Absolute continuity in periodic thin tubes and strongly coupled leaky wires

Using a perturbative argument, we show that in any finite region containing the lowest transverse eigenmode, the spectrum of a periodically curved smooth Dirichlet tube in two or three dimensions is absolutely continuous provided the tube is sufficiently thin. In a similar way we demonstrate absolute continuity at the bottom of the spectrum for generalized Schr\"odinger operators with a sufficiently strongly attractive $\delta$ interaction supported by a periodic curve in $\mathbb{R}^d, d=2,3$.Comment: LaTeX 2e, 10 page

Economic Isolation, Inequality, and the Suits Index of Progressivity

We present a class of social evaluation functions and inequality indices that obey standard axioms of welfare economics and that can be intuitively linked to measures of relative deprivation and economic isolation. From this, associated classes of indices of tax departure from proportionality and tax redistribution are derived. A special case of these indices is the popular Suits index of progressivity, for which no social welfare foundation has previously been provided. We illustrate the application of these indices using the British regime of personal income taxes and National Insurance contributions.Progressivity, redistribution, inequality, economic isolation.

Equity and Equality

Is horizontal equity (HE) the "most widely accepted principle of equity"? Or does it stand in "opposition to the advancement of human welfare"? This paper argues that the case for the HE principle is not as straightforward as is usually thought and that it requires advanced notions of justice and well-being. The most likely ethical basis HE appears to combine a Rawlsian maximin principle and a view of well-being that allows for relative local comparison effects. The paper also explores some of the dimensions of equality and well-being along which the HE principle can be applied and presents a number of examples showing how HE considerations can provide an important input into policy analysis.Horizontal equity, vertical equity, redistribution, equality, social justice

Topologically non-trivial quantum layers

Given a complete non-compact surface embedded in R^3, we consider the Dirichlet Laplacian in a layer of constant width about the surface. Using an intrinsic approach to the layer geometry, we generalise the spectral results of an original paper by Duclos et al. to the situation when the surface does not possess poles. This enables us to consider topologically more complicated layers and state new spectral results. In particular, we are interested in layers built over surfaces with handles or several cylindrically symmetric ends. We also discuss more general regions obtained by compact deformations of certain layers.Comment: 15 pages, 6 figure

Relative Performance, Relative Deprivation and Generalised Gini Indices of Inequality and Horizontal Inequity

Il est possible d'interpréter une classe d'indices d'iniquité horizontale comme représentant un regroupement de moyennes normatives de sentiments individuels de performance relative dans la distribution des taxes et des transferts. Un membre très connu de cette classe est l'indice d'iniquité horizontale d'Atkinson et de Plotnick. On peut interpréter de façon similaire des indices généralisés de Gini comme étant des moyennes normatives de sentiments individuels de privation relative. La combinaison de ces deux classes d'indices peut nous permettre de mieux soupeser les objectifs concurrents de réduction d'inégalité et d'équité horizontale. Les résultats portant sur l'équité horizontale sont illustrés à l'aide de la distribution des taxes et des transferts au Canada en 1981 et en 1990. Nous trouvons que les sentiments de mauvaise performance relative ainsi que l'iniquité horizontale ont augmenté de façon statistiquement significative dans les années 1980.

Kitaev's Z_d-Codes Threshold Estimates

We study the quantum error correction threshold of Kitaev's toric code over the group Z_d subject to a generalized bit-flip noise. This problem requires novel decoding techniques, and for this purpose we generalize the renormalization group method we previously introduced for Z_2 topological codes.Comment: 5 pages, 5 figure

Measuring Progressivity and Inequality

A general class of progressivity indices is proposed which is consistent with the well-developed theory of the measurement of inequality and social welfare. In particular, we show that the more progressive a tax system, the more equal the distribution of net income and the greater the progressivity index. For an additive social welfare function and a progressive tax system, the greater the degree of relative inequality aversion, the greater the progressivity index. We also discuss the link between inequality of gross income and tax progressivity. A by-product is the derivation of a general class of inequality measures that are invariant to equi-proportionate changes in incomes. We illustrate the analysis using the British tax and benefit system.Progressivity, redistribution, inequality, social welfare

Limiting absorption principle and perfectly matched layer method for Dirichlet Laplacians in quasi-cylindrical domains

We establish a limiting absorption principle for Dirichlet Laplacians in quasi-cylindrical domains. Outside a bounded set these domains can be transformed onto a semi-cylinder by suitable diffeomorphisms. Dirichlet Laplacians model quantum or acoustically-soft waveguides associated with quasi-cylindrical domains. We construct a uniquely solvable problem with perfectly matched layers of finite length. We prove that solutions of the latter problem approximate outgoing or incoming solutions with an error that exponentially tends to zero as the length of layers tends to infinity. Outgoing and incoming solutions are characterized by means of the limiting absorption principle.Comment: to appear in SIAM Journal on Mathematical Analysi

A State Distillation Protocol to Implement Arbitrary Single-qubit Rotations

An important task required to build a scalable, fault-tolerant quantum computer is to efficiently represent an arbitrary single-qubit rotation by fault-tolerant quantum operations. Traditionally, the method for decomposing a single-qubit unitary into a discrete set of gates is Solovay-Kitaev decomposition, which in practice produces a sequence of depth O(\log^c(1/\epsilon)), where c~3.97 is the state-of-the-art. The proven lower bound is c=1, however an efficient algorithm that saturates this bound is unknown. In this paper, we present an alternative to Solovay-Kitaev decomposition employing state distillation techniques which reduces c to between 1.12 and 2.27, depending on the setting. For a given single-qubit rotation, our protocol significantly lowers the length of the approximating sequence and the number of required resource states (ancillary qubits). In addition, our protocol is robust to noise in the resource states.Comment: 10 pages, 18 figures, 5 table