Recently, a distributed middleware application called contract automata runtime environment (CARE) has been introduced to realise service applications specified using a dialect of finite-state automata. In this paper, we detail the formal modelling, verification and testing of CARE. We provide a formalisation as a network of stochastic timed automata. The model is verified against the desired properties with the tool Uppaal, utilising exhaustive and statistical model checking techniques. Abstract tests are generated from the Uppaal models that are concretised for testing CARE. This research emphasises the advantages of employing formal modelling, verification and testing processes to enhance the dependability of an open-source distributed application. We discuss the methodology used for modelling the application and generating concrete tests from the abstract model, addressing the issues that have been identified and fixed.

In this paper we prove disintegration results for self-conformal measures and affinely irreducible self-similar measures. The measures appearing in the disintegration resemble self-conformal/self-similar measures for iterated function systems satisfying the strong separation condition. As an application of our results, we prove the following Diophantine statements: 1. Using a result of Pollington and Velani, we show that if $μ$ is a self-conformal measure in $\mathbb{R}$ or an affinely irreducible self-similar measure, then there exists $α>0$ such that for all $β>α$ we have $$μ\left(\left\{\mathbf{x}\in \mathbb{R}^{d}:\max_{1\leq i\leq d}|x_{i}-p_i/q|\leq \frac{1}{q^{\frac{d+1}{d}}(\log q)^β}\textrm{ for i.m. }(p_1,\ldots,p_d,q)\in \mathbb{Z}^{d}\times \mathbb{N}\right\}\right)=0.$$ 2. Using a result of Kleinbock and Weiss, we show that if $μ$ is an affinely irreducible self-similar measure, then $μ$ almost every $\mathbf{x}$ is not a singular vector.

Nonlocal quantum realizations, certified by the violation of a Bell inequality, are core resources for device-independent quantum information processing. Although proof-of-principle experiments demonstrating device-independent quantum information processing have already been reported, identifying physical platforms that are realistically closer to practical, viable devices remains a significant challenge. In this work, we present an automated framework for designing photonic implementations of nonlocal realizations using homodyne detections and quantum state heralding. Combining deep reinforcement learning and efficient simulations of quantum optical processes, our method generates photonic circuits that achieve significant violations of the Clauser-Horne-Shimony-Holt inequality. In particular, we find an experimental setup, robust to losses, that yields a CHSH violation of $2.068$ with $3.9$ dB and $0.008$ dB squeezed light sources and two beam splitters.

In this article, we study the pressure at infinity of potentials defined over countable Markov shifts. We establish an upper semi-continuity result concerning the limiting behaviour of the pressure of invariant probability measures, where the escape of mass is controlled by the pressure at infinity. As a consequence, we establish criteria for the existence of equilibrium states and maximizing measures for uniformly continuous potentials. Additionally, we study the pressure at infinity of suspension flows defined over countable Markov shifts and prove an upper semi-continuity result for the pressure map.

We deploy the new Arkenstone galactic wind model in cosmological simulations for the first time, allowing us to robustly resolve the evolution and impact of high specific energy winds. In a (25 $h^{-1}$ Mpc)$^3$ box we perform a set of numerical experiments that systematically vary the mass and energy loadings of such winds, finding that their energy content is the key parameter controlling the stellar to dark matter mass ratio. Increasing the mass loading, at fixed energy, actually results in mildly enhanced star formation, counter to prevailing wisdom, due to the wind becoming cooler. Of the simple parametrisations that we test, we find that an energy loading that scales inversely with halo mass best matches a wide range of observations and can do so with mass loadings drastically lower than those in most previous cosmological simulations. In this scenario, much less material is ejected from the interstellar medium. Instead, winds both heat gas in the circumgalactic medium, slowing infall onto the galaxy, and also drive shocks beyond the virial radius, decreasing the halo-scale accretion rate. We can also report that a much lower fraction of the available supernova energy is needed in preventative galaxy regulation than required by ejective wind feedback models such as IllustrisTNG. This is a Learning the Universe collaboration publication.

We construct a 2-functor from the Kac-Moody 2-category for the extended quantum affine sl(3) to the homotopy 2-category of bounded chain complexes with values in the Kac-Moody 2-category for quantum gl(3), categorifying the evaluation map between the corresponding quantum Kac-Moody algebras.

