Arbitrary Density Research Articles

Approximate analytical methods for stationary response probability density of a SDOF system with a nonlinear spring under non-Gaussian random excitation

Approximate analytical methods for finding the stationary response probability density of a SDOF system with a wide class of nonlinear springs under non-Gaussian random excitation are proposed. The non-Gaussian excitation is a stationary stochastic process prescribed by an arbitrary probability density with finite variance and a power spectral density with a bandwidth parameter. In order to obtain the response probability distribution of the system, two approximate methods are developed by theoretically extending the Monte Carlo simulation observations on the properties of the response distribution obtained in the previous study. One of the methods is applied when the bandwidth of the excitation power spectrum is narrower than that of the primary resonance peak of the system. In this case, the system becomes quasi-static, and its equation of motion reduces to the equilibrium equation for the stiffness and excitation terms. This equilibrium equation is used to derive the approximate solution of the response distribution. The only requirement of this method is that the nonlinear spring of the system is continuous and monotonically increasing. The other method is proposed for the case of the wide excitation bandwidth compared to the system bandwidth. In this method, the exact solution of the response distribution under Gaussian white noise is utilized. The value of the white noise spectral density parameter in the solution is determined by the equivalent linearization. The second method is applicable to systems with arbitrary integrable nonlinear springs. In numerical examples, we consider three nonlinear systems whose spring characteristics are quite different. Besides, two types of non-Gaussian excitation probability densities with differences in shape are used. It is demonstrated that the proposed methods can reproduce the unique shape of the response probability density caused by the spring nonlinearity and the excitation non-Gaussianity. From the results, the range of the bandwidth ratio between the excitation and the system where the response distribution is accurately obtainable is also examined for each method.

X-ray attenuation of bone, soft and adipose tissue in CT from 70 to 140 kV and comparison with 3D printable additive manufacturing materials

Additive manufacturing and 3D printing are widely used in medical imaging to produce phantoms for image quality optimization, imaging protocol definition, comparison of image quality between different imaging systems, dosimetry, and quality control. Anthropomorphic phantoms mimic tissues and contrasts in real patients with regard to X-ray attenuation, as well as dependence on X-ray spectra. If used with different X-ray energies, or to optimize the spectrum for a certain procedure, the energy dependence of the attenuation must replicate the corresponding energy dependence of the tissues mimicked, or at least be similar. In the latter case the materials’ Hounsfield values need to be known exactly to allow to correct contrast and contrast to noise ratios accordingly for different beam energies. Fresh bovine and porcine tissues including soft and adipose tissues, and hard tissues from soft spongious bone to cortical bone were scanned at different energies, and reference values of attenuation in Hounsfield units (HU) determined. Mathematical model equations describing CT number dependence on kV for bones of arbitrary density, and for adipose tissues are derived. These data can be used to select appropriate phantom constituents, compare CT values with arbitrary phantom materials, and calculate correction factors for phantoms consisting of materials with an energy dependence different to the tissues. Using data on a wide number of additive manufacturing and 3D printing materials, CT numbers and their energy dependence were compared to those of the tissues. Two commercially available printing filaments containing calcium carbonate powder imitate bone tissues with high accuracy at all kV values. Average adipose tissue can be duplicated by several off-the-shelf printing polymers. Since suitable printing materials typically exhibit a too high density for the desired attenuation of especially soft tissues, controlled density reduction by underfilling might improve tissue equivalence.

Few-mode field quantization for multiple emitters.

The control of the interaction between quantum emitters using nanophotonic structures holds great promise for quantum technology applications, while its theoretical description for complex nanostructures is a highly demanding task as the electromagnetic (EM) modes form a high-dimensional continuum. We here introduce an approach that permits a quantized description of the full EM field through a small number of discrete modes. This extends the previous work in ref. (I. Medina, F. J. García-Vidal, A. I. Fernández-Domínguez, and J. Feist, “Few-mode field quantization of arbitrary electromagnetic spectral densities,” Phys. Rev. Lett., vol. 126, p. 093601, 2021) to the case of an arbitrary number of emitters, without any restrictions on the emitter level structure or dipole operators. The low computational demand of this method makes it suitable for studying dynamics for a wide range of parameters. We illustrate the power of our approach for a system of three emitters placed within a hybrid metallodielectric photonic structure and show that excitation transfer is highly sensitive to the properties of the hybrid photonic–plasmonic modes.

Existence of solutions for a viscous hybrid quantum system for arbitrary large current density

In this paper, we continue to analyse the properties of the hybrid model introduced in the works by Chiarelli et al. and Di Michele et al. In particular, we consider a quantum hybrid model with a special viscous functional, as introduced in the work by Gamba and Jüngel, for the quantum hydrodynamical model in its standard form (QHD). We show that the hybrid QHD (H-QHD) model admits weak solutions also for arbitrary large value of the current density J under a suitable pressure function.

Divergence of magnetic susceptibility in the SU( N ) Nagaoka-Thouless ferromagnet

Using finite-temperature strong-coupling expansions for the SU($N$) Hubbard model, we calculate the thermodynamic properties of the model in the infinite-$U$ limit for arbitrary density $0\ensuremath{\le}\ensuremath{\rho}\ensuremath{\le}1$ and all $N$. We express the ferromagnetic susceptibility of the model as a Curie term plus $\mathrm{\ensuremath{\Delta}}\ensuremath{\chi}$, an excess susceptibility above the Curie behavior. We show that, on a bipartite lattice, graph by graph the contributions to $\mathrm{\ensuremath{\Delta}}\ensuremath{\chi}$ are non-negative in the limit that the hole density $\ensuremath{\delta}=1\ensuremath{-}\ensuremath{\rho}$ goes to zero. By summing the contributions from all graphs consisting of closed loops we find that the low-hole-density ferromagnetic susceptibility diverges exponentially as $exp\mathrm{\ensuremath{\Delta}}/T$ as $T\ensuremath{\rightarrow}0$ in two and higher dimensions. This demonstrates that the Nagaoka-Thouless ferromagnetic state exists as a thermodynamic state of matter at low enough density of holes and sufficiently low temperatures. The constant $\mathrm{\ensuremath{\Delta}}$ scales with the SU($N$) parameter $N$ as $1/N$ implying that ferromagnetism is gradually weakened with increasing $N$ as the characteristic temperature scale for ferromagnetic order goes down.

A general analytical PBEM for solving three-dimensional transient inhomogeneous heat conduction problems with spatially varying heat generation

In the present work, a polygonal boundary element method (PBEM) for solving transient inhomogeneous heat conduction problems with spatially-varying heat generation is developed for the first time. A new general analytical method is proposed and employed in the present PBEM, by which the radial integrals associated with arbitrary spatially-varying density, specific heat, and thermal conductivity can be analytically computed, and the efficiency of the PBEM for solving the transient inhomogeneous problems can be significantly improved. The transient term is dealt with by a finite difference scheme. Three examples are designed to investigate the performance of the proposed method, and the results show that the present method could accurately and efficiently solve transient heat conduction problems with arbitrarily-varying thermal physical properties.

Broken axial symmetry as essential feature for a consistent modelling of various observables in heavy nuclei

Although most nuclear spectroscopy as well as atomic hyperfine structure data do not deliver accurate information on nuclear axiality the ad-hoc assumption of symmetry about one axis found widespread use in nuclear model calculations. In the theoretical interpretation of nuclear properties as well as in the analysis of experimental data triaxiality was considered – if at all – only for some, often exotic, nuclides. A breaking of axial symmetry combined to a spin-independent moment of inertia results in a surprisingly simple heuristic triaxial parametrization of the yrast sequence in all heavy nuclei, including well deformed ones. No additional fit parameters are needed in detailed studies of the mass and charge dependence of the electric dipole strength in the range of and outside of giant dipole resonances. Allowing triaxiality also avoids the introduction of an arbitrary level density parameter ã to fit the accurate values observed in n-capture experiments and ã can be taken from nuclear matter studies. A combination of this value to the yrast energies no longer based on axiality and the related I(I+1) rule results in agreement to data independent of spin. And predictions for radiative neutron capture as derived on the basis of non-axiality are improved as well. The experimentally favoured broken axial symmetry is in accord to HFB and MC-shell model calculations already for nuclei in the valley of stability.

ONE-BIT QUANTIZER PARAMETRIZATION FOR ARBITRARY LAPLACIAN SOURCES

In this paper we suggest an exact formula for the total distortion of one-bit quantizer and for the arbitrary Laplacian probability density function (pdf). Suggested formula additionally extends normalized case of zero mean and unit variance, which is the most applied quantization case not only in traditional quantization rather in contemporary solutions that involve quantization. Additionally symmetrical quantizer’s representation levels are calculated from minimal distortion criteria. Note that one-bit quantization is the most sensitive quantization from the standpoint of accuracy degradation and quantization error, thus increasing importance of the suggested parameterization of one-bit quantizer.

Influence of particle shape and material on the acoustic radiation force and microstreaming in a standing wave.

In view of its influence on the acoustic radiation force, we investigate the microstreaming around a small solid elastic particle in an ultrasonic standing wave in dependence of its material properties and shape. The configuration is axisymmetric, making it accessible to numerical methods, such as the finite element method. The results reveal a transition from viscous scattering- to microstreaming-dominated acoustic radiation force that depends on the particle density. When a deviation of the particle shape from a sphere becomes smaller than the viscous boundary layer thickness, we show that the influence of the shape on the viscous contributions to the acoustic radiation force diminishes, allowing the use of theoretical models for a spherical particle. However, extreme asymmetric shape perturbations, such as crowns with sharp edges, can give rise to noticeable viscous contributions for a dense particle that is larger than the viscous boundary layer thickness. We also introduce a hybrid analytical model for the acoustic radiation force on a spherical particle that accounts for the microstreaming and particle compressibility and shows a good agreement with numerical simulations for an arbitrary particle size and density.

Aligned signed-rank tests of a linear autoregressive model against an exponential autoregressive one

In this work, we have established the aligned signed-rank tests for testing the ordinary dependence against exponential autoregression [] model. The considered tests are asymptotically valid under an arbitrary symmetric innovation density, and locally asymptotically most stringent against the EXPAR alternatives associated with some predetermined symmetric density type. Several simulation was curried out to show the good performance of these tests and an application for modeling the annual sunspots number is provided.

Quantitative theory for spikes and bubbles in the Richtmyer−Meshkov instability at arbitrary density ratios in three dimensions

It is known that conducting numerical simulations and experiments for the shock-induced Richtmyer–Meshkov instability in three dimensions is much more difficult and time-consuming than that in two dimensions. Therefore, theories can play a more important role in the study of three-dimensional Richtmyer–Meshkov instability. We present analytical formulas for predicting the behavior of growth rate and amplitude of fingers at the unstable Richtmyer–Meshkov interface. Our theory is for both spikes and bubbles, for the arbitrary density ratio between the two fluids, and for the entire development process from early to late times.

A clustering algorithm based on density decreased chain for data with arbitrary shapes and densities

A clustering algorithm based on density decreased chain for data with arbitrary shapes and densities

A model of mechanics of growing solids for the stress-strain analysis of wound composites reinforced with thin unidirectional fibers

A mathematical model of the mechanics of growing solids has been developed that allows us to retrace the elastic stress-strain state progressing for cylindrical composites in the process of their slow layer-by-layer manufacturing by force winding of elementary material layers reinforced with sufficiently thin fibers in the circumferential direction, with an arbitrary density of their laying and an arbitrary pre-tension force. The obtained analytical relationships enable to set different control problems for a composite layer being formed including those for the characteristics of its contact interaction with the mandrel used.

Lipschitz retractions and complementation properties of Banach spaces

In the present paper we introduce and study the Lipschitz retractional structure of metric spaces. This topic was motivated by the analogous projectional structure of Banach spaces, a topic that has been thoroughly investigated. The more general metric setting fits well with the currently active theory of Lipschitz free spaces and spaces of Lipschitz functions. Among our applications we show that the Lipschitz free space F(X) is a Plichko space whenever X is a Plichko Banach space. Our main results include two examples of metric spaces. The first one M contains two points {0,1} such that no separable subset of M containing these points is a Lipschitz retract of M. The second example fails the analogous property for arbitrary infinite density. Finally, we introduce the metric version of the concept of locally complemented Banach subspace, and prove some metric analogues to the linear theory.

Study of nonhomogeneous linear second‐order discrete dynamical systems with uncertainties: Solution and stability with applications

We study, from a probabilistic standpoint, a full randomization of nonhomogeneous second‐order linear difference equations assuming that its data (initial conditions, coefficients, and forcing term) are random variables. Our analysis consists of computing the so‐called first probability density function of the solution, which is a stochastic process, and then analyzing the stability of the solution assuming that all data have an arbitrary joint probability density function. To achieve these goals, we take extensive advantage of the so‐called random variable transformation technique. The theoretical results extend their deterministic counterpart, and then, they have many applications in real‐world problems where uncertainty plays a key role. Our findings are first illustrated by means of several numerical examples, where different simulations are carried out, and, second, by means of a model belonging to biomathematics.

Standing Sausage Perturbations in Solar Coronal Loops with Diffuse Boundaries: An Initial Value Problem Perspective

Abstract Working in pressureless magnetohydrodynamics, we examine the consequences of some peculiar dispersive properties of linear fast sausage modes (FSMs) in one-dimensional cylindrical equilibria with a continuous radial density profile (ρ 0(r)). As recognized recently on solid mathematical grounds, cutoff axial wavenumbers may be absent for FSMs when ρ 0(r) varies sufficiently slowly outside the nominal cylinder. Trapped modes may therefore exist for arbitrary axial wavenumbers and density contrasts, their axial phase speeds in the long-wavelength regime differing little from the external Alfvén speed. If these trapped modes indeed show up in the solutions to the associated initial value problem (IVP), then FSMs have a much better chance to be observed than expected with classical theory and can be invoked to account for a considerably broader range of periodicities than practiced. However, with axial fundamentals in active region loops as an example, we show that this long-wavelength expectation is not seen in our finite-difference solutions to the IVP, the reason for which is then explored by superposing the necessary eigenmodes to construct solutions to the same IVP. At least for the parameters we examine, the eigenfunctions of trapped modes are characterized by a spatial extent well exceeding the observationally reasonable range of the spatial extent of initial perturbations, meaning a negligible fraction of energy that a trapped mode can receive. We conclude that the absence of cutoff wavenumbers for FSMs in the examined equilibrium does not guarantee a distinct temporal behavior.

Acoustic Insulation Mechanism of Membrane-Type Acoustic Metamaterials Loaded with Arbitrarily Shaped Mass Blocks of Variable Surface Density.

Membrane-type acoustic metamaterials (MAMs) have recently received widespread attention due to their good low-frequency sound-transmission-loss (STL) performance. A fast prediction method for the STL of rectangular membranes loaded with masses of arbitrary shapes and surface density values is proposed as a semi-analytical model for calculating the STL of membrane-type acoustic metamaterials. Through conformal mapping theory, the mass blocks of arbitrary shapes were transformed into regular shapes, which simplified the calculation model of acoustic propagation loss prediction, and the prediction results were verified by finite element simulations. The results show that the change in mass surface density was closely related to the size and frequency distribution of STL. The influence of the mass center on the STL and characteristic frequency of the film metamaterial is discussed.

