the series being uniformly and absolutely convergent in (x,y). A number of generalisations to Mercer’s theorem may be found in the literature, in particular dealing with kernels K : Y × Y → C for various choices of Y . However there would appear to have been (to the best of the author’s knowledge) no attempts made to extend Mercer’s theorem to cover non-complex valued kernels. In the present pa...

Recently, Huang showed that every ( 2 n − 1 + ) -vertex induced subgraph of the n-dimensional hypercube has maximum degree at least . In this paper, we discuss subgraphs Cartesian product graphs and semistrong to generalize Huang's result. Let Γ be a connected signed bipartite graph order m. By defining two kinds , denoted by □ ˜ ⋈ show if have exactly distinct adjacency eigenvalues ± θ respect...

The rays # and 4 are said to be orthogonal (denoted by (I,$ 4) = 0) if (I,$ 4) = 0 and non-orthogonal (denoted by (9 4) # 0) if (+, 4) # 0. This definition is independent of the choice of I/I E +Q and #J E 4. The rays Ilr,, . . . , a,%k are said to be independent if and only if I,!J~, . . . , +!I~ are linearly independent. This definition does not depend on the choice of I,!+ E +&, i = 1,. . . ...

Suppose Gi, G2, . . . , Gk are graphs of order at most n. We say that there is a packing of Gi, G2, . . . , G, into the complete graph K,, if there exist injections q : V(G,)+ V(K,), i = 1, 2, . . . , k such that aT(E(Gi)) fl (Y/*(E(G~)) = $ for i #j, where the map ar: : E(G,)-+ E(K,) . IS induced by a;. Similarly, suppose G is a graph of order m and H is a graph of order n 2 m and there exists...

A two player game (or more correctly, a two player normal-form game) is specified by two m × n payoff matrices R and C corresponding to the row and column player respectively. Each of these matrices has m rows corresponding to the m strategies of the row player and n columns corresponding to the n strategies of the column payer. The row player picks a row i ∈ [m], and the column player picks a ...

The aim of this paper is to promote web geometry and, especially, the Reidemeister closure condition as a powerful and intuitive tool characterizing associativity of the Archimedean triangular norms. In order to demonstrate its possible applications, we provide the full solution to the problem of convex combinations of nilpotent triangular norms. Keywords— Archimedean triangular norm, web geome...