Isotropic Solution Research Articles

Anisotropic and isotropic persistent singularities of solutions of the fast diffusion equation

The aim of this paper is to study a class of positive solutions of the fast diffusion equation with specific persistent singular behavior. First, we construct new types of solutions with anisotropic singularities. Depending on parameters, either these solutions solve the original equation in the distributional sense, or they are not locally integrable in space-time. We show that the latter also holds for solutions with snaking singularities, whose existence has been proved recently by M. Fila, J.R. King, J. Takahashi, and E. Yanagida. Moreover, we establish that in the distributional sense, isotropic solutions whose existence was proved by M. Fila, J. Takahashi, and E. Yanagida in 2019, actually solve the corresponding problem with a moving Dirac source term. Last, we discuss the existence of solutions with anisotropic singularities in a critical case.

Extended minimal theories of massive gravity

In this work, we introduce a class of extended Minimal Theories of Massive Gravity (eMTMG), without requiring a priori that the theory should admit the same homogeneous and isotropic cosmological solutions as the de Rham-Gabadadze-Tolley massive gravity. The theory is constructed as to have only two degrees of freedom in the gravity sector. In order to perform this step we first introduce a precursor theory endowed with a general graviton mass term, to which, at the level of the Hamiltonian, we add two extra constraints as to remove the unwanted degrees of freedom, which otherwise would typically lead to ghosts and/or instabilities. On analyzing the number of independent constraints and the properties of tensor mode perturbations, we see that the gravitational waves are the only propagating gravitational degrees of freedom which do acquire a non-trivial mass, as expected. In order to understand how the effective gravitational force works for this theory we then investigate cosmological scalar perturbations in the presence of a pressureless fluid. We then restrict the whole class of models by imposing the following conditions at all times: 1) it is possible to define an effective gravitational constant, $G_{{\rm eff}}$; 2) the value $G_{\text{eff}}/G_{N}$ is always finite but not always equal to unity (as to allow some non-trivial modifications of gravity, besides the massive tensorial modes); and 3) the square of mass of the graviton is always positive. These constraints automatically make also the ISW-effect contributions finite at all times. Finally we focus on a simple subclass of such theories, and show they already can give a rich and interesting phenomenology.

Facile determination of matter’s optical properties by combining Monte Carlo and Berreman 4x4 matrix approaches: Dispersion in nematic and emerging ferroelectric nematic liquid crystals

Determination of material and optical properties in structured films and fluids, e.g., refractive index and material thickness, usually requires special optical geometries and instruments. This prevents one from studying optical properties especially in soft matter systems with nonuniform hierarchical and orientational structures. Here we develop a data measurement and analysis scheme by combining Monte Carlo and Berreman 4x4 matrix approaches for obtaining the dispersion of refractive index, material thickness, and the optical anisotropy from a single shot of simple transmission or reflection spectroscopy data at a time scale of seconds. We show the validity of the method by determining Cauchy coefficients and sample thickness of the well-known reference materials of both isotropic toluene solution and anisotropic material of a nematic liquid crystal (4-cyano-4′-pentyl biphenyl). This enables us to further investigate the unknown optical properties of an emerging ferroelectric nematic material, the so-called RM734.

Förster resonance energy transfer from Hoechst 33258 to SYBR Green I in cholesteric liquid-crystalline DNA

In the present work Förster resonance energy transfer (FRET) between DNA-specific fluorescent dyes Hoechst 33258 (Ht58; donor) and SYBR Green I (SG; acceptor) is compared in isotropic DNA solution and in its cholesteric liquid-crystalline dispersion (DNA CLCD). At a fixed donor concentration of 3.57 × 10−6 М in the spatially ordered system, FRET starts at lower acceptor concentration than in the isotropic system and at the SG concentration below 1.05 × 10−7 M, its efficiency in DNA CLCD is up to ≈4-fold higher than in DNA solution (3D-FRET vs. 1D-FRET). However, the maximum FRET efficiency is higher in isotropic solution (≈70% vs. ≈50%, respectively) apparently due to the absence of steric restrictions on intermolecular interactions imposed by the dense packaging of the DNA molecules. In isotropic DNA solution FRET from Ht58 to SG is accompanied by the displacement of Ht58 from the minor-groove complex as the SG concentration increased, while the displacement of Ht58 in DNA CLCD seems to be limited. Using FRET between Ht58 and SG, the formation of DNA CLCD particles via stopped-flow method was investigated and for the first time the subsecond stage of the DNA CLCD particle formation, generation of “precholesteric” optically inactive nuclei, was demonstrated.

Life of the homogeneous and isotropic universe in dynamical string tension theories

Cosmological solutions are studied in the context of the modified measure formulation of string theory , then the string tension is a dynamical variable and the string the tension is an additional dynamical degree of freedom and its value is dynamically generated. These tensions are then not universal, rather each string generates its own tension which can have a different value for each of the string world sheets and in an ensemble of strings. The values of the tensions can have a certain dispersion in the ensemble. We consider a new background field that can couple to these strings, the “tension scalar” which is capable of changing locally along the world sheet and then the value of the tension of the string changes accordingly. When many types of strings probing the same region of space are considered this tension scalar is constrained by the requirement of quantum conformal invariance. For the case of two types of strings probing the same region of space with different dynamically generated tensions, there are two different metrics, associated to the different strings. Each of these metrics have to satisfy vacuum Einstein’s equations and the consistency of these two Einstein’s equations determine the tension scalar. The universal metric, common to both strings generically does not satisfy Einstein’s equation . The two string dependent metrics considered here are flat space in Minkowski space and Minkowski space after a special conformal transformation. The limit where the two string tensions are the same is studied, it leads to a well defined solution. If the string tension difference between the two types of strings is very small but finite, the approximately homogeneous and isotropic cosmological solution lasts for a long time, inversely proportional to the string tension difference and then the homogeneity and and isotropy of the cosmological disappears and the solution turns into an expanding braneworld where the strings are confined between two expanding bubbles separated by a very small distance at large times. The same principle is applied to the static end of the universe wall solution that lasts a time inversely proportional to the dispersion of string tensions. This suggest a scenario where quantum fluctuations of the cosmological or static solutions induce the evolution towards braneworld scenarios and decoherence between the different string tension states.

Phase-field model of grain boundary diffusion in nanocrystalline solids: Anisotropic fluctuations, anomalous diffusion, and precipitation

The anisotropic phase-filed model of grain boundary diffusion and precipitation of solute in nanocrystalline solids has been developed. In this model, the Cahn–Hilliard equation is generalized for the anisotropic phase-field diffusion of solute and anisotropic compositional fluctuations. It is found that dynamics of solute concentration profile demonstrates the anomalous diffusion behavior with scaling parameters depending on the mobility ratio and microstructure of a solid solution. It is noteworthy that the increase in source concentration can slow down the concentration front propagation due to uphill diffusion or formation of a new phase. Parameters of grain boundary diffusion control the precipitation dynamics. In particular, a decrease in transverse diffusion coefficient is responsible for longer incubation time, and lower rates of nucleation and nuclei growth in comparison with the case of isotropic solute transport near grain boundaries. Transport properties of boundary and bulk are responsible for the formation of the bimodal size distribution function of second phase particles and specific kinetics of average radius and number density.

Charged anisotropic decoupled compact objects in f(R) gravity

In this paper, we aim to formulate anisotropic solutions that represent charged self-gravitating static spheres in [Formula: see text] gravity. We introduce the effects of anisotropy in the domain of an isotropic solution by including an extra matter source. The spherical configuration is deformed by transforming radial as well as temporal metric functions. As a result of these transformations, the isotropic and anisotropic matter sources can be decoupled from each other. The system of field equations is disintegrated into two arrays such that the first set corresponds to the isotropic source while the second comprises the new gravitational source. The charged Krori–Barua spacetime is employed to specify the first set in the background of [Formula: see text] Starobinsky model. Two solutions of the other system are determined by incorporating a barotropic equation of state along with additional constraints on the anisotropic source. The matching conditions are imposed at the hypersurface to determine the values of unknown constants. Different features of the constructed models are inspected by using the mass and radius of Her X-I. One of the constructed solutions agrees with the viability and stability conditions for specific values of charge and the decoupling parameter.

