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Let \begin{document}$ \mathbb{F}_{q} $\end{document} be a finite field with character \begin{document}$ p $\end{document} . In this paper, the multiplicative group \begin{document}$ \mathbb{F}_{q}^{*} = \mathbb{F}_{q}\setminus\{0\} $\end{document} is decomposed into a mutually disjoint union of \begin{document}$ \gcd(6l^mp^n,q-1) $\end{document} cosets over subgroup \begin{document}$ $\end{document} , where \begin{document}$ \xi $\end{document} is a primitive element of \begin{document}$ \mathbb{F}_{q} $\end{document} . Based on the decomposition, the structure of constacyclic codes of length \begin{document}$ 6l^mp^n $\end{document} over finite field \begin{document}$ \mathbb{F}_{q} $\end{document} and their duals is established in terms of their generator polynomials, where \begin{document}$ p\neq{3} $\end{document} and \begin{document}$ l\neq{3} $\end{document} are distinct odd primes, \begin{document}$ m $\end{document} and \begin{document}$ n $\end{document} are positive integers. In addition, we determine the characterization and enumeration of all linear complementary dual(LCD) negacyclic codes and self-dual constacyclic codes of length \begin{document}$ 6l^mp^n $\end{document} over \begin{document}$ \mathbb{F}_{q} $\end{document} .

We study the diffusive generalized logistic problem with constant yield harvesting: { u ″ ( x ) + λ g ( u ) − μ = 0 , − 1 x 1 , u ( − 1 ) = u ( 1 ) = 0 , where λ , μ > 0 . We assume that g satisfies g ( 0 ) = g ( 1 ) = 0 , g ( u ) > 0 on ( 0 , 1 ) , and g is either convex-concave or convex-concave-convex on ( 0 , 1 ) and satisfies certain conditions. We prove that, for any fixed μ > 0 , on the ( λ , ‖ u ‖ ∞ ) -plane, the bifurcation diagram always consists of a continuous, ⊂-shaped curve, and we study the structures and evolution of bifurcation diagrams for varying μ > 0 . We also prove that, for any fixed λ > λ 0 ⁎ for some λ 0 ⁎ > 0 , on the ( μ , ‖ u ‖ ∞ ) -plane, the bifurcation diagram consists of a reversed ⊂-shaped curve with possibly the disjoint union of a strictly increasing curve, and we study the structures and evolution of bifurcation diagrams for varying λ > λ 0 ⁎ . Our results complement those of Hung, Suen, and Wang (2020) [10] in which problem g is either concave or concave-convex on ( 0 , 1 ) .

The maximal ideal space of the algebra of bounded holomorphic functions on the countable disjoint union of open unit disks D ⊂ C \mathbb {D}\subset \mathbb {C} is studied from a topological point of view. The results are similar to those for the maximal ideal space of the algebra H ∞ ( D ) H^\infty (\mathbb {D}) .

Let Kn be a complete graph with n vertices. For graphs G and H, the Ramsey number R(G, H) is the smallest positive integer n such that in every red-blue coloring on the edges of Kn, there is a red copy of graph G or a blue copy of graph H in Kn. Determining the Ramsey number R(Cn, tWm) for any integers t ≥ 1, n ≥ 3 and m ≥ 4 in general is a challenging problem, but we conjecture that for any integers t ≥ 1 and m ≥ 4, there exists n0 = f(t, m) such that cycle Cn is tWm–good for any n ≥ n0. In this paper, we provide some evidence for the conjecture in the case of m = 4 that if n ≥ n0 then the Ramsey number R(Cn, tW4)=2n + t − 2 with n0 = 15t2 − 4t + 2 and t ≥ 1. Furthermore, if G is a disjoint union of cycles then the Ramsey number R(G, tW4) is also derived.

Let Z2 denote a cyclic group of 2 order and Z 2 2 = Z2 × Z2 the direct product of groups. Suppose that (M, Φ) is a closed and smooth manifold M with a smooth Z 2 2 -action whose fixed point set is the disjoint union of two real projective spaces with the same dimension. In this paper, the authors give a sufficient condition on the fixed data of the action for (M, Φ) bounding equivariantly.

The adjacency matrix of a simple and undirected graph G is denoted by $${\mathcal {A}}(G)$$ and $${\mathcal {D}}_{G}$$ is the degree diagonal matrix of G. The Laplacian matrix of G is $${\mathcal {L}}(G)={\mathcal {D}}_{G}-{\mathcal {A}}(G)$$ and the signless Laplacian matrix of G is $${\mathcal {Q}}(G)={\mathcal {D}}_{G}+{\mathcal {A}}(G) $$ . The star graph of order n is denoted by $$S_{n}$$ . The double starlike tree $${\mathcal {G}}_{p,n,q}$$ is obtained by attaching p pendant vertices to one pendant vertex of the path $$P_n$$ and q pendant vertices to the other pendant vertex of $$P_n$$ . In this paper, we first investigate the disjoint union of double starlike graphs $${\mathcal {G}}_{p,2,q}$$ and the star graphs $$S_{n}$$ for Laplacian (signless) spectral characterization. Also, the signless Laplacian spectral determination of the disjoint union of odd unicyclic graphs and star graphs is studied. Abdian et al. [AKCE Int. J. Graphs Combin. (2018) https://doi.org/10.1016/j.akcej.2018.06.009] proved that if G is a $$\mathcal {DQS}$$ connected non-bipartite graph with $$n\ge 3$$ vertices, then $$G\cup rK_{1}\cup sK_{2}$$ is $$\mathcal {DQS}$$ . Here we give a counterexample for the claim and also we study the graph $$G\cup rK_{1}\cup sK_{2}$$ for signless Laplacian charcterization when G has at least $$((n-2)(n-3)+10)/2$$ edges and $$s=1$$ . It is shown that the graph $$K_{n}\cup K_{2}\cup rK_{1}$$ is $$\mathcal {DQS}$$ for $$n\ge 4$$ . We also prove that the complement graph of $$K_{n}\cup K_{2}\cup rK_{1}$$ is $$\mathcal {DQS}$$ for $$r>1$$ and $$n\ne 3$$ .

