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Abstract The duality principle for group representations developed in Dutkay et al. (J Funct Anal 257:1133–1143, 2009), Han and Larson (Bull Lond Math Soc 40:685–695, 2008) exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties in Gabor analysis: the biorthogonality and the fundamental identity of Gabor analysis. The main purpose of this this paper is to show that these two fundamental properties remain to be true for general projective unitary group representations. Moreover, we also present a general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pair of group representations. Applying it to the Gabor representations, we obtain that $$\{\pi _{\Lambda }(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda }(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ ( m , n ) g k } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})\oplus \cdots \oplus L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) ⊕ ⋯ ⊕ L 2 ( R d ) if and only if $$\cup _{i=1}^{k}\{\pi _{\Lambda ^{o}}(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ o ( m , n ) g i } m , n ∈ Z d is a Riesz sequence, and $$\cup _{i=1}^{k} \{\pi _{\Lambda }(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ ( m , n ) g i } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) if and only if $$\{\pi _{\Lambda ^{o}}(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda ^{o}}(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ o ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ o ( m , n ) g k } m , n ∈ Z d is a Riesz sequence, where $$\pi _{\Lambda }$$ π Λ and $$\pi _{\Lambda ^{o}}$$ π Λ o is a pair of Gabor representations restricted to a time–frequency lattice $$\Lambda $$ Λ and its adjoint lattice $$\Lambda ^{o}$$ Λ o in $${\mathbb {R}}\,^{d}\times {\mathbb {R}}\,^{d}$$ R d × R d . 

Because electron transfer reactions are fundamental to life processes, such as respiration, vision, and energy catabolism, it is critically important to understand the relationship between functional states of individual redox enzymes and the macroscopically observed phenotype, which results from averaging over all copies of the same enzyme. To address this problem, we have developed a new technology, based on a bifunctional nanoelectrochemical-nanophotonic architecture - the electrochemical zero mode waveguide (E-ZMW) - that can couple biological electron transfer reactions to luminescence, making it possible to observe single electron transfer events in redox enzymes. Here we describe E-ZMW architectures capable of supporting potential-controlled redox reactions with single copies of the oxidoreductase enzyme, glutathione reductase, GR, and extend these capabilities to electron transfer events where reactive oxygen species are synthesized within the  100 zL volume of the nanopore. 

Photoresponsive hydrogels have become invaluable 3D culture matrices for mimicking aspects of the extracellular matrix. Recent efforts have focused on using ultraviolet (UV) light exposure and multifunctional macromers to induce secondary hydrogel crosslinking and dynamic matrix stiffening in the presence of cells. This contribution reports the design of a novel yet simple dynamic poly(ethylene glycol)–peptide hydrogel system through flavin mononucleotide (FMN) induced di‐tyrosine crosslinking. These di‐tyrosine linkages effectively increase hydrogel crosslinking density and elastic modulus. In addition, the degree of stiffening in hydrogels at a fixed PEG macromer content can be readily tuned by controlling FMN concentration or the number of tyrosine residues built‐in to the peptide linker. Furthermore, tyrosine‐bearing pendant biochemical motifs can be spatial‐temporally patterned in the hydrogel network via controlling light exposure through a photomask. The visible light and FMN‐induced tyrosine dimerization process produces a cytocompatible and physiologically relevant degree of stiffening, as shown by changes of cell morphology and gene expression in pancreatic cancer and stromal cells. This new dynamic hydrogel scheme should be highly desirable for researchers seeking a photoresponsive hydrogel system without complicated chemical synthesis and secondary UV light irradiation.