1,251 research outputs found
Bi-stochastic kernels via asymmetric affinity functions
In this short letter we present the construction of a bi-stochastic kernel p
for an arbitrary data set X that is derived from an asymmetric affinity
function {\alpha}. The affinity function {\alpha} measures the similarity
between points in X and some reference set Y. Unlike other methods that
construct bi-stochastic kernels via some convergent iteration process or
through solving an optimization problem, the construction presented here is
quite simple. Furthermore, it can be viewed through the lens of out of sample
extensions, making it useful for massive data sets.Comment: 5 pages. v2: Expanded upon the first paragraph of subsection 2.1. v3:
Minor changes and edits. v4: Edited comments and added DO
Sur l'unicité des solutions de l'équation d'abel-schröder et l'itération continue
Soit f(x) continue strictement croissante pour x [0, a0] et telle que 0 < f(x) < x pour x 0, a0]. Il est connu que l'équation fonctionnelle d'Abel ainsi que l'équation de Schröder possèdent une infinité de solutions continues strictement croissante
Lp Fourier multipliers on compact Lie groups
In this paper we prove Lp multiplier theorems for invariant and non-invariant
operators on compact Lie groups in the spirit of the well-known
Hormander-Mikhlin theorem on Rn and its variants on tori Tn. We also give
applications to a-priori estimates for non-hypoelliptic operators. Already in
the case of tori we get an interesting refinement of the classical multiplier
theorem.Comment: 22 pages; minor correction
Asymptotics and numerics of a family of two-dimensional generalized surface quasi-geostrophic equations
We study the generalised 2D surface quasi-geostrophic (SQG) equation, where the active scalar is given by a fractional power α of Laplacian applied to the stream function. This includes the 2D SQG and Euler equations as special cases. Using Poincaré’s successive approximation to higher α-derivatives of the active scalar, we derive a variational equation for describing perturbations in the generalized SQG equation. In particular, in the limit α → 0, an asymptotic equation is derived on a stretched time variable τ = αt, which unifies equations in the family near α = 0. The successive approximation is also discussed at the other extreme of the 2D Euler limit α = 2–0. Numerical experiments are presented for both limits. We consider whether the solution behaves in a more singular fashion, with more effective nonlinearity, when α is increased. Two competing effects are identified: the regularizing effect of a fractional inverse Laplacian (control by conservation) and cancellation by symmetry (nonlinearity depletion). Near α = 0 (complete depletion), the solution behaves in a more singular fashion as α increases. Near α = 2 (maximal control by conservation), the solution behave in a more singular fashion, as α decreases, suggesting that there may be some α in [0, 2] at which the solution behaves in the most singular manner. We also present some numerical results of the family for α = 0.5, 1, and 1.5. On the original time t, the H 1 norm of θ generally grows more rapidly with increasing α. However, on the new time τ, this order is reversed. On the other hand, contour patterns for different α appear to be similar at fixed τ, even though the norms are markedly different in magnitude. Finally, point-vortex systems for the generalized SQG family are discussed to shed light on the above problems of time scale
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