A combinatorial approach to the power of 2 in the number of involutions

We provide a combinatorial approach to the largest power of $p$ in the number of permutations $\pi$ with $\pi^p=1$, for a fixed prime number $p$. With this approach, we find the largest power of $2$ in the number of involutions, in the signed sum of involutions and in the numbers of even or odd involutions.Comment: 13 page

Investigating Sustainable Fuel Effects on Mixing and Combustion through Design and Development of a Gasoline Direct Injection Optically Accessible Engine

Due to ever-growing sustainability issues, more demanding exhaust emission regulations are imposed on internal combustion engines. There is growing introduction of full electrification but, as there are practical issues regarding the full electrification, internal combustion engines are proven to be still useful and often coupled with electric motors. It is, therefore, vital to establish detailed understandings of in-cylinder combustion processes so that the release of greenhouse gas and production of pollutant emissions can be reduced and minimised. Therefore, novel fuels, such as second-generation biofuels, are thoroughly studied to explore possible use as future fuels for hybrid gasoline direct injection powertrains which are derived from sustainable feedstock and provide efficient energy release. For this project, a novel optical engine was designed that facilitate easy access to the piston and rapid cleaning of the piston crown window. A state-of-the-art gasoline direct injection engine was selected for hybrid applications. The initial design of the optical engine was modified to resolve the slackness in the extended timing chains. As the optical engine adopted the Bowditch system and only number 1 cylinder operated, various auxiliary components were also designed and developed to accommodate optical systems and oil circulation, and consider the change in the crankshaft balancing and volume of the air intake. Furthermore, an external fuel supply system was designed to enable a use of different fuels, while minimising a risk of damaging or contaminating the conventional fuel supply lines, by allowing easy cleaning processes. To quantitatively compare the difference in both spray and combustion images between various fuels, MATLAB codes were developed to process the captured images from a high-speed camera. Seven fuels were tested namely gasoline, ethanol, acetic acid, anisole, guaiacol, 2-MF and 2-MTHF; one a reference fossil fuel, one a first-generation biofuel and five second-generation biofuels, respectively. Three different injection timings were applied to simulate stratified, quasi-homogeneous and homogeneous states at low and high injection pressures for combustion studies, with only the injection pressure varied for the constant injection timing at the stratified spray studies. In general, injection pressure did not have a significant effect on soot formation, with exception for anisole, with injection timing found to be the dominant factor. Ethanol showed a similar spray development pattern to that of gasoline but displayed narrower sprays around the injector tip and became wider towards the spark plug. Acetic acid showed an indistinctive spray pattern and all six sprays merged together to form a cloud of fuel. Anisole showed wider sprays than gasoline and ethanol, but exhibited a similar penetration rate. Guaiacol exhibited similar spray characteristics to that of acetic acid, in that it formed a fuel cloud rather than maintaining distinct fuel sprays. Both 2-MF and 2-MTHF showed wide sprays

Experimental Investigations for Heat Transfer Characteristics of Under-expanded Impinging Jet

Department of Mechanical EngineeringIn this study, under-expanded impinging gas jets are investigated experimentally for understanding heat transfer characteristics of the jets. As working fluids, nitrogen (N2) and carbon dioxide (CO2) are selected in order to observe heat transfer effect changed by different working fluid. CO2 jet has a potential way to enhance the heat transfer effect which is sublimation. The novel concept of dry-ice assisted jet impingement cooling is proposed in this study. When carbon-dioxide (CO2) passes through a tiny orifice gap or jet nozzle, it experiences a rapid temperature drop as well as a pressure decrease via the Joule-Thomson effect. Joule-Thomson coefficient of CO2 is proven to be higher than the coefficient of other gases??? such as nitrogen, hydrogen, air. This temperature drop causes the formation of small CO2 dry-ice particles. In addition to the enhanced cooling performance caused by lowered bulk-jet temperature, heat transfer is improved by the additional sublimation effect between the dry-ice particles and the cooling target surface. A comparison of the cooling performance between the suggested CO2 solid-gas two-phase jet and a single-phase nitrogen (N2) jet was performed experimentally as well. In order to form dry-ice particles, high pressure and velocity of jet fluid are inevitably required, which is enough for compressible effect to appear. Both jets have differences not only in phase change but also in jet flow structures. As jet velocity increased, shock structures at jet downstream appeared and surface temperature is changed as well. The structures are detected more clearly in N2 jet than CO2 jet because of the difference in total pressure at jet boundary, therefore, relationship between shock structure of N2 jet flow and heat transfer is investigated in this study. In case of high Reynolds number impinging jet which is enough to reach supersonic flow regime, stagnation temperature of impinged surface is affected by jet structure as well as other factors such as nozzle-plate distance or radial distance does. When the jet flow velocity becomes supersonic, shock structures are constructed at downstream of the nozzle exit. Complicated shock structure such as Mach shock disk, plate shock is highly expected to affect to the heat transfer behavior of impingement surface. In this study, a cooling performance of supersonic N2 jet is investigated by measuring the impinged surface temperature and the flow of the jet is visualized by Schlieren image system. Visualized image and surface temperature are compared to clarify the flow structure-related heat transfer characteristics. In all the experiments of present study, jet fluids are expanded through a circular nozzle and impinged on an electrically heated flat heater surface, and their heat transfer coefficients are measured. The performances of the impinging jet for both fluids are also evaluated via the variance of flow parameters, for example, the Reynolds number, and the jet geometry configurations.ope

Transparentizing Black Holes to Eternal Traversable Wormholes

We present the gravity description of evaporating black holes that end up with eternal traversable wormholes where every would-be behind horizon degree is available in asymptotic regions. The transition is explicitly realized by a time-dependent bulk solution in the two-dimensional Einstein-dilaton gravity. In this solution, the initial AdS$_2$ black hole is evolved into an eternal traversable wormhole free of any singularity, which may be dubbed as transparentization of black holes to eternal traversable wormholes. The bulk construction completely matches with the boundary description governed by the Schwarzian boundary theory. We also obtain solutions describing eternal traversable wormholes as well as excitations by an additional matter and graviton oscillations on eternal traversable wormholes, which show that the eternal traversable wormhole states are gapped and non-chaotic. Embedding the 2d solution into a 4d traversable wormhole connecting two magnetically charged holes, we discuss 4d scattering of a wave incident upon one end of the traversable wormhole.Comment: 1+34 pages, 5 figures, 4 tables; v2: various typos fixed, Ref. added; v3: a few typos fixed, published versio

Three-Dimensional Simulations of the Parker Instability in a Uniformly-rotating Disk

We investigate the nonlinear effects of uniform rotation on the Parker instability in an exponentially-stratified disk through high-resolution simulations. During the linear stage, the speed of gas motion is subsonic and the evolution with the rotation is not much different from that without the rotation. This is because the Coriolis force is small. During the nonlinear stage, oppositely-directed supersonic flows near a magnetic valley are under the influence of the Coriolis force with different directions, resulting in twisted magnetic field lines near the valley. Sheet-like structures, which are tilted with respect to the initial field direction, are formed with an 1.5 enhancement of column density with respect to its initial value. Even though uniform rotation doesn't give much impact on density enhancement, it generates helically twisted field lines, which may become an additional support mechanism of clouds.Comment: 3 pages, uses rmaa.cls, to appear in Proc. of the Conference on "Astrophysical Plasmas: Codes, Models and Observations", Eds. J. Franco, J. Arthur, N. Brickhouse, Rev.Mex.AA Conf. Serie