An eternal $m$-secure set of a graph $G = (V,E)$ is aset $S_0subseteq V$ that can defend against any sequence ofsingle-vertex attacks by means of multiple-guard shifts along theedges of $G$. A suitable placement of the guards is called aneternal $m$-secure set. The eternal $m$-security number$sigma_m(G)$ is the minimum cardinality among all eternal$m$-secure sets in $G$. An edge $uvin E(G)$ is ...

Let S be a Heegaard splitting surface of a compact orientable 3-manifold M. If S is strongly irreducible, the manner in which it can intersect a ball or a solid torus in M is very constrained and the allowable configurations are simple and useful. Splitting surfaces not conforming to these simple local pictures must be weakly reducible.

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. In this paper we introduce a new graph associated to modules over commutative rings. We study the relationship between the algebraic properties of modules and their associated graphs. A topological characterization for the completeness of the special subgraphs is presented. Also modules whose associated graph is complete...

The recent measurement by CDF M(Σb)−M(Λb) = 192MeV is in striking agreement with our theoretical prediction M(Σb)−M(Λb) = 194MeV. In addition, the measured splitting M(Σb)−M(Σb) = 21 MeV agrees well with the predicted splitting of 22 MeV. We point out the connection between these predictions and an effective supersymmetry between mesons and baryons related by replacing a light antiquark by a li...

The aim of this study is investigation on effect of stem end splitting and Geranium essential oil on vase life on quality of cut chrysanthemum (Dendranthema grandiflorum L.). This experiment arranged as factorial based on RCD with 2 factors of stem end splitting at 2 levels (with splitting and without splitting) and Geranium essential oil at 6 levels (0, 1, 2, 4, 8 and 10 %), with 12 treatments...

The incremental planarity testing problem is to perform the following operations on a biconnected planar graph G of at most n vertices: test if an edge can be added between two vertices while preserving planarity; add edges and vertices that preserve planarity. Let m be the total number of operations. We present fast data structures for this problem that can be used in conjunction with the prev...

The paper describes a natural splitting in the rational homology and homotopy of the spaces of long knots. This decomposition presumably arises from the cabling maps in the same way as a natural decomposition in the homology of loop spaces arises from power maps. The generating function for the Euler characteristics of the terms of this splitting is presented. Based on this generating function ...

M. Mahowald, in his work on bo-resolutions, constructed a bomodule splitting of the spectrum bo ∧ bo into a wedge of summands related to integral Brown-Gitler spectra. In this paper, a similar splitting of bo ∧ tmf is constructed. This splitting is then used to understand the bo∗-algebra structure of bo∗tmf and allows for a description of bo∗tmf.

we consider a dynamic domination problem for graphs in which an infinitesequence of attacks occur at vertices with guards and the guard at theattacked vertex is required to vacate the vertex by moving to a neighboringvertex with no guard. other guards are allowed to move at the same time, andbefore and after each attack and the resulting guard movements, the verticescontaining guards form a dom...