Eisenstein series and automorphic representations

We provide an introduction to the theory of Eisenstein series and automorphic forms on real simple Lie groups G, emphasising the role of representation theory. It is useful to take a slightly wider view and define all objects over the (rational) adeles A, thereby also paving the way for connections to number theory, representation theory and the Langlands program. Most of the results we present are already scattered throughout the mathematics literature but our exposition collects them together and is driven by examples. Many interesting aspects of these functions are hidden in their Fourier coefficients with respect to unipotent subgroups and a large part of our focus is to explain and derive general theorems on these Fourier expansions. Specifically, we give complete proofs of the Langlands constant term formula for Eisenstein series on adelic groups G(A) as well as the Casselman--Shalika formula for the p-adic spherical Whittaker function associated to unramified automorphic representations of G(Q_p). In addition, we explain how the classical theory of Hecke operators fits into the modern theory of automorphic representations of adelic groups, thereby providing a connection with some key elements in the Langlands program, such as the Langlands dual group LG and automorphic L-functions. Somewhat surprisingly, all these results have natural interpretations as encoding physical effects in string theory. We therefore also introduce some basic concepts of string theory, aimed toward mathematicians, emphasising the role of automorphic forms. In particular, we provide a detailed treatment of supersymmetry constraints on string amplitudes which enforce differential equations of the same type that are satisfied by automorphic forms. Our treatise concludes with a detailed list of interesting open questions and pointers to additional topics which go beyond the scope of this book.Comment: 326 pages. Detailed and example-driven exposition of the subject with highlighted applications to string theory. v2: 375 pages. Substantially extended and small correction

A reduction principle for Fourier coefficients of automorphic forms

In this paper we analyze a general class of Fourier coefficients of automorphic forms on reductive adelic groups $\mathbf{G}(\mathbb{A}_\mathbb{K})$ and their covers. We prove that any such Fourier coefficient is expressible through integrals and sums involving 'Levi-distinguished' Fourier coefficients. By the latter we mean the class of Fourier coefficients obtained by first taking the constant term along the nilradical of a parabolic subgroup, and then further taking a Fourier coefficient corresponding to a $\mathbb{K}$-distinguished nilpotent orbit in the Levi quotient. In a follow-up paper we use this result to establish explicit formulas for Fourier expansions of automorphic forms attached to minimal and next-to-minimal representations of simply-laced reductive groups.Comment: 35 pages. v2: Extended results and paper split into two parts with second part appearing soon. New title to reflect new focus of this part. v3: Minor corrections and updated reference to the second part that has appeared as arXiv:1908.08296. v4: Minor corrections and reformulation

Enhancement of noncontact friction between closely spaced bodies by two-dimensional systems

. We consider the effect of an external bias voltage and the spatial variation of the surface potential, on the damping of cantilever vibrations. The electrostatic friction is due to energy losses in the sample created by the electromagnetic field from the oscillating charges induced on the surface of the tip by the bias voltage and spatial variation of the surface potential. A similar effect arises when the tip is oscillating in the electrostatic field created by charged defects in a dielectric substrate. The electrostatic friction is compared with the van der Waals friction originating from the fluctuating electromagnetic field due to quantum and thermal fluctuation of the current density inside the bodies. We show that the electrostatic and van der Waals friction can be greatly enhanced if on the surfaces of the sample and the tip there are two-dimension (2D) systems, e.g. a 2D-electron system or incommensurate layers of adsorbed ions exhibiting acoustic vibrations. We show that the damping of the cantilever vibrations due to the electrostatic friction may be of similar magnitude as the damping observed in recent experiments of Stipe \textit{et al} [B.C.Stipe, H.J.Mamin, T.D.Stowe, T.W.Kenny, and D.Rugar, Phys.Rev. Lett.% \textbf{87}, 0982001]. We also show that at short separation the van der Waals friction may be large enough to be measured experimentally.Comment: 11 pages, 2 figure

