Asymptotic Behavior of Beta-Integers

Beta-integers (“β-integers”) are those numbers which are the counterparts of integers when real numbers are expressed in an irrational base β > 1. In quasicrystalline studies, β-integers supersede the “crystallographic” ordinary integers. When the number β is a Parry number, the corresponding β-integers realize only a finite number of distances between consecutive elements and are in this sense the most comparable to ordinary integers. In this paper, we point out the similarity of β-integers and ordinary integers in the asymptotic sense, in particular for a subclass of Parry numbers – Pisot numbers for which their Parry and minimal polynomial coincide.     

Secondary structure prediction of beta-hairpin peptide tryptophan zipper-I

We have investigated the folding properties of tryptophan zipper-I molecule which folds into a stable β-hairpin motif in aqueous solution as suggested by nuclear magnetic resonance (NMR) experiments. An all-atom presentation, including hydrogen, was used with an implicit solvent. As a simulation technique, simulated tempering algorithm was used to obtain equilibrium conformations of the molecule at ten distinct temperatures. Our minimum energy configuration obtained from simulated tempering algorithm is a β-hairpin motif with 1.30 Å backbone root-mean-square deviation from the reference PDB structure (1le0.pdb). Several quantities and exhaustive folding free energy landscapes were determined and discussed in order to clarify the folding behavior.     

Non-Equilibrium Steady States of the XY Chain

We study the non-equilibrium statistical mechanics of the two-sided XY chain. We start from an initial state in which the left and right part of the lattice, $$\mathbb{Z}_{\text{L}} = \{ x \in \mathbb{Z}|x < - M\} ,{\text{ }}\mathbb{Z}_{\text{R}} = \{ x \in \mathbb{Z}|x >M\} ,$$ are at inverse temperatures β _L and β _R. Using a simple scattering theoretic analysis, we construct the unique non-equilibrium steady state (NESS). This state depends on β _L and β _R, but not on the choice of the decoupling parameter M. We prove that in the non-equilibrium case, β _L≠β _R, this state has strictly positive entropy production.     

Glueball masses and string tension Smeared loop-loop correlation functions

By using results obtained by the Ape Collaboration [1] we study the glueball spectrum and the string tension in lattice QCD by analyzing correlation functions between operators. We use and discuss the smearing method. We consider a 5³ × 16 lattice at β = 5.6 as a test case, and a 12³ × 32 lattice at β = 5.9 in order to compare with the old wall results of ref. [3]. We study the structure of the excited states.     

Splitting of folded strings in AdS 4×CP 3

We investigate the dynamics of generalized tachyon field in FRW spacetime. We obtain the autonomous dynamical system for the general case. Because the general autonomous dynamical system cannot be solved analytically, we discuss two cases in detail: β=1 and β=2. We find the critical points and study their stability. At these critical points, we also consider the stability of the generalized tachyon field, which is as important as the stability of critical points. The possible final states of the universe are discussed.     

Deposition of particles under gravity in a kinetic lattice model

We study the growth through particle deposition of the surface of a discrete two-dimensional system, in which the motion of particles is affected by infinite gravity and the Kob-Andersen kinetic rule. Computer simulation results are found to be consistent with previous results in literature, showing that this particular case belongs to the same universality class as Ballistic Deposition, the Eden model, one step solid-on-solid (SOS) deposition and Kardar-Parisi-Zhang (KPZ) characterized by scaling exponents α=0.5, β=1/3=0.33 and z=α/β=1.5.     

The Spectrum of the Cubic Oscillator

We prove the simplicity and analyticity of the eigenvalues of the cubic oscillator Hamiltonian, $$\begin{array}{ll}H(\beta)=-\frac{d^2}{dx^2}+x^2+i\sqrt{\beta}x^3\end{array}$$ , for β in the cut plane ${\mathcal{C}_c:=\mathcal{C}\backslash \mathcal{R}_-}$ . Moreover, we prove that the spectrum consists of the perturbative eigenvalues {E _n (β)} _{n ≥ 0} labeled by the constant number n of nodes of the corresponding eigenfunctions. In addition, for all ${\beta \in \mathcal{C}_c, E_n(\beta)}$ can be computed as the Stieltjes-Padé sum of its perturbation series at β = 0. This also gives an alternative proof of the fact that the spectrum of H(β) is real when β is a positive number. This way, the main results on the repulsive PT-symmetric and on the attractive quartic oscillators are extended to the cubic case.     

Instanton zero modes and β-functions in gauged supergravity

We prove that one loop infinities in N?3 gauged O(N) supergravities are produced only by zero modes both on de Sitter and gauge instanton backgrounds. This strongly suggests (in analogy with the super Yang-Mills case) that instantons may be used for establishing exact β-functions in supergravities. We also derive the general formula for the number of gauge instanton zero modes for arbitrary spin and find that the “magnetic moment” part of the contribution in the gauge β-function is completely due to zero modes.     

