We bridge sheaves of rings over a topological space with common meadows (algebraic structures where the inverse for multiplication is a total operation). More specifically, we show that, given a topological space X, the subclass of pre-meadows with a , coming from the lattice of open sets of X, and the class of presheaves over X are the same structure. Furthermore, we provide a construction that, given a sheaf of rings F on X, produces a common meadow as a disjoint union of elements of the form F ( U ) indexed over the open subsets of X. As a consequence, we see that the process of going from a presheaf to a sheaf (called sheafification) allows us to get a way to construct a common meadow from a given pre-meadow as above.

Let G be a finite, non-abelian group of the form G = AN , where A ≤ G is abelian, and N ⊴ G is cyclic. We prove that the commuting graph Γ ( G ) of G is either a connected graph of diameter at most four, or the disjoint union of | G ′ | + 1 complete graphs. These results apply to all finite metacyclic groups, and to groups of square-free order in particular.

ABSTRACT We offer a new, gradual approach to the largest girth problem for cubic graphs . It is easily observed that the largest possible girth of all ‐vertex cubic graphs is attained by a 2‐connected graph . By Petersen's graph theorem, is the disjoint union of a 2‐factor and a perfect matching . We refer to the edges of as ribs and classify the cycles in by their number of ribs. We define to be the smallest integer such that every cubic ‐vertex graph with a given perfect matching has a cycle of length at most with at most ribs. Here, we determine this function up to small additive constant for and up to a small multiplicative constant for larger .

For a graph [Formula: see text], a restrained Roman dominating function [Formula: see text] has the property that every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] for which [Formula: see text] and at least one vertex [Formula: see text] for which [Formula: see text]. The weight of a restrained Roman dominating function is the sum [Formula: see text]. The minimum weight of a restrained Roman dominating function is called the restrained Roman domination number and is denoted by [Formula: see text]. Let the external neighborhood set [Formula: see text] of a set [Formula: see text] be the set of vertices in [Formula: see text] that have a neighbor in the vertex set [Formula: see text]. The restrained external neighborhood of [Formula: see text] is defined as [Formula: see text]. The restrained differential of [Formula: see text] is defined as [Formula: see text] and the restrained differential of a graph is defined as [Formula: see text]}. The theory of restrained differential is perfectly integrated into the theory of restrained Roman domination for development by the use of a Gallai-type theorem proven recently. The complementary prism [Formula: see text] of [Formula: see text] arises from the disjoint union of [Formula: see text] and its complement [Formula: see text] by adding the edges of a perfect matching between the corresponding vertices of [Formula: see text] and [Formula: see text]. This paper is devoted to the computation of restrained differential and restrained Roman domination of complementary prisms, and results are obtained for complementary prims of specified family of graphs.

A multiple group rack is a rack which is a disjoint union of groups equipped with a binary operation satisfying some conditions. It is used to define invariants of spatial surfaces, i.e. oriented compact surfaces with boundaries embedded in the [Formula: see text]-sphere [Formula: see text]. A [Formula: see text]-family of racks is a set with a family of binary operations indexed by the elements of a group [Formula: see text]. There are two known methods for constructing multiple group racks. One is via a [Formula: see text]-family of racks. The resulting multiple group rack is called the associated multiple group rack of the [Formula: see text]-family of racks. The other is by taking an abelian extension of a multiple group rack. In this paper, we introduce a new method for constructing multiple group racks by using a [Formula: see text]-family of racks and a normal subgroup [Formula: see text] of [Formula: see text]. We show that this construction yields multiple group racks that are neither the associated multiple group racks of any [Formula: see text]-family of racks nor their abelian extensions when the right conjugation action of [Formula: see text] on [Formula: see text] is nontrivial. As an application, we present a pair of spatial surfaces that cannot be distinguished by invariants derived from the associated multiple group racks of any [Formula: see text]-family of racks, yet can be distinguished using invariants obtained from a multiple group rack introduced in this paper.

Abstract Let $\mathbf{\Sigma }:=(\Sigma ,\mathbb{M},\mathbb{P})$ be a surface with marked points $\mathbb{M}\subset \partial \Sigma \neq \emptyset $ on the boundary, and punctures $\mathbb{P}\subset \Sigma \setminus \partial \Sigma $, and $T$ an arbitrary tagged triangulation of $\mathbf{\Sigma }$ in the sense of Fomin–Shapiro–Thurston. The Jacobian algebra $A(T):=\mathcal{P}(Q(T), W(T))$ corresponding to the non-degenerate potential $W(T)$ of $T$ defined by Cerulli Irelli and the second author is tame, as shown by Schröer and the first two authors. In this paper, we show that there is a natural isomorphism $\pi _{T}\colon \operatorname{Lam}(\mathbf{\Sigma })\rightarrow \operatorname{DecIrr}^\tau (A(T))$ of tame partial KRS-monoids that intertwines dual shear coordinates with respect to $T$, and generic $g$-vectors of irreducible components. Here, $\operatorname{Lam}(\mathbf{\Sigma })$ is the set of laminations of $\mathbf{\Sigma }$, considered by Musiker–Schiffler–Williams, with the disjoint union of non-intersecting laminations as partial monoid operation. On the other hand, $\operatorname{DecIrr}^\tau (A(T))$ denotes the set of generically $\tau $- regular irreducible components of the decorated representation varieties of $A(T )$, with the direct sum of generically $E$-orthogonal irreducible components as partial monoid operation, where $E$ is the symmetrized $E$-invariant of Derksen–Weyman–Zelevinsky, $E(-,\bullet )=\dim \operatorname{Hom}_{A(T)}(-,\tau (\bullet ))+\dim \operatorname{Hom}_{A(T)}(\bullet ,\tau (-))$.

