An abelian processor is an automaton whose output is independent of the order of its inputs. Bond and Levine have proved that a network of abelian processors performs the same computation regardless of processing order (subject only to a halting condition). We prove that any finite abelian processor can be emulated by a network of certain very simple abelian processors, which we call gates. The most fundamental gate is a toppler, which absorbs input particles until their number exceeds some given threshold, at which point it topples, emitting one particle and returning to its initial state. With the exception of an adder gate, which simply combines two streams of particles, each of our gates has only one input wire, which sends letters (‘particles’) from a unary alphabet. Our results can be reformulated in terms of the functions computed by processors, and one consequence is that any increasing function from ℕk to ℕℓ that is the sum of a linear function and a periodic function can be expressed in terms of (possibly nested) sums of floors of quotients by integers.

A tree functional is called additive if it satisfies a recursion of the form $F(T) = \sum_{j=1}^k F(B_j) + f(T)$, where B1, …, Bk are the branches of the tree T and f (T) is a toll function. We prove a general central limit theorem for additive functionals of d-ary increasing trees under suitable assumptions on the toll function. The same method also applies to generalized plane-oriented increasing trees (GPORTs). One of our main applications is a log-normal law that we prove for the size of the automorphism group of d-ary increasing trees, but other examples (old and new) are covered as well.

For an edge-coloured graph G, the minimum colour degree of G means the minimum number of colours on edges which are incident to each vertex of G. We prove that if G is an edge-coloured graph with minimum colour degree at least 5, then V(G) can be partitioned into two parts such that each part induces a subgraph with minimum colour degree at least 2. We show this theorem by proving amuch stronger form. Moreover, we point out an important relationship between our theorem and Bermond and Thomassen’s conjecture in digraphs.

We study a restricted form of list colouring, for which every pair of lists that correspond to adjacent vertices may not share more than one colour. The optimal list size such that a proper list colouring is always possible given this restriction, we call separation choosability. We show for bipartite graphs that separation choosability increases with (the logarithm of) the minimum degree. This strengthens results of Molloy and Thron and, partially, of Alon. One attempt to drop the bipartiteness assumption precipitates a natural class of Ramsey-type questions, of independent interest. For example, does every triangle-free graph of minimum degree d contain a bipartite induced subgraph of minimum degree Ω(log d) as d→∞?

An n × n partial Latin square P is called α-dense if each row and column has at most αn non-empty cells and each symbol occurs at most αn times in P. An n × n array A where each cell contains a subset of {1,…, n} is a (βn, βn, βn)-array if each symbol occurs at most βn times in each row and column and each cell contains a set of size at most βn. Combining the notions of completing partial Latin squares and avoiding arrays, we prove that there are constants α, β > 0 such that, for every positive integer n, if P is an α-dense n × n partial Latin square, A is an n × n (βn, βn, βn)-array, and no cell of P contains a symbol that appears in the corresponding cell of A, then there is a completion of P that avoids A; that is, there is a Latin square L that agrees with P on every non-empty cell of P, and, for each i, j satisfying 1 ≤ i, j ≤ n, the symbol in position (i, j) in L does not appear in the corresponding cell of A.

As a strengthening of Hadwiger’s conjecture, Gerards and Seymour conjectured that every graph with no odd Kt minor is (t − 1)-colourable. We prove two weaker variants of this conjecture. Firstly, we show that for each t ⩾ 2, every graph with no odd Kt minor has a partition of its vertex set into 6t − 9 sets V1, …, V6t−9 such that each Vi induces a subgraph of bounded maximum degree. Secondly, we prove that for each t ⩾ 2, every graph with no odd Kt minor has a partition of its vertex set into 10t −13 sets V1,…, V10t−13 such that each Vi induces a subgraph with components of bounded size. The second theorem improves a result of Kawarabayashi (2008), which states that the vertex set can be partitioned into 496t such sets.

We prove a generalmulti-dimensional central limit theorem for the expected number of vertices of a given degree in the family of planar maps whose vertex degrees are restricted to an arbitrary (finite or infinite) set of positive integers D. Our results rely on a classical bijection with mobiles (objects exhibiting a tree structure), combined with refined analytic tools to deal with the systems of equations on infinite variables that arise. We also discuss possible extensions to maps of higher genus and to weighted maps.

We consider a problem introduced by Mossel and Ross (‘Shotgun assembly of labeled graphs’, arXiv:1504.07682). Suppose a random n × n jigsaw puzzle is constructed by independently and uniformly choosing the shape of each ‘jig’ from q possibilities. We are given the shuffled pieces. Then, depending on q, what is the probability that we can reassemble the puzzle uniquely? We say that two solutions of a puzzle are similar if they only differ by a global rotation of the puzzle, permutation of duplicate pieces, and rotation of rotationally symmetric pieces. In this paper, we show that, with high probability, such a puzzle has at least two non-similar solutions when 2 ⩽ q ⩽ 2e−1/2n, all solutions are similar when q ⩾ (2+ϵ)n, and the solution is unique when q = ω(n).

We answer a challenge posed in Booker [$L$-functions as distributions. Math. Ann. 363(1–2) (2015), 423–454, §1.3] by proving a version of Weil’s converse theorem [Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967), 149–156] that assumes a functional equation for character twists but allows their root numbers to vary arbitrarily.

We generalize a method of Conrey and Ghosh [Simple zeros of the Ramanujan $\unicode[STIX]{x1D70F}$-Dirichlet series. Invent. Math. 94(2) (1988), 403–419] to prove quantitative estimates for simple zeros of modular form $L$-functions of arbitrary conductor.

