Trees and languages with periodic signature

The signature of a labelled tree (and hence of its prefix-closed branch language) is the sequence of the degrees of the nodes of the tree in the breadth-first traversal. In a previous work, we have characterised the signatures of the regular languages. Here, the trees and languages that have the simplest possible signatures, namely the periodic ones, are characterised as the sets of representations of the integers in rational base numeration systems.For any pair of co-prime integers $p$ and $q$, $p>q>1$, the language  $L_{\frac{p}{q}}$ of representations of the integers in base  $\frac{p}{q}$ looks chaotic and does not fit in the classical Chomsky hierarchy of formal languages. On the other hand, the most basic example given by  $L_{\frac{3}{2}}$, the set of representations in base  $\frac{3}{2}$, exhibits a remarkable regularity: its signature is the infinite periodic sequence: $2,1,2,1,2,1,\dots$We first show that  $L_{\frac{p}{q}}$ has a periodic signature and the period (a sequence of $q$ integers whose sum is $p$) is directly derived from the Christoffel word of slope  $\frac{p}{q}$. Conversely, we give a canonical way to label a tree generated by any periodic signature; its branch language then proves to be the set of representations of the integers in a rational base (determined by the period) and written with a non-canonical alphabet of digits. This language is very much of the same kind as a  $L_{\frac{p}{q}}$ since rational base numeration systems have the key property that, even though  $L_{\frac{p}{q}}$ is not regular, normalisation is realised by a finite letter-to-letter transducer.     

Finite-time blow-up for quasilinear degenerate Keller–Segel systems of parabolic–parabolic type

This paper deals with the quasilinear degenerate Keller–Segel systems of parabolic–parabolic type in a ball of ${\mathbb{R}}^N$ ($N\ge 2$). In the case of non-degenerate diffusion, Cie?lak–Stinner [3], [4] proved that if $q>m+\frac{2}{N}$, where m denotes the intensity of diffusion and q denotes the nonlinearity, then there exist initial data such that the corresponding solution blows up in finite time. As to the case of degenerate diffusion, it is known that a solution blows up if $q>m+\frac{2}{N}$ (see Ishida–Yokota [13]); however, whether the blow-up time is finite or infinite has been unknown. This paper gives an answer to the unsolved problem. Indeed, the finite-time blow-up of energy solutions is established when $q>m+\frac{2}{N}$.     

Sharp well-posedness of the cauchy problem for the higher-order dispersive equation

This current paper is devoted to the Cauchy problem for higher order dispersive equation u_t+ ?_x~(2n+1)u = ?_x(u?_x~nu) + ?_x~(n-1)(u_x~2), n ≥ 2, n ∈ N~+.By using Besov-type spaces, we prove that the associated problem is locally well-posed in H~(-n/2+3/4,-1/(2n))(R). The new ingredient is that we establish some new dyadic bilinear estimates. When n is even, we also prove that the associated equation is ill-posed in H~(s,a)(R) with s -n/2+3/4 and all a∈R.     

Rough path properties for local time of symmetric α stable process

In this paper, we first prove that the local time associated with symmetric $\alpha$-stable processes is of bounded $p$-variation for any $p>\frac{2}{\alpha ?1}$ partly based on Barlow’s estimation of the modulus of the local time of such processes.  The fact that the local time is of bounded $p$-variation for any $p>\frac{2}{\alpha ?1}$ enables us to define the integral of the local time ${\int }_{?\infty }^{\infty }{?}_{?}^{\alpha ?1}f\left(x\right)d_x{L}_t^x$ as a Young integral for less smooth functions being of bounded $q$-variation with $1\le q<\frac{2}{3?\alpha }$. When $q\ge \frac{2}{3?\alpha }$, Young’s integration theory is no longer applicable. However, rough path theory is useful in this case. The main purpose of this paper is to establish a rough path theory for the integration with respect to the local times of symmetric $\alpha$-stable processes for $\frac{2}{3?\alpha }\le q<4$.     

Boundary estimates in homogenization of systems of elasticity on Lipschitz domains: Lp theory

The primary purpose of this paper is to investigate a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients, arising in the theory of homogenization in Lipschitz domains. As a consequence, for $d\ge 4$, we prove that the $L^p$ Neumann and $L^p$ Dirichlet boundary value problems for systems of second order linear elasticity are uniquely solvable for $\frac{2(d?1)}{d+1}?\delta <p<2+\delta$ and $2?\delta <p<\frac{2(d?1)}{d?3}+\delta$ respectively.     

EXISTENCE AND BLOW-UP BEHAVIOR OF CONSTRAINED MINIMIZERS FOR SCHRDINGER-POISSON-SLATER SYSTEM

In this article,we study constrained minimizers of the following variational problem e(p):=inf{u∈H1(R3),||u||22=p}E(u),p〉0,where E(u)is the Schrdinger-Poisson-Slater(SPS)energy functional E(u):=1/2∫R3︱▽u(x)︱2dx-1/4∫R3∫R3u2(y)u2(x)/︱x-y︱dydx-1/p∫R3︱u(x)︱pdx in R3 and p∈(2,6).We prove the existence of minimizers for the cases 2p10/3,ρ0,and p=10/3,0ρρ~*,and show that e(ρ)=-∞for the other cases,whereρ~*=||φ||_2~2 andφ(x)is the unique(up to translations)positive radially symmetric solution of-△u+u=u~(7/3)in R~3.Moreover,when e(ρ~*)=-∞,the blow-up behavior of minimizers asρ↗ρ~*is also analyzed rigorously.     

