scholarly journals Asymptotic enumeration of orientations

2010 ◽  
Vol Vol. 12 no. 2 ◽  
Author(s):  
Stefan Felsner ◽  
Eric Fusy ◽  
Marc Noy
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Singularity Analysis ◽  
Linear Differential Equations ◽  
Asymptotic Enumeration ◽  
International Audience ◽  
Asymptotic Number ◽  
Complex Linear ◽  
Linear Differential ◽  
Prime 2

International audience We find the asymptotic number of 2-orientations of quadrangulations with n inner faces, and of 3-orientations of triangulations with n inner vertices. We also find the asymptotic number of prime 2-orientations (no separating quadrangle) and prime 3-orientations (no separating triangle). The estimates we find are of the form c . n(-alpha)gamma(n), for suitable constants c, alpha, gamma with alpha = 4 for 2-orientations and alpha = 5 for 3-orientations. The proofs are based on singularity analysis of D-finite generating functions, using the Fuchsian theory of complex linear differential equations.

scholarly journals On [p,q]-order of growth of solutions of complex linear differential equations near a singular point

Filomat ◽  
2019 ◽  
Vol 33 (13) ◽  
pp. 4013-4020
Author(s):  
Jianren Long ◽  
Sangui Zeng
Keyword(s):  
Differential Equation ◽  
Singular Point ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Linear Differential Equations ◽  
Order Of Growth ◽  
Complex Linear ◽  
Linear Differential ◽  
Growth Of Solutions

We investigate the [p,q]-order of growth of solutions of the following complex linear differential equation f(k)+Ak-1(z) f(k-1) + ...+ A1(z) f? + A0(z) f = 0, where Aj(z) are analytic in C? - {z0}, z0 ? C. Some estimations of [p,q]-order of growth of solutions of the equation are obtained, which is generalization of previous results from Fettouch-Hamouda.

scholarly journals V. On the calculus of symbols, with applications to the theory of differential equations

1861 ◽  
Vol 151 ◽  
pp. 69-82 ◽  
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Variable Coefficients ◽  
Linear Differential Equations ◽  
New Methods ◽  
Methods Of Calculation ◽  
Great Utility ◽  
Long Time ◽  
Constant Coefficients ◽  
Linear Differential

The calculus of generating functions, discovered by Laplace, was, as is well known, highly instrumental in calling the attention of mathematicians to the analogy which exists between differentials and powers. This analogy was perceived at length to involve an essential identity, and several analysts devoted themselves to the improvement of the new methods of calculation which were thus called into existence. For a long time the modes of combination assumed to exist between different classes of symbols were those of ordinary algebra; and this sufficed for investigations respecting functions of differential coefficients and constants, and consequently for the integration of linear differential equations, with constant coefficients. The laws of combination of ordinary algebraical symbols may be divided into the commutative and distributive laws; and the number of symbols in the higher branches of mathematics, which are commutative with respect to one another, is very small. It became then necessary to invent an algebra of non-commutative symbols. This important step was effected by Professor Boole, for certain classes of symbols, in his well-known and beautiful memoir published in the Transactions of this Society for the year 1844, and the object of the paper which I have now the honour to lay before the Society is to perfect and develope the methods there employed. For this purpose I have constructed systems of multiplication and division for functions of non-commutative symbols, subject to the same laws of combination as those assumed in Professor Boole’s memoir, and I thus arrive at equations of great utility in the integration of linear differential equations with variable coefficients.

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