For n ≥ 1, let an count the number of ways one can tile a 1 × n chessboard using 1× 1 square tiles, which come in w colors, and 1× 2 rectangular tiles, which come in t colors. The results for an generalize the Fibonacci numbers and provide generalizations of many of the properties satisfied by the Fibonacci and Lucas numbers. We count the total number of 1× 1 square tiles and 1× 2 rectangular t...

We consider a class of generating functions that appear in the context of Carlitz compositions. In order to combinatorially interpret them, we introduce a combinatorial structures that we name generalized compositions and p-Carlitz compositions of integers. We explain their connection to Carlitz compositions, the relation to other combinatorial structures, and we describe their basic properties...

A tree-composition is a tree-like family that serves to describe the obstacles to k-edge-connected orientability of mixed graphs. Here we derive a structural result on tree-compositions that gives rise to a simple algorithm for computing an obstacle when the orientation does not exist. As another application, we show a min–max theorem on the minimal in-degree of a given node set in a k-edge-con...

Noting that the usual ‘propositionally’ based way of composing retrenchments can yield many ‘junk’ cases, alternative approaches to composition are introduced (via notions of tidy, neat, and fastidious retrenchments) that behave better in this regard. These alternatives do however make other issues such as associativity harder. The technical details are presented for vertical composition of ret...

The correlation between structure and electrical properties of lead-free (1−x)(Bi1/2Na1/2)TiO3–xBaTiO3 (BNT-100xBT) polycrystalline piezoceramics was investigated systematically by in situ synchrotron diffraction technique, combined with electrical property characterization. It was found that the morphotropic phase boundary (MPB) between a rhombohedral and a tetragonal phase evolved into a morp...

The surface area of electrodeposited thin films of Ni, Co, and NiCo was evaluated using electrochemical double-layer capacitance, electrochemical area measurements using the [Ru(NH[Formula: see text])[Formula: see text]][Formula: see text]/[Ru(NH[Formula: see text])[Formula: see text]][Formula: see text] redox couple, and topographic atomic force microscopy (AFM) imaging. These three methods we...

A fundamental question in cometary science is whether the different dynamical classes of comets have different chemical compositions, which would reflect different initial conditions. From the ground or Earth orbit, radio and infrared spectroscopic observations of a now significant sample of comets indeed reveal deep differences in the relative abundances of cometary ices. However, no obvious c...

Using renewable energy to recycle CO2 provides an opportunity to both reduce net CO2 emissions and synthesize fuels and chemical feedstocks. It is of central importance to design electrocatalysts that both are efficient and can access a tunable spectrum of products. Syngas, a mixture of carbon monoxide (CO) and hydrogen (H2), is an important chemical precursor that can be converted downstream i...

As a partner for the DOE Metal Hydride Center of Excellence (MHCoE), GE Global Research in Niskayuna, NY developed a robust approach for metal hydride discovery as schematically shown in Figure 1. The approach is used on a daily basis at GE and it has streamlined the metal hydride discovery process. Our research allows us to investigate large numbers of compositions simultaneously, quickly scre...

The ability to control and direct self-assembly of nanostructures into specific geometries with new functionalities, while preserving their original optical and electronic properties, is an attractive research endeavor. We have fabricated liquid crystal (LC) based matrices into which chemically synthesized nanostructures of varied morphologies and compositions are uniformly dispersed. Using hig...