Exact mathematical formulas for wall-heat flux in compressible turbulent channel flows

In this paper,several exact expressions for the mean heat flux at the wall(qw)for the compressible turbulent channel flows are derived by using the internal energy equation or the total energy equation.Two different routes,including the FIK method and the RD method,can be applied.The direct numerical simulation data of compressible channel flows at different Reynolds and Mach numbers verify the correctness of the derived formulas.Discussions related to the different energy equations,and different routes are carried out,and we may arrive at the conclusion that most of the formulas derived in the present work are just mathematical ones and that they generally are lacking in clear physical interpretation in our opinion.They can be used to estimate qw,but might not be suitable for exploring the underlying physics.

Mathematical Approach on the GDPR Complexity

The objective of this paper work is to create a mathematical approach that can quantify the complexity of the General Data Protection Regulation(GDPR)and,at the same time,the implementing of the rules in a company according to the actual benefits of doing so.The scope is to study the rules and regulations imposed by the law,the steps and requirements needed for implementing and to define the indicators that can be used in the mathematical definition of the model.Also,it checks the impact of each indicator in the system and identifies the factors that determine vulnerabilities,what damages are caused by these factors,the risk and impact level of the factors.It proposes a model to evaluate the indicators and the assignment of weights in formula evaluation of each indicator,so the risks of implementing the rules in the business will be smaller as well as the evaluation of the data protection terms of a company will be more balanced and optimal.The approach is from the point of view of the law imposed in implementing the model and the easiness and costs for the companies to do so,including the advantages or disadvantages and the risks they can expose to by doing so.

Volume, Side-Area, and Force Direction of Berkovich and Cubecorner Indenters, Novel Important Insights

The iteration-free physical description of pyramidal indentations with closed mathematical equations is comprehensively described and extended for creating new insights in this important field of research and applications. All calculations are easily repeatable and should be programmed by instrument builders for even easier general use. Formulas for the volumes and side-areas of Berkovich and cubecorner as a function of depth are deduced and provided, as are the resulting forces and force directions. All of these allow for the detailed comparison of the different indenters on the mathematical reality. The pyramidal values differ remarkably from the ones of so-called “equivalent cones”. The worldwide use of such pseudo-cones is in severe error. The earlier claimed and used 3 times higher displaced volume with cube corner than with Berkovich is disproved. Both displace the same amount at the same applied force. The unprecedented mathematical results are experimentally confirmed for the physical indentation hardness and for the sharp-onset phase-transi- tions with calculated transition energy. The comparison of both indenters provides novel basic insights. Isotropic materials exhibit the same phase transition onset force, but the transition energy is larger with the cube corner, due to higher force and flatter force direction. This qualifies the cube corner for fracture toughness studies. Pile-up is not from the claimed “friction with the indenter”. Anisotropic materials with cleavage planes and channels undergo sliding along these under pressure, both to the surface and internally. Their volumes add to the depression volume. These volumes are essential for the exemplified pile-up management. Phase-transitions produce polymorph interfaces that are nucleation sites for cracks. Technical materials must be developed with onset forces higher than the highest thinkable stresses (at airliners, bridges, etc.). This requires urgent revision of ISO 14577-ASTM standards.

An Only-Once-Sorting Algorithm

This paper provides a new sorting algorithm called 'Only-Once-Sorting' algorithm a mathemati cal formula,this algorithm can put elements in the positions they should be stored only once,then compacts them.The algorithm completes sorting a sequence of n elements in a calculation time of O(n ).

The Distribution Search:An O(n) Expected Time Search

Provided an algorithm for the distribution search and proves the time complexity of the algorithm. This algorithm uses a mathematical formula to search n elements in the sequence of n elements in O(n)expected time,and experimental reesult proves that distribution search is superior to binary search.