122,751 research outputs found
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 be an abelian group, and a downward directed family of subsets of
. The finest topology on under which converges to
has been described by I.Protasov and E.Zelenyuk. In particular, their
description yields a criterion for to be Hausdorff. They then
show that if is the filter of cofinite subsets of a countable subset
, there is a simpler criterion: is Hausdorff if and
only if for every and positive integer , there is an
such that does not lie in the n-fold sum .
In this note, their proof is adapted to a larger class of families . In
particular, if is any infinite subset of , any regular infinite
cardinal , and the set of complements in of
subsets of cardinality , then the above criterion holds.
We then give some negative examples, including a countable downward directed
set of subsets of not of the above sort which satisfies the
"" condition, but does not induce a Hausdorff
topology.
We end with a version of our main result for noncommutative .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
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