Painlev\'{e}-Gullstrand-type coordinates for the five-dimensional Myers-Perry black hole

The Painleve-Gullstrand coordinates provide a convenient framework for presenting the Schwarzschild geometry because of their flat constant-time hypersurfaces, and the fact that they are free of coordinate singularities outside r=0. Generalizations of Painleve-Gullstrand coordinates suitable for the Kerr geometry have been presented by Doran and Natario. These coordinate systems feature a time coordinate identical to the proper time of zero-angular-momentum observers that are dropped from infinity. Here, the methods of Doran and Natario are extended to the five-dimensional rotating black hole found by Myers and Perry. The result is a new formulation of the Myers-Perry metric. The properties and physical significance of these new coordinates are discussed.Comment: 15 page

Coordinate families for the Schwarzschild geometry based on radial timelike geodesics

We explore the connections between various coordinate systems associated with observers moving inwardly along radial geodesics in the Schwarzschild geometry. Painlev\'e-Gullstrand (PG) time is adapted to freely falling observers dropped from rest from infinity; Lake-Martel-Poisson (LMP) time coordinates are adapted to observers who start at infinity with non-zero initial inward velocity; Gautreau-Hoffmann (GH) time coordinates are adapted to observers dropped from rest from a finite distance from the black hole horizon. We construct from these an LMP family and a proper-time family of time coordinates, the intersection of which is PG time. We demonstrate that these coordinate families are distinct, but related, one-parameter generalizations of PG time, and show linkage to Lema\^itre coordinates as well.Comment: 13 pages and 10 figures. New title and abstract; expanded discussion on Gautreau-Hoffmann coordinates, and coordinate classifications; figures added; references added. Submitted for publicatio

Computing the Mertens and Meissel-Mertens constants for sums over arithmetic progressions

We give explicit numerical values with 100 decimal digits for the Mertens constant involved in the asymptotic formula for $\sum\limits_{\substack{p\leq x p\equiv a \bmod{q}}}1/p$ and, as a by-product, for the Meissel-Mertens constant defined as $\sum_{p\equiv a \bmod{q}} (\log(1-1/p)+1/p)$, for $q \in \{3$, ..., $100\}$ and $(q, a) = 1$.Comment: 12 pages, 6 table

Solutions of the Yang-Baxter equation: descendants of the six-vertex model from the Drinfeld doubles of dihedral group algebras

The representation theory of the Drinfeld doubles of dihedral groups is used to solve the Yang-Baxter equation. Use of the 2-dimensional representations recovers the six-vertex model solution. Solutions in arbitrary dimensions, which are viewed as descendants of the six-vertex model case, are then obtained using tensor product graph methods which were originally formulated for quantum algebras. Connections with the Fateev-Zamolodchikov model are discussed.Comment: 34 pages, 2 figure

EFFECTS OF PROLONGED BENCH STEPPING ON IMPACT FORCES

INTRODUCTION - This study examined the effects of prolonged bench stepping on the vertical ground reaction forces (GRF). Ten college-aged females experienced in bench step aerobics performed a 40 minute bench step aerobic routine on a 20 cm bench using an arm and leg crossover, alternate lead leg routine at a 100 bpm cadence. Twenty seconds of force platform data was collected at 0 min, 10 min, 20 min, 30 min and 40 min of the aerobics activity. The vertical GRFs were collected by an Ariel APAS system using a Kistler force plate at 1000 Hz sampling rate. Three right and 3 left stride impacts were selected from the middle of the 20 second interval for analysis. Each stride was delineated into 3 phases 1) landing contact, 2) full weight double support, and 3) toe push-off. - RESULTS - The time duration for the 3 phases for each stride was measured and an 5x3~3A NOVA (Time x Steps x Phase) with repeated measures on all factors was used for analysis. The analysis found significant differences in the time of contact over the exercise duration. No differences in the contact time were found between the right and left strides. The mean contact time was 1.262 seconds at the beginning of the workout and decreased to 1.19 seconds over 30 minutes of step bench aerobics. This change of time was the result of the subjects standing on the bench longer and using a more ballistic stepping technique on :he platform as the exercise was prolonged. Analysis of the GRF data found a significant phase factor, where the contact phase's forces were 946.7 Nt(l62%BWT), the support phase was 1044.1 Nt (183%BVVT), and the push-off phase was 816 Nt (134%BVVT). These vertical GRFs were of similar magnitudes reported by Humphries & Newton (1991 ), Johnson et al., (1991) and Ricard & Veatch (1990). Differences in the GRFs in the 40 minute workout were marginally significant (p=.10). These GRF forces recorded would place the impact loading in the range of easy jogging. CONCLUSION - In summary, the timing of the step bench technique was altered as the exercise was prolonged and the GRFs during the 3 phases of stepping were significantly different. An 8% increase in the double support phase GRFs after 20 to 30 min of step aerobics was the result of greater loading due to fatigue. The GRFs observed during the 40 minute stepping exercise would be classified as a mild impact activity. The combination of increased GRFs during fatigue and thousands of foot impacts have the potential for the development of musculoskeletal injuries if sufficient recovery time is not provided between workouts. REFERENCES - Humphries, B. & Newton, R. (1991). Peak ground reaction forces during step aerobics, walking and jogging. Biomechanics in Sports 1X:67-71. Johnson, B., Berry, S., Rupp, D. & Rupp, J. (1991). Peak vertical ground reaction forces and time to peak force in bench step aerobics and other activities. Research Quarterly for Exercise & Sport, 22:370-376. Ricard, M. & Veatch, S. (1990). Comparison of impact forces in high and low impact aerobic dance movements. International Journal of Sports Biomechanics, 6:67-77

