Topological finiteness properties of monoids. Part 1: Foundations

We initiate the study of higher dimensional topological finiteness properties of monoids. This is done by developing the theory of monoids acting on CW complexes. For this we establish the foundations of $M$-equivariant homotopy theory where $M$ is a discrete monoid. For projective $M$-CW complexes we prove several fundamental results such as the homotopy extension and lifting property, which we use to prove the $M$-equivariant Whitehead theorems. We define a left equivariant classifying space as a contractible projective $M$-CW complex. We prove that such a space is unique up to $M$-homotopy equivalence and give a canonical model for such a space via the nerve of the right Cayley graph category of the monoid. The topological finiteness conditions left-$\mathrm{F}_n$ and left geometric dimension are then defined for monoids in terms of existence of a left equivariant classifying space satisfying appropriate finiteness properties. We also introduce the bilateral notion of $M$-equivariant classifying space, proving uniqueness and giving a canonical model via the nerve of the two-sided Cayley graph category, and we define the associated finiteness properties bi-$\mathrm{F}_n$ and geometric dimension. We explore the connections between all of the these topological finiteness properties and several well-studied homological finiteness properties of monoids which are important in the theory of string rewriting systems, including $\mathrm{FP}_n$, cohomological dimension, and Hochschild cohomological dimension. We also develop the corresponding theory of $M$-equivariant collapsing schemes (that is, $M$-equivariant discrete Morse theory), and among other things apply it to give topological proofs of results of Anick, Squier and Kobayashi that monoids which admit presentations by complete rewriting systems are left-, right- and bi-$\mathrm{FP}_\infty$.Comment: 59 pages, 1 figur

Method of measuring the thickness of radioactive thin films

Thickness monitor consists of proportional X-ray counter coupled to pulse counting system, copper filter over face of counter, rotatable collimator containing radioactive source, and rotatable shutter. Monitor can be used as integral part of neutron generator. It has been used to measure titanium tritide film thicknesses from 0.1 to 30 micrometers

A superior process for forming titanium hydrogen isotopic films

Process forms stoichiometric, continuous, strongly bonded titanium hydrogen isotopic films. Films have thermal and electrical conductivities approximately the same as bulk pure titanium, ten times greater than those of usual thin films

An X-ray monitor for measurement of a titanium tritide target thickness

An X-ray device capable of measuring titanium tritide film thickness from 0.1 to 30 micrometers has been built and tested. The monitor was designed for use in a rotating target system which used thick targets and incorporated a sputtering electrode to remove depleted layers from the target surface. The thickness measurement can be done in the presence of an intense background of bremsstrahlung and characteristic titanium X-radiation. A measurement can be accomplished in situ in two hours with reasonable accuracy

A proposed method for regeneration of neutron producing targets, within an accelerator, by ion sputtering techniques

Sputtering electrode system design for regeneration of targets within accelerato

Access to uncombined titanium through an inhibiting film in sublimation pumping of deuterium

It was demonstrated, through a series of experiments, that it is possible (by the addition of a thin layer of titanium to an apparently occluded surface) to gain access to previously deposited sublayers of uncombined titanium in spite of the presence of an inhibiting film (such as an oxide) on the surface

A high yield neutron target

Target, in cylinder form, rotates rapidly in front of beam. Titanium tritide film is much thicker than range of accelerated deutron. Sputtering electrode permits full use of thick film. Stream of high-velocity coolant provides efficient transfer of heat from target

Passage of a Bessel beam through a classically forbidden region

The motion of an electromagnetic wave, through a classically-forbidden region, has recently attracted renewed interest because of its implication with regard to the theoretical and experimental problems of superluminality. From an experimental point of view, many papers provide an evidence of superluminality in different physical systems. Theoretically, the problem of a passage through a forbidden gap has been treated by considering plane waves at oblique incidence into a plane parallel layer of a medium with a refractive index smaller than the index of the surrounding medium, and also confined (Gaussian) beams, still at oblique incidence. In the present paper the case of a Bessel beam is examined, at normal incidence into the layer (Secs. II and III), in the scalar approximation (Sec. IV) and by developing also a vectorial treatment (Sec. V). Conclusions are reported in Sic. VI