The unified transform method for linear initial-boundary value problems: a spectral interpretation

It is known that the unified transform method may be used to solve any well-posed initial-boundary value problem for a linear constant-coefficient evolution equation on the finite interval or the half-line. In contrast, classical methods such as Fourier series and transform techniques may only be used to solve certain problems. The solution representation obtained by such a classical method is known to be an expansion in the eigenfunctions or generalised eigenfunctions of the self-adjoint ordinary differential operator associated with the spatial part of the initial-boundary value problem. In this work, we emphasise that the unified transform method may be viewed as the natural extension of Fourier transform techniques for non-self-adjoint operators. Moreover, we investigate the spectral meaning of the transform pair used in the new method; we discuss the recent definition of a new class of spectral functionals and show how it permits the diagonalisation of certain non-self-adjoint spatial differential operators.Comment: 3 figure

CRA goes global: a good idea in the United States could use a makeover and a bigger audience

Community development

New Alignment Methods for Discriminative Book Summarization

We consider the unsupervised alignment of the full text of a book with a human-written summary. This presents challenges not seen in other text alignment problems, including a disparity in length and, consequent to this, a violation of the expectation that individual words and phrases should align, since large passages and chapters can be distilled into a single summary phrase. We present two new methods, based on hidden Markov models, specifically targeted to this problem, and demonstrate gains on an extractive book summarization task. While there is still much room for improvement, unsupervised alignment holds intrinsic value in offering insight into what features of a book are deemed worthy of summarization.Comment: This paper reflects work in progres

Schleiermacher and Otto on religion : a reappraisal

An interpretation of the work of Schleiermacher and Otto recently offered by Andrew Dole, according to which these two thinkers differed over the extent to which religion can be explained naturalistically, and over the sense in which the supernatural can be admitted, is examined and refuted. It is argued that there is no difference between the two thinkers on this issue. It is shown that Schleiermacher's claim that a supernatural event is at the same time a natural event does not invite, but rather forecloses the possibility of, a naturalistic explanation of the event. It is further demonstrated that Otto, like Schleiermacher, denied the existence of supernatural events interpreted as events that infringe the laws of nature

Nonlocal and multipoint boundary value problems for linear evolution equations

We derive the solution representation for a large class of nonlocal boundary value problems for linear evolution PDEs with constant coefficients in one space variable. The prototypical such PDE is the heat equation, for which problems of this form model physical phenomena in chemistry and for which we formulate and prove a full result. We also consider the third order case, which is much less studied and has been shown by the authors to have very different structural properties in general. The nonlocal conditions we consider can be reformulated as \emph{multipoint conditions}, and then an explicit representation for the solution of the problem is obtained by an application of the Fokas transform method. The analysis is carried out under the assumption that the problem being solved is well posed, i.e.\ that it admits a unique solution. For the second order case, we also give criteria that guarantee well-posedness.Comment: 28 pages, 4 figure

Evolution PDEs and augmented eigenfunctions. I finite interval

The so-called unified method expresses the solution of an initial-boundary value problem for an evolution PDE in the finite interval in terms of an integral in the complex Fourier (spectral) plane. Simple initial-boundary value problems, which will be referred to as problems of type~I, can be solved via a classical transform pair. For example, the Dirichlet problem of the heat equation can be solved in terms of the transform pair associated with the Fourier sine series. Such transform pairs can be constructed via the spectral analysis of the associated spatial operator. For more complicated initial-boundary value problems, which will be referred to as problems of type~II, there does \emph{not} exist a classical transform pair and the solution \emph{cannot} be expressed in terms of an infinite series. Here we pose and answer two related questions: first, does there exist a (non-classical) transform pair capable of solving a type~II problem, and second, can this transform pair be constructed via spectral analysis? The answer to both of these questions is positive and this motivates the introduction of a novel class of spectral entities. We call these spectral entities augmented eigenfunctions, to distinguish them from the generalised eigenfunctions introduced in the sixties by Gel'fand and his co-authors

Well-posed two-point initial-boundary value problems with arbitrary boundary conditions

We study initial-boundary value problems for linear evolution equations of arbitrary spatial order, subject to arbitrary linear boundary conditions and posed on a rectangular 1-space, 1-time domain. We give a new characterisation of the boundary conditions that specify well-posed problems using Fokas' transform method. We also give a sufficient condition guaranteeing that the solution can be represented using a series. The relevant condition, the analyticity at infinity of certain meromorphic functions within particular sectors, is significantly more concrete and easier to test than the previous criterion, based on the existence of admissible functions.Comment: 21 page

Flow stabilization with active hydrodynamic cloaks

We demonstrate that fluid flow cloaking solutions based on active hydrodynamic metamaterials exist for two-dimensional flows past a cylinder in a wide range of Reynolds numbers, up to approximately 200. Within the framework of the classical Brinkman equation for homogenized porous flow, we demonstrate using two different methods that such cloaked flows can be dynamically stable for $Re$ in the range 5-119. The first, highly efficient, method is based on a linearization of the Brinkman-Navier-Stokes equation and finding the eigenfrequencies of the least stable eigen-perturbations; the second method is a direct, numerical integration in the time domain. We show that, by suppressing the Karman vortex street in the weekly turbulent wake, porous flow cloaks can raise the critical Reynolds number up to about 120, or five times greater than for a bare, uncloaked cylinder.Comment: 5 pages, 3 figure