Logarithmic Oscillations Research Articles

Power Laws and Logarithmic Oscillations in Diffusion Processes on Discrete Ultrametric Spaces

Power Laws and Logarithmic Oscillations in Diffusion Processes on Discrete Ultrametric Spaces

Log-periodic gravitational-wave background beyond Einstein gravity

Periodic patterns of logarithmic oscillations can arise in primordial curvature perturbation spectra and in the associated gravitational-wave background via different mechanisms. We show that, in the presence of log oscillations, the spectral shape of the stochastic background has a unique parametrization independent of its physical origin. We also show that this log-periodic modulation can be generated in any scenario beyond Einstein gravity endowed with an approximate discrete scale invariance, a symmetry typical of deterministic fractal spacetimes that could emerge in quantum gravity under certain conditions. We discuss how a log-oscillatory spectral shape arises from concrete inflationary models beyond Einstein gravity and the prospects for detection in Einstein Telescope and other next-generation gravitational-wave observatories. We find that these instruments will be sensitive to log-periodic features if the detection is made with a high signal-to-noise ratio (SNR) and that the error scales as .

Coexistence of logarithmic and SdH quantum oscillations in ferromagnetic Cr-doped tellurium single crystals

We report the synthesis of transition-metal-doped ferromagnetic elemental single-crystal semiconductors with quantum oscillations using the physical vapor transport method. The 7.7 atom% Cr-doped Te crystals (Cr:Te) show ferromagnetism, butterfly-like negative magnetoresistance in the low temperature (<3.8 K) and low field (<0.15 T) region, and high Hall mobility, e.g. 1320 cm2 V−1 s−1 at 30 K and 350 cm2 V−1 s−1 at 300 K, implying that Cr:Te crystals are ferromagnetic elemental semiconductors. When B // [001] // I, the maximum negative MR is ∼−27% at T = 20 K and B = 8 T. In the low temperature semiconducting region, Cr:Te crystals show strong discrete scale invariance dominated logarithmic quantum oscillations when the direction of the magnetic field B is parallel to the [100] crystallographic direction (B // [100]) and show Landau quantization dominated Shubnikov-de Haas oscillations for B // [210] direction, which suggests the broken rotation symmetry of the Fermi pockets in the Cr:Te crystals. The findings of coexistence of multiple quantum oscillations and ferromagnetism in such an elemental quantum material may inspire more study of narrow bandgap semiconductors with ferromagnetism and quantum phenomena.

EXPERIMENTAL METHOD FOR STRUCTURAL CONCRETE DAMPING PROPERTIES EVALUATION

This article proposes a solution for structural materials such as concrete and cement mortars dynamic properties investigation using experimental modal analysis technique. The studied dynamic characteristics of structural materials include the dynamic modulus of elasticity and the loss factor or its derivatives: the logarithmic oscillation decrement or the relative damping coefficient. Closed expressions are presented for determining the loss factor of mechanical vibrations, obtained on the basis of solving the differential equation for vibrations of a single-mass dynamic system. A method for calculating the loss factor based on the analysis of the spectrum of the transfer function of an oscillatory system loaded with an impulsive dynamic force is presented, in which the results of measuring accelerations at various points of the sample are used as a response. The experiments were carried out on short and long samples made from samples of structural materials - cement mortars with a density of 1500 - 1900 kg/m3 with special aggregates. Based on the solution of the equation of oscillations of a beam with distributed masses, a formula is presented for determining the dynamic modulus of elasticity of the beam material.

Holomorphic semigroups and Sarason’s characterization of vanishing mean oscillation

It is a classical theorem of Sarason that an analytic function of bounded mean oscillation (BMOA) is of vanishing mean oscillation if and only if its rotations converge in norm to the original function as the angle of the rotation tends to zero. In a series of two papers, Blasco et al. have raised the problem of characterizing all semigroups of holomorphic functions (\varphi_{t}) that can replace the semigroup of rotations in Sarason’s theorem. We give a complete answer to this question, in terms of a logarithmic vanishing oscillation condition on the infinitesimal generator of the semigroup (\varphi_{t}) . In addition, we confirm the conjecture of Blasco et al. that all such semigroups are elliptic. We also investigate the analogous question for the Bloch and the little Bloch spaces, and surprisingly enough, we find that the semigroups for which the Bloch version of Sarason’s theorem holds are exactly the same as in the BMOA case.

