We characterize the completeness and frame/basis property of a union of under-sampled windowed exponentials of the form
$$ {\mathcal{F}}(g): =\bigl\{ e^{2\pi i n x}: n\ge 0\bigr\} \cup \bigl\{ g(x)e^{2\pi i nx}: n< 0\bigr\} $$
for \(L^{2}[-1/2,1/2]\) by the spectra of the Toeplitz operators with the symbol \(g\). Using this characterization, we classify all real-valued functions \(g\) such that \({\mathcal{F}}(g)\) is complete or forms a frame/basis. Conversely, we use the classical non-harmonic Fourier series theory to determine all \(\xi \) such that the Toeplitz operators with the symbol \(e^{2\pi i \xi x}\) is injective or invertible. These results demonstrate an elegant interaction between frame theory of windowed exponentials and Toeplitz operators. Finally, we use our results to answer some open questions in dynamical sampling, and derivative samplings on Paley-Wiener spaces of bandlimited functions.
A series of monodisperse six-armed conjugated starbursts ( Tr1F , Tr2F , and Tr3F ) containing a truxene core and multibranched oligofluorene bridges capped with diphenylamine (DPA) units has been designed, synthesized, and investigated as robust gain media for organic semiconductor lasers (OSLs). The influence of electron-rich DPA end groups on their optoelectronic characteristics has been discussed at length. DPA cappers effectively raise HOMO levels of the starbursts, thus enhancing the hole injection and transport ability. Solution-processed electroluminescence devices based on the resulting six-armed starbursts exhibited efficient deep-blue electroluminescence with clear reduced turn-on voltages (3.2–3.5 V). Moreover, the resulting six-armed molecules showed stabilized electroluminescence and amplified spontaneous emission with low thresholds (27.4–63.9 nJ pulse−1), high net gain coefficients (80.1–101.3 cm−1), and small optical loss (2.6–4.4 cm−1). Distributed feedback OSLs made from Tr3F exhibited a low lasing threshold of 0.31 kW cm−2 (at 465 nm). The results suggest that the construction of truxene-centered six-armed conjugated starbursts with the incorporation of DPA units can effectively enhance EL properties by precisely regulating the HOMO energy levels, and further optimizing their optical gain properties. 相似文献
Water-soluble triplet sensitizer with permethyl-β-cyclodextrin(PMCD) grafting on a Schiff-base Pt(Ⅱ)complex(Pt-2),in which PMCD unit serves as a host for binding the acceptors and the Schiff-base Pt(Ⅱ)complex serves as a triplet sensitizer,was synthesized to investigate the effect of supramolecular complexation and assembly on the triplet-triplet annihilation upconversion emission in water.9,10-Diphenylanthracence(DPA) carboxylate(A-1) and its dimer(A-2) in which two DPA carboxylate were covalently linked with an alkyl chain were synthesized as triplet acceptors which also play a role of guest molecules for PMCD.A-1 and A-2 showed high affinity with PMCD,and A-2 can readily aggregate in water and form micron sized assemblies due to the hydrophobic effect and π-π stacking of anthracene core in A-2.The efficiency of TTA-UC was demonstrated to be enhanced by a synergistic effect of host-guest complexation of Pt-2 with A-2 and the self-aggregation of the acceptor A-2,which facilitated the energy transfer and energy fusion among donor and acceptor. 相似文献
The sensitive and selective detection of dopamine (DA) is very important for the early diagnosis of DA-related diseases. In this study, we reported the colorimetric detection of DA using Ganoderma lucidum polysaccharide (GLP) stabilized platinum nanoclusters (Ptn-GLP NCs). When Pt600-GLP NCs was added, 3,3’,5,5’-tetramethylbenzidine (TMB) was rapidly catalyzed and oxidized to blue oxTMB, indicating the peroxidase-like activity of Pt600-GLP NCs. The catalytic reaction on the substrate TMB followed the Michaelis-Menton kinetics with the ping-pong mechanism. The mechanism of the colorimetric reaction was mainly due to the formation of hydroxyl radical (•OH). Furthermore, the catalytic reaction of Pt600-GLP NCs was used in the colorimetric detection of DA. The linear range for DA was 1–100 μM and the detection limit was 0.66 μM. The sensitive detection of DA using Pt-GLP NCs with peroxidase-like activity offers a simple and practical method that may have great potential applications in the biotechnology field. 相似文献
We consider a generating function of the domino tilings of an Aztec rectangle with several unit squares removed from the boundary. Our generating function involves two statistics: the rank of the tiling and half number of vertical dominoes as in the Aztec diamond theorem by Elkies, Kuperberg, Larsen and Propp. In addition, our work deduces a combinatorial explanation for an interesting connection between the number of lozenge tilings of a semihexagon and the number of domino tilings of an Aztec rectangle. 相似文献
A series of air‐stable spiro‐fused ladder‐type boron(III) compounds has been designed, synthesized, and the electrochemistry and photophysical behavior have been characterized. By simply varying the substituents on the pyridine ring and extending the π‐conjugation of the spiro framework, the emission color of these compounds can be easily fine‐tuned spanning the visible spectrum from blue to red. All compounds exhibit a broad and structureless emission band across the entire visible region, assigned as an intramolecular charge‐transfer transition originating from the thiophene of the spiro framework to the pyridine‐borane moieties. In addition, these compounds demonstrate high photoluminescence quantum yields of up to 0.81 in dichloromethane solution and 0.86 in doped thin films. Some of the compounds have also been employed as emissive materials, in which solution‐processed organic light‐emitting devices (OLEDs) with tunable emission colors spanning the visible spectrum from blue, green to red have been realized, demonstrating the potential applications of these boron compounds in OLEDs. 相似文献
A phenylacetylene macrocycle (PAM) derivative containing triphenylamine as the framework was synthesized in one-step Sonogashira coupling. The photophysical and electrochemical properties were investigated in details. This hexamer shows significant enhancement in two-photon absorption cross-section relative to reported PAM derivatives. 相似文献
To elucidate the structure of a compound is a necessary step for its practical applications. To study the structure and properties of metal sulfide fullerene Sc2S@C88 detected by mass spectrometry, 11 194 isomers of C88 and 33 isomers of Sc2S@C88 were systematically examined by density functional theory calculations. The calculations show that the two lowest‐energy isomers are Sc2S@C88:81 738 (IPR‐35) and Sc2S@C88:81 735 (IPR‐32), followed by Sc2S@C88:81 729 (IPR‐26), Sc2S@C88:81 712 (IPR‐9), and Sc2S@C88:81 733 (IPR‐30). Structural analysis shows that the first two energetically favored isomers are bridged by the third and fifth energetically favored isomers, which can transfer into each other via direct Stone–Wales rotation. The calculations of temperature effect show that the first two favored isomers become dominant forms of Sc2S@C88 with decreasing temperature and may coexist in the soot. This structural convertibility among favored isomers of Sc2S@C88 suggest a hidden rule that birds of a feather flock together in metal sulfide fullerenes. This rule may decrease the range of candidate cages for the structural identification of a metal sulfide fullerene. IR spectra are simulated for helping the future experimental identification of Sc2S@C88. 相似文献