International Mathematics Research Notices

doi: 10.1093/imrn/rnv148pmid: N/A

doi: 10.1093/imrn/rnv146pmid: N/A

doi: 10.1093/imrn/rnv147pmid: N/A

Shao, Xuancheng

doi: 10.1093/imrn/rnu004pmid: N/A

We study the following generalization of Roths theorem for 3-term arithmetic progressions. For s2, define a nontrivial s-configuration to be a set of s(s1)/2 integers consisting of s distinct integers x1,,xs as well as the averages (xixj)/2 (1i<js). Our main result states that if a set A[N] has density $\delta \gg (\log N)^{-c(s)}$ for some positive constant c(s)>0 depending on s, then A contains a nontrivial s-configuration. This improves on the previous bound of the form $\delta \gg (\log \log N)^{-c(s)}$ due to Dousse [3]. We also deduce, as a corollary, an improvement of a problem involving sum-free subsets.

Katz, Nicholas M.

doi: 10.1093/imrn/rnu007pmid: N/A

We give what is arguably a simple (though certainly not elementary, cf. [12]) proof of the Riemann Hypothesis for (projective, smooth, geometrically connected) curves and hypersurfaces over finite fields, by an argument which reduces us to checking a few examples.

Blatt, Simon; Struwe, Michael

doi: 10.1093/imrn/rnt359pmid: N/A

The natural setting for the LaneEmden equation uup2u on a domain $\Omega \subset \mathbb {R}^n$, n3, for supercritical exponents p>22n/(n2) is identified as the space of functions $u\in H^1_0\cap L^p(\Omega)$ with finite scale-invariant Morrey norms. We show that this Morrey regularity is propagated by the heat flow associated with this equation, and we study the blow-up profiles.

Majdoub, Mohamed; Masmoudi, Nader

doi: 10.1093/imrn/rnu002pmid: N/A

In a recent paper [39], Struwe considered the Cauchy problem for a class of nonlinear wave and Schrdinger equations. Under some assumptions on the nonlinearities, it was shown that uniqueness of classical solutions can be obtained in the much larger class of distribution solutions satisfying the energy inequality. As pointed out in [39], the conditions on the nonlinearities are satisfied for any polynomial growth but they fail to hold for higher growth (for example eu2). Our aim here is to improve Struwe's result by showing that uniqueness holds for more general nonlinearities including higher growth or oscillations.

Lester, Stephen J.

doi: 10.1093/imrn/rnt356pmid: N/A

We investigate the proportion of the nontrivial roots of the equation (s)a, which lie on the line ${\mathfrak {R}\,} s=\frac {1}{2}$ for $a \in {\mathbb C}$ not equal to zero. We show that at most one-half of these points lie on the line ${\mathfrak {R}} s=\frac {1}{2}$. Moreover, assuming a spacing condition on the ordinates of zeros of the Riemann zeta-function, we prove that zero percent of the nontrivial solutions to (s)a lie on the line ${\mathfrak {R}} s=\frac {1}{2}$ for any nonzero complex number a.

Wang, Feng; Zhu, Xiaohua

doi: 10.1093/imrn/rnu006pmid: N/A

In this paper, we prove that a sequence of weak almost KhlerRicci solitons under further suitable conditions converges to a KhlerRicci soliton with complex codimension of singularities at least 2 in the GromovHausdorff topology. As a corollary, we show that on a Fano manifold with the modified K-energy bounded below, there exists a sequence of weak almost KhlerRicci solitons which converges to a KhlerRicci soliton with complex codimension of singularities at least 2 in the GromovHausdorff topology.

Ye, Deping

doi: 10.1093/imrn/rnu009pmid: N/A

In this paper, we introduce the Lp geominimal surface area for all np<1, which extends the classical geominimal surface area (p1) by Petty and the Lp geominimal surface area by Lutwak (p>1). Our extension of the Lp geominimal surface area is motivated by recent work on the extension of the Lp affine surface areaa fundamental object in (affine) convex geometry. We prove some properties for the Lp geominimal surface area and its related inequalities, such as, the affine isoperimetric inequality and a Santal style inequality. Cyclic inequalities are established to obtain the monotonicity of the Lp geominimal surface areas. Comparison between the Lp geominimal surface area and the p-surface area is also provided.