Global well-posedness of the Kirchhoff equation and Kirchhoff systems

This article is devoted to review the known results on global well-posedness for the Cauchy problem to the Kirchhoff equation and Kirchhoff systems with small data. Similar results will be obtained for the initial-boundary value problems in exterior domains with compact boundary. Also, the known results on large data problems will be reviewed together with open problems.Comment: arXiv admin note: text overlap with arXiv:1211.300

Local well-posedness for the space-time Monopole equation in Lorenz gauge

It is known from the work of Czubak that the space-time Monopole equation is locally well-posed in the Coulomb gauge for small initial data in $H^s(\mathbb{R}^2)$ for $s>1/4$. Here we prove local well-posedness for arbitrary initial data in $H^s(\mathbb{R^2})$ with $s>1/4$ in the Lorenz gauge.Comment: To appear in NoDE

Dispersive estimates for Schr\"odinger operators with point interactions in R3\mathbb{R}^3

The study of dispersive properties of Schr\"odinger operators with point interactions is a fundamental tool for understanding the behavior of many body quantum systems interacting with very short range potential, whose dynamics can be approximated by non linear Schr\"odinger equations with singular interactions. In this work we proved that, in the case of one point interaction in $\mathbb{R}^3$, the perturbed Laplacian satisfies the same $L^p-L^q$ estimates of the free Laplacian in the smaller regime $q\in[2,3)$. These estimates are implied by a recent result concerning the $L^p$ boundedness of the wave operators for the perturbed Laplacian. Our approach, however, is more direct and relatively simple, and could potentially be useful to prove optimal weighted estimates also in the regime $q\geq 3$.Comment: To appear on: "Advances in Quantum Mechanics: Contemporary Trends and Open Problems", G. Dell'Antonio and A. Michelangeli eds., Springer-INdAM series 201

Strichartz Estimates for the Vibrating Plate Equation

We study the dispersive properties of the linear vibrating plate (LVP) equation. Splitting it into two Schr\"odinger-type equations we show its close relation with the Schr\"odinger equation. Then, the homogeneous Sobolev spaces appear to be the natural setting to show Strichartz-type estimates for the LVP equation. By showing a Kato-Ponce inequality for homogeneous Sobolev spaces we prove the well-posedness of the Cauchy problem for the LVP equation with time-dependent potentials. Finally, we exhibit the sharpness of our results. This is achieved by finding a suitable solution for the stationary homogeneous vibrating plate equation.Comment: 18 pages, 4 figures, some misprints correcte

L^p boundedness of the wave operator for the one dimensional Schroedinger operator

Given a one dimensional perturbed Schroedinger operator H=-(d/dx)^2+V(x) we consider the associated wave operators W_+, W_- defined as the strong L^2 limits as s-> \pm\infty of the operators e^{isH} e^{-isH_0} We prove that the wave operators are bounded operators on L^p for all 1<p<\infty, provided (1+|x|)^2 V(x) is integrable, or else (1+|x|)V(x) is integrable and 0 is not a resonance. For p=\infty we obtain an estimate in terms of the Hilbert transform. Some applications to dispersive estimates for equations with variable rough coefficients are given.Comment: 26 page

‘In it together’? The political consequences of perceived discommunions of interest in British politics

© The Author(s) 2015. The presence of shared interests between politicians and citizens is central to many accounts of political representation, yet there has been little empirical research into how citizens perceive these interests and whether they think a ‘communion of interests’ exists between them and their elected representatives. Using new survey data, this paper explores the extent to which Britons think their MPs share the same everyday experiences as they do. It further explores the relationship between these perceptions and indicators of specific and diffuse political support. We show that British citizens generally say that politicians are less affected by social and economic conditions and less reliant on public services than they are. The size of this perceived ‘discommunion’ affects voting behaviour and satisfaction with democracy.The ESRC (grant number RES-000-22-3459) and British Academy (grant numbers SG-52322)

Interpolation Theorems for Self-adjoint Operators

We prove a complex and a real interpolation theorems on Besov spaces and Triebel-Lizorkin spaces associated with a selfadjoint operator $L$, without assuming the gradient estimate for its spectral kernel. The result applies to the cases where $L$ is a uniformly elliptic operator or a Schr\"odinger operator with electro-magnetic potential.Comment: 8 pages. Submitte

A time-frequency analysis perspective on Feynman path integrals

The purpose of this expository paper is to highlight the starring role of time-frequency analysis techniques in some recent contributions concerning the mathematical theory of Feynman path integrals. We hope to draw the interest of mathematicians working in time-frequency analysis on this topic, as well as to illustrate the benefits of this fruitful interplay for people working on path integrals.Comment: 26 page

Michael Gove’s war on professional historical expertise : conservative curriculum reform, extreme Whig history and the place of imperial heroes in modern multicultural Britain

Six years of continuously baiting his opponents within the history profession eventually amounted to little where it mattered most. UK Secretary of State for Education, Michael Gove, finally backtracked in 2013 on his plans to impose a curriculum for English schools based on a linear chronology of the achievements of British national heroes. His ‘history as celebration’ curriculum was designed to instil pride amongst students in a supposedly shared national past, but would merely have accentuated how many students in modern multicultural Britain fail to recognise themselves in what is taught in school history lessons. Now that the dust has settled on Gove’s tenure as Secretary of State, the time is right for retrospective analysis of how his plans for the history curriculum made it quite so far. How did he construct an ‘ideological’ conception of expertise which allowed him to go toe-to-toe for so long with the ‘professional’ expertise of academic historians and history teachers? What does the content of this ideological expertise tell us about the politics of race within Conservative Party curriculum reforms? This article answers these questions to characterise Gove as a ‘whig historian’ of a wilfully extreme nature in his attachment to imperial heroes as the best way to teach national history in modern multicultural Britain