We aim to re-prove a theorem conjectured by Gauss, namely there are exactly nine imaginary quadratic fields $\mathbf{Q}(\sqrt{-q})$ with class number one: specifically the list is $q\in \{3,4,7,8,11,19,43,67,163\}$. Our method initially follows an idea of Goldfeld, but rather than using an elliptic curve of analytic rank three (provided by the Gross–Zagier theorem), we instead use an elliptic curve of analytic rank two, where this $L$-function vanishing can be proven by modular symbols rather than a difficult height formula. It is already clear that Goldfeld’s work yields a constant lower bound for the class number by such means, but unfortunately it seems that even for the best choice of elliptic curve this numerical constant is less than 1, unless one can show non-trivial cancellation in the $L$-function coefficients restricted to values taken by quadratic forms. To show the latter, we consider a specific analytic rank-two elliptic curve with complex multiplication by $\mathbf{Q}(\sqrt{-1})$, and then by adapting a result of Hooley’s regarding equi-distrbution of roots of a quadratic polynomial to varying moduli, are able to show that there is indeed sufficient coefficient cancellation, giving an effective resolution of class number one. As we use various aspects of the principal form, our proof seems inapplicable for larger class numbers. We also comment on the possibility of using spectral techniques (following Templier and Tsimerman) to show the desired coefficient cancellation, though postpone the details of this to elsewhere.

We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Gröbner basis techniques, half-open decompositions and methods for interlacing polynomials we provide an explicit formula for the $h^{\ast }$-polynomial in case of complete bipartite graphs. In particular, we show that the $h^{\ast }$-polynomial is $\unicode[STIX]{x1D6FE}$-positive and real-rooted. This proves Gal’s conjecture for arbitrary flag unimodular triangulations in this case, and, beyond that, we prove a strengthening due to Nevo and Petersen [On $\unicode[STIX]{x1D6FE}$-vectors satisfying the Kruskal–Katona inequalities. Discrete Comput. Geom.45(3) (2011), 503–521].

We establish Diophantine inequalities for the fractional parts of generalized polynomials, in particular for sequences $\unicode[STIX]{x1D708}(n)=\lfloor n^{c}\rfloor +n^{k}$ with $c>1$ a non-integral real number and $k\in \mathbb{N}$, as well as for $\unicode[STIX]{x1D708}(p)$ where $p$ runs through all prime numbers. This is related to classical work of Heilbronn and to recent results of Bergelson et al.

In this paper we study the number of rational points on curves in an ensemble of abelian covers of the projective line: let $\ell$ be a prime, $q$ a prime power and consider the ensemble ${\mathcal{H}}_{g,\ell }$ of $\ell$-cyclic covers of $\mathbb{P}_{\mathbb{F}_{q}}^{1}$ of genus $g$. We assume that $q\not \equiv 0,1~\text{mod}~\ell$. If $2g+2\ell -2\not \equiv 0~\text{mod}~(\ell -1)\operatorname{ord}_{\ell }(q)$, then ${\mathcal{H}}_{g,\ell }$ is empty. Otherwise, the number of rational points on a random curve in ${\mathcal{H}}_{g,\ell }$ distributes as $\sum _{i=1}^{q+1}X_{i}$ as $g\rightarrow \infty$, where $X_{1},\ldots ,X_{q+1}$ are independent and identically distributed random variables taking the values $0$ and $\ell$ with probabilities $(\ell -1)/\ell$ and $1/\ell$, respectively. The novelty of our result is that it works in the absence of a primitive $\ell$th root of unity, the presence of which was crucial in previous studies.

We generalize the work of Sarnak and Tsimerman to twisted sums of Kloosterman sums and thus give evidence towards the twisted Linnik–Selberg conjecture.

Motivated by the Erdős–Szekeres convex polytope conjecture in $\mathbb{R}^{d}$, we initiate the study of the following induced Ramsey problem for hypergraphs. Given integers $n>k\geqslant 5$, what is the minimum integer $g_{k}(n)$ such that any$k$-uniform hypergraph on $g_{k}(n)$ vertices with the property that any set of $k+1$ vertices induces 0, 2, or 4 edges, contains an independent set of size $n$. Our main result shows that $g_{k}(n)>2^{cn^{k-4}}$, where $c=c(k)$.

In this paper various analytic techniques are combined in order to study the average of a product of a Hecke $L$-function and a symmetric square $L$-function at the central point in the weight aspect. The evaluation of the second main term relies on the theory of Maaß forms of half-integral weight and the Rankin–Selberg method. The error terms are bounded using the Liouville–Green approximation.

We prove that for every sufficiently large integer $n$, the polynomial $1+x+x^{2}/11+x^{3}/111+\cdots +x^{n}/111\ldots 1$ is irreducible over the rationals, where the coefficient of $x^{k}$ for $1\leqslant k\leqslant n$ is the reciprocal of the decimal number consisting of $k$ digits which are each $1$. Similar results following from the same techniques are discussed.

We provide a unified approach that encompasses some integral formulas for functions of the visual angle of a compact convex set due to Crofton, Hurwitz and Masotti. The basic tool is an integral formula that also allows us to integrate new functions of the visual angle. Also, we establish some upper and lower bounds for the considered integrals, generalizing, in particular, those obtained by Santaló for Masotti’s integral.

We discuss 1-factorizations of complete graphs that “match” a given Hadamard matrix. We prove the existence of these factorizations for two families of Hadamard matrices: Walsh matrices and certain Paley matrices.