Let $$G_n$$ and $$G'_{n}$$ be two nonisomorphic graphs on n vertices with spectra $$\begin{aligned} \lambda _{1} \geqslant \lambda _{2} \geqslant \cdots \geqslant \lambda _{n} \ \ \mathrm {and} \ \ \lambda '_{1} \geqslant \lambda '_{2} \geqslant \cdots \geqslant \lambda '_{n}, \end{aligned}$$ respectively. Define the distance between the spectra of $$G_{n}$$ and $$G'_{n}$$ as $$\begin{aligned} \lambda (G_{n}, G'_{n})= \sum \limits _{i = 1}^n (\lambda _{i}- \lambda _{i}')^{2} \ \mathrm { \big( or \ use \ } \sum \limits _{i = 1}^n \vert \lambda _{i}- \lambda _{i}' \vert \big). \end{aligned}$$ Let $$\begin{aligned} \mathrm {cs}(G_{n})= \mathrm {min} \{ \lambda (G_n, G'_{n}): G'_{n} \ \mathrm { not \ isomorphic \ to} \ G_{n} \}, \end{aligned}$$ and $$\begin{aligned} \text{cs}_{n}=\max \{\text{cs}(G_{n}): G_{n} \text{ a graph on } n \text{ vertices}\}. \end{aligned}$$ Richard Brualdi in [9] proposed the following problems: Problem A. Investigate cs( $$G_{n}$$ ) for special classes of graphs. Problem B. Find a good upper bound on $$\mathrm {cs}_n$$ . In this paper, we investigate problem A and determine cs( $$G_{n}$$ ), when $$G_n$$ is a graph of order n with size two or three. Let $$K_{n}+ K_{1}$$ be a disjoint union of the complete graph $$K_n$$ with one isolated vertex, and $$K_{n+1}^{1}$$ be a complete graph $$K_{n+1}$$ by deleting $$n-1$$ edges from one vertex of $$K_{n+1}$$ . We show that if $$\lambda (K_{n}+ K_{1}, G_{n}')=$$ cs( $$G_{n}$$ ), then $$G_{n}'$$ is isomorphic to $$K_{n+1}^{1}$$ .

The energy of a graph G, E ( G ) , is the sum of absolute values of the eigenvalues of its adjacency matrix. This concept was extended by Nikiforov to arbitrary complex matrices. Recall that the trace norm of a digraph D is defined as, N ( D ) = ∑ i = 1 n σ i , where σ 1 ≥ ⋯ ≥ σ n are the singular values of the adjacency matrix of D. In this paper we would like to present some lower and upper bounds for N ( D ) . For any digraph D it is proved that N ( D ) ≥ rank ( D ) and the equality holds if and only if D is a disjoint union of directed cycles and directed paths. Finally, we present a lower bound on σ 1 and N ( D ) in terms of the size of digraph D.

Hyperplane arrangements dissect $\mathbb{R}^n$ into connected components called chambers, and a well-known theorem of Zaslavsky counts chambers as a sum of nonnegative integers called Whitney numbers of the first kind. His theorem generalizes to count chambers within any cone defined as the intersection of a collection of halfspaces from the arrangement, leading to a notion of Whitney numbers for each cone. This paper focuses on cones within the braid arrangement, consisting of the reflecting hyperplanes $x_i=x_j$ inside $\mathbb{R}^n$ for the symmetric group, thought of as the type $A_{n-1}$ reflection group. Here cones correspond to posets, chambers within the cone correspond to linear extensions of the poset, and the Whitney numbers of the cone interestingly refine the number of linear extensions of the poset. We interpret this refinement for all posets as counting linear extensions according to a statistic that generalizes the number of left-to-right maxima of a permutation. When the poset is a disjoint union of chains, we interpret this refinement differently, using Foata's theory of cycle decomposition for multiset permutations, leading to a simple generating function compiling these Whitney numbers., Historical references added; version to appear in ORDER

The general position number ${\rm gp}(G)$ of a graph $G$ is the cardinality of a largest set of vertices $S$ such that no element of $S$ lies on a geodesic between two other elements of $S$. The complementary prism $G\overline{G}$ of $G$ is the graph formed from the disjoint union of $G$ and its complement $\overline{G}$ by adding the edges of a perfect matching between them. It is proved that ${\rm gp}(G\overline{G})\le n(G) + 1$ if $G$ is connected and ${\rm gp}(G\overline{G})\le n(G)$ if $G$ is disconnected. Graphs $G$ for which ${\rm gp}(G\overline{G}) = n(G) + 1$ holds, provided that both $G$ and $\overline{G}$ are connected, are characterized. A sharp lower bound on ${\rm gp}(G\overline{G})$ is proved. If $G$ is a connected bipartite graph or a split graph then ${\rm gp}(G\overline{G})\in \{n(G), n(G)+1\}$. Connected bipartite graphs and block graphs for which ${\rm gp}(G\overline{G})=n(G)+1$ holds are characterized. A family of block graphs is constructed in which the ${\rm gp}$-number of their complementary prisms is arbitrary smaller than their order.

