We give a $q$ -analogue of the following congruence: for any odd prime $p$ ,

$$\begin{eqnarray}\mathop{\sum }_{k=0}^{(p-1)/2}(-1)^{k}(6k+1)\frac{(\frac{1}{2})_{k}^{3}}{k!^{3}8^{k}}\mathop{\sum }_{j=1}^{k}\biggl(\frac{1}{(2j-1)^{2}}-\frac{1}{16j^{2}}\biggr)\equiv 0\;(\text{mod}\;p),\end{eqnarray}$$

$q$

The mean skin-friction drag in a wall-bounded turbulent flow can be decomposed into different physics-informed contributions based on the mean and statistical turbulence quantities across the wall layer. Following Renard & Deck’s study (J. Fluid Mech., vol. 790, 2016, pp. 339–367) on the skin-friction drag decomposition of incompressible wall-bounded turbulence, we extend their method to a compressible form and use it to investigate the effect of density and viscosity variations on skin-friction drag generation, using direct numerical simulation data of compressible turbulent channel flows. We use this novel decomposition to study the skin-friction contributions associated with the molecular viscous dissipation and the turbulent kinetic energy production and we investigate their dependence on Reynolds and Mach number. We show that, upon application of the compressibility transformation of Trettel & Larsson (Phys. Fluids, vol. 28, 2016, 026102), the skin-friction drag contributions can be only partially transformed into the equivalent incompressible ones, as additional terms appear representing deviations from the incompressible counterpart. Nevertheless, these additional contributions are found to be negligible at sufficiently large equivalent Reynolds number and low Mach number. Moreover, we derive an exact relationship between the wall heat flux coefficient and the skin-friction drag coefficient, which allows us to relate the wall heat flux to the skin-friction generation process.

Severe phase coarsening and separation in Sn–Bi alloys have brought increasing reliability concern in microelectronic packages. In this study, a phase field model is developed to simulate the microstructural evolution and evaluate the change in macroscopic physical properties of the flip chip Cu/Sn58Bi/Cu joint under the conditions of isothermal aging, as well as the coupled loads of elastic stress and electric current stressing. Results show that large-sized Bi-rich phase particles grow up at the expense of small-sized ones. Under the coupled loads, Bi atoms migrate along the electron flow direction, consequently Bi-rich phase segregates to form a Bi-rich phase layer at the anode. The current crowding ratio in the solder decreases rapidly first and then fluctuates slightly with time. Current density and von Mises stress exhibit inhomogeneous distribution, and both of them are higher in the Sn-rich phase than in the Bi-rich phase. Electric current transfers through the Sn-rich phase and detours the Bi-rich phase. As time proceeds, the resistance of the solder joint increases, and the average von Mises stress of the solder joint decreases. The Bi-rich phase coarsens much faster under the coupled loads than under the conditions of isothermal aging.

Research has demonstrated that trusting belief in one's ability is critical to coproducer selection; however, the importance of trusting belief in dedication has been ignored. This study aims to explore how reputation (i.e., word-of-mouth, certification, and recommendation) affects trusting belief in a potential coproducer's ability and dedication and examines its mediating effects in coproducing with a potential partner.

Empirical results show significant mediating effects of trusting belief in a potential coproducer's ability from certification, word-of-mouth, and recommendation, which in turn motivate coproduction, whereas certification leads to the motivation for coproduction through trusting belief in a potential coproducer's dedication. The findings refer to a unitary acceptance of ability but a divergent recognition of dedication. The focal party may regard certification as factual without personal distortion, while the recommendation is an evaluation worth considering. In considering dedication, the focal party may be unable to assess the extent of distortion from second-hand information (e.g., word-of-mouth).

Studies have shown that voice could be utilized as an effective method to improve organizational effectiveness. This study explores the relationship between ethical leadership and employee voice behavior by focusing on the mediating role of the error management climate and the moderating role of the employee's organizational commitment. Analysis of data collected in three phases in China indicates that the error management climate partly mediates the relationship between ethical leadership and voice behavior. Also, organizational commitment is found to moderate the relationship between the error management climate and voice behavior. Theoretical and practical implications of these results are discussed.

