This paper shows how Hodge’s theory of harmonic p-sets (a discrete version of his theory of harmonic forms) allows a new approach to be taken to the problem of providing a combinatorial definition of the Pontrjagin classes of a compact manifold. This approach is then related to the author’s definition of flag vectors for hypergraphs, and other objects constructed out of vertices and cells. It i...

When 1 < p < ∞ and p 6= 2 the p-harmonic measure on the boundary of the half plane R+ is not subadditive. In fact, there are finitely many sets E1, E2,...,Eκ on R, of p-harmonic measure zero, such that E1 ∪ E2 ∪ ... ∪ Eκ = R.

This paper aims to introduce a measure of the non-univalency harmonic mapping. By using it, we find best approximation an analytic function by univalent

Let Ω be a bounded simply connected domain in the complex plane, C. Let N be a neighborhood of ∂Ω, let p be fixed, 1 < p < ∞, and let û be a positive weak solution to the p Laplace equation in Ω ∩N. Assume that û has zero boundary values on ∂Ω in the Sobolev sense and extend û to N \ Ω by putting û ≡ 0 on N \ Ω. Then there exists a positive finite Borel measure μ̂ on C with support contained in ...

The well-known Wolstenholme’s Theorem says that for every prime p > 3 the (p−1)-st partial sum of the harmonic series is congruent to 0 modulo p2. If one replaces the harmonic series by ∑ k≥1 1/n for k even, then the modulus has to be changed from p2 to just p. One may consider generalizations of this to multiple harmonic sums (MHS) and alternating multiple harmonic sums (AMHS) which are partia...

The surface-matching problem is investigated in this paper using a mathematical tool called harmonic maps. The theory of harmonic maps studies the mapping between different metric manifolds from the energyminimization point of view. With the application of harmonic maps, a surface representation called harmonic shape images is generated to represent and match 3D freeform surfaces. The basic ide...

the harmonic index h(g) , of a graph g is defined as the sum of weights 2/(deg(u)+deg(v)) of all edges in e(g), where deg (u) denotes the degree of a vertex u in v(g). in this paper we define the harmonic polynomial of g. we present explicit formula for the values of harmonic polynomial for several families of specific graphs and we find the lower and upper bound for harmonic index in caterpill...

This work presents a novel surface matching and registration method based on the landmark curve-driven canonical surface quasiconformal mapping, where an open genus zero surface decorated with landmark curves is mapped to a canonical domain with horizontal or vertical straight segments and the local shapes are preserved as much as possible. The key idea of the canonical mapping is to minimize t...

The ideals of the ring Zp are {0} and pZp, n ≥ 0. From this it follows that Zp is a discrete valuation ring, a principal ideal domain with exactly one maximal ideal, namely pZp; Zp is the valuation ring of Qp with the valuation vp. For n ≥ 1, Zp/pZp is isomorphic as a ring with Z/pZ. |x|p = p−vp(x), dp(x, y) = |x− y|p. With the topology induced by the metric dp, Qp is a locally compact abelian ...