We consider Bethe subalgebras B(C) in the Yangian $${\mathrm {Y}}({\mathfrak {gl}}_2)$$ with C regular $$2\times 2$$ matrix. We study the action of Bethe subalgebras of $${\mathrm {Y}}({\mathfrak {gl}}_2)$$ on finite-dimensional representations of $${\mathrm {Y}}({\mathfrak {gl}}_2)$$ . We prove that B(C) with real diagonal C has simple spectrum on any irreducible $${\mathrm {Y}}({\mathfrak {gl}}_2)$$ -module corresponding to a disjoint union of real strings. We extend this result to limits of Bethe algebras. Our main tool is the computation of Shapovalov-type determinant for the nilpotent degeneration of B(C).

For a closed oriented smooth 4–manifold X with b+2(X)>0, the Seiberg–Witten invariants are well-defined. Taubes’ “ SW= Gr” theorem asserts that if X carries a symplectic form then these invariants are equal to well-defined counts of pseudoholomorphic curves, Taubes’ Gromov invariants. In the absence of a symplectic form, there are still nontrivial closed self-dual 2–forms which vanish along a disjoint union of circles and are symplectic elsewhere. This paper and its sequel describe well-defined integral counts of pseudoholomorphic curves in the complement of the zero set of such near-symplectic 2–forms, and it is shown that they recover the Seiberg–Witten invariants over ℤ∕2ℤ. This is an extension of “ SW= Gr” to nonsymplectic 4–manifolds. The main result of this paper asserts the following. Given a suitable near-symplectic form ω and tubular neighborhood 𝒩 of its zero set, there are well-defined counts of pseudoholomorphic curves in a completion of the symplectic cobordism (X−𝒩,ω) which are asymptotic to certain Reeb orbits on the ends. They can be packaged together to form “near-symplectic” Gromov invariants as a function of spin-c structures on X.

We characterize classes of graphs closed under taking vertex-minors and having no $P_n$ and no disjoint union of $n$ copies of the $1$-subdivision of $K_{1,n}$ for some $n$. Our characterization is described in terms of a tree of radius $2$ whose leaves are labelled by the vertices of a graph $G$, and the width is measured by the maximum possible cut-rank of a partition of $V(G)$ induced by splitting an internal node of the tree to make two components. The minimum width possible is called the depth-$2$ rank-brittleness of $G$. We prove that for all $n$, every graph with sufficiently large depth-$2$ rank-brittleness contains $P_n$ or disjoint union of $n$ copies of the $1$-subdivision of $K_{1,n}$ as a vertex-minor., 23 pages, 4 figures

We decompose the copositive cone $$\mathcal {COP}^{n}$$ into a disjoint union of a finite number of open subsets $$S_{{\mathcal {E}}}$$ of algebraic sets $$Z_{{\mathcal {E}}}$$ . Each set $$S_{{\mathcal {E}}}$$ consists of interiors of faces of $$\mathcal {COP}^{n}$$ . On each irreducible component of $$Z_{{\mathcal {E}}}$$ these faces generically have the same dimension. Each algebraic set $$Z_{{\mathcal {E}}}$$ is characterized by a finite collection $${{\mathcal {E}}} = \{(I_{\alpha },J_{\alpha })\}_{\alpha = 1,\dots ,|\mathcal{E}|}$$ of pairs of index sets. Namely, $$Z_{{\mathcal {E}}}$$ is the set of symmetric matrices A such that the submatrices $$A_{J_{\alpha } \times I_{\alpha }}$$ are rank-deficient for all $$\alpha $$ . For every copositive matrix $$A \in S_{{\mathcal {E}}}$$ , the index sets $$I_{\alpha }$$ are the minimal zero supports of A. If $$u^{\alpha }$$ is a corresponding minimal zero, then $$J_{\alpha }$$ is the set of indices j such that $$(Au^{\alpha })_j = 0$$ . We call the pair $$(I_{\alpha },J_{\alpha })$$ the extended support of the zero $$u^{\alpha }$$ , and $${{\mathcal {E}}}$$ the extended minimal zero support set of A. We provide some necessary conditions on $${{\mathcal {E}}}$$ for $$S_{{\mathcal {E}}}$$ to be non-empty, and for a subset $$S_{{{\mathcal {E}}}'}$$ to intersect the boundary of another subset $$S_{{\mathcal {E}}}$$ .

For given r-uniform hypergraphs $${\mathcal {G}}$$ and $${\mathcal {H}}$$ , the Ramsey number $$R({\mathcal {G}},{\mathcal {H}};r)$$ is defined as the smallest positive integer p such that any red/blue coloring of the edges of the complete r-uniform hypergraph on p vertices contains either a red copy of $${\mathcal {G}}$$ or a blue copy of $${\mathcal {H}}$$ . The hypergraph $${\mathcal {G}}$$ is called $${\mathcal {H}}$$ -good, if $$R({\mathcal {G}},{\mathcal {H}};r)=(|V({\mathcal {G}})|-1) (\chi ({\mathcal {H}})-1)+s({\mathcal {H}}),$$ where $$s({\mathcal {H}})$$ is the chromatic surplus number of $${\mathcal {H}}$$ , i.e., the minimum cardinality of color classes over all colorings of $$V({\mathcal {H}})$$ with $$\chi ({\mathcal {H}})$$ colors. In this note, for a given r-uniform hypergraph $${\mathcal {H}}$$ , the exact value of the Ramsey number of the disjoint union of $${\mathcal {H}}$$ -good hypergraphs versus $${\mathcal {H}}$$ will be determined exactly.

For simple graphs G and H the size multipartite Ramsey number mj ( G , H ) is the smallest natural number t such that any arbitrary red-blue coloring on the edges of Kjxt contains a red G or a blue H as a subgraph. We studied the size tripartite Ramsey numbers m 3( G , H ) where G=mK1,n and H=P3 . In this paper, we generalize this result. We determine m3(G,H) where G is a star forest, namely a disjoint union of heterogeneous stars, and H=P3 . Moreover, we also determine m2(G,H) for this pair of graphs G and H .

