scholarly journals Bijections between Łukasiewicz Walks and Generalized Tandem Walks

10.37236/8261 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Frédéric Chyzak ◽  
Karen Yeats
Keyword(s):  
Recent Work ◽  
Half Plane ◽  
Square Lattice ◽  
Plane Model ◽  
North West ◽  
Quarter Plane ◽  
Upper Half Plane

In this article, we study the enumeration by length of several walk models on the square lattice. We obtain bijections between walks in the upper half-plane returning to the $x$-axis and walks in the quarter plane. A recent work by Bostan, Chyzak, and Mahboubi has given a bijection for models using small north, west, and south-east steps. We adapt and generalize it to a bijection between half-plane walks using those three steps in two colours and a quarter-plane model over the symmetrized step set consisting of north, north-west, west, south, south-east, and east. We then generalize our bijections to certain models with large steps: for given $p\geq1$, a bijection is given between the half-plane and quarter-plane models obtained by keeping the small south-east step and replacing the two steps north and west of length 1 by the $p+1$ steps of length $p$ in directions between north and west. This model is close to, but distinct from, the model of generalized tandem walks studied by Bousquet-Mélou, Fusy, and Raschel.

scholarly journals Sorting with two stacks in parallel

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Michael Albert ◽  
Mireille Bousquet-Mélou
Keyword(s):  
Generating Function ◽  
Square Lattice ◽  
North West ◽  
International Audience ◽  
Asymptotic Number ◽  
Upper Half Plane ◽  
Recent Activity ◽  
The 1960S ◽  
The Moment ◽  
Forbidden Patterns

International audience At the end of the 1960s, Knuth characterised in terms of forbidden patterns the permutations that can be sorted using a stack. He also showed that they are in bijection with Dyck paths and thus counted by the Catalan numbers. Subsequently, Pratt and Tarjan asked about permutations that can be sorted using two stacks in parallel. This question is significantly harder, and the associated counting question has remained open for 40 years. We solve it by giving a pair of equations that characterise the generating function of such permutations. The first component of this system describes the generating function $Q(a,u)$ of square lattice loops confined to the positive quadrant, counted by the length and the number of North-West and East-South factors. Our analysis of the asymptotic number of sortable permutations relies at the moment on two intriguing conjectures dealing with this series. Given the recent activity on walks confined to cones, we believe them to be attractive $\textit{per se}$. We prove these conjectures for closed walks confined to the upper half plane, or not confined at all. Nous énumérons les permutations triables par deux piles en parallèle. Cette question était restée ouverte depuis les travaux de Knuth, Pratt et Tarjan dans les années 70. Notre solution consiste en une paire d’équations qui caractérisent la série génératrice. La première composante de ce système décrit la série $Q(a,u)$ des chemins fermés confinés dans le quart de plan positif, comptés selon leur longueur et le nombre de facteurs Nord-Ouest ou Est-Sud. Notre analyse du comportement asymptotique du nombre de permutations triables repose à ce stade sur deux conjectures remarquables portant sur $Q(a; u)$. Nous les prouvons pour les chemins fermés non confinés, ou confinés au demi-plan supérieur.

scholarly journals Construction with ruler and compass, van Hiele model for geometry thinking and Project-based learning

2021 ◽  
Vol 9 (9) ◽  
pp. 335-341
Author(s):  
Nafaa Chbili
Keyword(s):  
Half Plane ◽  
Project Based Learning ◽  
Geometric Construction ◽  
Plane Model ◽  
Van Hiele ◽  
Upper Half Plane ◽  
Van Hiele Model

We describe how problems of geometric construction using straightedge and compass can be introduced to students through project-based learning. We discuss how these problems can be extended to the upper half-plane model. Furthermore, we discuss the use of these problems to assess advanced levels in van Hiele model for geometry thinking.

scholarly journals On Complex Explicit Formulae Connected with the Möbius Function of an Elliptic Curve

2014 ◽  
Vol 57 (2) ◽  
pp. 381-389
Author(s):  
Adrian Łydka
Keyword(s):  
Functional Equation ◽  
Meromorphic Function ◽  
Elliptic Curve ◽  
Explicit Formula ◽  
Half Plane ◽  
Complex Plane ◽  
Möbius Function ◽  
Mobius Function ◽  
Upper Half Plane ◽  
Explicit Formulae

AbstractWe study analytic properties function m(z, E), which is defined on the upper half-plane as an integral from the shifted L-function of an elliptic curve. We show that m(z, E) analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for m(z, E) in the strip |ℑz| < 2π.

Exact determination of gravity stresses in finite elastic slopes: Part I. Theoretical considerations

1983 ◽  
Vol 20 (1) ◽  
pp. 47-54 ◽  
Author(s):  
V. Silvestri ◽  
C. Tabib
Keyword(s):  
Plane Strain ◽  
Half Plane ◽  
Inclination Angle ◽  
Complex Variable ◽  
Earth Pressure ◽  
Exact Distributions ◽  
Exact Determination ◽  
Upper Half Plane ◽  
Theoretical Considerations

The exact distributions of gravity stresses are obtained within slopes of finite height inclined at various angles, −β (β = π/2, π/3, π/4, π/6, and π/8), to the horizontal. The solutions are obtained by application of the theory of a complex variable. In homogeneous, isotropic, and linearly elastic slopes under plane strain conditions, the gravity stresses are independent of Young's modulus and are a function of (a) the coordinates, (b) the height, (c) the inclination angle, (d) Poisson's ratio or the coefficient of earth pressure at rest, and (e) the volumetric weight. Conformal applications that transform the planes of the various slopes studied onto the upper half-plane are analytically obtained. These solutions are also represented graphically.

scholarly journals Revisiting the Siegel upper half plane II

2004 ◽  
Vol 376 ◽  
pp. 45-67 ◽  
Author(s):  
Pedro J. Freitas ◽  
Shmuel Friedland
Keyword(s):  
Half Plane ◽  
Upper Half Plane

An AF Algebra Associated with the Farey Tessellation

2008 ◽  
Vol 60 (5) ◽  
pp. 975-1000 ◽  
Author(s):  
Florin P. Boca
Keyword(s):  
Half Plane ◽  
Path Algebra ◽  
Cutting Sequences ◽  
Upper Half Plane ◽  
Af Algebras ◽  
Algebra Model

AbstractWe associate with the Farey tessellation of the upper half-plane an AF algebra encoding the “cutting sequences” that define vertical geodesics. The Effros–Shen AF algebras arise as quotients of . Using the path algebra model for AF algebras we construct, for each τ ∈ ( 0, ¼], projections (En) in such that EnEn±1En ≤ τ En.

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