In this paper, we use the Minkowski functional to introduce an SL-type property or condition and then define a SL-type integral for a function taking values in a locally convex topological vector space (LCTVS). We show that this integral is equivalent to the SH1 integral, a version of the Henstock-Kurzweil integral in a LCTVS.
The main objective of this article is to find some vortex solutions of finite core size for plane Boussinesq equations under the radial gravity, coupled with a diffusive equation of temperature in a weighted subspace of L-2(R-2). Solutions are expanded into series of Hermite eigenfunctions. We find the coefficients of the series and show the convergence of them.
For a finite group G, the Hurwitz space H-(in)(r,g)(G) is the space of genus g covers of the Riemann sphere P-1 with r branch points and the monodromy group G. In this paper, we study the connectedness of the Hurwitz spaceH-(in)(r,g)(G) where G is almost simple groups of Lie rank two, with at least four branch points and genus two. Our approach uses computational tools, relying on the computer algebra system GAP and the MAPCLASS package, to find the connected components of H-(in)(r,g)(G). This work gives us the complete classification of G.
Let sigma = {sigma(i)|(i) is an element of I} be a partition of the set of all primes P and G a finite group. A set H of subgroups of G is said to be a complete Hall a-set of G if every member =/ 1 of H is a Hall sigma(i)-subgroup of G for some i is an element of I and H contains exactly one Hall sigma(i)-subgroup of G for every i such that sigma(i)boolean AND pi (G) =/ & empty;. In this paper, we study the structure of G based on the notion of sigma-conditionally permutable subgroups.
A Roman dominating function (RDF) of a graph G = (V, E) is a function f : V (G) -> {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f(v) = 2. A vertex u with f (u) = 0 is said to be undefended if it is not adjacent to a vertex with f(v) > 0. For a graph G, a function f : V(G) -> {0, 1, 2} is said to be a weak Roman dominating function (WRDF) if each vertex u with f (u) = 0 is adjacent to a vertex v with f(v) > 0 such that the function f ': V(G) -> {0, 1, 2} defined by f '(u) = 1, f '(v) = f(v)-1 and f '(w) = f(w) if w is an element of V-{u,v}, has no undefended vertex. The weight of f is defined to be the value f(V) = Sigma(u is an element of V) f(u). The minimum weight of a weak Roman dominating function of a graph G is called the weak Roman domination number of G and is denoted by gamma(r)(G). A WRDF with weight gamma(r)(G) is called a gamma(r)(G )-function. A set S subset of V is a global dominating set if S dominates both G and its complement G(-). The global domination number gamma(g)(G) of a graph G is the minimum cardinality of a global dominating set S. We extend the idea of global domination to weak Roman domination as follows: For a graph G, the function f : V(G) -> {0, 1, 2} is a global weak Roman dominating function (GWRDF) if f is a WRDF for both G and its complement G. The weight of a global weak Roman dominating function is the value f(V) = Sigma(u is an element of V)f(u). The minimum weight of a global weak Roman dominating function of a graph G is called the global weak Roman domination number of G and is denoted by gamma(gr)(G). In this paper, we initiate a study of this parameter.
