Related papers: Operads in It{\^o} calculus

We give a general account of family algebras over a finitely presented linear operad, this operad together with its presentation naturally defining an algebraic structure on the set of parameters.

We study the enumerative geometry of orbits of multidimensional toric action on projective algebraic varieties and develop a new cyclic differential-graded operad, conjecturally governing the real version of the enumerative geometry of…

We provide a representation of the $C^*$-algebra generated by multidimensional integral operators with piecewise constant kernels and discrete ergodic operators. This representation allows us to find the spectrum and to construct the…

It is clarified how cohomologies and Gerstenhaber algebras can be associated with linear pre-operads (comp algebras). Their relation to mechanics and operadic physics is concisely discussed.

The purpose of this paper is to give a characterisation of divided power algebras over a reduced operad. Such a characterisation is given in terms of polynomial operations, following the classical example of divided power algebras. We…

The constructive martingale representation theorem of functional It\^o calculus is extended, from the space of square integrable martingales, to the space of local martingales. The setting is that of an augmented filtration generated by a…

We develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics. The geometric approach consists of using suitable Lagrangian and isotropic submanifolds. We also…

In this paper we give an attempt to extend some arithmetic properties such as multiplicativity, convolution products to the setting of operators theory. We provide a significant examples which are of interest in number theory. We also give…

Let P be a quadratic operad. We determine an associated operad ~P such that for any P-algebra A and any ~P-algebra B then the tensor product $A \otimes B$ is a P-algebra.

We present a general construction of the derived category of an algebra over an operad and establish its invariance properties. A central role is played by the enveloping operad of an algebra over an operad.

In this paper, we study the consequences of the fundamental theorem of calculus from an algebraic point of view. For functions with singularities, this leads to a generalized notion of evaluation. We investigate properties of such…

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…

Let $K$ be an imaginary quadratic field. For an order $\mathcal{O}$ in $K$ and a positive integer $N$, let $K_{\mathcal{O},\,N}$ be the ray class field of $\mathcal{O}$ modulo $N\mathcal{O}$. We deal with various subjects related to…

There are versions of "calculus" in many settings, with various mixtures of algebra and analysis. In these informal notes we consider a few examples that suggest a lot of interesting questions.

We begin with a deformation of a differential graded algebra by adding time and using a homotopy. It is shown that the standard formulae of It\^o calculus are an example, with four caveats: First, it says nothing about probability. Second,…

We construct an operational calculus supported on the algebraic operational calculus introduced by Bengochea and Verde. With this operational calculus we study the solution of certain Bessel type equations.

We introduce a new diagrammatic notation for representing the result of (algebraic) effectful computations. Our notation explicitly separates the effects produced during a computation from the possible values returned, this way simplifying…

A calcular algebra is a subalgebra of $H^\infty(\Omega)$ with norm given by $\| \phi \| = \sup \| \phi(T) \|$ as $T$ ranges over a given class of commutative $d$-tuples of operators with Taylor spectrum in $\O$. We discuss what algebras…

In this paper we consider the $f$-orthomorphisms and $f$-linear operators on the order dual of an $f$-algebra. In particular, when the $f$-algebra has the factorization property (not necessarily unital), we prove that the orthomorphisms,…

Lectures notes (in italian) of some arguments of classical analysis, with exercises. A particular emphasis to functional analysis and elementary operator algebra theory is given, by means of exercises and examples.