A metric graph is a pair (G, d), where G and $$d:E(G) \rightarrow \mathbb {R}_{\ge 0}$$ distance function. Let $$p \in [1,\infty ]$$ be fixed. An isometric embedding of the d) in $$\ell _p^k = (\mathbb {R}^k, d_p)$$ map $$\phi :V(G) {R}^k$$ such that $$d_p(\phi (v), \phi (w)) d(vw)$$ for all edges $$vw\in E(G)$$ . The _p$$ -dimension least integer k there exists an _p^k$$ functions d has _p^K$$...

We investigate a problem about certain walks in the ring of Gaussian integers. Let n and d be two natural numbers. Does there exist a sequence of Gaussian integers zj , such that |zj+1 zj | = 1 and a pair of indices r and s, such that zr zs = n and for all indices t and u, zt zu 6= d? If there exists such a sequence, we say that n is d avoidable. Let An be the set of all d 2 N such that n is no...

We give a new proof of the four-color theorem by exhibiting an unavoidable set of 2822 D-reducible configurations. The existence of such a set had been conjectured by several researchers including Stromquist [18], Appel and Haken [2] and Robertson, Sanders, Seymour and Thomas [15].

We prove that, for each positive integer k, every sufficiently large 3-connected regular matroid has a parallel minor isomorphic to M∗(K3,k), M(Wk), M(Kk), the cycle matroid of the graph obtained from K2,k by adding paths through the vertices of each vertex class, or the cycle matroid of the graph obtained from K3,k by adding a complete graph on the vertex class with three vertices.

We introduce the notion of unavoidable (complete) sets of word patterns, which is a refinement for that of words, and study certain numerical characteristics for unavoidable sets of patterns. In some cases we employ the graph of pattern overlaps introduced in this paper, which is a subgraph of the de Bruijn graph and which we prove to be Hamiltonian. In other cases we reduce a problem under con...

A parallel minor is obtained from a graph by any sequence of edge contractions and parallel edge deletions. We prove that, for any positive integer k, every internally 4-connected graph of sufficiently high order contains a parallel minor isomorphic to a variation of K4,k with a complete graph on the vertices of degree k, the k-partition triple fan with a complete graph on the vertices of degre...

An improved one-dimensional (1-D) analysis of CMOS photodiode has been derived in which the effect of the substrate, which forms a high–low junction with the epitaxial layer, has been included. The analytical solution was verified with numerical simulations based on parameters extracted from a standard 0.35 m CMOS process. Two empirical parameters are suggested to offset the unavoidable inaccur...

The unavoidable diet change in emerging countries, projected for the coming years, will significantly increase the global consumption of animal protein. It is expected that Brazilian livestock production, responsible for close to 15% of global production, be prepared to answer to the increasing demand of beef. Consequently, the evaluation of pasture quality at regional scale is important to inf...

TThe present paper evaluates the plan of combustion air pre-heater installation on the fired heater from thermodynamics and thermos-economics point of view. As a real case study, one of the fired heaters (H_101) of Distillation unit in Tehran Oil Refinery, Iran, is intended. With applying an air pre-heater in this study, flue gases temperature falls down from 430 ºC to 200ºCand combustion a...

Simple permutations are the building blocks of permutation classes. As such, classes with only finitely many simple permutations, e.g., the class of 132-avoiding permutations, have nice properties. To name three: these classes have algebraic generating functions (as established by Albert and Atkinson [1]; see Brignall, Huczynska, and Vatter [9] for extensions), are partially well-ordered (see t...