Norm of the Hilbert matrix on Bergman spaces

It is well known that the Hilbert matrix operator H is a bounded operator from the Bergman space $A^p$ into $A^p$ if and only if $2<p<\infty$. In [5] it was shown that the norm of the Hilbert matrix operator H on the Bergman space $A^p$ is equal to $\frac{\pi }{\sin ?\frac{2\pi }{p}}$, when $4\le p<\infty$, and it was also conjectured that

${6H6}_{A^p\to A^p}=\frac{\pi }{\sin ?\frac{2\pi }{p}},$

when $2<p<4$. In this paper we prove this conjecture.     

On L-functions of certain exponential sums

Let ${\mathbb{F}}_q$ denote the finite field of order q of characteristic p. We study the p-adic valuations for zeros of L-functions associated with exponential sums of the following family of Laurent polynomials$f(x)=a_1{x}_{n+1}(x_1+\frac{1}{x_1})+\cdots +a_n{x}_{n+1}(x_n+\frac{1}{x_n})+a_{n+1}{x}_{n+1}+\frac{1}{x_{n+1}}$ where $a_i\in {\mathbb{F}}_q^{⁎}$, $i=1,2,\dots ,n+1$. When $n=2$, the estimate of the associated exponential sum appears in Iwaniecʼs work on small eigenvalues of the Laplace–Beltrami operator acting on automorphic functions with respect to the group ${\Gamma }_0(p)$, and Adolphson and Sperber gave complex absolute values for zeros of the corresponding L-function. Using the decomposition theory of Wan, we determine the generic Newton polygon (q-adic values of the reciprocal zeros) of the L-function. Working on the chain level version of Dworkʼs trace formula and using Wanʼs decomposition theory, we are able to give an explicit Hasse polynomial for the generic Newton polygon in low dimensions, i.e., $n⩽3$.     

Multiplicity of positive solutions for a class of concave-convex elliptic equations with critical growth

In this article, the following concave and convex nonlinearities elliptic equations involving critical growth is considered,{-△u=g(x)|u|2*-2u+λf(x)|u|q-2u,x∈Ω,u=0,x∈■Ω where Ω■R~N(N≥3) is an open bounded domain with smooth boundary, 1 q 2, λ 0.2*=2 N/(N-2)is the critical Sobolev exponent,f∈L2~*/(2~*-q)(Ω)is nonzero and nonnegative,and g ∈ C(■) is a positive function with k local maximum points. By the Nehari method and variational method,k+1 positive solutions are obtained. Our results complement and optimize the previous work by Lin [MR2870946, Nonlinear Anal. 75(2012) 2660-2671].     

Small Kakeya sets in non-prime order planes

The finite field analog of the classical Kakeya problem asks the smallest possible size for a set of points in the Desarguesian affine plane which contains a line in every direction. This problem has been definitively solved by Blokhuis and Mazzocca (2008), who show that in $AG(2,s)$, $s$ a prime power, the smallest possible size of such a set is $\frac{1}{2}s(s+1)$. In this paper we examine a new construction of an infinite family of sets in $AG(2,s)$ containing a line in every direction, where $s=q^e$ with $q$ a prime power and $e$ an integer with $e>1$. These sets have size $\frac{1}{2}{q}^{2e}+O\left(q^{2e-1}\right)$, which is small in the sense that the lower bound is also $\frac{1}{2}{q}^{2e}$ plus smaller order terms. In addition, we discuss the minimality of our sets, showing that they contain no proper subset containing a line in every direction.     

Sparse univariate polynomials with many roots over finite fields

Suppose q is a prime power and $f\in {\mathbb{F}}_q[x]$ is a univariate polynomial with exactly t monomial terms and degree $<q-1$. To establish a finite field analogue of Descartes' Rule, Bi, Cheng, and Rojas (2013) proved an upper bound of $2{(q-1)}^{\frac{t-2}{t-1}}$ on the number of cosets in ${\mathbb{F}}_q^{⁎}$ needed to cover the roots of f in ${\mathbb{F}}_q^{⁎}$. Here, we give explicit f with root structure approaching this bound: When q is a perfect $(t-1)$-st power we give an explicit t-nomial vanishing on $q^{\frac{t-2}{t-1}}$ distinct cosets of ${\mathbb{F}}_q^{⁎}$. Over prime fields ${\mathbb{F}}_p$, computational data we provide suggests that it is harder to construct explicit sparse polynomials with many roots. Nevertheless, assuming the Generalized Riemann Hypothesis, we find explicit trinomials having $\mathrm{\Omega }\left(\frac{\log p}{\log \log p}\right)$ distinct roots in ${\mathbb{F}}_p$.