Modelling underground excavations in rock masses with anisotropic time-dependent behaviour

When modelling rock masses that behave anisotropically and in addition present a time dependent behaviour, it is relevant to select a constitutive model able to represent their actual behaviour realistically. This article presents an alternative anisotropic time-dependent constitutive model able to predict the coupling between anisotropic behaviour and time-dependent (or viscous) behaviour. The viscous behaviour is simulated with the Burgers model and all elastic springs and viscous dashpots are considered to exhibit transversely isotropic properties. The proposed constitutive model has been implemented in the finite element method software CODE_BRIGHT. To verify the basic anisotropic elastic solution, it has been compared with that of PLAXIS results. And to verify the isotropic viscoelastic solution, it has been compared with analytical solutions. Furthermore, the proposed constitutive model has been used to predict the behaviour of samples from laboratory tests. Finally, parametric analyses have been carried out to investigate the influence of different factors on tunnelling responses, including the selection of different constitutive models, anisotropy of initial stresses and anisotropy of material properties. The proposed model provides an alternative method for the preliminary design of geotechnical engineering works involving geomaterials that exhibit anisotropic time-dependent behaviour.Article highlights• An anisotropic time-dependent model has been implemented in CODE_BRIGHT and validated.• The model can predict the coupled anisotropic time-dependent behaviour of geomaterials.• Parametric analyses have been performed to study the influence of different factors in ground response.

Integration of the Lane–Emden equation for relativistic anisotropic polytropes through gravitational decoupling: a novel approach

In this work we propose a novel approach to integrate the Lane–Emden equations for relativistic anisotropic polytropes. We take advantage of the fact that Gravitational Decoupling allows to decrease the number of degrees of freedom once a well known solution of the Einstein field equations is provided as a seed so after demanding the polytropic equation for the radial pressure the system is automatically closed. The approach not only allows to extend both isotropic or anisotropic known solutions but simplifies the computation of the Tolman mass whenever the Minimal Geometric Deformation is considered given that the g_{tt} component of the metric remains unchanged. We illustrate how the the method works by analyzing the solutions obtained from Tolman IV, Durgapal IV and Wymann IIa isotropic systems as a seed for the integration.

De-homogenization of optimal 2D topologies for multiple loading cases

This work presents an extension of the highly efficient de-homogenization method for obtaining high-resolution, near-optimal 2D topologies optimized for minimum compliance subjected to multiple load cases. We perform a homogenization-based topology optimization based on stiffness optimal Rank-N microstructure parameterizations to obtain stiffness optimal multi-scale designs on relatively coarse grids. To avoid relatively thin microstructure features, we regularize the design by introducing a material indicator field which results in well-defined widths and structural boundaries. In order to avoid singularities from the multiple load case problem, the orientations of the microstructures are further regularized. Subsequently, we derive a single-scale interpretation of stiffness optimal multi-scale designs on a fine grid using de-homogenization. The single-scale interpretation can be derived without costly postprocessing analysis on the fine grid, as an implicit boundary formulation is used.The effect of starting guesses is discussed, as they are non-trivial for Rank-N microstructures. Different numerical examples verify the performance of the inexpensive high-resolution solutions, both in comparison to the Rank-N based homogenization solutions, to equivalent density-based large-scale solutions, as well as to strict isotropic microstructure solutions. Depending on starting guesses, the approach consistently delivers structural performance values within a few percent of density-based large-scale solutions with a CPU time reduction factor of more than 300. Finally, we confirm that isotropic as well as orthogonal Rank-2 microstructure models are inferior to stiffness optimal anisotropic microstructure models for minimum compliance problems subjected to multiple load cases.

