A Note on Flips in Diagonal Rectangulations

Rectangulations are partitions of a square into axis-aligned rectangles. A number of results provide bijections between combinatorial equivalence classes of rectangulations and families of pattern-avoiding permutations. Other results deal with local changes involving a single edge of a rectangulation, referred to as flips, edge rotations, or edge pivoting. Such operations induce a graph on equivalence classes of rectangulations, related to so-called flip graphs on triangulations and other families of geometric partitions. In this note, we consider a family of flip operations on the equivalence classes of diagonal rectangulations, and their interpretation as transpositions in the associated Baxter permutations, avoiding the vincular patterns { 3{14}2, 2{41}3 }. This complements results from Law and Reading (JCTA, 2012) and provides a complete characterization of flip operations on diagonal rectangulations, in both geometric and combinatorial terms

Calculation of the relativistic Bethe logarithm in the two-center problem

We present a variational approach to evaluate relativistic corrections of order \alpha^2 to the Bethe logarithm for the ground electronic state of the Coulomb two center problem. That allows to estimate the radiative contribution at m\alpha^7 order in molecular-like three-body systems such as hydrogen molecular ions H_2^+ and HD^+, or antiprotonic helium atoms. While we get 10 significant digits for the nonrelativistic Bethe logarithm, calculation of the relativistic corrections is much more involved especially for small values of bond length R. We were able to achieve a level of 3-4 significant digits starting from R=0.2 bohr, that will allow to reach 10^{-10} relative uncertainty on transition frequencies.Comment: 19 pages, 5 tables, 7 figure

Flat zones filtering, connected operators, and filters by reconstruction

This correspondence deals with the notion of connected operators. Starting from the definition for operator acting on sets, it is shown how to extend it to operators acting on function. Typically, a connected operator acting on a function is a transformation that enlarges the partition of the space created by the flat zones of the functions. It is shown that from any connected operator acting on sets, one can construct a connected operator for functions (however, it is not the unique way of generating connected operators for functions). Moreover, the concept of pyramid is introduced in a formal way. It is shown that, if a pyramid is based on connected operators, the flat zones of the functions increase with the level of the pyramid. In other words, the flat zones are nested. Filters by reconstruction are defined and their main properties are presented. Finally, some examples of application of connected operators and use of flat zones are described.Peer ReviewedPostprint (published version

Music enrichment for gifted children in the first grade

Thesis (Ed.M.)--Boston Universit

Effect of reinforcing submicron SiC particles on the wear of electrolytic NiP coatings Part 2: Bi-directional sliding

As-plated and heat-treated electrodeposited NiP and composite NiP-SiC coatings were investigated in bi-directional ball-on-disc sliding tests. All tests were performed under gross slip conditions. Heat treatment decreases the wear volume loss during fretting in ambient air for all coatings investigated. Heat-treated NiP coating has a lower wear volume loss compared to composite NiP-SiC coatings for all sliding tests. The wear rate at the bi-directional sliding test was found to be lower relative to the wear rate at uni-directional sliding test

Effect of reinforcing submicron SiC particles on the wear of electrolytic NiP coatings Part 1. Uni-directional sliding

As-plated and annealed NiP coatings and composite NiP-SiC coatings were investigated in uni-directional ball-on-disc sliding tests. Abrasive wear was noticed in the case of composite NiP coatings containing submicron SiC particles, whereas in NiP coatings oxidational wear was active. The addition of submicron SiC particles not only increases the hardness of these electrolytic coatings but also hinders the formation of an oxide film in the sliding wear track. As a consequence, the wear loss on as-plated NiP coatings is not markedly reduced by the addition of SiC particles. On the contrary, a heat treatment at 420 °C for 1 h decreases the wear loss on both pure NiP and composite NiP-SiC coatings. During that heat treatment, Ni3P precipitates are formed in the NiP matrix and owing to this fact, the hardness of both pure NiP and composite NiP-SiC coatings increases. However, the heat treatment of composite NiP-SiC coatings induces the sensitivity for crack formation in the NiP matrix around these SiC particles. As a result, the pull out of SiC particles in the wear track occurs easily during sliding, and the wear loss of composite NiP-SiC coatings remains above the wear loss on NiP coatings