Self-Policing: Dissemination and Adoption of Police Eyewitness Policies in Virginia

Professional policing organizations emphasize the importance of the adoption of sound police policies and procedures, but traditionally doing so has been left to individual agencies. State and local government typically does not closely regulate police, and neither federal constitutional rulings nor state law typically sets out in any detail the practices that police should follow. Thus, law enforcement agencies must themselves draft and disseminate policy. This paper presents the results of studies used to assess the adoption of eyewitness identification policies by law enforcement agencies in Virginia. Policymakers were focused on this problem because Virginia experienced a series of DNA exonerations in cases involving eyewitness misidentifications. In 2005, lawmakers enacted a law that required agencies to have some written policy in place. However, there was little guidance on what that policy should be. To remedy this problem, the state law enforcement policy agency, the Virginia Department of Criminal Justice Services (DCJS) promulgated, in 2011, a detailed model policy on eyewitness procedure. Nevertheless, as reported in a 2013 study, those model practices were only haltingly adopted. In particular, many agencies did not use blind or blinded lineups, in which the administrator does not know which photo is that of a suspect or cannot view which photo the eyewitness is examining. In Fall 2018, all of the over-three hundred law enforcement agencies in Virginia had their policies on this subject requested, using the state freedom of information law. The results show that there has now been widespread adoption of the DCJS model policy. Improved eyewitness identification practices have been adopted by the vast majority of agencies, including large and small agencies. This Article concludes by asking what contributed to the extensive dissemination of a model police policy, and what its implications are for improving police policy and practice without the use of regulation

Friction Laws for Elastic Nano-Scale Contacts

The effect of surface curvature on the law relating frictional forces F with normal load L is investigated by molecular dynamics simulations as a function of surface symmetry, adhesion, and contamination. Curved, non-adhering, dry, commensurate surfaces show a linear dependency, F proportional to L, similar to dry flat commensurate or amorphous surfaces and macroscopic surfaces. In contrast, curved, non-adhering, dry, amorphous surfaces show F proportional to L^(2/3) similar to friction force microscopes. In our model, adhesive effects are most adequately described by the Hertz plus offset model, as the simulations are confined to small contact radii. Curved lubricated or contaminated surfaces show again different behavior; details depend on how much of the contaminant gets squeezed out of the contact. Also, it is seen that the friction force in the lubricated case is mainly due to atoms at the entrance of the tip.Comment: 7 pages, 5 figures, submitted to Europhys. Let

What is the influence on water quality in temperate eutrophic lakes of a reduction of planktivorous and benthivorous fish? A systematic review protocol

Background: In lakes that have become eutrophic due to sewage discharges or nutrient runoff from land, problems such as algal blooms and oxygen deficiency often persist even when nutrient supplies have been reduced. One reason is that phosphorus stored in the sediments can exchange with the water. There are indications that the high abundance of phytoplankton, turbid water and lack of submerged vegetation seen in many eutrophic lakes may represent a semi-stable state. For that reason, a shift back to more natural clear-water conditions could be difficult to achieve. In some cases, though, temporary mitigation of eutrophication-related problems has been accomplished through biomanipulation: stocks of zooplanktivorous fish have been reduced by intensive fishing, leading to increased populations of phytoplankton-feeding zooplankton. Moreover, reduction of benthivorous fish may result in lower phosphorus fluxes from the sediments. An alternative to reducing the dominance of planktivores and benthivores by fishing is to stock lakes with piscivorous fish. These two approaches have often been used in combination. The implementation of the EU Water Framework Directive has recently led to more stringent demands for measures against eutrophication, and a systematic review could clarify whether biomanipulation is efficient as a measure of that kind. Methods: The review will examine primary field studies of how large-scale biomanipulation has affected water quality and community structure in eutrophic lakes or reservoirs in temperate regions. Such studies can be based on comparison between conditions before and after manipulation, on comparison between treated and non-treated water bodies, or both. Relevant outcomes include Secchi depth, concentrations of oxygen, nutrients, suspended solids and chlorophyll, abundance and composition of phytoplankton, zooplankton and fish, and coverage of submerged macrophytes.A Systematic review to this article was published on 22 May 2015: ernes, C., Carpenter, S.R., Gårdmark, A. et al. What is the influence of a reduction of planktivorous and benthivorous fish on water quality in temperate eutrophic lakes? A systematic review. Environ Evid 4, 7 (2015). DOI: 10.1186/s13750-015-0032-9Mistr

