Valid constraints for the Point Packing in a Square problem

We consider the problem of placing n$n$ points in the unit square in such a way as to maximize their minimum pairwise distance m$m$. Starting from two properties of the optimal solution presented by Locatelli and Raber in [Discrete Applied Mathematics 122 (1–3) (2002) 139–166], and using the known theoretical lower and upper bounds, we derive some constraints for tightening the original formulation of the problem.     

ECO:a methodology for the enumeration of combinatorial objects

In this Paper, we illustrate a method (called the ECO method) for enumerating some classes of combinatorial objects. The basic idea of this method is the following: by means of an operator that performs a "local expansion" on the objects, we give some recursive constructions of these classes. We use these constructions to deduce some new funtional equations verified by classes' generating functions. By solving the functional equations, we enumerate the combinatorial objects according to various parameters. We show some applications of the method referring to some classical combinatorial objects, such as: trees, paths, polyminoes and permutations     

Goldie dimension of a sum of modules

The formula dim(A+B)=dim(A)+dim(B)-dim(A∩B) works when ‘dim’ stands for the dimension of subspaces A,B of any vector space. In general, however, it does no longer hold if 'dim' means the uniform (or Goldie) dimension of submodules A,B of a module M over a ring R, and in fact the left hand side may be infinite while the right hand side is finite. In this paper we shall give a characterization of those modules M in which the formula holds for any two submodules A,B, as well as some conditions in the ring R which guarantee that dim(A+B) is finite whenever A and B are finite dimensional R-modules.     

Wave propagation in nonlocal elastic continua modelled by a fractional calculus approach

In the present paper, the wave propagation in one-dimensional elastic continua, characterized by nonlocal interactions modeled by fractional calculus, is investigated. Spatial derivatives of non-integer order 1 < α < 2 are involved in the governing equation, which is solved by fractional finite differences. The influence of long-range interactions is then analyzed as α varies: the resonant frequencies and the standing waves of a nonlocal bar are evaluated and the deviations from the classical (local) ones, recovered by imposing α = 2, are discussed.     

Planar Hamiltonian systems at resonance: the Ahmad–Lazer–Paul condition

We consider the planar Hamiltonian system $$Ju^{\prime} = \nabla F(u) + \nabla_u R(t,u), \quad t \in [0,T], \,u \in \mathbb{R}^2,$$ with F(u) positive and positively 2-homogeneous and ${\nabla_{u}R(t, u)}$ sublinear in u. By means of an Ahmad-Lazer-Paul type condition, we prove the existence of a T-periodic solution when the system is at resonance. The proof exploits a symplectic change of coordinates which transforms the problem into a perturbation of a linear one. The relationship with the Landesman–Lazer condition is analyzed, as well.     

Third-order right-focal multi-point problems on time scales

We are concerned with the existence and form of positive solutions to a third-order multi-point boundary-value problem on time scales with mixed derivatives. We find and utilize the Green function for the corresponding homogeneous right-focal problem as the kernel of an integral equation of Hammerstein-type. Two examples are included to illustrate the results.     

Uniqueness of vortexless Ginzburg-Landau type minimizers in two dimensions

In a simply connected two dimensional domain Ω, we consider Ginzburg-Landau minimizers u with zero degree Dirichlet boundary condition ${g \in H^{1/2}(\partial \Omega; \mathbb{S}^1)}$ . We prove uniqueness of u whenever either the energy or the Ginzburg-Landau parameter are small. This generalizes a result of Ye and Zhou requiring smoothness of g. We also obtain uniqueness when Ω is multiply connected and the degrees of the vortexless minimizer u are prescribed on the components of the boundary, generalizing a result of Golovaty and Berlyand for annular domains. The proofs rely on new global estimates connecting the variation of |u| to the Ginzburg-Landau energy of u. These estimates replace the usual global pointwise estimates satisfied by ${\nabla u}$ when g is smooth, and apply to fairly general potentials. In a related direction, we establish new uniqueness results for critical points of the Ginzburg-Landau energy.     

An MPI parallel level-set algorithm for propagating front curvature dependent detonation shock fronts in complex geometries

We present a parallel, two-dimensional, grid-based algorithm for solving a level-set function PDE that arises in Detonation Shock Dynamics (DSD). In the DSD limit, the detonation shock propagates at a speed that is a function of the curvature of the shock surface, subject to a set of boundary conditions applied along the boundaries of the detonating explosive. Our method solves for the full level-set function field, φ(x, y, t), that locates the detonation shock with a modified level-set function PDE that continuously renormalises the level-set function to a distance function based off of the locus of the shock surface, φ(x, y, t)=0. The boundary conditions are applied with ghost nodes that are sorted according to their connectivity to the interior explosive nodes. This allows the boundary conditions to be applied via a local, direct evaluation procedure. We give an extension of this boundary condition application method to three dimensions. Our parallel algorithm is based on a domain-decomposition model which uses the Message-Passing Interface (MPI) paradigm. The computational order of the full level-set algorithm, which is O(N ⁴), where N is the number of grid points along a coordinate line, makes an MPI-based algorithm an attractive alternative. This parallel model partitions the overall explosive domain into smaller sub-domains which in turn get mapped onto processors that are topologically arranged into a two-dimensional rectangular grid. A comparison of our numerical solution with an exact solution to the problem of a detonation rate stick shows that our numerical solution converges at better than first-order accuracy as measured by an L1-norm. This represents an improvement over the convergence properties of narrow-band level-set function solvers, whose convergence is limited to a floor set by the width of the narrow band. The efficiency of the narrow-band method is recovered by using our parallel model.     

Synthesis and characterization of graphene-based nanocomposites with potential use for biomedical applications

In this paper, GaN nanoparticles were synthesized from the complex Ga(H₂NCONH₂)₆Cl₃ in the flow of NH₃ at a mild temperature (350 °C). Further purification was performed by the ethanol-thermal method. The ethanol-thermal method also prompted the GaN nanoparticles to grow into an anisotropic morphology. XRD patterns reveal that GaN nanoparticles have crystallized in a hexagonal wurtzite structure. TEM observation shows that the average size of the as-prepared nanoparticles is about 5–10 nm. The photoluminescence spectrum exhibits a broad green emission band with a peak at 510 nm. It can be known from the first-principle theoretic simulation by the TDDFT method that this fluorescence emission band is attributed to the hydride defects of V _N-H on the surface of GaN nanoparticles.     

Emergence of acronyms in a community of language users

Language is a complex system that evolves over time, due to several phenomena. In recent years, new communication media are affecting interpersonal written communication. In particular, mobile phones and internet-based communication media are leading people to use a small number of characters when writing messages. Hence, acronyms or abbreviations are used in most cases. In particular, a mobile phone message is usually composed by short phrases, the social network Twitter only allows 140 characters per message and in many online forums users have limited space for questions and answers. Although the use of acronyms dates back to ancient times, nowadays this type of linguistic sign is gaining prestige. In this work, we study the introduction of acronyms in social systems. In particular, we define a simple game for the purpose of analyzing how the use of an acronym spreads in a population, considering its ability to create a shared meaning. We performed many numerical simulations according to the proposed model, showing the creation of acronyms to be the result of collective dynamics in a population.