Exceptional sets related to the largest digits in Lüroth expansions

Let [Formula: see text] denote the largest digit of the first [Formula: see text] terms in the Lüroth expansion of [Formula: see text]. Shen, Yu and Zhou, A note on the largest digits in Luroth expansion, Int. J. Number Theory 10 (2014) 1015–1023 considered the level sets [Formula: see text] and proved that each [Formula: see text] has full Hausdorff dimension. In this paper, we investigate the Hausdorff dimension of the following refined exceptional set: [Formula: see text] and show that [Formula: see text] has full Hausdorff dimension for each pair [Formula: see text] with [Formula: see text]. Combining the two results, [Formula: see text] can be decomposed into the disjoint union of uncountably many sets with full Hausdorff dimension.

Another Antimagic Conjecture

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.

Convex Obstacles from Travelling Times

We consider situations where rays are reflected according to geometrical optics by a set of unknown obstacles. The aim is to recover information about the obstacles from the travelling-time data of the reflected rays using geometrical methods and observations of singularities. Suppose that, for a disjoint union of finitely many strictly convex smooth obstacles in the Euclidean plane, no Euclidean line meets more than two of them. We then give a construction for complete recovery of the obstacles from the travelling times of reflected rays.

Enumeration of the Edge Weights of Symmetrically Designed Graphs

The idea of super a , 0 -edge-antimagic labeling of graphs had been introduced by Enomoto et al. in the late nineties. This article addresses super a , 0 -edge-antimagic labeling of a biparametric family of pancyclic graphs. We also present the aforesaid labeling on the disjoint union of graphs comprising upon copies of C 4 and different trees. Several problems shall also be addressed in this article.

Remarks on Stationary and Uniformly-rotating Vortex Sheets: Rigidity Results

In this paper, we show that the only solution of the vortex sheet equation, either stationary or uniformly rotating with negative angular velocity $$\Omega $$ Ω , 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.

Independent Sets in ($$P_4+P_4$$,Triangle)-Free Graphs

The Maximum Weight Independent Set Problem (WIS) is a well-known NP-hard problem. A popular way to study WIS is to detect graph classes for which WIS can be solved in polynomial time, with particular reference to hereditary graph classes, i.e., defined by a hereditary graph property or equivalently by forbidding one or more induced subgraphs. Given two graphs G and H, $$G+H$$ G + H denotes the disjoint union of G and H. This manuscript shows that (i) WIS can be solved for ($$P_4+P_4$$ P 4 + P 4 , Triangle)-free graphs in polynomial time, where a $$P_4$$ P 4 is an induced path of four vertices and a Triangle is a cycle of three vertices, and that in particular it turns out that (ii) for every ($$P_4+P_4$$ P 4 + P 4 , Triangle)-free graph G there is a family $${{\mathcal {S}}}$$ S of subsets of V(G) inducing (complete) bipartite subgraphs of G, which contains polynomially many members and can be computed in polynomial time, such that every maximal independent set of G is contained in some member of $${\mathcal {S}}$$ S . These results seem to be harmonic with respect to other polynomial results for WIS on [subclasses of] certain $$S_{i,j,k}$$ S i , j , k -free graphs and to other structure results on [subclasses of] Triangle-free graphs.

Certain partitions on a set and their applications to different classes of graded algebras

Let (𝔘, ɛu ) and (𝔅, ɛb ) be two pointed sets. Given a family of three maps ℱ = {f 1 : 𝔘 → 𝔘; f 2 : 𝔘 × 𝔘 → 𝔘; f 3 : 𝔘 × 𝔘 → 𝔅}, this family provides an adequate decomposition of 𝔘 \ {ɛu } as the orthogonal disjoint union of well-described ℱ-invariant subsets. This decomposition is applied to the structure theory of graded involutive algebras, graded quadratic algebras and graded weak H *-algebras.