Neutrinoless double-beta decay and double-electron capture

The fundamental importance of searching for neutrinoless double-beta decay is widely recognized. Observation of the decay would tell us that the total lepton number is not conserved and that, consequently, neutrinos are massive Majorana fermions. The same statement could be made in the case of observing neutrinoless double-electron capture. We address the question of the sensitivity of the 0νεε decay to the effective mass of the Majorana neutrino. According to our estimates, in the case of ¹⁵²Gd and ¹⁶⁴Er the sensitivity can be comparable to the favored 0νββ decays of nuclei. The main uncertainty in the prediction of half-lives of the 0νεε decay stems from the lack of sufficient precision in measuring the mass difference between the parent and daughter atoms. More accurate measurements can be accomplished using the modern high-precision ion traps.     

Prediction of accidental cancellation of different deformation components and optimum fusion orientations

In view of the role of nuclear deformations in the fusion between spherical and deformed nuclei to form super-heavy elements, we try to understand how a cancellation of the different nuclear deformations could arise. We first investigated the correlation between the orientation variation of the deformed nucleus radius and the orientation Coulomb barrier distribution in presence of the higher order deformation components, β₆ and β₈, in addition to the lower order ones. This correlation has been reported in our previous work (Ismail and Seif, 2010) [1] in presence of the lower order (β₂, β₃ and β₄) deformations. Even if there are higher deformations, we found here that the simple expression which describes the deformed target nucleus can be used to predict with good accuracy the behavior of the fusion Coulomb barrier with both orientation and deformation as well as the optimum (cold or hot) fusion configurations. It can predict the orientations of compact and elongated configurations of the interaction and whether they are equatorial or polar or none of them. The value and sign of the deformation parameters ratios with respect to one of them have been used to classify these configurations. We applied the same correlation to predict successfully the mutual cancellation effects between the different deformation components up to β₈. Illustrative examples are given in which the cancellation, at some orientations, brings the fusion barrier back to the spherical case or keeps only the effect of quadrupole deformation, or the effects of both β₂ and β₄.     

Self-avoiding and planar random surfaces on the lattice

We study models of self-avoiding (SARS) and of planar (PRS) random surfaces on a (hyper-) cubic lattice. If N_γ(A) is the number of such surfaces with given boundary γ and area A, then N_γ(A) = exp(β₀A + o(A)), where β₀ is independent of γ. We prove that, for β > β₀, the string tension is finite for the SARS model and strictly positive for the PRS model and that in both models the correlation length (inverse mass) is positive and finite. We discuss the possibility of the existence of a critical point and of a roughening transition. Estimates on intersection probabilities for random surfaces and connections with lattice gauge theories are sketched.     

Activated Random Walkers: Facts, Conjectures and Challenges

We study a particle system with hopping (random walk) dynamics on the integer lattice ? ^d . The particles can exist in two states, active or inactive (sleeping); only the former can hop. The dynamics conserves the number of particles; there is no limit on the number of particles at a given site. Isolated active particles fall asleep at rate λ>0, and then remain asleep until joined by another particle at the same site. The state in which all particles are inactive is absorbing. Whether activity continues at long times depends on the relation between the particle density ζ and the sleeping rate λ. We discuss the general case, and then, for the one-dimensional totally asymmetric case, study the phase transition between an active phase (for sufficiently large particle densities and/or small λ) and an absorbing one. We also present arguments regarding the asymptotic mean hopping velocity in the active phase, the rate of fixation in the absorbing phase, and survival of the infinite system at criticality. Using mean-field theory and Monte Carlo simulation, we locate the phase boundary. The phase transition appears to be continuous in both the symmetric and asymmetric versions of the process, but the critical behavior is very different. The former case is characterized by simple integer or rational values for critical exponents (β=1, for example), and the phase diagram is in accord with the prediction of mean-field theory. We present evidence that the symmetric version belongs to the universality class of conserved stochastic sandpiles, also known as conserved directed percolation. Simulations also reveal an interesting transient phenomenon of damped oscillations in the activity density.     

Surface-plasmon-polariton waves guided by the uniformly moving planar interface of a metal film and dielectric slab

We explored the effects of relative motion on the excitation of surface-plasmon-polariton (SPP) waves guided by the planar interface of a metal film and a dielectric slab, both materials being isotropic and homogeneous. Electromagnetic phasors in moving and non-moving reference frames were related directly using the corresponding Lorentz transformations. Our numerical studies revealed that, in the case of a uniformly moving dielectric slab, the angle of incidence for SPP-wave excitation is highly sensitive to (i) the ratio β of the speed of motion to speed of light in free space and (ii) the direction of motion. When the direction of motion is parallel to the plane of incidence, the SPP wave is excited by p-polarized (but not s-polarized) incident plane waves for low and moderate values of β, while at higher values of β the total reflection regime breaks down. When the direction of motion is perpendicular to the plane of incidence, the SPP wave is excited by p-polarized incident plane waves for low values of β, but s-polarized incident plane waves at moderate values of β, while at higher values of β the SPP wave is not excited. In the case of a uniformly moving metal film, the sensitivity to β and the direction of motion is less obvious.     