Because of its unique mechanical, chemical, and biological properties, 3D-printed polyether ether ketone (PEEK) has great potential as customized bone replacement and other metal alloy implant replacement. PEEK samples were printed using fused deposition modeling (FDM) and evaluated in terms of their dimensional accuracy, crystallinity, and mechanical properties. Crystallinity and mechanical properties increased with elevated chamber temperature and post-printing annealing. Variations of material properties from three printers are evident. Many factors affect the quality of 3D-printed PEEK. Future FDA regulations for 3D-printed products are needed for this highly customizable manufacturing process to ensure safety and effectiveness for biomedical applications.

We present large-eddy simulation (LES) of flow past different airfoils with $Re_{c}$ , based on the free-stream velocity and airfoil chord length, ranging from $10^{4}$ to $2.1\times 10^{6}$ . To avoid the challenging resolution requirements of the near-wall region, we develop a virtual wall model in generalized curvilinear coordinates and incorporate the non-equilibrium effects via proper treatment of the momentum equations. It is demonstrated that the wall model dynamically captures the instantaneous skin-friction vector field on arbitrary curved surfaces at the resolved scale. By combining the present wall model with the stretched-vortex subgrid-scale model, we apply the wall-modelled LES approach to three different airfoil cases, spanning different geometrical parameters, different attack angles and low to high $Re_{c}$ . The numerical results are verified with direct numerical simulation (DNS) at low $Re_{c}$ , and validated with experiment data at higher $Re_{c}$ , including typical aerodynamic properties such as pressure coefficient distributions, velocity components and also more challenging measurements such as skin-friction coefficient and Reynolds stresses. All comparisons show reasonable agreement, providing a measure of validity that enables us to further probe simulation results into aspects of flow physics that are not available from experiments. Two techniques to quantify hitherto unexplored physics of flows past airfoils are employed: one is the construction of the anisotropy invariant map, and the second is skin-friction portraits with emphasis on flow transition and unsteady separation along the airfoil surface. The anisotropy maps for all three $Re_{c}$ cases, show clearly that a portion of the flow field is aligned along the axisymmetric expansion line, corresponding to the turbulent boundary layer log-law behaviour and the appearance of turbulent transition. The instantaneous skin-friction portraits reveal a monotonic shrinking of the near wall structure scale. At $Re_{c}=10^{4}$ , the interaction between the primary separation bubble and the secondary separation bubble contributes to turbulent transition, similar to the case of flow past a cylinder. At higher $Re_{c}=10^{5}$ , the primary separation breaks into several small separation bubbles. At even higher $Re_{c}=2.1\times 10^{6}$ , near the turbulent separation, the skin-friction lines show small-scale reversal flows that are similar to those observed in DNS of the flat plate turbulent separation. A notable feature of turbulent separation in flow past an airfoil is the appearance of turbulence structures and small-scale reversal flows in the spanwise direction due to the vortex shedding behaviour.

The determination of low boron concentrations in silicate glasses by electron probe microanalysis (EPMA) remains a significant challenge. The internal interferences from the diffraction crystal, i.e. the Mo-B4C large d-spacing layered synthetic microstructure crystal, can be thoroughly diminished by using an optimized differential mode of pulse height analysis (PHA). Although potential high-order spectral interferences from Ca, Fe, and Mn on the BKα peak can be significantly reduced by using an optimized differential mode of PHA, a quantitative calibration of the interferences is required to obtain accurate boron concentrations in silicate glasses that contain these elements. Furthermore, the first-order spectral interference from ClL-lines is so strong that they hinder reliable EPMA of boron concentrations in Cl-bearing silicate glasses. Our tests also indicate that, due to the strongly curved background shape on the high-energy side of BKα, an exponential regression is better than linear regression for estimating the on-peak background intensity based on measured off-peak background intensities. We propose that an optimal analytical setting for low boron concentrations in silicate glasses (≥0.2 wt% B2O3) would best involve a proper boron-rich glass standard, a low accelerating voltage, a high beam current, a large beam size, and a differential mode of PHA.