Reciprocal and Non‐reciprocal Chiroptical Features in Thin Films of Organic Dyes

AbstractThe study of chiroptical properties in thin films of chiral organic π‐conjugated dyes is more complex than isotropic solutions, due to the interference of macroscopic anisotropies often present in the solid state. The experimental circularly polarized (CP) absorption (electronic circular dichroism, ECD) and CP emission (circularly polarized luminescence, CPL) of a thin film can be expressed as the sum of several terms, some of which are non‐reciprocal upon film flipping. In this review, we shall discuss the fundamental theoretical concepts about reciprocal and non‐reciprocal chiroptical features in thin films of organic dyes, describing their correlation with the solid‐state structural organization and suggesting a practical toolbox to quantify their relative weight on an experimental ECD or CPL spectrum. Moreover, we shall also take into account the literature on thin films of organic π‐conjugated dyes showing strong non‐reciprocal CP absorption and/or CP emission, due to their potential high interest in the flourishing field of chiral photonics and electronics.

Transition to Liquid-Crystalline Phase During Coagulation of Hydroxypropylcellulose Isotropic Solutions

Transition to Liquid-Crystalline Phase During Coagulation of Hydroxypropylcellulose Isotropic Solutions

Vacuum Einstein field equations in smooth metric measure spaces: the isotropic case* *Supported by projects PID2019-105138GB-C21/AEI/10.13039/501100011033 (Spain) and ED431C 2019/10 (Xunta de Galicia, Spain).

On a smooth metric measure spacetime (M, g, e −f dvol g ), we define a weighted Einstein tensor. It is given in terms of the Bakry–Émery Ricci tensor as a tensor which is symmetric, divergence-free, concomitant of the metric and the density function. We consider the associated vacuum weighted Einstein field equations and show that isotropic solutions have nilpotent Ricci operator. Moreover, the underlying manifold is a Brinkmann wave if it is two-step nilpotent and a Kundt spacetime if it is three-step nilpotent. More specific results are obtained in dimension 3, where all isotropic solutions are given in local coordinates as plane waves or Kundt spacetimes.

Anisotropic tolman V solutions by decoupling approach in f(R,T 2) gravity

This paper investigates the behavior of anisotropic static spheres that are constructed by employing a minimal geometric deformation in the framework of gravity ( is the Ricci scalar and is the energy-momentum tensor). We consider a spherical setup with two sources: seed and additional. It is assumed that the seed source is isotropic whereas the new source is responsible for inducing anisotropy. We deform the component to split the field equations into two sets. The first array corresponds to the isotropic solution whereas the second set contains the effect of the anisotropic source. The system related to isotropic source is determined by the metric potentials of Tolman V solution while three solutions of the second set are constructed corresponding to three different constraints. The physical acceptability of all solutions is checked through energy conditions by employing the radius and mass of PSR J1614–2230 star. We also examine the stability, mass, compactness and redshift of the obtained solutions. We conclude that first two solutions satisfy the viability and stability criteria only for small values of the decoupling parameter while third solution is stable for its all possible values.

SU(N) natural inflation

We study SU($N$) gauge fields that couple to the inflaton through the Chern-Simons term. We provide a general procedure to construct homogeneous, isotropic, and attractor solutions of the gauge fields during inflation. The gauge fields develop various VEVs corresponding to different spontaneous symmetry breaking patterns of SU($N$) where embedded SU($2$) subgroups are broken with the spatial rotation SO($3$) symmetry. As specific examples, we construct the stable solutions for $N = 3$ and $4$. We also numerically solve the gauge field dynamics and confirm that our analytic solutions are complete and attractor. It is straightforward to apply our procedure to the other simple Lie groups.

Reaction field induction self-amplification optical activity during polymerization in liquid crystal

A monomer having both chiral induction from nematic liquid crystal (LC) to cholesteric LC and polymerization activity was prepared. Addition of the compound to nematic LC induced the formation of a cholesteric liquid crystal in its chiral form. Subsequent electrochemical polymerization of the monomer produced an optically active polymer with a helical structure as confirmed by circular dichroism (CD) and optical rotatory dispersion measurements. While, the addition of the chiral monomer to an isotropic non-LC solution, and subsequent polymerization resulted a product with no clear optical activity. These results suggest that polymerization of the monomer in a cholesteric environment allows the formation of a polymer with high optical activity. Morphological imprinting from the matrix LC was confirmed by surface observation through polarizing optical microscopy. The chiral monomer changes the surrounding environment to grow into a more highly ordered helical structure. The matrix LC as the environment amplifies the optical activity with helical growth of the monomer during polymerization.

