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Working in Zermelo-Fraenkel Set Theory with Atoms over an $\omega$-categorical $\omega$-stable structure, we show how \emph{infinite} constructions over definable sets can be encoded as \emph{finite} constructions over the Stone-\v{C}ech compactification of the sets. In particular, we show that for a definable set $X$ with its Stone-\v{C}ech compactification $\overline{X}$ the following holds: a) the powerset $\mathcal{P}(X)$ of $X$ is isomorphic to the finite-powerset $\mathcal{P}_{\textit{fin}}(\overline{X})$ of $\overline{X}$, b) the vector space $\mathcal{K}^X$ over a field $\mathcal{K}$ is the free vector space $F_{\mathcal{K}}(\overline{X})$ on $\overline{X}$ over $\mathcal{K}$, c) every measure on $X$ is tantamount to a \emph{discrete} measure on $\overline{X}$. Moreover, we prove that the Stone-\v{C}ech compactification of a definable set is still definable, which allows us to obtain some results about equivalence of certain formalizations of register machines.

In this contribution, we consider a zero-dimensional polynomial system in $n$ variables defined over a field $\mathbb{K}$. In the context of computing a Rational Univariate Representation (RUR) of its solutions, we address the problem of certifying a separating linear form and, once certified, calculating the RUR that comes from it, without any condition on the ideal else than being zero-dimensional. Our key result is that the RUR can be read (closed formula) from lexicographic Groebner bases of bivariate elimination ideals, even in the case where the original ideal that is not in shape position, so that one can use the same core as the well known FGLM method to propose a simple algorithm. Our first experiments, either with a very short code (300 lines) written in Maple or with a Julia code using straightforward implementations performing only classical Gaussian reductions in addition to Groebner bases for the degree reverse lexicographic ordering, show that this new method is already competitive with sophisticated state of the art implementations which do not certify the parameterizations.

The proper conflict-free chromatic number, $\chi_{pcf}(G)$, of a graph $G$ is the least $k$ such that $G$ has a proper $k$-coloring in which for each non-isolated vertex there is a color appearing exactly once among its neighbors. The proper odd chromatic number, $\chi_{o}(G)$, of $G$ is the least $k$ such that $G$ has a proper coloring in which for every non-isolated vertex there is a color appearing an odd number of times among its neighbors. We say that a graph class $\mathcal{G}$ is $\chi_{pcf}$-bounded ($\chi_{o}$-bounded) if there is a function $f$ such that $\chi_{pcf}(G) \leq f(\chi(G))$ ($\chi_{o}(G) \leq f(\chi(G))$) for every $G \in \mathcal{G}$. Caro et al. (2022) asked for classes that are linearly $\chi_{pcf}$-bounded ($\chi_{pcf}$-bounded), and as a starting point, they showed that every claw-free graph $G$ satisfies $\chi_{pcf}(G) \le 2\Delta(G)+1$, which implies $\chi_{pcf}(G) \le 4\chi(G)+1$. In this paper, we improve the bound for claw-free graphs to a nearly tight bound by showing that such a graph $G$ satisfies $\chi_{pcf}(G) \le \Delta(G)+6$, and even $\chi_{pcf}(G) \le \Delta(G)+4$ if it is a quasi-line graph. These results also give evidence for a conjecture by Caro et al. Moreover, we show that convex-round graphs and permutation graphs are linearly $\chi_{pcf}$-bounded. For these last two results, we prove a lemma that reduces the problem of deciding if a hereditary class is linearly $\chi_{pcf}$-bounded to deciding if the bipartite graphs in the class are $\chi_{pcf}$-bounded by an absolute constant. This lemma complements a theorem of Liu (2022) and motivates us to study boundedness in bipartite graphs. In particular, we show that biconvex bipartite graphs are $\chi_{pcf}$-bounded while convex bipartite graphs are not even $\chi_o$-bounded, and exhibit a class of bipartite circle graphs that is linearly $\chi_o$-bounded but not $\chi_{pcf}$-bounded.

We consider the \emph{exact plurality consensus} problem for \emph{population protocols}. Here, $n$ anonymous agents start each with one of $k$ opinions. Their goal is to agree on the initially most frequent opinion (the \emph{plurality opinion}) via random, pairwise interactions. The case of $k = 2$ opinions is known as the \emph{majority problem}. Recent breakthroughs led to an always correct, exact majority population protocol that is both time- and space-optimal, needing $O(\log n)$ states per agent and, with high probability, $O(\log n)$ time~[Doty, Eftekhari, Gasieniec, Severson, Stachowiak, and Uznanski; 2021]. We know that any always correct protocol requires $\Omega(k^2)$ states, while the currently best protocol needs $O(k^{11})$ states~[Natale and Ramezani; 2019]. For ordered opinions, this can be improved to $O(k^6)$~[Gasieniec, Hamilton, Martin, Spirakis, and Stachowiak; 2016]. We design protocols for plurality consensus that beat the quadratic lower bound by allowing a negligible failure probability. While our protocols might fail, they identify the plurality opinion with high probability even if the bias is $1$. Our first protocol achieves this via $k-1$ tournaments in time $O(k \cdot \log n)$ using $O(k + \log n)$ states. While it assumes an ordering on the opinions, we remove this restriction in our second protocol, at the cost of a slightly increased time $O(k \cdot \log n + \log^2 n)$. By efficiently pruning insignificant opinions, our final protocol reduces the number of tournaments at the cost of a slightly increased state complexity $O(k \cdot \log\log n + \log n)$. This improves the time to $O(n / x_{\max} \cdot \log n + \log^2 n)$, where $x_{\max}$ is the initial size of the plurality. Note that $n/x_{\max}$ is at most $k$ and can be much smaller (e.g., in case of a large bias or if there are many small opinions).