Quantum state discrimination plays a central role in quantum information and communication. For the discrimination of optical quantum states, the two most widely adopted measurement techniques are photon detection, which produces discrete outcomes, and homodyne detection, which produces continuous outcomes. While various protocols using photon detection have been proposed for optimal and near-optimal discrimination between two coherent states, homodyne detection is known to have higher error rates, with its minimum achievable error rate often referred to as the Gaussian limit. In this work, we demonstrate that, despite the fundamental differences between discretely labelled and continuously labelled measurements, continuously labelled non-Gaussian measurements can also achieve near-optimal coherent state discrimination. We design two discrimination protocols that surpass the Gaussian limit: one using non-Gaussian unitary operations with homodyne detection, and another based on orthogonal polynomials. Our results show that photon detection is not required for near-optimal coherent state discrimination and that we can achieve error rates close to the Helstrom bound at low energies with continuously labelled measurements. We also find that our schemes maintain an advantage over the photon detection-based Kennedy receiver for a moderate range of coherent state amplitudes.

Levels of cancellativity in commutative monoids $M$, determined by stable rank values in $\mathbb{Z}_{> 0} \cup \{\infty\}$ for elements of $M$, are investigated. The behavior of the stable ranks of multiples $ka$, for $k \in \mathbb{Z}_{> 0}$ and $a \in M$, is determined. In the case of a refinement monoid $M$, the possible stable rank values in archimedean components of $M$ are pinned down. Finally, stable rank in monoids built from isomorphism or other equivalence classes of modules over a ring is discussed.

In this paper, we introduce the deformed homogeneous polynomials $\mathrm{R}_{n}(x,y;u|q)$. These polynomials generalize some classical polynomials: the Rogers-Szegö polynomials $\mathrm{h}_{n}(x|q)$, the generalized Rogers-Szegö polynomials $\mathrm{r}_{n}(x,y)$, the Stieltjes-Wigert polynomials $\mathrm{S}_{n}(x;q)$, among others. Basic properties of the polynomial $\mathrm{R}_{n}$ are given, along with recurrence relations, its $q$-difference equation, and representations. Generating functions for the polynomials $\mathrm{R}_{n}(x,y;u|q)$ are given. These functions include generalizations of the Mehler and Rogers formulas. In addition, generalizations of the $q$-binomial formula and the Heine transformation formula are obtained. These results are obtained via the $u$-deformed $q$-exponential operator $\mathrm{E}(yD_{q}|u)$, defined here. From this operator, we obtain for free the operators T$(yD_{q})$ the Chen, $\mathrm{R}(yD_{q})$ of Saad, $\mathcal{E}(yD_{q})$ of Exton, and $\mathcal{R}(yD_{q})$ of Rogers-Ramanujan when $u=1,q,\sqrt{q},q^2$, respectively. We introduce the deformed basic hypergeometric series ${}_{r}Φ_{s}$, a generalization of the classical basic hypergeometric series. New transformation formulas for basic hypergeometric series are obtained.

We introduce a generating function approach to the affine Brauer and Kauffman categories and show how it allows one to efficiently recover important sets of relations in these categories. We use this formalism to deduce restrictions on possible categorical actions and show how this recovers admissibility results that have appeared in the literature on cyclotomic Birman-Murakami-Wenzl (BMW) algebras and their degenerate versions, also known as cyclotomic Nazarov-Wenzl algebras or VW algebras.

The purpose of this paper is to analyze the degree index and clustering index in random graphs. The degree index in our setup is a certain measure of degree irregularity whose basic properties are well studied in the literature, and the corresponding theoretical analysis in a random graph setup turns out to be tractable. On the other hand, the clustering index, based on a similar reasoning, is first introduced in this manuscript. Computing exact expressions for the expected clustering index turns out to be more challenging even in the case of Erdős-Rényi graphs, and our results are on obtaining relevant upper bounds. These are also complemented with observations based on Monte Carlo simulations. Besides the Erdős-Rényi case, we also do simulation-based analysis for random regular graphs, the Barabási-Albert model and the Watts-Strogatz model.

Let $Δ,q\geq 3$ be integers. We prove that there exists $η\geq 0.002$ such that if $q\geq (2-η)Δ$, then there exists an open set $\mathcal{U}\subset \mathbb{C}$ that contains the interval $[0,1]$ such that for each $w\in \mathcal{U}$ and any graph $G=(V,E)$ of maximum degree at most $Δ$, the partition function of the anti-ferromagnetic $q$-state Potts model evaluated at $w$ does not vanish. This provides a (modest) improvement on a result of Liu, Sinclair, and Srivastava, and breaks the $q=2Δ$-barrier for this problem. As a direct consequence we obtain via Barvinok's interpolation method a deterministic polynomial time algorithm to approximate the number of proper $q$-colorings of graphs of maximum degree at most $Δ$, provided $q\geq (2-η)Δ$.