abstract: For a rigid space $X$, we answer two questions of de~Jong about the category $\mathbf{Cov}^\mathrm{adm}_X$ of coverings which are locally in the admissible topology on $X$ the disjoint union of finite \'etale coverings: we show that this class is different from the one used by de~Jong, but still gives a tame infinite Galois category. In addition, we prove that the objects of $\mathbf{Cov}^{\text{\'et}}_X$ (with the analogous definition) correspond precisely to locally constant sheaves for the pro-\'etale topology defined by Scholze.

In this paper we are concerned with elliptic equations in divergence form with a potential, posed in a bounded domain $\Omega$. We allow the coefficients of the diffusion matrix $A(x)$ and the potential $Q(x)$ to diverge at the boundary; in addition, we permit that $Q(x)$ vanishes inside $\Omega$, and $A(x)$ loses ellipticity at $\partial\Omega$. The boundary $\partial \Omega$ is assumed to be the (disjoint) union of a finite number $p$ of submanifolds of dimension $\kappa_i\in \{0, \ldots, n-1\}\, (i=1, \ldots, p$). Under suitable assumptions on the behavior of $Q(x)$ and $A(x)$, which also depend on $\kappa_i$, we prove the validity of a Liouville-type theorem. Finally, we show an example for which our hypotheses on $Q$ and $A$ are sharp.

Abstract We investigate the interplay between (−1)-form symmetries and their quantum-dual (d − 1)-form counterparts within the framework of Symmetry Topological Field Theories (SymTFTs). In this framework the phenomenon of decomposition — a d-dimensional quantum field theory with (d − 1)-form symmetry being the disjoint union of other theories (or “universes”) — arises naturally from manipulations of topological boundary conditions of the SymTFT. We corroborate our findings with various examples, including a generalization of “instanton-restricted” 4d Yang-Mills theories with no sum over instanton sectors. Furthermore, we construct a 3d SymTFT with a non-invertible (−1)-form symmetry. The absolute 2d quantum field theory includes a 0-form global symmetry that depends on a parameter whose value gets shifted by the action of the (−1)-form symmetry, and we show that the non-invertibility of the latter is needed to encode this modification of the 0-form symmetry.

In this paper, we prove an upper bound on the second nonzero Laplacian eigenvalue on n -dimensional real projective space. The sharp result for 2-dimensions was shown by Nadirashvili and Penskoi and later by Karpukhin when the metric degenerates to that of the disjoint union of a round projective space and a sphere. That conjecture is open in higher dimensions, but this paper proves it up to a constant factor that tends to 1 as the dimension tends to infinity. Also, we introduce a topological argument that deals with the orthogonality conditions in a single step proof.

This paper proves that, in a four-dimensional spherically symmetric spacetime manifold, one can consider coordinate transformations expressed by fractional linear maps which give rise to isometries and are the simplest example of coordinate transformation used to bring infinity down to a finite distance. The projective boundary of spherically symmetric spacetimes here studied is the disjoint union of three points: future timelike infinity, past timelike infinity, spacelike infinity, and the three-dimensional products of half-lines with a [Formula: see text]-sphere. Geodesics are then studied in the projectively transformed [Formula: see text] coordinates for Schwarzschild spacetime, with special interest in their way of approaching our points at infinity. Next, Nariai, de Sitter and Gödel spacetimes are studied with our projective method. Since the kinds of infinity here defined depend only on the symmetry of interest in a spacetime manifold, they have a broad range of applications, which motivate the innovative analysis of Schwarzschild, Nariai, de Sitter and Gödel spacetimes.

In this article, we classify (non-compact) 3 3 -manifolds with uniformly positive scalar curvature. Precisely, we show that an orientable 3-manifold has a complete metric with uniformly positive scalar curvature if and only if it is homeomorphic to a connected sum of spherical 3 3 -manifolds and some copies of S 2 × S 1 \mathbb {S}^2\times \mathbb {S}^1 . Further, we study a 3 3 -manifold with mean convex boundary and with uniformly positive scalar curvature. If the boundary is a disjoint union of closed surfaces, then the manifold is a connected sum of spherical 3 3 -manifolds, some copies of S 1 × S 2 \mathbb {S}^1\times \mathbb {S}^2 and some handlebodies.

