Approximation orders of shift-invariant subspaces of W2s(Rd)W^s_2({\Bbb R}^d)

We extend the existing theory of approximation orders provided by shift-invariant subspaces of $L_2$ to the setting of Sobolev spaces, provide treatment of $L_2$ cases that have not been covered before, and apply our results to determine approximation order of solutions to a refinement equation with a higher-dimensional solution space.Comment: 49 page

Long-Term Follow-Up In Paroxysmal Atrial Fibrillation Patients With Documented Isolated Trigger

AimsSupraventricular tachycardias may trigger atrial fibrillation (AF). The aim of the study was to evaluate the prevalence of supraventricular tachycardia (SVT) inducibility in patients referred for AF ablation and to evaluate the effects of SVT ablation on AF recurrences.Methods and results249 patients (mean age: 54 ± 14 years) referred for paroxysmal AF ablation were studied. In all patients, only AF relapses had been documented in the clinical history. 47 patients (19%; mean age: 42 ± 11 years) had inducible SVT during the electrophysiological study and underwent an ablation targeted only at SVT suppression. Ablation was successful in all 47 patients. The ablative procedures were: 11 slow-pathway ablations for atrioventricular nodal re-entrant tachycardia; 6 concealed accessory pathway ablations for atrioventricular re-entrant tachycardia; 17 focal ectopic atrial tachycardia ablations; 13 with only one arrhythmogenic pulmonary vein. No recurrences of SVT were observed during the follow-up (32 ± 18 months). 4 patients (8.5%) showed recurrence of at least one episode of AF. Patients with inducible SVT had less structural heart disease and were younger than those without inducible SVT.ConclusionA significant proportion of candidates for AF ablation are inducible for an SVT. SVT ablation showed a preventive effect on AF recurrences. Those patients should be selected for simpler ablation procedures tailored only to the triggering arrhythmia suppression

Analysis of the archetypal functional equation in the non-critical case

We study the archetypal functional equation of the form $y(x)=\iint_{R^2} y(a(x-b))\,\mu(da,db)$ ($x\in R$), where $\mu$ is a probability measure on $R^2$; equivalently, $y(x)=E\{y(\alpha (x-\beta))\}$, where $E$ is expectation with respect to the distribution $\mu$ of random coefficients $(\alpha,\beta)$. Existence of non-trivial (i.e. non-constant) bounded continuous solutions is governed by the value $K:=\iint_{R^2}\ln |a| \mu(da,db) =E \{\ln |\alpha|\}$; namely, under mild technical conditions no such solutions exist whenever $K0$ (and $\alpha>0$) then there is a non-trivial solution constructed as the distribution function of a certain random series representing a self-similar measure associated with $(\alpha,\beta)$. Further results are obtained in the supercritical case $K>0$, including existence, uniqueness and a maximum principle. The case with $P(\alpha0$ is drastically different from that with $\alpha>0$; in particular, we prove that a bounded solution $y(\cdot)$ possessing limits at $\pm\infty$ must be constant. The proofs employ martingale techniques applied to the martingale $y(X_n)$, where $(X_n)$ is an associated Markov chain with jumps of the form $x\rightsquigarrow\alpha (x-\beta)$

Stability of Localized Operators

Let $\ell^p, 1\le p\le \infty$, be the space of all $p$-summable sequences and $C_a$ be the convolution operator associated with a summable sequence $a$. It is known that the $\ell^p$- stability of the convolution operator $C_a$ for different $1\le p\le \infty$ are equivalent to each other, i.e., if $C_a$ has $\ell^p$-stability for some $1\le p\le \infty$ then $C_a$ has $\ell^q$-stability for all $1\le q\le \infty$. In the study of spline approximation, wavelet analysis, time-frequency analysis, and sampling, there are many localized operators of non-convolution type whose stability is one of the basic assumptions. In this paper, we consider the stability of those localized operators including infinite matrices in the Sj\"ostrand class, synthesis operators with generating functions enveloped by shifts of a function in the Wiener amalgam space, and integral operators with kernels having certain regularity and decay at infinity. We show that the $\ell^p$- stability (or $L^p$-stability) of those three classes of localized operators are equivalent to each other, and we also prove that the left inverse of those localized operators are well localized

Activation of m1 muscarinic acetylcholine receptor induces surface transport of KCNQ channel via CRMP-2 mediated pathway

Neuronal excitability is strictly regulated by various mechanisms, including modulation of ion channel activity and trafficking. Stimulation of m1 muscarinic acetylcholine receptor (also known as CHRM1) increases neuronal excitability by suppressing the M-current generated by the Kv7/KCNQ channel family. We found that m1 muscarinic acetylcholine receptor stimulation also triggers surface transport of KCNQ subunits. This receptor-induced surface transport was observed with KCNQ2 as well as KCNQ3 homomeric channels, but not with Kv3.1 channels. Deletion analyses identified that a conserved domain in a proximal region of the N-terminal tail of KCNQ protein is crucial for this surface transport - the translocation domain. Proteins that bind to this domain were identified as alpha-and beta-tubulin and collapsin response mediator protein 2 (CRMP-2; also known as DPYSL2). An inhibitor of casein kinase 2 (CK2) reduced tubulin binding to the translocation domain, whereas an inhibitor of glycogen synthase kinase 3 (GSK3) facilitated CRMP-2 binding to the translocation domain. Consistently, treatment with the GSK3 inhibitor enhanced receptor-induced KCNQ2 surface transport. M-current recordings from neurons showed that treatment with a GSK3 inhibitor shortened the duration of muscarinic suppression and led to over-recovery of the M-current. These results suggest that m1 muscarinic acetylcholine receptor stimulates surface transport of KCNQ channels through a CRMP-2-mediated pathway.National Institutes of Health [R01NS067288, R01GM074830]; National Natural Science Foundation of China [81473235, 81020108031]SCI(E)[email protected]