Relation between vibrotactile perception thresholds and reductions in finger blood flow induced by vibration of the hand at frequencies in the range 8–250 Hz

Purpose: this study investigated how the vasoconstriction induced by vibration depends on the frequency of vibration when the vibration magnitude is defined by individual thresholds for perceiving vibration [i.e. sensation levels (SL)].Methods: fourteen healthy subjects attended the laboratory on seven occasions: for six vibration frequencies (8, 16, 31.5, 63, 125, or 250 Hz) and a static control condition. Finger blood flow (FBF) was measured in the middle fingers of both hands at 30-second intervals during five successive periods: (i) no force or vibration, (ii) 2-N force, no vibration, (iii) 2-N force, vibration, (iv) 2-N force, no vibration, (v) no force or vibration. During period (iii), vibration was applied to the right thenar eminence via a 6-mm diameter probe during ten successive 3-min periods as the vibration magnitude increased in ten steps (?10 to +40 dB SL).Results: with vibration at 63, 125, and 250 Hz, there was vasoconstriction on both hands when the vibration magnitude reached 10 dB SL. With vibration at 8, 16, and 31.5 Hz, there was no significant vasoconstriction until the vibration reached 25 dB SL. At all frequencies, there was greater vasoconstriction with greater magnitudes of vibration.Conclusions: it is concluded that at the higher frequencies (63, 125, and 250 Hz), the Pacinian channel mediates vibrotactile sensations near threshold and vasoconstriction occurs when vibration is perceptible. At lower frequencies (8, 16, and 31.5 Hz), the Pacinian channel does not mediate sensations near threshold and vasoconstriction commences at greater magnitudes when the Pacinian channel is activate

When are Multiples of Polygonal Numbers again Polygonal Numbers?

Euler showed that there are infinitely many triangular numbers that are three times other triangular numbers. In general, it is an easy consequence of the Pell equation that for a given square-free m > 1, the relation P=mP' is satisfied by infinitely many pairs of triangular numbers P, P'. After recalling what is known about triangular numbers, we shall study this problem for higher polygonal numbers. Whereas there are always infinitely many triangular numbers which are fixed multiples of other triangular numbers, we give an example that this is false for higher polygonal numbers. However, as we will show, if there is one such solution, there are infinitely many. We will give conditions which conjecturally assure the existence of a solution. But due to the erratic behavior of the fundamental unit in quadratic number fields, finding such a solution is exceedingly difficult. Finally, we also show in this paper that, given m > n > 1 with obvious exceptions, the system of simultaneous relations P = mP', P = nP'' has only finitely many possibilities not just for triangular numbers, but for triplets P, P', P'' of polygonal numbers, and give examples of such solutions.Comment: 17 pages, 1 figure, 2 tables. New version added a table of solutions to the second proble

The ammonolysis of esters in liquid ammonia

The rates of ammonolysis of alkyl benzoate and phenylacetate esters in liquid ammonia increase with the acidity of the leaving group alcohol and show relatively large Brønsted βlg values of −1.18 and −1.34, respectively, when plotted against the aqueous pKa of the alcohol. The Brønsted βlg obtained using the pKa of the leaving group alcohol in liquid ammonia is significantly reduced to ~ −0.7, which indicates that the rate-limiting step involves a reaction of the tetrahedral intermediate with little C–OR bond fission in the transition state. The solvolysis reaction is subject to significant catalysis by ammonium ion, which, surprisingly, generates a similar Brønsted βlg indicating little interaction between the ammonium ion and the leaving group. It is concluded that the rate-limiting step for the ammonium-ion-catalysed solvolysis of alkyl esters in liquid ammonia is the diffusion-controlled protonation of the zwitterionic tetrahedral intermediate T+- to give T+, which is rapidly deprotonated to give T0 which is compatible with the rate-limiting step for the uncatalysed reaction being the formation of the neutral T0 by a ‘proton switc

Jensen polynomials for the Riemann zeta function and other sequences

In 1927 P\'olya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for the Riemann zeta function $\zeta(s)$ at its point of symmetry. This hyperbolicity has been proved for degrees $d\leq 3$. We obtain an asymptotic formula for the central derivatives $\zeta^{(2n)}(1/2)$ that is accurate to all orders, which allows us to prove the hyperbolicity of a density $1$ subset of the Jensen polynomials of each degree. Moreover, we establish hyperbolicity for all $d\leq 8$. These results follow from a general theorem which models such polynomials by Hermite polynomials. In the case of the Riemann zeta function, this proves the GUE random matrix model prediction in derivative aspect. The general theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function.Comment: 11 page

Proof of the Umbral Moonshine Conjecture

The Umbral Moonshine Conjectures assert that there are infinite-dimensional graded modules, for prescribed finite groups, whose McKay-Thompson series are certain distinguished mock modular forms. Gannon has proved this for the special case involving the largest sporadic simple Mathieu group. Here we establish the existence of the umbral moonshine modules in the remaining 22 cases.Comment: 56 pages, to appear in Research in the Mathematical Science