A supramolecular and liquid crystalline water-based alignment medium based on azobenzene-substituted 1,3,5-benzenetricarboxamides.

A supramolecular, lyotropic liquid crystalline alignment medium based on an azobenzene-containing 1,3,5-benzenetricarboxamide (BTA) building block is described and investigated. As we demonstrate, this water-based system is suitable for the investigation of various water-soluble analytes and allows for a scaling of alignment strength through variation of temperature. Additionally, alignment is shown to reversibly collapse above a certain temperature, yielding an isotropic solution. This collapse allows for isotropic reference measurements, which are typically needed in addition to those in an anisotropic environment, to be performed using the same sample just by varying the temperature. The medium described thus provides easy access to anisotropic NMR observables and simplifies structure elucidation techniques based thereon.

Modelling the gamma-ray morphology of HESSJ1804−216 from two supernova remnants in a hadronic scenario

ABSTRACT HESS J1804−216 is one of the brightest yet most mysterious TeV γ-ray sources discovered to date. Previous arc-minute scale studies of the interstellar medium (ISM) surrounding this TeV γ-ray source revealed HESS J1804−216 is likely powered by a mature supernova remnant (SNR) or pulsar, hence its origin remains uncertain. In this paper, we focus on the diffusive escape of cosmic ray protons from potential SNR accelerators. These cosmic rays interact with the ISM to produce TeV γ-rays. We utilize the isotropic diffusion equation solution for particles escaping from a shell, to model the energy-dependent escape and propagation of protons into the ISM. This work is the first attempt at modelling the spatial morphology of γ-rays towards HESS J1804−216, using arc-minute ISM observations from both Mopra and the Southern Galactic Plane Survey. The spectral and spatial distributions of γ-rays for the two nearby potential SNR counterparts, SNR G8.7−0.1 and the progenitor SNR of PSR J1803−2137, are presented here. We vary the diffusion parameters and particle spectrum and use a grid search approach to find the best combination of model parameters. We conclude that moderately slow diffusion is required for both candidates. The most promising candidate to be powering the TeV γ-rays from HESS J1804−216 in a hadronic scenario is the progenitor SNR of PSR J1803−2137.

Calculating the isotropic and anisotropic roughness by using MW model: A variable thermal conductivity study

In this paper, we analyze the mathematical and theoretical studies of flow over a heated rough rotating disk. By using the Carreau fluid model we examine the dilatant and pesudoplastic behavior along with mean flow under the partial slip conditions. The variable fluid property is considered for heat phenomena. This study has stimulated a great deal of interest in applied engineering due to its enhanced theoretical and mathematical advantages. We analyzed the isotropic and anisotropic solutions for Carreau fluid model with variable fluid property. The similarity transformations used here for the Carreau fluid was initially presented by von Kàrmàn, it reduces the boundary layer equations into ordinary differential equations. Shooting method is used to solve the dimensionless ordinary differential equations. All of the obtained outcomes are analyzed for the mean flow velocity and temperature profiles and physical characteristics are shown in the form of graphs and table.

Study of Decoupled Anisotropic Solutions in f(R, T, RρηTρη) Theory

In this paper, we consider isotropic solution and extend it to two different exact well-behaved spherical anisotropic solutions through minimal geometric deformation method in f(R,T,RρηTρη) gravity. We only deform the radial metric component that separates the field equations into two sets corresponding to their original sources. The first set corresponds to perfect matter distribution while the other set exhibits the effects of additional source, i.e., anisotropy. The isotropic system is resolved by assuming the metric potentials proposed by Krori-Barua while the second set needs one constraint to be solved. The physical acceptability and consistency of the obtained solutions are analyzed through graphical analysis of effective matter components and energy bounds. We also examine mass, surface redshift and compactness of the resulting solutions. For particular values of the decoupling parameter, our both solutions turn out to be viable and stable. We conclude that this curvature-matter coupling gravity provides more stable solutions corresponding to a self-gravitating geometry.