Related papers: Formal calculus and umbral calculus
We use high girth, high chromatic number hypergraphs to show that there are finite models of the equational theory of the semiring of nonnegative integers whose equational theory has no finite axiomatisation, and show this also holds if…
This course, intended for undergraduates familiar with elementary calculus and linear algebra, introduces the extension of differential calculus to functions on more general vector spaces, such as functions that take as input a matrix and…
We generalize the tensor product theory for modules for a vertex operator algebra previously developed in a series of papers by the first two authors to suitable module categories for a ``conformal vertex algebra'' or even more generally,…
A Baxter algebra is a commutative algebra $A$ that carries a generalized integral operator. In the first part of this paper we review past work of Baxter, Miller, Rota and Cartier in this area and explain more recent work on explicit…
In the references [HL1]--[HL5] and [H1], a theory of tensor products of modules for a vertex operator algebra is being developed. To use this theory, one first has to verify that the vertex operator algebra satisfies certain conditions. We…
Classical mechanics is presented here in a unary operator form, constructed using the binary multiplication and Poisson bracket operations that are given in a phase space formalism, then a Gibbs equilibrium state over this unary operator…
At the first part of the paper we show how specific umbral extensions of the Stirling numbers of the second kind result in new type of Dobinski-like formulas. In the second part among others one recovers how and why Ward solution of…
We present a new approach to the question of when the commutativity of operator exponentials implies that of the operators. This is proved in the setting of bounded normal operators on a complex Hilbert space. The proofs are based on some…
The concept of $\lambda$-differential operators is a natural generalization of differential operators and difference operators. In this paper, we determine the $\lambda$-differential Lie algebraic structure on the Witt algebra and the…
In this paper, we study averaging operators from an algebraic and combinatorial point of view. We first construct free averaging algebras in terms of a class of bracketed words called averaging words. We next apply this construction to…
In the theory of species, differential as well as integral operators are known to arise in a natural way. In this paper, we shall prove that they precisely fit together in the algebraic framework of integro-differential rings, which are…
Given a weighted $\ell^2$ space with weights associated to an entire function, we consider pairs of weighted shift operators, whose commutators are diagonal operators, when considered as operators over a general Fock space. We establish a…
In this paper we study the representations of loop Affine-Virasoro Algebras. As they have canonical triangular decomposition, we define Verma modules and its irreducible quotients. We give necessary and sufficient condition for an…
Modular operads are a special type of operad: in fact, they bear the same relationship to operads that graphs do to trees (i.e. simply connected graphs). One of the basic examples of a modular operad is the collection of…
Using functional equations, we define functors that generalize standard examples from calculus of one variable. Examples of such functors are discussed and their Taylor towers are computed. We also show that these functors factor through…
In this paper, mirror extensions of vertex operator algebras is considered via tensor categories. The mirror extension conjecture is proved.
We investigate and characterize the dynamics of operator growth in irrational two-dimensional conformal field theories. By employing the oscillator realization of the Virasoro algebra and CFT states, we systematically implement the Lanczos…
In this paper, extensions of nonunitary rational Virasoro vertex operator algebras corresponding to some exceptional modular invariants are constructed. The uniqueness of these extensions is also established.
Using $q$-calculus we study a family of reproducing kernel Hilbert spaces which interpolate between the Hardy space and the Fock space. We give characterizations of these spaces in terms of classical operators such as integration and…
We introduce the Umbral calculus into Clifford analysis starting from the abstract of the Heisenberg commutation relation $[\frac{d}{dx}, x] = {\bf id}$. The Umbral Clifford analysis provides an effective framework in continuity and…