Let us fix a prime $\Gamma <p$
We show that certain singular structures (Hölderian cusps and mild divergences) are transported by the flow of homeomorphisms generated by an Osgood velocity field. The structure of these singularities is related to the modulus of continuity of the velocity and the results are shown to be sharp in the sense that slightly more singular structures cannot generally be propagated. For the 2D Euler equation, we prove that certain singular structures are preserved by the motion, e.g. a system of
 NSFPAR ID:
 10378137
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Mathematische Annalen
 Volume:
 387
 Issue:
 34
 ISSN:
 00255831
 Format(s):
 Medium: X Size: p. 16911718
 Size(s):
 p. 16911718
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract p and a homogeneous system ofm linear equations for$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$ ${a}_{j,1}{x}_{1}+\cdots +{a}_{j,k}{x}_{k}=0$ with coefficients$$j=1,\dots ,m$$ $j=1,\cdots ,m$ . Suppose that$$a_{j,i}\in \mathbb {F}_p$$ ${a}_{j,i}\in {F}_{p}$ , that$$k\ge 3m$$ $k\ge 3m$ for$$a_{j,1}+\dots +a_{j,k}=0$$ ${a}_{j,1}+\cdots +{a}_{j,k}=0$ and that every$$j=1,\dots ,m$$ $j=1,\cdots ,m$ minor of the$$m\times m$$ $m\times m$ matrix$$m\times k$$ $m\times k$ is nonsingular. Then we prove that for any (large)$$(a_{j,i})_{j,i}$$ ${\left({a}_{j,i}\right)}_{j,i}$n , any subset of size$$A\subseteq \mathbb {F}_p^n$$ $A\subseteq {F}_{p}^{n}$ contains a solution$$A> C\cdot \Gamma ^n$$ $\leftA\right>C\xb7{\Gamma}^{n}$ to the given system of equations such that the vectors$$(x_1,\dots ,x_k)\in A^k$$ $({x}_{1},\cdots ,{x}_{k})\in {A}^{k}$ are all distinct. Here,$$x_1,\dots ,x_k\in A$$ ${x}_{1},\cdots ,{x}_{k}\in A$C and are constants only depending on$$\Gamma $$ $\Gamma $p ,m andk such that . The crucial point here is the condition for the vectors$$\Gamma in the solution$$x_1,\dots ,x_k$$ ${x}_{1},\cdots ,{x}_{k}$ to be distinct. If we relax this condition and only demand that$$(x_1,\dots ,x_k)\in A^k$$ $({x}_{1},\cdots ,{x}_{k})\in {A}^{k}$ are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a nondiagonal tensor in combination with combinatorial and probabilistic arguments.$$x_1,\dots ,x_k$$ ${x}_{1},\cdots ,{x}_{k}$ 
Abstract Let
M (x ) denote the largest cardinality of a subset of on which the Euler totient function$$\{n \in \mathbb {N}: n \le x\}$$ $\{n\in N:n\le x\}$ is nondecreasing. We show that$$\varphi (n)$$ $\phi \left(n\right)$ for all$$M(x) = \left( 1+O\left( \frac{(\log \log x)^5}{\log x}\right) \right) \pi (x)$$ $M\left(x\right)=\left(1,+,O,\left(\frac{{(loglogx)}^{5}}{logx}\right)\right)\pi \left(x\right)$ , answering questions of Erdős and Pollack–Pomerance–Treviño. A similar result is also obtained for the sum of divisors function$$x \ge 10$$ $x\ge 10$ .$$\sigma (n)$$ $\sigma \left(n\right)$ 
Abstract We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight
for$$y^a$$ ${\lefty\right}^{a}$ . Such problem arises as the local extension of the obstacle problem for the fractional heat operator$$a \in (1,1)$$ $a\in (1,1)$ for$$(\partial _t  \Delta _x)^s$$ ${({\partial}_{t}{\Delta}_{x})}^{s}$ . Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove AlmgrenPoon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation ($$s \in (0,1)$$ $s\in (0,1)$ ).$$a=0$$ $a=0$ 
Abstract For a smooth projective variety
