61 research outputs found
Algorithms for the Numerical Solution of a Finite-Part Integral Equation
The authors investigate a hypersingular integral equation which arises in the study of acoustic wave scattering by moving objects. A Galerkin method and two collocation methods are presented for solving the problem numerically. These numerical techniques are compared and contrasted in three test problems
Asymptotic expansions and fast computation of oscillatory Hilbert transforms
In this paper, we study the asymptotics and fast computation of the one-sided
oscillatory Hilbert transforms of the form where the bar indicates the Cauchy principal value and is a
real-valued function with analytic continuation in the first quadrant, except
possibly a branch point of algebraic type at the origin. When , the
integral is interpreted as a Hadamard finite-part integral, provided it is
divergent. Asymptotic expansions in inverse powers of are derived for
each fixed , which clarify the large behavior of this
transform. We then present efficient and affordable approaches for numerical
evaluation of such oscillatory transforms. Depending on the position of , we
classify our discussion into three regimes, namely, or
, and . Numerical experiments show that the convergence
of the proposed methods greatly improve when the frequency increases.
Some extensions to oscillatory Hilbert transforms with Bessel oscillators are
briefly discussed as well.Comment: 32 pages, 6 figures, 4 table
Remarks on nonlocal trace expansion coefficients
In a recent work, Paycha and Scott establish formulas for all the Laurent
coefficients of Tr(AP^{-s}) at the possible poles. In particular, they show a
formula for the zero'th coefficient at s=0, in terms of two functions
generalizing, respectively, the Kontsevich-Vishik canonical trace density, and
the Wodzicki-Guillemin noncommutative residue density of an associated
operator. The purpose of this note is to provide a proof of that formula
relying entirely on resolvent techniques (for the sake of possible
generalizations to situations where powers are not an easy tool).
- We also give some corrections to transition formulas used in our earlier
works.Comment: Minor corrections. To appear in a proceedings volume in honor of K.
Wojciechowski, "Analysis and Geometry of Boundary Value Problems", World
Scientific, 19 page
The mechanics of a chain or ring of spherical magnets
Strong magnets, such as neodymium-iron-boron magnets, are increasingly being
manufactured as spheres. Because of their dipolar characters, these spheres can
easily be arranged into long chains that exhibit mechanical properties
reminiscent of elastic strings or rods. While simple formulations exist for the
energy of a deformed elastic rod, it is not clear whether or not they are also
appropriate for a chain of spherical magnets. In this paper, we use
discrete-to-continuum asymptotic analysis to derive a continuum model for the
energy of a deformed chain of magnets based on the magnetostatic interactions
between individual spheres. We find that the mechanical properties of a chain
of magnets differ significantly from those of an elastic rod: while both
magnetic chains and elastic rods support bending by change of local curvature,
nonlocal interaction terms also appear in the energy formulation for a magnetic
chain. This continuum model for the energy of a chain of magnets is used to
analyse small deformations of a circular ring of magnets and hence obtain
theoretical predictions for the vibrational modes of a circular ring of
magnets. Surprisingly, despite the contribution of nonlocal energy terms, we
find that the vibrations of a circular ring of magnets are governed by the same
equation that governs the vibrations of a circular elastic ring
Gelfand-Shilov and Gevrey smoothing effect for the spatially inhomogeneous non-cutoff Kac equation
We consider the spatially inhomogeneous non-cutoff Kac's model of the
Boltzmann equation. We prove that the Cauchy problem for the fluctuation around
the Maxwellian distribution enjoys Gelfand-Shilov regularizing properties with
respect to the velocity variable and Gevrey regularizing properties with
respect to the position variable.Comment: 47 page
On lightest baryon and its excitations in large-N 1+1-dimensional QCD
We study baryons in multicolour 1+1D QCD via Rajeev's gauge-invariant
reformulation as a non-linear classical theory of a bilocal meson field
constrained to lie on a Grassmannian. It is known to reproduce 't Hooft's meson
spectrum via small oscillations around the vacuum, while baryons arise as
topological solitons. The lightest baryon has zero mass per colour in the
chiral limit; we find its form factor. It moves at the speed of light through a
family of massless states. To model excitations of this baryon, we linearize
equations for motion in the tangent space to the Grassmannian, parameterized by
a bilocal field U. A redundancy in U is removed and an approximation is made in
lieu of a consistency condition on U. The baryon spectrum is given by an
eigenvalue problem for a hermitian singular integral operator on such tangent
vectors. Excited baryons are like bound states of the lightest one with a
meson. Using a rank-1 ansatz for U in a variational formulation, we estimate
the mass and form factor of the first excitation.Comment: 26 pages, 3 figures, shorter published version, added remarks on
parit
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