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A basic open question motivated by the study of ribbon knot diagrams asks whether every ribbon knot can be presented as a symmetric union. In this article, we give a negative answer to this question by exhibiting a ribbon Montesinos knot which does not admit a symmetric union presentation.

Resolving in a strong sense an old problem of Gyárfás from the 1980s on the union of two perfect graphs, we prove that for every pair of positive integers $d$ and $k$, there is a graph $G$ with clique number $k$ and chromatic number $k^d$ that is the union of $d$ comparability graphs.

Labor unions influence economic outcomes not only through bargaining with employers over work contracts but also via political activities that can profoundly shape political systems. In unionized workplaces, they may mobilize and change the ideological positions of both unionizing workers and their non-unionizing management. In this paper, we analyze the workplace-level impact of unionization on workers' and managers' political campaign contributions. We link establishment-level union election data with transaction-level campaign contributions to federal and local candidates in the United States. Using a difference-in-differences design, validated through regression discontinuity tests and a novel instrumental variable approach, we find that unionization leads to a leftward shift of campaign contributions. Unionization increases support for Democrats relative to Republicans not only among workers but especially among managers, suggesting that managers converge toward workers' political preferences. The effects are stronger in settings with more cooperative union-employer interactions, such as when union elections are not contested by an unfair labor practice charge and result in a

Automatic C-to-Rust translation is a promising way to enhance the reliability of legacy system software. However, C2Rust, an industrially developed translator, generates Rust code with unsafe features, undermining the translation's objective. While researchers have proposed techniques to remove unsafe features in C2Rust-generated code, these efforts have targeted only a limited subset of unsafe features. One important unsafe feature remaining unaddressed is a union, a type consisting of multiple fields sharing the same memory storage. Programmers often place a union with a tag in a struct to record the last-written field, but they can still access wrong fields. In contrast, Rust's tagged unions combine tags and unions at the language level, ensuring correct value access. In this work, we propose techniques to replace unions with tagged unions during C-to-Rust translation. We develop a static analysis that facilitates such replacement by identifying tag fields and the corresponding tag values. The analysis involves a must-points-to analysis computing struct field values and a heuristic interpreting these results. To enhance efficiency, we adopt intraprocedural function-wise analysis

We show that for any ground term equation systems $E$ and $F$, (1) the union of the generated congruences by $E$ and $F$ is a congruence on the ground term algebra if and only if there exists a ground term equation system $H$ such that the congruence generated by $H$ is equal to the union of the congruences generated by $E$ and $F$ if and only if the congruence generated by the union of $E $ and $F$ is equal to the union of the congruences generated by $E $ and $F$, and (2) it is decidable in square time whether the congruence generated by the union of $E$ and $F$ is equal to the union of the congruences generated by $E $ and $F$, where the size of the input is the number of occurrences of symbols in $E$ plus the number of occurrences of symbols in $F$.

'A semigroup is completely regular if and only if it is a union of groups'- an analogue of this structure theorem of completely regular semigroup has been obtained in the setting of seminearrings in [[16], Mukherjee (Pal) et al., Semigroup Forum (2018)]. In it, a class of seminearrings (called generalized left completely regular seminearrings, abbreviated as GLCR) has been characterized as a union of near-rings. This work has been extended in the present article to characterize the seminearrings which are union of various types (regular, completely regular, inverse, Clifford) of regular near-rings.

Data scientists often draw on multiple relational data sources for analysis. A standard assumption in learning and approximate query answering is that the data is a uniform and independent sample of the underlying distribution. To avoid the cost of join and union, given a set of joins, we study the problem of obtaining a random sample from the union of joins without performing the full join and union. We present a general framework for random sampling over the set union of chain, acyclic, and cyclic joins, with sample uniformity and independence guarantees. We study the novel problem of the union of joins size evaluation and propose two approximation methods based on histograms of columns and random walks on data. We propose an online union sampling framework that initializes with cheap-to-calculate parameter approximations and refines them on the fly during sampling. We evaluate our framework on workloads from the TPC-H benchmark and explore the trade-off of the accuracy of union approximation and sampling efficiency.

