Related papers: Euler Calculus with Applications to Signals and Se…
In this article we study certain properties of the image of Euler's totient function; we also consider the structure of the preimage of certain elements of the image of this function.
In this paper, we solve in the convergence set, the fractional logistic equation making use of Euler's numbers. To our knowledge, the answer is still an open question. The key point is that the coefficients can be connected with Euler's…
We discuss the notion of linearization through examples, which include the Price map, PageRank, representation theory, the Euler characteristic and quantum invariants. We also review categorification, which adds an additional layer of…
Euler's solution in 1734 of the Basel problem, which asks for a closed form expression for the sum of the reciprocals of all perfect squares, is one of the most celebrated results of mathematical analysis. In the modern era, numerous proofs…
We propose to use neural networks to estimate the rates of coherent and incoherent processes in quantum systems from continuous measurement records. In particular, we adapt an image recognition algorithm to recognize the patterns in…
In this paper we consider the problem of the calculus of variations for a functional which is the composition of a certain scalar function $H$ with the delta integral of a vector valued field $f$, i.e., of the form…
We stratify families of projective and very affine hypersurfaces according to their topological Euler characteristic. Our new algorithms compute all strata using algebro-geometric techniques. For very affine hypersurfaces, we investigate…
Many aspects of Schubert calculus are easily modeled on a computer. This enables large-scale experimentation to investigate subtle and ill-understood phenomena in the Schubert calculus. A well-known web of conjectures and results in the…
This short survey presents the essential features of what is called Painlev\'e analysis, i.e. the set of methods based on the singularities of differential equations in order to perform their explicit integration. Full details can be found…
We propose the use of algebras of generalized functions for the analysis of certain highly singular problems in the calculus of variations. After a general study of extremal problems on open subsets of Euclidean space in this setting we…
We give a diagrammatic summary of the connections between various theorems and conjectures about the vanishing of the Euler characteristic.
At a crossroads of calculus and combinatorics, the generating function of secant and tangent numbers (Euler numbers) provides enumeration of alternating permutations. In this article, we present a new refinement of Euler numbers to answer…
We derive an Ehrhart function for symbols from the Euler-MacLaurin formula with remainder.
This paper describes several applications in astronomy and cosmology that are addressed using probabilistic modelling and statistical inference.
Series involving hypergeometric functions are used to derive, extend and evaluate integrals involving the product of two Bessel functions of the first kind $J_{u}(a z)$ $J_{v}(b z)$ with order $u,v$, studied by Landau et al. The method used…
Bases, mappings, projections and metrics, natural for Neural network training, are introduced. Graph-theoretical interpretation is offered. Non-Gaussianity naturally emerges, even in relatively simple datasets. Training statistics,…
Euler starts with a hypergeometric series F(a, b, c, x), and differentiates it to get a functional relation. This relation is today known as Euler's identity. Then he integrates to get another and ends up with something like Legendre…
The emergence and continued reliance on the Internet and related technologies has resulted in the generation of large amounts of data that can be made available for analyses. However, humans do not possess the cognitive capabilities to…
Functional integrals are central to modern theories ranging from quantum mechanics and statistical thermodynamics to biology, chemistry, and finance. In this work we present a new method for calculating functional integrals based on a…
The proposed system of integer functions is logically fully independent from the traditional mathematical analysis of the real functions, but there is a well-defined mutual correspondence between the two disciplines. The system of integer…