Embedding Formulae Research Articles

Filling the gap between Continuous and Discrete Time Dynamics of Autoregressive Processes

Discretization of continuous time autoregressive (AR) processes driven by a Brownian motion and embedding of discrete time AR sequences driven by a Gaussian white noise are classical issues. The article aims at establishing and using such discretization and embedding formulae between extended AR continuous time processes and discrete time sequences. The continuous‐time processes are driven by either Brownian or jump processes, and may have random coefficients depending on time; Lévy‐driven processes are also considered. The innovation of the discrete time processes may be of many types – including Gaussian. In one way, observing the continuous time AR process at discrete times leads the AR dynamics of the discretized process to be characterized. The other way round, AR sequences can be embedded, in the almost sure sense, into continuous time AR processes with the same dynamics. Illustration is provided through many examples and simulation.

Embedding FLRW geometries in pseudo-Euclidean and anti–de Sitter spaces

Contrary to the general consensus in the literature that Friedmann--Lema\^{i}tre--Robertson--Walker (FLRW) geometries are of embedding class one (i.e.,\ embeddable in one higher dimensional pseudo-Euclidean spaces), we show that the most general $k=0$ and $k=-1$ FLRW geometries are of embedding class two, and their corresponding pseudo-Euclidean spaces have strictly one and two negative eigenvalues, respectively. These are particular results that follow from the new perspective on FLRW embedding that we develop in this paper, namely that these embeddings are equivalent to unit-speed parametrized curves in two or three dimensions. A careful analysis of appropriate tensor fields then gives identical results and further explains why the class-two geometries remained hidden. However, the signatures of the embedding spaces, as well as the explicit embedding formulae, follow only from the curve picture. This also streamlines the comparatively difficult $k=0$ class and provides new explicit embedding formulae for it and reproduces known embedding formulae for the $k=1,-1$ classes. Embedding into anti-de Sitter space in one higher dimension can likewise be done by constructing associated curves. In particular, we find that all $k=1$ and mildly restricted subclasses of $k=0, -1$ geometries are embeddable in anti-de Sitter space in one higher dimension.

Embedding formulae for scattering in a waveguide containing polygonal obstacles

For certain wave scattering problems embedding formulae can be derived, which express the solution, or far-field behaviour of the solution, for arbitrary plane wave incident angle in terms of the corresponding quantities for a finite number of other related problems. Their scope has so far been limited to scattering in R^2, and to a lesser extent R^3; in this paper we derive embedding formulae for wave scattering in a class of two-dimensional waveguide. The waveguide is straight and of uniform width outside a finite length region within which the boundaries are piecewise-linear and the waveguide can contain polygonal obstacles, a restriction being that all boundaries of the waveguide and obstacles must be inclined at a rational angle to the axis of the waveguide. Once solutions are determined for a finite set of incident propagating modes, the embedding formulae provide expressions for reflection and transmission coefficients for all remaining incident propagating modes. The precise number of solutions required is a function of the number and nature of the corners of the boundaries and obstacles. The formulae are illustrated for a particular waveguide geometry for which the problem can be formulated as an integral equation and approximate numerical solutions determined using the Galerkin method.

Embedding formulae for wave diffraction by a circular arc

For certain wave diffraction problems, embedding formulae can be derived, which represent the solution (or far-field behaviour of the solution) for all plane wave incident angles in terms of solutions of a (typically small) set of other auxiliary problems. Thus a complete characterisation of the scattering properties of an obstacle can be determined by only determining the solutions of the auxiliary problems, and then implementing the embedding formula. The class of scatterers for which embedding formulae can be derived has previously been limited to obstacles with piecewise linear boundaries; here this class is extended to include a simple curved obstacle, consisting of a thin circular arc. Approximate numerical calculations demonstrate the accuracy of the new embedding formulae.

Embedding formulae for Laplace–Beltrami problems on the sphere with a cut

We consider the problem of diffraction of a plane wave by a quarter-plane (a flat cone) with Dirichlet boundary conditions. The most efficient approach to this problem is the technique of Smyshlyaev’s formulae or a modified Smyshlyaev’s formula, both of which are representations of the diffraction coefficient as contour integrals over a complex parameter. These representations have been proven independently. Here we are demonstrating a link between these classes of formulae. The link is established by developing an embedding procedure (in the very special sense of diffraction theory) on the unit sphere.