Modeling of In0.17Ga0.83N/InxGa1–xN/AlyGa1–yN light emitting diode structure on ScAlMgO4 (0001) substrate for high intensity red emission

An In0.17Ga0.83N light emitting diode (LED) structure on ScAlMgO4 (0001) substrate is modeled for high intensity red emission. The high indium composition (In &gt; 15%) inside the c-plane polar quantum well (QW) for longer wavelength emission degrades the structural and optical properties of LEDs because of induced strain energy and quantum confinement Stark effect. To compensate these effects, it has been demonstrated by simulation that an AlyGa1–yN cap layer of 2 nm thick and Al composition of 17% deposited onto QW of 3 nm thick and In composition of 35% will allow to have less defect density and higher intensity red emission at 663 nm than that of In0.17Ga0.83N/InxGa1–xN LEDs grown on ScAlMgO4 (0001) substrate. This LED structure has perfect in-plane equilibrium lattice parameter (αeq = 3.249 Å) and higher logarithmic oscillator strength (Γ = –0.93) values.

Non-linear damping of superimposed primordial oscillations on the matter power spectrum in galaxy surveys

Galaxy surveys are an important probe for superimposed oscillations on the primordial power spectrum of curvature perturbations, which are predicted in several theoretical models of inflation and its alternatives. In order to exploit the full cosmological information in galaxy surveys it is necessary to study the matter power spectrum to fully non-linear scales. We therefore study the non-linear clustering in models with superimposed linear and logarithmic oscillations to the primordial power spectrum by running high-resolution dark-matter-only N-body simulations. We fit a Gaussian envelope for the non-linear damping of superimposed oscillations in the matter power spectrum to the results of the N-body simulations for k ≲ 0.6 h/Mpc at 0 ⩽ z ⩽ 5 with an accuracy below the percent. We finally use this fitting formula to forecast the capabilities of future galaxy surveys, such as Euclid and Subaru, to probe primordial oscillation down to non-linear scales alone and in combination with the information contained in CMB anisotropies.

Dynamic phase transition in the contact process with spatial disorder: Griffiths phase and complex persistence exponents.

We present a model which displays the Griffiths phase, i.e., algebraic decay of density with continuously varying exponents in the absorbing phase. In the active phase, the memory of initial conditions is lost with continuously varying complex exponents in this model. This is a one-dimensional model where a fraction r of sites obey rules leading to the directed percolation class and the rest evolve according to rules leading to the compact directed percolation class. For infection probability p≤p_{c}, the fraction of active sites ρ(t)=0 asymptotically. For p>p_{c},ρ(∞)>0. At p=p_{c},ρ(t), the survival probability from a single seed and the average number of active sites starting from single seed decay logarithmically. The local persistence P_{l}(∞)>0 for p≤p_{c} and P_{l}(∞)=0 for p>p_{c}. For p≥p_{s}, local persistence P_{l}(t) decays as a power law with continuously varying exponents. The persistence exponent is clearly complex as p→1. The complex exponent implies logarithmic periodic oscillations in persistence. The wavelength and the amplitude of the logarithmic periodic oscillations increase with p. We note that the underlying lattice or disorder does not have a self-similar structure.

Shubin Regularity for the Radially Symmetric Spatially Homogeneous Boltzmann Equation with Debye-Yukawa Potential

In this work, we study the Cauchy problem for the radially symmetric spatially homogeneous Boltzmann equation with Debye-Yukawa potential. We prove that this Cauchy problem enjoys the same smoothing effect as the Cauchy problem defined by the evolution equation associated to a fractional logarithmic harmonic oscillator. To be specific, we can prove the solution of the Cauchy problem belongs to Shubin spaces.

Power-law modulation of the scalar power spectrum from a heavy field with a monomial potential

The effects of heavy fields modulate the scalar power spectrum during inflation. We analytically calculate the modulations of the scalar power spectrum from a heavy field with a separable monomial potential, i.e. V(ϕ)∼ ϕn. In general the modulation is characterized by a power-law oscillation which is reduced to the logarithmic oscillation in the case of n=2.

Violation of the virial theorem and generalized equipartition theorem for logarithmic oscillators serving as a thermostat

A logarithmic oscillator has been proposed to serve as a thermostat recently since it has a peculiar property of infinite heat capacity according to the virial theorem. In order to examine its feasibility in numerical simulations, a modified logarithmic potential has been applied in previous studies to eliminate the singularity at the origin. The role played by the modification has been elucidated in the present study. We argue that the virial theorem is practically violated in finite-time simulations of the modified log-oscillator illustrated by a linear dependence of kinetic temperature on energy. Furthermore, as far as a thermalized log-oscillator is concerned, our calculation based on the canonical ensemble average shows that the generalized equipartition theorem is broken if the temperature is higher than a critical temperature. Finally, we show that log-oscillators fail to serve as thermostats for their incapability of maintaining a nonequilibrium steady state even though their energy is appropriately assigned.

