Related papers: An operadic approach to operator-valued free cumul…
We investigate operator-valued monotone independence, a noncommutative version of independence for conditional expectation. First we introduce operator-valued monotone cumulants to clarify the whole theory and show the moment-cumulant…
We discuss free probability theory and free harmonic analysis from a categorical perspective. In order to do so, we extend first the set of analytic convolutions and operations and then show that the comonadic structure governing free…
Wall-crossing phenomena are ubiquitous in many problems of algebraic geometry and theoretical physics. Various ways to encode the relevant information and the need to track the changes under the variation of parameters lead to rather…
We consider various systematic ways of defining unbounded operator valued integrals of complex functions with respect to (mostly) positive operator measures and positive sesquilinear form measures, and investigate their relationships to…
We establish the functional relations between generating series of higher-order free cumulants and moments in higher-order free probability, solving an open problem posed fifteen years ago by Collins, Mingo, \'Sniady and Speicher. We…
An analytically derived 'integral operator' approach is introduced to estimate the expectation value of a quantum operator for an evolving state weighted with an exponential function. This allows to compute quantities useful in Nuclear…
The operad of moulds is realized in terms of an operational calculus of formal integrals (continuous formal power series). This leads to many simplifications and to the discovery of various suboperads. In particular, we prove a conjecture…
We study quotients of the magmatic operad, that is the free nonsymmetric operad over one binary generator. In the linear setting, we show that the set of these quotients admits a lattice structure and we show an analog of the Grassmann…
In this paper, we investigate the power of nearly purely operational techniques in the study of umbral calculus. We present a concise reconstruction of the theory based on a systematic use of linear operators, with particular attention to…
The functional equation defining the free cumulants in free probability is lifted successively to the noncommutative Fa\`a di Bruno algebra, and then to the group of a free operad over Schr\"oder trees. This leads to new combinatorial…
This is a survey on Drinfeld associators and their generalizations, where we focus on operadic aspects.
Operads are algebraic devices offering a formalization of the concept of operations with several inputs and one output. Such operations can be naturally composed to form bigger and more complex ones. Coming historically from algebraic…
This paper is an introduction to a series of papers in which we give combinatorial models for certain important operads (including A-infinity and E-infinity operads, the little n-cubes operads, and the framed little disks operad) and…
This paper introduces a simple and computationally efficient algorithm for conversion formulae between moments and cumulants. The algorithm provides just one formula for classical, boolean and free cumulants. This is realized by using a…
A unitary operator which relates the system of a particle in a linear potential with time-dependent parameters to that of a free particle, has been given. This operator, closely related to the one which is responsible for the existence of…
We give an explicit description, via analytic subordination, of free multiplicative convolution of operator-valued distributions. In particular, the subordination function is obtained from an iteration process. This algorithm is easily…
We establish and explore a relationship between two approaches to moment-cumulant relations in free probability theory: on one side the main approach, due to Speicher, given in terms of M\"obius inversion on the lattice of noncrossing…
In this paper, we introduce the notion of conditionally bi-free independence in an amalgamated setting. We define operator-valued conditionally bi-multiplicative pairs of functions and construct operator-valued conditionally bi-free moment…
We give a formula for the derivatives of a correlation function of composite operators with respect to the parameters (i.e., the strong fine structure constant and the quark mass) of QCD in four-dimensional euclidean space. The formula is…
A general dynamical invariant operator for three coupled time-dependent oscillators is derived. Although the obtained invariant operator satisfies the Liouville-von Neumann equation, its mathematical formula is somewhat complicated due to…