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In this article, we deal with the blow-up phenomenon of a pseudo-parabolic equation with memory and convection terms. By constructing some appropriate auxiliary functions and using the modified concavity method, the blow-up criterion of the weak solution is given.

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.

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.

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.

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.

For the independent reference model with popularity vector \\(pın_N^\\), let \\(H_C(p)\\) denote the exact stationary hit rate of an LRU cache of capacity \\(C\\). We prove that, for every \\(1 C

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