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The cutoff phenomenon was recently shown to systematically follow from non-negative curvature and the product condition, for all Markov diffusions. The proof crucially relied on a classical \emph{chain rule} satisfied by the carr\'e du…
We study incommensurate fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives and generalized fractional integrals and derivatives. We obtain necessary optimality…
This paper is a summary of the theory of discrete embeddings introduced in [5]. A discrete embedding is an algebraic procedure associating a numerical scheme to a given ordinary differential equation. Lagrangian systems possess a…
A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of…
A method of calculation for the variational derivatives for gravitational actions in the pseudo-Riemannian case is proposed as a practical variant of the first order formalism with constraints. The method is then used to derive the metric…
Most of theoretical physics is based on the mathematics of functions of a real or a complex variable; yet we frequently are drawn to try extending our reach to include quaternions. The non-commutativity of the quaternion algebra poses…
We establish the weak large deviations principle for empirical measures of Markov chains on $\mathbb R^d$ under mild assumptions. In particular, no irreducibility is assumed and the initial measure may be arbitrary. The proof is entirely…
Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as…
We discuss alternative iteration methods for differential equations. We provide a convergence proof for exactly solvable examples and show more convenient formulas for nontrivial problems.
This chapter presents some numerical methods to solve problems in the fractional calculus of variations and fractional optimal control. Although there are plenty of methods available in the literature, we concentrate mainly on approximating…
In 2012 Chen and Singer introduced the notion of discrete residues for rational functions as a complete obstruction to rational summability. More explicitly, for a given rational function f(x), there exists a rational function g(x) such…
The chain rule for derivatives of a function of a function is extended to a function of a statistical functional, and applied to obtain approximations to the cumulants, distribution and quantiles of functions of sample moments, and so to…
Geoffrion's theorem is a fundamental result from mathematical programming assessing the quality of Lagrangian relaxation, a standard technique to get bounds for integer programs. An often implicit condition is that the set of feasible…
Computational methods for fractional differential equations exhibit essential instability. Even a minor modification of the coefficients or other entry data may switch good results to the divergent. The goal of this paper is to suggest the…
The Hurwitz chain gives a sequence of pairs of Farey approximations to an irrational real number. Minkowski gave a criterion for a number to be algebraic by using a certain generalization of the Hurwitz chain. We apply Minkowski's…
The purpose of this note is to present a formulation of a given nonlinear ordinary differential equation into an equivalent system of linear ordinary differential equations. It is evident that the easiness of a such procedure would be able…
We introduce a new fractional derivative which obeys classical properties including: linearity, product rule, quotient rule, power rule, chain rule, vanishing derivatives for constant functions, the Rolle's Theorem and the Mean Value…
The paper concerns the second-order generalized differentiation theory of variational analysis and new applications of this theory to some problems of constrained optimization in finitedimensional spaces. The main attention is paid to the…
We prove a chain rule for the Goodwillie calculus of functors from spectra to spectra. We show that the (higher) derivatives of a composite functor $FG$ at a base object $X$ are given by taking the composition product (in the sense of…
Existential rules have been proposed for representing ontological knowledge, specifically in the context of Ontology-Based Query Answering. Entailment with existential rules is undecidable. We focus in this paper on conditions that ensure…