X over an algebraic number fieldk a conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map ofX is a torsion group. In this article we consider a product of smooth projective curves and show that if the conjecture is true for any subproduct of two curves, then it is true for$$X=C_1\times \cdots \times C_d$$ $X={C}_{1}\times \cdots \times {C}_{d}$X . For a product of two curves over$$X=C_1\times C_2$$ $X={C}_{1}\times {C}_{2}$ with positive genus we construct many nontrivial examples that satisfy the weaker property that the image of the natural map$$\mathbb {Q} $$ $Q$ is finite, where$$J_1(\mathbb {Q})\otimes J_2(\mathbb {Q})\xrightarrow {\varepsilon }{{\,\textrm{CH}\,}}_0(C_1\times C_2)$$ ${J}_{1}\left(Q\right)\otimes {J}_{2}\left(Q\right)\stackrel{\epsilon}{\to}{\phantom{\rule{0ex}{0ex}}\text{CH}\phantom{\rule{0ex}{0ex}}}_{0}({C}_{1}\times {C}_{2})$ is the Jacobian variety of$$J_i$$ ${J}_{i}$ . Our constructions include many new examples of nonisogenous pairs of elliptic curves$$C_i$$ ${C}_{i}$ with positive rank, including the first known examples of rank greater than 1. Combining these constructions with our previous result, we obtain infinitely many nontrivial products$$E_1, E_2$$ ${E}_{1},{E}_{2}$ for which the analogous map$$X=C_1\times \cdots \times C_d$$ $X={C}_{1}\times \cdots \times {C}_{d}$ has finite image.$$\varepsilon $$ $\epsilon $ 
Abstract Approximate integer programming is the following: For a given convex body
, either determine whether$$K \subseteq {\mathbb {R}}^n$$ $K\subseteq {R}^{n}$ is empty, or find an integer point in the convex body$$K \cap {\mathbb {Z}}^n$$ $K\cap {Z}^{n}$ which is$$2\cdot (K  c) +c$$ $2\xb7(Kc)+c$K , scaled by 2 from its center of gravityc . Approximate integer programming can be solved in time while the fastest known methods for exact integer programming run in time$$2^{O(n)}$$ ${2}^{O\left(n\right)}$ . So far, there are no efficient methods for integer programming known that are based on approximate integer programming. Our main contribution are two such methods, each yielding novel complexity results. First, we show that an integer point$$2^{O(n)} \cdot n^n$$ ${2}^{O\left(n\right)}\xb7{n}^{n}$ can be found in time$$x^* \in (K \cap {\mathbb {Z}}^n)$$ ${x}^{\ast}\in (K\cap {Z}^{n})$ , provided that the$$2^{O(n)}$$ ${2}^{O\left(n\right)}$remainders of each component for some arbitrarily fixed$$x_i^* \mod \ell $$ ${x}_{i}^{\ast}\phantom{\rule{0ex}{0ex}}mod\phantom{\rule{0ex}{0ex}}\ell $ of$$\ell \ge 5(n+1)$$ $\ell \ge 5(n+1)$ are given. The algorithm is based on a$$x^*$$ ${x}^{\ast}$cuttingplane technique , iteratively halving the volume of the feasible set. The cutting planes are determined via approximate integer programming. Enumeration of the possible remainders gives a algorithm for general integer programming. This matches the current best bound of an algorithm by Dadush (Integer programming, lattice algorithms, and deterministic, vol. Estimation. Georgia Institute of Technology, Atlanta, 2012) that is considerably more involved. Our algorithm also relies on a new$$2^{O(n)}n^n$$ ${2}^{O\left(n\right)}{n}^{n}$asymmetric approximate Carathéodory theorem that might be of interest on its own. Our second method concerns integer programming problems in equationstandard form . Such a problem can be reduced to the solution of$$Ax = b, 0 \le x \le u, \, x \in {\mathbb {Z}}^n$$ $Ax=b,0\le x\le u,\phantom{\rule{0ex}{0ex}}x\in {Z}^{n}$ approximate integer programming problems. This implies, for example that$$\prod _i O(\log u_i +1)$$ ${\prod}_{i}O(log{u}_{i}+1)$knapsack orsubsetsum problems withpolynomial variable range can be solved in time$$0 \le x_i \le p(n)$$ $0\le {x}_{i}\le p\left(n\right)$ . For these problems, the best running time so far was$$(\log n)^{O(n)}$$ ${(logn)}^{O\left(n\right)}$ .$$n^n \cdot 2^{O(n)}$$ ${n}^{n}\xb7{2}^{O\left(n\right)}$