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Background:In current literature, left atrial (LA) function assessed by 2D speckle tracking echocardiography (2DSTE) has shown increasing clinical and prognostic significance. We sought to establish age- and sex-based normative values of LA strain measures in the general population and to assess the diagnostic and prognostic yield of the prediction intervals (PI) in relation to baseline and future atrial fibrillation (AF).Methods:We performed 2DSTE in members of the general population included in the 5thround of the Copenhagen City Heart Study. We determined normative values of peak atrial longitudinal strain (PALS), peak atrial contraction strain (PACS), and left atrial conduit strain (LACS) based on age and sex in 1,757 healthy participants. In a secondary analysis, 3,657 participants, regardless of health status, were included to assess the diagnostic and prognostic value of the established reference values.Results:In the healthy cohort, median age was 46 years (IQR: 33, 57), 62% were female. Median normative values of PALS, PACS, and LACS in the healthy participants were 39.1% (PI: 22.7-67.4%), 15.5% (PI: 6.3-28.0%), and 23.5% (PI: 8.6-44.8%), respectively. We observed higher values and reference limits of PALS and LACS in females compared to males, while no differences were observed for PACS. Across four different age categories, all three LA strain parameters differed significantly in both females and males. PALS and LACS decreased, while PACS increased with advancing age (Figure 1).The lower reference limits of all three LA strain parameters showed high specificity (range 96-98%) in relation to baseline and future AF, implying a low risk of AF in participants with LA strain above the lower prediction interval in their respective sex and age group.Conclusion:We report normal values for LA strain measures stratified by sex and age. The established lower reference limits showed high specificity regarding baseline and future AF diagnosis.

Background:Despite improvement in treatment strategies of atrial fibrillation (AF), a considerable number of patients still experience recurrence of atrial tachyarrhythmia (ATA) following catheter ablation (CA). We hypothesized that baseline myocardial impairment of the left atrium (LA) is useful in identifying patients at high risk of ATA recurrence.Methods:This retrospective study included 921 patients with AF who underwent CA. Echocardiography including two-dimensional speckle tracking echocardiography (2DSTE) was performed in all patients prior to CA. Logistic regression analysis was used to assess the association between ATA recurrence and peak atrial longitudinal strain (PALS), peak atrial contraction strain (PACS), and left atrial conduit strain (LACS).Results:During one-year follow-up, 375 (41%) experienced ATA recurrence. The median age of the included study population was 63.5 years (IQR: 55.6, 69.6) and 665 (72%) were male. Patients with recurrence had lower PALS (19.6% vs. 22.0%, p=0.004) and PACS (10.7% vs. 12.4%, p < 0.001). No difference in LACS was observed. After adjusting for potential clinical and echocardiographic confounders PALS (OR = 1.04, CI95% [1.01; 1.06], p = 0.011, per 1% decrease) and PACS (OR = 1.06, CI95% [1.02; 1.11], p = 0.007, per 1% decrease) remained independent predictors of recurrence. However, in patients with a normal-sized LA (LA volume index < 34 mL/m2), only PACS remained an independent predictor of recurrence (OR = 1.06, CI95% [1.01; 1.12], p = 0.013, per 1% decrease). Figure 1 displays the increasing risk of recurrence with decreasing PALS, PACS, and LACS.Conclusion:In patients undergoing CA for AF, LA deformation analysis by 2DSTE could be of use in risk stratification in clinical practice regarding ATA recurrence, even in patients with a normal-sized LA.

Consider the random variable Lndefined as the length of a longest common subsequence of two random strings of length nand whose random characters are independent and identically distributed over a finite alphabet. Chvátal and Sankoff showed that the limit ?=limn?8E[Ln]/nis well defined. The exact value of this constant is not known, but various methods for the computation of upper and lower bounds have been discussed in the literature. Even so, high-precision bounds are hard to come by. In this paper we discuss how large deviation theory can be used to derive a consistent sequence of upper bounds, (qm)m?N, on ?, and how Monte Carlo simulation can be used in theory to compute estimates, q^m, of the qmsuch that, for given ? > 0 and ? ? (0,1), we have P[? < q^< ? + ?] = ?. In other words, with high probability the result is an upper bound that approximates ? to high precision. We establish O((1 - ?)-1?-(4+e)) as a theoretical upper bound on the complexity of computing q^mto the given level of accuracy and confidence. Finally, we discuss a practical heuristic based on our theoretical approach and discuss its empirical behavior.

Self-scaled barrier functions on self-scaled cones were axiomatically introduced by Nesterov and Todd in 1994 as a tool for the construction of primal—dual long-step interior point algorithms. This paper provides firm foundations for these objects by exhibiting their symmetry properties, their close ties with the symmetry groups of their domains of definition, and subsequently their decomposition into irreducible parts and their algebraic classification theory. In the first part we recall the characterization of the family of self-scaled cones as the set of symmetric cones and develop a primal—dual symmetric viewpoint on self-scaled barriers, results that were first discovered by the second author. We then show in a short, simple proof that any pointed, convex cone decomposes into a direct sum of irreducible components in a unique way, a result which can also be of independent interest. We then proceed to showing that any self-scaled barrier function decomposes, in an essentially unique way, into a direct sum of self-scaled barriers defined on the irreducible components of the underlying symmetric cone. Finally, we present a complete algebraic classification of self-scaled barrier functions using the correspondence between symmetric cones and Euclidean—Jordan algebras.

Let A be a $ {C^ \ast }$A then admits a smallest faithful representation if and only if the ideal of compact elements is an essential ideal in A.