Pascal's Triangle Fractal Symmetries.

Divisibility and cellular automata

The mathematical characterization of patterns in Pascal’s triangle is useful and can be achieved by a variety of brute-force computations, but sometimes at a cost of the loss of an intuitive feeling for the regularities and irregularities in the structure. The graphic and mathematical approach described here reveals a visually striking and intricate class of patterns that make up a family of regular fractal networks known as Sierpinski gaskets. The figures indicate self-similarity of the gasket structures for several orders of dilational invariance.

We analyze chiral symmetry breaking in quark matter in an external magnetic field at zero and finite temperature and quark chemical potential. We first give a brief overview of analytic results within the mean-field approximation. There the critical temperature for chiral restoration is increased by the magnetic field effect. Then we investigate the effects of matter and quantum fluctuations on the Magnetic Catalysis. More specifically, we compute the critical coupling as a function of the magnetic field and the temperature for zero and finite quark chemical potential in the presence of quantum fluctuations. As soon as a non-zero temperature and/or density is turned on, long-range correlations are screened and the critical coupling is no longer vanishing. We extend our dynamical results beyond the leading-order bubble resummation which results in a non-local four-Fermi coupling. This includes in-medium meson effects on the more quantitative level.

The present paper focuses on the order-disorder transition of an Ising model on a self-similar lattice. We present a detailed numerical study, based on the Monte Carlo method in conjunction with the finite-size scaling method, of the critical properties of the Ising model on some two-dimensional deterministic fractal lattices with different Hausdorff dimensions. Those with finite ramification order do not display ordered phases at any finite temperature, whereas the lattices with infinite connectivity show genuine critical behavior. In particular, we considered two Sierpinski carpets constructed using different generators and characterized by Hausdorff dimensions ${d}_{H}=\mathrm{ln}8/\mathrm{ln}3=1.8927\dots{}$ and ${d}_{H}=\mathrm{ln}12/\mathrm{ln}4=1.7924\dots{},$ respectively. The data show in a clear way the existence of an order-disorder transition at finite temperature in both Sierpinski carpets. By performing several Monte Carlo simulations at different temperatures and on lattices of increasing size in conjunction with a finite-size scaling analysis, we were able to determine numerically the critical exponents in each case and to provide an estimate of their errors. Finally, we considered the hyperscaling relation and found indications that it holds, if one assumes that the relevant dimension in this case is the Hausdorff dimension of the lattice.

A new set of fractal multiband antennas called mod-p Sierpinski gaskets is presented. Mod-p Sierpinski fractal antennas derive from the Pascal triangle and present a log-periodic behavior, which is a consequence of their self-similarity properties. Mod-p Sierpinski fractal antennas constitute a generalization of the classical Sierpinski antenna.

Meson spectral functions at finite temperature

We review the recent results on the energy level statistics in disordered conductors near the Anderson localization transition and other critical systems with multifractal eigenstates. We focus on the critical random matrix ensemble (CRMT) suggested by Mirlin and Fyodorov as a possible generic description of the critical parametric spectral statistics (CPSS). We show that in the region of large energy and/or parameter separations and weak multifractality this CRMT is equivalent to the Luttinger liquid of energy levels at a finite temperature with the interaction between the fictitious 1D fermions (energy levels) being dependent on the Dyson symmetry parameter β = 1, 2, 4 and the temperature being proportional to the multifractality exponent η = 1 — D2. We show that the CRMT is the simplest extension of the classical Wigner-Dyson random matrix ensemble for the case of finite dimensionless conductance g ≫ 1 and suggest the form of the two-level correlation functions for all the symmetry classes and all energy separations. We argue that these correlation functions coincide with the density correlation functions for the Calogero-Sutherland model at finite temperatures.

Broadly, fractals are sets that exhibit a repeating pattern at multiple scales. One important fractal set is the Sierpinski Gasket (SG) which is made up of nested equilateral triangles. A variation of the classic Sierpinski gasket is to create n-levels with the equilateral triangles. Another variation is to stretch the points of intersection for the triangles in SG into line segments of length 0 < α < 1/3. When one combines these variations, one arrives at the stretched level-n Sierpinski gaskets (SSGn) which are the focus of this work. We give an introduction to iterated function systems (IFS) and determine an IFS which generates SSG3. We then describe how one can acquire the IFS for SSGn in general, and conclude with a theorem which determines the Hausdorff dimension for SSGn.

