On a spectral theorem of Weyl

We give a new proof of a theorem of Weyl on the continuous part of the spectrum of Sturm–Liouville operators on the half-line with asymptotically constant coefficients. Earlier arguments, due to Weyl and Kodaira, depended on particular features of Green’s functions for linear ordinary differential operators. We use a concept of asymptotic containment of $C^1$-algebra representations that has geometric origins. We apply the concept elsewhere to the Plancherel formula for spherical functions on reductive groups.     

On Yau's problem of evolving one curve to another: Convex case

A revised Yau's Curvature Difference Flow is considered to deform one convex curve $X_0$ to another one $\stackrel{?}{X}$. It is proved that this flow exists globally on time interval $[0,+\infty )$ and the evolving curve, preserving its convexity and bounded area A, converges to a fixed limiting curve $X_{\infty }$ (congruent to $\sqrt{A/\stackrel{?}{A}}\stackrel{?}{X}$) as time tends to infinity, where $\stackrel{?}{A}$ is the area bounded by the target curve $\stackrel{?}{X}$.     

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.     

Semi-classical analysis of piecewise quasi-polynomial functions and applications to geometric quantization

Motivated by applications to multiplicity formulas in index theory, we study a family of distributions $\Theta (m;k)$ associated to a piecewise quasi-polynomial function $m$. The family is indexed by an integer $k\in {\mathbb{Z}}_{>0}$, and admits an asymptotic expansion as $k\to \infty$, which generalizes the expansion obtained in the Euler–Maclaurin formula. When $m$ is the multiplicity function arising from the quantization of a symplectic manifold, the leading term of the asymptotic expansion is the Duistermaat–Heckman measure. Our main result is that $m$ is uniquely determined by a collection of such asymptotic expansions. We also show that the construction is compatible with pushforwards. As an application, we describe a simpler proof that formal quantization is functorial with respect to restrictions to a subgroup.     

Compressible subsonic jet flows issuing from a nozzle of arbitrary cross-section

We are concerned with the well-posedness theory of two-dimensional compressible subsonic jet flow issuing from a semi-infinitely long nozzle of arbitrary cross-section. Given any atmospheric pressure $p_0$, we show that there exists a critical mass flux $m_{cr}$ depending on $p_0$ and Ω, such that if the incoming mass flux $m_0$ is less than the critical value, then there exists a unique smooth subsonic jet flow, issuing from the given nozzle. The jet boundary is a free streamline, which initiates from the end point of the nozzle smoothly and extends to the infinity. One of the key observations in this paper is that the restriction of the incoming mass flux guarantees completely the subsonicity of the compressible jet in the whole flow field, which coincides with the observation on the compressible subsonic flows in an infinitely long nozzle without free boundary in [8].     

Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos

We develop the asymptotic expansion theory for vector-valued sequences ${\left(F_N\right)}_{N\ge 1}$ of random variables in terms of the convergence of the Stein–Malliavin matrix associated with the sequence $F_N$. Our approach combines the classical Fourier approach and the recent Stein–Malliavin theory. We find the second order term of the asymptotic expansion of the density of $F_N$ and we illustrate our results by several examples.     

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.     

Identities from representation theory

We give a new Jacobi–Trudi-type formula for characters of finite-dimensional irreducible representations in type $C_n$ using characters of the fundamental representations and non-intersecting lattice paths. We give equivalent determinant formulas for the decomposition multiplicities for tensor powers of the spin representation in type $B_n$ and the exterior representation in type $C_n$. This gives a combinatorial proof of an identity of Katz and equates such a multiplicity with the dimension of an irreducible representation in type $C_n$. By taking certain specializations, we obtain identities for $q$-Catalan triangle numbers, a slight modification of the $q,t$-Catalan number of Stump, $q$-triangle versions of Motzkin and Riordan numbers, and generalizations of Touchard’s identity. We use (spin) rigid tableaux and crystal base theory to show some formulas relating Catalan, Motzkin, and Riordan triangle numbers.     

Non parametric estimation of the diffusion coefficients of a diffusion with jumps

In this article, we consider a jump diffusion process ${\left(X_t\right)}_{t\ge 0}$, with drift function $b$, diffusion coefficient $\sigma$ and jump coefficient ${\xi }^2$. This process is observed at discrete times $t=0,\Delta ,\dots ,n\Delta$. The sampling interval $\Delta$ tends to 0 and the time interval $n\Delta$ tends to infinity. We assume that ${\left(X_t\right)}_{t\ge 0}$ is ergodic, strictly stationary and exponentially $\beta$-mixing. We use a penalized least-square approach to compute adaptive estimators of the functions ${\sigma }^2+{\xi }^2$ and ${\sigma }^2$. We provide bounds for the risks of the two estimators.     

Deterministic particle approximation for nonlocal transport equations with nonlinear mobility

We construct a deterministic, Lagrangian many-particle approximation to a class of nonlocal transport PDEs with nonlinear mobility arising in many contexts in biology and social sciences. The approximating particle system is a nonlocal version of the follow-the-leader scheme. We rigorously prove that a suitable discrete piece-wise density reconstructed from the particle scheme converges strongly in $L_{loc}^1$ towards the unique entropy solution to the target PDE as the number of particles tends to infinity. The proof is based on uniform BV estimates on the approximating sequence and on the verification of an approximated version of the entropy condition for large number of particles. As part of the proof, we also prove uniqueness of entropy solutions. We further provide a specific example of non-uniqueness of weak solutions and discuss the interplay of the entropy condition with the steady states. Finally, we produce numerical simulations supporting the need of a concept of entropy solution in order to get a well-posed semigroup in the continuum limit, and showing the behaviour of solutions for large times.