Applications of operator identities to the multiple q-binomial theorem and q-Gauss summation theorem

In this paper, we first give an interesting operator identity. Furthermore, using the q-exponential operator technique to the multiple q-binomial theorem and q-Gauss summation theorem, we obtain some transformation formulae and summation theorems of multiple basic hypergeometric series.     

Bartholdi zeta functions of some graphs

We give a decomposition formula for the Bartholdi zeta function of a graph G which is partitioned into some irregular coverings. As a corollary, we obtain a decomposition formula for the Bartholdi zeta function of G which is partitioned into some regular coverings.     

Sloane's box stacking problem

Recently, Sloane suggested the following problem: We are given n boxes, labeled 1,2,…,n. For i=1,…,n, box i weighs (m-1)i grams (where m?2 is a fixed integer) and box i can support a total weight of i grams. What is the number of different ways to build a single stack of boxes in which no box will be squashed by the weight of the boxes above it? Prior to this generalized problem, Sloane and Sellers solved the case m=2. More recently, Andrews and Sellers solved the case m?3. In this note we give new and simple proofs of the results of Sloane and Sellers and of Andrews and Sellers, using a known connection with m-ary partitions.     

Darcy's law as long-time limit of adiabatic porous media flow

It is conjectured that Darcy's law governs the motion of compressible porous media flow in large time. This has been justified for one-dimensional isentropic flows. In this work, we show the conjecture is true for one-dimensional adiabatic flows with generic small smooth initial data.     

Ordered K-groups associated to substitutional dynamics

The Matsumoto K₀-group is an interesting invariant of flow equivalence for symbolic dynamical systems. Because of its origin as the K-theory of a certain C^∗-algebra, which is also a flow invariant, this group comes equipped with a flow invariant order structure. We emphasize this order structure and demonstrate how methods from operator algebra and symbolic dynamics combine to allow a computation of it in certain cases, including Sturmian and primitive substitutional shifts. In the latter case we show by example that the ordered group is a strictly finer invariant than the group itself.     

Independence polynomials of k-tree related graphs

An independent set of a graph G is a set of pairwise non-adjacent vertices. Let α(G) denote the cardinality of a maximum independent set and f_s(G) for 0≤s≤α(G) denote the number of independent sets of s vertices. The independence polynomial defined first by Gutman and Harary has been the focus of considerable research recently. Wingard bounded the coefficients f_s(T) for trees T with n vertices: for s≥2. We generalize this result to bounds for a very large class of graphs, maximal k-degenerate graphs, a class which includes all k-trees. Additionally, we characterize all instances where our bounds are achieved, and determine exactly the independence polynomials of several classes of k-tree related graphs. Our main theorems generalize several related results known before.     

The lattice of embedded subsets

In cooperative game theory, games in partition function form are real-valued function on the set of the so-called embedded coalitions, that is, pairs (S,π) where S is a subset (coalition) of the set N of players, and π is a partition of N containing S. Despite the fact that many studies have been devoted to such games, surprisingly nobody clearly defined a structure (i.e., an order) on embedded coalitions, resulting in scattered and divergent works, lacking unification and proper analysis. The aim of the paper is to fill this gap, thus to study the structure of embedded coalitions (called here embedded subsets), and the properties of games in partition function form.     

On the packing chromatic number of some lattices

For a positive integer k, a k-packing in a graph G is a subset A of vertices such that the distance between any two distinct vertices from A is more than k. The packing chromatic number of G is the smallest integer m such that the vertex set of G can be partitioned as V₁,V₂,…,V_m where V_i is an i-packing for each i. It is proved that the planar triangular lattice T and the three-dimensional integer lattice Z³ do not have finite packing chromatic numbers.     

On k-networks and weak bases for spaces

In this paper, we answer some problems on k-networks and weak bases posed by Liu and Tanaka (2007) [19].     

Charge dynamics and optical properties of layered cobaltates

We study the charge dynamics and electronic structure by optical spectroscopy technique. Here we focus on the following four issues: (1) the evolution of optical spectra with Na content; (2) the spectral features specific to different regions in the phase diagram; (3) the c-axis optical response for crystal at the A-type antiferromagnetic region; (4) the optical response of misfit-layered Bi₂M₂Co₂O_y (M=Ba, Sr, Ca) and Ca₃Co₄O_y single crystals.