Protected points in k-ary trees

Following Cheon and Shapiro (2008) [3], in this note we compute the average number of protected points in all k-ary trees with n internal points.     

Pressure drop caused by flow area changes in capillaries under low flow conditions

Single-phase and two-phase flow pressure drops caused by flow area expansion and contraction were measured using air and water. The test section consisted of two capillaries with 0.84 mm and 1.6 mm diameters. For single-phase flow, the Reynolds numbers defined based on the smaller diameter capillary covered the range 160–11,000. For two-phase flow, the all-liquid Reynolds number based on the smaller capillary varied in the 410–1020 range, and the flow quality varied in the 0.018–0.2 range. The single-phase flow loss coefficients for both flow area expansion and contraction were empirically correlated. For two-phase flow, the data indicated the occurrence of significant velocity slip, and the one-dimensional homogeneous flow model utterly disagreed with the data. For flow area expansion the one-dimensional slip flow model along with an Armand-type slip ratio correlation could predict the data well. For flow area contraction, the one-dimensional slip flow model along with the slip ratio expression of Zivi agreed with the data very well, provided that no vena-contracta was considered.     

Longest alternating subsequences of k-ary words

Recently, Stanley [Longest alternating subsequences of permutations, preprint, arXiv/0511419v1] studied the length of the longest alternating subsequence of a permutation in the symmetric group, where a sequence a,b,c,d,… is alternating if a>b<c>d<?. In this paper, we extend this result to the case of k-ary words. More precisely, we find an explicit formula for the generating function of the number of k-ary words of length n according to the length of the longest alternating subsequence.     

Embedding Distributions and Chebyshev Polynomials

The history of genus distributions began with J. Gross et?al. in 1980s. Since then, a lot of study has given to this parameter, and the explicit formulas are obtained for various kinds of graphs. In this paper, we find a new usage of Chebyshev polynomials in the study of genus distribution, using the overlap matrix, we obtain homogeneous recurrence relation for rank distribution polynomial, which can be solved in terms of Chebyshev polynomials of the second kind. The method here can find explicit formula for embedding distribution of some other graphs. As an application, the well known genus distributions of closed-end ladders and cobblestone paths (Furst et?al. in J Combin Ser B 46:22–36, 1989) are derived. The explicit formula for non-orientable embedding distributions of closed-end ladders and cobblestone paths are also obtained.     

Compositions of n as alternating sequences of weakly increasing and strictly decreasing partitions

Compositions and partitions of positive integers are often studied in separate frameworks where partitions are given by q-series generating functions and compositions exhibiting specific patterns are designated by generating functions for these patterns. Here, we view compositions as alternating sequences of weakly increasing and strictly decreasing partitions (i.e. alternating blocks). We obtain generating functions for the number of such partitions in terms of the size of the composition, the number of parts and the total number of “valleys” and “peaks”. From this, we find the total number of “peaks” and “valleys” in the composition of n which have the mentioned pattern. We also obtain the generating function for compositions which split into just two partition blocks. Finally, we obtain the two generating functions for compositions of n that start either with a weakly increasing partition or a strictly decreasing partition.     

Effect of temperature and silicon resistivity on the elaboration of silicon nanowires by electroless etching

The morphology of silicon nanowire (SiNW) layers formed by Ag-assisted electroless etching in HF/H₂O₂ solution was studied. Prior to the etching, the Ag nanoparticles were deposited on p-type Si(1 0 0) wafers by electroless metal deposition (EMD) in HF/AgNO₃ solution at room temperature. The effect of etching temperature and silicon resistivity on the formation process of nanowires was studied. The secondary ion mass spectra (SIMS) technique is used to study the penetration of silver in the etched layers. The morphology of etched layers was investigated by scanning electron microscope (SEM).     

An arithmetical property of powers of Salem numbers

Let ζ be a nonzero real number and let α be a Salem number. We show that the difference between the largest and smallest limit points of the fractional parts of the numbers ζαn, when n runs through the set of positive rational integers, can be bounded below by a positive constant depending only on α if and only if the algebraic integer α−1 is a unit.     

Counting occurrences of 231 in an involution

We study the generating function for the number of involutions on n letters containing exactly r?0 occurrences of 231. It is shown that finding this function for a given r amounts to a routine check of all involutions of length at most 2r+2.