New results of intersection numbers on moduli spaces of curves

We present a series of new results we obtained recently about the intersection numbers of tautological classes on moduli spaces of curves, including a simple formula of the n-point functions for Witten's $\tau$ classes, an effective recursion formula to compute higher Weil-Petersson volumes, several new recursion formulae of intersection numbers and our proof of a conjecture of Itzykson and Zuber concerning denominators of intersection numbers. We also present Virasoro and KdV properties of generating functions of general mixed $\kappa$ and $\psi$ intersections.Comment: 9 pages, a brief surve

Quarter-fraction factorial designs constructed via quaternary codes

The research of developing a general methodology for the construction of good nonregular designs has been very active in the last decade. Recent research by Xu and Wong [Statist. Sinica 17 (2007) 1191--1213] suggested a new class of nonregular designs constructed from quaternary codes. This paper explores the properties and uses of quaternary codes toward the construction of quarter-fraction nonregular designs. Some theoretical results are obtained regarding the aliasing structure of such designs. Optimal designs are constructed under the maximum resolution, minimum aberration and maximum projectivity criteria. These designs often have larger generalized resolution and larger projectivity than regular designs of the same size. It is further shown that some of these designs have generalized minimum aberration and maximum projectivity among all possible designs.Comment: Published in at http://dx.doi.org/10.1214/08-AOS656 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Finding The Sign Of A Function Value By Binary Cellular Automaton

Given a continuous function $f(x)$, suppose that the sign of $f$ only has finitely many discontinuous points in the interval $[0,1]$. We show how to use a sequence of one dimensional deterministic binary cellular automata to determine the sign of $f(\rho)$ where $\rho$ is the (number) density of 1s in an arbitrarily given bit string of finite length provided that $f$ satisfies certain technical conditions.Comment: Revtex, uses amsfonts, 10 page

Modulo Three Problem With A Cellular Automaton Solution

An important global property of a bit string is the number of ones in it. It has been found that the parity (odd or even) of this number can be found by a sequence of deterministic, translational invariant cellular automata with parallel update in succession for a total of O(N^2) time. In this paper, we discover a way to check if this number is divisible by three using the same kind of cellular automata in O(N^3) time. We also speculate that the method described here could be generalized to check if it is divisible by four and other positive integers.Comment: 10 pages in revtex 4.0, using amsfont

Higher-spin Realisations of the Bosonic String

It has been shown that certain $W$ algebras can be linearised by the inclusion of a spin--1 current. This provides a way of obtaining new realisations of the $W$ algebras. Recently such new realisations of $W_3$ were used in order to embed the bosonic string in the critical and non-critical $W_3$ strings. In this paper, we consider similar embeddings in $W_{2,4}$ and $W_{2,6}$ strings. The linearisation of $W_{2,4}$ is already known, and can be achieved for all values of central charge. We use this to embed the bosonic string in critical and non-critical $W_{2,4}$ strings. We then derive the linearisation of $W_{2,6}$ using a spin--1 current, which turns out to be possible only at central charge $c=390$. We use this to embed the bosonic string in a non-critical $W_{2,6}$ string.Comment: 8 pages. CTP TAMU-10/95

Liouville and Toda Solitons in M-theory

We study the general form of the equations for isotropic single-scalar, multi-scalar and dyonic $p$-branes in superstring theory and M-theory, and show that they can be cast into the form of Liouville, Toda (or Toda-like) equations. The general solutions describe non-extremal isotropic $p$-branes, reducing to the previously-known extremal solutions in limiting cases. In the non-extremal case, the dilatonic scalar fields are finite at the outer event horizon.Comment: Latex, 10 pages. Minor corrections to text and titl

A trigonometric approach to quaternary code designs with application to one-eighth and one-sixteenth fractions

The study of good nonregular fractional factorial designs has received significant attention over the last two decades. Recent research indicates that designs constructed from quaternary codes (QC) are very promising in this regard. The present paper shows how a trigonometric approach can facilitate a systematic understanding of such QC designs and lead to new theoretical results covering hitherto unexplored situations. We focus attention on one-eighth and one-sixteenth fractions of two-level factorials and show that optimal QC designs often have larger generalized resolution and projectivity than comparable regular designs. Moreover, some of these designs are found to have maximum projectivity among all designs.Comment: Published in at http://dx.doi.org/10.1214/10-AOS815 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Quantising Higher-spin String Theories

In this paper, we examine the conditions under which a higher-spin string theory can be quantised. The quantisability is crucially dependent on the way in which the matter currents are realised at the classical level. In particular, we construct classical realisations for the $W_{2,s}$ algebra, which is generated by a primary spin-$s$ current in addition to the energy-momentum tensor, and discuss the quantisation for $s\le8$. From these examples we see that quantum BRST operators can exist even when there is no quantum generalisation of the classical $W_{2,s}$ algebra. Moreover, we find that there can be several inequivalent ways of quantising a given classical theory, leading to different BRST operators with inequivalent cohomologies. We discuss their relation to certain minimal models. We also consider the hierarchical embeddings of string theories proposed recently by Berkovits and Vafa, and show how the already-known $W$ strings provide examples of this phenomenon. Attempts to find higher-spin fermionic generalisations lead us to examine the whether classical BRST operators for $W_{2,{n\over 2}}$ ($n$ odd) algebras can exist. We find that even though such fermionic algebras close up to null fields, one cannot build nilpotent BRST operators, at least of the standard form.Comment: CTP TAMU-24/94, KUL-TF-94/11, SISSA-135/94/E