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

Structural Antennas for 3cm Radar Onboard Multi-Rotor UAV

A series of 3cm amateur band radar antennas suitable for installation on a cinematic grade multi-rotor UAV were considered. A wideband open waveguide mouth antenna was developed that can be made from the existing arms of a multi-rotor UAV without any increase in weight for side-looking wall detection ranging radar. For downward looking radio altimeter, cutting slots in the arms to form slotted waveguide antennas was shown in simulation to be possible both in terms of covering the entire 3cm band from 10 to 10.5GHz and without overly weakening the arms as structural members

Extreme paths in oriented 2D Percolation

A useful result about leftmost and rightmost paths in two dimensional bond percolation is proved. This result was introduced without proof in \cite{G} in the context of the contact process in continuous time. As discussed here, it also holds for several related models, including the discrete time contact process and two dimensional site percolation. Among the consequences are a natural monotonicity in the probability of percolation between different sites and a somewhat counter-intuitive correlation inequality

Solar cell angular position transducer

An angular position transducer utilizing photocells and a light source is disclosed. The device uses a fully rotatable baffle which is connected via an actuator shaft to the body whose rotational displacement is to be measured. The baffle blocks the light path between the light source and the photocells so that a constant semicircular beam of light reaches the photocells. The current produced by the photocells is fed through a resistor, a differential amplifier measures the voltage drop across the resistor which indicates the angular position of the actuator shaft and hence of the object

Hydrodynamic air lubricated compliant surface bearing for an automotive gas turbine engine. 2: Materials and coatings

Material coatings for an air-lubricated, compliant journal bearing for an automotive gas turbine engine were exposed to service test temperatures of 540 C or 650 C for 300 hours, and to 10 temperature cycles from room temperatures to the service test temperatures. Selected coatings were then put on journal and partial-arc foils and tested in start-stop cycle tests at 14 kPa (2 psi) loading for 2000 cycles. Half of the test cycles were performed at a test chamber service temperature of 540 C (1000 F) or 650 C (1200 F); the other half were performed at room temperature. Based on test results, the following combinations and their service temperature limitations are recommended: HL-800 TM (CdO and graphite) on foil versus chrome carbide on journal up to 370 C (700 F); NASA PS 120 (Tribaloy 400, silver and CaF2 on journal versus uncoated foil up to 540 C (1000 F); and Kaman DES on journal and foil up to 640 C (1200 F). Kaman DES coating system was further tested successfully at 35 kPa (5 psi) loading for 2000 start-stop cycles

Approximate infrared emissivity calculations for HCl at elevated temperatures

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FMCW rail-mounted SAR: Porting spotlight SAR imaging from MATLAB to FPGA

In this work, a low-cost laptop-based radar platform derived from the MIT open courseware has been implemented. It can perform ranging, Doppler measurement and SAR imaging using MATLAB as the processor. In this work, porting the signal processing algorithms onto a FPGA platform will be addressed as well as differences between results obtained using MATLAB and those obtained using the FPGA platform. The target FPGA platforms were a Virtex6 DSP kit and Spartan3A starter kit, the latter was also low-cost to further reduce the cost for students to access radar technology