Silent emergency alarm system for schools and the like

An emergency alert system is described. In a school each classroom (or other area) is instrumented with a hidden microphone and receiver tuned to a non-audible frequency. The receivers' outputs are connected to a central display unit in the school's administrative office. Each instructor is provided with a small concealable transmitter which, when hand activated by the instructor upon the occurrance of any emergency, generates a non-audible signal at the receiver's tuned frequency

The in-vacuo torque performance of dry-lubricated ball bearings at cryogenic temperatures

The performance of dry-lubricated, angular contact ball bearings in vacuum at a temperature of 20 degrees K has been investigated, and is compared with the in-vacuo performance at room temperatures. Bearings were lubricated using dry-lubricant techniques which have been previously established for space applications involving operations at or near room temperature. Comparative tests were undertaken using three lubricants: molybdenum disulphide, lead, and PTFE. Results obtained using the three lubricants are presented

On the size of approximately convex sets in normed spaces

Let X be a normed space. A subset A of X is approximately convex if $d(ta+(1-t)b,A) \le 1$ for all $a,b \in A$ and $t \in [0,1]$ where $d(x,A)$ is the distance of $x$ to $A$. Let \Co(A) be the convex hull and \diam(A) the diameter of $A$. We prove that every $n$-dimensional normed space contains approximately convex sets $A$ with \mathcal{H}(A,\Co(A))\ge \log_2n-1 and \diam(A) \le C\sqrt n(\ln n)^2, where $\mathcal{H}$ denotes the Hausdorff distance. These estimates are reasonably sharp. For every $D>0$, we construct worst possible approximately convex sets in $C[0,1]$ such that \mathcal{H}(A,\Co(A))=\diam(A)=D. Several results pertaining to the Hyers-Ulam stability theorem are also proved.Comment: 32 pages. See also http://www.math.sc.edu/~howard

Extremal Approximately Convex Functions and Estimating the Size of Convex Hulls

A real valued function $f$ defined on a convex $K$ is anemconvex function iff it satisfies $$f((x+y)/2) \le (f(x)+f(y))/2 + 1.$$ A thorough study of approximately convex functions is made. The principal results are a sharp universal upper bound for lower semi-continuous approximately convex functions that vanish on the vertices of a simplex and an explicit description of the unique largest bounded approximately convex function~$E$ vanishing on the vertices of a simplex. A set $A$ in a normed space is an approximately convex set iff for all $a,b\in A$ the distance of the midpoint $(a+b)/2$ to $A$ is $\le 1$. The bounds on approximately convex functions are used to show that in $\R^n$ with the Euclidean norm, for any approximately convex set $A$, any point $z$ of the convex hull of $A$ is at a distance of at most $[\log_2(n-1)]+1+(n-1)/2^{[\log_2(n-1)]}$ from $A$. Examples are given to show this is the sharp bound. Bounds for general norms on $R^n$ are also given.Comment: 39 pages. See also http://www.math.sc.edu/~howard

MEXIT: Maximal un-coupling times for stochastic processes

Classical coupling constructions arrange for copies of the \emph{same} Markov process started at two \emph{different} initial states to become equal as soon as possible. In this paper, we consider an alternative coupling framework in which one seeks to arrange for two \emph{different} Markov (or other stochastic) processes to remain equal for as long as possible, when started in the \emph{same} state. We refer to this "un-coupling" or "maximal agreement" construction as \emph{MEXIT}, standing for "maximal exit". After highlighting the importance of un-coupling arguments in a few key statistical and probabilistic settings, we develop an explicit \MEXIT construction for stochastic processes in discrete time with countable state-space. This construction is generalized to random processes on general state-space running in continuous time, and then exemplified by discussion of \MEXIT for Brownian motions with two different constant drifts.Comment: 28 page

Update of MRST parton distributions.

We discuss the latest update of the MRST parton distributions in response to the most recent data. We discuss the areas where there are hints of difficulties in the global fit, and compare to some other updated sets of parton distributions, particularly CTEQ6. We briefly discuss the issue of uncertainties associated with partons