The existence of Hall polynomials for x2-bounded invariant subspaces of nilpotent linear operators

We prove the existence of Hall polynomials for $x^2$-bounded invariant subspaces of nilpotent linear operators.     

Additive G-codes over Fq and their dualities

We define additive G-codes over finite fields. We prove that if C is an additive G-code over ${\mathbb{F}}_q$ with duality M then its dual with respect to this duality $C^M$ is an additive G-code. We prove that if M and $M^{\prime }$ are two dualities, then $C^M$ and $C^{M^{\prime }}$ are equivalent codes. Finally, we study the existence of self-dual codes for a variety of dualities and relate them to formally self-dual and linear self-dual codes.     

Schmidt representation of bilinear operators on Hilbert spaces

Current work defines Schmidt representation of a bilinear operator $T:H_1\times H_2\to K$, where $H_1,H_2$ and $K$ are separable Hilbert spaces. Introducing the concept of singular value and ordered singular value, we prove that if $T$ is compact, and its singular values are ordered, then $T$ has a Schmidt representation on real Hilbert spaces. We prove that the hypothesis of existence of ordered singular values is fundamental.     

Finite dimensional global attractor for a dissipative anisotropic fourth order Schrödinger equation

We study the long-time behavior of the solutions to a nonlinear damped anisotropic fourth order Schrödinger type equation in ${\mathbb{R}}^2$. We prove that this behavior is described by the existence of regular finite-dimensional global attractor in the energy space.     

Almost perfect nonlinear families which are not equivalent to permutations

An important problem on almost perfect nonlinear (APN) functions is the existence of APN permutations on even-degree extensions of ${\mathbb{F}}_2$ larger than 6. Browning et al. (2010) gave the first known example of an APN permutation on the degree-6 extension of ${\mathbb{F}}_2$. The APN permutation is CCZ-equivalent to the previously known quadratic Kim κ-function (Browning et al. (2009)). Aside from the computer based CCZ-inequivalence results on known APN functions on even-degree extensions of ${\mathbb{F}}_2$ with extension degrees less than 12, no theoretical CCZ-inequivalence result on infinite families is known. In this paper, we show that Gold and Kasami APN functions are not CCZ-equivalent to permutations on infinitely many even-degree extensions of ${\mathbb{F}}_2$. In the Gold case, we show that Gold APN functions are not equivalent to permutations on any even-degree extension of ${\mathbb{F}}_2$, whereas in the Kasami case we are able to prove inequivalence results for every doubly-even-degree extension of ${\mathbb{F}}_2$.     

Forbidden arithmetic progressions in permutations of subsets of the integers

Permutations of the positive integers avoiding arithmetic progressions of length 5 were constructed in Davis et al. (1977), implying the existence of permutations of the integers avoiding arithmetic progressions of length 7. We construct a permutation of the integers avoiding arithmetic progressions of length 6. We also prove a lower bound of $\frac{1}{2}$ on the lower density of subsets of positive integers that can be permuted to avoid arithmetic progressions of length 4, sharpening the lower bound of $\frac{1}{3}$ from LeSaulnier and Vijay (2011).     

The lifespan of classical solutions for the inviscid Surface Quasi-geostrophic equation

We consider classical solutions of the inviscid Surface Quasi-geostrophic equation that are a small perturbation ϵ from a radial stationary solution $\theta =|x|$. We use a modified energy method to prove the existence time of classical solutions from $\frac{1}{ϵ}$ to a time scale of $\frac{1}{{ϵ}^4}$. Moreover, by perturbing in a suitable direction we construct global smooth solutions, via bifurcation, that rotate uniformly in time and space.     

Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions

In this article we develop the local wellposedness theory for quasilinear Maxwell equations in $H^m$ for all $m\ge 3$ on domains with perfectly conducting boundary conditions. The macroscopic Maxwell equations with instantaneous material laws for the polarization and the magnetization lead to a quasilinear first order hyperbolic system whose wellposedness in $H^3$ is not covered by the available results in this case. We prove the existence and uniqueness of local solutions in $H^m$ with $m\ge 3$ of the corresponding initial boundary value problem if the material laws and the data are accordingly regular and compatible. We further characterize finite time blowup in terms of the Lipschitz norm and we show that the solutions depend continuously on their data. Finally, we establish the finite propagation speed of the solutions.     

