scholarly journals A tight colored Tverberg theorem for maps to manifolds (extended abstract)

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Pavle V. M. Blagojević ◽  
Benjamin Matschke ◽  
Günter M. Ziegler
Keyword(s):  
Prime Power ◽  
Dimensional Manifold ◽  
Dimensional Simplex ◽  
Continuous Map ◽  
Tverberg Theorem ◽  
International Audience ◽  
Special Case

International audience Any continuous map of an $N$-dimensional simplex $Δ _N$ with colored vertices to a $d$-dimensional manifold $M$ must map $r$ points from disjoint rainbow faces of $Δ _N$ to the same point in $M$, assuming that $N≥(r-1)(d+1)$, no $r$ vertices of $Δ _N$ get the same color, and our proof needs that $r$ is a prime. A face of $Δ _N$ is called a rainbow face if all vertices have different colors. This result is an extension of our recent "new colored Tverberg theorem'', the special case of $M=ℝ^d$. It is also a generalization of Volovikov's 1996 topological Tverberg theorem for maps to manifolds, which arises when all color classes have size 1 (i.e., without color constraints); for this special case Volovikov's proofs, as well as ours, work when r is a prime power. Étant donné un simplex $Δ _N$ de dimension $N$ ayant les sommets colorés, une face de $Δ _N$ est dite arc-en-ciel, si tous les sommets de cette face ont des couleurs différentes. Toute fonction continue d'un simplex $Δ _N$ de dimension $N$ aux sommets colorés vers une variété $d$-dimensionnelle $M$ doit envoyer $r$ points provenant de faces arc-en-ciel disjointes de $Δ _N$ au mêmes points dans $M$ ; en supposant que $N ≥(r-1)(d +1)$, un ensemble de $r$ sommets de $Δ _N$ doit être coloré à l'aide d'au moins deux couleurs. Notre démonstration requiert que $r$ soit un nombre premier. Ce résultat est une extension de notre "nouveau théorème de Tverberg coloré'', le cas particulier où $M = ℝ^d$. Il est également une généralisation du théorème de Tverberg topologique de Volovikov datant de 1996, pour les fonctions vers une variété, dont les classes de couleurs sont de taille 1 (c'est-à-dire sans contraintes de couleur). Dans ce cas particulier, la démonstration de Volovikov et la nôtre fonctionnent lorsque r est une puissance d'un premier.

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 On the monotone hook hafnian conjecture

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Mirkó Visontai
Keyword(s):  
Graph Polynomials ◽  
Real Roots ◽  
Threshold Graphs ◽  
Multivariate Generalization ◽  
International Audience ◽  
Special Case

International audience We investigate a conjecture of Haglund that asserts that certain graph polynomials have only real roots. We prove a multivariate generalization of this conjecture for the special case of threshold graphs. Nous étudions une conjecture de Haglund qui affirme que certaines polynômes des graphes ont uniquement des racines réelles. Nous prouvons une généralisation multivariée de cette conjecture pour le cas particulier des graphes à seuil.

scholarly journals Enumerating (2+2)-free posets by the number of minimal elements and other statistics

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Sergey Kitaev ◽  
Jeffrey Remmel
Keyword(s):  
Generating Function ◽  
Minimal Elements ◽  
Nous Montrons ◽  
International Audience ◽  
Ascent Sequences ◽  
Special Case