Lorentz violations in multifractal spacetimes

Using the recent observation of gravitational waves (GW) produced by a black-hole merger, we place a lower bound on the energy above which a multifractal spacetime would display an anomalous geometry and, in particular, violations of Lorentz invariance. In the so-called multifractional theory with q-derivatives, we show that the deformation of dispersion relations is much stronger than in generic quantum-gravity approaches (including loop quantum gravity) and, contrary to the latter, present observations on GWs can place very strong bounds on the characteristic scales at which spacetime deviates from standard Minkowski. The energy at which multifractal effects should become apparent is E_{*}>10^{14}~{mathrm{GeV}} (thus improving previous bounds by 12 orders of magnitude) when the exponents in the measure are fixed to their central value 1 / 2. We also estimate, for the first time, the effect of logarithmic oscillations in the measure (corresponding to a discrete spacetime structure) and find that they do not change much the bounds obtained in their absence, unless the amplitude of the oscillations is fine tuned. This feature, unavailable in known quantum-gravity scenarios, may help the theory to avoid being ruled out by gamma-ray burst (GRB) observations, for which E_{*}> 10^{17}~{mathrm{GeV}} or greater.

Geometries Induced by Logarithmic Oscillations as Examples of Gromov Hyperbolic Spaces

We explore the connection between the geometries generated by logarithmic oscillations and the class of metric spaces satisfying the Gromov hyperbolicity condition. We investigate the most fundamental examples, inspired from classical geometries, e.g. the Euclidean distance on the infinite strip or Hilbert’s distance on the unit disk. We continue our study with the Barbilian’s distance, which historically appeared as a natural extension of a model of hyperbolic geometry. We introduce and investigate a new metric, called the stabilizing metric. In a natural development, we explore a class of extensions of this distance which, under some analytic conditions, produces infinitely many new examples of Gromov hyperbolic metric spaces. Using similar procedures, we construct Vuorinen’s stabilizing metric and its extensions, and we discuss their Gromov hyperbolicity.

Single-file diffusion on self-similar substrates

.We study the single file diffusion problem on a one-dimensional lattice with a self-similar distribution of hopping rates. We find that the time dependence of the mean-square displacement of both a tagged particle and the center of mass of the system present anomalous power laws modulated by logarithmic periodic oscillations. The anomalous exponent of a tagged particle is one half of the exponent of the center of mass, and always smaller than 1/4. Using heuristic arguments, the exponents and the periods of oscillation are analytically obtained and confirmed by Monte Carlo simulations.

Logarithmic Mean Oscillation on the Polydisc, Multi-Parameter Paraproducts and Iterated Commutators

We introduce another notion of bounded logarithmic mean oscillation in the \(N\)-torus and give an equivalent definition in terms of boundedness of multi-parameter paraproducts from the dyadic little \(\mathrm {BMO}\), \(\mathrm {bmo}^d(\mathbb {T}^N)\) to the dyadic product \(\mathrm {BMO}\) space, \(\mathrm {BMO}^d(\mathbb {T}^N)\). We also obtain a sufficient condition for the boundedness of the iterated commutators from the subspace of \(\mathrm {bmo}(\mathbb {R}^N)\) consisting of functions with support in \([0,1]^N\) to \(\mathrm {BMO}(\mathbb {R}^N)\).

Searching for oscillations in the primordial power spectrum. I. Perturbative approach

In this first of two papers, we present a new method for searching for oscillatory features in the primordial power spectrum. A wide variety of models predict these features in one of two different flavors: logarithmically spaced oscillations and linearly spaced oscillations. The proposed method treats the oscillations as perturbations on top of the scale-invariant power spectrum, allowing us to vary all cosmological parameters. This perturbative approach reduces the computational requirements for the search as the transfer functions and their derivatives can be precomputed. We show that the most significant degeneracy in the analysis is between the distance to last scattering and the overall amplitude at low frequencies. For models with logarithmic oscillations, this degeneracy leads to an uncertainty in the phase. For linear-spaced oscillations, it affects the frequency of the oscillations. In this first of two papers, we test our code on simulated Planck-like data and show that we are able to recover fiducial input oscillations with an amplitude of a few times $\mathcal{O}({10}^{\ensuremath{-}2})$. We apply the code to WMAP9-year data and confirm the existence of two intriguing resonant frequencies for log-spaced oscillations. For linear-spaced oscillations, we find a single resonance peak. We use numerical simulations to assess the significance of these features and conclude that the data do not provide compelling evidence for the existence of oscillatory features in the primordial spectrum.