Energy conservation for the weak solutions to the ideal inhomogeneous magnetohydrodynamic equations in a bounded domain

In this paper, we prove the energy conservation for the weak solutions of the three-dimensional ideal inhomogeneous magnetohydrodynamic (MHD) equations in a bounded domain. Two types of sufficient conditions on the regularity of the weak solutions are provided to ensure the energy conservation. Due to the presence of the boundary, we need to impose the boundedness in $L^p$ and the continuity in $L^{\frac{p}{p-2}}$ for the velocity and magnetic fields near the boundary.     

Extremal functions for sharp Moser–Trudinger type inequalities in the whole space RN

In this paper, we prove the existence of maximizers for the sharp Moser–Trudinger type inequalities in whole space ${\mathbb{R}}^N$, $N\ge 2$ with more general nonlinearity. The main key in our proof is a precise estimate of the concentrating level of the Moser–Trudinger functional associated with our inequalities on the normalized concentrating sequences. This estimate solves a heavily non-trivial and open problem related to the sharp Moser–Trudinger inequality. Our method gives an alternative proof of the existence of maximizers for the Moser–Trudinger inequality and singular Moser–Trudinger inequality in whole space ${\mathbb{R}}^N$ due to Li and Ruf [30] and Li and Yang [31] without using blow-up analysis argument.     

Highly nonlinear functions over finite fields

We consider a generalisation of a conjecture by Patterson and Wiedemann from 1983 on the Hamming distance of a function from ${\mathbb{F}}_q^n$ to ${\mathbb{F}}_q$ to the set of affine functions from ${\mathbb{F}}_q^n$ to ${\mathbb{F}}_q$. We prove the conjecture for each q such that the characteristic of ${\mathbb{F}}_q$ lies in a subset of the primes with density 1 and we prove the conjecture for all q by assuming the generalised Riemann hypothesis. Roughly speaking, we show the existence of functions for which the distance to the affine functions is maximised when n tends to infinity. This also determines the asymptotic behaviour of the covering radius of the $[q^n,n+1]$ Reed-Muller code over ${\mathbb{F}}_q$ and so answers a question raised by Leducq in 2013. Our results extend the case $q=2$, which was recently proved by the author and which corresponds to the original conjecture by Patterson and Wiedemann. Our proof combines evaluations of Gauss sums in the semiprimitive case, probabilistic arguments, and methods from discrepancy theory.     

Uniqueness of certain Fourier-Jacobi models over finite fields

In this paper, we prove the uniqueness of certain Fourier-Jacobi models for the split exceptional group ${\mathrm{G}}_2$ over finite fields with odd characteristic. Similar results are also proved for ${\mathrm{Sp}}_4$ and ${\mathrm{U}}_4$.     

Representation and factorization theorems for almost-Lp-spaces

We extend the notions of $p$-convexity and $p$-concavity for Banach ideals of measurable functions following an asymptotic procedure. We prove a representation theorem for the spaces satisfying both properties as the one that works for the classical case: each almost $p$-convex and almost $p$-concave space is order isomorphic to an almost-$L^p$-space. The class of almost-$L^p$-spaces contains, in particular, direct sums of (infinitely many) $L^p$-spaces with different norms, that are not in general $p$-convex – nor $p$-concave –. We also analyze in this context the extension of the Maurey–Rosenthal factorization theorem that works for $p$-concave operators acting in $p$-convex spaces. In this way we provide factorization results that allow to deal with more general factorization spaces than $L^p$-spaces.     

Prescribed scalar curvatures for homogeneous toric bundles

In this paper, we study the generalized Abreu equation on a Delzant ploytope $\mathrm{\Delta }\subset {\mathbb{R}}^2$ and prove the existence of metrics with prescribed scalar curvatures of homogeneous toric bundles under the assumption of an appropriate stability.