We consider planar lattice walks that start from (0, 0), remain in the first quadrant i, j ≥ 0, and are made of three types of steps: North-East, West and South. These walks are known to have remarkable enumerative and probabilistic properties: – they are counted by nice numbers (Kreweras 1965), – the generating function of these numbers is algebraic (Gessel 1986), – the stationary distribution...

The paper investigates relationship between algebraic expressions and graphs. We consider a digraph called a full square rhomboid that is an example of non-series-parallel graphs. Our intention is to simplify the expressions of full square rhomboids and eventually find their shortest representations. With that end in view, we describe two decomposition methods for generating expressions of full...

In this survey we present an analytic approach to solve problems concerning (deterministic or random) walks in the quarter plane. We illustrate the recent breakthroughs in that domain with two examples. The first one is about the combinatorics of walks confined to the quarter plane, and more precisely about the numbers of walks evolving in the quarter plane and having given length, starting and...

where the q-shifted factorial is defined by (x; q)0 = 1 and (x; q)n = (1− x)(1 − qx) · · · (1− q x) for n = 1, 2, · · · with the following abbreviated multiple parameter notation [α, β, · · · , γ; q]∞ = (α; q)∞(β; q)∞ · · · (γ; q)∞. This identity has several important applications in combinatorial analysis, number theory and special functions. For the historical note, we refer the reader to the...

In order to improve Mean Square Error of its reliance on reference images when evaluating image sharpness, the no-reference metric based on algebraic multi-grid is proposed. The proposed metric first reconstructs the original image by Algebraic Multi-grid (AMG), then compute the Mean Square Error between original image and reconstructed image, the result represents image sharpness. Experiments ...

A novel compact wide bandpass filter is proposed by cascading two square-ring resonators, a quarter wave length short-stub and a quarter wavelength open-stub. This filter offers wide passband, with high out-of-band rejection and suppression of second spurious harmonic. Design procedure of a filter at 1GHz with 50% fractional bandwidth is presented showing good agreement between simulation and e...

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. In this paper we introduce a new graph associated to modules over commutative rings. We study the relationship between the algebraic properties of modules and their associated graphs. A topological characterization for the completeness of the special subgraphs is presented. Also modules whose associated graph is complete...

We produced morph sequences between identities at a variety of viewpoints, ranging from the three quarter leftward facing view, to the three quarter rightward facing view. We measured the strength of identity adaptation as a function of changing test viewpoint whilst keep the adaptation viewpoint constant, and as a function of adaptation viewpoint whilst keeping test viewpoint constant. Our res...