We investigate groups that act amenably on their Higson corona (also known as bi-exact groups) and we provide reformulations of this in relation to the stable Higson corona, nuclearity of crossed products and to positive type kernels. We further investigate implications of this in relation to the Baum-Connes conjecture, and prove that Gromov hyperbolic groups have isomorphic equivariant K-theories of their Gromov boundary and their stable Higson corona.

Polychoric correlation is often an important building block in the analysis of rating data, particularly for structural equation models. However, the commonly employed maximum likelihood (ML) estimator is highly susceptible to misspecification of the polychoric correlation model, for instance through violations of latent normality assumptions. We propose a novel estimator that is designed to be robust against partial misspecification of the polychoric model, that is, when the model is misspecified for an unknown fraction of observations, such as careless respondents. To this end, the estimator minimizes a robust loss function based on the divergence between observed frequencies and theoretical frequencies implied by the polychoric model. In contrast to existing literature, our estimator makes no assumption on the type or degree of model misspecification. It furthermore generalizes ML estimation, is consistent as well as asymptotically normally distributed, and comes at no additional computational cost. We demonstrate the robustness and practical usefulness of our estimator in simulation studies and an empirical application on a Big Five administration. In the latter, the polychoric correlation estimates of our estimator and ML differ substantially, which, after further inspection, is likely due to the presence of careless respondents that the estimator helps identify.

This paper introduces a generic framework that provides sufficient conditions for guaranteeing polynomial-time decidability of fixed-negation fragments of first-order theories that adhere to certain fixed-parameter tractability requirements. It enables deciding sentences of such theories with arbitrary existential quantification, conjunction and a fixed number of negation symbols in polynomial time. It was recently shown by Nguyen and Pak [SIAM J. Comput. 51(2): 1--31 (2022)] that an even more restricted such fragment of Presburger arithmetic (the first-order theory of the integers with addition and order) is NP-hard. In contrast, by application of our framework, we show that the fixed negation fragment of weak Presburger arithmetic, which drops the order relation from Presburger arithmetic in favour of equality, is decidable in polynomial time. We give two further examples of instantiations of our framework, showing polynomial-time decidability of the fixed negation fragments of weak linear real arithmetic and of the restriction of Presburger arithmetic in which each inequality contains at most one variable.

A \emph{trace} of a sequence is generated by deleting each bit of the sequence independently with a fixed probability. The well-studied \emph{trace reconstruction} problem asks how many traces are required to reconstruct an unknown binary sequence with high probability. In this paper, we study the multivariate version of this problem for matrices and hypermatrices, where a trace is generated by deleting each row/column of the matrix or each slice of the hypermatrix independently with a constant probability. Previously, Krishnamurthy et al. showed that $\exp(\widetilde{O}(n^{d/(d+2)}))$ traces suffice to reconstruct any unknown $n\times n$ matrix (for $d=2$) and any unknown $n^{\times d}$ hypermatrix. By developing a dimension reduction procedure and establishing a multivariate version of the Littlewood-type result, we improve this upper bound by showing that $\exp(\widetilde{O}(n^{3/7}))$ traces suffice to reconstruct any unknown $n\times n$ matrix, and $\exp(\widetilde{O}(n^{3/5}))$ traces suffice to reconstruct any unknown $n^{\times d}$ hypermatrix. This breaks the tendency to trivial $\exp(O(n))$ as the dimension $d$ grows.

The generating series of a number of different objects studied in arithmetic statistics can be built out of Euler products. Euler products often have very nice analytic properties, and by constructing a meromorphic continuation one can use complex analytic techniques, including Tauberian theorems to prove asymptotic counting theorems for these objects. One standard technique for producing a meromorphic continuation is to factor out copies of the Riemann zeta function, for which a meromorphic continuation is already known. This paper is an exposition of the "Factorization Method" for meromorphic continuation. We provide the following three resources with an eye towards research in arithmetic statistics: (1) an introduction to this technique targeted at new researchers, (2) exposition of existing works, with self-contained proofs, that give a continuation of Euler products with constant or Frobenain coefficients to the right halfplane ${\rm Re}(s)>0$ (away from an isolated set of singularities), and (3) explicit statements on the locations and orders of all singularities for these Euler products.

