Distribution of first-reaction times with target regions on boundaries of shell-like domains

We study the probability density function (PDF) of the first-reaction times between a diffusive ligand and a membrane-bound, immobile imperfect target region in a restricted "onion-shell" geometry bounded by two nested membranes of arbitrary shapes. For such a setting, encountered in diverse molecular signal transduction pathways or in the narrow escape problem with additional steric constraints, we derive an exact spectral form of the PDF, as well as present its approximate form calculated by help of the so-called self-consistent approximation. For a particular case when the nested domains are concentric spheres, we get a fully explicit form of the approximated PDF, assess the accuracy of this approximation, and discuss various facets of the obtained distributions. Our results can be straightforwardly applied to describe the PDF of the terminal reaction event in multi-stage signal transduction processes.

A Future of Current Flow Modelling for Transcranial Electrical Stimulation?

Purpose of Review Transcranial electrical stimulation (tES) is used to non-invasively modulate brain activity in health and disease. Current flow modeling (CFM) provides estimates of where and how much electrical current is delivered to in the brain during tES. It therefore holds promise as a method to reduce commonplace variability in tES delivery and, in turn, the outcomes of stimulation. However, the adoption of CFM has not yet been widespread and its impact on tES outcome variability is unclear. Here, we discuss the potential barriers to effective, practical CFM-informed tES use. Recent Findings CFM has progressed from models based on concentric spheres to gyri-precise head models derived from individual MRI scans. Users can now estimate the intensity of electrical fields (E-fields), their spatial extent, and the direction of current flow in a target brain region during tES. Here. we consider the multi-dimensional challenge of implementing CFM to optimise stimulation dose: this requires informed decisions to prioritise E-field characteristics most likely to result in desired stimulation outcomes, though the physiological consequences of the modelled current flow are often unknown. Second, we address the issue of a disconnect between predictions of E-field characteristics provided by CFMs and predictions of the physiological consequences of stimulation which CFMs are not designed to address. Third, we discuss how ongoing development of CFM in conjunction with other modelling approaches could overcome these challenges while maintaining accessibility for widespread use. Summary The increasing complexity and sophistication of CFM is a mandatory step towards dose control and precise, individualised delivery of tES. However, it also risks counteracting the appeal of tES as a straightforward, cost-effective tool for neuromodulation, particularly in clinical settings.

Evaluation of Secondary Flow between Rotating Concentric Spheres Using a Perturbation Method

A new approach to evaluate the Newtonian flow between concentric rotating spheres is introduced in this paper. A general analytic solution to the problem is deduced using a perturbation method that takes into account the primary and secondary flows produced between the spheres, as well as an alternative analytical method. In order to exemplify the results of the previous analysis, six particular cases were studied. The results of the perturbation method show that under certain circumstances the secondary flow is no negligible, as is usually considered, but it is comparable to the value of the primary one. While the analytical method allows us to simulate the flow with results very similar to those of other authors.

Morphology-Dependent Resonances in Two Concentric Spheres with Variable Refractive Index in the Outer Layer: Analytic Solutions

In many applications constant or piecewise constant refractive index profiles are used to study the scattering of plane electromagnetic waves by a spherical object. When the structured media has variable refractive indices, this is more of a challenge. In this paper, we investigate the morphology dependent resonances for the scattering of electromagnetic waves from two concentric spheres when the outer shell has a variable refractive index. The resonance analysis is applied to the general solutions of the radial Debye potential for both transverse magnetic and transverse electric modes. Finally, the analytic conditions to determine the resonance locations for this system are derived in the closed form of both modes. Our numerical results are provided with discussion.

