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...

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...

The paper is devoted to the solution of the inverse boundary problem for the heat equation. Let Ω be a connected bounded domain in R n (n ≥ 2) with C l (l ≥ 2) boundary Γ. Consider the mixed problem for the heat equation (1.1)        (ρ(x)∂ t − −)u f (t, x) = 0 in (0, +∞) × Ω, u f (t, x) = f (t, x) o n(0 , +∞) × Γ, u f (0, x) = 0 on Ω. The density ρ(x) is a C l+σ , 0 < σ < 1, function on...

The paper is devoted to the solution of the inverse boundary problem for the heat equation. Let Ω be a connected bounded domain in R n (n ≥ 2) with C l (l ≥ 2) boundary Γ. Consider the mixed problem for the heat equation (1.1)        (ρ(x)∂ t − −)u f (t, x) = 0 in (0, +∞) × Ω, u f (t, x) = f (t, x) o n(0 , +∞) × Γ, u f (0, x) = 0 on Ω. The density ρ(x) is a C l+σ , 0 < σ < 1, function on...

We study the distribution of complex zeros of Gaussian harmonic polynomials with independent complex coefficients. The expected number of zeros is evaluated by applying a formula of independent interest for the expected absolute value of quadratic forms of Gaussian random variables.

For real polynomials in two indeterminates a classical polynomial harmonic decomposition (cf. (1) below) is extended from square-norm divisors to conic ones. The main result is then applied to obtain a full polynomial harmonic decomposition, and to solve a Dirichlet problem with polynomial boundary data. Harmonic functions are of utmost importance in analysis, geometry, and mathematical physics...

In this paper, parametric polynomial minimal surfaces of degree six with isothermal parameter are discussed. We firstly propose the sufficient and necessary condition of a harmonic polynomial parametric surface of degree six being a minimal surface. Then we obtain two kinds of new minimal surfaces from the condition. The new minimal surfaces have similar properties as Enneper’s minimal surface,...