Related papers: On the Gap Structure of Generalized Stirling Numbe…

In this undergraduate thesis, we expand on the study of statistics on restricted growth functions avoiding patterns initiated by Campbell, et. al. Restricted growth functions are of interest because they are in bijection with set…

It is well-known that for a quickly increasing sequence $(n_k)_{k \geq 1}$ the functions $(\cos 2 \pi n_k x)_{k \geq 1}$ show a behavior which is typical for sequences of independent random variables. If the growth condition on $(n_k)_{k…

We study orbits of semigroups of $\text{SL}(2,\mathbb{Z})$, and demonstrate reciprocity obstructions: we show that certain such orbits avoid squares, but not as a consequence of obstructions inherited from an algebraic set, and not as a…

Let k be an algebraically closed field of characteristic p>0. We compute the Weyl filtration multiplicities in indecomposable tilting modules and the decomposition numbers for the general linear group over k in terms of cap diagrams under…

We consider the sequential composite binary hypothesis testing problem in which one of the hypotheses is governed by a single distribution while the other is governed by a family of distributions whose parameters belong to a known set…

The separation dimension of a graph $G$, written $\pi(G)$, is the minimum number of linear orderings of $V(G)$ such that every two nonincident edges are "separated" in some ordering, meaning that both endpoints of one edge appear before…

Let $s_1, s_2, \ldots$ be the sequence of positive integers, arranged in increasing order, that are representable by any binary quadratic form of fixed discriminant $D$. We show that \[ \limsup_{n \rightarrow \infty} \frac{s_{n+1}-s_n}{\log…

Combining the mollifiers, we exhibit other choices of coefficients that improve the results on large gaps between the zeros of the Riemann zeta-function. Precisely, assuming the Generalized Riemann Hypothesis (GRH), we show that there exist…

In this paper we study the problem of long gaps between values of binary quadratic forms. Let $D_{1}$, $D_{2},\ldots ,D_{r}$ be negative integers and $(s_{n})_{n=1}^{\infty}$ be the sequence of all the numbers representable by any binary…

A seminal open question of Pisier and Mendel--Naor asks whether every degree-regular graph which satisfies the classical discrete Poincar\'e inequality for scalar functions, also satisfies an analogous inequality for functions taking values…

In Chapter 1 we fully characterise pairs of finite graphs which form a gap in the full homomorphism order. This leads to a simple proof of the existence of generalised duality pairs. We also discuss how such results can be carried to…

A value of a CSP instance is typically defined as a fraction of constraints that can be simultaneously met. We propose an alternative definition of a value of an instance and show that, for purely combinatorial reasons, a value of an…

We develop an obstruction theory for the extension of truncated minimal $A$-infinity bimodule structures over truncated minimal $A$-infinity algebras. Obstructions live in far-away pages of a (truncated) fringed spectral sequence of…

Given a field $F$, an \'etale extension $L/F$ and an Azumaya algebra $A/L$, one knows that there are extensions $E/F$ such that $A \otimes_F E$ is a split algebra over $L \otimes_F E$. In this paper we bound the degree of a minimal…

Suppose $a^2 (a^2 + 1)$ divides $b^2 (b^2 + 1)$ with $b > a$. In this paper, we improve a previous result and prove a gap principle, without any additional assumptions, namely $b \gg a (\log a)^{1/8} / (\log \log a)^{12}$. We also obtain $b…

The Three Gap Theorem states that for any $\alpha \in \mathbb{R}$ and $N \in \mathbb{N}$, the fractional parts of $\{ 0\alpha, 1\alpha, \dots, (N - 1)\alpha \}$ partition the unit circle into gaps of at most three distinct lengths. We prove…

Various specifiable combinatorial structures, with d extensive parameters, can be exactly sampled both by the recursive method, with linear arithmetic complexity if a heavy preprocessing is performed, or by the Boltzmann method, with…

Let $\omega$ be a factor of Doubling sequence $D_\infty=x_1x_2\cdots$, then it occurs in the sequence infinitely many times. Let $\omega_p$ be the $p$-th occurrence of $\omega$ and $G_p(\omega)$ be the gap between $\omega_p$ and…

We consider sets of real numbers in $[0,1)$ with prescribed frequencies of partial quotients in their regular continued fraction expansions. It is shown that the Hausdorff dimensions of these sets, always bounded from below by $1/2$, are…

We provide new limit theory for functionals of a general class of processes lying at the boundary between stationarity and nonstationarity -- what we term weakly nonstationary processes (WNPs). This includes, as leading examples, fractional…