Hawking Radiation is Nothing: Developed Correlation of Entropy with Black Hole Area

Read is the first word and divine order in the Holy Quran. My lovely physics does not answer many mysteries in the universe including black holes. The observed ordinary matter in the universe is only ~5 %. The remaining is 27% dark matter and approximately 68% dark energy of. This paper introduces a developed model and concept to black hole anatomy and entropy-surface area correlation. It considers the stationary black hole a one entity of 4 concentric spheres around the singularity. They are the event horizon, photon sphere at 1.5Rs, unstable light sphere at 2.6Rs and innermost stable particle sphere at 3Rs. An equation (Sayed Ts formula) was derived excluded heuristics of Professor S. Hawking. This formula is based on different proportional constant between entropy and black hole surface area; not ¼ as He conjectured. In spite of my respect and humanity sympathy with Prof. Hawking, his tombstone equation should be corrected. There is neither radiation escape from black holes nor any signal detected due to micro black holes evaporation. Finally, it can be stated that Hawking radiation is Nothing.

Scattering Theory for 2N Parameter Models of Finitely Many Relativistic δ-Sphere and δ-Sphere plus Coulomb Interactions Supported by Concentric Spheres in Quantum Mechanics

We study scattering theory for 2N parameter models of finitely many relativistic δ-sphere and δ-sphere plus Coulomb interactions. We provide the mathematical definitions of the Hamiltonians, solve the resolvent equations, and compute the nonrelativistic limits for both models. We obtain new results related to spectral properties and scattering data.

Stewartson-layer instability in a wide-gap spherical Couette experiment: Rossby number dependence

Instabilities of a viscous fluid between two fast but differentially rotating concentric spheres, the so-called spherical Couette flow, with a fixed radius ratio of $\unicode[STIX]{x1D702}=r_{i}/r_{o}=1/3$ are studied, where $r_{i}$ is the inner and $r_{o}$ the outer radius of the spherical shell. Of particular interest is the difference between cases where the Rossby number $Ro=(\unicode[STIX]{x1D6FA}_{i}-\unicode[STIX]{x1D6FA}_{o})/\unicode[STIX]{x1D6FA}_{o}>0$ and cases with $Ro<0$, where $\unicode[STIX]{x1D6FA}_{i}$ and $\unicode[STIX]{x1D6FA}_{o}$ are the inner- and outer-sphere angular velocities. The basic state in both situations is an axisymmetric shear flow with a Stewartson layer situated on the tangent cylinder. The tangent cylinder is given by a cylinder that touches the equator of the inner sphere with an axis parallel to the axis of rotation. The experimental results presented fully confirm earlier numerical results obtained by Hollerbach (J. Fluid Mech., vol. 492, 2003, pp. 289–302) showing that for $Ro>0$ a progression to higher azimuthal wavenumbers $m$ can be seen as the rotation rate $\unicode[STIX]{x1D6FA}_{0}$ increases, but $Ro<0$ gives $m=1$ over a large range of rotation rates. It is further found that in the former case the modes have spiral structures radiating away from Stewartson layer towards the outer shell whereas for $Ro<0$ the modes are trapped in the vicinity of the Stewartson layer. Further, the mean flow excited by inertial mode self-interaction and its correlation with the mode’s amplitudes are investigated. The scaling of the critical $Ro$ with Ekman number $E=\unicode[STIX]{x1D708}/(\unicode[STIX]{x1D6FA}_{o}\,d^{2})$, where $\unicode[STIX]{x1D708}$ is the kinematic viscosity and $d$ the gap width, is well within the bounds that have been established in a number of experimental studies using cylindrical geometries and numerical studies using spherical cavities. However, the present work is the first that experimentally examines Stewartson-layer instabilities as a function of the sign of $Ro$ for the true spherical-shell geometry.

On the Moving Trajectory of a Ball in a Viscous Liquid between Two Concentric Rigid Spheres

In the paper, the dynamic motion of a point ball with a mass of m , sliding in a viscous liquid between two concentric spheres under the influence of gravity and viscous and dry resistance, is investigated. In addition, it is considered that the ball starts its motion from some arbitrary point M 0 = M ( θ 0 , φ 0 ) . A system of nonlinear differential equations in a spheroidal coordinate system is obtained for the angular variables θ and φ to account for all the forces acting on the ball. The dependence of the reaction force on the angular variables is found, and the solution of the resulting system of equations is numerically analyzed. The projections of the trajectories on the plane x − y , y − z , x − z are found.