ImportanceIn patients with severe aortic valve stenosis at intermediate surgical risk, transcatheter aortic valve replacement (TAVR) with a self-expanding supra-annular valve was noninferior to surgery for all-cause mortality or disabling stroke at 2 years. Comparisons of longer-term clinical and hemodynamic outcomes in these patients are limited.ObjectiveTo report prespecified secondary 5-year outcomes from the Symptomatic Aortic Stenosis in Intermediate Risk Subjects Who Need Aortic Valve Replacement (SURTAVI) randomized clinical trial.Design, Setting, and ParticipantsSURTAVI is a prospective randomized, unblinded clinical trial. Randomization was stratified by investigational site and need for revascularization determined by the local heart teams. Patients with severe aortic valve stenosis deemed to be at intermediate risk of 30-day surgical mortality were enrolled at 87 centers from June 19, 2012, to June 30, 2016, in Europe and North America. Analysis took place between August and October 2021.InterventionPatients were randomized to TAVR with a self-expanding, supra-annular transcatheter or a surgical bioprosthesis.Main Outcomes and MeasuresThe prespecified secondary end points of death or disabling stroke and other adverse events and hemodynamic findings at 5 years. An independent clinical event committee adjudicated all serious adverse events and an independent echocardiographic core laboratory evaluated all echocardiograms at 5 years.ResultsA total of 1660 individuals underwent an attempted TAVR (n = 864) or surgical (n = 796) procedure. The mean (SD) age was 79.8 (6.2) years, 724 (43.6%) were female, and the mean (SD) Society of Thoracic Surgery Predicted Risk of Mortality score was 4.5% (1.6%). At 5 years, the rates of death or disabling stroke were similar (TAVR, 31.3% vs surgery, 30.8%; hazard ratio, 1.02 [95% CI, 0.85-1.22]; P = .85). Transprosthetic gradients remained lower (mean [SD], 8.6 [5.5] mm Hg vs 11.2 [6.0] mm Hg; P < .001) and aortic valve areas were higher (mean [SD], 2.2 [0.7] cm2 vs 1.8 [0.6] cm2; P < .001) with TAVR vs surgery. More patients had moderate/severe paravalvular leak with TAVR than surgery (11 [3.0%] vs 2 [0.7%]; risk difference, 2.37% [95% CI, 0.17%- 4.85%]; P = .05). New pacemaker implantation rates were higher for TAVR than surgery at 5 years (289 [39.1%] vs 94 [15.1%]; hazard ratio, 3.30 [95% CI, 2.61-4.17]; log-rank P < .001), as were valve reintervention rates (27 [3.5%] vs 11 [1.9%]; hazard ratio, 2.21 [95% CI, 1.10-4.45]; log-rank P = .02), although between 2 and 5 years only 6 patients who underwent TAVR and 7 who underwent surgery required a reintervention.Conclusions and RelevanceAmong intermediate-risk patients with symptomatic severe aortic stenosis, major clinical outcomes at 5 years were similar for TAVR and surgery. TAVR was associated with superior hemodynamic valve performance but also with more paravalvular leak and valve reinterventions.

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Equiangular tight frames (ETFs) and biangular tight frames (BTFs) -sets of unit vectors with basis-like properties whose pairwise absolute inner products admit exactly one or two values, respectivelyare useful for many applications. A well-understood class of ETFs are those which manifest as harmonic frames -vector sets defined in terms of the characters of finite abelian groups -because they are characterized by combinatorial objects called difference sets.This work is dedicated to the study of the underlying combinatorial structures of harmonic BTFs. We show that if a harmonic frame is generated by a divisible difference set, a partial difference set or by a special structure with certain Gauss summing properties -all three of which are generalizations of difference sets that fall under the umbrella term "bidifference set" -then it is either a BTF or an ETF. However, we also show that the relationship between harmonic BTFs and bidifference sets is not as straightforward as the correspondence between harmonic ETFs and difference sets, as there are examples of bidifference sets that do not generate harmonic BTFs. In addition, we study another class of combinatorial structures, the nested divisible difference sets, which yields an example of a harmonic BTF that is not generated by a bidifference set.

A fundamental inequality for Hilbert spaces is the ℓ1 − ℓ2norm inequality which gives that for any x ∈ H n , x 1 ≤ √ n x 2. But this is a strict inequality for all but vectors with constant modulus for their coefficients. We will give a trivial method to compute, for each x, the constant c for which x 1 = c √ n x 2. Since this inequality is one of the most used results in Hilbert space theory, we believe this will have unlimited applications in the field. We will also show some variations of this result.

We resolve a longstanding open problem by reformulating the Grassmannian fusion frames to the case of mixed dimensions and show that this satisfies the proper properties for the problem. In order to compare elements of mixed dimension, we use a classical embedding to send all fusion frame elements to points on a higher dimensional Euclidean sphere, where they are given "equal footing". Over the embedded images -a compact subset in the higher dimensional embedded sphere -we define optimality in terms of the corresponding restricted coding problem. We then construct infinite families of solutions to the problem by using maximal sets of mutually unbiased bases and block designs. Finally, we show that using Hadamard 3-designs in this construction leads to infinite examples of maximal orthoplectic fusion frames of constant-rank. Moreover, any such fusion frames constructed by this method must come from Hadamard 3-designs.