The properties of the vector meson current-current correlation function and its spectral representation are investigated in details with and without isoscalar-vector interaction within the framework of effective QCD approach, namely Nambu-Jona-Lasinio (NJL) model and its Polyakov Loop extended version (PNJL), at finite temperature and finite density. The influence of the isoscalarvector interaction on the vector meson correlator is obtained using the ring resummation known as the Random Phase Approximation (RPA). The spectral as well as the correlation function in PNJL model show that the vector meson retains its bound property up to a moderate value of temperature above the phase transition. Using the vector meson spectral function we, for the first time, obtained the dilepton production rate from a hot and dense medium within the framework of PNJL model that takes into account the nonperturbative effect through the Polyakov Loop fields. The dilepton production rate in PNJL model is enhanced compared to NJL and Born rate in the deconfined phase due to the suppression of color degrees of freedom at moderate temperature. The presence of isoscalar-vector interaction further enhances the dileption rate over the Born rate in the low mass region. Further, we also have computed the Euclidean correlation function in vector channel and the conserved density fluctuation associated with temporal correlation function appropriate for a hot and dense medium. The dilepton rate and the Euclidean correlator are also compared with available lattice data and those quantities in PNJL model are found to agree well in certain domain.

Towards the application of the maximum entropy method to finite temperature upsilon spectroscopy

This paper presents the development of a multiband fractal antenna based on the Sierpinski triangle. By changing the structure, upper bands may be selected over a number of multiple frequencies. This property is based on Pascal triangle properties. The paper also develops an explanation of why fractal antennas have constant radiation properties at multiband frequencies. Finally, some impedance and radiation measurements are presented.

Wilson-Polyakov loops for critical strings and superstrings at finite temperature

Results are presented for the static and time-dependent behavior of the classical one-dimensional XY-model at finite temperatures. Equilibrium properties were determined using an importance sampling Monte Carlo method. A spin-dynamics computer simulation method was then used to evaluate spin-spin and energy-energy time-displaced correlation functions out to time t=10J−1 using initial states chosen from the equilibrium distribution. The transverse spin-spin auto correlation function decreases smoothly as a function of time. The longitudinal spin-spin correlation functions, however, show temperature dependent oscillatory behavior. At moderately high temperature the energy-energy correlations show diffusive t−1/2 behavior at long times with only a slow temperature dependence of the diffusion constant.

We consider the one-dimensional delta-interacting electron gas in the case of infinite repulsion. We use determinant representations to study the long time, large distance asymptotics of correlation functions of local fields in the gas phase. We derive differential equations which drive the correlation functions. Using a related Riemann–Hilbert problem we obtain formulae for the asymptotics of the correlation functions, which are valid at all finite temperatures. At low temperatures these formulae lead to explicit asymptotic expressions for the correlation functions, which describe power law behavior and exponential decay as functions of temperature, magnetic field and chemical potential.

In this paper we compare the construction of equilibrium states at finite temperature for self-interacting massive scalar quantum field theories on Minkowski spacetime proposed by Fredenhagen and Lindner with results obtained in ordinary thermal field theory, by means of real time and Matsubara formalisms. In the construction of this state, even if the adiabatic limit is considered, the interaction Lagrangian is multiplied by a smooth time cut-off. In this way the interaction starts adiabatically and the correlation functions are free from divergences. The corresponding interaction Hamiltonian is a local interacting field smeared over the interval of time where the chosen cut-off is not constant. In order to cope with this smearing, the Matsubara propagator needs to be modified. We obtain an expansion of the correlation functions of the interacting equilibrium state as a sum over certain type of graphs with mixed edges, some of them correspond to modified Matsubara propagators and others to propagators of the real time formalism. An integration over the adiabatic time cut-off is present in every vertex. However, at every order in perturbation theory, the final result does not depend on the particular form of the cut-off function. The obtained graphical expansion contains in it both the real time formalism and the Matsubara formalism as particular cases. Finally, we show that a particular factorisation which is used to derive the ordinary real time formalism holds only in special cases and we present a counterexample. We conclude with the analysis of certain correlation functions and we notice that corrections to the self-energy in a $\lambda\phi^4$ at finite temperature theory are expected.