The takeoff-mass of a two-stage-to-orbit Rocket-Based Combined Cycle Engine-Rocket (RBCC-RKT) launch vehicle is a crucial factor in its comprehensive performance. This paper optimizes the takeoff-mass together with the trajectory by reformulating it to a nonlinear optimal control problem. The range of the second stage rocket mass is considered as a process constraint. When the scopes of initial and terminal states are specified, the problem can be solved by using the Gauss pseudo-spectral method (GPM). In order to reduce the convergent difficulty caused by using table data, the data in different stages are utilized by employing an integrated interpolation strategy through the optimization. Simulation results show that the mass can be effectively optimized to meet the inertia mass ratio constraint of the first-stage, and the separation of Mach number and altitude can be optimized at the same time.

Nutritional geometry (NG) is a novel dietary analysis approach that considers nutrient balance, rather than single nutrient effects, on health and behaviour. Through NG, recent animal experiments have found that lifespan and reproduction are differentially altered by dietary macronutrient distribution. Epidemiological research using NG report similar findings for human ageing. Yet, the relation of macronutrient balance to human reproduction, especially reproductive maturation, remains undefined. We studied the impact of childhood macronutrient intake on pubertal maturation, by applying NG to an Australian longitudinal adolescent dataset. Food records, collected at age 8y from 142 pre-pubertal children (F:92; M:50), were analysed for absolute energy, percent energy and energy-adjusted residuals from protein, carbohydrate and fat. Pubertal stage change (assessed at 8y, 13y and 15y) was modelled to obtain individual mathematical estimates of pubertal timing and tempo. Timing of menarche was recorded. The association of macronutrients to pubertal timing/tempo was assessed via NG, involving generalised additive models and heat maps to aid interpretation. Results showed lower dietary protein (relative to carbohydrate and fat) in girls consistently predicted earlier pubertal timing and menarche, and was related to faster pubertal tempo (all p<0.05). No significant associations were identified in boys for both timing and tempo. Results suggest a role of non-protein macronutrients in facilitating female maturation; corroborating feeding and reproductive behaviour patterns observed in earlier NG studies of primates. Application of NG to other adolescent datasets is required to confirm current findings. Such work would advance understanding of how nutrient balance shapes human development and health.

We report on an initial long-term study of dissolved inorganic and organic carbon (DIC) from Sabino Creek, located in Sabino Canyon, Pima County, Arizona. The purpose of this study was to monitor changes in dissolved radiocarbon (14C) with time and to understand the processes contributing to these variations. Our results span the period 2009–2016 and show a mixing trend between dissolved inorganic and organic carbon modern end-members with an older component. This study provides preliminary information for more detailed research on recycling of organic components in this stream system.

Annexin A2 (ANXA2) is reported to be associated with cancer development. To investigate the roles ANXA2 plays during the development of cancer, the RNAi method was used to inhibit the ANXA2 expression in caco2 (human colorectal cancer cell line) and SMMC7721 (human hepatocarcinoma cell line) cells. The results showed that when the expression of ANXA2 was efficiently inhibited, the growth and motility of both cell lines were significantly decreased, and the development of the motility relevant microstructures, such as pseudopodia, filopodia, and the polymerization of microfilaments and microtubules were obviously inhibited. The cancer cell apoptosis was enhanced without obvious significance. The possible regulating pathway in the process was also predicted and discussed. Our results suggested that ANXA2 plays important roles in maintaining the malignancy of colorectal and hepatic cancer by enhancing the cell proliferation, motility, and development of the motility associated microstructures of cancer cells based on a possible complicated signal pathway.