On Gibbs Measures of Models with Competing Ternary and Binary Interactions and Corresponding von Neumann Algebras

In the present paper a model with competing ternary (J ₂) and binary (J ₁) interactions with spin values ±1, on a Cayley tree is considered. One studies the structure of Gibbs measures for the model considered. It is known, that under some conditions on parameters J ₁,J ₂ (resp. in the opposite case) there are three (resp. a unique) translation-invariant Gibbs measures. We prove, that two of them (minimal and maximal) are extreme in the set of all Gibbs measures and also construct two periodic (with period 2) and uncountable number of distinct non-translation-invariant Gibbs measures. One shows that they are extreme. Besides, types of von Neumann algebras, generated by GNS-representation associated with diagonal states corresponding to extreme periodic Gibbs measures, are determined. Namely, it is shown that an algebra associated with the unordered phase is a factor of type III _λ , where λ=exp{?2βJ ₂}, β>0 is the inverse temperature. We find conditions, which ensure that von Neumann algebras, associated with the periodic Gibbs measures, are factors of type III _δ , otherwise they have type III₁.     

Shifts of X-ray characteristic L lines of free atoms of rare-earth metals

Results of the study of X-ray characteristic spectra of free atoms of a number of rare-earth metals from Ce to Ho are reported. L _α1 and L _β1 lines were studied, and the shifts of the peaks of emission lines of free atoms relative to their positions in the case of emission by the solid aggregate state were measured. The data obtained enable one to use the corresponding X-ray characteristic lines as references with the accuracy Δλ/λ≈10^?5. The Z-dependences of shifts are plotted.     

High statistics study of the chiral condensate in quenched lattice QCD

We present the results of a high statistics study of the chiral condensate in quenched lattice QCD on an 8⁴ lattice at β = 5.4, 5.5, 5.6, 5.7, 5.8 and 6.0. We see clear evidence for deviation from asymptotic scaling in the range of β considered. Our results are in agreement with the behaviour anticipated from recent Monte Carlo renormalisation group studies of the β-function. We find indications of a common scaling behaviour for the condensate, the string tension and the deconfinement temperature.     

Fractional fermion number on a thermal ø4 soliton backgroud field at finite temperature

We employ a thermal kink soliton field as the background field and discuss the effect of the field on fractional fermion number 〈N〉_β at finite temperature. It is found that the fractional fermion charge will be restored at critical temperature T_c. The expressions of 〈N〉_β at the high-temperarute limit and the low-temperature limit are given.     

Thin Static Charged Dust Majumdar–Papapetrou Shells with High Symmetry in <Emphasis Type="Italic">D</Emphasis>≥4

We present a systematical study of static D≥4 space-times of high symmetry with the matter source being a thin charged dust hypersurface shell. The shell manifold is assumed to have the following structure \(\mathbb{S}_{\beta}\times\mathbb{R}^{D-2-\beta}\), β∈{0,…,D?2} is dimension of a sphere \(\mathbb{S}_{\beta}\). In case of β=0, we assume that there are two parallel hyper–plane shells instead of only one.The space-time has Majumdar–Papapetrou form and it inherits the symmetries of the shell manifold—it is invariant under both rotations of the \(\mathbb {S}_{\beta}\) and translations along ? ^D?2?β .We find a general solution to the Einstein–Maxwell equations with a given shell. Then, we examine some flat interior solutions with special attention paid to D=4. A connection to D=4 non-relativistic theory is pointed out. We also comment on a straightforward generalisation to the case of Kastor–Traschen space-time, i.e. adding a non-negative cosmological constant to the charged dust matter source.     

Maximum entropy with fluctuating constraints: The example of K-distributions

We indicate that in a maximum entropy setting, the thermodynamic β and the observation constraint are linked, so that fluctuations of the latter imposes fluctuations of the former. This gives an alternate viewpoint to ‘superstatistics’. While a Gamma model for fluctuations of the β parameter gives the so-called Tsallis distributions, we work out the case of a Gamma model for fluctuations of the observable, and show that this leads to K-distributions. We draw attention to the fact that these heavy-tailed distributions have high interest in physical applications, and we discuss them in some details.     

Approximately analytical solutions of the Manning-Rosen potential with the spin-orbit coupling term and spin symmetry

In this Letter the approximately analytical bound state solutions of the Dirac equation with the Manning-Rosen potential for arbitrary spin-orbit coupling quantum number k are carried out by taking a properly approximate expansion for the spin-orbit coupling term. In the case of exact spin symmetry, the associated two-component spinor wave functions of the Dirac equation for arbitrary spin-orbit quantum number k are presented and the corresponding bound state energy equation is derived. We study briefly two special cases; the general s-wave problem and the equal scalar and vector Manning-Rosen potential.