A (set) partition of [n] = {1, 2, . . . , n} is a collection of mutually disjoint nonempty subsets, called blocks, of [n] whose union is [n]. We will write a partition as a sequence of blocks (B1, B2, . . . , Bk) such that min(B1) < min(B2) < · · · < min(Bk). There are two natural representations of a partition. Let π = (B1, B2, . . . , Bk) be a partition of [n]. The partition diagram of π is t...

The canonical partition function of a system of rotators (classical X-Y spins) on a lattice, coupled by terms decaying as the inverse of their distance to the power alpha, is analytically computed. It is also shown how to compute a rescaling function that allows us to reduce the model, for any d-dimensional lattice and for any alpha<d, to the mean-field (alpha=0) model.

Theorems on zeroes of the truncated generating function in the complex plane are reviewed. When examined in the framework of a statistical model of high energy collisions based on the negative binomial (Pascal) multiplicity distribution, these results lead to maps of zeroes of the grand canonical partition function which allow to interpret in a novel way different classes of events in pp collis...

Abstract. Starting from the defining transformations of complex matrices for the SO(4, R) group, we construct the fundamental representation and the tensor and spinor representations of the group SO(4, R). Given the commutation relations for the corresponding algebra, the unitary representations of the group in terms of the generalized Euler angles are constructed. The crucial step for the Barr...

We present a path integral formalism for quantising the spectral action of noncommutative geometry, based on the principle of summing over all Dirac operators. The approach is demonstrated on two simple finite noncommutative geometries. On the first, the graviton is described by a Higgs field, and on the second, it is described by a gauge field. We start with the partition function, and calcula...

The simplest possible noncommutative harmonic oscillator in two dimensions is used to quantize the free closed bosonic string in two flat dimensions. The partition function is not deformed by the introduction of noncommutativity, if we rescale the time and change the compactification radius appropriately. The four point function is deformed, preserving, nevertheless, the sl(2,C) invariance. Fin...

The graphical calculus method is generalized to study the relation between covariant and canonical dynamics of loop quantum gravity. On one hand, a derivation partition function Euclidean Engle-Pereira-Rovelli-Livine (EPRL) spin-foam model presented. other action Hamiltonian constraint operator on certain spin network states calculated by method. It turns out that EPRL can provide rigging map s...