Dendric shifts are defined by combinatorial restrictions of the extensions of the words in their languages. This family generalizes well-known families of shifts such as Sturmian shifts, Arnoux–Rauzy shifts and codings of interval exchange transformations. It is known that any minimal dendric shift has a primitive $\mathcal {S}$ -adic representation where the morphisms in $\mathcal {S}$ are positive tame automorphisms of the free group generated by the alphabet. In this paper, we investigate those $\mathcal {S}$ -adic representations, heading towards an $\mathcal {S}$ -adic characterization of this family. We obtain such a characterization in the ternary case, involving a directed graph with two vertices.

In this paper we analyse the limiting conditional distribution (Yaglom limit) for stochastic fluid models (SFMs), a key class of models in the theory of matrix-analytic methods. So far, only transient and stationary analyses of SFMs have been considered in the literature. The limiting conditional distribution gives useful insights into what happens when the process has been evolving for a long time, given that its busy period has not ended yet. We derive expressions for the Yaglom limit in terms of the singularity˜$s^*$ such that the key matrix of the SFM, ${\boldsymbol{\Psi}}(s)$, is finite (exists) for all $s\geq s^*$ and infinite for $s<s^*$. We show the uniqueness of the Yaglom limit and illustrate the application of the theory with simple examples.

We present a new manifestation of Gödel’s second incompleteness theorem and discuss its foundational significance, in particular with respect to Hilbert’s program. Specifically, we consider a proper extension of Peano arithmetic ($\mathbf {PA}$) by a mathematically meaningful axiom scheme that consists of $\Sigma ^0_2$-sentences. These sentences assert that each computably enumerable ($\Sigma ^0_1$-definable without parameters) property of finite binary trees has a finite basis. Since this fact entails the existence of polynomial time algorithms, it is relevant for computer science. On a technical level, our axiom scheme is a variant of an independence result due to Harvey Friedman. At the same time, the meta-mathematical properties of our axiom scheme distinguish it from most known independence results: Due to its logical complexity, our axiom scheme does not add computational strength. The only known method to establish its independence relies on Gödel’s second incompleteness theorem. In contrast, Gödel’s theorem is not needed for typical examples of $\Pi ^0_2$-independence (such as the Paris–Harrington principle), since computational strength provides an extensional invariant on the level of $\Pi ^0_2$-sentences.

We apply the power-of-two-choices paradigm to a random walk on a graph: rather than moving to a uniform random neighbour at each step, a controller is allowed to choose from two independent uniform random neighbours. We prove that this allows the controller to significantly accelerate the hitting and cover times in several natural graph classes. In particular, we show that the cover time becomes linear in the number n of vertices on discrete tori and bounded degree trees, of order $${\mathcal O}(n\log \log n)$$ on bounded degree expanders, and of order $${\mathcal O}(n{(\log \log n)^2})$$ on the Erdős–Rényi random graph in a certain sparsely connected regime. We also consider the algorithmic question of computing an optimal strategy and prove a dichotomy in efficiency between computing strategies for hitting and cover times.

We bound the number of distinct minimal subsystems of a given transitive subshift of linear complexity, continuing work of Ormes and Pavlov [On the complexity function for sequences which are not uniformly recurrent. Dynamical Systems and Random Processes (Contemporary Mathematics, 736). American Mathematical Society, Providence, RI, 2019, pp. 125--137]. We also bound the number of generic measures such a subshift can support based on its complexity function. Our measure-theoretic bounds generalize those of Boshernitzan [A unique ergodicity of minimal symbolic flows with linear block growth. J. Anal. Math.44(1) (1984), 77–96] and are closely related to those of Cyr and Kra [Counting generic measures for a subshift of linear growth. J. Eur. Math. Soc.21(2) (2019), 355–380].

A connected graph G is $\mathcal {CF}$ -connected if there is a path between every pair of vertices with no crossing on its edges for each optimal drawing of G. We conjecture that a complete bipartite graph $K_{m,n}$ is $\mathcal {CF}$ -connected if and only if it does not contain a subgraph of $K_{3,6}$ or $K_{4,4}$ . We establish the validity of this conjecture for all complete bipartite graphs $K_{m,n}$ for any $m,n$ with $\min \{m,n\}\leq 6$ , and conditionally for $m,n\geq 7$ on the assumption of Zarankiewicz’s conjecture that $\mathrm {cr}(K_{m,n})=\big \lfloor \frac {m}{2} \big \rfloor \big \lfloor \frac {m-1}{2} \big \rfloor \big \lfloor \frac {n}{2} \big \rfloor \big \lfloor \frac {n-1}{2} \big \rfloor $ .

