Error Reduction by Parallel Repetition—A Negative Result

We show that no xed number of parallel repetitions suces in order to reduce the error in two-prover one-round proof systems from one constant to another. Our results imply that the recent bounds proven by Ran Raz, showing that the number of rounds that suffice is inversely proportional to the answer length, are nearly best possible.* Incumbent of the Joseph and Celia Reskin Career Development Chair. Supported in part by grant No. MQT 300 from the International Science Foundation and by an AMS-FSU grant.     

Weighted norm inequalities for integral operators

We consider a large class of positive integral operators acting on functions which are defined on a space of homogeneous type with a group structure. We show that any such operator has a discrete (dyadic) version which is always essentially equivalent in norm to the original operator. As an application, we study conditions of ``testing type,' like those initially introduced by E. Sawyer in relation to the Hardy-Littlewood maximal function, which determine when a positive integral operator satisfies two-weight weak-type or strong-type estimates. We show that in such a space it is possible to characterize these estimates by testing them only over ``cubes'. We also study some pointwise conditions which are sufficient for strong-type estimates and have applications to solvability of certain nonlinear equations.

     

Workflow to Investigate Subtle Differences in Wine Volatile Metabolome Induced by Different Root Systems and Irrigation Regimes

To allow for a broad survey of subtle metabolic shifts in wine caused by rootstock and irrigation, an integrated metabolomics-based workflow followed by quantitation was developed. This workflow was particularly useful when applied to a poorly studied red grape variety cv. Chambourcin. Allowing volatile metabolites that otherwise may have been missed with a targeted analysis to be included, this approach allowed deeper modeling of treatment differences which then could be used to identify important compounds. Wines produced on a per vine basis, over two years, were analyzed using SPME-GC-MS/MS. From the 382 and 221 features that differed significantly among rootstocks in 2017 and 2018, respectively, we tentatively identified 94 compounds by library search and retention index, with 22 confirmed and quantified using authentic standards. Own-rooted Chambourcin differed from other root systems for multiple volatile compounds with fewer differences among grafted vines. For example, the average concentration of β-Damascenone present in own-rooted vines (9.49 µg/L) was significantly lower in other rootstocks (8.59 µg/L), whereas mean Linalool was significantly higher in 1103P rootstock compared to own-rooted. β-Damascenone was higher in regulated deficit irrigation (RDI) than other treatments. The approach outlined not only was shown to be useful for scientific investigation, but also in creating a protocol for analysis that would ensure differences of interest to the industry are not missed.     

Angular and energy dependences of cross sections for deuteron electrodisintegration at subbarrier energies

A method is proposed for analyzing a large set of spectra calculated for the Coulomb breakup of deuterons. The regularities in the behavior of the respective cross section as a function of the emission angles and energies of particles constituting the deuteron are studied on the basis of this method. The deuteron-breakup reaction on lead nuclei at an energy of E _d = 8 MeV is considered by way of example. It is shown that the approximate Landau-Lifshitz formulas for this process are unable to predict some of its features. An experiment is proposed for studying the physical nature of the Nemets effect.     

Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry

A hypercomplex manifold is a manifold equipped with a triple of complex structures I, J, K satisfying the quaternionic relations. We define a quaternionic analogue of plurisubharmonic functions on hypercomplex manifolds, and interpret these functions geometrically as potentials of HKT (hyperkähler with torsion) metrics. We prove a quaternionic analogue of A. D. Aleksandrov and ChernLevine-Nirenberg theorems.     

Coherent control of rotational wave-packet dynamics via fractional revivals

We show (i) how the evolution of a wave packet created from an initial thermal ensemble can be controlled by manipulating interferences during the wave packet's fractional revivals and (ii) how the wave-packet evolution can be mapped onto the dynamics of a few-state system, where the number of states is determined by the amount of information one wants to track about the wave packet in the phase space. We illustrate our approach by (i) switching off and on field-free molecular axis alignment induced by a strong laser pulse and (ii) converting alignment into field-free orientation, starting with rotationally cold or hot systems.     

Tunable optimal compression of ultrabroadband pulses by cross-phase modulation

We show how cross-phase modulation between two pulses, combined with optimal pulse shaping at the input of a dielectric medium, can be used to generate nearly single-cycle pulses that are tunable from the ultraviolet to the mid-infrared at the output of the medium, precompensating for dispersion to all orders.     

Wirtinger numbers and holomorphic symplectic immersions

For any subvariety of a compact holomorphic symplectic Kähler manifold, we define the symplectic Wirtinger number W(X). We show that and the equality is reached if and only if the subvariety is trianalytic, i.e. compatible with the hyperkähler structure on M. For a sequence of immersions of simple holomorphic symplectic manifolds, we show that      

Kapitza-Dirac diffraction without standing waves: diffraction without a grating?

We discuss electron diffraction from two counterpropagating light waves with two different frequencies. We show that, even though these waves do not form a standing wave, electron diffraction similar to the conventional Kapitza-Dirac effect, i.e., scattering on a standing wave, is still possible. The nonlinear response of the electron to the laser fields creates a stationary diffraction grating from which the same electron scatters.