We study the index of the APS boundary value problem for a strongly Callias-type operator $${\mathcal {D}}$$ on a complete Riemannian manifold M. We show that this index is equal to an index on a simpler manifold whose boundary is a disjoint union of two complete manifolds $$N_0$$ and $$N_1$$ . If the dimension of M is odd we show that the latter index depends only on the restrictions $${\mathcal {A}}_0$$ and $${\mathcal {A}}_1$$ of $${\mathcal {D}}$$ to $$N_0$$ and $$N_1$$ and thus is an invariant of the boundary. We use this invariant to define the relative $$\eta $$ -invariant $$\eta ({\mathcal {A}}_1,{\mathcal {A}}_0)$$ . We show that even though in our situation the $$\eta $$ -invariants of $${\mathcal {A}}_1$$ and $${\mathcal {A}}_0$$ are not defined, the relative $$\eta $$ -invariant behaves as if it were the difference $$\eta ({\mathcal {A}}_1)-\eta ({\mathcal {A}}_0)$$ .

The $k$-Colouring problem is to decide if the vertices of a graph can be coloured with at most $k$ colours for a fixed integer $k$ such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list $L(u) \subseteq \{1,\cdots, k\}$, then we obtain the List $k$-Colouring problem. A graph $G$ is $H$-free if $G$ does not contain $H$ as an induced subgraph. We continue an extensive study into the complexity of these two problems for $H$-free graphs. The graph $P_r+P_s$ is the disjoint union of the $r$-vertex path $P_r$ and the $s$-vertex path $P_s$. We prove that List $3$-Colouring is polynomial-time solvable for $(P_2+P_5)$-free graphs and for $(P_3+P_4)$-free graphs. Combining our results with known results yields complete complexity classifications of $3$-Colouring and List $3$-Colouring on $H$-free graphs for all graphs $H$ up to seven vertices., 20 pages, 6 figures. An extended abstract of this paper appeared in the proceedings of ISAAC 2018

A face irregular total k -labeling λ : V ∪ E → { 1 , 2 , … , k } of a 2-connected plane graph G is a labeling of vertices and edges such that their face-weights are pairwise distinct. The weight of a face f under a labeling λ is the sum of the labels of all vertices and edges surrounding f . The minimum value k for which G has a face irregular total k -labeling is called the total face irregularity strength of G , denoted by t f s ( G ) . The lower bound of t f s ( G ) is provided along with the exact value of two certain plane graphs. Improving the results, this paper deals with the total face irregularity strength of the disjoint union of multiple copies of a plane graph G . We estimate the bounds of t f s ( G ) and prove that the lower bound is sharp for G isomorphic to a cycle, a book with m polygonal pages, or a wheel.

Based on the generic construction of linear codes, we construct linear codes over the ring $\Bbb Z_4$ via posets of the disjoint union of two chains. We determine the Lee weight distributions of the quaternary codes. Moreover, we obtain some new linear quaternary codes and good binary codes by using the Gray map., Comment: 5 pages. Tp appear in IEEE Communications Letters

An antimagic labeling of a graph G is a bijection f:E(G)→{1,…,|E(G)|} such that the weights w(x)=∑y∼xf(y) distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1990) is that every connected graph other than K2 admits an antimagic labeling. For a set of distances D, a D-antimagic labeling of a graph G is a bijection f:V(G)→{1,…,|V(G)|} such that the weightω(x)=∑y∈ND(x)f(y) is distinct for each vertex x, where ND(x)={y∈V(G)|d(x,y)∈D} is the D-neigbourhood set of a vertex x. If ND(x)=r, for every vertex x in G, a graph G is said to be (D,r)-regular. In this paper, we conjecture that a graph admits a D-antimagic labeling if and only if it does not contain two vertices having the same D-neighborhood set. We also provide evidence that the conjecture is true. We present computational results that, for D={1}, all graphs of order up to 8 concur with the conjecture. We prove that the set of (D,r)-regular D-antimagic graphs is closed under union. We provide examples of disjoint union of symmetric (D,r)-regular that are D-antimagic and examples of disjoint union of non-symmetric non-(D,r)-regular graphs that are D-antimagic. Furthermore, lastly, we show that it is possible to obtain a D-antimagic graph from a previously known distance antimagic graph.

In the theory of semigroups there exists a construction of some semilattices from a special family of semigroups. It is the so-called ‘strong semilattice of semigroups’. Modifying this idea we can present a new construction method for arbitrary pseudocomplemented semilattice (= PCS) L using ‘full triples’. This construction centers around the classical Glivenko-Frink congruence $$\Gamma (L)$$ . This fact plays an important role. Namely, PCS L is a disjoint union of all congruence classes of $$\Gamma (L)$$ (or of GF-blocks for short). In order to get a ‘full triple’ of L, the so-called ‘associate’ full triple of L, we need the Boolean algebra of closed elements B(L), the whole family of GF-blocks $$\{\,\Gamma _a\mid a\in B(L)\,\}$$ and a suitable semilattice homomorphism $$\varphi _{a,b}:\Gamma _a\rightarrow \Gamma _b$$ for any $$a\ge b$$ in B(L). There is also a definition of an ‘abstract’ full triple, which we use by a construction of a PCS. The notion of a full triple is an extension of the ‘classical’ triple, which do work only with just one GF-block D(L) satisfying $$1\in D(L)$$ . It is known that there exist PCS’s which cannot be constructed by using a classical triple method. In addition, we explore in some detail the homomorphisms and the subalgebras of PCS’s.