International audience A poset is said to be (2+2)-free if it does not contain an induced subposet that is isomorphic to 2+2, the union of two disjoint 2-element chains. In a recent paper, Bousquet-Mélou et al. found, using so called ascent sequences, the generating function for the number of (2+2)-free posets: $P(t)=∑_n≥ 0 ∏_i=1^n ( 1-(1-t)^i)$. We extend this result by finding the generating function for (2+2)-free posets when four statistics are taken into account, one of which is the number of minimal elements in a poset. We also show that in a special case when only minimal elements are of interest, our rather involved generating function can be rewritten in the form $P(t,z)=∑_n,k ≥0 p_n,k t^n z^k = 1+ ∑_n ≥0\frac{zt}{(1-zt)^n+1}∏_i=1^n (1-(1-t)^i)$ where $p_n,k$ equals the number of (2+2)-free posets of size $n$ with $k$ minimal elements. Un poset sera dit (2+2)-libre s'il ne contient aucun sous-poset isomorphe à 2+2, l'union disjointe de deux chaînes à deux éléments. Dans un article récent, Bousquet-Mélou et al. ont trouvé, à l'aide de "suites de montées'', la fonction génératrice des nombres de posets (2+2)-libres: c'est $P(t)=∑_n≥ 0 ∏_i=1^n ( 1-(1-t)^i)$. Nous étendons ce résultat en trouvant la fonction génératrice des posets (\textrm2+2)-libres rendant compte de quatre statistiques, dont le nombre d'éléments minimaux du poset. Nous montrons aussi que lorsqu'on ne s'intéresse qu'au nombre d'éléments minimaux, notre fonction génératrice assez compliquée peut être simplifiée en$P(t,z)=∑_n,k ≥0 p_n,k t^n z^k = 1+ ∑_n ≥0\frac{zt}{(1-zt)^n+1}∏_i=1^n (1-(1-t)^i)$, où $p_n,k$ est le nombre de posets (2+2)-libres de taille $n$ avec $k$ éléments minimaux.

scholarly journals On the minimal distance of a polynomial code

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Peter Pal Pach ◽  
Csaba Szabo
Keyword(s):  
Finite Subset ◽  
60Th Birthday ◽  
Theoretical Problem ◽  
Minimal Distance ◽  
Distribution Of Primes ◽  
Special Issue ◽  
Theoretical Methods ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Special Case

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience For a polynomial f(x) is an element of Z(2)[x] it is natural to consider the near-ring code generated by the polynomials f circle x, f circle x(2) ,..., f circle x(k) as a vectorspace. It is a 19 year old conjecture of Gunter Pilz that for the polynomial f (x) - x(n) broken vertical bar x(n-1) broken vertical bar ... broken vertical bar x the minimal distance of this code is n. The conjecture is equivalent to the following purely number theoretical problem. Let (m) under bar = \1, 2 ,..., m\ and A subset of N be an arbitrary finite subset of N. Show that the number of products that occur odd many times in (n) under bar. A is at least n. Pilz also formulated the conjecture for the special case when A = (k) under bar. We show that for A = (k) under bar the conjecture holds and that the minimal distance of the code is at least n/(log n)(0.223). While proving the case A = (k) under bar we use different number theoretical methods depending on the size of k (respect to n). Furthermore, we apply several estimates on the distribution of primes.

scholarly journals On the class number formula of certain real quadratic fields

2013 ◽  
Vol Volume 36 ◽  
Author(s):  
K Chakraborty ◽  
S Kanemitsu ◽  
T Kuzumaki
Keyword(s):  
Class Number ◽  
Character Sums ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Classical Class ◽  
Class Number Formula ◽  
Number Formula ◽  
International Audience ◽  
Special Case ◽  
Real Quadratic

International audience In this note we give an alternate expression of class number formula for real quadratic fields with discriminant $d \equiv 5\, {\rm mod}\, 8$. %Dirichlet's classical class number formula for real quadratic fields expresses `class number' in somewhat `transcend' manner, which was simplified by P. Chowla in the special case when the discriminant $d = p \equiv 5\,{\rm mod}\, 8$ is a prime. We use another form of class number formula and transform it using Dirichlet's $1/4$-th character sums. Our result elucidates other generalizations of the class number formula by Mitsuhiro, Nakahara and Uhera for general real quadratic fields.

scholarly journals Dual combinatorics of zonal polynomials

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Valentin Féray ◽  
Piotr Sniady
Keyword(s):  
Irreducible Character ◽  
Symmetric Functions ◽  
Symmetric Groups ◽  
Combinatorial Formula ◽  
Zonal Polynomials ◽  
Power Sums ◽  
Power Sum ◽  
Recent Developments ◽  
International Audience ◽  
Special Case

