The concept of soft set theory can be used as a mathematical tool for dealing with problems that contain uncertainty. Then, new mixed model called the bipolar is created by merging sets and bipolarity, which gave binary grading. Bipolar characterized two sets, one provides positive information other negative. generalized topology generalization topology. importance limit points in all branches ...

A classical limit theorem of stochastic process theory concerns the sample cumulative distribution function (CDF) from independent random variables. If the variables are uniformly distributed then these centered CDFs converge in a suitable sense to the sample paths of a Brownian Bridge. The so-called Hungarian construction of Komlos, Major and Tusnady provides a strong form of this result. In t...

We study quantum field theories with sextic interactions in $3\ensuremath{-}\ensuremath{\epsilon}$ dimensions, where the scalar fields ${\ensuremath{\phi}}^{ab}$ form irreducible representations under $O(N{)}^{2}$ or $O(N)$ global symmetry group. calculate beta functions up to four-loop order and find renormalization group (RG) fixed points. In an example of large $N$ equivalence, parent theory...

Statistical mechanical lattice models are called solvable if their associated Boltzmann weights satisfy the factorization or star-triangle equations of McGuire [1], Yang [2] and Baxter [3]. For such models the free energy per site and the one-point correlations in the thermodynamic limit are expressible in closed form [4]. There exists a deep mathematical structure [4, 5] underlying these solva...