Rational points near planar curves and Diophantine approximation

In this paper, we establish asymptotic formulae with optimal errors for the number of rational points that are close to a planar curve, which unify and extend the results of Beresnevich–Dickinson–Velani [6] and Vaughan–Velani [22]. Furthermore, we complete the Lebesgue theory of Diophantine approximation on weakly non-degenerate planar curves that was initially developed by Beresnevich–Zorin [5] in the divergence case.     

Metric Diophantine approximation over a local field of positive characteristic

The conjectures of Sprind?uk in the metric theory of Diophantine approximation are established over a local field of positive characteristic. In the real case, these were settled by D. Kleinbock and G.A. Margulis using a new technique which involved nondivergence estimates for quasi-polynomial flows on the space of lattices. We extend their technique to the positive characteristic setting.     

Explicit bounds for rational points near planar curves and metric Diophantine approximation

The primary goal of this paper is to complete the theory of metric Diophantine approximation initially developed in Beresnevich et al. (2007) [10] for C³ non-degenerate planar curves. With this goal in mind, here for the first time we obtain fully explicit bounds for the number of rational points near planar curves. Further, introducing a perturbational approach we bring the smoothness condition imposed on the curves down to C¹ (lowest possible). This way we broaden the notion of non-degeneracy in a natural direction and introduce a new topologically complete class of planar curves to the theory of Diophantine approximation. In summary, our findings improve and complete the main theorems of Beresnevich et al. (2007) [10] and extend the celebrated theorem of Kleinbock and Margulis (1998) [20] in dimension 2 beyond the notion of non-degeneracy.     

Modified Schmidt games and Diophantine approximation with weights

We show that the sets of weighted badly approximable vectors in Rn are winning sets of certain games, which are modifications of (α,β)-games introduced by W.M. Schmidt in 1966. The latter winning property is stable with respect to countable intersections, and is shown to imply full Hausdorff dimension.     

Eigenvalue asymptotics, inverse problems and a trace formula for the linear damped wave equation

We determine the general form of the asymptotics for Dirichlet eigenvalues of the one-dimensional linear damped wave operator. As a consequence, we obtain that given a spectrum corresponding to a constant damping term this determines the damping term in a unique fashion. We also derive a trace formula for this problem.     

Diophantine approximation on rational quadrics

We compute the Hausdorff dimension of sets of very well approximable vectors on rational quadrics. We use ubiquitous systems and the geometry of locally symmetric spaces. As a byproduct we obtain the Hausdorff dimension of the set of rays with a fixed maximal singular direction, which move away into one end of a locally symmetric space at linear depth, infinitely many times.     

Stationary isothermic surfaces and some characterizations of the hyperplane in the N-dimensional Euclidean space

We consider the entire graph S of a continuous real function over RN−1 with N?3. Let Ω be a domain in RN with S as a boundary. Consider in Ω the heat flow with initial temperature 0 and boundary temperature 1. The problem we consider is to characterize S in such a way that there exists a stationary isothermic surface in Ω. We show that S must be a hyperplane under some general conditions on S. This is related to Liouville or Bernstein-type theorems for some elliptic Monge-Ampère-type equation.     

Weighted branching and a pathwise renewal equation

This paper is devoted to the study of a pathwise renewal equation for stochastic processes which are functions of a weighted tree defined in a general weighted branching model. Motivated by applications in the analysis of certain stochastic fixed-point equations and in the theory of general (Crump–Mode–Jagers) branching processes, we analyze the solutions to the equation under several conditions, the main result being a characterization of the set of solutions satisfying appropriate integrability conditions.     

On a problem of K. Mahler: Diophantine approximation and Cantor sets

Let K denote the middle third Cantor set and . Given a real, positive function ψ let denote the set of real numbers x in the unit interval for which there exist infinitely many such that |x − p/q| < ψ(q). The analogue of the Hausdorff measure version of the Duffin–Schaeffer conjecture is established for . One of the consequences of this is that there exist very well approximable numbers, other than Liouville numbers, in K—an assertion attributed to K. Mahler. Explicit examples of irrational numbers satisfying Mahler’s assertion are also given. Dedicated to Maurice Dodson on his retirement—finally!     

A parabolic two-phase obstacle-like equation

For the parabolic obstacle-problem-like equation

Δu−t_∂u=λ₊χ{u>0}−λ₋χ{u<0},     

A note on simultaneous Diophantine approximation on planar curves

Let denote the set of simultaneously - approximable points in and denote the set of multiplicatively ψ-approximable points in . Let be a manifold in . The aim is to develop a metric theory for the sets and analogous to the classical theory in which is simply . In this note, we mainly restrict our attention to the case that is a planar curve . A complete Hausdorff dimension theory is established for the sets and . A divergent Khintchine type result is obtained for ; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on of is full. Furthermore, in the case that is a rational quadric the convergent Khintchine type result is obtained for both types of approximation. Our results for naturally generalize the dimension and Lebesgue measure statements of Beresnevich et al. (Mem AMS, 179 (846), 1–91 (2006)). Moreover, within the multiplicative framework, our results for constitute the first of their type. The research of Victor V. Beresnevich was supported by an EPSRC Grant R90727/01. Sanju L. Velani is a Royal Society University Research Fellow. For Iona and Ayesha on No. 3.     

Symmetry of integral equation systems with Bessel kernel on bounded domains

In this paper, we investigate the symmetry of integral equation systems with Bessel kernel on bounded domains. Under some natural integrability conditions, we prove that the domains are balls and all positive solutions are radially symmetric and monotonic decreasing.     

On the Cauchy problem for a reaction-diffusion equation with a singular nonlinearity

We consider the following Cauchy problem with a singular nonlinearity

(P)

     

Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping

We consider the wave equation with supercritical interior and boundary sources and damping terms. The main result of the paper is local Hadamard well-posedness of finite energy (weak) solutions. The results obtained: (1) extend the existence results previously obtained in the literature (by allowing more singular sources); (2) show that the corresponding solutions satisfy Hadamard well-posedness conditions during the time of existence. This result provides a positive answer to an open question in the area and it allows for the construction of a strongly continuous semigroup representing the dynamics governed by the wave equation with supercritical sources and damping.     

A global subsonic solution to an interacting transonic shock for the self-similar nonlinear wave equation

We present the existence of the subsonic solution to a two-dimensional Riemann problem governed by a self-similar nonlinear wave equation where the boundary of the subsonic region consists of a transonic shock and the sonic circle. Thus the governing equation becomes a free boundary problem on the transonic shock and degenerates on the sonic circle. By utilizing the barrier methods and iterative methods, we show the well-posedness of the transonic shock in the entire subsonic region and thus establish the global solution. This result does not rely on any smallness of Riemann data.     

An integro-differential equation for surface ocean waves with finite depth

An explicit integro-differential equation formulation is derived for surface ocean waves with finite depth. The equation involves only 2D surface variables. For this equation, we establish the stability and existence of solutions, and explain the effect of depth on surface wave properties.     

Ground states of a prescribed mean curvature equation

We study the existence of radial ground state solutions for the problem