Metaplectic flavor symmetries from magnetized tori

We revisit the flavor symmetries arising from compactifications on tori with magnetic background fluxes. Using Euler’s Theorem, we derive closed form analytic expressions for the Yukawa couplings that are valid for arbitrary flux parameters. We discuss the modular transformations for even and odd units of magnetic flux, M, and show that they give rise to finite metaplectic groups the order of which is determined by the least common multiple of the number of zero-mode flavors involved. Unlike in models in which modular flavor symmetries are postulated, in this approach they derive from an underlying torus. This allows us to retain control over parameters, such as those governing the kinetic terms, that are free in the bottom-up approach, thus leading to an increased predictivity. In addition, the geometric picture allows us to understand the relative suppression of Yukawa couplings from their localization properties in the compact space. We also comment on the role supersymmetry plays in these constructions, and outline a path towards non-supersymmetric models with modular flavor symmetries.

Different forms of Euler’s theorem forhomogeneous functions to solve partial differential equation

In this paper we will discuss Euler’s theorem for homogenous functions to solve different order partial differential equations. We will see that how we can predict the solution of partial differential Equation using different approaches of this theorem. In fact we also consider the case when more than two independent variables will be involved in the partial differential equation whenever dependent functions will be homogenous functions. We will throw a light on one method called Ajayous rules to predict the solution of homogenous partial differential equation.

Quasilocal Smarr relation for an asymptotically flat spacetime

We investigate the thermodynamics of Einstein–Maxwell (-dilaton) theory for an asymptotically flat spacetime in a quasilocal frame. We firstly define a quasilocal thermodynamic potential via the Euclidean on-shell action and formulate a quasilocal Smarr relation from Euler’s theorem. Then we calculate the quasilocal energy and surface pressure by employing a Brown–York quasilocal method along with Mann–Marolf counterterm and find entropy from the quasilocal thermodynamic potential. These quasilocal variables are consistent with the Tolman temperature and the entropy in a quasilocal frame turns out to be same as the Bekenstein–Hawking entropy. As a result, we found that a surface pressure term and its conjugate variable, a quasilocal area, do not participate in a quasilocal thermodynamic potential, but should be present in a quasilocal Smarr relation and the quasilocal first law of black hole thermodynamics. For dyonic black hole solutions having dynamic dilaton field, a non-trivial dilaton contribution should occur in the quasilocal first law but not in the quasilocal Smarr relation.

A theorized new class of polyhedral hydrocarbons of molecular formula CnHn and their bottom-up scaffold expansions into hyperstructures

We address the use of Euler's theorem and topological algorithms to design 18 polyhedral hydrocarbons of general formula CnHn that exist up to 28 vertexes containing four- and six-membered rings only; compounds we call “nuggets”. Subsequently, we evaluated their energies to verify the likelihood of their chemical existence. Among these compounds, 13 are novel systems, of which 3 exhibit chirality. Further, the ability of all nuggets to perform fusion reactions either through their square faces, or through their hexagonal faces was evaluated. Indeed, they are potentially able to form bottom-up derived molecular hyperstructures with great potential for several applications. By considering these fusion abilities, the growth of the nuggets into 1D, 2D, and 3D-scaffolds was studied. The results indicate that nugget24a (C24H24) is predicted to be capable of carrying out fusion reactions. From nugget24a, we then designed 1D, 2D, and 3D-scaffolds that are predicted to be formed by favorable fusion reactions. Finally, a 3D-scaffold generated from nugget24a exhibited potential to be employed as a voxel with a chemical structure remarkably similar to that of MOF ZIF-8. And, such a voxel, could in principle be employed to generate any 3D sculpture with nugget24a as its level of finest granularity.

Icosadeltahedral Geometry of Geodesic Domes, Fullerenes and Viruses: A Tutorial on the T-Number

The Caspar–Klug (CK) classification of viruses is discussed by parallel examination of geometry of icosahedral geodesic domes, fullerenes, and viruses. The underlying symmetry of all structures is explained and thoroughly visually represented. Euler’s theorem on polyhedra is used to calculate the number of vertices, edges, and faces in domes, number of atoms, bonds, and pentagonal and hexagonal rings in fullerenes, and number of proteins and protein–protein contacts in viruses. The T-number, the characteristic for the CK classification, is defined and discussed. The superposition of fullerene and dome designs is used to obtain a representation of a CK virus with all the proteins indicated. Some modifications of the CK classifications are sketched, including elongation of the CK blueprint, fusion of two CK blueprints, dodecahedral view of the CK shapes, and generalized CK designs without a clearly visible geometry of the icosahedron. These are compared to cases of existing viruses.

The new integer factorization algorithm based on Fermat’s Factorization Algorithm and Euler’s theorem

Although, Integer Factorization is one of the hard problems to break RSA, many factoring techniques are still developed. Fermat’s Factorization Algorithm (FFA) which has very high performance when prime factors are close to each other is a type of integer factorization algorithms. In fact, there are two ways to implement FFA. The first is called FFA-1, it is a process to find the integer from square root computing. Because this operation takes high computation cost, it consumes high computation time to find the result. The other method is called FFA-2 which is the different technique to find prime factors. Although the computation loops are quite large, there is no square root computing that included into the computation. In this paper, the new efficient factorization algorithm is introduced. Euler’s theorem is chosen to apply with FFA to find the addition result between two prime factors. The advantage of the proposed method is that almost of square root operations are left out from the computation while loops are not increased, they are equal to the first method. Therefore, if the proposed method is compared with the FFA-1, it implies that the computation time is decreased, because there is no the square root operation and the loops are same. On the other hand, the loops of the proposed method are less than the second method. Therefore, time is also reduced. Furthermore, the proposed method can be also selected to apply with many methods which are modified from FFA to decrease more cost.

A Note on Integrating Factors of a Conformable Fractional Differential Equation

Recently exact fractional differential equations have been introduced, using the conformable fractional derivative. In this paper, we propose and prove some new results on the integrating factor. We introduce a conformable version of several classical special cases for which the integrating factor can be determined. Specifically, the cases we will consider are where there is an integrating factor that is a function of only x, or a function of only y, or a simple formula of x and y. In addition, using the Conformable Euler's Theorem on homogeneous functions, an integration factor for the conformable homogeneous differential equations is established. Finally, the above results apply in some interesting examples.