A $r$-role assignment of a simple graph $G$ is an assignment of $r$ distinct roles to the vertices of $G$, such that two vertices with the same role have the same set of roles assigned to related vertices. Furthermore, a specific $r$-role assignment defines a role graph, in which the vertices are the distinct $r$ roles, and there is an edge between two roles whenever there are two related vertices in the graph $G$ that correspond to these roles. We consider complementary prisms, which are graphs formed from the disjoint union of the graph with its respective complement, adding the edges of a perfect matching between their corresponding vertices. In this work, we characterize the complementary prisms that do not admit a $3$-role assignment. We highlight that all of them are complementary prisms of disconnected bipartite graphs. Moreover, using our findings, we show that the problem of deciding whether a complementary prism has a $3$-role assignment can be solved in polynomial time.

For a state $\rho_{A_1^n B}$, we call a sequence of states $(\sigma_{A_1^k B}^{(k)})_{k=1}^n$ an approximation chain if for every $1 \leq k \leq n$, $\rho_{A_1^k B} \approx_\epsilon \sigma_{A_1^k B}^{(k)}$. In general, it is not possible to lower bound the smooth min-entropy of such a $\rho_{A_1^n B}$, in terms of the entropies of $\sigma_{A_1^k B}^{(k)}$ without incurring very large penalty factors. In this paper, we study such approximation chains under additional assumptions. We begin by proving a simple entropic triangle inequality, which allows us to bound the smooth min-entropy of a state in terms of the R\'enyi entropy of an arbitrary auxiliary state while taking into account the smooth max-relative entropy between the two. Using this triangle inequality, we create lower bounds for the smooth min-entropy of a state in terms of the entropies of its approximation chain in various scenarios. In particular, utilising this approach, we prove approximate versions of the asymptotic equipartition property and entropy accumulation. In our companion paper, we show that the techniques developed in this paper can be used to prove the security of quantum key distribution in the presence of source correlations.

Given a graph~$G$, the domination number, denoted by~$\gamma(G)$, is the minimum cardinality of a dominating set in~$G$. Dual to the notion of domination number is the packing number of a graph. A packing of~$G$ is a set of vertices whose pairwise distance is at least three. The packing number~$\rho(G)$ of~$G$ is the maximum cardinality of one such set. Furthermore, the inequality~$\rho(G) \leq \gamma(G)$ is well-known. Henning et al.\ conjectured that~$\gamma(G) \leq 2\rho(G)+1$ if~$G$ is subcubic. In this paper, we progress towards this conjecture by showing that~${\gamma(G) \leq \frac{120}{49}\rho(G)}$ if~$G$ is a bipartite cubic graph. We also show that $\gamma(G) \leq 3\rho(G)$ if~$G$ is a maximal outerplanar graph, and that~$\gamma(G) \leq 2\rho(G)$ if~$G$ is a biconvex graph. Moreover, in the last case, we show that this upper bound is tight.

Let $\mathbf{G}:=(G_1, G_2, G_3)$ be a triple of graphs on the same vertex set $V$ of size $n$. A rainbow triangle in $\mathbf{G}$ is a triple of edges $(e_1, e_2, e_3)$ with $e_i\in G_i$ for each $i$ and $\{e_1, e_2, e_3\}$ forming a triangle in $V$. The triples $\mathbf{G}$ not containing rainbow triangles, also known as Gallai colouring templates, are a widely studied class of objects in extremal combinatorics. In the present work, we fully determine the set of edge densities $(\alpha_1, \alpha_2, \alpha_3)$ such that if $\vert E(G_i)\vert> \alpha_i n^2$ for each $i$ and $n$ is sufficiently large, then $\mathbf{G}$ must contain a rainbow triangle. This resolves a problem raised by Aharoni, DeVos, de la Maza, Montejanos and \v{S}\'amal, generalises several previous results on extremal Gallai colouring templates, and proves a recent conjecture of Frankl, Gy\"ori, He, Lv, Salia, Tompkins, Varga and Zhu.

Classical Krylov subspace projection methods for the solution of linear problem $Ax = b$ output an approximate solution $\widetilde{x}\simeq x$. Recently, it has been recognized that projection methods can be understood from a statistical perspective. These probabilistic projection methods return a distribution $p(\widetilde{x})$ in place of a point estimate $\widetilde{x}$. The resulting uncertainty, codified as a distribution, can, in theory, be meaningfully combined with other uncertainties, can be propagated through computational pipelines, and can be used in the framework of probabilistic decision theory. The problem we address is that the current probabilistic projection methods lead to the poorly calibrated posterior distribution. We improve the covariance matrix from previous works in a way that it does not contain such undesirable objects as $A^{-1}$ or $A^{-1}A^{-T}$, results in nontrivial uncertainty, and reproduces an arbitrary projection method as a mean of the posterior distribution. We also propose a variant that is numerically inexpensive in the case the uncertainty is calibrated a priori. Since it usually is not, we put forward a practical way to calibrate uncertainty that performs reasonably well, albeit at the expense of roughly doubling the numerical cost of the underlying projection method.

We expound on some known lower bounds of the quadratic Wasserstein distance between random vectors in $\mathbb{R}^n$ with an emphasis on affine transformations that have been used in manifold learning of data in Wasserstein space. In particular, we give concrete lower bounds for rotated copies of random vectors in $\mathbb{R}^2$ by computing the Bures metric between the covariance matrices. We also derive upper bounds for compositions of affine maps which yield a fruitful variety of diffeomorphisms applied to an initial data measure. We apply these bounds to various distributions including those lying on a 1-dimensional manifold in $\mathbb{R}^2$ and illustrate the quality of the bounds. Finally, we give a framework for mimicking handwritten digit or alphabet datasets that can be applied in a manifold learning framework.