Let X be a finite subset of Euclidean space R. We define for each x ∈ X, B(x) := {(x, y) | y ∈ X, x 6= y, (x, x) ≥ (y, y)} where (, ) denotes the standard inner product. X is called an inside s-inner product set if |B(x)| ≤ s for all x ∈ X. In this paper, we prove that the cardinalities of inside s-inner product sets have the Fisher type upper bound. An inside s-inner product set is said to be ...

with a symmetric and positive definite operator A defined on a Hilbert space V with dimV = N and its preconditioned version PCG. We derive the CG from the projection in A-inner product point of view. We shall use (·, ·) for the standard inner product and (·, ·)A for the inner product introduced by the SPD operator A. When we say ‘orthogonal’ vectors, we refer to the default (·, ·) inner product...

We prove that the universal Teichmüller space T (1) carries a new structure of a complex Hilbert manifold. We show that the connected component of the identity of T (1), the Hilbert submanifold T0(1), is a topological group. We define a Weil-Petersson metric on T (1) by Hilbert space inner products on tangent spaces, compute its Riemann curvature tensor, and show that T (1) is a Kähler -Einstei...

In 1964, Gähler 1 introduced the concept of 2-norm and 2-inner product spaces as generalization of norm and inner product spaces. A systematic presentation of the results related to the theory of 2-inner product spaces can be found in the book in 2, 3 and in list of references in it. Generalization of 2-inner product space for n ≥ 2 was developed by Misiak 4 in 1989. Gunawan and Mashadi 5 in 20...

Let K be an imaginary quadratic field and f an integral ideal. Denote by Cl(f) the ray class group of f. For every non-trivial character χ of Cl(f), we show that L(1, χ)/π is transcendental. If f = f, then complex conjugation acts on the character group of Cl(f). Denoting by Ĉl(f) + the orbits of the group of characters, we show that the values L(1, χ) as χ ranges over elements of Ĉl(f) + are l...

The classical Weil-Petersson metric on the Teichmüller space of compact Riemann surfaces is a Kähler metric, which is complete only in the case of elliptic curves [Wo]. It has a natural generalization to the deformation spaces of higher dimansional polarized Kähler-Einstein manifolds. It is still Kähler, and in the case of abelian varieties and K3 surfaces, the Weil-Petersson metric turns out t...

In this paper, we investigate the geometric properties of random hyperbolic surfaces of large genus. We describe the relationship between the behavior of lengths of simple closed geodesics on a hyperbolic surface and properties of the moduli space of such surfaces. First, we study the asymptotic behavior of Weil-Petersson volume Vg,n of the moduli spaces of hyperbolic surfaces of genus g with n...

The aim of this note is to prove a rather explicit formula for the generating function of the Weil-Petersson volumes of the moduli spaces Mg,n of smooth npointed curves of genus g that was conjectured in [12]. E. Getzler noticed that the conecture in [12] is identical to the formulas in [5], Sect. 5, and [2], Sect. 3 and Appendix D, for the genus expansion of the free energy in Witten’s two dim...