara’s prin...

For any oblique incidence and arbitrarily high order, lattice sums for one-dimensional gratings can be expressed in terms of exponentially convergent series. The scattering Green's function can be efficiently evaluated also in the grating plane. Numerical implementation of the method is 200 times faster than for the previous best result.

A hybrid nonlinear differentiator being fit for rapid convergence is presented, which is based on singular perturbation technique and consists of linear and nonlinear parts. This differentiator is simple and can converge rapidly at all times, and no chattering phenomena happen. The theoretical results are confirmed by computer simulations and an experiment.

Frank Rosenblatt invented the Perceptron algorithm in 1957 as part of an early attempt to build “brain models” – artificial neural networks. In this paper, we apply tools from symbolic logic – dependent type theory as implemented in the interactive theorem prover Coq – to prove that one-layer perceptrons for binary classification converge when trained on linearly separable datasets (the Percept...

We give conclusive answers to some questions about definability in analytic languages that arose shortly after the work by Denef and van den Dries, [DD], on p-adic subanalytic sets, and we continue the study of nonarchimedean fields with analytic structure of [LR3], [CLR1] and [CL1]. We show that the language LK consisting of the language of valued fields together with all strictly convergent p...

In convergent-beam electron diffraction (CBED) a highly convergent electron beam is focussed on to a small (~< 50 nm) area of the sample. Instead of the diffraction spots that are obtained in the back focal plane of the objective lens with parallel illumination in conventional selected-area electron diffraction, CBED produces discs of intensity. The point group can be determined uniquely from t...

Use of divergent (or inverse) perspective in pictures is often regarded as arbitrary or even as erroneous in spite of the fact that entire schools of art exist in which this kind of perspective is regularly used. An experiment is reported which shows that a significant trend towards divergent perspective is experienced by subjects viewing laterally displaced three-dimensional arrays. Centrally ...

If a function f has finite Henstock integral on the boundary of the unit disk of R 2 then its Poisson integral exists for |z| < 1 and is o((1 − |z|) −1) as |z| → 1 −. It is shown that this is the best possible uniform pointwise estimate. For an L 1 measure the best estimate is O((1 − |z|) −1). In this paper we consider estimates of Poisson integrals on the unit circle with respect to Alexiewicz...