A quantitative improvement for Roth's theorem on arithmetic progressions

We improve the quantitative estimate for Roth's theorem on three-term arithmetic progressions, showing that if $A\subset\{1,\ldots,N\}$ contains no non-trivial three-term arithmetic progressions then $\lvert A\rvert\ll N(\log\log N)^4/\log N$. By the same method we also improve the bounds in the analogous problem over $\mathbb{F}_q[t]$ and for the problem of finding long arithmetic progressions in a sumset

IMPROVING THE PERFORMANCE OF THE FOOD DISTRIBUTION INDUSTRY

Discusses the "productivity crisis" in the food industry and suggests positive and negative fluences on the situation during the 1970's.Productivity Analysis,

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

The V5A13.1 envelope glycoprotein deletion mutant of mouse hepatitis virus type-4 is neuroattenuated by its reduced rate of spread in the central nervous system.

Following intracerebral inoculation of adult Balb/c Byj mice, the MHV-4 strain of mouse hepatitis virus (MHV) had an LD50 of less than 0.1 PFU, whereas its monoclonal antibody resistant variant V5A13.1 had an LD50 of 10(4.2) PFU. To determine the basis for this difference in neurovirulence we have studied the acute central nervous system (CNS) infection of these two viruses by in situ hybridization. Both viruses infected the same, specific neuroanatomical areas, predominantly neurons, and spread via the cerebrospinal fluid, along neuronal pathways and between adjacent cells. The neuronal nuclei infected and the spread of virus within the brain are described. The main difference between the parental and variant viruses was the rate at which the infection spread. MHV-4 spread rapidly, destroying large numbers of neurons and the animals died within 4 days of infection. The variant virus spread to the same areas of the brain but at a slower rate. This difference in the rate of virus spread was also apparent from the brain virus titers. The slower rate of spread of the variant virus appears to allow intervention by the immune response. Consistent with this, the variant virus spread slowly in athymic nu/nu mice, but in the absence of an intact immune response, infection and destruction of neurons eventually reached the same extent as that of the parental virus and the mice died within 6 days of infection. We conclude that the V5A13.1 variant of MHV-4 is neuroattenuated by its slower rate of spread in the CNS