Connected sums of lens spaces which smoothly bound a rational homology ball are classified by P Lisca. In the classification, there is a phenomenon that a connected sum of a pair of lens spaces [math] appears in one of the typical cases of rational homology cobordisms. We consider smooth negative-definite cobordisms among several disjoint union of lens spaces and a rational homology [math] –sphere to give a topological condition for the cobordism to admit the above “pairing” phenomenon. By using Donaldson theory, we show that if [math] has a certain minimality condition concerning the Chern–Simons invariants of the boundary components, then any [math] must have a counterpart [math] in negative-definite cobordisms with a certain condition only on homology. In addition, we show an existence of a reducible flat connection through which the pair is related over the cobordism. As an application, we give a sufficient condition for a closed smooth negative-definite [math] –orbifold with two isolated singular points whose neighborhoods are homeomorphic to the cones over lens spaces [math] and [math] to admit a finite uniformization.

Star selection principles first introduced and studied by Kocinac et al. in [10] and [2] are investigated. In particular we consider the relationship between the star-Menger, strongly star-Menger as well as the relationship between star covering properties of the Niemytzki plane with the dominating number and the bounding number. Furthermore, we give a characterization of the strongly star-Menger property in terms of games on a strongly star-Lindelof space which consist of the disjoint union of a closed discrete set with a σ-compact subspace. A consistent example of a metacompact star-Menger not strongly star-Menger space is given. In addition, we show some results related to weaker properties than paracompactness and we study particular examples. We pose several problems about these properties.

We consider a complete Riemannian manifold M whose boundary is a disjoint union of finitely many complete connected Riemannian manifolds. We compute the index of a local boundary value problem for a strongly Callias-type operator on M. Our result extends an index theorem of D. Freed to non-compact manifolds, thus providing a new insight on the Horava-Witten anomaly., 13 pages

Let $C_b(X)$ be the C*-algebra of bounded continuous functions on some non-compact, but locally compact Hausdorff space $X$. Moreover, let $A_0$ be some ideal and $A_1$ be some unital C*-subalgebra of $C_b(X)$. For $A_0$ and $A_1$ having trivial intersection, we show that the spectrum of their vector space sum equals the disjoint union of their individual spectra, whereas their topologies are nontrivially interwoven. Indeed, they form a so-called twisted-sum topology which we will investigate before. Within the whole framework, e.g., the one-point compactification of $X$ and the spectrum of the algebra of asymptotically almost periodic functions can be described., 12 pages, LaTeX. Extension of Section 3 in arXiv:1010.0449 (version v1)

We introduce a new family of classical r-matrices for the Lie algebra sl n that lies in the Zariski boundary of the Belavin–Drinfeld space M of quasi-triangular solutions to the classical Yang–Baxter equation. In this setting M is a finite disjoint union of components; exactly ϕ ( n ) of these components are S L n -orbits of single points. These points are the generalized Cremmer–Gervais r-matrices r i , n which are naturally indexed by pairs of positive coprime integers, i and n, with i n . A conjecture of Gerstenhaber and Giaquinto states that the boundaries of the Cremmer–Gervais components contain r-matrices having maximal parabolic subalgebras p i , n ⊆ sl n as carriers. We prove this conjecture in the cases when n ≡ ± 1 (mod i). The subprime linear functionals f ∈ p i , n ⁎ and the corresponding principal elements H ∈ p i , n play important roles in our proof. Since the subprime functionals are Frobenius precisely in the cases when n ≡ ± 1 (mod i), this partly explains our need to require these conditions on i and n. We conclude with a proof of the GG boundary conjecture in an unrelated case, namely when ( i , n ) = ( 5 , 12 ) , where the subprime functional is no longer a Frobenius functional.

A relational structure is called reversible iff every bijective endomorphism of that structure is an automorphism. We give several equivalents of that property in the class of disconnected binary structures and some its subclasses. For example, roughly speaking and denoting the set of integers by ${\mathbb Z}$, a structure having reversible components is reversible iff its components can not be "merged" by condensations (bijective homomorphisms) and each ${\mathbb Z}$-sequence of condensations between different components must be, in fact, a sequence of isomorphisms. We also give equivalents of reversibility in some special classes of structures. For example, we characterize CSB linear orders of a limit type and show that a disjoint union of such linear orders is a reversible poset iff the corresponding sequence of order types is finite-to-one., Comment: 16 pages

The Ramsey number $R_X(p,q)$ for a class of graphs $X$ is the minimum $n$ such that every graph in $X$ with at least $n$ vertices has either a clique of size $p$ or an independent set of size $q$. We say that Ramsey numbers are linear in $X$ if there is a constant $k$ such that $R_{X}(p,q) \leq k(p+q)$ for all $p,q$. In the present paper we conjecture that if $X$ is a hereditary class defined by finitely many forbidden induced subgraphs, then Ramsey numbers are linear in $X$ if and only if $X$ excludes a forest, a disjoint union of cliques and their complements. We prove the "only if" part of this conjecture and verify the "if" part for a variety of classes. We also apply the notion of linearity to bipartite Ramsey numbers and reveal a number of similarities and differences between the bipartite and non-bipartite case., 24 pages

Recently Cutler and Radcliffe proved that the graph on $n$ vertices with maximum degree at most $r$ having the most cliques is a disjoint union of $\lfloor n/(r+1)\rfloor$ cliques of size $r+1$ together with a clique on the remainder of the vertices. It is very natural also to consider this question when the limiting resource is edges rather than vertices. In this paper we prove that among graphs with $m$ edges and maximum degree at most $r$, the graph that has the most cliques of size at least two is the disjoint union of $\bigl\lfloor m \bigm/\binom{r+1}{2} \bigr\rfloor$ cliques of size $r+1$ together with the colex graph using the remainder of the edges.

