The XX-model with boundaries. Part III:Magnetization profiles and boundary bound states

We calculate the magnetization profiles of the $\sigma_j^x$ and $\sigma_j^z$ operators for the XX-model with hermitian boundary terms. We study the profiles on the finite chain and in the continuum limit. The results are discussed in the context of conformal invariance. We also discuss boundary excitations and their effect on the magnetization profiles.Comment: 30 pages, 3 figure

Zero modes, entropy bounds and partition functions

Some recent finite temperature calculations arising in the investigation of the Verlinde-Cardy relation are re-analysed. Some remarks are also made about temperature inversion symmetry.Comment: 12 pages, JyTe

Near-zero humidities on Ben Nevis, Scotland, revealed by pioneering 19th-century observers and modern volunteers

The weather on Ben Nevis – the highest mountain in the British Isles, 1345 m AMSL – sometimes shows episodes of remarkably low relative humidity (RH) with few precedents anywhere else in the British Isles. We are able to quantify this for the first time using a high-quality series of hourly dry- and wet-bulb observations, made on the summit. These observations were made between 1883 and 1904, but have only just become available to modern science, thanks to thousands of volunteers who worked to rescue this unique and exemplary dataset from published volumes. Careful examination and analysis of the original observations using modern psychrometric theory revealed several occasions where we are confident that the summit RH fell close to zero as a result of anticyclonic subsidence. Three case histories are examined in some detail. The nineteenth-century Ben Nevis humidity records are also compared with contemporary automatic weather station (AWS) data from two high-altitude Scottish mountain sites

Nonlinear Differential Equations Satisfied by Certain Classical Modular Forms

A unified treatment is given of low-weight modular forms on \Gamma_0(N), N=2,3,4, that have Eisenstein series representations. For each N, certain weight-1 forms are shown to satisfy a coupled system of nonlinear differential equations, which yields a single nonlinear third-order equation, called a generalized Chazy equation. As byproducts, a table of divisor function and theta identities is generated by means of q-expansions, and a transformation law under \Gamma_0(4) for the second complete elliptic integral is derived. More generally, it is shown how Picard-Fuchs equations of triangle subgroups of PSL(2,R) which are hypergeometric equations, yield systems of nonlinear equations for weight-1 forms, and generalized Chazy equations. Each triangle group commensurable with \Gamma(1) is treated.Comment: 40 pages, final version, accepted by Manuscripta Mathematic

Super congruences and Euler numbers

Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\binom{2k}{k}/2^k=(-1)^{(p-1)/2}-p^2E_{p-3} (mod p^3),$$ $$\sum_{k=1}^{(p-1)/2}\binom{2k}{k}/k=(-1)^{(p+1)/2}8/3*pE_{p-3} (mod p^2),$$ $$\sum_{k=0}^{(p-1)/2}\binom{2k}{k}^2/16^k=(-1)^{(p-1)/2}+p^2E_{p-3} (mod p^3)$$, where E_0,E_1,E_2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for $\pi^2$, $\pi^{-2}$ and the constant $K:=\sum_{k>0}(k/3)/k^2$ (with (-) the Jacobi symbol), two of which are $$\sum_{k=1}^\infty(10k-3)8^k/(k^3\binom{2k}{k}^2\binom{3k}{k})=\pi^2/2$$ and $$\sum_{k>0}(15k-4)(-27)^{k-1}/(k^3\binom{2k}{k}^2\binom{3k}k)=K.$

Projective dynamics and classical gravitation

Given a real vector space V of finite dimension, together with a particular homogeneous field of bivectors that we call a "field of projective forces", we define a law of dynamics such that the position of the particle is a "ray" i.e. a half-line drawn from the origin of V. The impulsion is a bivector whose support is a 2-plane containing the ray. Throwing the particle with a given initial impulsion defines a projective trajectory. It is a curve in the space of rays S(V), together with an impulsion attached to each ray. In the simplest example where the force is identically zero, the curve is a straight line and the impulsion a constant bivector. A striking feature of projective dynamics appears: the trajectories are not parameterized. Among the projective force fields corresponding to a central force, the one defining the Kepler problem is simpler than those corresponding to other homogeneities. Here the thrown ray describes a quadratic cone whose section by a hyperplane corresponds to a Keplerian conic. An original point of view on the hidden symmetries of the Kepler problem emerges, and clarifies some remarks due to Halphen and Appell. We also get the unexpected conclusion that there exists a notion of divergence-free field of projective forces if and only if dim V=4. No metric is involved in the axioms of projective dynamics.Comment: 20 pages, 4 figure

Supersymmetric QCD corrections to e+e−→tbˉH−e^+e^-\to t\bar{b}H^- and the Bernstein-Tkachov method of loop integration

The discovery of charged Higgs bosons is of particular importance, since their existence is predicted by supersymmetry and they are absent in the Standard Model (SM). If the charged Higgs bosons are too heavy to be produced in pairs at future linear colliders, single production associated with a top and a bottom quark is enhanced in parts of the parameter space. We present the next-to-leading-order calculation in supersymmetric QCD within the minimal supersymmetric SM (MSSM), completing a previous calculation of the SM-QCD corrections. In addition to the usual approach to perform the loop integration analytically, we apply a numerical approach based on the Bernstein-Tkachov theorem. In this framework, we avoid some of the generic problems connected with the analytical method.Comment: 14 pages, 6 figures, accepted for publication in Phys. Rev.