We answer four questions from a recent paper of Rao and Shinkar [17] on Lipschitz bijections between functions from {0, 1}n to {0, 1}. (1) We show that there is no O(1)-bi-Lipschitz bijection from Dictator to XOR such that each output bit depends on O(1) input bits. (2) We give a construction for a mapping from XOR to Majority which has average stretch $O(\sqrt{n})$ , matching a previously known lower bound. (3) We give a 3-Lipschitz embedding $\phi \colon \{0,1\}^n \to \{0,1\}^{2n+1}$ such that $${\rm{XOR }}(x) = {\rm{ Majority }}(\phi (x))$$ for all $x \in \{0,1\}^n$ . (4) We show that with high probability there is an O(1)-bi-Lipschitz mapping from Dictator to a uniformly random balanced function.

The aim of this paper is to shed light on our understanding of large scale properties of infinite strings. We say that one string $\alpha $ has weaker large scale geometry than that of $\beta $ if there is color preserving bi-Lipschitz map from $\alpha $ into $\beta $ with small distortion. This definition allows us to define a partially ordered set of large scale geometries on the classes of all infinite strings. This partial order compares large scale geometries of infinite strings. As such, it presents an algebraic tool for classification of global patterns. We study properties of this partial order. We prove, for instance, that this partial order has a greatest element and also possess infinite chains and antichains. We also investigate the sets of large scale geometries of strings accepted by finite state machines such as Büchi automata. We provide an algorithm that describes large scale geometries of strings accepted by Büchi automata. This connects the work with the complexity theory. We also prove that the quasi-isometry problem is a $\Sigma _2^0$ -complete set, thus providing a bridge with computability theory. Finally, we build algebraic structures that are invariants of large scale geometries. We invoke asymptotic cones, a key concept in geometric group theory, defined via model-theoretic notion of ultra-product. Partly, we study asymptotic cones of algorithmically random strings, thus connecting the topic with algorithmic randomness.

A k-permutation family on n vertices is a set-system consisting of the intervals of k permutations of the integers 1 to n. The discrepancy of a set-system is the minimum over all red–blue vertex colourings of the maximum difference between the number of red and blue vertices in any set in the system. In 2011, Newman and Nikolov disproved a conjecture of Beck that the discrepancy of any 3-permutation family is at most a constant independent of n. Here we give a simpler proof that Newman and Nikolov’s sequence of 3-permutation families has discrepancy $\Omega (\log \,n)$. We also exhibit a sequence of 6-permutation families with root-mean-squared discrepancy $\Omega (\sqrt {\log \,n} )$; that is, in any red–blue vertex colouring, the square root of the expected squared difference between the number of red and blue vertices in an interval of the system is $\Omega (\sqrt {\log \,n} )$.

Given a positive integer M and $q \in (1, M+1]$ we consider expansions in base q for real numbers $x \in [0, {M}/{q-1}]$ over the alphabet $\{0, \ldots , M\}$ . In particular, we study some dynamical properties of the natural occurring subshift $(\boldsymbol{{V}}_q, \sigma )$ related to unique expansions in such base q. We characterize the set of $q \in \mathcal {V} \subset (1,M+1]$ such that $(\boldsymbol{{V}}_q, \sigma )$ has the specification property and the set of $q \in \mathcal {V}$ such that $(\boldsymbol{{V}}_q, \sigma )$ is a synchronized subshift. Such properties are studied by analysing the combinatorial and dynamical properties of the quasi-greedy expansion of q. We also calculate the size of such classes as subsets of $\mathcal {V}$ giving similar results to those shown by Blanchard [ 10 ] and Schmeling in [ 36 ] in the context of $\beta $ -transformations.

In this work we consider three well-studied broadcast protocols: push, pull and push&pull. A key property of all these models, which is also an important reason for their popularity, is that they are presumed to be very robust, since they are simple, randomized and, crucially, do not utilize explicitly the global structure of the underlying graph. While sporadic results exist, there has been no systematic theoretical treatment quantifying the robustness of these models. Here we investigate this question with respect to two orthogonal aspects: (adversarial) modifications of the underlying graph and message transmission failures.

