217 research outputs found
On the degree of regularity of generalized van der Waerden triples
Let be integers. A triple of the form ,
where are positive integers is called an {\em (a,b)-triple}. The {\em
degree of regularity} of the family of all -triples, denoted dor(,
is the maximum integer such that every -coloring of admits
a monochromatic -triple. We settle, in the affirmative, the conjecture
that dor for all . We also disprove the
conjecture that dor( for all .Comment: 5 page
Enumeration of three term arithmetic progressions in fixed density sets
Additive combinatorics is built around the famous theorem by Szemer\'edi
which asserts existence of arithmetic progressions of any length among the
integers. There exist several different proofs of the theorem based on very
different techniques. Szemer\'edi's theorem is an existence statement, whereas
the ultimate goal in combinatorics is always to make enumeration statements. In
this article we develop new methods based on real algebraic geometry to obtain
several quantitative statements on the number of arithmetic progressions in
fixed density sets. We further discuss the possibility of a generalization of
Szemer\'edi's theorem using methods from real algebraic geometry.Comment: 62 pages. Update v2: Corrected some references. Update v3:
Incorporated feedbac
Volume of the set of unistochastic matrices of order 3 and the mean Jarlskog invariant
A bistochastic matrix B of size N is called unistochastic if there exists a
unitary U such that B_ij=|U_{ij}|^{2} for i,j=1,...,N. The set U_3 of all
unistochastic matrices of order N=3 forms a proper subset of the Birkhoff
polytope, which contains all bistochastic (doubly stochastic) matrices. We
compute the volume of the set U_3 with respect to the flat (Lebesgue) measure
and analytically evaluate the mean entropy of an unistochastic matrix of this
order. We also analyze the Jarlskog invariant J, defined for any unitary matrix
of order three, and derive its probability distribution for the ensemble of
matrices distributed with respect to the Haar measure on U(3) and for the
ensemble which generates the flat measure on the set of unistochastic matrices.
For both measures the probability of finding |J| smaller than the value
observed for the CKM matrix, which describes the violation of the CP parity, is
shown to be small. Similar statistical reasoning may also be applied to the MNS
matrix, which plays role in describing the neutrino oscillations. Some
conjectures are made concerning analogous probability measures in the space of
unitary matrices in higher dimensions.Comment: 33 pages, 6 figures version 2 - misprints corrected, explicit
formulae for phases provide
Random strategies are nearly optimal for generalized van der Waerden Games
In a (1 : q) Maker-Breaker game, one of the central questions is to find (or at least estimate) the maximal value of q that allows Maker to win the game. Based on the ideas of Bednarska and Luczak [Bednarska, M., and T. Luczak, Biased positional games for which random strategies are nearly optimal, Combinatorica, 20 (2000), 477–488], who studied biased H-games, we prove general winning criteria for Maker and Breaker and a hypergraph generalization of their result. Furthermore, we study the biased version of a strong generalization of the van der Waerden games introduced by Beck [Beck, J., Van der Waerden and Ramsey type games, Combinatorica, 1 (1981), 103–116] and apply our criteria to determine the threshold bias of these games up to constant factor. As in the result of [Bednarska, M., and T. Luczak, Biased positional games for which random strategies are nearly optimal, Combinatorica, 20 (2000), 477–488], the random strategy for Maker is again the best known strategy.Postprint (updated version
Non-Rectangular Convolutions and (Sub-)Cadences with Three Elements
The discrete acyclic convolution computes the 2n-1 sums sum_{i+j=k; (i,j) in
[0,1,2,...,n-1]^2} (a_i b_j) in O(n log n) time. By using suitable offsets and
setting some of the variables to zero, this method provides a tool to calculate
all non-zero sums sum_{i+j=k; (i,j) in (P cap Z^2)} (a_i b_j) in a rectangle P
with perimeter p in O(p log p) time.
This paper extends this geometric interpretation in order to allow arbitrary
convex polygons P with k vertices and perimeter p. Also, this extended
algorithm only needs O(k + p(log p)^2 log k) time.
Additionally, this paper presents fast algorithms for counting sub-cadences
and cadences with 3 elements using this extended method
A sagbi basis for the quantum Grassmannian
The maximal minors of a p by (m + p) matrix of univariate polynomials of
degree n with indeterminate coefficients are themselves polynomials of degree
np. The subalgebra generated by their coefficients is the coordinate ring of
the quantum Grassmannian, a singular compactification of the space of rational
curves of degree np in the Grassmannian of p-planes in (m + p)-space. These
subalgebra generators are shown to form a sagbi basis. The resulting flat
deformation from the quantum Grassmannian to a toric variety gives a new
`Gr\"obner basis style' proof of the Ravi-Rosenthal-Wang formulas in quantum
Schubert calculus. The coordinate ring of the quantum Grassmannian is an
algebra with straightening law, which is normal, Cohen-Macaulay, Gorenstein and
Koszul, and the ideal of quantum Pl\"ucker relations has a quadratic Gr\"obner
basis. This holds more generally for skew quantum Schubert varieties. These
results are well-known for the classical Schubert varieties (n=0). We also show
that the row-consecutive p by p-minors of a generic matrix form a sagbi basis
and we give a quadratic Gr\"obner basis for their algebraic relations.Comment: 18 pages, 3 eps figure, uses epsf.sty. Dedicated to the memory of
Gian-Carlo Rot
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