We give a self-contained introduction to isolated points on curves and their counterpoint, parameterized points, that situates these concepts within the study of the arithmetic of curves. In particular, we show how natural geometric constructions of infinitely many degree d points on curves motivate the definitions of $\mathbb{P}^1$- and AV-parameterized points and explain how a result of Faltings implies that there are only finitely many isolated points on any curve. We use parameterized points to deduce properties of the density degree set and show that parameterized points of very low degree arise for a unique geometric reason. The paper includes several examples that illustrate the possible behaviors of degree d points.

For all $d \geq 3$ we show that the cardinality of $ \mathbb{R} $ is at most $\aleph_n $ if and only if $ \mathbb{R}^d $ can be covered with $ ( n + 1 ) ( d - 1 ) + 1 $ sprays whose centers are in general position in a hyperplane. This extends previous results by Schmerl when $ d = 2 $.

We classify all Polish semigroup topologies on the symmetric inverse monoid on the natural numbers. This result answers a question of Elliott et al. There are countably infinitely many such topologies. Under containment, these Polish semigroup topologies form a join-semilattice with infinite descending chains, no infinite ascending chains, and arbitrarily large finite anti-chains. Also, we show that the monoid endowed with any second countable T_1 semigroup topology is homeomorphic to the Baire space.

In this paper, we give a complete list of strongly tempered hyperspherical Hamiltonian spaces. We show that the period integrals attached to the list contains many previously studied Rankin-Selberg integrals and period integrals, thus give a new conceptual understanding of these integrals. The list also proposes many new interesting period integrals to study.

Using 3D-3D correspondence, we construct 3D dual bulk field theories for general Virasoro minimal models $M(P,Q)$. These theories correspond to Seifert fiber spaces $S^2 ((P,P-R),(Q,S),(3,1))$ with two integers $(R,S)$ satisfying $PS-QR =1$. In the unitary case, where $|P-Q|=1$, the bulk theory has a mass gap and flows to a unitary topological field theory (TQFT) in the IR, which is expected to support the chiral Virasoro minimal model at the boundary under an appropriate boundary condition. For the non-unitary case, where $|P-Q|>1$, the bulk theory flows to a 3D $\mathcal{N}=4$ rank-0 superconformal field theory, whose topologically twisted theory supports the chiral minimal model at the boundary. We also provide a concrete field theory description of the 3D bulk theory using $T[SU(2)]$ theories. Our proposals are supported by various consistency checks using 3D-3D relations and direct computations of various partition functions.

In this paper, we consider induced subgraphs of the Hamming graph $H(n,3)$. We show that if $U \subseteq \mathbb{Z}_3^n$ and $U$ induces a subgraph of $H(n,3)$ with maximum degree at most $1$ then 1. If $U$ is disjoint from a maximum size independent set of $H(n,3)$ then $|U| \leq 3^{n-1}+1$. Moreover, all such $U$ with size $3^{n-1}+1$ are isomorphic to each other. 2. For $n \geq 6$, there exists such a $U$ with size $|U| = 3^{n-1}+18$ and this is optimal for $n = 6$. 3. If $U \cap \{x, x+e_1, x+2e_1\} \ne ϕ$ for all $x \in \mathbb{Z}_3^n$ then $|U| \leq 3^{n-1} + 81$.

When using graphs and graph transformations to model systems, consistency is an important concern. While consistency has primarily been viewed as a binary property, i.e., a graph is consistent or inconsistent with respect to a set of constraints, recent work has presented an approach to consistency as a graduated property. This allows living with inconsistencies for a while and repairing them when necessary. For repairing inconsistencies in a graph, we use graph transformation rules with so-called {\em impairment-indicating and repair-indicating application conditions} to understand how much repair gain certain rule applications would bring. Both types of conditions can be derived from given graph constraints. Our main theorem shows that the difference between the number of actual constraint violations before and after a graph transformation step can be characterised by the difference between the numbers of violated impairment-indicating and repair-indicating application conditions. This theory forms the basis for algorithms with look-ahead that rank graph transformations according to their potential for graph repair. An evaluation shows that graph repair can be well-supported by rules with these new types of application conditions in terms of effectiveness and scalability.