Real-time motion of open quantum systems: Structure of entanglement, renormalization group, and trajectories

In this paper, we provide a complete description of the life cycle of entanglement during the real-time motion of open quantum systems. The quantum environment can have arbitrary (e.g., structured) spectral density. The entanglement can be seen constructively as a Lego: its bricks are the modes of the environment. These bricks are connected to each other via operator transforms. The central result is that during each infinitesimal time interval one new (incoming) mode of the environment gets coupled (entangled) to the open system, and one new (outgoing) mode gets irreversibly decoupled (disentangled from the future). Moreover, each moment of time, only a few relevant modes (three to four in the considered cases) are non-negligibly coupled to the future quantum motion. These relevant modes change (flow or renormalize) with time. As a result, the temporal entanglement has the structure of a matrix-product operator. This allows us to pose a number of questions and to answer them in this paper: What is the intrinsic quantum complexity of a real time motion? Does this complexity saturate with time or grow without bounds? How does one do the real-time renormalization group in a justified way? How do the classical Brownian stochastic trajectories emerge from the quantum evolution? How does one construct the few-mode representations of non-Markovian environments? We provide illustrative simulations of the spin-boson model for various spectral densities of the environment: semicircle, subohmic, ohmic, and superohmic.

Effects of Electron Correlation inside Disordered Crystals

We propose a novel approach for characterising the electron spectrum of disordered crystals constructed from a Hamiltonian of electrons as well as phonons and a diagram approach for Green’s function. The system’s electronic states were modelled by means of the multi-band, tight-binding approach. The system’s Hamiltonian is described based on the electron wave functions at the field of the atom nucleus. Our novel approach incorporates the long-range Coulomb interplay of electrons located in different lattice positions. Explicit interpretations of Green’s functions are derived using a diagram method. Equations are obtained for the vertex components for the mass operators of the electron–electron as well aselectron–phonon interplays. A system of equations for the spectrum of elementary excitations in the crystal is obtained, in which the vertex components for the mass operators of electron–electron as well as electron–phonon interplays are renormalised. Thismakes it possible to perform numerical computationsfor the system’s energy spectrum with a predetermined accuracy. In contrast to other approaches in which electron correlations are only taken into account in the limiting cases of an infinitely large and infinitesimal electron density, in this method, electron correlations are described in the general case of an arbitrary density. We obtained the cluster expansion of the density of states (DOS) of the disordered systems. We demonstrate that the addition of the electron-scattering mechanismsto the clusters is decreasing. This happens due to a growing number of positions in the cluster, which hang ontothe small parameter. The computing exactness is fixed by a small parameter for cluster expansion of Green’s functions of electrons as well as phonons.

A multiscale methodology for small-scale bubble dynamics in turbulence

We formulate in this paper a multiscale numerical framework that handles small-scale bubble dynamics in turbulence. Our framework involves bubbles with arbitrary density ratios with the carrier phase. We use a Moving Reference Frame method that follows a bubble to deal with a fast rising of bubbles present at high density ratios between the phases. We use a Proportional Integral Derivative controller to handle an additional acceleration term in the governing equations that stems from the change of a coordinate system from a fixed to a non-inertial one. Our framework accounts for the fact that the dynamics of bubbles are significantly influenced by the unsteadiness of the small-scale turbulent liquid fluctuations that modify the bubble shapes and alter their motion. In addition, we improve and speed up, with at least two orders of magnitude in computational time, the numerical framework recently proposed by Milan et al. (2020). The developed numerical framework can capture processes occurring at time scales even smaller than the Kolmogorov times. It can be applied to droplets, bubbles or particle systems in both laminar and turbulent flows, using any general DNS technique that handles two-phase flows.