On the diffraction of acoustic waves by a quarter-plane

This paper follows the work of A.V. Shanin on diffraction by an ideal quarter-plane. Shanin’s theory, based on embedding formulae, the acoustic uniqueness theorem and spherical edge Green’s functions, leads to three modified Smyshlyaev formulae, which partially solve the far-field problem of scattering of an incident plane wave by a quarter-plane in the Dirichlet case. In this paper, we present similar formulae in the Neumann case, and describe a numerical method allowing a fast computation of the diffraction coefficient using Shanin’s third modified Smyshlyaev formula. The method requires knowledge of the eigenvalues of the Laplace–Beltrami operator on the unit sphere with a cut, and we also describe a way of computing these eigenvalues. Numerical results are given for different directions of incident plane wave in the Dirichlet and the Neumann cases, emphasising the superiority of the third modified Smyshlyaev formula over the other two.

Embedding formulae for scattering by three-dimensional structures

The far field diffraction behaviour for canonical scattering problems involving corners or sectors, in three-dimensions, are considered. The far-field results are obtained using ideas based upon embedding formulae and therefore complement and extend existing results. Specific geometries such as the flat cone and a corner formed by a solid octant are considered in detail. The formulae derived for the diffraction behaviour are also computed in special cases and compared with known results.

Pseudo-differential operators for embedding formulae

A new method is proposed for deriving embedding formulae in 2D diffraction problems. In contrast to the approach developed in Craster and Shanin (2005) [7], which is based on a differential operator, here a pseudo-differential, i.e., a non-local operator is applied to the wave field. Using this non-local operator a new embedding formula is derived for scattering by a single wedge. The formula has uniform structure for all opening angles, including angles irrational with respect to π ; the earlier theory, Craster and Shanin (2005) [7], was valid only for rational angles.

Embedding formulae for diffraction by non-parallel slits

Embedding formulae for diffraction theory encode the diffraction coefficients for some given wave incidence on a scatterer in terms of the directivity from a single or reduced number of scattering problems. If one deduces the relation between these directivities, then the resulting formulae enable rapid computations and allow one to concentrate computational resources accordingly. Unfortunately, the range of applicability of embedding formulae is currently rather restricted. In this article, we demonstrate how embedding is applied to plane-wave scattering by non-parallel strips or slits. Primarily, we concentrate upon the problem of a line crack, or strip, inclined to a flat infinite surface and we derive and implement the embedding formula. Various other generalizations are possible given these formulae and we outline them.

A new family of embedding formulae for diffraction by wedges and polygons

Embedding formulae represent the solution to certain linear plane-wave scattering problems in terms of solutions to other auxiliary scattering problems. Here the method of [R.V. Craster, A.V. Shanin, E.M. Doubravsky, Embedding formulae in diffraction theory, Proc. R. Soc. Lond. A 459 (2003) 2475–2496] and, in particular, [R.V. Craster, A.V. Shanin, Embedding formulae for diffraction by rational wedge and angular geometries, Proc. R. Soc. Lond. A 461 (2005) 2227–2242], whose auxiliary problems are forced by multipoles located at the corners of the scatterer, is adapted and extended to determine formulae relating solutions corresponding to different incident plane wave angles. As in [R.V. Craster, A.V. Shanin, Embedding formulae for diffraction by rational wedge and angular geometries, Proc. R. Soc. Lond. A 461 (2005) 2227–2242], wedge geometries of rational angle and polygons with rational internal angle are considered and appropriate embedding formulae derived. The method is also shown to extend to the more general class of polygon whose internal angles are all multiples of the same rational angle.

Embedding formulae for diffraction by rational wedge and angular geometries

Embedding is the process of taking the far-field directivity from a diffraction problem (or problems) involving line sources or multipoles placed at a sharp edge, and then constructing the far field, for the same geometry, for more general incidence using only this canonical problem(s). Thus far, embedding has been limited to planar, parallel scattering surfaces, for instance, collections of parallel cracks or slits; it appeared that there was a fundamental limitation to embedding, disallowing its use for angular structures. In this article, we overcome this limitation and demonstrate the use of embedding upon wedge diffraction problems and upon a simple polygonal shape; a limitation is that the final formulae are for rational wedge angles.

