scholarly journals Double Hurwitz Numbers and Multisingularity Loci in Genus 0

2021 ◽  
Author(s):  
Maxim Kazarian ◽  
Sergey Lando ◽  
Dimitri Zvonkine
Keyword(s):  
Differential Equation ◽  
Recursion Relation ◽  
Rational Functions ◽  
Hurwitz Numbers ◽  
Hurwitz Space

Abstract In the Hurwitz space of rational functions on ${{\mathbb{C}}}\textrm{P}^1$ with poles of given orders, we study the loci of multisingularities, that is, the loci of functions with a given ramification profile over 0. We prove a recursion relation on the Poincaré dual cohomology classes of these loci and deduce a differential equation on Hurwitz numbers.

Exact Solutions for Time-Fractional Ramani and Jimbo—Miwa Equations by Direct Algebraic Method

2020 ◽  
Vol 12 (7) ◽  
pp. 982-988
Author(s):  
Serbay Duran
Keyword(s):  
Differential Equation ◽  
Three Dimensional ◽  
Rational Functions ◽  
Algebraic Method ◽  
Fractional Partial Differential Equations ◽  
Fractional Partial Differential Equation ◽  
Partial Differential ◽  
Higher Dimensional ◽  
Liouville Derivative ◽  
Three Dimensional Simulations

In this study, the analytical solutions of some nonlinear time-fractional partial differential equations are investigated by the direct algebraic method. The nonlinear fractional partial differential equation (NLfPDE) which is based on the fractional derivative (fd) in the sense of modified Riemann-Liouville derivative is transformed to the nonlinear non-fractional ordinary differential equation. The hyperbolic and rational functions which are contained solutions are obtained for the sixth-order time-fractional Ramani equation and time-fractional Jimbo—Miwa equation (JME) with the help of this technique. In addition, this method can be applied to higher order and higher dimensional NLfPDEs. Finally, three dimensional simulations of some solutions are given.

scholarly journals Monotone Hurwitz Numbers in Genus Zero

2013 ◽  
Vol 65 (5) ◽  
pp. 1020-1042 ◽  
Author(s):  
I. P. Goulden ◽  
Mathieu Guay-Paquet ◽  
Jonathan Novak
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Asymptotic Expansion ◽  
Symmetric Functions ◽  
Initial Conditions ◽  
Riemann Sphere ◽  
Branched Covers ◽  
Genus Zero ◽  
Hurwitz Numbers ◽  
Arbitrary Genus

AbstractHurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers related to the expansion of complete symmetric functions in the Jucys–Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra–Itzykson–Zuber integral. In this paper we begin a detailed study of monotone Hurwitz numbers. We prove two results that are reminiscent of those for classical Hurwitz numbers. The first is the monotone join-cut equation, a partial differential equation with initial conditions that characterizes the generating function for monotone Hurwitz numbers in arbitrary genus. The second is our main result, in which we give an explicit formula for monotone Hurwitz numbers in genus zero.

scholarly journals X. On linear differential equations.—No. V

1871 ◽  
Vol 19 (123-129) ◽  
pp. 526-528
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Rational Functions ◽  
Linear Differential Equations ◽  
Linear Differential

Let us now endeavour to ascertain under what circumstance a linear differential equation admits a solution of the form P log e Q, where P and Q are rational functions of ( x ).

scholarly journals Counting quadrant walks via Tutte's invariant method (extended abstract)

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Olivier Bernardi ◽  
Mireille Bousquet-Mélou ◽  
Kilian Raschel
Keyword(s):  
Differential Equation ◽  
Functional Equation ◽  
Linear Differential Equation ◽  
Generating Function ◽  
Algebraic Approach ◽  
Rational Functions ◽  
The Past ◽  
Lattice Walks ◽  
International Audience ◽  
Linear Differential

Extended abstract presented at the conference FPSAC 2016, Vancouver. International audience In the 1970s, Tutte developed a clever algebraic approach, based on certain " invariants " , to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks confined to the first quadrant is governed by similar equations, and has led in the past decade to a rich collection of attractive results dealing with the nature (algebraic, D-finite or not) of the associated generating function, depending on the set of allowed steps. We first adapt Tutte's approach to prove (or reprove) the algebraicity of all quadrant models known or conjectured to be algebraic (with one small exception). This includes Gessel's famous model, and the first proof ever found for one model with weighted steps. To be applicable, the method requires the existence of two rational functions called invariant and decoupling function respectively. When they exist, algebraicity comes out (almost) automatically. Then, we move to an analytic viewpoint which has already proved very powerful, leading in particular to integral expressions of the generating function in the non-D-finite cases, as well as to proofs of non-D-finiteness. We develop in this context a weaker notion of invariant. Now all quadrant models have invariants, and for those that have in addition a decoupling function, we obtain integral-free expressions of the generating function, and a proof that this series is differentially algebraic (that is, satisfies a non-linear differential equation).

