Influence of corrugation on light-scattering properties of capsule and finite-cylinder particles: Analytic solution using Sh-matrices

We use the Sh-matrices to derive an analytic T-matrix solution for the light scattering from capsule and finite-cylinder particles with corrugated surfaces. The solution is tested by comparison with discrete-dipole-approximation (DDA) calculations. We present results from finite circular cylinders and capsules whose diameter/length ratio is 1/2. Such particles have interest as simulants of aerosolized spores. We analyze and compare their two-dimensional scattering patterns. The effect of corrugated surfaces of relatively small amplitude is quite significant on the resulting scattering patterns.     

Light scattering by a finite cylinder containing a spherical cavity using Sh-matrices

We use the Sh-matrix formalism that contains the shape-dependent parameters of the T-matrix to derive an analytical solution for the light scattering from a finite cylinder containing a spherical cavity. The integral expressions for the Sh-matrix elements are simpler than those of the T-matrix elements and the case of a sphere embedded in a finite cylinder these integrals can be solved analytically.     

Analytic solution of Percus-Yevick equation for fluid of hard spheres

A new method of analytic solution of the Percus-Yevick equation for the radial distribution functiong(r) of hard-sphere fluid is proposed. The original non-linear integral equation is reduced to non-homogeneous linear integral equation of Volterra's type of the second order. The kernel of this new equation has a polynomial form which allows to find analytic expression forg(r) itself without using the Laplace transformation. In addition, the first three moments of the total correlation function can be found.     

Polarized light-scattering measurements of dielectric spheres upon a silicon surface

The polarization of light scattered into directions out of the plane of incidence by polystyrene latex spheres upon a silicon substrate was measured for p -polarized incident light. The experimental data show good agreement with theoretical predictions for three sizes of spheres. These results demonstrate that the polarization of light scattered by particles can be used to determine the size of particulate contaminants on silicon wafers. Theoretical models, based on successive degrees of approximation, indicate that the mean distance of a particle from the surface is the primary determinant of the scattered light polarization for small out-of-plane scattering angles.     

Analytic study of self-gravitating polytropic spheres with light rings

Ultra-compact objects describe horizonless solutions of the Einstein field equations which, like black-hole spacetimes, possess null circular geodesics (closed light rings). We study analytically the physical properties of spherically symmetric ultra-compact isotropic fluid spheres with a polytropic equation of state. It is shown that these spatially regular horizonless spacetimes are generally characterized by two light rings \(\{r^{\text {inner}}_{\gamma },r^{\text {outer}}_{\gamma }\}\) with the property \(\mathcal{C}(r^{\text {inner}}_{\gamma })\le \mathcal{C}(r^{\text {outer}}_{\gamma })\), where \(\mathcal{C}\equiv m(r)/r\) is the dimensionless compactness parameter of the self-gravitating matter configurations. In particular, we prove that, while black-hole spacetimes are characterized by the lower bound \(\mathcal{C}(r^{\text {inner}}_{\gamma })\ge 1/3\), horizonless ultra-compact objects may be characterized by the opposite dimensionless relation \(\mathcal{C}(r^{\text {inner}}_{\gamma })\le 1/4\). Our results provide a simple analytical explanation for the interesting numerical results that have recently presented by Novotný et al. (Phys Rev D 95:043009, 2017).     

Capacitance coefficients of two spheres

Compact and exact expressions are obtained for the capacitance coefficients C_aa, C_bb and C_ab of two conducting spheres of radii a and b, for any distance c between the sphere centres. The results are equivalent to those of Maxwell, Russell and Jeffery, but enable rapid calculation of the coefficients in the limit of close approach, which is otherwise computationally difficult. Erroneous results published by several authors are corrected.     

Polarizability of two conducting spheres

Exact series expressions are obtained for the longitudinal and transverse polarizabilities of a system of two conducting spheres, for arbitrary sphere radii and separation. In the important case of close approach, where the exact series converge slowly, approximate analytical expressions are obtained. These analytical expressions become more accurate as the distance between the spheres gets small compared to the sphere radii, and reduce to the known analytic values at sphere contact.     

