Characterization and computation of canonical tight windows for Gabor frames

Let $(g_{nm})_{n,m\in Z}$ be a Gabor frame for $L_2(R)$ for given window $g$. We show that the window $h^0=S^{-1/2} g$ that generates the canonically associated tight Gabor frame minimizes $\|g-h\|$ among all windows $h$ generating a normalized tight Gabor frame. We present and prove versions of this result in the time domain, the frequency domain, the time-frequency domain, and the Zak transform domain, where in each domain the canonical $h^0$ is expressed using functional calculus for Gabor frame operators. Furthermore, we derive a Wiener-Levy type theorem for rationally oversampled Gabor frames. Finally, a Newton-type method for a fast numerical calculation of \ho is presented. We analyze the convergence behavior of this method and demonstrate the efficiency of the proposed algorithm by some numerical examples

Scaling regimes and critical dimensions in the Kardar-Parisi-Zhang problem

We study the scaling regimes for the Kardar-Parisi-Zhang equation with noise correlator R(q) ~ (1 + w q^{-2 \rho}) in Fourier space, as a function of \rho and the spatial dimension d. By means of a stochastic Cole-Hopf transformation, the critical and correction-to-scaling exponents at the roughening transition are determined to all orders in a (d - d_c) expansion. We also argue that there is a intriguing possibility that the rough phases above and below the lower critical dimension d_c = 2 (1 + \rho) are genuinely different which could lead to a re-interpretation of results in the literature.Comment: Latex, 7 pages, eps files for two figures as well as Europhys. Lett. style files included; slightly expanded reincarnatio

On Lerch's transcendent and the Gaussian random walk

Let $X_1,X_2,...$ be independent variables, each having a normal distribution with negative mean $-\beta<0$ and variance 1. We consider the partial sums $S_n=X_1+...+X_n$, with $S_0=0$, and refer to the process $\{S_n:n\geq0\}$ as the Gaussian random walk. We present explicit expressions for the mean and variance of the maximum $M=\max\{S_n:n\geq0\}.$ These expressions are in terms of Taylor series about $\beta=0$ with coefficients that involve the Riemann zeta function. Our results extend Kingman's first-order approximation [Proc. Symp. on Congestion Theory (1965) 137--169] of the mean for $\beta\downarrow0$. We build upon the work of Chang and Peres [Ann. Probab. 25 (1997) 787--802], and use Bateman's formulas on Lerch's transcendent and Euler--Maclaurin summation as key ingredients.Comment: Published at http://dx.doi.org/10.1214/105051606000000781 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Short- and long-term experience in pulmonary vein segmental ostial ablation for paroxysmal atrial fibrillation*

Introduction: Segmental ostial pulmonary vein isolation (PVI) is considered a potentially curative therapeutic approach in the treatment of paroxysmal atrial fibrillation (PAF). There is only limited data available on the long-term effect of this procedure. Methods: Patients (Pts) underwent a regular clinical follow up visit at 3, 6 and 24 months after PVI. Clinical success was classified as complete (i.e. no arrhythmia recurrences, no antiarrhythmic drug), partial (i.e. no/only few recurrences, on drug) or as a failure (no benefit). The clinical responder rate (CRR) was determined by combining complete and partial success. Results: 117 patients (96 male, 21 female), aged 51±11 years (range 25 to 73) underwent a total of 166 procedures (1.4/patient) in 2-4 pulmonary veins (PV). 115 patients (98%) had AF, 2 patients presented with regular PV atrial tachycardia. ,109/115 patients. exhibited PAF as the primary arrhythmia (versus persistent AF). A total of 113 patients with PVI in the years 2001 to 2003 were evaluated for their CRR after 6 (3) months. A single intervention was carried out in 63 patients (55.8%), two interventions were performed in 45 patients (39.8%) and three interventions in 5 patients (4.4%). The clinical response demonstrated a complete success of 52% (59 patients), a partial success of 26% (29 patients) and a failure rate of 22% (25 patients), leading to a CRR of 78% (88 patients). Ostial PVI in all 4 PVs exhibited a tendency towards higher curative success rates (54% versus 44% in patients with 3 PVs ablated for the 6 month follow up). Long-term clinical outcome was evaluated in 39 patients with an ablation attempt at 3 PVs only (excluding the right inferior PV in our early experience) and a mean clinical follow up of 21±6 months. At this point in time the success rate was 41% (complete, 16 patients) and 21% (partial, 8 patients), respectively, adding up to a CRR of 62% (24 patients). In total, 20 patients (17.1%) had either a single or 2 (3 patients, 2.6%) complications independent of the number of procedures performed with PV stenosis as the leading cause (7.7%). Conclusion: The CRR of patients with medical refractory PAF in our patient cohort is 78% at the 6 month follow up. PV stenosis is the main cause for procedure-related complications. Ablation of all 4 PV exhibits a tendency towards higher complete success rates despite equal CRR. Calculation of the clinical response after a mid- to long-term follow of 21±6 months in those patients with an ostial PVI in only 3 pulmonary veins (sparing the right inferior PV) shows a further reduction to 62%, exclusively caused by a drop in patients with a former partial success. To evaluate the long-term clinical benefit of segmental ostial PVI in comparison with other ablation techniques, more extended follow up periods are mandatory, including a larger study cohort and a detailed description of procedural parameters

Successful paediatric HIV treatment in rural primary care in Africa

&lt;p&gt;Objective: Clinical outcomes of HIV-infected children on antiretroviral treatment (ART) in a decentralised, nurse/counsellor-led programme.&lt;/p&gt; &lt;p&gt;Design: Clinical cohort.&lt;/p&gt; &lt;p&gt;Setting: KwaZulu-Natal, South Africa.&lt;/p&gt; &lt;p&gt;Patients: HIV-infected children aged &lt;= 15 years on ART, June 2004-2008.&lt;/p&gt; &lt;p&gt;Main outcome measures: Survival according to baseline characteristics including age, WHO clinical stage, haemoglobin and CD4%, was assessed in Kaplan-Meier analyses. Hazard ratios for mortality were estimated using Cox proportional hazards regression and changes in laboratory parameters and weight-for-age z scores after 6-12 months' treatment were calculated.&lt;/p&gt; &lt;p&gt;Results: 477 HIV-infected children began ART at a median age of 74 months (range 4-180), median CD4 count (CD4%) of 433 cells/mm(3) (17%) and median HIV viral load of log 4.2 copies/ml; 105 (22%) were on treatment for tuberculosis and 317 (76.6%) were WHO stage 3/4. There were significant increases after ART initiation in CD4% (17% vs 22%; p&lt;0.001), haemoglobin (9.9 vs 11.7 g/l; p &lt;= 0.001) and albumin (30 vs 36 g/l; p &lt;= 0.001). 32 (6.7%) children died over 732 child-years of follow-up (43.7 deaths/1000 child-years; 95% CI 32.7 to 58.2), 17 (53.1%) within 90 days of treatment initiation; median age of death was 84 (IQR 10-181) months. Children with baseline haemoglobin &lt;= 8 g/l were more likely to die (adjusted HR 4.5; 95% CI 1.6 to 12.3), as were those aged &lt;18 months compared with &gt;60 months (adjusted HR 3.2; 95% CI 1.2 to 9.1).&lt;/p&gt; &lt;p&gt;Conclusions Good clinical outcomes in HIV-infected children on ART are possible in a rural, decentralised service. Few young children are on ART, highlighting the urgent need to identify HIV-exposed infants.&lt;/p&gt