This paper studies the dynamics of two incompressible immiscible fluids in 2D modeled by inhomogeneous Navier-Stokes equations. We prove that if initially viscosity contrast is small then there global-in-time regularity. result has been proved recently [32] for $H^{5/2}$ Sobolev regularity interface. Here we provide a new approach which allows to obtain preservation natural $C^{1+\gamma}$ H\old...

A quick gradient training algorithm for a specific neural network structure called an extra reduced size lattice-ladder multilayer perceptron is introduced. Presented derivation of the algorithm utilizes recently found by author simplest way of exact computation of gradients for rotation parameters of lattice-ladder filter. Developed neural network training algorithm is optimal in terms of mini...

In this paper, we consider the modified quasi-geostrophic equation ∂tθ + (u · ∇) θ + κΛθ = 0 u = Λα−1R⊥θ. with κ > 0, α ∈ (0, 1] and θ0 ∈ L2(R2). We remark that the extra Λα−1 is introduced in order to make the scaling invariance of this system similar to the scaling invariance of the critical quasi-geostrophic equations. In this paper, we use Besov space techniques to prove global existence an...

In this paper, we consider the modified quasi-geostrophic equation ∂tθ + (u · ∇) θ + κΛ θ = 0 u = ΛRθ. with κ > 0, α ∈ (0, 1] and θ0 ∈ L(R). We remark that the extra Λ is introduced in order to make the scaling invariance of this system similar to the scaling invariance of the critical quasi-geostrophic equations. In this paper, we use Besov space techniques to prove global existence and regula...

For a broad class of jackknife statistics, it is shown that the Tukey estimator of the variance converges almost surely to its population counterpart. Moreover, the usual invariance principles (relating to the Wiener process approximations) usually filter through jackknifing under no extra regularity conditions. These results are then incorporated in providing a bounded-length (sequential) conf...

We consider the Schrödinger equation on a compact manifold, in the presence of a nonlinear damping term, which is homogeneous and sublinear. For initial data in the energy space, we construct a weak solution, defined for all positive time, which is shown to be unique. In the one-dimensional case, we show that it becomes zero in finite time. In the two and three-dimensional cases, we prove the s...

Abstract: In this paper we consider a control problem governed by a semilinear elliptic equation with pointwise control and state constraints. We analyze the existence of an exact penalization of the state constraints. In particular, we prove that the first and second order optimality conditions imply the existence of such a penalization. Finally, we prove some extra regularity of the strict lo...

A quick gradient training algorithm for a specific neural network structure called an extra reduced size lattice–ladder multilayer perceptron is introduced. Presented derivation of the algorithm utilizes recently found by author simplest way of exact computation of gradients for rotation parameters of lattice–ladder filter. Developed neural network training algorithm is optimal in terms of mini...

We revisit regularity of SLE trace, for all κ 6= 8, and establish Besov regularity under the usual half-space capacity parametrization. With an embedding theorem of Garsia–Rodemich–Rumsey type, we obtain finite moments (and hence almost surely) optimal variation regularity with index min(1 + κ/8, 2), improving on previous works of Werness, and also (optimal) Hölder regularity à la Johansson Vik...

Green [Geometric and Functional Analysis 15 (2005), 340–376] established a version of the Szemerédi Regularity Lemma for abelian groups and derived the Removal Lemma for abelian groups as its corollary. We provide another proof of his Removal Lemma that allows us to extend its statement to all finite groups. We also discuss possible extensions of the Removal Lemma to systems of equations.