A sum-product theorem in function fields

Let $A$ be a finite subset of \ffield, the field of Laurent series in $1/t$ over a finite field $\mathbb{F}_q$. We show that for any $\epsilon>0$ there exists a constant $C$ dependent only on $\epsilon$ and $q$ such that $\max\{|A+A|,|AA|\}\geq C |A|^{6/5-\epsilon}$. In particular such a result is obtained for the rational function field $\mathbb{F}_q(t)$. Identical results are also obtained for finite subsets of the $p$-adic field $\mathbb{Q}_p$ for any prime $p$.Comment: Simplification of argument and note that methods also work for the p-adic

Gauge Coupling Unification in MSSM + 5 Flavors

We investigate gauge coupling unification at 2-loops for theories with 5 extra vectorlike SU(5) fundamentals added to the MSSM. This is a borderline case where unification is only predicted in certain regions of parameter space. We establish a lower bound on the scale for the masses of the extra flavors, as a function of the sparticle masses. Models far outside of the bound do not predict unification at all (but may be compatible with unification), and models outside but near the boundary cannot reliably claim to predict it with an accuracy comparable to the MSSM prediction. Models inside the boundary can work just as well as the MSSM.Comment: 28 pages, 13 figures. Added references, fixed minor typos. No changes to content. Page count was incorrect in v1 Comment

A process model of children's early verb use

The verb-island hypothesis (Tomasello, 1992) states that children’s early grammars consist of sets of lexically-specific predicate structures (or verb-islands). However, Pine, Lieven and Rowland (1998) have found that children’s early language can also be built around lexical items other than verbs, such as pronouns (this contradicts a strict version of the verb-island hypothesis). This paper presents a computational model (called MOSAIC), which constructs a network of nodes and links based on a performance-limited distributional analysis of the input (mother’s speech). The results show that utterances generated from MOSAIC: (1) more closely resemble the child’s data than the child’s mother’s data on which MOSAIC is trained; and (2) can readily simulate both the verb-island and other-island phenomena which exist in the child’s data

Learning novel sound patterns

The acquisition of vocabulary represents a key phenomenon in language acquisition, yet it is still poorly understood. Gathercole and colleagues have recently provided a rigorous test of vocabulary knowledge (the nonword repetition test, Gathercole, Willis, Baddeley, & Emslie, 1994) and have adapted the phonological loop part of the working memory model (Baddeley & Hitch, 1974) to explain the nonword repetition findings (e.g. Gathercole & Baddeley, 1989). However, there are two major failings in their explanation: there is no description of how words are learned, and no definition of how the phonological loop interacts with long-term memory. We present an EPAM based computational model which overcomes these problems by combining the phonological loop approach with the EPAM/chunking approach (Feigenbaum & Simon, 1984). Trained on naturalistic phonemically coded speech (from mother’s utterances to 2-3 year old children), the model provides a good match to the nonword repetition data from 2-3 year old children. The model is also able to show the effect on nonword repetition when the model is trained using different sets of input. Implementing the phonological loop within EPAM represents a parsimonious approach to learning novel sound patterns and provides a more precise definition of how vocabulary acquisition may occur

Improved bounds on the set A(A+1)

For a subset A of a field F, write A(A + 1) for the set {a(b + 1):a,b\in A}. We establish new estimates on the size of A(A+1) in the case where F is either a finite field of prime order, or the real line. In the finite field case we show that A(A+1) is of cardinality at least C|A|^{57/56-o(1)} for some absolute constant C, so long as |A| < p^{1/2}. In the real case we show that the cardinality is at least C|A|^{24/19-o(1)}. These improve on the previously best-known exponents of 106/105-o(1) and 5/4 respectively

Fracture behavior of unidirectional boron/aluminum composite laminates

An experiment was conducted to verify the results of mathematical models which predict the stresses and displacements of fibers and the amount of damage growth in a center-notched lamina as a function of the applied remote stress and the matrix and fiber material properties. A brittle lacquer coating was used to detect the yielding in the matrix while X-ray techniques were used to determine the number of broken fibers in the laminate. The notched strengths and the amounts of damage found in the specimens agree well with those predicted by the mathematical model. It is shown that the amount of damage and the crack opening displacement does not depend strongly on the number of plies in the laminate for a given notch width. By heat-treating certain laminates to increase the yield stress of the alumina matrix, the effect of different matrix properties on the fracture behavior was investigated. The stronger matrix is shown to weaken the notched laminate by decreasing the amount of matrix damage, thereby making the laminate more notch sensitive

Acceleration Rates and Injection Efficiencies in Oblique Shocks

The rate at which particles are accelerated by the first-order Fermi mechanism in shocks depends on the angle, \teq{\Tbone}, that the upstream magnetic field makes with the shock normal. The greater the obliquity the greater the rate, and in quasi-perpendicular shocks rates can be hundreds of times higher than those seen in parallel shocks. In many circumstances pertaining to evolving shocks (\eg, supernova blast waves and interplanetary traveling shocks), high acceleration rates imply high maximum particle energies and obliquity effects may have important astrophysical consequences. However, as is demonstrated here, the efficiency for injecting thermal particles into the acceleration mechanism also depends strongly on obliquity and, in general, varies inversely with \teq{\Tbone}. The degree of turbulence and the resulting cross-field diffusion strongly influences both injection efficiency and acceleration rates. The test particle \mc simulation of shock acceleration used here assumes large-angle scattering, computes particle orbits exactly in shocked, laminar, non-relativistic flows, and calculates the injection efficiency as a function of obliquity, Mach number, and degree of turbulence. We find that turbulence must be quite strong for high Mach number, highly oblique shocks to inject significant numbers of thermal particles and that only modest gains in acceleration rates can be expected for strong oblique shocks over parallel ones if the only source of seed particles is the thermal background.Comment: 24 pages including 6 encapsulated figures, as a compressed, uuencoded, Postscript file. Accepted for publication in the Astrophysical Journa