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

Photonic measurements of the longitudinal expansion dynamics in Heavy-Ion collisions

Due to the smallness of the electromagnetic coupling, photons escape from the hot and dense matter created in an heavy-ion collision at all times, in contrast to hadrons which are predominantly emitted in the final freeze-out phase of the evolving system. Thus, the thermal photon yield carries an imprint from the early evolution. We suggest how this fact can be used to gain information about where between the two limiting cases of Bjorken (boost-invariant expansion) and Landau (complete initial stopping and re-expansion) hydrodynamics the actual evolution can be found. We argue that both the rapidity dependence of the photon yield and photonic HBT radii are capable of answering this question.Comment: 10 pages, 3 figure

Topological finiteness properties of monoids. Part 2: special monoids, one-relator monoids, amalgamated free products, and HNN extensions

We show how topological methods developed in a previous article can be applied to prove new results about topological and homological finiteness properties of monoids. A monoid presentation is called special if the right-hand side of each relation is equal to $1$. We prove results which relate the finiteness properties of a monoid defined by a special presentation with those of its group of units. Specifically we show that the monoid inherits the finiteness properties $F_n$ and $FP_n$ from its group of units. We also obtain results which relate the geometric and cohomological dimensions of such a monoid to those of its group of units. We apply these results to prove a Lyndon's Identity Theorem for one-relator monoids of the form $\langle A \mid r=1 \rangle$. In particular we show that all such monoids are of type $F_{\infty}$ (and $FP_{\infty}$), and that when $r$ is not a proper power, then the monoid has geometric and cohomological dimension at most $2$. The first of these results resolves an important case of a question of Kobayashi from 2000 on homological finiteness properties of one-relator monoids. We also show how our topological approach can be used to prove results about the closure properties of various homological and topological finiteness properties for amalgamated free products and HNN-extensions of monoids. To prove these results we introduce new methods for constructing equivariant classifying spaces for monoids, as well as developing a Bass-Serre theory for free constructions of monoids.Comment: 36 pages, Major revision: final section extracted as a separate short not

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

Extracting joint weak values with local, single-particle measurements

Weak measurement is a new technique which allows one to describe the evolution of postselected quantum systems. It appears to be useful for resolving a variety of thorny quantum paradoxes, particularly when used to study properties of pairs of particles. Unfortunately, such nonlocal or joint observables often prove difficult to measure weakly in practice (for instance, in optics -- a common testing ground for this technique -- strong photon-photon interactions would be needed). Here we derive a general, experimentally feasible, method for extracting these values from correlations between single-particle observables.Comment: 6 page

Characteristic spatial scale of vesicle pair interactions in a plane linear flow

We report the experimental studies on interaction of two vesicles trapped in a microfluidic analog of four-roll mill, where a plane linear flow is realized. We found that the dynamics of a single vesicle is significantly altered by the presence of another vesicle at separation distances up to about 3.2 \div 3.7 times of effective radius of the vesicles. This is supported by direct measurements of a single vesicle back-reaction on the velocity field. Thus, the experiment provides the lower bound for the interaction scale of vesicles and so the corresponding upper bound for the volume fraction \phi=0.08 \div 0.13 of non-interacting vesicle suspensions.Comment: 5 pages, 8 figures, PRE accepted for publicatio

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

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