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In a 2001 paper, S. Montgomery-Smith provides a one-dimensional model for the three-dimensional, incompressible Navier-Stokes equations, for which he proves the blow-up of solutions associated with a class of large initial data, while the same global existence results as for the Navier-Stokes equations hold for small data. In this paper the model is adapted to the cases of two and three space dimensions, with the additional feature that the divergence-free condition is preserved. It is checked that a family of initial data constructed by Chemin and Gallagher, which is arbitrarily large yet generates a global solution to the Navier-Stokes equations in three space dimensions, actually causes blow-up for the toy model --- meaning that the precise structure of the nonlinear term is crucial to understanding the dynamics of large solutions to the Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear version of the Extrapolation Theorem is proved and, as a corollary, a nonlinear version of Grothendieck's Theorem is presented. Finally, we prove that if $ T:X\to H$ being a pointed metric space and $ T(0)=0$ $ T^\char93 \vert _{H^*}$-summing $ (1\le q<\infty)$ is Lipschitz 1-summing. [ABSTRACT FROM AUTHOR]

In this paper, we use the Perron method to prove the existence of viscosity solutions with asymptotic behavior at infinity to Hessian equations. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove that there exist measures $ d\mu(x)=\gamma(x)dx$ $ \gamma(x)=\gamma_{0}(\vert x\vert)$ $ \gamma_{0}$ $ \mathcal{M}_{\mu}$ does not map $ L^{p}_{\mu}(\mathbb{R}^{n})$ $ L^{p}_{\mu}(\mathbb{R}^{n})$. This result answers an open question of P. Sjögren and F. Soria. [ABSTRACT FROM AUTHOR]

It is well known that the average position or barycenter of generic orbits for the standard tent map is $ 0.5$. In this paper we prove that for any positive integer $ n$ distinct periodic orbits for the standard tent map with the same barycenter. We also provide some patterns of periodic orbits with the same barycenter. [ABSTRACT FROM AUTHOR]

Let $N(a,m)$ be the least integer $n$ (if it exists) such that $varphi (n)equiv apmod m$. Friedlander and Shparlinski proved that for any $varepsilon >0$ there exists $A=A(varepsilon )>0$ such that for any positive integer $m$ which has no prime divisors $p<(log m)^A$ and any integer $a$ with $gcd (a,m)=1,$ we have the bound $N(a,m)ll m^{3+varepsilon }.$ In the present paper we improve this bound to $N(a,m)ll m^{2+varepsilon }.$ [ABSTRACT FROM AUTHOR]

In this paper we prove a weighted version of the Khintchine inequalities. [ABSTRACT FROM AUTHOR]