This chapter discusses the Casson invariant of a closed three-manifold for a knot in a closed three-manifold. This is a chord diagram version of the method noted in the end of by Lickorish. It also discusses the extension of the construction of the Casson invariant from the universal Vassiliev-Kontsevich invariant by adding the 3T relation and some natural relations. The semi-simple quotient of this representation is isomorphic to a disjoint union of two copies of the natural representation of SL(2, Z). The contribution from the extension of SL(2, Z) appears in the Levi part of this representation, and this part seems to contain information for the Casson invariant. The Kirby moves correspond to Kirby’s handle slide move, where any component can be changed only along a thin line component since the thick line component is a knot and is not applied the surgery.

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for \textsc{Maximum Clique} on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time $2^{\tilde{O}(n^{2/3})}$ for \textsc{Maximum Clique} on disk and unit ball graphs. We then obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. This, in combination with our structural results, yields a randomized EPTAS for \textsc{Max Clique} on disk and unit ball graphs. \textsc{Max Clique} on unit ball graphs is equivalent to finding, given a collection of points in $\mathbb R^3$, a maximum subset of points with diameter at most some fixed value. In stark contrast, \textsc{Maximum Clique} on ball graphs and unit $4$-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have such algorithms. Indeed, we show that, for all those problems, there is a constant ratio of approximation which cannot be attained even in time $2^{n^{1-\varepsilon}}$, unless the Exponential Time Hypothesis fails., arXiv admin note: substantial text overlap with arXiv:1712.05010, arXiv:1803.01822

Let \(\overrightarrow{G} = (V, A)\) be an oriented graph and G the underlying graph of \(\overrightarrow{G}\). An oriented k-coloring of \(\overrightarrow{G}\) is a partition of V into k subsets such that there are no two adjacent vertices belonging to the same subset, and all the arcs between a pair of subsets have the same orientation. The oriented chromatic number \(\chi _o({\overrightarrow{G}})\) of \(\overrightarrow{G}\) is the smallest k, such that \({\overrightarrow{G}}\) admits an oriented k-coloring. The oriented chromatic number of G, denoted by \(\chi _o(G)\), is the maximum of \(\chi _o(\overrightarrow{G})\) for all orientations \(\overrightarrow{G}\) of G. Oriented chromatic number of product of graphs were widely studied, but the disjoint union has not being considered. In this article we study oriented coloring for the disjoint union of graphs. We establish the exact values of the union: of two complete graphs, of one complete with a forest graph, and of one complete and one cycle. Given a positive integer k, we denote by \(\mathcal {C}\mathcal {N}_k\) the class of graphs G such that \(\chi _o(G) \le k\). We use those results to characterize the class of graphs \(\mathcal {CN}_3\). We evaluate, as far as we know for the first time, the value of \(\chi _o(W_n)\) and we yield with this value an upper bound for the union of one complete and one wheel graph \(W_n\).

A function \(c:V(G)\rightarrow \{1,2,\ldots ,k\}\) is a k-colouring of a graph G if \(c(u)\ne c(v)\) whenever u and v are adjacent. If any two colour classes induce the disjoint union of vertices and edges, then c is called injective. Injective colourings are also known as L(1, 1)-labellings and distance 2-colourings. The corresponding decision problem is denoted Injective Colouring. A graph is H-free if it does not contain H as an induced subgraph. We prove a dichotomy for Injective Colouring for graphs with bounded independence number. Then, by combining known with further new results, we determine the complexity of Injective Colouring on H-free graphs for every H except for one missing case.

Ehrenborg noted that all tilings of a bipartite planar graph are encoded by its cubical matching complex and claimed that this complex is collapsible. We point out to an oversight in his proof and explain why these complexes can be the disjoint union of two or more collapsible complexes. We also prove that all links in these complexes are suspensions up to homotopy. Furthermore, we extend the definition of a cubical matching complex to planar graphs that are not necessarily bipartite, and show that these complexes are either contractible or a disjoint union of contractible complexes. For a simple connected region that can be tiled with dominoes ( 2 × 1 and 1 × 2 ) and 2 × 2 squares, let f i denote the number of tilings with exactly i squares. We prove that f 0 − f 1 + f 2 − f 3 + ⋯ = 1 (established by Ehrenborg) is the only linear relation for the numbers f i .

In this paper, we show that the only solution of the vortex sheet equation, either stationary or uniformly rotating with negative angular velocity $\Omega$, such that it has positive vorticity and is concentrated in a finite disjoint union of smooth curves with finite length is the trivial one: constant vorticity amplitude supported on a union of nested, concentric circles. The proof follows a desingularization argument and a calculus of variations flavor., Comment: 31 pages, 4 figures

We show that for compact Riemannian manifolds of dimension at least $3$ with nonempty boundary, we can modify the manifold by performing surgeries of codimension $2$ or higher, while keeping the Steklov spectrum nearly unchanged. This shows that certain changes in the topology of a domain do not have an effect when considering shape optimization questions for Steklov eigenvalues in dimensions $3$ and higher. Our result generalizes the 1-dimensional surgery in [FS2] to higher dimensional surgeries and to higher eigenvalues. It is proved in [FS2] that the unit ball does not maximize the first nonzero normalized Steklov eigenvalue among contractible domains in $\mathbb{R}^n$, for $n \geq 3$. We show that this is also true for higher Steklov eigenvalues. Using similar ideas we show that in $\mathbb{R}^n$, for $n\geq 3$, the $j$-th normalized Steklov eigenvalue is not maximized in the limit by a sequence of contractible domains degenerating to the disjoint union of $j$ unit balls, in contrast to the case in dimension $2$ [GP1]., 17 pages

