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Given a sequence α = ( a k ) k ≥ 0 of nonnegative numbers, define a new sequence L ( α ) = ( b k ) k ≥ 0 by b k = a k 2 - a k - 1 a k + 1 . The sequence α is called r - log-concave if L i ( α ) = L ( L i - 1 ( α ) ) is a nonnegative sequence for all 1 ≤ i ≤ r . In this paper, we study the r -log-concavity and its q -analogue for r = 2 , 3 using total positivity of matrices. We show the 6-log-concavity of the Taylor coefficients of the Riemann ξ -function. We give some criteria for r - q -log-concavity for r = 2 , 3 . As applications, we get 3- q -log-concavity of q -binomial coefficients and different q -Stirling numbers of two kinds, which extends results for q -log-concavity. We also present some results for r - q -log-concavity from the linear transformations. Finally, we pose an interesting question.

Purpose Surgical treatment of shoulder instability caused by anterior glenoid bone loss is based on a critical threshold of the defect size. Recent studies indicate that the glenoid concavity is essential for glenohumeral stability. However, biomechanical proof of this principle is lacking. The aim of this study was to evaluate whether glenoid concavity allows a more precise assessment of glenohumeral stability than the defect size alone. Methods The stability ratio (SR) is a biomechanical estimate of glenohumeral stability. It is defined as the maximum dislocating force the joint can resist related to a medial compression force. This ratio was determined for 17 human cadaveric glenoids in a robotic test setup depending on osteochondral concavity and anterior defect size. Bony defects were created gradually, and a 3D measuring arm was used for morphometric measurements. The influence of defect size and concavity on the SR was examined using linear models. In addition, the morphometrical-based bony shoulder stability ratio (BSSR) was evaluated to prove its suitability for estimation of glenohumeral stability independent of defect size. Results Glenoid concavity is a significant predictor for the SR, while the defect size provides minor informative value. The linear model featured a high goodness of fit with a determination coefficient of R 2  = 0.98, indicating that 98% of the SR is predictable by concavity and defect size. The low mean squared error (MSE) of 4.2% proved a precise estimation of the SR. Defect size as an exclusive predictor in the linear model reduced R 2 to 0.9 and increased the MSE to 25.7%. Furthermore, the loss of SR with increasing defect size was shown to be significantly dependent on the initial concavity. The BSSR as a single predictor for glenohumeral stability led to highest precision with MSE = 3.4%. Conclusion Glenoid concavity is a crucial factor for the SR. Independent of the defect size, the computable BSSR is a precise biomechanical estimate of the measured SR. The inclusion of glenoid concavity has the potential to influence clinical decision-making for an improved and personalised treatment of glenohumeral instability with anterior glenoid bone loss.

In view of the existing incremental forming machine, there is a problem that the tool needs to be replaced when machining parts with different degrees of sidewall concavity. The incremental forming machine with a tiltable cutter was designed, and the feasibility of a cutter tilting structure was discussed. The forming results show that the tiltable structure meets the design requirements.

Stability selection was recently introduced by Meinshausen and Bühlmann as a very general technique designed to improve the performance of a variable selection algorithm. It is based on aggregating the results of applying a selection procedure to subsamples of the data. We introduce a variant, called complementary pairs stability selection, and derive bounds both on the expected number of variables included by complementary pairs stability selection that have low selection probability under the original procedure, and on the expected number of high selection probability variables that are excluded. These results require no (e.g. exchangeability) assumptions on the underlying model or on the quality of the original selection procedure. Under reasonable shape restrictions, the bounds can be further tightened, yielding improved error control, and therefore increasing the applicability of the methodology.

The concept of strong F -convexity is a natural generalization of strong convexity. Although strongly concave functions are rarely mentioned and used, we show that in more effective and specific analysis this concept is very useful, and especially its generalization, namely strong F -concavity. Using this concept, refinements of the Young inequality are given as a model case. A general form of the self-improving property for Jensen type inequalities is presented. We show that a careful choice of control functions for convex or concave functions can give a control over these refinements and produce refinements of the power mean inequalities.

Let ${\\overline{p}}(n)$ denote the overpartition function. In this paper, we study the asymptotic higher-order log-concavity property of the overpartition function in a similar framework done by Hou and Zhang for the partition function. This will enable us to move on further in order to prove log-concavity of overpartitions, explicitly by studying the asymptotic expansion of the quotient ${\\overline{p}}(n-1){\\overline{p}}(n+1)/{\\overline{p}}(n)^2$ up to a certain order. This enables us to additionally prove 2-log-concavity and higher Turán inequalities with a unified approach.

This paper presents a study of blow-up of solutions to a quasilinear p ( x )-Laplacian problem related to the equation z t ( x , t ) = Δ p ( x ) z ( x , t ) + g ( z ( x , t ) ) We use a condition on the nonlinear function g ( z ) given by, ς ∫ 0 z g ( s ) d s ⩽ z g ( z ) + η z p ( x ) + μ , z > 0 We extend the existing results on blow-up for a nonlinear heat equation to variable exponent case and establish an upper bound for the blow-up time with the help of concavity method.

More than half a century ago, it was proved that the increasing failure rate (IFR) property is preserved under the formation of k -out-of- n systems (order statistics) when the lifetimes of the components are independent and have a common absolutely continuous distribution function. However, this property has not yet been proved in the discrete case. Here we give a proof based on the log-concavity property of the function $f({{\\mathrm{e}}}^x)$ . Furthermore, we extend this property to general distribution functions and general coherent systems under some conditions.

Purpose Glenohumeral joint injuries frequently result in shoulder instability. However, the biomechanical effect of cartilage loss on shoulder stability remains unknown. The aim of the current study was to investigate biomechanically the effect of two severity stages of cartilage loss in different dislocation directions on shoulder stability. Methods Joint dislocation was provoked in 11 human cadaveric glenoids for 7 different directions between 3 o'clock (anterior) and 9 o'clock (posterior). Shoulder stability ratio (SSR) and concavity gradient were assessed in three states: intact, 3 mm and 6 mm simulated cartilage loss. The influence of cartilage loss on SSR and concavity gradient was statistically evaluated. Results Both SSR and concavity gradient decreased significantly between intact state and 6 mm cartilage loss in every dislocation direction ( p ≤ 0.038), except concavity gradient in 4 o'clock direction. Thereby, anterior–inferior dislocation directions were associated with the highest decrease in both SSR and concavity gradient of up to 59.0% and 49.4%, respectively, being significantly bigger for SSR compared with all other dislocation directions ( p ≤ 0.040). Correlations between concavity gradient and SSR for pooled dislocation directions were significant in each separate specimen's state ( p < 0.001). Conclusion From a biomechanical perspective, articular cartilage of the glenoid contributes significantly to the concavity gradient, correlating strongly with the associated loss in glenohumeral joint stability. The biggest effect of cartilage loss is observed in the most frequently occurring anterior–inferior dislocation directions, suggesting that surgical interventions to restore cartilage's surface and concavity should be considered for recurrent shoulder dislocations in presence of cartilage loss.