Schatten classes of integration operators on Dirichlet spaces

We address the problem of determining membership in Schatten-Von Neumann ideals S _p of integration operators (T _g f)(z) = ∫ ₀ ^z = ∫ ₀ ^z f(ξ)g′(ξ)dξ acting on Dirichlet type spaces. We also study this problem for multiplication, Hankel and Toeplitz operators. In particular, we provide an extension of Luecking's result on Toeplitz operators [10, p. 347].     

How to measure long-range proton-carbon coupling constants from 1H-selective HSQMBC experiments

Heteronuclear long-range scalar coupling constants (ⁿJ_CH) are a valuable tool for solving problems in organic chemistry and are especially suited for stereochemical and configurational analyses of small molecules and natural products. This tutorial will focus on the step-by-step implementation of several 2D ¹H frequency selective HSQMBC experiments for the easy and accurate measurement of either the magnitude or both the magnitude and the sign of long-range ⁿJ_CH couplings. The performance of these experiments will be showcased with several scenarios in a range of different experimental conditions.     

LR-selHSQMBC: Simultaneous Detection and Quantification of Very Weak Long-Range Heteronuclear NMR Correlations

The optimum detection and accurate measurement of longer-range (⁴J and higher) heteronuclear NMR correlations is described. The magnitude and/or the sign of a wide range of large and small long-range couplings can be simultaneously determined for protonated and non-protonated ¹³C and ¹⁵N nuclei using the LR-selHSQMBC experiment.     

Enantioselective approach to indolizidine and quinolizidine scaffolds. Application to the synthesis of peptide mimics

An enantioselective approach to substituted indolizidine and quinolizidine frameworks has been developed. Key steps of the synthesis are the enantioselective, palladium-catalyzed N-allylation of an imide, the nucleophilic allylation of an acyliminium ion and a ring closing metathesis. This general strategy has been applied to the synthesis of indolizidine peptide mimics, starting from a chiral imide derived from l-aspartic acid. It was observed that the preexisting stereogenic center of this substrate has a moderate influence on the stereoselectivity of the electrophilic allylation, which is mainly determined by the sense of chirality of the catalyst.     

Expanding the Range of Bioorthogonal Tags for Multiplex Stimulated Raman Scattering Microscopy

Multiplex optical detection in live cells is challenging due to overlapping signals and poor signal-to-noise associated with some chemical reporters. To address this, the application of spectral phasor analysis to stimulated Raman scattering (SRS) microscopy for unmixing three bioorthogonal Raman probes within cells is reported. Triplex detection of a metallacarborane using the B−H stretch at 2480–2650 cm⁻¹, together with a bis-alkyne and deuterated fatty acid can be achieved within the cell-silent region of the Raman spectrum. When coupled to imaging in the high-wavenumber region of the cellular Raman spectrum, nine discrete regions of interest can be spectrally unmixed from the hyperspectral SRS dataset, demonstrating a new capability in the toolkit of multiplexed Raman imaging of live cells.     

Optimum spin-state selection for all multiplicities in the acquisition dimension of the HSQC experiment

Most conventional heteronuclear spin-state-selective (S(3)) NMR experiments only work for a specific multiplicity, typically IS spin systems. Here, we introduce a general and efficient IPAP strategy to achieve S(3) editing simultaneously for all multiplicities in the acquisition dimension of the HSQC experiment. Complementary in-phase (HSQC-IP) and anti-phase (HSQC-AP) data are separately recorded with a simple phase exchange of two 90 degrees proton pulses involved in the mixing process of the F2-coupled sensitivity-improved HSQC pulse sequence. Additive and subtractive linear combination of these IP/AP data generates simplified F2-alpha/beta-spin-edited HSQC subspectra for all IS, I(2)S, and I(3)S spin systems and combines enhanced and optimized sensitivity with excellent tolerance to unwanted cross-talk contributions over a considerable range of coupling constants. Practical aspects such as pulse phase settings, transfer efficiency dependence, inter-pulse delay optimization, and percentage of cross-talk are theoretically analyzed and discussed as a function of each I(n)S multiplicity. Particular emphasis on the features associated to spin-editing in diastereotopic I(2)S spin systems and application to the measurement of long-range proton-carbon coupling constants are also provided.     

Extreme value laws in dynamical systems under physical observables

Extreme value theory for chaotic deterministic dynamical systems is a rapidly expanding area of research. Given a system and a real function (observable) defined on its phase space, extreme value theory studies the limit probabilistic laws obeyed by large values attained by the observable along orbits of the system. Based on this theory, the so-called block maximum method is often used in applications for statistical prediction of large value occurrences. In this method, one performs statistical inference for the parameters of the Generalised Extreme Value (GEV) distribution, using maxima over blocks of regularly sampled observable values along an orbit of the system. The observables studied so far in the theory are expressed as functions of the distance with respect to a point, which is assumed to be a density point of the system’s invariant measure. However, at least with respect to the ambient (usually Euclidean) metric, this is not the structure of the observables typically encountered in physical applications, such as windspeed or vorticity in atmospheric models. In this paper we consider extreme value limit laws for observables which are not expressed as functions of the distance (in the ambient metric) from a density point of the dynamical system. In such cases, the limit laws are no longer determined by the functional form of the observable and the dimension of the invariant measure: they also depend on the specific geometry of the underlying attractor and of the observable’s level sets. We present a collection of analytical and numerical results, starting with a toral hyperbolic automorphism as a simple template to illustrate the main ideas. We then formulate our main results for a uniformly hyperbolic system, the solenoid map. We also discuss non-uniformly hyperbolic examples of maps (Hénon and Lozi maps) and of flows (the Lorenz63 and Lorenz84 models). Our purpose is to outline the main ideas and to highlight several serious problems found in the numerical estimation of the limit laws.