On the near periodicity of eigenvalues of Toeplitz matrices

Let $A$ be an infinite Toeplitz matrix with a real symbol $f$ defined on $[-\pi, \pi]$. It is well known that the sequence of spectra of finite truncations $A_N$ of $A$ converges to the convex hull of the range of $f$. Recently, Levitin and Shargorodsky, on the basis of some numerical experiments, conjectured, for symbols $f$ with two discontinuities located at rational multiples of $\pi$, that the eigenvalues of $A_N$ located in the gap of $f$ asymptotically exhibit periodicity in $N$, and suggested a formula for the period as a function of the position of discontinuities. In this paper, we quantify and prove the analog of this conjecture for the matrix $A^2$ in a particular case when $f$ is a piecewise constant function taking values $-1$ and $1$.

Equitability revisited: why the “equitable threat score” is not equitable

In the forecasting of binary events, verification measures that are “equitable” were defined by Gandin and Murphy to satisfy two requirements: 1) they award all random forecasting systems, including those that always issue the same forecast, the same expected score (typically zero), and 2) they are expressible as the linear weighted sum of the elements of the contingency table, where the weights are independent of the entries in the table, apart from the base rate. The authors demonstrate that the widely used “equitable threat score” (ETS), as well as numerous others, satisfies neither of these requirements and only satisfies the first requirement in the limit of an infinite sample size. Such measures are referred to as “asymptotically equitable.” In the case of ETS, the expected score of a random forecasting system is always positive and only falls below 0.01 when the number of samples is greater than around 30. Two other asymptotically equitable measures are the odds ratio skill score and the symmetric extreme dependency score, which are more strongly inequitable than ETS, particularly for rare events; for example, when the base rate is 2% and the sample size is 1000, random but unbiased forecasting systems yield an expected score of around −0.5, reducing in magnitude to −0.01 or smaller only for sample sizes exceeding 25 000. This presents a problem since these nonlinear measures have other desirable properties, in particular being reliable indicators of skill for rare events (provided that the sample size is large enough). A potential way to reconcile these properties with equitability is to recognize that Gandin and Murphy’s two requirements are independent, and the second can be safely discarded without losing the key advantages of equitability that are embodied in the first. This enables inequitable and asymptotically equitable measures to be scaled to make them equitable, while retaining their nonlinearity and other properties such as being reliable indicators of skill for rare events. It also opens up the possibility of designing new equitable verification measures.

A statistical mechanical approach for the computation of the climatic response to general forcings

The climate belongs to the class of non-equilibrium forced and dissipative systems, for which most results of quasi-equilibrium statistical mechanics, including the fluctuation-dissipation theorem, do not apply. In this paper we show for the first time how the Ruelle linear response theory, developed for studying rigorously the impact of perturbations on general observables of non-equilibrium statistical mechanical systems, can be applied with great success to analyze the climatic response to general forcings. The crucial value of the Ruelle theory lies in the fact that it allows to compute the response of the system in terms of expectation values of explicit and computable functions of the phase space averaged over the invariant measure of the unperturbed state. We choose as test bed a classical version of the Lorenz 96 model, which, in spite of its simplicity, has a well-recognized prototypical value as it is a spatially extended one-dimensional model and presents the basic ingredients, such as dissipation, advection and the presence of an external forcing, of the actual atmosphere. We recapitulate the main aspects of the general response theory and propose some new general results. We then analyze the frequency dependence of the response of both local and global observables to perturbations having localized as well as global spatial patterns. We derive analytically several properties of the corresponding susceptibilities, such as asymptotic behavior, validity of Kramers-Kronig relations, and sum rules, whose main ingredient is the causality principle. We show that all the coefficients of the leading asymptotic expansions as well as the integral constraints can be written as linear function of parameters that describe the unperturbed properties of the system, such as its average energy. Some newly obtained empirical closure equations for such parameters allow to define such properties as an explicit function of the unperturbed forcing parameter alone for a general class of chaotic Lorenz 96 models. We then verify the theoretical predictions from the outputs of the simulations up to a high degree of precision. The theory is used to explain differences in the response of local and global observables, to define the intensive properties of the system, which do not depend on the spatial resolution of the Lorenz 96 model, and to generalize the concept of climate sensitivity to all time scales. We also show how to reconstruct the linear Green function, which maps perturbations of general time patterns into changes in the expectation value of the considered observable for finite as well as infinite time. Finally, we propose a simple yet general methodology to study general Climate Change problems on virtually any time scale by resorting to only well selected simulations, and by taking full advantage of ensemble methods. The specific case of globally averaged surface temperature response to a general pattern of change of the CO2 concentration is discussed. We believe that the proposed approach may constitute a mathematically rigorous and practically very effective way to approach the problem of climate sensitivity, climate prediction, and climate change from a radically new perspective.

