A note on Marino-Vafa formula

Hodge integrals over moduli spaces of curves appear naturally during the localization procedure in computation of Gromov-Witten invariants. A remarkable formula of Marino-Vafa expresses a generation function of Hodge integrals via some combinatorial and algebraic data seemingly unrelated to these apriori algebraic geometric objects. We prove in this paper by directly expanding the formula and estimating the involved terms carefully that except a specific type all the other Hodge integrals involving up to three Hodge classes can be calculated from this formula. This implies that amazingly rich information about moduli spaces and Gromov-Witten invariants is encoded in this complicated formula. We also give some low genus examples which agree with the previous results in literature. Proofs and calculations are elementary as long as one accepts Mumford relations on the reductions of products of Hodge classes.     

A brief note on the coarea formula

In this note we consider a special case of the famous Coarea Formula whose initial proof (for functions from any Riemannian manifold of dimension 2 into \({\mathbb {R}}\)) is due to Kronrod (Uspechi Matem Nauk 5(1):24–134, 1950) and whose general proof (for Lipschitz maps between two Riemannian manifolds of dimensions n and p) is due to Federer (Am Math Soc 93:418–491, 1959). See also Maly et al. (Trans Am Math Soc 355(2):477–492, 2002), Fleming and Rishel (Arch Math 11(1):218–222, 1960) and references therein for further generalizations to Sobolev mappings and BV functions respectively. We propose two counterexamples which prove that the coarea formula that we can find in many references (for example Bérard (Spectral geometry: direct and inverse problems, Springer, 1987), Berger et al. (Le Spectre d’une Variété Riemannienne, Springer, 1971) and Gallot (Astérisque 163(164):31–91, 1988), is not valid when applied to \(C^\infty \) functions. The gap appears only for the non generic set of non Morse functions.     

A note on Mertens' formula for arithmetic progressions

We study the Mertens product over primes in arithmetic progressions, and find a uniform version of previous results.     

A note on the Voronoï summation formula

In this paper, we derive the Voronoï summation formula from the standpoint of harmonic analysis.This work was done at Harvard University (in 1973).     

A note on the Dixon formula for a finite hypergeometric series

This is a note on the paper [4, B.M. Brown, M.S.P. Eastham, Extensions of the Kummer evaluation of a finite hypergeometric series, J. Comput. Appl. Math., to appear, doi:10.1016/j.cam.2005.06.018].     

A note on a generalized circular summation formula of theta functions

In this note, we make a correction of the imaginary transformation formula of Chan and Liu?s circular formula of theta functions. We also get the imaginary transformation formulaes for a type of generalized cubic theta functions.     

A note on the Cochrane sum and its hybrid mean value formula

In this paper, we use the properties of Gauss sums, primitive characters and the mean value theorems of Dirichlet L-functions to study the hybrid mean value of Cochrane sums and general Kloosterman sums, and give two sharp asymptotic formulae.     

A note on one-point quadrature formula for Daubechies scale function with partial support

This note is concerned with an efficient computation of integrals of products of a smooth function and Daubechies scale function with partial support by using a one-point quadrature rule. The error estimate is obtained. The rule is illustrated by considering an example from the literature.     

A note on merging

For a finite poset P and x, yP let pr(x>y) be the fraction of linear extensions which put x above y. N. Linial has shown that for posets of width 2 there is always a pair x, y with 1/3 pr(x>y)2/3. The disjoint union C ₁C ₂ of a 1-element chain with a 2-element chain shows that the bound 1/3 cannot be further increased. In this paper the extreme case is characterized: If P is a poset of width 2 then the bound 1/3 is exact iff P is an ordinal sum of C ₁C ₂'s and C ₁'s.     

A note on disconnectedness

The purpose of this note is to characterize various disconnectedness in terms of (star) refinement of covers.     

A note on p-curvatures

In this note, we give an arithmetic criterion for the Lie-irreducibility of linear differential equations based on p-curvatures.     

A note on list-colorings

In this article we discuss the current results on the list chromatic conjecture and prove that if G is a triangle free graph with maximum degree Δ then χapos;_|(G) ?; 9Δ/5.     

A note on percolation

Summary An improvement of Harris' theorem on percolation is obtained; it implies relations between critical points of matching graphs of the type of the one stated by Essam and Sykes. As another consequence, it is proved that the percolation probability, as a function of the probability of occupation of a given site, is infinitely differentiable, except at most in the critical point.     

A note on pseudocompactifications

There exists a countable spaceV _ω such that:

1. V _ω has a single non-isolated point,
2. V _ω has no pseudocompactificationX witht(X)=ω.

     

A note on rank-additivity

A new criterion is given for rank additivity of a collection of m × n complex matrices.