We explore in particular the following notion of local resilience: beginning with a graph, we investigate up to which fraction of the edges an adversary may delete at each vertex, so that the protocols need significantly more rounds to broadcast the information. Our main findings establish a separation among the three models. On one hand, pull is robust with respect to all parameters that we consider. On the other hand, push may slow down significantly, even if the adversary may modify the degrees of the vertices by an arbitrarily small positive fraction only. Finally, push&pull is robust when no message transmission failures are considered, otherwise it may be slowed down.

On the technical side, we develop two novel methods for the analysis of randomized rumour-spreading protocols. First, we exploit the notion of self-bounding functions to facilitate significantly the round-based analysis: we show that for any graph the variance of the growth of informed vertices is bounded by its expectation, so that concentration results follow immediately. Second, in order to control adversarial modifications of the graph we make use of a powerful tool from extremal graph theory, namely Szemerédi’s Regularity Lemma.

In this paper we propose a polynomial-time deterministic algorithm for approximately counting the k-colourings of the random graph G(n, d/n), for constant d>0. In particular, our algorithm computes in polynomial time a $(1\pm n^{-\Omega(1)})$ -approximation of the so-called ‘free energy’ of the k-colourings of G(n, d/n), for $k\geq (1+\varepsilon) d$ with probability $1-o(1)$ over the graph instances.

Our algorithm uses spatial correlation decay to compute numerically estimates of marginals of the Gibbs distribution. Spatial correlation decay has been used in different counting schemes for deterministic counting. So far algorithms have exploited a certain kind of set-to-point correlation decay, e.g. the so-called Gibbs uniqueness. Here we deviate from this setting and exploit a point-to-point correlation decay. The spatial mixing requirement is that for a pair of vertices the correlation between their corresponding configurations becomes weaker with their distance.

Furthermore, our approach generalizes in that it allows us to compute the Gibbs marginals for small sets of nearby vertices. Also, we establish a connection between the fluctuations of the number of colourings of G(n, d/n) and the fluctuations of the number of short cycles and edges in the graph.

There has been substantial interest in estimating the value of a graph parameter, i.e. of a real-valued function defined on the set of finite graphs, by querying a randomly sampled substructure whose size is independent of the size of the input. Graph parameters that may be successfully estimated in this way are said to be testable or estimable, and the sample complexity qz = qz(ε) of an estimable parameter z is the size of a random sample of a graph G required to ensure that the value of z(G) may be estimated within an error of ε with probability at least 2/3. In this paper, for any fixed monotone graph property $\mathcal{P}= \text{Forb}\!(\mathcal{F}),$ we study the sample complexity of estimating a bounded graph parameter z that, for an input graph G, counts the number of spanning subgraphs of G that satisfy$\mathcal{P}$. To improve upon previous upper bounds on the sample complexity, we show that the vertex set of any graph that satisfies a monotone property $\mathcal{P}$ may be partitioned equitably into a constant number of classes in such a way that the cluster graph induced by the partition is not far from satisfying a natural weighted graph generalization of $\mathcal{P}$. Properties for which this holds are said to be recoverable, and the study of recoverable properties may be of independent interest.

We present a complete characterisation of the radial asymptotics of degree-one Mahler functions as $z$ approaches roots of unity of degree $k^{n}$, where $k$ is the base of the Mahler function, as well as some applications concerning transcendence and algebraic independence. For example, we show that the generating function of the Thue–Morse sequence and any Mahler function (to the same base) which has a nonzero Mahler eigenvalue are algebraically independent over $\mathbb{C}(z)$. Finally, we discuss asymptotic bounds towards generic points on the unit circle.

We study multidimensional minimal and quasiperiodic shifts of finite type. We prove for these classes several results that were previously known for the shifts of finite type in general, without restriction. We show that some quasiperiodic shifts of finite type admit only non-computable configurations; we characterize the classes of Turing degrees that can be represented by quasiperiodic shifts of finite type. We also transpose to the classes of minimal/quasiperiodic shifts of finite type some results on subdynamics previously known for effective shifts without restrictions: every effective minimal (quasiperiodic) shift of dimension $d$ can be represented as a projection of a subdynamics of a minimal (respectively, quasiperiodic) shift of finite type of dimension $d+1$.

