Abstract:
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 that head off to infinity. It begins by defining Archimedean classes of infinite numbers. Each class is denoted by a prototype sequence. These prototypes are used as asymptotes for determining leading term limits of sequences. By subtracting off leading term limits and repeating, limits are obtained for a subset of sequences called here ``smooth sequences". $\mathbb{I}_n$ is defined as the set of ratios of limits of smooth sequences. It is shown that $\mathbb{I}_n$ is an ordered field that includes real, infinite and infinitesimal numbers.

Abstract:
It is widely held that irrational numbers can be represented by infinite digit-sequences. We will show that this is not possible. A digit sequence is only an abbreviated notation for an infinite sequence of rational partial sums. As limits of sequences, irrational numbers are incommensurable with any grid of decimal fractions.

Abstract:
The oscillatory behavior of the solutions of the second-order linear nonautonomous equation where , is studied. Under the assumption that the sequence dominates somehow , the amplitude of the oscillations and the asymptotic behavior of its solutions are also analized. 1. Introduction The aim of this note is to study the oscillatory behavior of the equation where . Equation (1.1) is the nonautonomous case of the so-called Lucas sequences, which are obtained through the recursive law: which corresponds to have in (1.1) both sequences and constant and equal, respectively, to real numbers and . Lucas sequences are well known in number theory as an extension of the Fibonacci sequence (see [1, Chapter 2, Section IV]). Several particular cases of (1.1) are considered in literature. This is the case of equation (see [2, Chapter 6]) with being arbitrary and being positive, corresponding to have in (1.1) The equation (see [3, 4] and references therein) is also a particular case of (1.1). In fact, (1.5) can be written as corresponding to have in (1.1), and constant and equal to 1. Other particular cases of (1.1) will be referred along the text. As usual, we will say that a solution of (1.1) is nonoscillatory if it is either eventually positive or eventually negative. Otherwise is called oscillatory. When all solutions of (1.1) are oscillatory, (1.1) is said oscillatory. It is well known that (1.2) is oscillatory if and only if the polynomial has no positive real roots. This is equivalent to have one of the following two cases: In order to obtain (1.1) oscillatory, this specific case seems to justify that the sequence be assumed positive. However, for for every , notice that if one has either eventually or oscillatory, then cannot be neither eventually positive nor eventually negative. That is, in such circumstances the equation is oscillatory. So hereafter we will assume that and are both positive sequences. Through a direct manipulation of the terms of the solutions of (1.1), we show in the following sections a few results which seem, as far as we know, uncommon in literature. We state that if dominates somehow the sequence , then all solutions of the equation exhibit a specific oscillatory and asymptotic behavior. Oscillations through the existence of periodic solutions will be studied in a sequel, but they cannot then happen under a similar relationship between and . 2. Oscillatory Behavior The oscillatory behavior of (1.1) can be stated through the application of some results already existent in literature. This is the case of the following theorem based upon

Abstract:
We examine the composition of the $L^{\infty}$ norm with weakly convergent sequences of gradient fields associated with the homogenization of second order divergence form partial differential equations with measurable coefficients. Here the sequences of coefficients are chosen to model heterogeneous media and are piecewise constant and highly oscillatory. We identify local representation formulas that in the fine phase limit provide upper bounds on the limit superior of the $L^{\infty}$ norms of gradient fields. The local representation formulas are expressed in terms of the weak limit of the gradient fields and local corrector problems. The upper bounds may diverge according to the presence of rough interfaces. We also consider the fine phase limits for layered microstructures and for sufficiently smooth periodic microsturctures. For these cases we are able to provide explicit local formulas for the limit of the $L^\infty$ norms of the associated sequence of gradient fields. Local representation formulas for lower bounds are obtained for fields corresponding to continuously graded periodic microstructures as well as for general sequences of oscillatory coefficients. The representation formulas are applied to problems of optimal material design.

Abstract:
Hyperfinite knots, or limits of equivalence classes of knots induced by a knot invariant taking values in a metric space, were introduced in a previous article by the author. In this article, we present new examples of hyperfinite knots stemming from sequences of torus knots.

Abstract:
In the present paper we consider $F_k(x)=x^{k}-\sum_{t=0}^{k-1}x^t,$ the characteristic polynomial of the $k$-th order Fibonacci sequence, the latter denoted $G(k,l).$ We determine the limits of the real roots of certain odd and even degree polynomials related to the derivatives and integrals of $F_k(x),$ that form infinite sequences of polynomials, of increasing degree. In particular, as $k \to \infty,$ the limiting values of the zeros are determined, for both odd and even cases. It is also shown, in both cases, that the convergence is monotone for sufficiently large degree. We give an upper bound for the modulus of the complex zeros of the polynomials for each sequence. This gives a general solution related to problems considered by Dubeau 1989, 1993, Miles 1960, Flores 1967, Miller 1971 and later by the second author in the present paper, and Narayan 1997.

Abstract:
We study the limiting object of a sequence of Markov chains analogous to the limits of graphs, hypergraphs, and other objects which have been studied. Following a suggestion of Aldous, we assign to a sequence of finite Markov chains with bounded mixing times a unique limit object: an infinite Markov chain with a measurable state space. The limits of the Markov chains we consider have discrete spectra, which makes the limit theory simpler than the general graph case, and illustrates how the discrete spectrum setting (sometimes called "random-free" or "product measurable") is simpler than the general case.

Abstract:
We show that if a sequence of dense graphs has the property that for every fixed graph F, the density of copies of F in these graphs tends to a limit, then there is a natural ``limit object'', namely a symmetric measurable 2-variable function on [0,1]. This limit object determines all the limits of subgraph densities. We also show that the graph parameters obtained as limits of subgraph densities can be characterized by ``reflection positivity'', semidefiniteness of an associated matrix. Conversely, every such function arises as a limit object. Along the lines we introduce a rather general model of random graphs, which seems to be interesting on its own right.

Abstract:
Motivated in part by various sequences of graphs growing under random rules (like internet models), convergent sequences of dense graphs and their limits were introduced by Borgs, Chayes, Lov\'asz, S\'os and Vesztergombi and by Lov\'asz and Szegedy. In this paper we use this framework to study one of the motivating class of examples, namely randomly growing graphs. We prove the (almost sure) convergence of several such randomly growing graph sequences, and determine their limit. The analysis is not always straightforward: in some cases the cut distance from a limit object can be directly estimated, in other case densities of subgraphs can be shown to converge.

Abstract:
A permutation sequence is said to be convergent if the density of occurrences of every fixed permutation in the elements of the sequence converges. We prove that such a convergent sequence has a natural limit object, namely a Lebesgue measurable function $Z:[0,1]^2 \to [0,1]$ with the additional properties that, for every fixed $x \in [0,1]$, the restriction $Z(x,\cdot)$ is a cumulative distribution function and, for every $y \in [0,1]$, the restriction $Z(\cdot,y)$ satisfies a "mass" condition. This limit process is well-behaved: every function in the class of limit objects is a limit of some permutation sequence, and two of these functions are limits of the same sequence if and only if they are equal almost everywhere. An ingredient in the proofs is a new model of random permutations, which generalizes previous models and might be interesting for its own sake.