Extreme k-families

AbstractLet P be a poset. A subset A of P is a k-family iff A contains no (k + 1)-element chain. For i ⩾ 1, let Ai be the set of elements of A at depth i − 1 in A. The k-families of P can be ordered by defining A ⩽ B iff, for all i, Ai is included in the order ideal generated by Bi. This paper examines minimal r-element k-families, defined as k-families A such that |A| = r and for every B < A, |B| < r. Minimal k-families are related to maximal r-antichains and an operation called Sperner closure, which have been used to obtain extremal results for families of sets with width restrictions. Let Mk,r be the set of minimal r-element k-families and let Mk = ∪r ≥ 0 Mk,r. It is shown that Mk is a join-subsemilattice by the lattice Ak of k-families. Mk is a lower semimodular lattice, where the rth rank is given by Mk,r. If wk is the maximum size of a k-family, then |Mk,r| ⩽ (wrk)and |∪Mk| ⩽ Σi = 1wk ⌈i/k⌉. Let D(A) = max{|B| − |A| | B is a k-family and B ⩽ A}. For k-families A and B, D(A v B) ⩽ D(A) + D(B). This result shows that {A | D(A) = 0} is also a join-subsemilattice of Ak

Extreme rays of the (N,k)(N, k)-Schur Cone

We discuss several partial results towards proving Dennis White's conjecture on the extreme rays of the $(N,2)$-Schur cone. We are interested in which vectors are extreme in the cone generated by all products of Schur functions of partitions with $k$ or fewer parts. For the case where $k =2$, White conjectured that the extreme rays are obtained by excluding a certain family of "bad pairs," and proved a special case of the conjecture using Farkas' Lemma. We present an alternate proof of the special case, in addition to showing more infinite families of extreme rays and reducing White's conjecture to two simpler conjectures.Comment: This paper has been withdrawn by the authors due to a misinterpretation of the generalized Littlewood-Richardson rule in several proof

How tangents solve algebraic equations, or a remarkable geometry of discriminant varieties

Let $\mathcal D_{d,k}$ denote the discriminant variety of degree $d$ polynomials in one variable with at least one of its roots being of multiplicity $\geq k$. We prove that the tangent cones to $\mathcal D_{d,k}$ span $\mathcal D_{d,k-1}$ thus, revealing an extreme ruled nature of these varieties. The combinatorics of the web of affine tangent spaces to $\mathcal D_{d,k}$ in $\mathcal D_{d,k-1}$ is directly linked to the root multiplicities of the relevant polynomials. In fact, solving a polynomial equation $P(z) = 0$ turns out to be equivalent to finding hyperplanes through a given point P(z)\in \mathcal D_{d,1} \approx \A^d which are tangent to the discriminant hypersurface $\mathcal D_{d,2}$. We also connect the geometry of the Vi\`{e}te map \mathcal V_d: \A^d_{root} \to \A^d_{coef}, given by the elementary symmetric polynomials, with the tangents to the discriminant varieties $\{\mathcal D_{d,k}\}$. Various $d$-partitions $\{\mu\}$ provide a refinement $\{\mathcal D_\mu^\circ\}$ of the stratification of \A^d_{coef} by the $\mathcal D_{d,k}$'s. Our main result, Theorem 7.1, describes an intricate relation between the divisibility of polynomials in one variable and the families of spaces tangent to various strata $\{\mathcal D_\mu^\circ\}$.Comment: 43 pages, 12 figure

Hidden tail chains and recurrence equations for dependence parameters associated with extremes of higher-order Markov chains

We derive some key extremal features for kth order Markov chains, which can be used to understand how the process moves between an extreme state and the body of the process. The chains are studied given that there is an exceedance of a threshold, as the threshold tends to the upper endpoint of the distribution. Unlike previous studies with k>1 we consider processes where standard limit theory describes each extreme event as a single observation without any information about the transition to and from the body of the distribution. The extremal properties of the Markov chain at lags up to k are determined by the kernel of the chain, through a joint initialisation distribution, with the subsequent values determined by the conditional independence structure through a transition behaviour. We study the extremal properties of each of these elements under weak assumptions for broad classes of extremal dependence structures. For chains with k>1, these transitions involve novel functions of the k previous states, in comparison to just the single value, when k=1. This leads to an increase in the complexity of determining the form of this class of functions, their properties and the method of their derivation in applications. We find that it is possible to find an affine normalization, dependent on the threshold excess, such that non-degenerate limiting behaviour of the process is assured for all lags. These normalization functions have an attractive structure that has parallels to the Yule-Walker equations. Furthermore, the limiting process is always linear in the innovations. We illustrate the results with the study of kth order stationary Markov chains based on widely studied families of copula dependence structures.Comment: 35 page

The EUVE point of view of AD Leo

All the Extreme Ultraviolet Explorer (EUVE) observations of AD Leo, totalling 1.1 Ms of exposure time, have been employed to analyze the corona of this single M dwarf. The light curves show a well defined quiescent stage, and a distribution of amplitude of variability following a power law with a ~-2.4 index. The flaring behavior exhibits much similarity with other M active stars like FK Aqr or YY Gem, and flares behave differently from late type active giants and subgiants. The Emission Measure Distribution (EMD) of the summed spectrum, as well as that of quiescent and flaring stages, were obtained using a line-based method. The average EMD is dominated by material at log T(K)~6.9, with a second peak around log T(K)~6.3, and a large increase in the amount of material with log T(K)>~7.1 during flares, material almost absent during quiescence. The results are interpreted as the combination of three families of loops with maximum temperatures at log T(K)~6.3, ~6.9 and somewhere beyond log T(K)>~7.1. A value of the abundance of [Ne/Fe]=1.05+-0.08 was measured at log T(K)~5.9. No significative increment of Neon abundance was detected between quiescence and flaring states.Comment: Full PS version can be found also at http://www.astropa.unipa.it/~jsanz/papers0002.htm

On extremal measures for conservative particle systems

ABSTRACT. – It is well known that the exclusion, zero-range and misanthrope particle systems possess families of invariant measures due to the mass conservation property. Although these families have been classified a great deal, a full characterization of their extreme points is not available. In this article, we consider an approach to the study of this classification. One of the results in this note is that the zero-range product invariant measures, ∏ i∈S µα(·), for an infinite countable set S, under mild conditions, are identified as extremal for α(·) ∈ HZR where µα(i)(k) = Z(α(i)) −1α(i) k /g(1) ···g(k) with g and Z the rate function and normalization respectively, and HZR is the set of invariant measures for the transition probability p. © 2001 Éditions scientifiques et médicales Elsevier SA