Factorizable supersymmetric S-matrices in two dimensions with general internal symmetry for massive excitations with vanishing central charge

Factorizable supersymmetric S-matrices for massive particles with vanishing central charge are shown to decouple into the product of two S-matrices. Only one carries the isospin content and the other is the S-matrix of the supersymmetric sine-Gordon model.     

Analytic solution of the multiloop Baxter equation

The asymptotic spectrum of anomalous dimensions of gauge-invariant operators in maximally supersymmetric Yang–Mills theory is believed to be described by a long-range integrable spin chain model. We focus in this study on its sl(2)$sl(2)$ subsector spanned by the twist-two single-trace Wilson operators, which are shared by all gauge theories, supersymmetric or not. We develop a formalism for the solution of the perturbative multiloop Baxter equation encoding their asymptotic anomalous dimensions, using Wilson polynomials as basis functions and Mellin transform technique. These considerations yield compact results which allow analytical calculations of multiloop anomalous dimensions bypassing the use of the principle of maximal transcendentality. As an application of our method we analytically confirm the known four-loop result. We also determine the dressing part of the five-loop anomalous dimensions.     

Analytic solution of the general anharmonic-oscillator problem

The regular and irregular solution of the differential three-dimensional Schrödinger equation with the confining potential , is presented. In full analogy with the recentq=1 conjecture of Singhet al. [1], the corresponding Green's function is constructed in terms of the extended continued fraction. For the simplest example withV(r)=r ²+² r ⁴, its convergence is proved and the resulting energy spectrum is numerically tested.     

Using the dynamic light-scattering method for the analysis of a blood-serum model solution

We used photon correlation spectroscopy to investigate various dynamic processes that occur in samples of a blood-serum solution. These processes were induced by directional changes of the surface charges of macromolecules and the concentration of the solution. We suggest that the obtained associations between the translational diffusion coefficient value (D _t ) and the medium pH value, as well as the protein concentration value, indicate the possibility of using the dynamic light scattering method for the effective diagnosis of common diseases, including cardiovascular lesions and cancers.     

Analytic solution of a static scale-free network model

We present a detailed analytical study of a paradigmatic scale-free network model, the Static Model. Analytical expressions for its main properties are derived by using the hidden variables formalism. We map the model into a canonic hidden variables one, and solve the latter. The good agreement between our predictions and extensive simulations of the original model suggests that the mapping is exact in the infinite network size limit. One of the most remarkable findings of this study is the presence of relevant disassortative correlations, which are induced by the physical condition of absence of self and multiple connections.     

Analytic complexity of functions of two variables

The definition of analytic complexity of an analytic function of two variables is given. It is proved that the class of functions of a chosen complexity is a differentialalgebraic set. A differential polynomial defining the functions of first class is constructed. An algorithm for obtaining relations defining an arbitrary class is described. Examples of functions are given whose order of complexity is equal to zero, one, two, and infinity. It is shown that the formal order of complexity of the Cardano and Ferrari formulas is significantly higher than their analytic complexity. The complexity classes turn out to be invariant with respect to a certain infinite-dimensional transformation pseudogroup. In this connection, we describe the orbits of the action of this pseudogroup in the jets of orders one, two, and three. The notion of complexity order is extended to plane (or “planar”) 3-webs. It is discovered that webs of complexity order one are the hexagonal webs. Some problems are posed. The research is completed under the financial support of the RFBR grants no. 05-01-0981 and NSh-2040.2003.1.     

Analytic solution of gauge field equations for Lorentz gravity

The exterior analytic solution for a static, spherically symmetric system is given by means of a set of gauge field equations from Lorentz gravity in the curvature coordinate. The correction contributed by the gravitational gauge field in the exterior of a static sphere is obtained for the gravity.     

On interaction of two closely spaced charged conducting spheres

Electrostatic interaction between two charged conducting spheres is analyzed in the case of a small spacing between them, when the polarization effects are significant. It is shown that short-range polarization forces result in the attraction of the like-charged spheres. At a sufficiently small spacing, this attraction replaces repulsive forces acting on like charges.