Related papers: The Moment of an Operator in the Free Group Factor…
In this paper, we will consider the free probability theory of free group factor $L(F_{N}),$ where $F_{k}$ is the free group with $k$-generators. We compute the moment series and the R-transform of the generating operator,…
In [13] and [14], we found so-called the reccurence diagram. This paper is the application of such recurrence diagram. As a first application for the reccurence diagram, we will re-compute the well-known moment series of the generating…
In this paper, we will consider the generating operator of the free group factor L(F_2). Then we can construct the group von Neumann algebra L(K), where K is the commutator group of F_2 and the conditional expectation E. Then (L(F_2), E) is…
064<p type="texpara" tag="Body Text" et="abstract" bin="clone" >In this paper, we will consider an example of a (scalar-valued) moment series, under the compatibility. Suppose that we have an amalgamated free product of free group algebras,…
In this paper, we will compute the moments of the block operators, induced by the generators of the group, in a group von Neumann algebra and we will observe the identically distributedness.
We show that for a fixed free group F and an arbitrary finitely generated subgroup H (as given above) we can perform the Stalling's folding process in time O(N log^*(N)), where N is the sum of the word lengths of the given generators of H.
The theory of operator integrals is used to determine the moment operators of the Cartesian margins of the phase space observables generated by the mixtures of the number states. The moments of the $x$-margin are polynomials of the position…
We give an explicit formula for the symbol of a function of an operator. Given an operator {\hat A} on L^2(R^N) with symbol A, and a function f, we obtain the symbol of f(\hat A) in terms of A. As an application, Bohr-Sommerfeld…
We calculate the moments of the Meyer-K\"onig and Zeller operators in terms of elementary functions and polylogarithms.
Relations between two definitions of (total) angular momentum operator, as a generator of rotations and in the Lagrangian formalism, are explored in quantum field theory. Generally, these definitions result in different angular momentum…
We explicitly calculate the moments t_n of general Heisenberg Hamiltonians up to sixth order. They have the form of finite sums of products of two factors, the first factor being represented by a multigraph and the second factor being a…
Let $F_2$ be the free group on two generators and let $H$ be a subgroup of $F_2$. We investigate a method for calculating the number of elements in a coset of $H$ that have a given length when written in reduced form. More specifically,…
We revisit the problem of deciding whether a finitely generated subgroup H is a free factor of a given free group F. Known algorithms solve this problem in time polynomial in the sum of the lengths of the generators of H and exponential in…
The goal of this paper is to generalize most of the moment formulae obtained in [Pri11]. More precisely, we consider a general point process \mu, and show that the relevant quantities to our problem are the so-called Papangelou intensities.…
For each sequence $\{c_n\}_n$ in $l_{1}(\N)$ we define an operator $A$ in the hyperfinite $\mathrm{II}_1$-factor $\mathcal{R}$. We prove that these operators are quasinilpotent and they generate the whole hyperfinite $\mathrm{II}_1$-factor.…
Let $A,B\in B(H)$. In the present paper, we establish simple and interesting facts on when we have $|A||B|=|B||A|$, $|AB|=|A||B|$, $|A\pm B|\leq |A|+|B|$, $||A|-|B||\leq |A\pm B|$ and $\||A|-|B|\|\leq \|A\pm B\|$, where $|\cdot|$ denotes…
In the renormalisation analysis of critical phenomena in quasi-periodic systems, a fundamental role is often played by fixed points of functional recurrences of the form \begin{equation*} f_{n}(x) = \sum_{i=1}^\ell a_i(x) f_{n_i}…
This paper describes the probabilistic behaviour of a random Sturmian word. It performs the probabilistic analysis of the recurrence function which can be viewed as a waiting time to discover all the factors of length $n$ of the Sturmian…
We compute the "moments" and its continuous analogue of the random variable $\N \ni l \mapsto \gcd(l,n_1) \gcd(l, n_2) ... \gcd(l, n_k) \in \N$ by a purely elementary method. This generalizes a result of Deitmar-Koyama-Kurokawa, which…
An operator mean is a binary operation assigned to each pair of positive operators satisfying monotonicity, continuity from above, the transformer inequality and the fixed-point property. It is well known that there are one-to-one…