Search

We study the spectral property of a class of Sierpinski-type self-affine measures μ M , D ( ⋅ ) = 1 3 ∑ d ∈ D μ M , D ( M ( ⋅ ) − d ) on R 2 , where M = [ ρ 1 − 1 a 0 ρ 2 − 1 ] is a real upper triangular expanding matrix and D = { ( 0 0 ) , ( d 1 0 ) , ( d 2 d 3 ) } is a three-element real digit set with d 1 d 3 ≠ 0 . A necessary and sufficient condition for μ M , D to be a spectral measure is established.

Let D n = { 0 , a n , b n } = { 0 , 1 , 2 } ( m o d 3 ) , p n ∈ 3 Z + , n ≥ 1 , satisfy sup n ≥ 1 ⁡ max ⁡ { | a n | , | b n | } p n ∞ . It is well-known that there exists a unique Borel probability measure μ { p n } , { D n } generated by the following infinite convolution product μ { p n } , { D n } = δ p 1 − 1 D 1 ⁎ δ ( p 1 p 2 ) − 1 D 2 ⁎ ⋯ in the weak convergence. In this paper, we give some conditions to ensure that there exists a discrete set Λ such that the exponential function system { e 2 π i λ x } λ ∈ Λ forms an orthonormal basis for L 2 ( μ { p n } , { D n } ) .

Let [Formula: see text] be an arithmetic digit set for each [Formula: see text], where [Formula: see text], and let [Formula: see text] be a sequence of integers larger than 1. In this paper, we prove that the Moran measure [Formula: see text] generated by infinite convolution of finite atomic measures [Formula: see text] is a spectral measure if [Formula: see text] and [Formula: see text].

Let [Formula: see text] be a prime and [Formula: see text] be a sequence of finite arithmetic digit sets in [Formula: see text] with [Formula: see text] uniformly bounded, and let [Formula: see text] be the discrete probability measure on the finite set [Formula: see text] with equal distribution. For [Formula: see text], the infinite Bernoulli convolution [Formula: see text] converges weakly to a Borel probability measure (Moran measure). In this paper, we study the existence of exponential orthonormal basis for [Formula: see text].

Let [Formula: see text],[Formula: see text][Formula: see text],[Formula: see text][Formula: see text], satisfy [Formula: see text]. Let [Formula: see text] be the infinite convolution of probability measures with finite support and equal distribution. In this paper, we show that if [Formula: see text], then there exists a discrete set [Formula: see text] such that [Formula: see text] is an orthonormal basis for [Formula: see text].

Long-term synaptic plasticity is the cellular and molecular basis of learning and memory, and the maintenance of their late phases requires both transcription and translation. How targeting gene products shipped from cell body to the few activated synapses in a vast dendritic tree is not yet fully understood. The recent researches demonstrated that the induction of long-term synaptic plasticity could mark an activated synapse by a synaptic tag to capture and utilize synaptic plasticity-related transcriptional products that then serve to stabilize early to late phase of long-term synaptic plasticity. In this review, we outline the advancement in research of synaptic tagging.