Let k be a field of finite characteristic p, and G a finite group acting on the left on a finite dimensional k-vector space V . Then the dual vector space V ∗ is naturally a right kG-module, and the symmetric algebra of the dual, R := Sym(V ∗), is a polynomial ring over k on which G acts naturally by graded algebra automorphisms, and if k is algebraically closed can be regarded as the space k[V...

in this paper, we investigate duality of modular g-riesz bases and g-riesz basesin hilbert c*-modules. first we give some characterization of g-riesz bases in hilbert c*-modules, by using properties of operator theory. next, we characterize the duals of a giveng-riesz basis in hilbert c*-module. in addition, we obtain sucient and necessary conditionfor a dual of a g-riesz basis to be again a g...

Given a finite graph G, the maximum length of sequence $$(v_1,\ldots ,v_k)$$ vertices in G such that each $$v_i$$ dominates vertex is not dominated by any $$\{v_1,\ldots ,v_{i-1}\}$$ called Grundy domination number, $$\gamma _\mathrm{gr}(G)$$ , G. A small modification definition yields Z-Grundy which dual invariant well-known zero forcing number. In this paper, we prove _\mathrm{gr}(G) \ge \fra...

The moment map is a mathematical expression of the concept of the conservation associated with the symmetries of a Hamiltonian system. The abstract moment map is defined from G-manifold M to dual Lie algebra of G. We will interested study maps from G-manifold M to spaces that are more general than dual Lie algebra of G. These maps help us to reduce the dimension of a manifold much more.

In this article, we show that for any simple, bridgeless graph G on n vertices, there is a family C of at most n−1 cycles which cover the edges of G at least twice. A similar, dual result is also proven for cocycles namely: for any loopless graph G on n vertices and edges having cogirth g∗≥3 and k(G) components, there is a family of at most −n+k(G) cocycles which cover the edges of G at least t...

The r-Regular Induced Subgraph problem asks, given a graph G and a non-negative integer k, whether G contains an r-regular induced subgraph of size at least k, that is, an induced subgraph in which every vertex has degree exactly r. In this paper we examine its parameterization k-Size r-Regular Induced Subgraph with k as parameter and prove that it is W [1]-hard. We also examine the parameteriz...

Let G = (V,E) be a graph with n vertices and m edges, with a designated set of σ sources S ⊆ V . The fault tolerant subgraph for any graph problem maintains a sparse subgraph H = (V,E) of G with E ⊆ E, such that for any set F of k failures, the solution for the graph problem on G\F is maintained in its subgraph H \F . We address the problem of maintaining a fault tolerant subgraph for computing...

The complexification of the compact group G is the group G whose Lie algebra is the complexification of the Lie algebra g of G and which satisfies π1(G ) = π1(G). The complexification G/K of G/K can be then identified (G-equivariantly) with the tangent bundle of G/K. We also remark that the Kähler form obtained in the Theorem is exact. This result has been proved in [9] for symmetric spaces of ...