Related papers: Banishing divergence Part 2: Limits of oscillatory…

Sequences diverge either because they head off to infinity or because they oscillate. Part 1 constructs a non-Archimedean framework of infinite numbers that is large enough to contain asymptotic limit points for non-oscillating sequences…

The calculus of finite differences is a solid foundation for the development of operations such as the derivative and the integral for infinite sequences. Here we showed a way to extend it for finite sequences. We could then define…

Lagrangian motions of fluid particles in a general velocity field oscillating in time are studied with the use of the two-timing method. Our aims are: (i) to calculate systematically the most general and practically usable asymptotic…

Integration at a point is a new kind of integration derived from integration over an interval in infinitesimal and infinity domains which are spaces larger than the reals. Consider a continuous monotonic divergent function that is…

Let $f$ be a permutation from $\mathbb{N}_0$ onto $\mathbb{N}_0$. Let $x\in\mathbb{N}_0$ and consider a (finite or infinite) sequence $s= (x,f(x),f^2(x),\cdots)$. We call $s$ a permutation sequence. Let $D$ be the set of elements of $s$. If…

In the classical literature on infinite series there are various tests to determine if a given infinite series converges, diverges, or oscillates. But unfortunately, for very many infinite series all the existing tests can fail to provide…

We explore the notion of degree of asymmetry for integer sequences and related combinatorial objects. The degree of asymmetry is a new combinatorial statistic that measures how far an object is from being symmetric. We define this notion…

Let $\mathscr{X}$ be the set of positive real sequences $x=(x_n)$ such that the series $\sum_n x_n$ is divergent. For each $x \in \mathscr{X}$, let $\mathcal{I}_x$ be the collection of all $A\subseteq \mathbf{N}$ such that the subseries…

``Orderly divergence'' deals with limit theorems for weighted stochastic Gamma integrals of otherwise nonintegrable functions. Although for monotonic functions this category usually coincides with the classical notion of weighted limit…

Based on the definition of Riemann definite integral,deleting items and disturbing mesh theorems on Riemann sums are given. After deleting some items or disturbing the mesh of partition, the limit of Riemann sums still converges to Riemann…

We introduce smooth sequences of integral domains as well-ordered ascending chains that behave well at limit ordinals. Subsequently, we use this notion to give some conditions on the freeness of kernels of extension maps between groups of…

We consider fluctuations of error terms $\Delta(x)$ appearing in the asymptotic formula for a summatory function of coefficients of the Dirichlet series. These are quantified via $\Omega$ and $\Omega_{\pm}$ estimates. We obtain $\Omega$…

This paper deals with a version of the two-timing method which describes various `slow' effects caused by externally imposed `fast' oscillations. Such small oscillations are often called \emph{vibrations} and the research area can be…

In this paper, we extend the slow divergence-integral from slow-fast systems, due to De Maesschalck, Dumortier and Roussarie, to smooth systems that limit onto piecewise smooth ones as $\epsilon\rightarrow 0$. In slow-fast systems, the slow…

The deleting items and disturbing mesh theorems of Riemann Integral are extended to multiple integral,line integral and surface integral respectively by constructing various of incomplete Riemann sum and non-Riemann sum sequences which…

Fractional derivatives are a well-studied generalization of integer order derivatives. Naturally, for optimization, it is of interest to understand the convergence properties of gradient descent using fractional derivatives. Convergence…

Classical computations can not capture the essence of infinite computations very well. This paper will focus on a class of infinite computations called convergent infinite computations}. A logic for convergent infinite computations is…

Let $f$ be an inner function with $f(0)=0$ which is not a rotation and let $f^{n}$ be its $n$-th iterate. Let $\{a_{n}\}$ be a sequence of complex numbers. We prove that the series $\sum a_{n}f^{n}(\xi)$ converges at almost every point…

We introduce the notion of level-$\delta$ limit linear series, which describe limits of linear series along families of smooth curves degenerating to a singular curve $X$. We treat here only the simplest case where $X$ is the union of two…

Khinchin proved that the arithmetic mean of continued fraction digits of Lebesgue almost every irrational number in $(0,1)$ diverges to infinity. Hence, none of the classical limit theorems such as the weak and strong laws of large numbers…