Consider the following gap cycle counting problem in the streaming model: The edges of a $2$-regular $n$-vertex graph $G$ are arriving one-by-one in a stream and we are promised that $G$ is a disjoint union of either $k$-cycles or $2k$-cycles for some small $k$; the goal is to distinguish between these two cases. Verbin and Yu [SODA 2011] introduced this problem and showed that any single-pass streaming algorithm solving it requires $n^{1-\Omega(\frac{1}{k})}$ space. This result and the technique behind it -- the Boolean Hidden Hypermatching communication problem -- has since been used extensively for proving streaming lower bounds for various problems. Despite its significance and broad range of applications, the lower bound technique of Verbin and Yu comes with a key weakness that is inherited by all subsequent results: the Boolean Hidden Hypermatching problem is hard only if there is exactly one round of communication and can be solved with logarithmic communication in two rounds. Therefore, all streaming lower bounds derived from this problem only hold for single-pass algorithms. We prove the first multi-pass lower bound for the gap cycle counting problem: Any $p$-pass streaming algorithm that can distinguish between disjoint union of $k$-cycles vs $2k$-cycles -- or even $k$-cycles vs one Hamiltonian cycle -- requires $n^{1-\frac{1}{k^{\Omega(1/p)}}}$ space. As a corollary of this result, we can extend many of previous lower bounds to multi-pass algorithms. For instance, we can now prove that any streaming algorithm that $(1+\epsilon)$-approximates the value of MAX-CUT, maximum matching size, or rank of an $n$-by-$n$ matrix, requires either $n^{\Omega(1)}$ space or $\Omega(\log{(\frac{1}{\epsilon})})$ passes. For all these problems, prior work left open the possibility of even an $O(\log{n})$ space algorithm in only two passes., Comment: Fixed a mistake in one of the technical lemmas, see Section 1.4

We work with a metric space which we understand as a complex plane, unless otherwise specified. The letter \(\Omega \) will denote an open set in the metric space. We introduce the notion of connectedness in Sect. 1.2, and we show in Theorem 1.1 that an open set in a metric space is a disjoint union of open connected sets which we call regions. Further, we prove Theorem 1.3 in Sect. 1.3 that the extended complex plane is a metric space homeomorphic to the Riemann sphere. We introduce curves and paths, complex integral over paths, index of a point with respect to a closed path, homotopic paths, simply connected regions and give their basic properties in Sects. 1.4 and 1.5. Then we prove in Theorem 1.6 that the index of a point with respect to two \(\Omega \)-homotopic closed paths in \(\Omega \) are equal whenever the point lies outside \(\Omega \) and we apply it to prove in Theorem 1.9 that the complement of a simply connected region in the extended complex plane is connected. We shall prove in Sect. 3.5 the Riemann mapping theorem which implies the converse of the above statement, i.e a region is simply connected if its complement in the extended complex plane is connected. Hence a region is simply connected if and only if its complement in the extended complex plane is connected. Thus a region is simply connected if and only if it is without holes. This is a very transparent criterion to determine whether a region is simply connected or not. For example, it implies that the unbounded strip \(\{z\mid a

Given a graph class \(\mathcal {H}\), the task of the \(\mathcal {H}\)-Square Root problem is to decide, whether an input graph G has a square root H that belongs to \(\mathcal {H}\). We are interested in the parameterized complexity of the problem for classes \(\mathcal {H}\) that are composed by the graphs at vertex deletion distance at most k from graphs of maximum degree at most one, that is, we are looking for a square root H such that there is a modulator S of size k such that \(H-S\) is the disjoint union of isolated vertices and disjoint edges. We show that different variants of the problems with constraints on the number of isolated vertices and edges in \(H-S\) are FPT when parameterized by k by providing algorithms with running time \(2^{2^{\mathcal {O}(k)}}\cdot n^{\mathcal {O}(1)}\). We further show that the running time of our algorithms is asymptotically optimal and it is unlikely that the double-exponential dependence on k could be avoided. In particular, we prove that the VC- \(k\) Root problem, that asks whether an input graph has a square root with vertex cover of size at most k, cannot be solved in time \(2^{2^{o(k)}}\cdot n^{\mathcal {O}(1)}\) unless the Exponential Time Hypothesis fails. Moreover, we point out that VC- \(k\) Root parameterized by k does not admit a subexponential kernel unless P=N P.

This chapter deals with refinement of the theory to Borel measure spaces when the underlying space X is locally compact and Hausdorff. Prototypical examples are open sets in Euclidian space with Lebesgue measure and the natural numbers with the discrete topology and counting measure. The key observation now is that \(\mathfrak G\) is the disjoint union of compacts sets \(\mathfrak G\)(x), where x is an element of the one-point compactification of X and elements of \(\mathfrak G\)(x) are zero outside every open neighbourhood of x. This localisation result leads to a characterization of weakly convergent sequences in terms of the pointwise behaviour of related sequences of functions in neighbourhoods of x. Here, “pointwise” has its usual measure-theoretic meaning, whereas “localisation” refers to the behaviour of Borel measures and functions restricted to open neighbourhoods of points in a topological space. Similarly, the essential range is localised to reflect the fine structure of u at points x which is related to the isometric isomorphism in Chap. 7.