Component graphs $\unicode[STIX]{x1D6E4}_{0}(F)$ are defined for arrays of sets $F$, and, in particular, for arrays of path components for Vietoris–Rips complexes and Lesnick complexes. The path components of $\unicode[STIX]{x1D6E4}_{0}(F)$ are the stable components of the array $F$. The stable components for the system of Lesnick complexes $\{L_{s,k}(X)\}$ for a finite data set $X$ decompose into layers, which are themselves path components of a graph. Combinatorial scoring functions are defined for layers and stable components.

Quasi-Sturmian words, which are infinite words with factor complexity eventually $n+c$ share many properties with Sturmian words. In this article, we study the quasi-Sturmian colorings on regular trees. There are two different types, bounded and unbounded, of quasi-Sturmian colorings. We obtain an induction algorithm similar to Sturmian colorings. We distinguish them by the recurrence function.

Let G(n,M) be a uniform random graph with n vertices and M edges. Let ${\wp_{n,m}}$ be the maximum block size of G(n,M), that is, the maximum size of its maximal 2-connected induced subgraphs. We determine the expectation of ${\wp_{n,m}}$ near the critical point M = n/2. When n − 2M ≫ n2/3, we find a constant c1 such that

$$c_1 = \lim_{n \rightarrow \infty} \left({1 - \frac{2M}{n}} \right) \,\E({\wp_{n,m}}).$$

$$c_2(\lambda) = \lim_{n \rightarrow \infty} \frac{\E{\left({\wp_{n,{{(n/2)}({1+\lambda n^{-1/3}})}}}\right)}}{n^{1/3}}.$$

$n^{-1/3}\,\E{\left({\wp_{n,{{(n/2)}({1+\lambda n^{-1/3}})}}}\right)}$

As usual, $P_{n}$ ($n\geq 1$) denotes the path on $n$ vertices. The gem is the graph consisting of a $P_{4}$ together with an additional vertex adjacent to each vertex of the $P_{4}$. A graph is called ($P_{5}$, gem)-free if it has no induced subgraph isomorphic to a $P_{5}$ or to a gem. For a graph $G$, $\unicode[STIX]{x1D712}(G)$ denotes its chromatic number and $\unicode[STIX]{x1D714}(G)$ denotes the maximum size of a clique in $G$. We show that $\unicode[STIX]{x1D712}(G)\leq \lfloor \frac{3}{2}\unicode[STIX]{x1D714}(G)\rfloor$ for every ($P_{5}$, gem)-free graph $G$.

For $\unicode[STIX]{x1D6FD}\in (1,2]$ the $\unicode[STIX]{x1D6FD}$-transformation $T_{\unicode[STIX]{x1D6FD}}:[0,1)\rightarrow [0,1)$ is defined by $T_{\unicode[STIX]{x1D6FD}}(x)=\unicode[STIX]{x1D6FD}x\hspace{0.6em}({\rm mod}\hspace{0.2em}1)$. For $t\in [0,1)$ let $K_{\unicode[STIX]{x1D6FD}}(t)$ be the survivor set of $T_{\unicode[STIX]{x1D6FD}}$ with hole $(0,t)$ given by

$$\begin{eqnarray}K_{\unicode[STIX]{x1D6FD}}(t):=\{x\in [0,1):T_{\unicode[STIX]{x1D6FD}}^{n}(x)\not \in (0,t)\text{ for all }n\geq 0\}.\end{eqnarray}$$

$E_{\unicode[STIX]{x1D6FD}}$

$t\in [0,1)$

$t\mapsto K_{\unicode[STIX]{x1D6FD}}(t)$

$E_{\unicode[STIX]{x1D6FD}}$

$\unicode[STIX]{x1D6FD}\in (1,2)$

$\unicode[STIX]{x1D6FD}\in (1,2)$

$E_{\unicode[STIX]{x1D6FD}}$

$\unicode[STIX]{x1D6FD}\in (1,2)$

$E_{\unicode[STIX]{x1D6FD}}$

$E_{2}$

$\unicode[STIX]{x1D6FD}\in (1,2)$

$\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}$

$K_{\unicode[STIX]{x1D6FD}}(t)$

$t<\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}$

$\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}\leq 1-(1/\unicode[STIX]{x1D6FD})$

$\unicode[STIX]{x1D6FD}\in (1,2)$