We have presented tilted cosmological models in f RT ( , ) theory of gravitation without taking any relation between density and pressure p .The model are expanding, shearing and rotating universe. The model starts with big bang at t c 1 and the expansion in the model decreases as time increases and the expansion in the model stop at t .There is a singularity in the model at t c 1 This singularity is
At Sout hampt on Solent Universit y t he inclusive concept of Advanced Scholarship is used t o recognise and support high-level academic endeavour, including research, and t hird st ream (engagement wit h business and communit ies). The ut ilit y of t his approach has been demonst rat ed f or a wide range of act ivit ies and subj ect disciplines. A key f eat ure is t hat Advanced Scholarship act ivit ies are expect ed t o enhance t he st udent experience, hence a t hird st ream-t eaching-research (T-T-R) nexus.
where S could be of a two-surface with arbitrary topology. In the simplest prac- tical case, however, S would have either planar, cylindrical, toroidal or spherical topology. In these cases, we may assume that there exists a smooth real function ρ : Σ → R whose ρ = const level sets give the S ρ leaves of the foliation and that its
We have successfully obtained exact solution of complex form equation for symmetrical deformation of toroidal shells. The research has confirmed that the deformation of all regular shell structure, such cylindrical shell, conical shell, spherical shell and toiroidal shell can be solved by hypergeometric functions. This supports the doctrine of Zurich school of shell theory [1, 2, 7], which predicted that bending deformation of all regular shells can be expressed by the hypergeometric functions. Through numerical comparison study, the mechanics of toroidal shells is quite sensitive to the radius ratio ε = R a . By slightly adjustment of the ratio might get a desired high performance shell structure.
In this chapter, we present a toric framework using Gorenstein cones that attempts to unify the two mirror construction frameworks of Batyrev-Borisov and BHK. There has been some work in the past on this approach (, , and ). We provide an alternate approach that is done by using the approach given by  and fitting it into an explicit context of toric vector bundles. We take dual Gorenstein cones σ and σ ∨ of index r. In this work, we find fans Σ and Σ ∨ which have corresponding toric varieties X Σ and X Σ ∨ that are vector bundles over two other
In this model, first introduced in , we allow only one rook per column but more than one rook in a given row. We will call an arrangement of rooks, with at most one rook in each column (but possibly more than one rook in a row), an α-rook placement. We will denote the set of all α-rook placements of k rooks on a board B by R (α) k (B). If there are u rooks in a given row, that row has weight
: Let’s start with a very simple problem: This prob- lem is resolvable by a high school student: Consider a mouse moving over a grid consisting of n rows and m columns. His only possible actions are to advance or step up without ever going back or down. it is well known that counting all the possible trajectories consists of counting the number of anagrams consisting of n times the letter A and m times the letter M. in this case the sum of all possible trajectories is a finite number is the combinations of n among n + m: N T =
In recent years the SM4, the upgraded version of SM3 has been in point of attraction by many physicists. After the last piece, Higgs Boson in the SM3, there are many questions in paucity about the masses of next Leptons and Quarks. In SM3, the mass of Higgs Boson is 125.18 ± 0.16 GeV , but there is quite an update that we will encounter with the Higgs masses. In SM4, the mass of Higgs Boson is around 115 GeV (we will examine it later) . With the extension of SM3 to SM4, we just add a couple of Leptons and Quarks, Quarks t’ and b’ with isopin of 1 2 and − 1 2 respectively (but that is not the case, we will see it later) and Lepton l’ and its corresponding Neutrino. There are a lot of discussions about the masses and flavor mixing of leptons and quarks. In our model, we will be looking at the current Koide Formula (Hypothesis) :