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Near-optimal perfectly matched layers for indefinite Helmholtz problems
No full textA new construction of an absorbing boundary condition for indefinite Helmholtz problems on unbounded domains is presented. This construction is based on a near-best uniform rational interpolant of the inverse square root function on the union of a negative and positive real interval, designed with the help of a classical result by Zolotarev. Using Krein's interpretation of a Stieltjes continued fraction, this interpolant can be converted into a three-term finite difference discretization of a perfectly matched layer (PML) which converges exponentially fast in the number of grid points. The convergence rate is asymptotically optimal for both propagative and evanescent wave modes. Several numerical experiments and illustrations are included
Chamber graphs of some geometries related to the Petersen graph
No full textIn this thesis we study the chamber graphs of the geometries Gamma(A(2n+1)), Gamma(3A7), Gamma(L(2,11)) and Gamma(L(2,25)) which are related to the Petersen graph (see reference [13]).
We find and prove the diameter of all these chamber graphs and ask what chambers look like if they are as far apart as possible. We find the full automorphism group of all these chamber graphs
The Dirac operator and the limit-of-discrete-series for the universal cover of SL_2(R)
No full textThe principal series of the universal cover of admits a limit-of-discrete-series. Starting with this representation, we construct certain spinor fields on which the Dirac operator is a multiplication operator. This creates an unbounded Kasparov triple which generates of the reduced C*-algebra
The spectrum of the Dirac operator for the universal cover of SL_2(R)
No full textUsing representation theory, we compute the spectrum of the Dirac operator on the universal covering group of SL_2(R), exhibiting it as the generator
of KK^1(C, A), where A is the reduced C*-algebra of the group. This yields a new and direct computation of the K-theory of A.
A fundamental role is played by the limit-of-discrete-series representation, which is the frontier between the discrete and the principal series of the group
Antiplane elastic wave cloaking using metamaterials, homogenization and hyperelasticity
Get PDFWe consider the problem of how to cloak objects from antiplane elastic waves using two alternative techniques. The first is the use of a layered metamaterial in the spirit of the work of Torrent and Sanchez-Dehesa (2008) who considered acoustic cloaks, motivated by homogenization theories, whilst the second is the use of a hyperelastic cloak in the spirit of the work of Parnell et al. (2012). We extend the hyperelastic cloaking theory to the case of
a Mooney�Rivlin material since this is often considered to be a more realistic constitutive model of rubber-like media than the neo-Hookean case studied by Parnell et al. (2012),
certainly at the deformations required to produce a significant cloaking effect. Although not perfect, the Mooney�Rivlin material appears to be a reasonable hyperelastic cloak. This is clearly encouraging for applications. We quantify the effectiveness of the various cloaks considered by plotting the scattering cross section as a function of frequency, noting that this would be zero for a perfect cloak
Smoothing non-smooth systems with low-pass filters
Get PDFLow pass filters, which are used to remove high frequency noise from time series data, smooth the signals they are applied to. In this paper we examine the action of low pass filters on discontinuous or non-differentiable signals from non-smooth dynamical systems, where this smoothing action can be thought of as a smoothing of the underlying system
An Improved Schur--Pade Algorithm for Fractional Powers of a Matrix and their Frechet Derivatives
Get PDFThe Schur--Pad�© algorithm [N. J. Higham and L. Lin, A Schur--Pad�© algorithm for fractional powers of a matrix, SIAM J. Matrix Anal. Appl., 32(3):1056--1078, 2011] computes arbitrary real powers of a matrix using the building blocks of Schur decomposition, matrix square roots, and Pad�© approximants. We improve the algorithm by basing the underlying error analysis on the quantities , for several small , instead of . We extend the algorithm so that it computes along with one or more Fr�©chet derivatives, with reuse of information when more than one Fr�©chet derivative is required, as is the case in condition number estimation. We also derive a version of the extended algorithm that works entirely in real arithmetic when the data is real. Our numerical experiments show the new algorithms to be superior in accuracy to, and often faster than, the original Schur--Pad�© algorithm for computing matrix powers and more accurate than several alternative methods for computing the Fr�©chet derivative. They also show that reliable estimates of the condition number of are obtained by combining the algorithms with a matrix norm estimator
Optimal regulatory control of early contract termination
No full textWe present a quantitative method to find jointly optimal strategies for an industry regulator and a firm,
who operate under exogenous uncertainty. The firm controls its operating policy in order to maximize
its expected future profits, whilst taking account of regulatory fines. The regulator aims to control the
probability of the firm terminating production, by imposing a closure fine which is as low as possible,
while achieving the required reduction in probability. Our method determines the level of fine which
establishes a Nash equilibrium in these nonzero-sum games, under uncertainty
Smoothing non-smooth systems with low-pass filters
No full textLow pass filters, which are used to remove high frequency noise from time series data, smooth the signals they are applied to. In this paper we examine the action of low pass filters on discontinuous or non-differentiable signals from non-smooth dynamical systems, where this smoothing action can be thought of as a smoothing of the underlying system
Depth and the local Langlands correspondence
No full textLet G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters