An improved lower bound for (1,<=2)-identifying codes in the king grid

We call a subset $C$ of vertices of a graph $G$ a $(1,\leq \ell)$-identifying code if for all subsets $X$ of vertices with size at most $\ell$, the sets $\{c\in C |\exists u \in X, d(u,c)\leq 1\}$ are distinct. The concept of identifying codes was introduced in 1998 by Karpovsky, Chakrabarty and Levitin. Identifying codes have been studied in various grids. In particular, it has been shown that there exists a $(1,\leq 2)$-identifying code in the king grid with density 3/7 and that there are no such identifying codes with density smaller than 5/12. Using a suitable frame and a discharging procedure, we improve the lower bound by showing that any $(1,\leq 2)$-identifying code of the king grid has density at least 47/111

Improved Bounds for rr-Identifying Codes of the Hex Grid

For any positive integer $r$, an $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and pairwise distinct. For a finite graph, the density of a code is $|C|/|V(G)|$, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We find a code of density less than $5/(6r)$, which is sparser than the prior best construction which has density approximately $8/(9r)$.Comment: 12p

Vertex identifying codes for the n-dimensional lattice

An $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and different. Here, we provide an overview on codes for the $n$-dimensional lattice, discussing the case of 1-identifying codes, constructing a sparse code for the 4-dimensional lattice as well as showing that for fixed $n$, the minimum density of an $r$-identifying code is $\Theta(1/r^{n-1})$.Comment: 10p

Minimum density of identifying codes of king grids

International audienceA set C ⊆ V (G) is an identifying code in a graph G if for all v ∈ V (G), C[v] = ∅, and for all distinct u, v ∈ V (G), C[u] = C[v], where C[v] = N [v] ∩ C and N [v] denotes the closed neighbourhood of v in G. The minimum density of an identifying code in G is denoted by d * (G). In this paper, we study the density of king grids which are strong product of two paths. We show that for every king grid G, d * (G) ≥ 2/9. In addition, we show this bound is attained only for king grids which are strong products of two infinite paths. Given k ≥ 3, we denote by K k the (infinite) king strip with k rows. We prove that d * (K 3) = 1/3, d * (K 4) = 5/16, d * (K 5) = 4/15 and d * (K 6) = 5/18. We also prove that 2 9 + 8 81k ≤ d * (K k) ≤ 2 9 + 4 9k for every k ≥ 7

Using a multifrontal sparse solver in a high performance, finite element code

We consider the performance of the finite element method on a vector supercomputer. The computationally intensive parts of the finite element method are typically the individual element forms and the solution of the global stiffness matrix both of which are vectorized in high performance codes. To further increase throughput, new algorithms are needed. We compare a multifrontal sparse solver to a traditional skyline solver in a finite element code on a vector supercomputer. The multifrontal solver uses the Multiple-Minimum Degree reordering heuristic to reduce the number of operations required to factor a sparse matrix and full matrix computational kernels (e.g., BLAS3) to enhance vector performance. The net result in an order-of-magnitude reduction in run time for a finite element application on one processor of a Cray X-MP

Relative efficiency and accuracy of two Navier-Stokes codes for simulating attached transonic flow over wings

Two codes which solve the 3-D Thin Layer Navier-Stokes (TLNS) equations are used to compute the steady state flow for two test cases representing typical finite wings at transonic conditions. Several grids of C-O topology and varying point densities are used to determine the effects of grid refinement. After a description of each code and test case, standards for determining code efficiency and accuracy are defined and applied to determine the relative performance of the two codes in predicting turbulent transonic wing flows. Comparisons of computed surface pressure distributions with experimental data are made

Proceedings of the Twentieth Conference of the Association of Christians in the Mathematical Sciences

The proceedings of the twentieth conference of the Associate of Christians in the Mathematical Sciences held at Redeemer University College from May 27-30, 2015

Accretion disc outbursts: a new version of an old model

We have developed 1D time-dependent numerical models of accretion discs, using an adaptive grid technique and an implicit numerical scheme, in which the disc size is allowed to vary with time. The code fully resolves the cooling and heating fronts propagating in the disc. We show that models in which the radius of the outer edge of the disc is fixed produce incorrect results, from which probably incorrect conclusions about the viscosity law have been inferred. In particular we show that outside-in outbursts are possible when a standard bimodal behaviour of the Shakura-Sunyaev viscosity parameter alpha is used. We also discuss to what extent insufficient grid resolutions have limited the predictive power of previous models. We find that the global properties (magnitudes, etc. ...) of transient discs can be addressed by codes using a high, but reasonable, number of fixed grid points. However, the study of the detailed physical properties of the transition fronts generally requires resolutions which are out of reach of fixed grid codes. It appears that most time-dependent models of accretion discs published in the literature have been limited by resolution effects, improper outer boundary conditions, or both.Comment: 13 pages, 12 figures; accepted for publication in MNRA

Evaluation of three turbulence models for the prediction of steady and unsteady airloads

Two dimensional quasi-three dimensional Navier-Stokes solvers were used to predict the static and dynamic airload characteristics of airfoils. The following three turbulence models were used: the Baldwin-Lomax algebraic model, the Johnson-King ODE model for maximum turbulent shear stress, and a two equation k-e model with law-of-the-wall boundary conditions. It was found that in attached flow the three models have good agreement with experimental data. In unsteady separated flows, these models give only a fair correlation with experimental data