scholarly journals Are even maps on surfaces likely to be bipartite?

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Guillaume Chapuy
Keyword(s):  
Asymptotic Behaviour ◽  
Planar Map ◽  
Generating Series ◽  
Finite Set ◽  
Positive Genus ◽  
International Audience ◽  
Special Case

International audience It is well known that a planar map is bipartite if and only if all its faces have even degree (what we call an even map). In this paper, we show that rooted even maps of positive genus $g$ chosen uniformly at random are bipartite with probability tending to $4^{−g}$ when their size goes to infinity. Loosely speaking, we show that each of the $2g$ fundamental cycles of the surface of genus $g$ contributes a factor $\frac{1}{2}$ to this probability.We actually do more than that: we obtain the explicit asymptotic behaviour of the number of even maps and bipartite maps of given genus with any finite set of allowed face degrees. This uses a generalisation of the Bouttier-Di Francesco-Guitter bijection to the case of positive genus, a decomposition inspired by previous works of Marcus, Schaeffer and the author, and some involved manipulations of generating series counting paths. A special case of our results implies former conjectures of Gao.

scholarly journals Asymptotic Enumeration of Constellations and Related Families of Maps on Orientable Surfaces

2009 ◽  
Vol 18 (4) ◽  
pp. 477-516 ◽  
Author(s):  
GUILLAUME CHAPUY
Keyword(s):  
Graph Theory ◽  
Asymptotic Formulas ◽  
Asymptotic Enumeration ◽  
Algebraic Graph Theory ◽  
Lattice Walks ◽  
Generating Series ◽  
Finite Set ◽  
Special Case ◽  
Orientable Surfaces

We perform the asymptotic enumeration of two classes of rooted maps on orientable surfaces:m-hypermaps andm-constellations. Form= 2 they correspond respectively to maps with even face degrees and bipartite maps. We obtain explicit asymptotic formulas for the number of such maps with any finite set of allowed face degrees.Our proofs combine a bijective approach, generating series techniques related to lattice walks, and elementary algebraic graph theory.A special case of our results implies former conjectures of Z. Gao.

SH pulse in an elastic wedge

1973 ◽  
Vol 63 (5) ◽  
pp. 1571-1582
Author(s):  
A. M. Abo-Zena ◽  
Chi-Yu King
Keyword(s):  
Integral Transform ◽  
Wedge Angle ◽  
Integral Form ◽  
Diffraction Effect ◽  
Elastic Wedge ◽  
Sh Waves ◽  
Wedge Surface ◽  
Line Parallel ◽  
Finite Set ◽  
Special Case

abstract This paper gives an analysis of the response of an elastic wedge of arbitrary angle to an impulsive SH source applied on the wedge surface along a line parallel to the edge of the wedge. A two-dimensional time-dependent Green's function for SH waves is constructed from an integral-transform approach. The result is given in a closed form for the incident and the reflected pulses and in an integral form for the diffracted pulse from the edge. For the special case that the wedge angle is an integral fraction of π, the result is interpretable in terms of a finite set of image sources with no diffraction effect. Numerical examples are given for illustration.

scholarly journals Counting occurrences for a finite set of words: an inclusion-exclusion approach

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Frédérique Bassino ◽  
Julien Clément ◽  
J. Fayolle ◽  
P. Nicodème
Keyword(s):  
Generating Function ◽  
Exclusion Principle ◽  
Complete Proof ◽  
Detailed Proof ◽  
Finite Set ◽  
International Audience ◽  
The One ◽  
Maple Package

International audience In this paper, we give the multivariate generating function counting texts according to their length and to the number of occurrences of words from a finite set. The application of the inclusion-exclusion principle to word counting due to Goulden and Jackson (1979, 1983) is used to derive the result. Unlike some other techniques which suppose that the set of words is reduced (<i>i..e.</i>, where no two words are factor of one another), the finite set can be chosen arbitrarily. Noonan and Zeilberger (1999) already provided a MAPLE package treating the non-reduced case, without giving an expression of the generating function or a detailed proof. We give a complete proof validating the use of the inclusion-exclusion principle and compare the complexity of the method proposed here with the one using automata for solving the problem.

scholarly journals Graphs with many vertex-disjoint cycles

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Dieter Rautenbach ◽  
Friedrich Regen
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Linear Time ◽  
Simple Extension ◽  
Disjoint Cycles ◽  
Cyclomatic Number ◽  
Finite Set ◽  
Extension Rule ◽  
International Audience ◽  
Vertex Disjoint

Graph Theory International audience We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.

scholarly journals Constrained exchangeable partitions

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Alexander Gnedin
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Partitions ◽  
Type Representation ◽  
International Audience ◽  
De Finetti ◽  
Infinite Set ◽  
Special Case

International audience For a class of random partitions of an infinite set a de Finetti-type representation is derived, and in one special case a central limit theorem for the number of blocks is shown.

