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Recently Mansour and Shattuck studied ( k , a ) -paths and gave formulas that related the total number of humps in all ( k , a ) -paths to the number of super ( k , a ) -paths. These results generalized earlier results of Regev on Dyck paths and Motzkin paths. Their proofs are based on generating functions and they asked for bijective proofs for their results. In this paper we first give bijective proofs of Mansour and Shattuck’s results, then we extend our study to ( n , m ) -Dyck paths. We give a bijection that relates the total number of peaks in all ( n , m ) -Dyck paths to certain free ( n , m ) -paths when n and m are coprime. From this bijection we get the number of ( n , m ) -Dyck paths with exactly j peaks, which is a generalization of the well-known result that the number Dyck paths of order n with exactly j peaks is the Narayana number 1 k n − 1 k − 1 n k − 1 . [ABSTRACT FROM AUTHOR]

Abstract: Let us call a lattice path in from to using , , and steps and never going below the -axis, a -path (of order ). A super -path is a -path which is permitted to go below the -axis. We relate the total number of humps in all of the -paths of order to the number of super -paths, where a hump is defined to be a sequence of steps of the form , . This generalizes recent results concerning the cases when and or . A similar relation may be given involving peaks (consecutive steps of the form ). [Copyright &y& Elsevier]

Abstract: In this paper we study the number of humps (peaks) in Dyck, Motzkin and Schröder paths. Recently A. Regev noticed that the number of peaks in all Dyck paths of order is one half of the number of super-Dyck paths of order . He also computed the number of humps in Motzkin paths and found a similar relation, and asked for bijective proofs. We give a bijection and prove these results. Using this bijection we also give a new proof that the number of Dyck paths of order with peaks is the Narayana number. By double counting super-Schröder paths, we also get an identity involving products of binomial coefficients. [Copyright &y& Elsevier]

In order to optimize the laser-tungsten inert arc (TIG) welding process of cemented carbide to steel, a novel combined model consisting of circular disk source, line source, and double ellipsoidal TIG heat source was put forward to simulate the deep penetration phenomena and bead profile. In contrast, laser heat source model was validated using macrostructure analysis results. The effects of the circular disk of radius and the material properties on the bead profile were discussed. For mechanism surface formation, surface depression or humps was discussed based on the high-speed photography technique to verify bead profile characterization. The results manifested that in the model, the circular disk radius played a key role in the formation of bead profile. And the penetration coefficient chiefly affected the root of weld and the keyhole formation. During the hybrid welding, the main forces influencing the surface depression or humping were intrinsic stress from coefficient of thermal expansion gradient, surface tension, buoyancy force, stirring action, and shock waves at the end of the keyhole. Before the solidification of the upper part of the melt, the melt was driven by the fluctuated plasma, plumes, or other forces. While the amount of melt filling the back welds metal of keyhole was larger than that caused by plume losses and the shrinkage of weld metal, the humps came into being. [ABSTRACT FROM AUTHOR]

The study of keyhole (KH) instability in deep penetration laser beam welding (LBW) is essential to understand welding process and appearance of weld seam defects. The main cause of keyhole collapse is the instability in KH dynamics during the LBW process. This is mainly due to the surface tension forces associated with the KH collapse and the stabilizing action of vapour pressure. A deep penetration high power CW CO laser was used to generate KH in mild steel (MS) in two different welding conditions i.e. ambient atmospheric welding (AAW) and under water welding (UWW). KH, formed in case of under water welding, was deeper and narrower than keyhole formed in ambient and atmospheric condition. The number and dimensions of irregular humps increased in case of ambient and under water condition due to larger and rapid keyhole collapse also studied. The thermocapillary convection is considered to explain KH instability, which in turn gives rise to irregular humps. [ABSTRACT FROM AUTHOR]