Narrow Results

Here, we prove an isoperimetric inequality for the harmonic mean of the first $N-1$ non-trivial Neumann eigenvalues of the Laplace–Beltrami operator for domains contained in a hemisphere of ${𝕊}^N$.

Uniform probability distributions on ${\ell }_p$ balls and spheres have been studied extensively and are known to behave like product measures in high dimensions. In this note we consider the uniform distribution on the intersection of a simplex and a sphere. Certain new and interesting features, such as phase transitions and localization phenomena emerge.

This study investigates the interaction between fluid dynamics and electromagnetic fields, a complex problem that has not been extensively studied. The Lorentz force, which arises due to the interaction between magnetic fields and currents in a fluid is considered in this study. This research investigates the effects of a magnetic field, couple stress, and slip velocity on the behavior of a squeeze film (SF) formed between a porous flat and spherical plate. The Stokes equation for couple stress fluids is used to produce a generalized version of the Reynolds equation, which is then used to determine the film pressure. Also, this study considers the impact of a constant magnetic field orthogonal to the plate. The fluid in the porous region is governed by modified Darcy law. The effect of a uniform magnetic field perpendicular to the plate is considered. The bearing characteristics pressure, squeeze film time, and load-carrying capacity are graphically presented. The results revealed that the load-carrying capacity, pressure, and squeeze film time are reduced with a rise in slip and porousness parameters. The slip parameter decreases the values of film pressure, squeeze time, and load-carrying capacity as related to the no-slip case.

There are two smooth functions $\sigma$ and $\rho$ associated to a nontrivial concircular vector field $v$ on a connected Riemannian manifold $(M,g)$, called potential function and connecting function. In this paper, we show that presence of a timelike nontrivial concircular vector field influences the geometry of generalized Robertson–Walker space-times. We use a timelike concircular vector field $v$ on an $n$ -dimensional connected conformally flat Lorentzian manifold, $n>2$, to find a characterization of generalized Robertson–Walker space-time with fibers Einstein manifolds. It is interesting to note that for $n=4$ the concircular vector field annihilates energy-momentum tensor and also that in this case the potential function $\sigma$ is harmonic. In the second part of this paper, we show that presence of a nontrivial concircular vector field $v$ with connecting function $\rho$ on a complete and connected $n$ -dimensional conformally flat Riemannian manifold $(M,g)$, $n>2$, with Ricci curvature $\mathrm{\text{Ric}}(v,v)$ non-negative, satisfying $n(n-1)\rho +\tau =0$, is necessary and sufficient for $(M,g)$ to be isometric to either a sphere $S^n(c)$ or to the Euclidean space $E^n$, where $\tau$ is the scalar curvature.

In this paper, we prove the long-time existence of the CR Yamabe flow on the compact strictly pseudoconvex CR manifold with positive CR invariant. We also prove the convergence of the CR Yamabe flow on the sphere by proving that: the contact form which is pointwise conformal to the standard contact form on the sphere converges exponentially to a contact form of constant pseudo-Hermitian sectional curvature. We also show that the eigenvalues of some geometric operators are non-decreasing under the unnormalized CR Yamabe flow provided that the pseudo-Hermitian scalar curvature satisfies certain conditions.

This paper addresses the issue of obtaining the optimal rotation to match two functions on the sphere by minimizing the squared error norm and the Kullback–Leibler information criteria. In addition, the accuracy in terms of the band-limited approximations in both cases are also discussed. Algorithms for fast and accurate rotational matching play a significant role in many fields ranging from computational biology to spacecraft attitude estimation. In electron microscopy, peaks in the so-called "rotation function" determine correlations in orientation between density maps of macromolecular structures when the correspondence between the coordinates of the structures is not known. In X-ray crystallography, the rotational matching of Patterson functions in Fourier space is an important step in the determination of protein structures. In spacecraft attitude estimation, a star tracker compares observed patterns of stars with rotated versions of a template that is stored in its memory. Many algorithms for computing and sampling the rotation function have been proposed over the years. These methods usually expand the rotation function in a bandlimited Fourier series on the rotation group. In some contexts the highest peak of this function is interpreted as the optimal rotation of one structure into the other, and in other contexts multiple peaks describe symmetries in the functions being compared. Prior works on rotational matching seek to maximize the correlation between two functions on the sphere. We also consider the use of the Kullback–Leibler information criteria. A gradient descent algorithm is proposed for obtaining the optimal rotation, and a measure is defined to compare the convergence of this procedure applied to the maximal correlation and Kullback–Leibler information criteria.

In this paper we compute the leading order of the Casimir energy for a free massless scalar field confined in a sphere in three spatial dimensions, with the Dirichlet boundary condition. When one tabulates all of the reported values of the Casimir energies for two closed geometries, cubical and spherical, in different space–time dimensions and with different boundary conditions, one observes a complicated pattern of signs. This pattern shows that the Casimir energy depends crucially on the details of the geometry, the number of the spatial dimensions, and the boundary conditions. The dependence of the sign of the Casimir energy on the details of the geometry, for a fixed spatial dimensions and boundary conditions has been a surprise to us and this is our main motivation for doing the calculations presented in this paper. Moreover, all of the calculations for spherical geometries include the use of numerical methods combined with intricate analytic continuations to handle many different sorts of divergences which naturally appear in this category of problems. The presence of divergences is always a source of concern about the accuracy of the numerical results. Our approach also includes numerical methods, and is based on Boyer's method for calculating the electromagnetic Casimir energy in a perfectly conducting sphere. This method, however, requires the least amount of analytic continuations. The value that we obtain confirms the previously established result.

In this paper, we are interested in studying the initial value problem for parabolic problem associated with the Caputo–Fabrizio derivative. We deal the problem in two cases: linear inhomogeneous case and nonlinearity source term. For the linear case, we derive the convergence result of the mild solution when the fractional order $\alpha \to 1^{-}$ under some various assumptions on the initial datum. For the nonlinear problem, we show the existence and uniqueness of the mild solution using Banach fixed point theory. We also prove the convergence result of the mild solution when the fractional order $\alpha \to 1^{-}$.

We present a unified approach for constructing Slepian functions — also known as prolate spheroidal wave functions — on the sphere for arbitrary tensor ranks including scalar, vectorial, and rank 2 tensorial Slepian functions, using spin-weighted spherical harmonics. For the special case of spherical cap regions, we derived commuting operators, allowing for a numerically stable and computationally efficient construction of the spin-weighted spherical-harmonic-based Slepian functions. Linear relationships between the spin-weighted and the classical scalar, vectorial, tensorial, and higher-rank spherical harmonics allow the construction of classical spherical-harmonic-based Slepian functions from their spin-weighted counterparts, effectively rendering the construction of spherical-cap Slepian functions for any tensorial rank a computationally fast and numerically stable task.