scholarly journals Asymptotic behaviour of ideals relative to injective modules over commutative Noetherian rings

1991 ◽  
Vol 34 (1) ◽  
pp. 155-160 ◽  
Author(s):  
H. Ansari Toroghy ◽  
R. Y. Sharp
Keyword(s):  
Asymptotic Behaviour ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Finite Set ◽  
Attached Prime ◽  
Secondary Representation

LetEbe an injective module over the commutative Noetherian ringA, and letabe an ideal ofA. TheA-module (0:Eα) has a secondary representation, and the finite set AttA(0:Eα) of its attached prime ideals can be formed. One of the main results of this note is that the sequence of sets (AttA(0:Eαn))n∈Nis ultimately constant. This result is analogous to a theorem of M. Brodmann that, ifMis a finitely generatedA-module, then the sequence of sets (AssA(M/αnM))n∈Nis ultimately constant.

scholarly journals Small sets containing any pattern

2018 ◽  
Vol 168 (1) ◽  
pp. 57-73
Author(s):  
URSULA MOLTER ◽  
ALEXIA YAVICOLI
Keyword(s):  
Hausdorff Measure ◽  
Analogous Result ◽  
Measure Zero ◽  
General Construction ◽  
Dimension Function ◽  
Perfect Set ◽  
Finite Set ◽  
Isolated Points ◽  
Special Case

AbstractGiven any dimension function h, we construct a perfect set E ⊆ ${\mathbb{R}}$ of zero h-Hausdorff measure, that contains any finite polynomial pattern.This is achieved as a special case of a more general construction in which we have a family of functions $\mathcal{F}$ that satisfy certain conditions and we construct a perfect set E in ${\mathbb{R}}^N$, of h-Hausdorff measure zero, such that for any finite set {f1,. . .,fn} ⊆ $\mathcal{F}$, E satisfies that $\bigcap_{i=1}^n f^{-1}_i(E)\neq\emptyset$.We also obtain an analogous result for the images of functions. Additionally we prove some related results for countable (not necessarily finite) intersections, obtaining, instead of a perfect set, an $\mathcal{F}_{\sigma}$ set without isolated points.

scholarly journals Unicity theorems for meromorphic or entire functions II

1995 ◽  
Vol 52 (2) ◽  
pp. 215-224 ◽  
Author(s):  
Hong-Xun Yi
Keyword(s):  
Meromorphic Functions ◽  
Entire Functions ◽  
Inverse Image ◽  
Finite Sets ◽  
Finite Set ◽  
Special Case

In 1976, Gross posed the question “can one find two (or possibly even one) finite sets Sj (j = 1, 2) such that any two entire functions f and g satisfying Ef(Sj) = Eg(Sj) for j = 1,2 must be identical?”, where Ef(Sj) stands for the inverse image of Sj under f. In this paper, we show that there exists a finite set S with 11 elements such that for any two non-constant meromorphic functions f and g the conditions Ef(S) = Eg(S) and Ef({∞}) = Eg({∞}) imply f ≡ g. As a special case this also answers the question posed by Gross.

On positive entire solutions of the elliptic equation Δu + K(x)up = 0 and its applications to Riemannian geometry

1996 ◽  
Vol 126 (2) ◽  
pp. 225-237 ◽  
Author(s):  
Changfeng Gui
Keyword(s):  
Elliptic Equation ◽  
Asymptotic Behaviour ◽  
Scalar Curvature ◽  
Riemannian Geometry ◽  
Semilinear Elliptic Equation ◽  
Entire Space ◽  
Entire Solutions ◽  
Curvature Equation ◽  
Semilinear Elliptic ◽  
Special Case

We study the existence and asymptotic behaviour of positive solutions of a semilinear elliptic equation in entire space. A special case of this equation is the scalar curvature equation which arises in Riemannian geometry.

The (p,q)-potent ranks of certain semigroups of transformations

2016 ◽  
Vol 16 (07) ◽  
pp. 1750138
Author(s):  
Ping Zhao ◽  
Taijie You ◽  
Huabi Hu
Keyword(s):  
Partial Transformation ◽  
Transformation Semigroups ◽  
Generating Set ◽  
Idempotent Rank ◽  
Finite Set ◽  
Special Case

Let [Formula: see text] and [Formula: see text] be the partial transformation and the strictly partial transformation semigroups on the finite set [Formula: see text]. It is well known that the ranks of the semigroups [Formula: see text] and [Formula: see text] are [Formula: see text], for [Formula: see text], and [Formula: see text], for [Formula: see text], respectively. The idempotent rank, defined as the smallest number of idempotents generating set, of the semigroup [Formula: see text] has the same value as the rank. Idempotent can be seen as a special case (with [Formula: see text]) of [Formula: see text]-potent. In this paper, we determine the [Formula: see text]-potent ranks, defined as the smallest number of [Formula: see text]-potents generating set, of the semigroups [Formula: see text], for [Formula: see text], and [Formula: see text], for [Formula: see text].

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