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.

Abstract:
In 1967, Ross and Str\"omberg published a theorem about pointwise limits of orbital integrals for the left action of a locally compact group $G$ onto $(G,\rho)$, where $\rho$ is the right Haar measure. In this paper, we study the same kind of problem, but more generally for left actions of $G$ onto any measured space $(X,\mu)$, which leaves the $\sigma$-finite measure $\mu$ relatively invariant, in the sense that $s\mu = \Delta(s)\mu$ for every $s\in G$, where $\Delta$ is the modular function of $G$. As a consequence, we also obtain a generalization of a theorem of Civin, relative to one-parameter groups of measure preserving transformations. The original motivation for the circle of questions treated here dates back to classical problems concerning pointwise convergence of Riemann sums relative to Lebesgue integrable functions.

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:
We give an exact value of the rank of an $\mathcal{F}$-Fubini sum of filters for the case where $\mathcal{F}$ is a Borel filter of rank $1$. We also consider $\mathcal{F}$-limits of filters $\mathcal{F}_i$, which are of the form $\lim_\mathcal{F}\mathcal{F}_i=\left\{A\subset X: \left\{i\in I: A\in\mathcal{F}_i\right\}\in\mathcal{F}\right\}$. We estimate the ranks of such filters; in particular we prove that they can fall to $1$ for $\mathcal{F}$ as well as for $\mathcal{F}_i$ of arbitrarily large ranks. At the end we prove some facts concerning filters of countable type and their ranks.

Abstract:
We consider a class of growing random graphs obtained by creating vertices sequentially one by one: at each step, we choose uniformly the neighbours of the newly created vertex; its degree is a random variable with a fixed but arbitrary distribution, depending on the number of existing vertices. Examples from this class turn out to be the ER random graph, a natural random threshold graph, etc. By working with the notion of graph limits, we define a kernel which, under certain conditions, is the limit of the growing random graph. Moreover, for a subclass of models, the growing graph on any given n vertices has the same distribution as the random graph with n vertices that the kernel defines. The motivation stems from a model of graph growth whose attachment mechanism does not require information about properties of the graph at each iteration.