In the study of ribbon knots, Lamm introduced symmetric unions inspired by earlier work of Kinoshita and Terasaka. We show an identity between the twisted Alexander polynomials of a symmetric union and its partial knot. As a corollary, we obtain an inequality concerning their genera. It is known that there exists an epimorphism between their knot groups, and thus our inequality provides a positive answer to an old problem of Jonathan Simon in this case. Our formula also offers a useful condition to constrain possible symmetric union presentations of a given ribbon knot. It is an open question whether every ribbon knot is a symmetric union.

Existing techniques for unionable table search define unionability using metadata (tables must have the same or similar schemas) or column-based metrics (for example, the values in a table should be drawn from the same domain). In this work, we introduce the use of semantic relationships between pairs of columns in a table to improve the accuracy of union search. Consequently, we introduce a new notion of unionability that considers relationships between columns, together with the semantics of columns, in a principled way. To do so, we present two new methods to discover semantic relationship between pairs of columns. The first uses an existing knowledge base (KB), the second (which we call a "synthesized KB") uses knowledge from the data lake itself. We adopt an existing Table Union Search benchmark and present new (open) benchmarks that represent small and large real data lakes. We show that our new unionability search algorithm, called SANTOS, outperforms a state-of-the-art union search that uses a wide variety of column-based semantics, including word embeddings and regular expressions. We show empirically that our synthesized KB improves the accuracy of union search by represe

The Union Closed Sets Conjecture states that in every finite, nontrivial set family closed under taking unions there is an element contained in at least half of all the sets of the family. We investigate two new directions with respect to the conjecture. Firstly, we investigate the frequencies of all elements among a union closed family and pose a question generalizing the Union Closed Sets Conjecture. Secondly, we investigate structures equivalent to union closed families and obtain a weakening of the Union Closed Sets Conjecture. We pose some new open questions about union closed families and related structures and hint at some further directions of research regarding the conjecture.

The Union-Closed Sets Conjecture, also known as Frankl's conjecture, asks whether, for any union-closed set family $\mathcal{F}$ with $m$ sets, there is an element that lies in at least $\frac{1}{2}\cdot m$ sets in $\mathcal{F}$. In 2022, Nagel posed a stronger conjecture that within any union-closed family whose ground set size is at least $k$, there are always $k$ elements in the ground set that appear in at least $\frac{1}{2^{k-1}+1}$ proportion of the sets in the family. Das and Wu showed that this conjecture is true for $k\geq 3$ and $k=2$ if $|\mathcal{F}|$ is outside a particular range. In this companion paper, we analyse further when $\mathcal{F}$ fails Nagel's conjecture for $k=2$ via linear programming.

Disjoint union is a partial binary operation returning the union of two sets if they are disjoint and undefined otherwise. A disjoint-union partial algebra of sets is a collection of sets closed under disjoint unions, whenever they are defined. We provide a recursive first-order axiomatisation of the class of partial algebras isomorphic to a disjoint-union partial algebra of sets but prove that no finite axiomatisation exists. We do the same for other signatures including one or both of disjoint union and subset complement, another partial binary operation we define. Domain-disjoint union is a partial binary operation on partial functions, returning the union if the arguments have disjoint domains and undefined otherwise. For each signature including one or both of domain-disjoint union and subset complement and optionally including composition, we consider the class of partial algebras isomorphic to a collection of partial functions closed under the operations. Again the classes prove to be axiomatisable, but not finitely axiomatisable, in first-order logic. We define the notion of pairwise combinability. For each of the previously considered signatures, we examine the class isomo

We demonstrate that when a graph exhibits a specific type of symmetry, it satisfies the Union Closed Conjecture(UCC). Additionally, we show that certain graph classes, such as Cylindrical Grid Graphs and Torus Grid Graphs also satisfy the conjecture. We prove the known result that the union closed family generated by cyclic translates of a fixed set satisfies the UCC, offering a simpler proof via symmetry arguments. Later, we show that the union closed family generated by the family obtained through cyclically shifting elements from selected translates also satisfies the conjecture.

