The force of gravity in Schwarzschild and Gullstrand-Painlev\'e coordinates

We derive the exact equations of motion (in Newtonian, F=ma, form) for test masses in Schwarzschild and Gullstrand-Painlev\'e coordinates. These equations of motion are simpler than the usual geodesic equations obtained from Christoffel tensors in that the affine parameter is eliminated. The various terms can be compared against tests of gravity. In force form, gravity can be interpreted as resulting from a flux of superluminal particles (gravitons). We show that the first order relativistic correction to Newton's gravity results from a two graviton interaction.Comment: 6 pages, Honorable mention in 2009 Gravity Essay Competition, submitted IJMPD, added reference

Some non-linear s.p.d.e.'s that are second order in time

We extend Walsh's theory of martingale measures in order to deal with hyperbolic stochastic partial differential equations that are second order in time, such as the wave equation and the beam equation, and driven by spatially homogeneous Gaussian noise. For such equations, the fundamental solution can be a distribution in the sense of Schwartz, which appears as an integrand in the reformulation of the s.p.d.e. as a stochastic integral equation. Our approach provides an alternative to the Hilbert space integrals of Hilbert-Schmidt operators. We give several examples, including the beam equation and the wave equation, with nonlinear multiplicative noise terms

Multiple points of the Brownian sheet in critical dimensions

It is well known that an $N$-parameter $d$-dimensional Brownian sheet has no $k$-multiple points when $(k-1)d>2kN$, and does have such points when $(k-1)d<2kN$. We complete the study of the existence of $k$-multiple points by showing that in the critical cases where $(k-1)d=2kN$, there are a.s. no $k$-multiple points.Comment: Published at http://dx.doi.org/10.1214/14-AOP912 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Digital second-order phase-locked loop

A digital second-order phase-locked loop is disclosed in which a counter driven by a stable clock pulse source is used to generate a reference waveform of the same frequency as an incoming waveform, and to sample the incoming waveform at zero-crossover points. The samples are converted to digital form and accumulated over M cycles, reversing the sign of every second sample. After every M cycles, the accumulated value of samples is hard limited to a value SGN = + or - 1 and multiplied by a value delta sub 1 equal to a number of n sub 1 of fractions of a cycle. An error signal is used to advance or retard the counter according to the sign of the sum by an amount equal to the sum

Asymptotic analysis on a pseudo-Hermitian Riemann-zeta Hamiltonian

The differential-equation eigenvalue problem associated with a recently-introduced Hamiltonian, whose eigenvalues correspond to the zeros of the Riemann zeta function, is analyzed using Fourier and WKB analysis. The Fourier analysis leads to a challenging open problem concerning the formulation of the eigenvalue problem in the momentum space. The WKB analysis gives the exact asymptotic behavior of the eigenfunction

Resonance Region Structure Functions and Parity Violating Deep Inelastic Scattering

The primary motive of parity violating deep inelastic scattering experiments has been to test the standard model, particularly the axial couplings to the quarks, in the scaling region. The measurements can also test for the validity of models for the off-diagonal structure functions $F_{1,2,3}^{\gamma Z}(x,Q^2)$ in the resonance region. The off-diagonal structure functions are important for the accurate calculation of the $\gamma Z$-box correction to the weak charge of the proton. Currently, with no data to determine $F_{1,2,3}^{\gamma Z}(x,Q^2)$ directly, models are constructed by modifying existing fits to electromagnetic data. We present the asymmetry value for deuteron and proton target predicted by several different $F_{1,2,3}^{\gamma Z}(x,Q^2)$ models, and demonstrate that there are notable disagreements.Comment: 6 pages, 3 figures. New version contains additional descriptions of competing structure function model

New Physics and the Proton Radius Problem

Background: The recent disagreement between the proton charge radius extracted from Lamb shift measurements of muonic and electronic hydrogen invites speculation that new physics may be to blame. Several proposals have been made for new particles that account for both the Lamb shift and the muon anomalous moment discrepancies. Purpose: We explore the possibility that new particles' couplings to the muon can be fine-tuned to account for all experimental constraints. Method: We consider two fine-tuned models, the first involving new particles with scalar and pseudoscalar couplings, and the second involving new particles with vector and axial couplings. The couplings are constrained by the Lamb shift and muon magnetic moments measurements while mass constraints are obtained by kaon decay rate data. Results: For the scalar-pseudoscalar model, masses between 100 to 200 MeV are not allowed. For the vector model, masses below about 200 MeV are not allowed. The strength of the couplings for both models approach that of electrodynamics for particle masses of about 2 GeV. Conclusions: New physics with fine tuned couplings may be entertained as a possible explanation for the Lamb shift discrepancy.Comment: 6 pages, 6 figures, v2 contains revised comment on competing model of Lamb Shift discrepanc

Operator-valued zeta functions and Fourier analysis

The Riemann zeta function $\zeta(s)$ is defined as the infinite sum $\sum_{n=1}^\infty n^{-s}$, which converges when ${\rm Re}\,s>1$. The Riemann hypothesis asserts that the nontrivial zeros of $\zeta(s)$ lie on the line ${\rm Re}\,s= \frac{1}{2}$. Thus, to find these zeros it is necessary to perform an analytic continuation to a region of complex $s$ for which the defining sum does not converge. This analytic continuation is ordinarily performed by using a functional equation. In this paper it is argued that one can investigate some properties of the Riemann zeta function in the region ${\rm Re}\,s<1$ by allowing operator-valued zeta functions to act on test functions. As an illustration, it is shown that the locations of the trivial zeros can be determined purely from a Fourier series, without relying on an explicit analytic continuation of the functional equation satisfied by $\zeta(s)$.Comment: 8 pages, version to appear in J. Pays.