Abstract:
Sequences diverge either because they head off to infinity or because they oscillate. Part 1 \cite{Part1} of this paper laid the pure mathematics groundwork by defining Archimedean classes of infinite numbers as limits of smooth sequences. Part 2 follows that with applied mathematics, showing that general sequences can usually be converted into smooth sequences, and thus have a well-defined limit. Each general sequence is split into the sum of smooth, periodic (including Lebesgue integrable), chaotic and random components. The mean of each of these components divided by a smooth sequence, or the mean of the mean, will usually be a smooth sequence, and so the oscillatory sequence will have at least a leading term limit. Examples of limits of oscillatory sequences with well-defined limits are given. Methodologies are included for a way to calculate limits on the reals and on complex numbers, a way to evaluate improper integrals by limit of a Riemann sum, and a way to square the Dirac delta function.

Abstract:
The infinite numbers of the set M of finite and infinite natural numbers are defined starting from the sequence 0\Phi, where 0 is the first natural number, \Phi is a succession of symbols S and xS is the successor of the natural number x. The concept of limit of the natural number n, when n tends to infinite, is examined. Definitions and theorems about operations with elements of M, equivalence and equality of natural numbers, distance between elements of M and the order of the elements are presented.

Abstract:
In order to study large variations or fluctuations of finite or infinite sequences (time series), we bring to light an 1868 paper of Crofton and the (Cauchy-)Crofton theorem. After surveying occurrences of this result in the literature, we introduce the inconstancy of a sequence and we show why it seems more pertinent than other criteria for measuring its variational complexity. We also compute the inconstancy of classical binary sequences including some automatic sequences and Sturmian sequences.

Abstract:
We present a new method to obtain infinite Sidon sequences, based on the discrete logarithm. We construct an infinite Sidon sequence A, with A(x)= x^{\sqrt 2-1+o(1)}. Ruzsa proved the existence of a Sidon sequence with similar counting function but his proof was not constructive. Our method generalizes to B_h sequences: For all h\ge 3, there is a B_h sequence A such that A(x)=x^{\sqrt{(h-1)^2+1}-(h-1)+o(1)}.

Abstract:
For each positive integer n greater than or equal to 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n=2 case is equivalent to the standard continued fraction algorithm. For n=3, it reduces to a new iteration of the triangle. Cubic irrationals that are roots of x^3 + k x^2 + x - 1 are shown to be precisely those numbers with purely periodic expansions of period length one. For general positive integers n, it reduces to a new iteration of an n dimensional simplex.

Abstract:
In this paper the concept of strongly (λ_p Cesáro summability ofa sequence of fuzzy numbers and strongly λ statistically convergent sequences of fuzzy numbers are introduced.

Abstract:
In this paper we use the Recursion Theorem to show the existence of various infinite sequences and sets. Our main result is that there is an increasing sequence e_0, e_1, e_2 .. such that W_{e_n}={e_{n+1}} for every n. Similarly, we prove that there exists an increasing sequence such that W_{e_n}={e_{n+1},e_{n+2},...} for every n. We call a nonempty computably enumerable set A self-constructing if W_e=A for every e in A. We show that every nonempty computable enumerable set which is disjoint from an infinite computable set is one-one equivalent to a self-constructing set

Abstract:
Harmonic numbers $H_k=\sum_{05 we have $$\sum_{k=0}^{p-1}u_{k+\delta}H_k/2^k=0 (mod p),$$ where $\delta=0$ if p=1,2,4,8 (mod 15), and $\delta=1$ otherwise.