Graph Neural Networks (GNNs) are widely used for graph representation learning in many application domains. The expressiveness of vanilla GNNs is upper-bounded by 1-dimensional Weisfeiler-Leman (1-WL) test as they operate on rooted subtrees through iterative message passing. In this paper, we empower GNNs by injecting neighbor-connectivity information extracted from a new type of substructure. We first investigate different kinds of connectivities existing in a local neighborhood and identify a substructure called union subgraph, which is able to capture the complete picture of the 1-hop neighborhood of an edge. We then design a shortest-path-based substructure descriptor that possesses three nice properties and can effectively encode the high-order connectivities in union subgraphs. By infusing the encoded neighbor connectivities, we propose a novel model, namely Union Subgraph Neural Network (UnionSNN), which is proven to be strictly more powerful than 1-WL in distinguishing non-isomorphic graphs. Additionally, the local encoding from union subgraphs can also be injected into arbitrary message-passing neural networks (MPNNs) and Transformer-based models as a plugin. Extensive exp

We introduce and investigate $d$-convex union representable complexes: the complexes that arise as the nerve of a finite collection of convex open sets in $\mathbb R^d$ whose union is also convex. Chen, Frick, and Shiu recently proved that such complexes are collapsible and asked if all collapsible complexes are convex union representable. We disprove this by showing that there exist shellable and collapsible complexes that are not convex union representable; there also exist non-evasive complexes that are not convex union representable. In the process we establish several necessary conditions for a complex to be convex union representable such as: that such a complex $Δ$ collapses onto the star of any face of $Δ$, that the Alexander dual of $Δ$ must also be collapsible, and that if $k$ facets of $Δ$ contain all free faces of $Δ$, then $Δ$ is $(k-1)$-representable. We also discuss some sufficient conditions for a complex to be convex union representable. The notion of convex union representability is intimately related to the study of convex neural codes. In particular, our results provide new families of examples of non-convex neural codes.

This paper introduces an extension of descriptive intersection and provides a framework for descriptive unions of nonempty sets. Fibre bundles provide structures that characterize spatially near as well as descriptively near sets, their descriptive intersection and their unions. The properties of four different forms of descriptive unions are given. A main result given in this paper is the equivalence between ordinary set intersection and a descriptive union. Applications of descriptive unions are given with respect to Jeffs-Novik convex unions and descriptive unions in digital images.

Given a finite collection $\{X_i\}_{i\in I}$ of metric spaces, each of which has finite Nagata dimension and Lipschitz free space isomorphic to $L^1$, we prove that their union has Lipschitz free space isomorphic to $L^1$. The short proof we provide is based on the Pelczyński decomposition method. A corollary is a solution to a question of Kaufmann about the union of two planar curves with tangential intersection. A second focus of the paper is on a special case of this result that can be studied using geometric methods. That is, we prove that the Lipschitz free space of a union of finitely many quasiconformal trees is isomorphic to $L^1$. These geometric methods also reveal that any metric quotient of a quasiconformal tree has Lipschitz free space isomorphic to $L^1$. Finally, we analyze Lipschitz light maps on unions and metric quotients of quasiconformal trees in order to prove that the Lipschitz dimension of any such union or quotient is equal to 1.

An arborescence in a digraph is an acyclic arc subset in which every vertex execpt a root has exactly one incoming arc. In this paper, we reveal the reconfigurability of the union of $k$ arborescences for fixed $k$ in the following sense: for any pair of arc subsets that can be partitioned into $k$ arborescences, one can be transformed into the other by exchanging arcs one by one so that every intermediate arc subset can also be partitioned into $k$ arborescences. This generalizes the result by Ito et al. (2023), who showed the case with $k=1$. Since the union of $k$ arborescences can be represented as a common matroid basis of two matroids, our result gives a new non-trivial example of matroid pairs for which two common bases are always reconfigurable to each other.

A set system is called union closed if for any two sets in the set system their union is also in the set system. Gilmer recently proved that in any union closed set system some element belongs to at least a $0.01$ fraction of sets, and conjectured that his technique can be pushed to the constant $\frac{3-\sqrt{5}}{2}$. We verify his conjecture; show that it extends to approximate union closed set systems, where for nearly all pairs of sets their union belong to the set system; and show that for such set systems this bound is optimal.