Failure of logarithmic oscillators to serve as a thermostat for small atomic clusters.

A logarithmic oscillator has the outstanding property that the expectation value of its kinetic energy is constant for all stationary states. Recently the ansatz that this property can be used to define a Hamiltonian thermostat has been put forward and a suggestion has been made that this logarithmic oscillator weakly coupled to a small system would serve as a thermostat as long as few degrees of freedom are involved as is the case in atomic clusters. We have applied these ideas to a cluster of four Lennard-Jones atoms and inspected two different models of coupling between the cluster and the logarithmic oscillator in three dimensions. In both cases we show that there is a clear generation of kinetic motion of the cluster center of mass, but that kinetic energy due to interatomic vibrations is not significantly affected by coupling to the logarithmic oscillator. This is a failure of the published ansatz, as the logarithmic oscillator is unable to modify the kinetic energy due to vibrations in small atomic clusters.

Angular correlation functions for models with logarithmic oscillations

There exist several theoretical motivations for primordial correlation functions (such as the power spectrum) to contain oscillations as a logarithmic function of comoving momentum k. While these features are commonly searched for in k-space, an alternative is to use angular space; that is, search for correlations between the directional vectors of observation. We develop tools to efficiently compute the angular correlations based on a stationary phase approximation and examine several example oscillations in the primordial power spectrum, bispectrum, and trispectrum. We find that logarithmically-periodic oscillations are essentially featureless and therefore difficult to detect using the standard correlator, though others might be feasible.

Multi-scale gravity and cosmology

The gravitational dynamics and cosmological implications of three classes of recently introduced multi-scale spacetimes (with, respectively, ordinary, weighted and q-derivatives) are discussed. These spacetimes are non-Riemannian: the metric structure is accompanied by an independent measure-differential structure with the characteristics of a multi-fractal, namely, different dimensionality at different scales and, at ultra-short distances, a discrete symmetry known as discrete scale invariance. Under this minimal paradigm, five general features arise: (a) the big-bang singularity can be replaced by a finite bounce, (b) the cosmological constant problem is reinterpreted, since accelerating phases can be mimicked by the change of geometry with the time scale, without invoking a slowly rolling scalar field, (c) the discreteness of geometry at Planckian scales can leave an observable imprint of logarithmic oscillations in cosmological spectra and (d) give rise to an alternative mechanism to inflation or (e) to a fully analytic model of cyclic mild inflation, where near scale invariance of the perturbation spectrum can be produced without strong acceleration. Various properties of the models and exact dynamical solutions are discussed. In particular, the multi-scale geometry with weighted derivatives is shown to be a Weyl integrable spacetime.

$BMO$ estimates for nonvariational operators with discontinuous coefficients structured on Hörmander's vector fields on Carnot groups

We consider the class of operators \[ Lu=\sum_{i,j=1}^{q}a_{ij}(x)X_{i}X_{j}u, \] where $X_{1},X_{2},\dots,X_{q}$ are homogeneous left-invariant Hörmander's vector fields on $\mathbb{R}^{N}$ with respect to a structure of Carnot group, $q\leq N,$ the matrix $\{ a_{ij}\} $ is symmetric and uniformly positive on $\mathbb{R}^{q},$ the coefficients $a_{ij} $ belong to $L^{\infty}\cap VLMO_{loc}( \Omega) $ ("vanishing logarithmic mean oscillation") with respect to the distance induced by the vector fields (in particular, they can be discontinuous), and $\Omega$ is a bounded domain of $\mathbb{R}^{N}$. We prove local estimates in $BMO_{loc}\cap L^{p}$ of the following kind: \begin{align*} & \Vert X_{i}X_{j}u\Vert _{BMO_{loc}^{p}( \Omega^{\prime }) }+\Vert X_{i}u\Vert _{BMO_{loc}^{p}( \Omega ^{\prime}) } \\ & \leq c\big\{ \Vert Lu\Vert _{BMO_{loc}^{p}( \Omega) }+\Vert u\Vert _{BMO_{loc}^{p}( \Omega) }\big\} \end{align*} for any $\Omega^{\prime}\Subset\Omega$, $1 < p < \infty$. Even in the uniformly elliptic case $X_{i}=\partial_{x_{i}}$, $q=N$ our estimates improve the known results.