Related papers: An operadic approach to operator-valued free cumul…
A new anticyclic operad Mould is introduced, on spaces of functions in several variables. It is proved that the Dendriform operad is an anticyclic suboperad of this operad. Many operations on the free Mould algebra on one generator are…
We consider the notions of operator-valued infinitesimal (OVI) free independence, OVI Boolean independence, and OVI monotone independence. For each notion of OVI independence, we introduce the corresponding infinitesimal transforms, and…
We study the positive-operator-valued measures on the projective real line covariant with respect to the projective group, assuming that the energy is a positive operator. This problem is similar to the more complicated problem of finding…
A new model of quantum computing has recently been proposed which, in analogy with a classical lambda-calculus, exploits quantum processes which operate on other quantum processes. One such quantum meta-operator takes N unitary…
We propose a new unified framework for Thompson-like groups using a well-known device called operads and category theory as language. We discuss examples of operad groups which have appeared in the literature before. As a first application,…
The \Delta-convolution of real probability measures, introduced by Bo\.zejko, generalizes both free and boolean convolutions. It is linearized by the \Delta-cumulants, and Yoshida gave a combinatorial formula for moments in terms of…
We propose an operadic framework suitable for describing algebraic structures with operations being multilinear differential operators of varying orders or, more generally, formal series of such operators. The framework is built upon the…
Operator learning offers a robust framework for approximating mappings between infinite-dimensional function spaces. It has also become a powerful tool for solving inverse problems in the computational sciences. This chapter surveys…
We study the mirror-field interaction in several frameworks: when it is driven, when it is affected by an environment and when a two-level atom is introduced in the cavity. By using operator techniques we show how these problems may be…
This paper outlines a mathematical framework of quantum probability in which the time asymmetry in describing measuring processes is avoided. The main objects of the framework are hyperfinite operations, which are constructed by using…
In this paper we develop an operational formulation of General Relativity similar in spirit to existing operational formulations of Quantum Theory. To do this we introduce an operational space (or op-space) built out of scalar fields. A…
We present a new formula for umbral operators that yields three main insights. First, it makes explicit a connection between umbral calculus and iteration theory. Second, it leads naturally to a definition of fractional exponents of umbral…
Process theories provide a powerful framework for describing compositional structures across diverse fields, from quantum mechanics to computational linguistics. Traditionally, they have been formalized using symmetric monoidal categories…
We will show that the Morrison-Walker blob complex appearing in Topological Quantum Field Theory is an operadic bar resolution of a certain operad composed of fields and local relations. As a by-product we develop the theory of unary…
We review "quantum" invariants of closed oriented 3-dimensional manifolds arising from operator algebras.
In this paper, a connection between bi-free probability and the theory of non-commutative stochastic processes is examined. Specifically it is demonstrated that the transition operators for non-commutative stochastic processes can be…
It is explained how the time evolution of the operadic variables may be introduced. As an example, an operadic Lax representation of the harmonic oscillator is considered.
We use here a recent idea of studying functions of free random variables using Boolean cumulants. We develop idea of explicit calculations of conditional expectation using Boolean cumulants. We demonstrate Boolean cumulants approach allows…
The suggested operator manifold formalism enables to develop an approach to the unification of the geometry and the field theory. We also elaborate the formalism of operator multimanifold yielding the multiworld geometry involving the…
We prove an implicit function theorem and an inverse function theorem for free noncommutative functions over operator spaces and on the set of nilpotent matrices. We apply these results to study dependence of the solution of the initial…