The symmetry of intersection numbers in group theory

For suitable subgroups of a finitely generated group, we define the intersection number of one subgroup with another subgroup and show that this number is symmetric. We also give an interpretation of this number.Comment: 19 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol2/paper2.abs.html . Includes erratum added to the original, published 19 Mar 199

Breaking the Cycle: How Nevada Can Effectuate Meaningful Criminal Justice Reform

Why does society punish criminals? This paper examines what Nevada is attempting to accomplish through enacting and enforcing its criminal laws. We examine the current state of, as well as the challenges facing, Nevada’s criminal justice system. Additionally, we identify and propose certain solutions to reduce both recidivism and the financial burden that incarceration imposes on the state by looking to best practices in other states, as well as certain mechanisms and provisions that were, for one reason or another, removed from Nevada Assembly Bill 236

Splittings of groups and intersection numbers

We prove algebraic analogues of the facts that a curve on a surface with self-intersection number zero is homotopic to a cover of a simple curve, and that two simple curves on a surface with intersection number zero can be isotoped to be disjoint.Comment: 40 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol4/paper6.abs.htm

Engaging the Digitally Engaged Student: Comparing Technology-Mediated Communication Use and Effects on Student Learning

The role of communication technologies in the learning process is both a dynamic and complex issue. Yet, we know surprisingly little about how the use of specific communication technologies may influence classroom performance, key learning outcomes, and other measures of course satisfaction. The research reported here attempts to add to our knowledge about the role of communication in the technology enhanced classroom (TEC) education and in technology-enhanced online (TEO) education through a direct comparison of two courses. Our findings indicate additional support for “The No Significant Difference Phenomenon.” Furthermore, we found that prior experiences lead students to gravitate towards their preferred learning environments, and that basic website elements are required in any learning environment to enhance student outcomes. Finally, we found that when used appropriately, the benefits of communication technology use in education outweigh many of the drawbacks

Using SCUBA to place upper limits on arcsecond scale CMB anisotropies at 850 microns

The SCUBA instrument on the James Clerk Maxwell Telescope has already had an impact on cosmology by detecting relatively large numbers of dusty galaxies at high redshift. Apart from identifying well-detected sources, such data can also be mined for information about fainter sources and their correlations, as revealed through low level fluctuations in SCUBA maps. As a first step in this direction we analyse a small SCUBA data-set as if it were obtained from a Cosmic Microwave Background (CMB) differencing experiment. This enables us to place limits on CMB anisotropy at 850 microns. Expressed as Q_{flat}, the quadrupole expectation value for a flat power spectrum, the limit is 152 microKelvin at 95 per cent confidence, corresponding to C_0^{1/2} < 355 microKelvin for a Gaussian autocorrelation function, with a coherence angle of about 20--25 arcsec; These results could easily be reinterpretted in terms of any other fluctuating sky signal. This is currently the best limit for these scales at high frequency, and comparable to limits at similar angular scales in the radio. Even with such a modest data-set, it is possible to put a constraint on the slope of the SCUBA counts at the faint end, since even randomly distributed sources would lead to fluctuations. Future analysis of sky correlations in more extensive data-sets ought to yield detections, and hence additional information on source counts and clustering.Comment: 12 pages, 9 postscript figures, uses mn.st

3D-printed Acoustic Directional Couplers

Acoustic Directional Couplers permit separation of forward and reverse sound pressure waves. This separation opens the way to traceable precision acoustic reflection measurements. In order to span the audio frequency range, multiple couplers will be required, as each operates over a frequency range of slightly more than one octave. To reach 20kHz or above requires vary small, mechanically precise construction. We achieve this by 3D printing techniques. We manufactured two otherwise-identical couplers, one made with a powder-type 3D printer with photopolymer support structure, the other made with an ABS-filament thermoplastic-type 3D printer. We compare the measured acoustic performance of these two couplers. The wavelength of sound at 20 kHz is comparable to that encountered at a microwave frequency of 18 GHz. We expect to be able to fabricate couplers that reach 55 kHz where the wavelength is 6 mm, corresponding to a frequency of 50 GHz in the electromagnetic spectrum

Hybrid Electro-Optically Modulated Microcombs

Optical frequency combs based on mode-locked lasers have proven to be invaluable tools for a wide range of applications in precision spectroscopy and metrology. A novel principle of optical frequency comb generation in whispering-gallery mode microresonators ("microcombs") has been developed recently, which represents a promising route towards chip-level integration and out-of-the-lab use of these devices. Presently, two families of microcombs have been demonstrated: combs with electronically detectable mode spacing that can be directly stabilized, and broadband combs with up to octave-spanning spectra but mode spacings beyond electronic detection limits. However, it has not yet been possible to achieve these two key requirements simultaneously, as will be critical for most microcomb applications. Here we present a key step to overcome this problem by interleaving an electro-optic comb with the spectrum from a parametric microcomb. This allows, for the first time, direct control and stabilization of a microcomb spectrum with large mode spacing (>140 GHz) with no need for an additional mode-locked laser frequency comb. The attained residual 1-second-instability of the microcomb comb spacing is 10^-15, with a microwave reference limited absolute instability of 10^-12 at a 140 GHz mode spacing.Comment: 8 pages, 4 figures; accepted for publication in Physical Review Letter