Related papers: Simplicial Differential Calculus, Divided Differen…
The Chevalley-Eilenberg differential calculus and differential operators over N-graded commutative rings are constructed. This is a straightforward generalization of the differential calculus over commutative rings, and it is the most…
Extending the model of the interval, we explicitly define for each $n\ge 0$ a free complete differential graded Lie algebra $\mathfrak{L}_n$ generated by the simplices of $\Delta^n$, with desuspended degrees, in which the vertices are…
Some formal analogies between the Differential Calculus in One Variable and the Differential Calculus in Several Variables are presented. It is studied and introduced the derivability of functions at several variables from the single…
Simplicial distributions provide a framework for studying quantum contextuality, a generalization of Bell's non-locality. Understanding extremal simplicial distributions is of fundamental importance with applications to quantum computing.…
We provide a complete geometric solution to the problem of differentiating simplicial manifolds, extending classical Lie theory and complementing existing homotopical and formal approaches within a unifying framework. First, we establish a…
Let X,Y be finite sets and T a set of functions from X -> Y which we will call "tableaux". We define a simplicial complex whose facets, all of the same dimension, correspond to these tableaux. Such "tableau complexes" have many nice…
We develop differential calculus and gauge theory on a finite set G. An elegant formulation is obtained when G is supplied with a group structure and in particular for a cyclic group. Connes' two-point model (which is an essential…
In this paper the analogy between differential forms arising from integrals in additive calculus and forms arising from the integrals in product calculus is investigated. It is found that with an appropriate definition of scalar…
Recent work of Biedermann and R\"ondigs has translated Goodwillie's calculus of functors into the language of model categories. Their work focuses on symmetric multilinear functors and the derivative appears only briefly. In this paper we…
We introduce a simple extension of the $\lambda$-calculus with pairs---called the distributive $\lambda$-calculus---obtained by adding a computational interpretation of the valid distributivity isomorphism $A \Rightarrow (B\wedge C)\ \…
We shall describe the divisor class group and the graded canonical module of the multi-section ring for a normal projective variety X and Weil divisors D_1,..., D_s on X under a mild condition. In the proof, we use the theory of Krull…
In this note we prove that the simplicial derived functors introduced by Tierney and Vogel [TV69] are naturally isomorphic to the cubical derived functors introduced by the author in [P09]. We also explain how this result generalizes the…
We propose a sheaf-theoretic approach to the theory of differential calculi on quantum principal bundles over non-affine bases. After recalling the affine case we define differential calculi on sheaves of comodule algebras as sheaves of…
The usual Weyl calculus is intimately associated with the choice of the standard symplectic structure on $\mathbb{R}^{n}\oplus\mathbb{R}^{n}$. In this paper we will show that the replacement of this structure by an arbitrary symplectic…
In this paper we develop a geometric approach to convex subdifferential calculus in finite dimensions with employing some ideas of modern variational analysis. This approach allows us to obtain natural and rather easy proofs of basic…
We develop a calculus of variations for functionals which are defined on a set of non differentiable curves. We first extend the classical differential calculus in a quantum calculus, which allows us to define a complex operator, called the…
A consistent functional calculus approach to the spectral theorem for strongly commuting normal operators on Hilbert spaces is presented. In contrast to the common approaches using projection-valued measures or multiplication operators,…
We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed.…
Subdifferentials (in the sense of convex analysis) of matrix-valued functions defined on $\mathbb{R}^d$ that are convex with respect to the L\"{o}wner partial order can have a complicated structure and might be very difficult to compute…
We generalize the differential space concept as a tool for developing differential geometry, and enrich this geometry with infinitesimals that allow us to penetrate into the superfine structure of space. This is achieved by Yoneda embedding…