31,328 research outputs found
High Order Methods for a Class of Volterra Integral Equations with Weakly Singular Kernels
The solution of the Volterra integral equation, where , and are smooth functions, can be represented as ,, where , are, smooth and satisfy a system of Volterra integral equations. In this paper, numerical schemes for the solution of (*) are suggested which calculate via , in a neighborhood of the origin and use (*) on the rest of the interval . In this way, methods of arbitrarily high order can be derived. As an example, schemes based on the product integration analogue of Simpson's rule are treated in detail. The schemes are shown to be convergent of order . Asymptotic error estimates are derived in order to examine the numerical stability of the methods
The norm of a Ree group
We give an explicit construction of the Ree groups of type as groups
acting on mixed Moufang hexagons together with detailed proofs of the basic
properties of these groups contained in the two fundamental papers of Tits on
this subject. We also give a short proof that the norm of a Ree group is
anisotropic
Hyperspectral pan-sharpening: a variational convex constrained formulation to impose parallel level lines, solved with ADMM
In this paper, we address the issue of hyperspectral pan-sharpening, which
consists in fusing a (low spatial resolution) hyperspectral image HX and a
(high spatial resolution) panchromatic image P to obtain a high spatial
resolution hyperspectral image. The problem is addressed under a variational
convex constrained formulation. The objective favors high resolution spectral
bands with level lines parallel to those of the panchromatic image. This term
is balanced with a total variation term as regularizer. Fit-to-P data and
fit-to-HX data constraints are effectively considered as mathematical
constraints, which depend on the statistics of the data noise measurements. The
developed Alternating Direction Method of Multipliers (ADMM) optimization
scheme enables us to solve this problem efficiently despite the non
differentiabilities and the huge number of unknowns.Comment: 4 pages, detailed version of proceedings of conference IEEE WHISPERS
201
General linearized theory of quantum fluctuations around arbitrary limit cycles
The theory of Gaussian quantum fluctuations around classical steady states in
nonlinear quantum-optical systems (also known as standard linearization) is a
cornerstone for the analysis of such systems. Its simplicity, together with its
accuracy far from critical points or situations where the nonlinearity reaches
the strong coupling regime, has turned it into a widespread technique, which is
the first method of choice in most works on the subject. However, such a
technique finds strong practical and conceptual complications when one tries to
apply it to situations in which the classical long-time solution is time
dependent, a most prominent example being spontaneous limit-cycle formation.
Here we introduce a linearization scheme adapted to such situations, using the
driven Van der Pol oscillator as a testbed for the method, which allows us to
compare it with full numerical simulations. On a conceptual level, the scheme
relies on the connection between the emergence of limit cycles and the
spontaneous breaking of the symmetry under temporal translations. On the
practical side, the method keeps the simplicity and linear scaling with the
size of the problem (number of modes) characteristic of standard linearization,
making it applicable to large (many-body) systems.Comment: Constructive suggestions and criticism are welcom
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The Stokes conjecture for waves with vorticity
We study stagnation points of two-dimensional steady gravity free-surface water waves with vorticity.
We obtain for example that, in the case where the free surface is an injective curve, the asymptotics at any stagnation point is given either by the “Stokes corner flow” where the free surface has a corner of 120°, or the free surface ends in a horizontal cusp, or the free surface is horizontally flat at the stagnation point. The cusp case is a new feature in the case with vorticity, and it is not possible in the absence of vorticity.
In a second main result we exclude horizontally flat singularities in the case that the vorticity is 0 on the free surface. Here the vorticity may have infinitely many sign changes accumulating at the free surface, which makes this case particularly difficult and explains why it has been almost untouched by research so far. Our results are based on calculations in the original variables and do not rely on structural assumptions needed in previous results such as isolated singularities, symmetry and monotonicity
Number of distinct sites visited by N random walkers on a Euclidean lattice
The evaluation of the average number S_N(t) of distinct sites visited up to
time t by N independent random walkers all starting from the same origin on an
Euclidean lattice is addressed. We find that, for the nontrivial time regime
and for large N, S_N(t) \approx \hat S_N(t) (1-\Delta), where \hat S_N(t) is
the volume of a hypersphere of radius (4Dt \ln N)^{1/2},
\Delta={1/2}\sum_{n=1}^\infty \ln^{-n} N \sum_{m=0}^n s_m^{(n)} \ln^{m} \ln N,
d is the dimension of the lattice, and the coefficients s_m^{(n)} depend on the
dimension and time. The first three terms of these series are calculated
explicitly and the resulting expressions are compared with other approximations
and with simulation results for dimensions 1, 2, and 3. Some implications of
these results on the geometry of the set of visited sites are discussed.Comment: 15 pages (RevTex), 4 figures (eps); to appear in Phys. Rev.
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