a b s t r a c tGenome rearrangement and homological recombination processes have been modeled by Angeleska et al. [A. Angeleska, N. Jonoska, M. Saito, DNA recombinations through assembly graphs, Discrete Appl. Math. 157 (2009) 3020-3037] as 4-regular spacial graphs with rigid vertices, called assembly graphs. These graphs can also be represented by double occurrence words called assembly words. The rearranged DNA segments are modeled by certain types of paths in the assembly graphs called polygonal paths. The minimum number of polygonal paths visiting all vertices in a graph is called an assembly number for the graph.In this paper, we give formulas for counting certain types of assembly graphs and assembly words. Some of these formulas produce sequences not previously reported at the Online Encyclopedia of Integer Sequences (http://oeis.org). We provide a sharp upper bound for the number of polygonal paths in Hamiltonian sets of polygonal paths, and present a family of graphs that achieves this bound. We investigate changes in the assembly numbers as a result of graph compositions. Finally, we introduce a polynomial invariant for assembly graphs and show properties of this invariant.
The Mulatu numbers were introduced in [1]. The numbers are sequences of numbers of the form: 4,1,5,6,11,17,28,45... The numbers have wonderful and amazing properties and patterns.In mathematical terms, the sequence ofthe Mulatu numbers is defined by the following recurrence relation:The double Angel Formulas for Fibonacci and Lucas numbers are given by the following formulasrespectively.
Mathematical Subject Classification: 11
In this paper, a mathematical model that describes the flow of gas in a pipe is formulated. The model is simplified by making some assumptions. It is considered that the natural gas flowing in a long horizontal pipe, no heat source occurs inside the volume, transfer of heat due to heat conduction is dominated by heat exchange with the surrounding. The flow equations are coupled with equation of state. Different types of equations of state, ranging from the simple Ideal gas law to the more complex equation of state Benedict Webb Rubin Starling (BWRS), are considered. The flow equations are solved numerically using the Godunov scheme with Roe solver. Some numerical results are also presented.
In this paper we discuss the uniqueness and existence of solution to a real gas flow network by employing graph theory. A directed graph is an efficient way to represent a gas network. We consider steady state real gas flow network that includes pipelines, compressors, and the connectors. The pipelines and compressors are represented as edges of the graph and the interconnecting points are represented as nodes of the graph representing the network. We show that a unique solution of such a system exists. We use monotonicity property of a mapping to proof uniqueness, and the contraction mapping theorem is used to prove existence.
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