Casimir entropy is an important aspect of casimir effect and at the nanoscale is visible. In this paper, we employ the path integral method to obtain a general relation for casimir entropy and internal energy of arbitrary shaped objects in the presence of two, three and four dimension Scalar Fields and the electromagnetic field. For this purpose, using Lagrangian and based on a perturbative approach, a series expansion in susceptibility function of the medium was obtained for the Casimir force between arbitrary shaped objects foliated in a Scalar or vector fluctuating field in arbitrary dimensions. The finite temperature corrections are derived and using it, we obtain the casimir entropy and internal energy of two nano ribbons immersed in the Scalar field and two nanospheres immersed in the Scalar field and the electromagnetic field. The casimir entropy of two nanospheres immersed in the electromagnetic field behave differently in small interval of temperature variations.

We discuss a generic form of the Scalar potential appearing in the geometric Scalar theory of gravity. We find the conditions on the potential by considering weak and strong gravity. The modified black hole solutions are obtained for generic potentials and the inverse problems on a black hole and on a spherical body (‘pseudogravastar’) are investigated.

In this work, the effect of rotation on the evolution of kinematic and passive Scalar Fields in two dimensional homogeneous sheared turbulence is studied using two different approaches. The first one is analytical and it consists on the resolution of differential linear equations governing the turbulence at high shear when the non linear effects are neglected. The second one is numerical and it consists on the modeling of governing equations using the most known second order models of turbulence and their numerical integration using the fourth order Runge-kutta method. In this second approach, the classic Launder Reece Rodi model, the Speziale Sarkar Gatski and the Shih Lumley models are retained for the pressure-strain correlation, pressure- Scalar gradient correlation and for the time evolution equations of the kinematic and Scalar dissipations. The evolution of turbulence is studied according to the dimensionless rotation number R which is varied from -0.75 to 0.5. The obtained results are compared to the recent results of the DNS of Brethouwer. Both methods have confirmed the existence of asymptotic equilibrium states for dimensionless kinematic and Scalar parameters.

Let R be a commutative ring with identity and M an R-module. The Scalar-Product Graph of M is defined as the graph GR(M) with the vertex set M and two distinct vertices x and y are adjacent if and only if there exist r or s belong to R such that x = ry or y = sx. In this paper, we discuss connectivity and planarity of these graphs and computing diameter and girth of GR(M). Also we show some of these graphs is weakly perfect.

In the current paper, we have studied the effect of dark energy on formation where dark energy exists in the background. For this purpose, we used both WMAP9 and Planck data to study how the radius changes with redshift in these models. We used different data sets to fix the cosmological parameters to obtain a solution for a spherical region under collapse. The mechanism of structure formation for dark and baryonic matter is different. When processed by gravitational instability, density perturbations have given rise to collapsed dark matter structures, called halos. These dark matter halos offer the backdrop for the subsequent formation of all collapsed baryonic structures, including stars, galaxies, and galaxy clusters. In Planck Data for Λ CDM, with the presence of dark energy in the background, the formation of baryonic matter is delayed. Therefore, it is a factor for the largening of the baryonic matter radius. Accompanying dark energy is entailing an increment of dark matter virial radius. For WACDM Data, dark energy alongside time-dependent parameter of state and baryon acoustic oscillations are the reasons for the delay of dark matter formation and the radius reduction. Due to the lack of data without baryonic acoustic waves in the background, we are left unable to delineate its impact on the structures. In WCDM(BAO+H0) and WCDM(H0), the lack of BAO shows a critical role in the delaying of baryonic matter structure formation. Respectively, it causes growing virial radius of dark matter. BAO, without taking dark energy into accounts, is the reason for the increasing and decresing of radius of dark and baryonic matter. It also delays baryonic matter formation. In Λ CDM(BAO+H0) and Λ CDM(H0), We have studied Λ CDM data for standard model under two circumstances: (a) Λ CDM(BAO + H0), (b) Λ CDM(H0) data. Dark energy in this data delays formation and intensifies virial radius of baryonic matter. Our studies show WCDM and Λ CDM have the same effect on formation if we do not consider dark energy in BG. Planck data, in comparing with WMAP, has important role in describing standard model.

In this paper, we examine the singlet Scalar dark matter annihilation to becoming the Standard Model particles in the non-commutative space. In the recent decades, many candidates of dark matter have been offered, but our information about the nature of dark matter is still limited. There are such particle candidates as Scalar matetr, fermion, boson, gauge boson, etc.; however, they have neither approved nor rejected. Some complicated models offer many parameters that must be determined. Among these models, the singlet Scalar dark matter model is the simplest one that contains just two free parameters to specify. In this paper, we study the singlet Scalar dark matter in the noncommutative space, give the Scalar dark matter and higgs lagrangian to find their coupling in noncommutative space-time and calculate the cross-section of the Scalar dark matter pair annihilation to becoming the standard model particles. It must be noted that there is one extra coupling to explain the dark matter annihilation to becoming the standard model particles in the noncommutative space-time. We investigate the parameter space using the constraint of the thermal average of annihilation cross-section by the velocity. This is the first research on the dark matter annihilation to becoming the standard model particles through the higgs channel in the noncommutative space.

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