Related papers: A Tasty Combination: Multivariable Calculus and Di…
Differential complexes such as the de Rham complex have recently come to play an important role in the design and analysis of numerical methods for partial differential equations. The design of stable discretizations of systems of partial…
We prove multidimensional integration by parts formulas for generalized fractional derivatives and integrals. The new results allow us to obtain optimality conditions for multidimensional fractional variational problems with Lagrangians…
Modeling of high order multivariate probability distribution is a difficult problem which occurs in many fields. Copula approach is a good choice for this purpose, but the curse of dimensionality still remains a problem. In this paper we…
It is well known that the validity of the so called Lenard-Magri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st variational Poisson cohomology for one of the…
Causal functions of sequences occur throughout computer science, from theory to hardware to machine learning. Mealy machines, synchronous digital circuits, signal flow graphs, and recurrent neural networks all have behaviour that can be…
Differential forms is a highly geometric formalism for physics used from field theories to General Relativity (GR) which has been a great upgrade over vector calculus with the advantages of being coordinate-free and carrying a high degree…
Consider a sequence of real-valued functions of a real variable given by a homogeneous linear recursion with differentiable coefficients. We show that if the functions in the sequence are differentiable, then the sequence of derivatives…
This is a series of lecture notes, with embedded problems, aimed at students studying differential topology. Many revered texts, such as Spivak's "Calculus on Manifolds" and Guillemin and Pollack's "Differential Topology" introduce forms by…
For any discrete group $\Gamma$ and any 2-dimensional complex representation $\rho$ of $\Gamma$, we introduce the notion of $\rho-$equivariant functions, and we show that they are parameterized by vector-valued modular forms. We also…
This paper is an introduction to the theory of multivector functions of a real variable. The notions of limit, continuity and derivative for these objects are given. The theory of multivector functions of a real variable, even being similar…
A differential form defined on a Riemannian manifold is said to harmonic if it is closed and co-closed. Harmonic differential forms are a natural multi-dimensional extension of the concept of analytic function of complex variable. In this…
Fractional calculus is a generalization of classical theories of integration and differentiation to arbitrary order (i.e., real or complex numbers). In the last two decades, this new mathematical modeling approach has been widely used to…
High-order derivatives of nested functions of a single variable can be computed with the celebrated Fa\`a di Bruno's formula. Although generalizations of such formula to multiple variables exist, their combinatorial nature generates an…
We obtain a new decomposition of the Riemann-Liouville operators of fractional integration as a series involving derivatives (of integer order). The new formulas are valid for functions of class $C^n$, $n \in \mathbb{N}$, and allow us to…
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…
We study non-linear differential equations on the punctured formal disc by considering the natural derived enhancements of their spaces of solutions. In particular, by appealing to results of the inverse theory in the calculus of…
Many different types of fractional calculus have been defined, which may be categorised into broad classes according to their properties and behaviours. Two types that have been much studied in the literature are the Hadamard-type…
Derivatives play a critical role in computational statistics, examples being Bayesian inference using Hamiltonian Monte Carlo sampling and the training of neural networks. Automatic differentiation is a powerful tool to automate the…
A rather simple natural outer derivation of the graded Lie algebra of all vector valued differential forms with the Fr\"olicher-Nijenhuis bracket turns out to be a differential and gives rise to a cohomology of the manifold, which is…
In this survey, I suggest to approach the problem of functorial properties of quantum cohomology by drawing lessons from several versions of Mirror duality involving deformation spaces.