18,739 research outputs found
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
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
Operator-valued zeta functions and Fourier analysis
The Riemann zeta function is defined as the infinite sum
, which converges when . The Riemann
hypothesis asserts that the nontrivial zeros of lie on the line
. Thus, to find these zeros it is necessary to
perform an analytic continuation to a region of complex 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
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
.Comment: 8 pages, version to appear in J. Pays.
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
A new proof of a Nordgren, Rosenthal and Wintrobe Theorem on universal operators
A striking result by Nordgren, Rosenthal and Wintrobe states that the Invariant Subspace Problem is equivalent to the fact that any minimal invariant subspace for a composition operator Cφ induced by a hyperbolic automorphism φ of the unit disc D acting on the classical Hardy space H² is one dimensional. We provide a completely different proof of Nordgren, Rosenthal and Wintrobe’s Theorem based on analytic Toeplitz operators
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