Operational calculus and integral transforms for groups with finite propagation speed

Let $A$ be the generator of a strongly continuous cosine family $(\cos (tA))_{t\in {\bf R}}$ on a complex Banach space $E$. The paper develops an operational calculus for integral transforms and functions of $A$ using the generalized harmonic analysis associated to certain hypergroups. It is shown that characters of hypergroups which have Laplace representations give rise to bounded operators on $E$. Examples include the Mellin transform and the Mehler--Fock transform. The paper uses functional calculus for the cosine family $\cos( t\sqrt {\Delta})$ which is associated with waves that travel at unit speed. The main results include an operational calculus theorem for Sturm--Liouville hypergroups with Laplace representation as well as analogues to the Kunze--Stein phenomenon in the hypergroup convolution setting.Comment: arXiv admin note: substantial text overlap with arXiv:1304.5868. Substantial revision to version

Routes and Roots

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Crossing over

Publisher PD

Hill's Spectral Curves and the Invariant Measure of the Periodic KdV Equation

This paper analyses the periodic spectrum of Schr\"odinger's equation $-f''+qf=\lambda f$ when the potential is real, periodic, random and subject to the invariant measure $\nu_N^\beta$ of the periodic KdV equation. This $\nu_N^\beta$ is the modified canonical ensemble, as given by Bourgain ({Comm. Math. Phys.} {166} (1994), 1--26), and $\nu_N^\beta$ satisfies a logarithmic Sobolev inequality. Associated concentration inequalities control the fluctuations of the periodic eigenvalues $(\lambda_n)$. For $\beta, N>0$ small, there exists a set of positive $\nu_N^\beta$ measure such that $(\pm \sqrt{2(\lambda_{2n}+\lambda_{2n-1})})_{n=0}^\infty$ gives a sampling sequence for Paley--Wiener space $PW(\pi )$ and the reproducing kernels give a Riesz basis. Let $(\mu_j)_{j=1}^\infty$ be the tied spectrum; then $(2\sqrt{\mu_j}-j)$ belongs to a Hilbert cube in $\ell^2$ and is distributed according to a measure that satisfies Gaussian concentration for Lipschitz functions. The sampling sequence $(\sqrt{\mu_j})_{j=1}^\infty$ arises from a divisor on the spectral curve, which is hyperelliptic of infinite genus. The linear statistics $\sum_j g(\sqrt{\lambda_{2j}})$ with test function $g\in PW(\pi)$ satisfy Gaussian concentration inequalities.Comment: 34 page

Selected Issues in the Rise of Income Inequality

macroeconomics, Income Inequality

Where Did Productivity Growth Go? Inflation Dynamics and the Distribution of Income

Starting from the standard Gordon inflation model, which explains price changes by inertia, demand shocks, and supply shocks but excludes wages, the first part of this paper returns wages to the analysis by developing a model that includes both price and wage equations. The model allows for feedback between the two and captures the effect of changes in trend productivity growth on inflation, nominal wages, and labor’s income share. In dynamic simulations, changes in the productivity growth trend strongly boosted inflation during 1965-79 and slowed it between 1995 and 2005. The paper’s second part links the productivity growth analysis to changes in the income distribution. It finds, using IRS data, that only the top decile experienced real wage and salary income growth equal to or above average economywide productivity growth. And increasing inequality within the top decile was as important a source of growing inequality as the gap between the top and bottom deciles.macroeconomics, Productivity Growth, Inflation Dynamics, Distribution of Income

Why did Europe’s productivity catch-up sputter out? a tale of tigers and tortoises

This paper takes a different approach to examining the sharp turnaround in EU relative to U. S. labor productivity growth since 1995. The vast majority of the literature focuses on the American growth revival. But close to half of the turnaround was caused by a European retardation. What caused that retardation? Our paper shows that none of the consensus explanations of the American revival provide any help at all in explaining the European retardation. It is sui generis and therein lies a tale that has not previously been told. ; Europe has faltered across the board. The deceleration of its productivity growth is not explained at all by ICT production and only to a small degree by retailing/wholesaling. Rather, the big European countries have failed in nearly every dimension. The retardation of Europe is illuminated by dividing up the EU-15 countries into Tigers, a Middle group, and Tortoises. ; Our first surprise in examining the data is to find that the EU-US turnaround in labor productivity growth was not just a reflection of an equivalent turnaround in total factor productivity growth. Capital deepening also faltered in Europe, indicting all the macroeconomic determinants of economy-wide investment as part of Europe’s problem. ; The most striking aspect of our results is that the failing of the Tortoise countries is widespread, spanning industries as diverse as agriculture, wholesale trade, mining, chemicals, construction, fabricated metals, and clothing. Our explanation is that Europe’s previous catch-up to the U. S. productivity level before 1995 was artificial – by making labor expensive, Europe raised its average product. But after 1995 as labor market reforms have helped to make labor cheaper, so the average product of labor has been pushed down and its growth rate has faltered. The falling behind of Europe is quite simple to explain; European regulations defied the laws of labor market equilibrium for decades, and a slight leak in the dam holding back market forces has already led to faster growth in European employment and slower growth in productivity.

Welfare Rankings From Multivariate Data, A Non-Parametric Approach

Economic and Social Welfare is inherently multidimensional. However choosing a measure which combines several indicators is difficult and may have unintended and undesireable effects on the incentives for policymakers. We develope a nonparametric empirical method for deriving welfare rankings based on data envelopment which avoids the need to specify a weighting scheme. The results are valid for all possible social welfare functions which share certain cannonical properties. We apply this method to data on human development.Welfare Rankings, Data Envelopment, Human development

Conversion of a mathematics course to CAL: A case study of a large‐scale rapid change of resources and organization

During 1994–95, first‐year maths for the BTechEd degree at the University of Glasgow was student‐centred, teacher‐supported A modular online maths course replaced a traditional, lecture‐based course. Students worked at their own pace, with timetabled and open access computer classes and/or paper handbooks. The course was evaluated by open‐ended measures, and study of examination outcomes, providing us with some pedagogical questions and some recommendations for change. With some adaptation, and with important questions still open, the new course will continue to run