On tight 6-cycle decompositions of complete 3-uniform hypergraphs

The complete 3-uniform hypergraph of order v has a vertex set V of size v and the set of all 3-element subsets of V as its edge set. A tight 6-cycle is a hypergraph with vertex set $\{a,b,c,d,e,f\}$ and edge set $\{\{a,b,c\},\{b,c,d\},\{c,d,e\},\{d,e,f\},\{e,f,a\},\{f,a,b\}\}$. We show that there exists a decomposition of the complete 3-uniform hypergraph of order v into isomorphic copies of a tight 6-cycle if and only if $v\equiv 1$, 2, 10, 20, 28, or $29(\mathrm{mod}36)$.     

Two solutions for a singular elliptic equation with critical growth at infinity

We look for positive solutions for the singular equation $-\Delta u-\frac{1}{2}\left(x\cdot \nabla u\right)=\mu h\left(x\right)u^{q-1}+\lambda u+u^{(N+2)/(N-2)},\text{in }{\mathbb{R}}^N,$ where $N\ge 3$, $\lambda >0$, $\mu >0$ is a parameter, $0<q<1$ and $h$ has some summability properties. By using a perturbation method and critical point theory, we obtain two solutions when $max\{1,N/4\}<\lambda <N/2$ and the parameter $\mu >0$ is small.     

Hjelmslev quadrilateral central configurations

A Hjelmslev quadrilateral is a quadrilateral with two right angles at opposite vertices. Using mutual distances as coordinates, we show that any four-body central configuration forming a Hjelmslev quadrilateral must be a right kite configuration.     

Multifold factorizations of cyclic groups into subsets

Let $(G,+)$ be an abelian group. A finite multiset A over G is said to give a λ-fold factorization of G if there exists a multiset B over G such that each element of G occurs λ times in the multiset $A+B:=\{a+b:a\in A,b\in B\}$. In this article, restricting G to a cyclic group, we will provide sufficient conditions on a given multiset A under which the exact value or an upper bound of the minimum multiplicity λ of a factorization of G can be given by introducing a concept of ‘lcm-closure’. Furthermore, a couple of properties on a given factor A will be shown when A has a prime or prime power order (cardinality). A relation to multifold factorizations of the set of integers will be also glanced at a general perspective.     

On the rate of convergence of solutions in free boundary problems via penalization

The rate of convergence of approximate solutions via penalization for free boundary problems are concerned. A key observation is to obtain global bounds of penalized terms which give necessary estimates on integrations by the nonlinear adjoint method by L.C. Evans.     

Domain decomposition methods in a geometrical multiscale domain using finite volume schemes

The heat equation is solved by using a finite volume discretization in a domain that consists of a two-dimensional central node and several one-dimensional outgoing branches. Several interface connection options to match the submodels set on the node and on the branches, with or without continuity, are looked at. For each of them, a monolithic scheme is defined, and existence and uniqueness of the solution is proved. New schemes are deduced, which are obtained through domain decomposition methods in the form of interface systems, with one or two unknowns per interface. A comparative systematic study is carried out from an algebraic and numerical point of view according to the interface conditions: Dirichlet, Neumann, or Robin. An efficient diagonal preconditioning is proposed.