III. On linear differential equations.—No. VI

1873 ◽  
Vol 21 (139-147) ◽  
pp. 14-19
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Explicit Solution ◽  
Rational Functions ◽  
Linear Differential Equations ◽  
Linear Differential

We now consider linear differential equations which are satisfied by the roots of an algebraical equation admitting of explicit solution. To determine in what cases the linear differential equation P d n y / dx n + Q d n -1 y / dx n -1 + . ..+ R y = 0 is satisfied by assuming y = m √X + r √Y + s √Z + ... when X, Y, Z are rational functions of ( x ).

scholarly journals Analytical Solutions of Fractional Klein-Gordon and Gas Dynamics Equations, via the (G′/G)-Expansion Method

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 566 ◽  
Author(s):  
Hassan Khan ◽  
Shoaib Barak ◽  
Poom Kumam ◽  
Muhammad Arif
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Gas Dynamics ◽  
Rational Functions ◽  
Expansion Method ◽  
Travelling Wave ◽  
Gas Dynamics Equations ◽  
Klein Gordon

In this article, the ( G ′ / G ) -expansion method is used for the analytical solutions of fractional-order Klein-Gordon and Gas Dynamics equations. The fractional derivatives are defined in the term of Jumarie’s operator. The proposed method is based on certain variable transformation, which transforms the given problems into ordinary differential equations. The solution of resultant ordinary differential equation can be expressed by a polynomial in ( G ′ / G ) , where G = G ( ξ ) satisfies a second order linear ordinary differential equation. In this paper, ( G ′ / G ) -expansion method will represent, the travelling wave solutions of fractional-order Klein-Gordon and Gas Dynamics equations in the term of trigonometric, hyperbolic and rational functions.

On One Class of Algebraic Differential Equations

Vestnik MEI ◽  
2021 ◽  
pp. 135-137
Author(s):  
Irina N. Dorofeeva ◽  
◽  
Viktoriya A. Podkopaeva ◽  
Aleksandr Ya. Yanchenko ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Part ◽  
Rational Functions ◽  
Second Order ◽  
Algebraic Differential Equations ◽  
All Solutions ◽  
Order Algebraic ◽  
Integer Solutions ◽  
Linear Differential

The article addresses second-order algebraic differential equations that have a separated linear part and admit a finite-order integer function as a solution. All possible integer solutions of such equations are described. It is shown that all solutions are the solutions of certain second-order linear differential equations the coefficients of which are represented by rational functions. It has been demonstrated that any such integer function y = f(z) is either a solution of the algebraic equation R(z, exp{Q(z)}, y) ≡ 0 (where R is a polynomial of three variables, and Q(z) is a polynomial of one variable), or a solution of a differential equation with separable variables y′ = a(z)y (for some rational function a(z)).

scholarly journals Liouvillian solutions of second order differential equation without Fuchsian singularities

1986 ◽  
Vol 103 ◽  
pp. 145-148 ◽  
Author(s):  
Michihiko Matsuda
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Rational Functions ◽  
Second Order ◽  
Complex Numbers ◽  
First Order ◽  
Independent Variable ◽  
Linear Differential ◽  
Homogeneous Linear ◽  
Homogeneous Linear Differential Equation

Consider a homogeneous linear differential equation of the second order whose coefficients are rational functions of the independent variable x over the field C of complex numbers. We assume that the coefficient of the first order derivative vanishes:

scholarly journals Permutable functions concerning differential equations

2007 ◽  
Vol 83 (3) ◽  
pp. 369-384 ◽  
Author(s):  
X. Hua ◽  
R. Vaillancourt ◽  
X. L. Wang
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Entire Functions ◽  
Julia Set ◽  
Rational Functions ◽  
Exponential Functions ◽  
Polynomial Coefficients ◽  
Transcendental Entire Functions ◽  
Linear Differential ◽  
Open Question

AbstractLet f and g be two permutable transcendental entire functions. Assume that f is a solution of a linear differential equation with polynomial coefficients. We prove that, under some restrictions on the coefficients and the growth of f and g, there exist two non-constant rational functions R1 and R2 such that R1 (f) = R(g). As a corollary, we show that f and g have the same Julia set: J(f) = J(g). As an application, we study a function f which is a combination of exponential functions with polynomial coefficients. This research addresses an open question due to Baker.

Sign in / Sign up

Export Citation Format

Share Document