The squared Commutativity degree of dihedral groups

The commutativity degree of a finite group is the probability that a random pair of elements in the group commute. Furthermore, the n-th power commutativity degree of a group is a generalization of the commutativity degree of a group which is defined as the probability that the n-th power of a random pair of elements in the group commute. In this paper, the n-th power commutativity degree for some dihedral groups is computed for the case n equal to 2, called the squared commutativity degree

Macroscopic Screening of Coulomb Potentials From UV/IR-Mixing

We compute the static potential in a non-commutative theory including a term due to UV/IR-mixing. As a result, the potential decays exponentially fast with distance rather than like a power law Coulomb type potential due to the exchange of massless particles. This shows that when quantum effects are taken into account the introduction of non-commutativity not only modifies physics at short distances but has dramatic macroscopic consequences as well. As a result, we give a lower bound on the scale of non-commutativity (if present at all) to be compatible with observations.Comment: 10 pages, V2 minor wording and reference

D=2, N=2, Supersymmetric theories on Non(anti)commutative Superspace

The classical action of a two dimensional N=2 supersymmetric theory, characterized by a general K\"{a}hler potential, is written down on a non(anti)commutative superspace. The action has a power series expansion in terms of the determinant of the non(anti)commutativity parameter $C^{\alpha\beta}$. The theory is explicitly shown to preserve half of the N=2 supersymmetry, to all orders in (det C)^n. The results are further generalized to include arbitrary superpotentials as well.Comment: 32 pages, Latex; v2:minor typos corrected and a reference adde

Non-commutative Power-law Inflation: Mode Equation and Spectra Index

Following an elegant approach that merge the effects of the stringy spacetime uncertainty relation into primordial perturbations suggested by Brandenberger and Ho, we show the mode equation up to the first order of non-commutative parameter. A new approximation is provided to calculate the mode functions analytically in the non-commutative power-law inflation models. It turns out that non-commutativity of spacetime can provide small corrections to the power spectrum of primordial fluctuations as the first-year results of WMAP indicate. Moreover, using the WMAP data, we obtain the value of expansion parameter, non-commutative parameter and find the approximation is viable. In addition, we determined the string scale $l_s \simeq 2.0\times 10^{-29}{cm}$.Comment: 10 pages, 1 figure, to appear in Phys. Lett.

Relative n-isoclinism classes and relative n-th nilpotency degree of finite groups

The purpose of the present paper is to consider the notion of isoclinism between two finite groups and its generalization to n-isoclinism, introduced by J. C. Bioch in 1976. A weaker form of n-isoclinism, called relative n-isoclinism, will be discussed. This will allow us to improve some classical results in literature. We will point out the connections between a relative n-isoclinism and the notions of commutativity degree, n-th nilpotency degree and relative n-th nilpotency degree, which arouse interest in the classification of groups of prime power order in the last years.Comment: 11 pages, to appear in Filomat with revision

Wilsonian Proof for Renormalizability of N=1/2 Supersymmetric Field Theories

We provide Wilsonian proof for renormalizability of four-dimensional quantum field theories with ${\cal N}=1/2$ supersymmetry. We argue that the non-hermiticity inherent to these theories permits assigning noncanonical scaling dimension both for the Grassman coordinates and superfields. This reassignment can be done in such a way that the non(anti)commutativity parameter is dimensionless, and then the rest of the proof ammounts to power counting. The renormalizability is also stable against adding standard four-dimensional soft-breaking terms to the theory. However, with the new scaling dimension assignments, some of these terms are not just relevant deformations of the theory but become marginal.Comment: 10 pages, no figure, v2: minor correctio