Gabor Frames in $${\ell }^2({\mathbb {Z}})$$ ℓ 2 ( Z ) and Linear Dependence

We prove that an overcomplete Gabor frame in \({\ell }^2({\mathbb {Z}})\) generated by a finitely supported sequence is always linearly dependent. This is a particular case of a general result about linear dependence versus independence for Gabor systems in \({\ell }^2({\mathbb {Z}})\) with modulation parameter 1 / M and translation parameter N for some \(M,N\in {\mathbb {N}},\) and generated by a finite sequence g in \({\ell }^2({\mathbb {Z}})\) with K nonzero entries.

     

Frame properties of systems arising via iterated actions of operators

Motivated by recent progress in dynamical sampling we prove that every frame which is norm-bounded below can be represented as a finite union of sequences ${\{{(T_j)}^n{\phi }_j\}}_{n=0}^{\infty },j=1,\dots ,J$ for some bounded operators $T_j$ and elements ${\phi }_j$ in the underlying Hilbert space. The result is optimal, in the sense that it turns out to be problematic to replace the collection of generators ${\phi }_1,\dots ,{\phi }_J$ by a singleton: indeed, for linearly independent frames we prove that we can represent the frame in terms of just one system ${\{T^n\phi \}}_{n=0}^{\infty }$, but unfortunately this representation often forces the operator T to be unbounded. Several examples illustrate the connection of the results to typical frames like Gabor frames and wavelet frames, as well as generic constructions in arbitrary separable Hilbert spaces.     

Stereocontrolled Organocatalytic Strategy for the Synthesis of Optically Active 2,3‐Disubstituted cis‐2,3‐Dihydrobenzofurans

An intramolecular, organocatalyzed Michael addition has been developed to obtain biologically important 2,3‐disubstituted cis‐2,3‐dihydrobenzofurans. By using mandelic acid salts of primary aminocatalysts, derived from cinchona alkaloids, the intramolecular cyclization reaction has been developed to proceed in high yield, with moderate to good diastereoselectivity, and up to 99 % ee. Based on the absolute configuration of the formed 2,3‐disubstituted‐cis‐2,3‐dihydrobenzofurans and by considering the observed substrate scope restrictions, a mechanistic rationalization has been presented.     

Graphene-based sensors for detection of heavy metals in water: a review

Graphene (G) is attracting significant attention because of its unique physical and electronic properties. The production of graphene through the reduction of graphene oxide (GO) is a low-cost method. The reduction of GO can further lead to electrically conductive reduced GO. These graphene-based nanomaterials are attractive for high-performance water sensors due to their unique properties, such as high specific surface areas, high electron mobilities, and exceptionally low electronic noise. Because of potential risks to the environment and human health arising from heavy-metal pollution in water, G-/GO-based water sensors are being developed for rapid and sensitive detection of heavy-metal ions. In this review, a general introduction to graphene and GO properties, as well as their syntheses, is provided. Recent advances in optical, electrochemical, and electrical detection of heavy-metal ions using graphene or GO are then highlighted. Finally, challenges facing G/GO-based water sensor development and outlook for future research are discussed.     

Formation and Structure of Fluorescent Silver Nanoclusters at Interfacial Binding Sites Facilitating Oligomerization of DNA Hairpins

Fluorescent, DNA‐stabilized silver nanoclusters (DNA‐AgNCs) are applied in a range of applications within nanoscience and nanotechnology. However, their diverse optical properties, mechanism of formation, and aspects of their composition remain unexplored, making the rational design of nanocluster probes challenging. Herein, a synthetic procedure is described for obtaining a high yield of emissive DNA‐AgNCs with a C‐loop hairpin DNA sequence, with subsequent purification by size‐exclusion chromatography (SEC). Through a combination of optical spectroscopy, gel electrophoresis, inductively coupled plasma mass spectrometry (ICP‐MS), and small‐angle X‐ray scattering (SAXS) in conjunction with the systematic study of various DNA sequences, the low‐resolution structure and mechanism of the formation of AgNCs were investigated. Data indicate that fluorescent DNA‐AgNCs self‐assemble by a head‐to‐head binding of two DNA hairpins, bridged by a silver nanocluster, resulting in the modelling of a dimeric structure harboring an Ag₁₂ cluster.     

Long-term optimal operation of series parallel reservoirs for critical period with specified monthly generation and average monthly storage

We present in this paper an efficient approach for solving the problem of planning the long-term (multiyear) operation of a multireservoir hydroelectric power system for the critical period with a monthly variable load. This load is equal to a certain percentage of the total generation at the end of the year, subject to satisfying a number of constraints on the hydrosystem, using the minimum norm formulation.The proposed method is efficient in computing time and in calculating the total expected benefits from the system during the critical period. Numerical results are reported for a real system in operation consisting of two rivers. Each river has two series reservoirs.This work was supported by the Natural Science and Engineering Research Council of Canada, Grant No. A4146.     

Perturbations of Banach Frames and Atomic Decompositions

Banach frames and atomic decompositions are sequences that have basis-like properties but which need not be bases. In particular, they allow elements of a Banach space to be written as linear combinations of the frame or atomic decomposition elements in a stable manner. In this paper we prove several functional — analytic properties of these decompositions, and show how these properties apply to Gabor and wavelet systems. We first prove that frames and atomic decompositions are stable under small perturbations. This is inspired by corresponding classical perturbation results for bases, including the Paley — Wiener basis stability criteria and the perturbation theorem el kato. We introduce new and weaker conditions which ensure the desired stability. We then prove quality properties of atomic decompositions and consider some consequences for Hilbert frames. Finally, we demonstrate how our results apply in the practical case of Gabor systems in weighted L² spaces. Such systems can form atomic decompositions for L²_w(IR), but cannot form Hilbert frames but L²_w(IR) unless the weight is trivial.     

Optimum load scheduling of nuclear-hydro-thermal power systems

This paper considers the problem of short-term optimal operation of nuclear-hydro-thermal electric power systems. The solution is obtained by use of a functional analytic optimization technique that employs the minimum norm formulation.A power system with an arbitrary number of generating stations is considered. The limited flexibility exhibited by the thermal nuclear reactors, when operating in a load-following mode, is accounted for by means of a model of the xenon concentration in their cores. The nonlinear effects induced by trapezoidal water reservoirs and the time delay of the water flow between upstream and downstream hydroplants is taken into consideration as well.A two-level iterative scheme of the feasible type is proposed for implementing the optimal solution.This work was supported in part by the National Research Council of Canada, Grant No. A-4146.