While discounted payoff games and classic games that reduce to them, like parity and mean-payoff games, are symmetric, their solutions are not. We have taken a fresh view on the properties that optimal solutions need to have, and devised a novel way to converge to them, which is entirely symmetric. We achieve this by building a constraint system that uses every edge to define an inequation, and update the objective function by taking a single outgoing edge for each vertex into account. These edges loosely represent strategies of both players, where the objective function intuitively asks to make the inequation to these edges sharp. In fact, where they are not sharp, there is an `error' represented by the difference between the two sides of the inequation, which is 0 where the inequation is sharp. Hence, the objective is to minimise the sum of these errors. For co-optimal strategies, and only for them, it can be achieved that all selected inequations are sharp or, equivalently, that the sum of these errors is zero. While no co-optimal strategies have been found, we step-wise improve the error by improving the solution for a given objective function or by improving the objective function for a given solution. This also challenges the gospel that methods for solving payoff games are either based on strategy improvement or on value iteration.

The heralded exact one-way distillable secret key is equal to the largest expected rate at which perfect secret key bits can be probabilistically distilled from a bipartite state by means of local operations and one-way classical communication. Here we define the set of super two-extendible states and prove that an arbitrary state in this set cannot be used for heralded exact one-way secret-key distillation. This broad class of states includes both erased states and all full-rank states. Comparing the heralded exact one-way distillable secret key with the more commonly studied approximate one-way distillable secret key, our results demonstrate an extreme gap between them for many states of interest, with the approximate one-way distillable secret key being much larger. Our findings naturally extend to heralded exact one-way entanglement distillation, with similar conclusions.

We prove an ergodic theorem for Markov chains indexed by the Ulam-Harris-Neveu tree over large subsets with arbitrary shape under two assumptions: with high probability, two vertices in the large subset are far from each other and have their common ancestor close to the root. The assumption on the common ancestor can be replaced by some regularity assumption on the Markov transition kernel. We verify that those assumptions are satisfied for some usual trees. Finally, with Markov-Chain Monte-Carlo considerations in mind, we prove when the underlying Markov chain is stationary and reversible that the Markov chain, that is the line graph, yields minimal variance for the empirical average estimator among trees with a given number of nodes.

On the one side, the formalism of Global Transformations comes with the claim of capturing any transformation of space that is local, synchronous and deterministic. The claim has been proven for different classes of models such as mesh refinements from computer graphics, Lindenmayer systems from morphogenesis modeling and cellular automata from biological, physical and parallel computation modeling. The Global Transformation formalism achieves this by using category theory for its genericity, and more precisely the notion of Kan extension to determine the global behaviors based on the local ones. On the other side, Causal Graph Dynamics describe the transformation of port graphs in a synchronous and deterministic way and has not yet being tackled. In this paper, we show the precise sense in which the claim of Global Transformations holds for them as well. This is done by showing different ways in which they can be expressed as Kan extensions, each of them highlighting different features of Causal Graph Dynamics. Along the way, this work uncovers the interesting class of Monotonic Causal Graph Dynamics and their universality among General Causal Graph Dynamics.

Like the notion of computation via (strong) monads serves to classify various flavours of impurity, including exceptions, non-determinism, probability, local and global store, the notion of guardedness classifies well-behavedness of cycles in various settings. In its most general form, the guardedness discipline applies to general symmetric monoidal categories and further specializes to Cartesian and co-Cartesian categories, where it governs guarded recursion and guarded iteration, respectively. Here, even more specifically, we deal with the semantics of call-by-value guarded iteration. It was shown by Levy, Power and Thielecke that call-by-value languages can be generally interpreted in Freyd categories, but in order to represent effectful function spaces, such a category must canonically arise from a strong monad. We generalize this fact by showing that representing guarded effectful function spaces calls for certain parameterized monads (in the sense of Uustalu). This provides a description of guardedness as an intrinsic categorical property of programs, complementing the existing description of guardedness as a predicate on a category.

Given a weakly almost additive sequence of continuous functions with bounded variation $\mathcal{F}=\{\log f_n\}_{n=1}^{\infty}$ on a subshift $X$ over finitely many symbols, we study properties of a function $f$ on $X$ such that $\lim_{n\to\infty}\frac{1}{n}\int \log f_n dμ=\int f dμ$ for every invariant measure $μ$ on $X$. Under some conditions we construct a function $f$ on $X$ explicitly and study a relation between the property of $\mathcal{F}$ and some particular types of $f$. As applications we study images of Gibbs measures for continuous functions under one-block factor maps. We investigate a relation between the almost additivity of the sequences associated to relative pressure functions and the fiber-wise sub-positive mixing property of a factor map. For a special type of one-block factor maps between shifts of finite type, we study necessary and sufficient conditions for the image of a one-step Markov measure to be a Gibbs measure for a continuous function.