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Mireille Bousquet-mélou - One of the best experts on this subject based on the ideXlab platform.
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An Elementary Solution of Gessel's walks in the quadrant
Advances in Mathematics, 2016Co-Authors: Mireille Bousquet-mélouAbstract:Abstract Around 2000, Ira Gessel conjectured that the number of lattice walks in the quadrant N 2 , starting and ending at the origin ( 0 , 0 ) and taking their steps in { → , ↗ , ← , ↙ } had a simple hypergeometric form. In the following decade, this problem was recast in the systematic study of walks with small steps (that is, steps in { − 1 , 0 , 1 } 2 ) confined to the quadrant. The generating functions of such walks are archetypal Solutions of partial discrete differential equations. A complete classification of quadrant walks according to the nature of their generating function (algebraic, D-finite or not) is now available, but Gessel's walks remained mysterious because they were the only model among the 23 D-finite ones that had not been given an Elementary Solution. Instead, Gessel's conjecture was first proved using an inventive computer algebra approach in 2008. A year later, the associated three-variate generating function was proved to be algebraic by a computer algebra tour de force. This was re-proved recently using elaborate complex analysis machinery. We give here an Elementary and constructive proof. Our approach also solves other quadrant models (with multiple steps) recently proved to be algebraic via computer algebra.
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An Elementary Solution of Gessel's walks in the quadrant
arXiv: Combinatorics, 2015Co-Authors: Mireille Bousquet-mélouAbstract:Around 2000, Ira Gessel conjectured that the number of lattice walks in the quadrant N^2, starting and ending at the origin (0,0) and taking their steps in {E,NE,W,SW} had a simple hypergeometric form. In the following decade, this problem was recast in the systematic study of walks with small steps (that is,steps in {-1,0,1}^2) confined to the quadrant. The generating functions of such walks are archetypal Solutions of partial discrete differential equations.A complete classification of quadrant walks according to the nature of their generating function(algebraic, D-finite or not) is now available, but Gessel'swalks remained mysterious because they were the only model among the 23D-finite ones that had not been given an ElementarySolution. Instead, Gessel's conjecture was first proved usingan inventive computer algebra approach in 2008. A year later, the associated three-variate generating function was proved to be algebraic by a computer algebra tour de force. This was re-proved recently using elaborate complex analysis machinery. We give here an Elementary and constructive proof. Our approach also solves other quadrant models (with multiple steps) recently proved to be algebraic via computer algebra.
Lennart Schmidt - One of the best experts on this subject based on the ideXlab platform.
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The non-abelian self-dual string
Letters in Mathematical Physics, 2020Co-Authors: Christian Sämann, Lennart SchmidtAbstract:We argue that the relevant higher gauge group for the non-abelian generalization of the self-dual string equation is the string 2-group. We then derive the corresponding equations of motion and discuss their properties. The underlying geometric picture is a string structure, i.e., a categorified principal bundle with connection whose structure 2-group is the string 2-group. We readily write down the explicit Elementary Solution to our equations, which is the categorified analogue of the ’t Hooft–Polyakov monopole. Our Solution passes all the relevant consistency checks; in particular, it is globally defined on $$\mathbb {R}^4$$ R 4 and approaches the abelian self-dual string of charge one at infinity. We note that our equations also arise as the BPS equations in a recently proposed six-dimensional superconformal field theory and we show that with our choice of higher gauge structure, the action of this theory can be reduced to four-dimensional supersymmetric Yang–Mills theory.
Andreas Seidelmorgenstern - One of the best experts on this subject based on the ideXlab platform.
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theoretical analysis of the influence of forced and inherent temperature fluctuations in an adiabatic chromatographic column
Chemical Engineering Science, 2017Co-Authors: Shamsul Qamar, Fouzia Abdul Sattar, Iqra Batool, Andreas SeidelmorgensternAbstract:Abstract A linearized non-isothermal equilibrium dispersive model (EDM) of liquid chromatography is investigated to quantify unavoidable thermal effects in adiabatic chromatographic columns. The considered model contains convection-diffusion partial differential equations (PDEs) for mass and energy balances in the mobile phase coupled with an algebraic equation for adsorption isotherm. The Solution process successively employ Laplace transformation and linear transformation steps to uncouple the governing set of coupled differential equations. The resulting uncoupled systems of ordinary differential equations are solved using an Elementary Solution technique. The Solutions are very useful to understand the speeds and shapes of concentration and thermal fronts in chromatographic columns. The moment generating property of the Laplace domain Solutions is utilized to derive analytical temporal moments of the concentration and temperature profiles. These moments are seen as useful to estimate unknown model parameters from measured profiles. For illustration several case studies of practical interest are provided. To evaluate the range of applicability of analytical Solutions, selected results are compared with numerical results applying a high reSolution finite volume scheme considering nonlinear isotherms.
Sudprathai Bupasiri - One of the best experts on this subject based on the ideXlab platform.
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on the Elementary Solution for the partial differential operator circledcirc_c k related to the wave equation
European Journal of Pure and Applied Mathematics, 2018Co-Authors: Sudprathai BupasiriAbstract:In this article, we defined the operator $\diamondsuit _{m,c}^{k}$ which is iterated $k$-times and is defined by $$\diamondsuit _{m,c}^{k}=\left[\left(\frac{1}{c^2}\sum_{i=1}^{p}\frac{\partial ^{2}}{\partial x_{i}^{2}} +\frac{m^{2}}{2}\right)^{2} - \left(\sum_{j=p+1}^{p+q}\frac{\partial ^{2}}{\partial x_{j}^{2}} - \frac{m^{2}}{2}\right)^{2}\right]^{k},$$ where $m$ is a nonnegative real number, $c$ is a positive real number and $p+q=n$ is the dimension of the $n$-dimensional Euclidean space $\mathbb{R}^{n}$, $x=(x_{1},\ldots x_{n})\in\mathbb{R}^{n}$ and $k$ is a nonnegative integer. We obtain a causal and anticausal Solution of the operator $\diamondsuit _{m,c}^{k}$, iterated $k$-times.
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On the Elementary Solution of the Operator ♦ k Related to the Ultra-hyperbolic Klein Gordon
2013Co-Authors: Sudprathai BupasiriAbstract:In this paper, we study the Elementary Solution of the operator ♦ k which is iterated k-times and is defined by
Louis Boutet De Monvel - One of the best experts on this subject based on the ideXlab platform.
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On the Holonomic Character of the Elementary Solution of a Partial Differential Operator
New Trends in Microlocal Analysis, 1997Co-Authors: Louis Boutet De MonvelAbstract:We describe an Elementary regular holonomic system of partial differential equations which should be satisfied by an Elementary Solution of a differential operator P(d) with constant coefficients and simple characteristics. This is heuristic, but it is exact for strictly hyperbolic operators, and exact mod. holomorphic functions for elliptic operators or operators with real principal part. For these this explains again why the Elementary Solution extends holomorphically, with the expected ramification
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On the Holonomic Character of the Elementary Solution of a Partial Differential Operator En l'honneur de H. Komatsu, pour son soixantieme anniversaire
1997Co-Authors: Louis Boutet De MonvelAbstract:We describe an Elementary regular holonomic system of partial dif ferential equations which should be satisfied by an Elementary Solution of a differential operator P(d) with constant coefficients and simple characteristics. This is heuristic, but it is exact for strictly hyperbolic operators, and exact mod. holomorphic functions for elliptic operators or operators with real principal part. For these this explains again why the Elementary Solution extends holomorphi cally, with the expected ramification