Fractal Strings and Multifractal Zeta Functions

For a Borel measure on the unit interval and a sequence of scales that tend to zero, we define a one-parameter family of zeta functions called multifractal zeta functions. These functions are a first attempt to associate a zeta function to certain multifractal measures. However, we primarily show that they associate a new zeta function, the topological zeta function, to a fractal string in order to take into account the topology of its fractal boundary. This expands upon the geometric information garnered by the traditional geometric zeta function of a fractal string in the theory of complex dimensions. In particular, one can distinguish between a fractal string whose boundary is the classical Cantor set, and one whose boundary has a single limit point but has the same sequence of lengths as the complement of the Cantor set. Later work will address related, but somewhat different, approaches to multifractals themselves, via zeta functions, partly motivated by the present paper.Comment: 32 pages, 9 figures. This revised version contains new sections and figures illustrating the main results of this paper and recent results from others. Sections 0, 2, and 6 have been significantly rewritte

Berry phases in superconducting transitions

I generalize the concept of Berry's geometrical phase for quasicyclic Hamiltonians to the case in which the ground state evolves adiabatically to an excited state after one cycle, but returns to the ground state after an integer number of cycles. This allows to extend the charge Berry phase gamma_c related to the macroscopic polarization, to many-body systems with fractional number of particles per site. Under certain conditions, gamma_c and the spin Berry phase gamma_s jump in pi at the boundary of superconducting phases. In the extended Hubbard chain with on-site attraction U and nearest-neighbor interaction V at quarter filling, the transitions detected agree very well with exact results in two limits solved by the Bethe ansatz, and with previous numerical studies. In chains with spin SU(2) symmetry, gamma_s jumps when a spin gap opens.Comment: 5 pages, 3 figures, accepted in Europhys. Let

Resummation Improved Rapidity Spectrum for Gluon Fusion Higgs Production

Gluon-induced processes such as Higgs production typically exhibit large perturbative corrections. These partially arise from large virtual corrections to the gluon form factor, which at timelike momentum transfer contains Sudakov logarithms evaluated at negative arguments $\ln^2(-1) = -\pi^2$. It has been observed that resumming these terms in the timelike form factor leads to a much improved perturbative convergence for the total cross section. We discuss how to consistently incorporate the resummed form factor into the perturbative predictions for generic cross sections differential in the Born kinematics, including in particular the Higgs rapidity spectrum. We verify that this indeed improves the perturbative convergence, leading to smaller and more reliable perturbative uncertainties, and that this is not affected by cancellations between resummed and unresummed contributions. Combining both fixed-order and resummation uncertainties, the perturbative uncertainty for the total cross section at N$^3$LO$+$N$^3$LL$^\prime_\varphi$ is about a factor of two smaller than at N$^3$LO. The perturbative uncertainty of the rapidity spectrum at NNLO$+$NNLL$^\prime_\varphi$ is similarly reduced compared to NNLO. We also study the analogous resummation for quark-induced processes, namely Higgs production through bottom quark annihilation and the Drell-Yan rapidity spectrum. For the former the resummation leads to a small improvement, while for the latter it confirms the already small uncertainties of the fixed-order predictions.Comment: 30 pages + 17 pages in Appendices, 10 figures; v2: journal version; references added, discussed individual partonic channels for Drell-Ya

CCD Photometry of Delta Scuti stars 7 Aql and 8 Aql

As a continuation of the study of the Delta Scuti stars 7 Aql and 8 Aql; new CCD photometric data were acquired in 2007. We present a period analysis on these data that confirm the dominant modes detected in each star in the framework of the STEPHI XII campaign in 2003.Comment: 9 pages, 7 figures, 2 tables; Accepted for publication in Communications in Asteroseismology, Vol 153, 200

Multi-site observations of Delta Scuti stars 7 Aql and 8 Aql (a new Delta Scuti variable): The twelfth STEPHI campaign in 2003

We present an analysis of the pulsation behaviour of the Delta Scuti stars 7 Aql (HD 174532) and 8 Aql (HD 174589) -- a new variable star -- observed in the framework of STEPHI XII campaign during 2003 June--July. 183 hours of high precision photometry were acquired by using four-channel photometers at three sites on three continents during 21 days. The light curves and amplitude spectra were obtained following a classical scheme of multi-channel photometry. Observations in different filters were also obtained and analyzed. Six and three frequencies have been unambiguously detected above a 99% confidence level in the range 0.090 mHz--0.300 mHz and 0.100 mHz-- 0.145 mHz in 7 Aql and 8 Aql respectively. A comparison of observed and theoretical frequencies shows that 7 Aql and 8 Aql may oscillate with p modes of low radial orders, typical among Delta Scuti stars. In terms of radial oscillations the range of 8 Aql goes from n=1 to n=3 while for 7 Aql the range spans from n=4 to n=7. Non-radial oscillations have to be present in both stars as well. The expected range of excited modes according to a non adiabatic analysis goes from n=1 to n=6 in both stars.Comment: 8 pages, 7 fugures, 5 tables, accepted for publication in Astronomical Journa

Inert states of spin-S systems

We present a simple but efficient geometrical method for determining the inert states of spin-S systems. It can be used if the system is described by a spin vector of a spin-S particle and its energy is invariant in spin rotations and phase changes. Our method is applicable to an arbitrary S and it is based on the representation of a pure spin state of a spin-S particle in terms of 2S points on the surface of a sphere. We use this method to find candidates for some of the ground states of spinor Bose-Einstein condensates.Comment: 4 pages, 2 figures, minor changes, references added, typos correcte

Single File Diffusion of particles with long ranged interactions: damping and finite size effects

We study the Single File Diffusion (SFD) of a cyclic chain of particles that cannot cross each other, in a thermal bath, with long ranged interactions, and arbitrary damping. We present simulations that exhibit new behaviors specifically associated to systems of small number of particles and to small damping. In order to understand those results, we present an original analysis based on the decomposition of the particles motion in the normal modes of the chain. Our model explains all dynamic regimes observed in our simulations, and provides convincing estimates of the crossover times between those regimes.Comment: 30 pages, 9 figure