Embedding formulae in diffraction theory

Embedding formulae are remarkable as they allow one to decompose scattering problems apparently dependent upon several angular variables (angles of incidence and observation) into those dependent upon fewer angular variables. In terms of facilitating rapid computations across considerable parameter regimes, this is a considerable advantage. Our aim is to derive embedding formulae for scattering and diffraction problems in acoustics, electromagnetism and elasticity. Here we construct a general approach to formulating and using embedding formulae. We do this using complementary approaches: overly singular states and a physical interpretation in terms of sources. The crucial point we identify is the form of the auxiliary state used in the reciprocal theorem; this is unphysically singular at the edge and is reminiscent of weight–function methods used in fracture mechanics. Illustrative implementations of our approach are given using Wiener–Hopf techniques for semi–infinite model problems in both elasticity and acoustics. We also demonstrate our approach using a numerical example from acoustics and we make connections with high–frequency asymptotic methods.

Wave Scattering by a Perforated Duct

Recently, a number of linear wave scattering problems have been studied in the context of embedding formulae. Such formulae typically express the solution for any incident wave angle in terms of the solutions corresponding to a finite number of angles; they are of interest from a theoretical point of view and because they lead to significant computational savings. Here we investigate the diffraction and radiation of plane waves by a straight, infinitely long, impermeable duct, the walls of which are punctuated by an arbitrary arrangement of finitely many gaps, each of finite extent. This problem is reduced to a matrix integral equation which may be regarded as the canonical form of a wide class of similar problems, and embedding formulae are derived for this extended class. The formulae are illustrated by developing a variational method for the matrix integral equation to generate the minimal solution set required.

Wave Diffraction Through a Perforated Barrier of Non-Zero Thickness

We consider the diffraction of a prescribed train of monochromatic plane surface water waves by a vertical‐sided, perfectly reflecting thick breakwater standing on a horizontal bed in water of uniform undisturbed depth. The breakwater is punctuated by a series of N gaps, arbitrarily arranged. The corresponding linearized boundary‐value problem is reduced to a pair of uncoupled first kind integral equations characterized by a combination of sum and difference kernels. This structure motivates the main purpose of the paper, the development of embedding formulae for a general integral equation of the type encountered, extending previous such results which exist for the simpler case of pure difference kernels. By setting the derivation in the context of the diffraction problem, the embedding results can be readily interpreted and practically exploited. Thus, for example, one of them gives the solution of the diffraction problem for any incident wave angle explicitly in terms of the solutions for 4N distinct angles, reducing to 2N distinct angles if the breakwater array is symmetric. We take advantage of the considerable saving in computational effort that follows when the wave problem is solved for a range of incident angles.

Wave diffraction through a perforated breakwater

Linear theory is used to examine the diffraction of surface water waves through an arbitrary arrangement of gaps in an infinite, straight breakwater standing in water of uniform quiescent depth. The governing boundary-value problem is reduced to an integral equation posed on the union of the gaps, for which a number of embedding formulae are derived. These formulae allow the solution of the diffraction problem for any incident wave angle to be expressed explicitly in terms of its solutions for 2N distinct incident angles, for example, where N is the number of gaps. If the breakwater arrangement is symmetric, solutions for only N distinct incident angles are required to generate the solution for any other angle. Numerical solutions of the integral equation, obtained by a variational method, confirm that computational saving obtained by the embedding approach is considerable. The techniques described may be applied to a range of similar problems.

Conformal and poincaré tensor calculi in N=1 supergravity

We present the superconformal tensor calculus for N=1 supergravity in a complete form; irreducible multiplets, their multiplication and embedding formulae and invariant action formulae. It is further clarified in detail how the various versions of N=1 Poincaré supergravity (i.e. with different sets of auxiliary fields) are reproduced from the unique superconformal theory. The tensor calculi for all the known versions of Poincaré supergravity are derived explicitly.