Suppose X is a compact Kähler manifold of dimension n and E is a holomorphic vector bundle. For every p ≤ dim C X we have a sheaf Ω p (E) whose sections are holomorphic (p, 0)-forms with coefficients in E. We set and we define the holomorphic Euler characteristics χ p (X, E) := q≥0 (−1) q h p,q (X, E). It is convenient to introduce the generating function of these numbers χ y (X, E) := p≥0 y p ...

We show that the arguments proposed by Park and Yee against our recent derivation of the statistical entropy of 2D black holes do not apply to the case under consideration PACS: 04.70. Dy, 04.50. +h E-Mail: [email protected] E-Mail: [email protected] In a recent Comment [1] Park and Yee claimed that derivation of the statistical entropy of 2D (two-dimensional) black holes proposed by us in Ref...

The systems of bases are constructed for the spaces of cusp forms Sk(Γ0(3), χ) (k≥6), Sk(Γ0(7), χ) (k≥3) and Sk(Γ0(11), χ) (k ≥ 3). Formulas are obtained for the number of representation of a positive integer by the sum of k binary quadratic forms of the kind x1 +x1x2 +x 2 2 (6 ≤ k ≤ 17), of the kind x 2 1 +x1x2 +2x 2 2 (3 ≤ k ≤ 11) and of the kind x1 + x1x2 + 3x 2 1 (3 ≤ k ≤ 7). Let Fk denote ...

An equitable coloring of a graph is a proper vertex coloring such that the sizes of any two color classes differ by at most one. The least positive integer k for which there exists an equitable coloring of a graph G with k colors is said to be the equitable chromatic number of G and is denoted by χ=(G). The least positive integer k such that for any k′ ≥ k there exists an equitable coloring of ...

The Hadwiger number η(G) of a graph G is the largest integer n for which the complete graph Kn on n vertices is a minor of G. Hadwiger conjectured that for every graph G, η(G) ≥ χ(G), where χ(G) is the chromatic number of G. In this paper, we study the Hadwiger number of the Cartesian product G H of graphs. As the main result of this paper, we prove that η(G1 G2) ≥ h √ l (1− o(1)) for any two g...

We consider locally finite graphs with vertex set N. A graph G is computable if the edge set is computable and highly computable if the neighborhood function NG (which given v outputs all of its adjacent vertices) is computable. Let χ(G) be the chromatic number of G and χ(G) be the computable chromatic number of G. Bean showed there is a computable graph G with χ(G) = 3 and χ(G) =∞, but if G is...

We prove that any odd dimensional geometric graph G = (V,E) has zero curvature everywhere. To do so, we prove that for every injective function f on the vertex set V of a simple graph the index formula 1 2 [1 − χ(S(x))/2 − χ(Bf (x))] = (if (x) + i−f (x))/2 = jf (x) holds, where if (x) is a discrete analogue of the index of the gradient vector field ∇f and where Bf (x) is a graph defined by G an...

We consider constraints on the structure formation model based on non-gaussian fluctuations generated during inflation, which have χ 2 m distributions. Using three data sets, the abundance of the clusters at z = 0, at moderate z and the correlation length, we show constraints on the non-gaussianity and the amplitude of fluctuations and the density parameter can be obtained. We obtain upper boun...