The systematic development of reduced low-dimensional stochastic climate models from observations or comprehensive high dimensional climate models is an important topic for atmospheric low-frequency variability, climate sensitivity, and improved extended range forecasting. Recently, techniques from applied mathematics have been utilized to systematically derive normal forms for reduced stochastic climate models for low-frequency variables. It was shown that dyad and multiplicative triad interactions combine with the climatological linear operator interactions to produce a normal form with both strong nonlinear cubic dissipation and Correlated Additive and Multiplicative (CAM) stochastic noise. The probability distribution functions (PDFs) of low frequency climate variables exhibit small but significant departure from Gaussianity but have asymptotic tails which decay at most like a Gaussian. Here, rigorous upper bounds with Gaussian decay are proved for the invariant measure of general normal form stochastic models. Asymptotic Gaussian lower bounds are also established under suitable hypotheses.

Let N be a sufficiently large even integer. Let p denote a prime and P ₂ denote an almost prime with at most two prime factors. In this paper, it is proved that the equation N = p + P ₂ (p ≤ N ^0.945) is solvable.

By an interpolation method, an intrinsic Harnack estimate and some sup-estimates are established for nonnegative solutions to a general singular parabolic equation.

The authors consider the homogenization of a class of nonlinear variational inequalities, which include rapid oscillations with respect to a parameter. The homogenization of the corresponding class of differential equations is also studied. The results are applied to some models for the pressure in a thin fluid film fluid between two surfaces which are in relative motion. This is an important problem in the lubrication theory. In particular, the analysis includes the effects of surface roughness on both faces and the phenomenon of cavitation. Moreover, the fluid can be modeled as Newtonian or non-Newtonian by using a Rabinowitsch fluid model.

Let M be a compact orientable irreducible 3-manifold, and F be an essential connected closed surface in M which cuts M into two manifolds M ₁ and M ₂. If M _i has a minimal Heegaard splitting M_i = $V_i \cup _{H_i }$W_i with d(H ₁) + d(H ₂) ≥ 2(g(M ₁) + g(M ₂) − g(F)) + 1, then g(M) = g(M 1) + g(M 2) − g(F).

The primary goal of this paper is to present a comprehensive study of the nonlinear Schrödinger equations with combined nonlinearities of the power-type and Hartree-type. Under certain structural conditions, the authors are able to provide a complete picture of how the nonlinear Schrödinger equations with combined nonlinearities interact in the given energy space. The method used in the paper is based upon the Morawetz estimates and perturbation principles.

Let Ω be a domain in ℝ ^N. It is shown that a generalized Poincaré inequality holds in cones contained in the Sobolev space W ^1,p(·)(ω), where p(·): $\bar \Omega $ → [1, ∞[ is a variable exponent. This inequality is itself a corollary to a more general result about equivalent norms over such cones. The approach in this paper avoids the difficulty arising from the possible lack of density of the space D(Ω) in the space {v ∈ W ^1,p(·)(Ω); tr v = 0 on ∂Ω}. Two applications are also discussed.

The authors study vanishing viscosity limits of solutions to the 3-dimensional incompressible Navier-Stokes system in general smooth domains with curved boundaries for a class of slip boundary conditions. In contrast to the case of flat boundaries, where the uniform convergence in super-norm can be obtained, the asymptotic behavior of viscous solutions for small viscosity depends on the curvature of the boundary in general. It is shown, in particular, that the viscous solution converges to that of the ideal Euler equations in C([0, T];H ¹(Ω)) provided that the initial vorticity vanishes on the boundary of the domain.

Curvature properties are studied for the Sasaki metric on the (1, 1) tensor bundle of a Riemannian manifold. As an application, examples of almost para-Nordenian and para-Kähler-Nordenian B-metrics are constructed on the (1, 1) tensor bundle by looking at the Sasaki metric. Also, with respect to the para-Nordenian B-structure, paraholomorphic conditions for the complete lifts of vector fields are analyzed.