International audience In this paper we establish a new combinatorial formula for zonal polynomials in terms of power-sums. The proof relies on the sign-reversing involution principle. We deduce from it formulas for zonal characters, which are defined as suitably normalized coefficients in the expansion of zonal polynomials in terms of power-sum symmetric functions. These formulas are analogs of recent developments on irreducible character values of symmetric groups. The existence of such formulas could have been predicted from the work of M. Lassalle who formulated two positivity conjectures for Jack characters, which we prove in the special case of zonal polynomials. Dans cet article, nous établissons une nouvelle formule combinatoire pour les polynômes zonaux en fonction des fonctions puissance. La preuve utilise le principe de l'involution changeant les signes. Nous en déduisons des formules pour les caractères zonaux, qui sont définis comme les coefficients des polynômes zonaux écrits sur la base des fonctions puissance, normalisés de manière appropriée. Ces formules sont des analogues de développements récents sur les caractères du groupe symétrique. L'existence de telles formules aurait pu être prédite à partir des travaux de M. Lassalle, qui a proposé deux conjectures de positivité sur les caractères de Jack, que nous prouvons dans le cas particulier des polynômes zonaux.

scholarly journals Rapidly mixing chain and perfect sampler for logarithmic separable concave distributions on simplex

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Shuji Kijima ◽  
Tomomi Matsui
Keyword(s):  
Probability Function ◽  
Discrete Distribution ◽  
Integral Point ◽  
Dimensional Simplex ◽  
Perfect Sampling ◽  
The Past ◽  
International Audience ◽  
Hit And Run ◽  
Multivariate Discrete Distribution ◽  
Dimensional Integral

International audience In this paper, we are concerned with random sampling of an n dimensional integral point on an $(n-1)$ dimensional simplex according to a multivariate discrete distribution. We employ sampling via Markov chain and propose two "hit-and-run'' chains, one is for approximate sampling and the other is for perfect sampling. We introduce an idea of <i>alternating inequalities </i> and show that a <i>logarithmic separable concave</i> function satisfies the alternating inequalities. If a probability function satisfies alternating inequalities, then our chain for approximate sampling mixes in $\textit{O}(n^2 \textit{ln}(Kɛ^{-1}))$, namely $(1/2)n(n-1) \textit{ln}(K ɛ^{-1})$, where $K$ is the side length of the simplex and $ɛ (0<ɛ<1)$ is an error rate. On the same condition, we design another chain and a perfect sampler based on monotone CFTP (Coupling from the Past). We discuss a condition that the expected number of total transitions of the chain in the perfect sampler is bounded by $\textit{O}(n^3 \textit{ln}(Kn))$.

scholarly journals Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Henri Mühle
Keyword(s):  
Coxeter Groups ◽  
Reflection Group ◽  
Sperner Property ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Complex Reflection Group ◽  
Noncrossing Partition ◽  
International Audience ◽  
Partition Lattices ◽  
Special Case

International audience We prove that the noncrossing partition lattices associated with the complex reflection groups G(d, d, n) for d, n ≥ 2 admit a decomposition into saturated chains that are symmetric about the middle ranks. A consequence of this result is that these lattices have the strong Sperner property, which asserts that the cardinality of the union of the k largest antichains does not exceed the sum of the k largest ranks for all k ≤ n. Subsequently, we use a computer to complete the proof that any noncrossing partition lattice associated with a well-generated complex reflection group is strongly Sperner, thus affirmatively answering a special case of a question of D. Armstrong. This was previously established only for the Coxeter groups of type A and B.

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.

The Homological Functors of Some Abelian Groups of Prime Power Order

2014 ◽  
Vol 71 (5) ◽  
Author(s):  
Rosita Zainal ◽  
Nor Muhainiah Mohd Ali ◽  
Nor Haniza Sarmin ◽  
Samad Rashid
Keyword(s):  
Tensor Product ◽  
Prime Power ◽  
Homotopy Theory ◽  
Abelian Groups ◽  
Prime Power Order ◽  
Schur Multiplier ◽  
Integer Coefficients ◽  
Nonabelian Tensor ◽  
Tensor Square ◽  
Special Case

The homological functors of a group were first introduced in homotopy theory. Some of the homological functors including the nonabelian tensor square and the Schur multiplier of abelian groups of prime power order are determined in this paper. The nonabelian tensor square of a group G introduced by Brown and Loday in 1987 is a special case of the nonabelian tensor product. Meanwhile, the Schur multiplier of G is the second cohomology with integer coefficients is named after Issai Schur. The aims of this paper are to determine the nonabelian tensor square and the Schur multiplier of abelian groups of order p5, where p is an odd prime

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