Rubber friction: role of the flash temperature

When a rubber block is sliding on a hard rough substrate, the substrate asperities will exert time-dependent deformations of the rubber surface resulting in viscoelastic energy dissipation in the rubber, which gives a contribution to the sliding friction. Most surfaces of solids have roughness on many different length scales, and when calculating the friction force it is necessary to include the viscoelastic deformations on all length scales. The energy dissipation will result in local heating of the rubber. Since the viscoelastic properties of rubber-like materials are extremely strongly temperature dependent, it is necessary to include the local temperature increase in the analysis. At very low sliding velocity the temperature increase is negligible because of heat diffusion, but already for velocities of order 0.01 m/s the local heating may be very important. Here I study the influence of the local heating on the rubber friction, and I show that in a typical case the temperature increase results in a decrease in rubber friction with increasing sliding velocity for v > 0.01 m/s. This may result in stick-slip instabilities, and is of crucial importance in many practical applications, e.g., for the tire-road friction, and in particular for ABS-breaking systems.Comment: 22 pages, 27 figure

Fluid flow at the interface between elastic solids with randomly rough surfaces

I study fluid flow at the interface between elastic solids with randomly rough surfaces. I use the contact mechanics model of Persson to take into account the elastic interaction between the solid walls and the Bruggeman effective medium theory to account for the influence of the disorder on the fluid flow. I calculate the flow tensor which determines the pressure flow factor and, e.g., the leak-rate of static seals. I show how the perturbation treatment of Tripp can be extended to arbitrary order in the ratio between the root-mean-square roughness amplitude and the average interfacial surface separation. I introduce a matrix D(Zeta), determined by the surface roughness power spectrum, which can be used to describe the anisotropy of the surface at any magnification Zeta. I present results for the asymmetry factor Gamma(Zeta) (generalized Peklenik number) for grinded steel and sandblasted PMMA surfaces.Comment: 16 pages, 14 figure

Transverse thermal depinning and nonlinear sliding friction of an adsorbed monolayer

We study the response of an adsorbed monolayer under a driving force as a model of sliding friction phenomena between two crystalline surfaces with a boundary lubrication layer. Using Langevin-dynamics simulation, we determine the nonlinear response in the direction transverse to a high symmetry direction along which the layer is already sliding. We find that below a finite transition temperature, there exist a critical depinning force and hysteresis effects in the transverse response in the dynamical state when the adlayer is sliding smoothly along the longitudinal direction.Comment: 4 pages, 4 figures, to appear in Phys. Rev. Let

Elastic contact between self-affine surfaces: Comparison of numerical stress and contact correlation functions with analytic predictions

Contact between an elastic manifold and a rigid substrate with a self-affine fractal surface is reinvestigated with Green's function molecular dynamics. Stress and contact autocorrelation functions (ACFs) are found to decrease algebraically. A rationale is provided for the observed similarity in the exponents for stress and contact ACFs. Both exponents differ substantially from analytic predictions over the range of Hurst roughness exponents studied. The effect of increasing the range of interactions from a hard sphere repulsion to exponential decay is analyzed. Results for exponential interactions are accurately described by recent systematic corrections to Persson's contact mechanics theory. The relation between the area of simply connected contact patches and the normal force is also studied. Below a threshold size the contact area and force are consistent with Hertzian contact mechanics, while area and force are linearly related in larger contact patches.Comment: 12 pages, 9 figure