On the multiplicativity of quantum cat maps

The quantum mechanical propagators of the linear automorphisms of the two-torus (cat maps) determine a projective unitary representation of the theta group, known as Weil's representation. We prove that there exists an appropriate choice of phases in the propagators that defines a proper representation of the theta group. We also give explicit formulae for the propagators in this representation.Comment: Revised version: proof of the main theorem simplified. 21 page

Quantum ergodicity for graphs related to interval maps

We prove quantum ergodicity for a family of graphs that are obtained from ergodic one-dimensional maps of an interval using a procedure introduced by Pakonski et al (J. Phys. A, v. 34, 9303-9317 (2001)). As observables we take the L^2 functions on the interval. The proof is based on the periodic orbit expansion of a majorant of the quantum variance. Specifically, given a one-dimensional, Lebesgue-measure-preserving map of an interval, we consider an increasingly refined sequence of partitions of the interval. To this sequence we associate a sequence of graphs, whose directed edges correspond to elements of the partitions and on which the classical dynamics approximates the Perron-Frobenius operator corresponding to the map. We show that, except possibly for subsequences of density 0, the eigenstates of the quantum graphs equidistribute in the limit of large graphs. For a smaller class of observables we also show that the Egorov property, a correspondence between classical and quantum evolution in the semiclassical limit, holds for the quantum graphs in question.Comment: 20 pages, 1 figur

Quantum Computing of Poincare Recurrences and Periodic Orbits

Quantum algorithms are built enabling to find Poincar\'e recurrence times and periodic orbits of classical dynamical systems. It is shown that exponential gain compared to classical algorithms can be reached for a restricted class of systems. Quadratic gain can be achieved for a larger set of dynamical systems. The simplest cases can be implemented with small number of qubits.Comment: revtex, 5 pages, research at Quantware MIPS Center (see http://www.quantware.ups-tlse.fr); minor changes and references adde

Bounding sup-norms of cusp forms of large level

Let f be an $L^2$-normalized weight zero Hecke-Maass cusp form of square-free level N, character $\chi$ and Laplacian eigenvalue $\lambda\geq 1/4$. It is shown that $\| f \|_{\infty} \ll_{\lambda} N^{-1/37}$, from which the hybrid bound $\|f \|_{\infty} \ll \lambda^{1/4} (N\lambda)^{-\delta}$ (for some $\delta > 0$) is derived. The first bound holds also for $f = y^{k/2}F$ where F is a holomorphic cusp form of weight k with the implied constant now depending on k.Comment: version 3: substantially revised versio

Anatomy of quantum chaotic eigenstates

The eigenfunctions of quantized chaotic systems cannot be described by explicit formulas, even approximate ones. This survey summarizes (selected) analytical approaches used to describe these eigenstates, in the semiclassical limit. The levels of description are macroscopic (one wants to understand the quantum averages of smooth observables), and microscopic (one wants informations on maxima of eigenfunctions, "scars" of periodic orbits, structure of the nodal sets and domains, local correlations), and often focusses on statistical results. Various models of "random wavefunctions" have been introduced to understand these statistical properties, with usually good agreement with the numerical data. We also discuss some specific systems (like arithmetic ones) which depart from these random models.Comment: Corrected typos, added a few references and updated some result

Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices

We study various aspects of the dynamics induced by integer matrices on the invariant rational lattices of the torus in dimension 2 and greater. Firstly, we investigate the orbit structure when the toral endomorphism is not invertible on the lattice, characterising the pretails of eventually periodic orbits. Next we study the nature of the symmetries and reversing symmetries of toral automorphisms on a given lattice, which has particular relevance to (quantum) cat maps.Comment: 29 pages, 3 figure

Spectral properties of noisy classical and quantum propagators

We study classical and quantum maps on the torus phase space, in the presence of noise. We focus on the spectral properties of the noisy evolution operator, and prove that for any amount of noise, the quantum spectrum converges to the classical one in the semiclassical limit. The small-noise behaviour of the classical spectrum highly depends on the dynamics generated by the map. For a chaotic dynamics, the outer spectrum consists in isolated eigenvalues (``resonances'') inside the unit circle, leading to an exponential damping of correlations. On the opposite, in the case of a regular map, part of the spectrum accumulates along a one-dimensional ``string'' connecting the origin with unity, yielding a diffusive behaviour. We finally study the non-commutativity between the semiclassical and small-noise limits, and illustrate this phenomenon by computing (analytically and numerically) the classical and quantum spectra for some maps.Comment: 35 pages, 6 .eps figures, to be published in Nonlinearity. I added some references and comment

The Telos of faith-based aid: Christian organising in development, humanitarianism and advocacy through the lens of institutional logics

This thesis is situated within the field of faith-based organisations (FBOs), but is concerned with a specific kind of FBO: large non-governmental organisations (NGOs) that operate within the same sphere of activity as those that do not have an explicit religious affiliation or faith basis. Since the turn of the century, the volume of funding passing through FBOs has increased, leading to a growing critical focus on such organisations: how should they be defined and categorised? In what ways are they distinct? Do they have comparative advantages? Rather than analysing how FBOs are distinct from other NGOs, the thesis explores what their faith orientation means in actual practice. It is also positioned within the field of religions and development, and thus the primary question it sets out to address concerns how faith influences the practice of faith-based development, humanitarian and advocacy NGOs. It focuses in particular on one UK-based FBO rooted in the Christian faith. The thesis makes two main contributions to the literature, the first of which concerns the theoretical perspective brought to the study of FBOs in religions and development. While various typologies have been constructed, the thesis employs a new perspective – that of institutional logics – and develops an analytical tool that can be adapted for use in future studies. The second relates to the emphasis placed on the ‘telos’ of each institutional logic. This is implicit within the perspective, but has not been a major area of focus to date. Within the case study organisation, the points of tension concerning the action of faith across organisational practice were found to connect to the telos of the logic of corporation, which is the ‘long-term sustainability of the organisation.’ This is because while the telos of the logic of religion (worship God) transcends the organisation, that of the logic of corporation is the organisation itself. Since ultimately, organisational practices must work to sustain the organisation, at the organisational level faith is restricted to certain spaces and forms, while at the individual level it is dominant and active. The argument advanced through the thesis is that the influence of faith at the organisational level predominantly relates to the process of organising. This brings a new perspective to religions and development. In Chapter 1, the research question is situated within its broader field. Chapter 2 then introduces the institutional logics perspective (ILP). The concept of the ‘field’ is also unpacked and defined, and its significance highlighted. The chapter ends with an overview of the methods of data collection and research instrument. Chapter 3 then focuses on historical research, using the ILP to highlight important dynamics in the history of development, humanitarianism and advocacy. Although connected, these spheres of activity have distinct historical paths, which are traced from their emergence up until the twentieth century, after which the post-WWII period is explored. The chapter highlights some of the tensions between dynamics related to various logics. Chapter 4 homes in on the contemporary UK-based sector. Bringing together the existing literature on typologies of institutional logics, insights from Chapter 3, and initial empirical research in a specified field, a tool for data analysis is developed: a field-level typology of institutional logics. In Chapter 5, this typology is used to process and analyse data collected within one FBO operating within the field identified in Chapter 4. Thus, the chapter demonstrates the use of the typology in action and value of the approach. Chapter 6 then explores the research findings, discussing these according to both the questions guiding the empirical research and the primary research question. Finally, Chapter 7 summarises the contribution of the thesis to the fields of religions and development and institutional logics respectively, and in particular, the study of FBOs within these fields