ABSTRACTGiven integers and , consider a graph of maximum degree and a partition of its vertices into blocks of size at least . By a seminal result of Haxell, there is an independent set of the graph that is transversal to the blocks, a so‐called independent transversal. We show that, if moreover , then every independent transversal can be transformed within the space of independent transversals to any other through a sequence of one‐vertex modifications, showing connectivity of the so‐called reconfigurability graph of independent transversals. This is sharp in that for (and ) the connectivity conclusion can fail. In this case, we show furthermore that in an essential sense it can only fail for the disjoint union of copies of the complete bipartite graph . This constitutes a qualitative refinement of Haxell's theorem.

Expected $$\mathcal {N}\mathcal {F}$$-number of disjoint union of finite copies of complete graph

Tree-width and path-width are well-known graph parameters. Many NP-hard graph problems admit polynomial-time solutions when restricted to graphs of bounded tree-width or bounded path-width. In this work, we study the behavior of tree-width and path-width under various unary and binary graph transformations. For considered transformations, we provide upper and lower bounds for the tree-width and path-width of the resulting graph in terms of those of the initial graphs or argue why such bounds are impossible to specify. Among the studied unary transformations are vertex addition, vertex deletion, edge addition, edge deletion, subgraphs, vertex identification, edge contraction, edge subdivision, minors, powers of graphs, line graphs, edge complements, local complements, Seidel switching, and Seidel complementation. Among the studied binary transformations, we consider the disjoint union, join, union, substitution, graph product, 1-sum, and corona of two graphs.

In this paper, we consider a compact Riemann surface [Formula: see text] with a complex of non-intersecting Jordan curves, whose complement is a pair of Riemann surfaces with boundary, each of which may be possibly disconnected. We investigate conformally invariant integral operators of Schiffer, which act on [Formula: see text] anti-holomorphic one-forms on one of these surfaces with boundary and produce holomorphic one-forms on the disjoint union. These operators arise in potential theory, boundary value problems, approximation theory, and conformal field theory, and are closely related to a kind of Cauchy operator. We develop an extensive calculus for the Schiffer and Cauchy operators, including a number of adjoint identities for the Schiffer operators. In the case that the Jordan curves are quasicircles, we derive a Plemelj–Sokhotski jump formula for Dirichlet-bounded functions. We generalize a theorem of Napalkov and Yulmukhametov, which shows that a certain Schiffer operator is an isomorphism for quasicircles. Finally, we characterize the kernels and images, and derive index theorems for the Schiffer operators, which will in turn connect conformal invariants to topological invariants.

ABSTRACTFor a multiset of positive integers, let . A ‐list assignment of is a list assignment of such that the colour set can be partitioned into the disjoint union of sets so that for each and each vertex of , . We say is ‐choosable if is ‐colourable for any ‐list assignment of . The concept of ‐choosability puts ‐colourability and ‐choosability in a same framework: If , then ‐choosability is equivalent to ‐choosability; if consists of copies of 1, then ‐choosability is equivalent to ‐colourability. If is ‐choosable, then is ‐colourable. On the other hand, there are ‐colourable graphs that are not ‐choosable, provided that contains an integer larger than 1. Let be the minimum number of vertices in a ‐colourable non‐‐choosable graph. This paper determines the value of for all .

We provide a new branching rule from the general linear group \mathrm{GL}_{2n}(\mathbb{C}) to the symplectic group \mathrm{Sp}_{2n}(\mathbb{C}) by establishing a simple algorithm which gives rise to a bijection from the set of semistandard tableaux of a fixed shape to a disjoint union of several copies of sets of symplectic tableaux of various shapes. The algorithm arises from representation theory of a quantum symmetric pair of type A\mathrm{II}_{2n-1} , which is a q -analogue of the classical symmetric pair (\mathfrak{gl}_{2n}(\mathbb{C}), \mathfrak{sp}_{2n}(\mathbb{C})) .

A poset X is said to be zigzag image-finite, if the least updownset (i.e., both an upset and a downset) containing x is finite, for all x∈X. We show that a bi-Heyting algebra is profinite if and only if it is isomorphic to the lattice of upsets of a zigzag image-finite poset. Zigzag image-finite posets have the property of being disjoint unions of finite connected posets. Because of this, we equivalently show that a bi-Heyting algebra is profinite if and only if it is isomorphic to a direct product of simple finite bi-Heyting algebras.

In reference to Werner's measure on self-avoiding loops on Riemann surfaces, Cardy conjectured a formula for the measure of all homotopically nontrivial loops in a finite type annular region with modular parameter $\rho$. Ang, Remy and Sun have announced a proof of this conjecture using random conformal geometry. Cardy's formula implies that the measure of the set of homotopically nontrivial loops in the punctured plane which intersect $S^1$ equals $\frac{2\pi}{\sqrt{3}}$. This set is the disjoint union of the set of loops which avoid a ray from the unit circle to infinity and its complement. There is an inclusion/exclusion sum which, in a limit, calculates the measure of the set of loops which avoid a ray. Each term in the sum involves finding the transfinite diameter of a slit domain. This is numerically accessible using the remarkable Schwarz-Christoffel package developed by Driscoll and Trefethen. Our calculations suggest this sum is around $\pi$, consistent with Cardy's formula.