Direct limits and fixed point sets

For which groups G is it true that whenever we form a direct limit of G-sets, dirlim_{i\in I} X_i, the set of its fixed points, (dirlim_I X_i)^G, can be obtained as the direct limit dirlim_I(X_i^G) of the fixed point sets of the given G-sets? An easy argument shows that this holds if and only if G is finitely generated. If we replace ``group G'' by ``monoid M'', the answer is the less familiar condition that the improper left congruence on M be finitely generated. Replacing our group or monoid with a small category E, the concept of set on which G or M acts with that of a functor E --> Set, and the concept of fixed point set with that of the limit of a functor, a criterion of a similar nature is obtained. The case where E is a partially ordered set leads to a condition on partially ordered sets which I have not seen before (pp.23-24, Def. 12 and Lemma 13). If one allows the {\em codomain} category Set to be replaced with other categories, and/or allows direct limits to be replaced with other kinds of colimits, one gets a vast area for further investigation.Comment: 28 pages. Notes on 1 Aug.'05 revision: Introduction added; Cor.s 9 and 10 strengthened and Cor.10 added; section 9 removed and section 8 rewritten; source file re-formatted for Elsevier macros. To appear, J.Al

Robotic tool change mechanism

An assembly of three major components is disclosed which included a wrist interface plate which is secured to the wrist joint of a robotic arm, a tool interface plate which is secured to each tool intended for use by the robotic arm, and a tool holster for each tool attached to the interface plate. The wrist interface plate and a selected tool interface plate are mutually connectable together through an opening or recess in the upper face of the interface plate by means of a notched tongue protruding from the front face of the wrist interface plate which engages a pair of spring-biased rotatable notched wheels located within the body of the tool interface plate. The tool holster captures and locks onto the tool interface plate by means of a pair of actuation claws including a locking tab and an unlocking wedge which operate respective actuation bosses on each of the notched wheels in response to a forward and backward motion of the tool interface plate as a result of motion of the robotic arm to either park the tool or use the tool

On group topologies determined by families of sets

Let $G$ be an abelian group, and $F$ a downward directed family of subsets of $G$. The finest topology $\mathcal{T}$ on $G$ under which $F$ converges to $0$ has been described by I.Protasov and E.Zelenyuk. In particular, their description yields a criterion for $\mathcal{T}$ to be Hausdorff. They then show that if $F$ is the filter of cofinite subsets of a countable subset $X\subseteq G$, there is a simpler criterion: $\mathcal{T}$ is Hausdorff if and only if for every $g\in G-\{0\}$ and positive integer $n$, there is an $S\in F$ such that $g$ does not lie in the n-fold sum $n(S\cup\{0\}\cup-S)$. In this note, their proof is adapted to a larger class of families $F$. In particular, if $X$ is any infinite subset of $G$, $\kappa$ any regular infinite cardinal $\leq\mathrm{card}(X)$, and $F$ the set of complements in $X$ of subsets of cardinality $<\kappa$, then the above criterion holds. We then give some negative examples, including a countable downward directed set $F$ of subsets of $\mathbb{Z}$ not of the above sort which satisfies the "$g\notin n(S\cup\{0\}\cup-S)$" condition, but does not induce a Hausdorff topology. We end with a version of our main result for noncommutative $G$.Comment: 10 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be updated more frequently than arXiv cop

More Abelian groups with free duals

In answer to a question of A. Blass, J. Irwin and G. Schlitt, a subgroup G of the additive group Z^{\omega} is constructed whose dual, Hom(G,Z), is free abelian of rank 2^{\aleph_0}. The question of whether Z^{\omega} has subgroups whose duals are free of still larger rank is discussed, and some further classes of subgroups of Z^{\omega} are noted.Comment: 9 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be updated more frequently than arXiv cop

Double-V block fingers with cruciform recess

In a robot having a gripper including a pair of fingers and a drive motor for driving the fingers toward and away from one another while the fingers remain parallel to each other, the fingers consist of finger pads, which interface with a handle on an object to be grasped, and a shank, which attaches the fingers to the robot gripper. The double-V finger has two orthogonal V-grooves forming in the center of the finger pads and recessed cruciform. The double-V finger is used with a handle on the object to be grasped which is the negative of the finger pads. The handle face consists of V-shaped pads capped with a rectangular cruciform. As the gripper is brought into place near the handle, the finger pads are lined up facing the handle pads. When the finger pad and the handle pad are in proper alignment, the rectangular ridges on the handle fall inside the rectangular grooves on the finger, and the grip is complete

A reservoir of test items for junior high school American history.

Thesis (Ed.M.)--Boston Universit