ALTERATIONS IN GROUND REACTION FORCES DURING TETHERED WALKING

This study examined t h e e f f e c t s of tethering conditions on ground forces while walking a t 2.5 mph. Six males w e r e unweighted 0%, 25%, 50% and 75% of t h e i r BWT while supported i n a Kinney upper body vest by an active traction prototype (Conva-Lift). They walked 10 min. a t each randomly selected unweighting condition in a 12 f t . dia. c i r c l e . Ten right and l e f t s t r i d e impacts w e r e collected a t 1000 Hz with an Ariel APAS system using a K i s t l e r force plate during each condition a t 2.5 mph (27%). Walking velocity was verified by an infra-red timing system. 5 right and l e f t s t r i d e s in which the velocity was closest t o 2.5 mph w e r e selected for analysis. The s t r i d e was delineated into the phases of heelstrike (HS), midfoot support (MFS), and toe-off (TO). The v e r t i c a l forces (F,) a t the 3 phases, the fore-aft forces (Fy) a t HS and TO and the contact time w e r e calculated. A 4x2x5 ANOVA (WtxFtxLeg) with repeated measures on a l l factors was used t o analyze (F,) forces a t HS, MFS, and TO, fore-aft forces (Fy) a t HS, TO, and contact t i m e . The v e r t i c a l forces (Fz) a t HS w e r e 902.0+112.9 Nt(76% BWT), and 389.3276.8 Nt(45% BWT) for the unweighting conditions. The i n i t i a l impact forces w e r e consistent with normal overland walking and the F, forces w e r e significantly reduced when the subject was unweighted a t 50% and 75% of t h e i r BWT. The forea f t forces representing the decelerative forces w e r e 16%, 16%, lo%, and 4% BWT for the weight conditions. Significant differences w e r e found t o exist between the i n i t i a l contact force a t 0% ( f u l l BWT) and 75% unweighting. The MFS v e r t i c a l forces (F,) w e r e 682.7±73.4, 595.7±73.3, 395.9±49.8, and 234.3±28.7 N t for the loading conditions. The midfoot forces represented about a 30% reduction or absorption of the i n i t i a l impact forces, while the 75% condition exhibited only a 18% reduction in the impact force which would suggest that weight bearing mechanism of the arch of the foot functioned differently when unloaded. The F, forces a t TO significantly differed from 869.1±118.5, 76.56±105.4, 537.9±91.8, and 281.4±78.2 f o r t h e conditions. The percentage of reduction of F, forces a t HS and TO for the unweighting conditions, and the v e r t i c a l forces a t TO w e r e decreased more for a similar degree of unweighting. This finding would suggest that t h i s active traction prototype might be beneficial in t h e r e h a b i l i t a t i o n of posterior compartment injuries. When examining the reduction i n the fore-aft accelerative forces a t TO, the % BWT w e r e similar t o the F, forces a t HS for similar unloading conditions. The s t r i d e contact times significantly differed between ,705±03 sec for t h e r i g h t s t r i d e s and ,68±.04 sec for the l e f t s t r i d e s and no differences existed for the weighting condition. Significant time differences w e r e attributable t o the outside leg having t o travel a greater distance in the same t i m e a s t h e inner leg, while walking in a c i r c l e . The Conva-Lift's travel direction is reversible t o prevent any imbalances due t o circular walking. The significant differences in the ground reaction forces in the v e r t i c a l and fore-aft direction as the subjects w e r e unweighted would indicate that the Conva-Lift is a viable means of reducing ground reaction forces while ambulating

Indication of electron neutrino appearance from an accelerator-produced off-axis muon neutrino beam

The T2K experiment observes indications of nu(mu) -> nu(mu) e appearance in data accumulated with 1.43 x 10(20) protons on target. Six events pass all selection criteria at the far detector. In a three-flavor neutrino oscillation scenario with |Delta m(23)(2)| = 2.4 x 10(-3) eV(2), sin(2)2 theta(23) = 1 and sin(2)2 theta(13) = 0, the expected number of such events is 1.5 +/- 0.3(syst). Under this hypothesis, the probability to observe six or more candidate events is 7 x 10(-3), equivalent to 2.5 sigma significance. At 90% C.L., the data are consistent with 0.03(0.04) < sin(2)2 theta(13) < 0.28(0.34) for delta(CP) = 0 and a normal (inverted) hierarchy