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Let R R be an expanding matrix with integer entries, and let B , L B,L be finite integer digit sets so that ( R , B , L ) (R,B,L) form a Hadamard triple on R d {\mathbb {R}}^d in the sense that the matrix 1 | det R | [ e 2 π i ⟨ R − 1 b , ℓ ⟩ ] ℓ ∈ L , b ∈ B \begin{equation*} \frac {1}{\sqrt {|\det R|}}\left [e^{2\pi i \langle R^{-1}b,\ell \rangle }\right ]_{\ell \in L,b\in B} \end{equation*} is unitary. We prove that the associated fractal self-affine measure μ = μ ( R , B ) \mu = \mu (R,B) obtained by an infinite convolution of atomic measures μ ( R , B ) = δ R − 1 B ∗ δ R − 2 B ∗ δ R − 3 B ∗ ⋯ \begin{equation*} \mu (R,B) = \delta _{R^{-1} B}\ast \delta _{R^{-2}B}\ast \delta _{R^{-3}B}\ast \cdots \end{equation*} is a spectral measure, i.e., it admits an orthonormal basis of exponential functions in L 2 ( μ ) L^2(\mu ) . This settles a long-standing conjecture proposed by Jorgensen and Pedersen and studied by many other authors. Moreover, we also show that if we relax the Hadamard triple condition to an almost-Parseval-frame condition, then we obtain a sufficient condition for a self-affine measure to admit Fourier frames.

The parasympathetic activity before falling asleep and the sympathetic activity before waking up change coincidentally with EEG frequency, and may respectively contain the messages of sleeping and waking drives.

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BackgroundPerinatal brain injury is the leading cause of subsequent neurological disability in both term and preterm baby. Glutamate excitotoxicity is one of the major factors involved in perinatal hypoxic-ischemic encephalopathy (HIE). Glutamate transporter GLT1, expressed mainly in mature astrocytes, is the major glutamate transporter in the brain. HIE induced excessive glutamate release which is not reuptaked by immature astrocytes may induce neuronal damage. Compounds, such as ceftriaxone, that enhance the expression of GLT1 may exert neuroprotective effect in HIE.MethodsWe used a neonatal rat model of HIE by unilateral ligation of carotid artery and subsequent exposure to 8% oxygen for 2 hrs on postnatal day 7 (P7) rats. Neonatal rats were administered three dosages of an antibiotic, ceftriaxone, 48 hrs prior to experimental HIE. Neurobehavioral tests of treated rats were assessed. Brain sections from P14 rats were examined with Nissl and immunohistochemical stain, and TUNEL assay. GLT1 protein expression was evaluated by Western blot and immunohistochemistry.ResultsPre-treatment with 200 mg/kg ceftriaxone significantly reduced the brain injury scores and apoptotic cells in the hippocampus, restored myelination in the external capsule of P14 rats, and improved the hypoxia-ischemia induced learning and memory deficit of P23-24 rats. GLT1 expression was observed in the cortical neurons of ceftriaxone treated rats.ConclusionThese results suggest that pre-treatment of infants at risk for HIE with ceftriaxone may reduce subsequent brain injury.

The spectral set conjecture, also known as the Fuglede conjecture, asserts that every bounded spectral set is a tile and vice versa. While this conjecture remains open on ${\mathbb R}^1$, there are many results in the literature that discuss the relations among various forms of the Fuglede conjecture on ${\mathbb Z}_n$, ${\mathbb Z}$ and ${\mathbb R}^1$ and also the seemingly stronger universal tiling (spectrum) conjectures on the respective groups. In this paper, we clarify the equivalences between these statements in dimension one. In addition, we show that if the Fuglede conjecture on ${\mathbb R}^1$ is true, then every spectral set with rational measure must have a rational spectrum. We then investigate the Coven-Meyerowitz property for finite sets of integers, introduced in \cite{CoMe99}, and we show that if the spectral sets and the tiles in ${\mathbb Z}$ satisfy the Coven-Meyerowitz property, then both sides of the Fuglede conjecture on ${\mathbb R}^1$ are true