It is shown that if $\lambda$ is a multiple of a fundamental weight of $\mathfrak{sl}_k$, the lower global basis of the irreducible $U_q(\mathfrak{sl}_k)$-representation $V^{\lambda}$ with highest weight $\lambda$ comprises the disjoint union of the lower global bases of the irreducible $U_q(\mathfrak{sl}_{k-1})$-representations appearing in the decomposition of the restriction of $V^{\lambda}$ to $U_q(\mathfrak{sl}_{k-1})$. Rhoades's description of the action of the long cycle on the dual canonical basis of $V^{\lambda}$ is then deduced from Berenstein--Zelevinsky's description of the action of the long element. This yields a short proof of Rhoades's result on tableaux fixed under promotion which directly relates it to Stembridge's result on tableaux fixed under evacuation., Comment: 17 pages

We determine the maximum number of edges that a chordal graph G can have if its degree, \(\Delta (G)\), and its matching number, \(\nu (G)\), are bounded. To do so, we show that for every \(d,\nu \in \mathbb {N}\), there exists a chordal graph G with \(\Delta (G)

Let $F$ be a totally real field and $p$ be an odd prime which splits completely in $F$. We prove that the eigenvariety associated to a definite quaternion algebra over $F$ satisfies the following property: over a boundary annulus of the weight space, the eigenvariety is a disjoint union of countably infinitely many connected components which are finite over the weight space; on each fixed connected component, the ratios between the $U_\mathfrak{p}$-slopes of points and the $p$-adic valuations of the $\mathfrak{p}$-parameters are bounded by explicit numbers, for all primes $\mathfrak{p}$ of $F$ over $p$. Applying Hansen's $p$-adic interpolation theorem, we are able to transfer our results to Hilbert modular eigenvarieties. In particular, we prove that on every irreducible component of Hilbert modular eigenvarieties, as a point moves towards the boundary, its $U_p$ slope goes to zero. In the case of eigencurves, this completes the proof of Coleman-Mazur's `halo' conjecture., Comment: 65 pages, 1 figure

The Turan number of a graph H, denoted by e x ( n , H ) , is the maximum number of edges of an n-vertex simple graph having no H as a subgraph. Let S l denote the star on l + 1 vertices, and let k ⋅ S l denote the disjoint union of k copies of S l . Erdős and Gallai determined e x ( n , k ⋅ S 1 ) for all positive integers k and n. Yuan and Zhang determined e x ( n , k ⋅ S 2 ) and characterized all extremal graphs for all positive integers k and n. Lidický et al. determined e x ( n , k ⋅ S l ) for k , l ≥ 1 and n sufficiently large. Lan et al. determined e x ( n , k ⋅ S l ) for k ≥ 2 , l ≥ 3 and n ≥ k ( l 2 + l + 1 ) − l 2 ( l − 3 ) . In this paper, we completely determine e x ( n , k ⋅ S l ) for all positive integers k, l and n.

The Turan number of hypergraphs has been studied extensively. Here we deal with a recent direction, the linear Turan number, and restrict ourselves to linear triple systems, a collection of triples on a set of points in which any two triples intersect in at most one point. For a fixed linear triple system F , the linear Turan number ex L ( n , F ) is the maximum number of triples in a linear triple system with n points that does not contain F as a subsystem. We initiate the study of the linear Turan number for an acyclic F . In this case ex L ( n , F ) is linear in n and we aim for good bounds. Since the case of trees is already difficult for graphs (Erdős–Sos conjecture), we concentrate on matchings, paths and small trees. In case of matchings, where M k is the set of k pairwise disjoint triples, we prove that for fixed k and large enough n , ex L ( n , M k ) = f ( n , k ) where f ( n , k ) is the maximum number of triples that can meet k − 1 points in a linear triple system on n points. This is an analogue of an old result of Erdős on hypergraph matchings. For the k -edge linear path P k we show (extending some standard path increasing methods used for graphs) that ex L ( n , P k ) ≤ 1 . 5 k n which is probably far from best possible. Finding ex L ( n , F ) relates to difficult problems on Steiner triple systems and interesting even for small trees. For example, for P 4 , the path with four triples, ex L ( n , P 4 ) ≤ 4 n 3 with equality only for disjoint union of affine planes of order 3. On the other hand, for E 4 , the tree having three pairwise disjoint triples and a fourth one meeting all of them, we have bounds only: 6 ⌊ n − 3 4 ⌋ ≤ ex L ( n , E 4 ) ≤ 2 n .

In this paper, we continue our study of strongly paracompact mappings, see [2]. More properties of strongly paracompact mappings are obtained. It is proved that paracompact locally compact regular mappings are strongly paracompact. And every paracompact locally compact mapping is a disjoint union of closed and open Lindel?f mappings.

We describe new irreducible components of the Gieseker–Maruyama moduli scheme M ( 3 ) of semistable rank 2 coherent sheaves with Chern classes c 1 = 0 , c 2 = 3 , c 3 = 0 on P 3 , general points of which correspond to sheaves whose singular loci contain components of dimensions both 0 and 1. These sheaves are produced by elementary transformations of stable reflexive rank 2 sheaves with c 1 = 0 , c 2 = 2 along a disjoint union of a projective line and a collection of points in P 3 . The constructed families of sheaves provide first examples of irreducible components of the Gieseker–Maruyama moduli scheme such that their general sheaves have singularities of mixed dimension.

The generalization of the Oberwolfach Problem, proposed by J. Liu in 2000, asks for a uniform $2$-factorization of the complete multipartite graph $K_{m\times n}$. Here we focus our attention on $2$-factorizations regular under the cyclic group $Z_{mn}$, whose $2$-factors are disjoint union of cycles all of even length $\ell$. In particular, we present a complete solution for the extremal cases $\ell=4$ and $\ell=mn$., Comment: 24 pages

The paper is devoted to studying the orthogonality graph of the matrix ring over a skew field. It is shown that for n ≥ 3 and an arbitrary skew field 𝔻, the orthogonality graph of the ring Mn(𝔻) of n × n matrices over a skew field 𝔻 is connected and has diameter 4. If n = 2, then the graph of the ring Mn(𝔻) is a disjoint union of connected components of diameters 1 and 2. As a corollary, the corresponding results on the orthogonality graphs of simple Artinian rings are obtained.