Related papers: Operads in It{\^o} calculus
We introduce an original approach to geometric calculus in which we define derivatives and integrals on functions which depend on extended bodies in space--that is, paths, surfaces, and volumes etc. Though this theory remains to be fully…
We construct generalized additions and multiplications, forming fields, and division algebras inspired by the Tsallis thermo-statistics. We also construct derivations and integrations in this spirit. These operations do not reduce to the…
We study some natural operators acting on configurations of points and lines in the plane and remark that many interesting configurations are fixed points for these operators. We review ancient and recent results on line or point…
Differential calculus on metric spaces is contained in the algebraic study of normed groupoids with $\delta$-structures. Algebraic study of normed groups endowed with dilatation structures is contained in the differential calculus on metric…
The task to construct parametrices of elliptic differential operators on a manifold with edges requires a calculus of operators with a two-component principal symbolic hierarchy, consisting of (edge-degenerate) interior and…
The Jacobian algebras are introduced and their various properties are studied.
We present geometric characterizations of the partial isometries, unitaries, and invertible operators in C*-algebras and von Neumann algebras.
In this paper we investigate the spectra and the ergodic properties of the multiplication operators and the convolution operators acting on the Schwartz space $\mathcal{S}(\mathbb{R})$ of rapidly decreasing functions, i.e., operators of the…
Some binary quadratic operads are endowed with anticyclic structures and their characteristic functions as anticyclic operads are determined, or conjectured in one case.
We present a treasure trove of open problems in matrix and operator inequalities, of a functional analytic nature, and with various degrees of hardness.
We use the methods of empirical mathematics to show that iterative logarithmic operations will result in an attractor point on the complex plane. Moreover, we demonstrate that different bases converge onto different attractors. Finally, we…
Unlike for $\ell_p$, $1<p\leq\infty$, the discrete Ces\`aro operator $C$ does not map $\ell_1$ into itself. We identify precisely those weights $w$ such that $C$ does map $\ell_1(w)$ continuously into itself. For these weights a complete…
We give a short direct proof of Agler's factorization theorem that uses the abstract characterization of operator algebras. the key ingredient of this proof is an operator algebra factorization theorem. Our proof provides some additional…
The isomorphism between the reduction algebra and the invariant differential operators on G/H is sketched.
We consider relationships between cubic algebras and implication algebras. We first exhibit a functorial construction of a cubic algebra from an implication algebra. Then we consider an collapse of a cubic algebra to an implication algebra…
A general notion of operad is given, which includes as instances, the operads originally conceived to study loop spaces, as well as the higher operads that arise in the globular approach to higher dimensional algebra. In the framework of…
In these notes we study several categorical generalizations of the M\"obius function and discuss the relations between the various approaches. We emphasize the topological and geometric meaning of these constructions.
We develop the integral calculus for quasi-standard smooth functions defined on the ring of Fermat reals. The approach is by proving the existence and uniqueness of primitives. Besides the classical integral formulas, we show the…
In a recent paper, the author introduced a rich class $NC^k(\mathbb{R})$ of "noncommutative $C^k$" functions $\mathbb{R} \to \mathbb{C}$ whose operator functional calculus is $k$-times differentiable and has derivatives expressible in terms…
We describe various structures of algebraic nature on the space of continuous valuations on convex sets, their properties (like versions of Poincar\'e duality and hard Lefschetz theorem), and their relations and applications to integral…