scholarly journals On Some Properties of the Star Graph

VLSI Design ◽  
1995 ◽  
Vol 2 (4) ◽  
pp. 389-396 ◽  
Author(s):  
Ke Qiu ◽  
Selim G. Akl
Keyword(s):  
Closed Form ◽  
Initial Conditions ◽  
Closed Form Solution ◽  
Recursive Formula ◽  
Form Solution ◽  
Star Graph ◽  
Star Graphs ◽  
Different Dimensions ◽  
Homogeneous Linear ◽  
Fixed Node

We derive some properties of the star graph in this paper. In particular, we compute the number of nodes at distance i from a fixed node e in a star graph. To this end, a recursive formula is first obtained. This recursive formula is, in general, hard to solve for a closed form solution. We then study the relations among the number of nodes at distance i to node e in star graphs of different dimensions. This study reveals a very interesting relation among these numbers, which leads to a simple homogeneous linear recursive formula whose characteristic equation is easy to solve. Thus, we get a systematic way to obtain a closed form solution with given initial conditions for any fixed i.

scholarly journals Use of a Strongly Nonlinear Gambier Equation for the Construction of Exact Closed Form Solutions of Nonlinear ODEs

2007 ◽  
Vol 2007 ◽  
pp. 1-25
Author(s):  
M. P. Markakis
Keyword(s):  
Closed Form ◽  
Initial Conditions ◽  
Closed Form Solution ◽  
Form Solution ◽  
Nonlinear Odes ◽  
Analytical Technique ◽  
Closed Form Solutions ◽  
Strongly Nonlinear ◽  
Parameter Values

We establish an analytical method leading to a more general form of the exact solution of a nonlinear ODE of the second order due to Gambier. The treatment is based on the introduction and determination of a new function, by means of which the solution of the original equation is expressed. This treatment is applied to another nonlinear equation, subjected to the same general class as that of Gambier, by constructing step by step an appropriate analytical technique. The developed procedure yields a general exact closed form solution of this equation, valid for specific values of the parameters involved and containing two arbitrary (free) parameters evaluated by the relevant initial conditions. We finally verify this technique by applying it to two specific sets of parameter values of the equation under consideration.

Pascal's recurrence relation and even-aged-forest stand dynamics

1992 ◽  
Vol 22 (12) ◽  
pp. 1996-1999
Author(s):  
Rolfe A. Leary ◽  
Hien Phan ◽  
Kevin Nimerfro
Keyword(s):  
Differential Equation ◽  
Relative Density ◽  
Closed Form ◽  
Initial Conditions ◽  
Closed Form Solution ◽  
Integral Form ◽  
Forest Stand ◽  
Form Solution ◽  
Density Change ◽  
Stand Dynamics

A common method of modelling forest stand dynamics is to use permanent growth plot remeasurements to calibrate a whole-stand growth model expressed as an ordinary differential equation. To obtain an estimate of future conditions, either the differential equation is integrated numerically or, if analytic, the differential equation is solved in closed form. In the latter case, a future condition is obtained simply by evaluating the integral form for the age of interest, subject to appropriate initial conditions. An older method of modelling forest stand dynamics was to use a normal or near-normal yield table as a density standard and calibrate a relative density change equation from permanent plot remeasurements. An estimate of a future stand property could be obtained by iterating from a known initial relative density. In this paper we show that when the relative density change equation has a particular form, the historical method also has a closed form solution, given by a sequence of polynomials with coefficients from successive rows of Pascal's arithmetic triangle.

scholarly journals Closed-Form Solutions for Gradient Elastic Beams with Geometric Discontinuities by Laplace Transform

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Mustafa Özgür Yayli
Keyword(s):  
Laplace Transform ◽  
Closed Form ◽  
Initial Conditions ◽  
Closed Form Solution ◽  
Elastic Beam ◽  
Form Solution ◽  
Static Deflection ◽  
Closed Form Solutions ◽  
Geometric Discontinuities ◽  
Algebraic Forms

The static bending solution of a gradient elastic beam with external discontinuities is presented by Laplace transform. Its utility lies in the ability to switch differential equations to algebraic forms that are more easily solved. A Laplace transformation is applied to the governing equation which is then solved for the static deflection of the microbeam. The exact static response of the gradient elastic beam with external discontinuities is obtained by applying known initial conditions when the others are derived from boundary conditions. The results are given in a series of figures and compared with their classical counterparts. The main contribution of this paper is to provide a closed-form solution for the static deflection of microbeams under geometric discontinuities.

scholarly journals Stochastic time-dependent enzyme kinetics: closed-form solution and transient bimodality

2020 ◽  
Author(s):  
James Holehouse ◽  
Augustinas Sukys ◽  
Ramon Grima
Keyword(s):  
Master Equation ◽  
Closed Form ◽  
Initial Conditions ◽  
Closed Form Solution ◽  
Delta Function ◽  
Product Formation ◽  
Form Solution ◽  
Chemical Master Equation ◽  
Time Dependent ◽  
Enzyme Molecules

AbstractWe derive an approximate closed-form solution to the chemical master equation describing the Michaelis-Menten reaction mechanism of enzyme action. In particular, assuming that the probability of a complex dissociating into enzyme and substrate is significantly larger than the probability of a product formation event, we obtain expressions for the time-dependent marginal probability distributions of the number of substrate and enzyme molecules. For delta function initial conditions, we show that the substrate distribution is either unimodal at all times or else becomes bimodal at intermediate times. This transient bimodality, which has no deterministic counterpart, manifests when the initial number of substrate molecules is much larger than the total number of enzyme molecules and if the frequency of enzyme-substrate binding events is large enough. Furthermore, we show that our closed-form solution is different from the solution of the chemical master equation reduced by means of the widely used discrete stochastic Michaelis-Menten approximation, where the propensity for substrate decay has a hyperbolic dependence on the number of substrate molecules. The differences arise because the latter does not take into account enzyme number fluctuations while our approach includes them. We confirm by means of stochastic simulation of all the elementary reaction steps in the Michaelis-Menten mechanism that our closed-form solution is accurate over a larger region of parameter space than that obtained using the discrete stochastic Michaelis-Menten approximation.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations

A Simple Closed-Form Solution of Bending Stiffness for Laminated Composite Tubes

2000 ◽  
Vol 19 (4) ◽  
pp. 278-291 ◽  
Author(s):  
WEN S. CHAN ◽  
KAZIM C. DEMIRHAN
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Laminated Composite ◽  
Bending Stiffness ◽  
Form Solution ◽  
Composite Tubes
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