From arrest to sentencing, cases in which the defendant is charged with capital murder in the United States take substantially longer to resolve than homicide cases in which prosecutors choose not to seek the death penalty. One might reasonably attribute the slowness of capital trials to heightened procedural safeguards that attend the potential deprivation of life. In this article, I suggest that this explanation, straightforward as it is, glosses over more probing and analytically interesting truths about the complex temporal dimensions of death penalty trials. Based on my experiences as both a former defense advocate and an ethnographic researcher of capital defense practices, the slowness of capital cases revolves in large measure around the investigative pursuits of sentencing mitigation. Mitigation investigation’s knowledge practices are informed by distinct temporal operations whose interrelations feed into a deeper logic to capital defense advocacy. This article parses out and traces the connections between these inner workings, using social theory on time to articulate the processes by which mitigation’s temporal logics produce the characteristically slow pace of death penalty cases. I conclude with brief thoughts speculating how the temporal analysis experimented with here might be extended to processes of US criminal adjudication more broadly.

Background: Previous studies have shown varied results with respect to the diagnostic utility of a positive nerve root sedimentation sign (SedSign) on MRI for symptomatic lumbar stenosis. The objective of this study was to analyze the clinical characteristics of SedSign utilizing a validated classification for low back and leg pain (Saskatchewan Spine Pathway classification; SSPc). Methods: This was a retrospective review of prospectively-collected data in 367 consecutive adult patients presenting to a spine surgeon with back and/or leg pain between January 1, 2012 and May 31, 2018. Inter- and intra-rater reliability for SedSign was 73% and 91%, respectively (3 examiners). Results: SedSign was positive in 111 (30.2%) and negative in 256 (69.8%) of patients. On the univariate analysis, a positive SedSign was correlated with age, male sex, several components of ODI, EQ5D mobility, cross-sectional area (CSA) of stenosis, and SSPc pattern 4 (intermitted leg dominant pain). On multivariate analysis, SedSign was associated with age, male sex, CSA stenosis and ODI sub-score for walking distance. The sensitivity and specificity of SedSign for neurogenic claudication was 50.3 and 82.9, respectively (positive predictive value 65.8%, negative predictive value 71.9%). Conclusions: The SedSign has high specificity for neurogenic claudication, but the sensitivity is poor.

Background: The nerve root sedimentation sign (SedSign) has been correlated with clinically significant lumbar spinal stenosis (LSS), and promoted as a possible prognostic indicator. However, diagnostic methods were not clearly defined in prior reports. In this study, the clinically validated Saskatchewan Spine Pathway enabled diagnosis of neurogenic claudication due to LSS. The objective was to compare the outcome of lumbar laminectomy for neurogenic claudication with respect to SedSign. Methods: This was a retrospective analysis of prospectively-collected data in patients with neurogenic claudication who underwent lumbar laminectomy. Outcome measures included Oswestry Disability Index, Visual Analogue Scale (VAS) for back and leg pain, and EuroQol 5-Dimension questionnaire. Results: Laminectomy was performed in 106 patients, and 60 were SedSign positive. Outcomes did not differ with respect to SedSign for all outcome measures, in non-instrumented and instrumented cohorts. Improvement in walking distance was associated with dural cross-sectional area of stenosis (p=0.02). VAS back and leg improvements were associated with back dominant (p=0.038) and leg dominant (p=0.0036) pain. Conclusions: This is the largest analysis of SedSign with respect to operative outcomes, and the only study with validated criteria for defining neurogenic claudication. Although other radiological and clinical factors are associated with improvements, SedSign did not correlate with laminectomy outcome.

Two-dimensional particle-in-cell (PIC) simulations have been used to investigate the interaction between a laser pulse and a foil exposed to an external strong longitudinal magnetic field. Compared with that in the absence of the external magnetic field, the divergence of proton with the magnetic field in radiation pressure acceleration (RPA) regimes has improved remarkably due to the restriction of the electron transverse expansion. During the RPA process, the foil develops into a typical bubble-like shape resulting from the combined action of transversal ponderomotive force and instabilities. However, the foil prefers to be in a cone-like shape by using the magnetic field. The dependence of proton divergence on the strength of magnetic field has been studied, and an optimal magnetic field of nearly 60 kT is achieved in these simulations.