Finite- N effects for ideal polymer chains near a flat impenetrable wall

This paper addresses the statistical mechanics of ideal polymer chains next to a hard wall. The principal quantity of interest, from which all monomer densities can be calculated, is the partition function, G N(z) , for a chain of N discrete monomers with one end fixed a distance z from the wall. It is well accepted that in the limit of infinite N , G N(z) satisfies the diffusion equation with the Dirichlet boundary condition, G N(0) = 0 , unless the wall possesses a sufficient attraction, in which case the Robin boundary condition, G N(0) = - x G N ′(0) , applies with a positive coefficient, x . Here we investigate the leading N -1/2 correction, D G N(z) . Prior to the adsorption threshold, D G N(z) is found to involve two distinct parts: a Gaussian correction (for z <~Unknown control sequence '\lesssim' aN 1/2 with a model-dependent amplitude, A , and a proximal-layer correction (for z <~Unknown control sequence '\lesssim' a described by a model-dependent function, B(z).

Investigation of ultra wideband multi-channel dichroic beamsplitters from 0.3 to 52µm

The development of a set of multi-channel dichroics which includes a 6 channel dichroic operating over the wavelength region from 0.3 to 52µm is described. In order to achieve the optimum performance, the optical constants of PbTe, Ge and CdTe coatings in the strongly absorptive region have been determined by use of a new iterative method using normal incidence reflectance measurement of the multilayer together with initial values of energy gap Eg and infinite refractive index n for the semiconductor model. The design and manufacture of the dichroics is discussed and the final results are presented.

Variational approach in weighted Sobolev spaces to scattering by unbounded rough surfaces

We consider the problem of scattering of time harmonic acoustic waves by an unbounded sound soft surface which is assumed to lie within a finite distance of some plane. The paper is concerned with the study of an equivalent variational formulation of this problem set in a scale of weighted Sobolev spaces. We prove well-posedness of this variational formulation in an energy space with weights which extends previous results in the unweighted setting [S. Chandler-Wilde and P. Monk, SIAM J. Math. Anal., 37 (2005), pp. 598–618] to more general inhomogeneous terms in the Helmholtz equation. In particular, in the two-dimensional case, our approach covers the problem of plane wave incidence, whereas in the three-dimensional case, incident spherical and cylindrical waves can be treated. As a further application of our results, we analyze a finite section type approximation, whereby the variational problem posed on an infinite layer is approximated by a variational problem on a bounded region.

Operator performance evaluation of controlled depth of field in a stereographically displayed virtual environment

Retinal blurring resulting from the human eye's depth of focus has been shown to assist visual perception. Infinite focal depth within stereoscopically displayed virtual environments may cause undesirable effects, for instance, objects positioned at a distance in front of or behind the observer's fixation point will be perceived in sharp focus with large disparities thereby causing diplopia. Although published research on incorporation of synthetically generated Depth of Field (DoF) suggests that this might act as an enhancement to perceived image quality, no quantitative testimonies of perceptional performance gains exist. This may be due to the difficulty of dynamic generation of synthetic DoF where focal distance is actively linked to fixation distance. In this paper, such a system is described. A desktop stereographic display is used to project a virtual scene in which synthetically generated DoF is actively controlled from vergence-derived distance. A performance evaluation experiment on this system which involved subjects carrying out observations in a spatially complex virtual environment was undertaken. The virtual environment consisted of components interconnected by pipes on a distractive background. The subject was tasked with making an observation based on the connectivity of the components. The effects of focal depth variation in static and actively controlled focal distance conditions were investigated. The results and analysis are presented which show that performance gains may be achieved by addition of synthetic DoF. The merits of the application of synthetic DoF are discussed.

