In this paper, we introduce ubergravity based on the idea of the ensemble average theory of f (R) gravity models. This model has interesting properties including its universality. Another property of this model is its curvature dependency: In high curvatures, ubergravity is reduced to standard Einstein-Hilbert action, while in low curvatures, it vanishes. This transition happens at a scale of R0. It is possible to show that this dimensionful scale plays the role of the cosmological constant at late times. To consider the behavior of the non-linear structure formation, we need to start with the spherical collapse; hence, here we study the spherically symmetric solutions. We show that the radius dependency of these solutions is totally different from the Schwarzchild solution. Interestingly, the solutions looks like the Reissner-Nordstrom in the absence of any electrical charge.

In this paper, exact closed-form solutions for displacement and stress components of thick-walled functionally graded (FG) spherical pressure vessels are presented. To this aim, linear variation of properties, as an important case of the known power-law function model is used to describe the FG material distribution in thickness direction. Unlike the previous studies, the vessels can have arbitrary inner to outer stiffness ratio without changing the function variation of FGM. After that, a closed-form solution is presented for displacement and stress components based on exponential model for variation of properties in radial direction. The accuracy of the present analyses is verified with known results. Finally, the effects of non-homogeneity and different values of inner to outer stiffness ratios on the displacement and stress distribution are discussed in detail. It can be seen that for FG vessels subjected to internal pressure, the variation of radial stress in radial direction becomes linear as the inner stiffness becomes five times higher than outer one. When the inner stiffness is half of the outer one, the distribution of the circumferential stress becomes uniform. For the case in which the external pressure is applied, as the inner to outer shear modulus becomes lower than 1/5, the value of the maximum radial stress is higher than external pressure.

In this study, an exact analytical solution for the convective heat transfer equation from a semi-spherical fin was presented. To obtain a mathematical model, the system was assumed to be a lump in the vertical direction and the governing equation in the Cartesian coordinate was transferred to the Mathieu equation. The exact solution was compared with numerical results such as the finite difference method and midpoint method with Richardson extrapolation (Midrich). Not surprisingly, the exact solution prevailed over the numerical solutions in terms of accuracy and ease of use. Furthermore, the effect of Biot number on the heat transfer of the fin and the fine performance was investigated. The relative error of the results obtained from the analytical and numerical solutions at the base, center, and tip of the fin was 0, 7. 72, and 40. 25 percent, respectively. The results showed that the relative error between the analytical and numerical solutions depends on the Biot number and varies as a function of the fin length. The obtained analytical solution could be encouraging from different mathematical and industrial applications' points of view.

Polishing is considered as the last and most important step in the manufacturing of optical components. Computer control polishing (CCP) methods are usually used to polish complex surfaces. In this method, material removal is controlled at each point, depending on error at that point. In contact polishing mechanism, tool feed rate is often controlled to eliminate local errors. It means that the higher the tool feed rate, the lower the material removal would be and vice versa. Tool influence function (TIF), which is defined as the instantaneous material removal under the polishing tool for a given tool motion, is the most important parameter in CCP and its predictability during the polishing process leads to reliable result. In this study, a new spherical tool which can polish complex surfaces by using a 3-axis CNC machine is presented. Because of spherical geometry of both tool and work piece, tool, material removal rate is variable because of changing the angle between tool axis and surface normal vector which leads to variation of relative speed. Tool influence function which depends on tool engagement’s angle was modeled based on Preston equation. Moreover, the simulation is modeled based on discretization of tool path. To evaluate the methodology, some polishing experimental tests were performed. The experimental results show that a 130 mm spherical convex lens with initial surface roughness of 1.114 micrometer for PV was decreased to 395 nm for PV using the CCP method developed in this study.

In this research article, new fuzzy K-algebras, namely, spherical fuzzy K-algebras and (∈ , ∈ ∨ q)-spherical fuzzy K-algebras are constructed. Certain properties of these spherical fuzzy K-structures are investigated. The behavior of spherical fuzzy K-algebras under homomorphism is characterized. The spherical fuzzy K-algebra with thresholds is also delineated.

Theory and experiment of a spherical probe-fed conformal antenna with a parasitic element mounted on a spherical multilayer structure are presented in this paper. Rigorous mathematical Method of Moments (MoMs) for analyzing various radiating spherical structures is presented in this paper by using Dyadic Green's Functions (DGFs) in conjunction with Mixed Potential Integral Equation (MPIE) formulation. Linear Rao-Wilton-Glisson (RWG) triangular basis functions are applied in MPIE formulation. Current distributions on coaxial probe and conformal radiating elements are computed by using spatial domain Dyadic Green's Function (DGF) and its asymptotic approximation. A prototype of such an antenna is fabricated and tested. The effect of the parasitic element on the input impedance and radiation patterns of the antenna is investigated. It is shown that the antenna characteristics are improved significantly with the presence of the conducting parasitic element. Good agreement is achieved between the results obtained from the proposed methods and the measurement results.

In this paper, we propose to extend the hierarchical bivariate Hermite Interpolant to the spherical case. Let T be an arbitrary spherical triangle of the unit sphere S and let u be a function defined over the triangle T. For kϵN, we consider a Hermite spherical Interpolant problem Hk defined by some data scheme Dk (u) and which admits a unique solution pk in the space Bnk (T) of homogeneous Bernstein-Bezier polynomials of degree nk=2 k (resp. nk=2k+1) defined on T. We discuss the case when the data scheme Dr(u) are nested, i.e., Dr-1 (u) ÌDr(u) for all 1£r£k. This, give a recursive formulae to compute the polynomialpk. Moreover, this decomposition give a new basis for the space Bnk (T), which are the hierarchical structure. The method is illustrated by a simple numerical example.

This paper presents an analytical solution for two-dimensional conductive heat transfer in spherical composite pressure vessels. The vessels are in a spherical shape and fibers are winded in circumferential direction. The vessel is made of one-layer reinforced composite material. The analytical solution is obtained under the general boundary conditions which consist of convection, conduction and radiation inside/outside of vessel. The heat transfer equation for orthotropic conduction in spherical coordinates is derived and solved using separation of variables method based on the Legendre and Euler functions. Here, the effect of fiber's angle on heat conduction in orthotropic spherical pressure vessels is investigated in detail. These results can be used extensively for analyzing the thermal stress in this kind of vessels.

The mean spherical approximation (MSA) has been applied to accurately correlate the activity coefficients of individual ions in aqueous solutions of single 1:1 electrolytes. In order to account for changes in the hydration of the ions, the size parameters of the ions are considered to be composition dependent. The values of the size parameters show that the hydration of the ions decreases as the electrolyte concentration increases. Results obtained from the full MSA and from a simplified version of MSA(SMSA), were compared. The use of the simplified version had a small effect on the final results and dramatically reduced the computation of the effort.