We demonstrate the existence of accelerating parabolic beams that constitute, together with the Airy beams, the only orthogonal and complete families of solutions of the two-dimensional paraxial wave equation that exhibit the unusual ability to remain diffraction-free and freely accelerate during propagation. Since the accelerating parabolic beams, like the Airy beams, carry infinite energy, we...

We introduce tessellation of the filled Julia sets for hyperbolic and parabolic quadratic maps. Then the dynamics inside their Julia sets are organized by tiles which work like external rays outside. We also construct continuous families of pinching semiconjugacies associated with hyperblic-to-parabolic degenerations without using quasiconformal deformation. Instead we use tessellation and inve...

We consider linear inhomogeneous non-autonomous parabolic problems associated to sesquilinear forms, with discontinuous dependence of time. We show that for these problems, the property of maximal parabolic regularity can be extrapolated to time integrability exponents r 6= 2. This allows us to prove maximal parabolic L r-regularity for discontinuous non-autonomous second-order divergence form ...

In this paper we study invariant differential operators on manifolds with a given parabolic structure. The model for the parabolic geometry is the quotient of the orthogonal group by a maximal parabolic subgroup corresponding to crossing of the k-th simple root of the Dynkin diagram. In particular, invariant differential operators discussed in the paper correspond (in a flat model) to the Dirac...

We consider an elliptic PDE in two variables. As one parameter approaches zero, this PDE collapses to a parabolic one, that is forward parabolic in a part of the domain and backward parabolic in the remainder. Such problems arise naturally in various stochastic models, such as fluid models for data-handling systems and Markovmodulated queues. We employ singular perturbation methods to study the...

In the present paper, stability and convergence properties of linear multistep methods are investigated. The attention is focused on parabolic problems and variable stepsizes. Under weak assumptions on the method and the stepsize sequence an asymptotic stability result is shown. Further, stability bounds for linear nonautonomous parabolic problems with Hölder continuous operator are given. With...

We establish a Lipschitz stability estimate for the inverse problem consisting in the determination of the coe cient σ(t), appearing in a Dirichlet initial-boundary value problem for the parabolic equation ∂tu − ∆xu + σ(t)f(x)u = 0, from Neumann boundary data. We extend this result to the same inverse problem when the previous linear parabolic equation is changed to the semi-linear parabolic eq...

We consider an elliptic PDE in two variables. As one parameter approaches zero, this PDE collapses to a parabolic one, that is forward parabolic in a part of the domain and backward parabolic in the remainder. Such problems arise naturally in various stochastic models, such as fluid models for data-handling systems and Markov-modulated queues. We employ singular perturbation methods to study th...

‎We first give bounds for domains where the unitarizabile subquotients can show up in the parabolically induced representations of classical $p$-adic groups‎. ‎Roughly‎, ‎they can show up only if the‎ ‎central character of the inducing irreducible cuspidal representation is dominated by the‎ ‎square root of the modular character of the minimal parabolic subgroup‎. ‎For unitarizable subquotients...