Self-assembled molecular complexes and coordination polymers of CdII, hexamine, and monocarboxylates: structural analysis and theoretical studies of supramolecular interactions

Four new cadmium(II) complexes [Cd-2(bz)(4)(H2O)(4)(mu 2-hmt)]center dot Hbz center dot H2O (1), [Cd-3(bz)(6)(H2O)(6)(mu 2-hmt)(2)]center dot 6H(2)O (2), [Cd(pa)(2)(H2O)(mu(2)-hmt)](n) (3), and {[Cd-3(ac)(6)(H2O)(3)(mu(3)-hmt)(2)]center dot 6H(2)O}(n) (4) with hexamine (hmt) and monocarboxylate ions, benzoate (bz), phenylacetate (pa), or acetate (ac) have been synthesized and characterized structurally. Structure determinations reveal that 1 is dinuclear, 2 is trinuclear, 3 is a one-dimensional (1D) infinite chain, and 4 is a two-dimensional (2D) polymer with fused hexagonal rings consisting of Cd-II and hmt. All the Cd-II atoms in the four complexes (except one CdII in 2) possess seven-coordinate pentagonal bipyramidal geometry with the various chelating bidentate carboxylate groups in equatorial sites. One of the CdII ions in 2, a complex that contains two monodentate carboxylates is in a distorted octahedral environment. The bridging mode of hmt is mu 2- in complexes 1-3 but is mu 3- in complex 4. In all complexes, there are significant numbers of H-bonds, C-H/pi, and pi-pi interactions which play crucial roles in forming the supramolecular networks. The importance of the noncovalent interactions in terms of energies and geometries has been analyzed using high level ab initio calculations. The effect of the cadmium coordinated to hmt on the energetic features of the C-H/pi interaction is analyzed. Finally, the interplay between C-H/pi and pi-pi interactions observed in the crystal structure of 3 is also studied.

Non-normal real estate return distributions by property type in the UK

Investment risk models with infinite variance provide a better description of distributions of individual property returns in the IPD UK database over the period 1981 to 2003 than normally distributed risk models. This finding mirrors results in the US and Australia using identical methodology. Real estate investment risk is heteroskedastic, but the characteristic exponent of the investment risk function is constant across time – yet it may vary by property type. Asset diversification is far less effective at reducing the impact of non‐systematic investment risk on real estate portfolios than in the case of assets with normally distributed investment risk. The results, therefore, indicate that multi‐risk factor portfolio allocation models based on measures of investment codependence from finite‐variance statistics are ineffective in the real estate context

Non-normal real estate return distributions by property type in the U.K.

Investment risk models with infinite variance provide a better description of distributions of individual property returns in the IPD database over the period 1981 to 2003 than Normally distributed risk models, which mirrors results in the U.S. and Australia using identical methodology. Real estate investment risk is heteroscedastic, but the Characteristic Exponent of the investment risk function is constant across time yet may vary by property type. Asset diversification is far less effective at reducing the impact of non-systematic investment risk on real estate portfolios than in the case of assets with Normally distributed investment risk. Multi-risk factor portfolio allocation models based on measures of investment codependence from finite-variance statistics are ineffectual in the real estate context.

