Related papers: Improved discrete restriction for the parabola

We study discrete versions of fractional integral operators along curves and surfaces. $l^p \to l^q$ estimates are obtained from upper bounds of the number of solutions of associated Diophantine systems. In particular, this relates the…

We consider the discretization y(t+\epsilon)=y(t-\epsilon)+2\epsilon\big(1-y(t)^{2}\big), $\epsilon>0$ a small parameter, of the logistic differential equation $y'=2\big(1-y^{2}\big)$, which can also be seen as a discretization of the…

We consider the optimal control of singular nonlinear partial differential equation which is the distributional formulation of the multiphase Stefan type free boundary problem for the general second order parabolic equation. Boundary heat…

Via operator theoretic methods, we formalize the concentration phenomenon for a given observable `$r$' of a discrete time Markov chain with `$\mu_{\pi}$' as invariant ergodic measure, possibly having support on an unbounded state space. The…

We consider the bifurcation problem $u'' + \lambda u = N(u)$ with two point boundary conditions where $N(u)$ is a general nonlinear term which may also depend on the eigenvalue $\lambda$. We give a variational characterization of the…

We prove bilinear $\ell^2$-decoupling and refined bilinear decoupling inequalities for the truncated hyperbolic paraboloid in $\mathbb{R}^3$. As an application, we prove the associated restriction estimate in the range $p>22/7$, matching an…

We present four frequently used finite difference methods and establish the error bounds for the discretization of the Dirac equation in the massless and nonrelativistic regime, involving a small dimensionless parameter $0< \varepsilon \ll…

We show that if $A$ is a set of mutually orthogonal exponentials with respect to the unit disk then $|A \cap [-R, R]^2| \lesssim_\varepsilon R^{3/5+\varepsilon}$ holds. This improves the previous bound of $R^{2/3}$ by…

For a $n$-dimensional spin manifold $M$ with a fixed spin structure and a spinor bundle $\Sigma M$, we prove an $\epsilon$-regularity theorem for weak solutions to the nonlinear Dirac equation of cubic nonlinearity. This, in particular,…

In this article we prove a differentiable rigidity result. Let $(Y, g)$ and $(X, g_0)$ be two closed $n$-dimensional Riemannian manifolds ($n\geqslant 3$) and $f:Y\to X$ be a continuous map of degree $1$. We furthermore assume that the…

We establish a mixed observability inequality for a class of degenerate hyperbolic equations on the cylindrical domain $\Omega = \mathbb{T} \times (0,1)$ with mixed Neumann Dirichlet boundary conditions. The degeneracy acts only in the…

This paper studies unique continuation for weakly degenerate parabolic equations in one space dimension. A new Carleman estimate of local type is obtained to deduce that all solutions that vanish on the degeneracy set, together with their…

For a prime number p greater than 6000, the Olson's constant for the group Z_p+Z_p is given by Ol(Z_p+Z_p)=p-1+Ol(Z_p).

We study the restriction estimates in a class of conical singular space $X=C(Y)=(0,\infty)_r\times Y$ with the metric $g=\mathrm{d}r^2+r^2h$, where the cross section $Y$ is a compact $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$.…

We provide a self-contained analysis, based entirely on pde methods, of the exponentially long time behavior of solutions to linear uniformly parabolic equations which are small perturbations of a transport equation with vector field having…

We consider a linear-quadratic pde constrained optimal control problem on an evolving surface with pointwise state constraints. We reformulate the optimization problem on a fixed surface and approximate the reformulated problem by a…

The celebrated Littlewood-Offord problem asks for an upper bound on the probability that the random variable $\epsilon_1 v_1 + \cdots + \epsilon_n v_n$ lies in the Euclidean unit ball, where $\epsilon_1, \ldots, \epsilon_n \in \{-1, 1\}$…

We report a robust constrain on the possible variation of fine-structure constant, alpha = e^2/(hbar*c), obtained using O III 4959,5007, nebular emission lines from QSOs. We find Delta-alpha/alpha=-(2.1 +/- 1.6) x 10^(-5) based on a well…

We propose a variational approach to solve Cauchy problems for parabolic equations and systems independently of regularity theory for solutions. This produces a universal and conceptually simple construction of fundamental solution…

We propose the difference discrete variational principle in discrete mechanics and symplectic algorithm with variable step-length of time in finite duration based upon a noncommutative differential calculus established in this paper. This…