Necessary and sufficient conditions for realizability of point processes

We give necessary and sufficient conditions for a pair of (generali- zed) functions 1(r1) and 2(r1, r2), ri 2X, to be the density and pair correlations of some point process in a topological space X, for ex- ample, Rd, Zd or a subset of these. This is an infinite-dimensional version of the classical “truncated moment” problem. Standard tech- niques apply in the case in which there can be only a bounded num- ber of points in any compact subset of X. Without this restriction we obtain, for compact X, strengthened conditions which are necessary and sufficient for the existence of a process satisfying a further re- quirement—the existence of a finite third order moment. We general- ize the latter conditions in two distinct ways when X is not compact.

Synthesis of water-soluble and ether-soluble tetra-μ-acetatodiruthenium(II,III) complexes and X-ray crystal structure of [Ru2(μ-O2CC6H5)4(C2H5OH)2][Ru2(μ-O2CC6H5)4(HSO4)2]

[Ru2(μ-O2CCH3)4Cl] reacts readily with aqueous Ag2SO4 (2: 1 molar ratio) to give the sulphate salt [Ru2(μ-O2CCH3)4(H2O)2]2(SO4) (1). Addition of NaBPh4 to an aqueous solution of 1 produces the ether-soluble tetraphenylborate salt [Ru2(μ-O2CCH3)4(H2O)2][BPh4] (2). A methanolic solution of 1 reacts with Ba(C6H5CCCO2)2 · H2O to give the tetraacetatemonophenylpropynoate complex [Ru2(μ-O2CCH3)4(O2CCCC6H5)] · H2O (3). The reaction of an ethanolic suspension of [Ru2(μ-O2CC6H5)4Cl] with Ag2SO4 and H2SO4 (2 : 1 : 1 molar ratio) leads to the tetra-μ-benzoatodiruthenium(II,III) double complex salt [Ru2(μ-O2CC6H5)4(C2H5OH)2][Ru2(μ-O2CC6H5)4(HSO4)2] (4). Complex 4 is also obtained by reacting an ethanolic solution of 1 with an excess of benzoic acid in the presence of H2SO4. The X-ray crystal structure of 4 shows it to consist of [Ru2(μ-O2CC6H5)4(C2H5OH)2]+ and [Ru2(μ-O2CC6H5)4(HSO4)2]− ions, which are linked together by hydrogen bonds into an infinite polymeric chain. The RuRu distances in the cation and anion are very similar [2.265(2) and 2.272(2) Å, respectively]. Spectroscopic, magnetic, conductivity and cyclic voltammetry data are given for the complexes.

Spectrum of a Feinberg-Zee random hopping matrix

This paper provides a new proof of a theorem of Chandler-Wilde, Chonchaiya, and Lindner that the spectra of a certain class of infinite, random, tridiagonal matrices contain the unit disc almost surely. It also obtains an analogous result for a more general class of random matrices whose spectra contain a hole around the origin. The presence of the hole forces substantial changes to the analysis.

The Mathematics of Control Theory

This text contains papers presented at the Institute of Mathematics and its Applications Conference on Control Theory, held at the University of Strathclyde in Glasgow. The contributions cover a wide range of topics of current interest to theoreticians and practitioners including algebraic systems theory, nonlinear control systems, adaptive control, robustness issues, infinite dimensional systems, applications studies and connections to mathematical aspects of information theory and data-fusion.

A double-decker bus, an elephant and 300 Girl Guides: analyzing 'space' and 'place' in Blue Peter

This article draws upon Karen Lury's definitions of 'space' and 'place' in relation to the BBC children's programme Blue Peter (1958–present). Through an analysis of the Blue Peter studio over the past 53 years, Amanda Beauchamp highlights its evolution from a 'space' to a 'place' within the history of children's television. Her article considers how the Blue Peter studio's 'infinite nature' was achieved, alongside the role it played in creating the programme institution. She addresses the impact of major changes in the studio layout since 2005, when the studio went from being 'tardis-like' to a 'cosy cubbyhole'. Amanda concludes by questioning the impact that this change has had on programme identity and whether the 'place' that pre-2005 Blue Peter took 47 years to create has been compromised.