Abstract:
Bisztriczky introduced the multiplex as a generalization of the simplex. A polytope is multiplicial if all its faces are multiplexes. In this paper it is proved that the flag vectors of multiplicial polytopes depend only on their face vectors. A special class of multiplicial polytopes, also discovered by Bisztriczky, is comprised of the ordinary polytopes. These are a natural generalization of the cyclic polytopes. The flag vectors of ordinary polytopes are determined. This is used to give a surprisingly simple formula for the h-vector of the ordinary d-polytope with n+1 vertices and characteristic k: h_i=binom{k-d+i}{i}+(n-k)binom{k-d+i-1}{i-1}, for i at most d/2. In addition, a construction is given for 4-dimensional multiplicial polytopes having two-thirds of their vertices on a single facet, answering a question of Bisztriczky.

Abstract:
A graded partially ordered set is Eulerian if every interval has the same number of elements of even rank and of odd rank. Face lattices of convex polytopes are Eulerian. For Eulerian partially ordered sets, the flag vector can be encoded efficiently in the cd-index. The cd-index of a polytope has all positive entries. An important open problem is to give the broadest natural class of Eulerian posets having nonnegative cd-index. This paper completely determines which entries of the cd-index are nonnegative for all Eulerian posets. It also shows that there are no other lower or upper bounds on cd-coefficients (except for the coefficient of c^n).

Abstract:
Ordinary polytopes were introduced by Bisztriczky as a (nonsimplicial) generalization of cyclic polytopes. We show that the colex order of facets of the ordinary polytope is a shelling order. This shelling shares many nice properties with the shellings of simplicial polytopes. We also give a shallow triangulation of the ordinary polytope, and show how the shelling and the triangulation are used to compute the toric h-vector of the ordinary polytope. As one consequence, we get that the contribution from each shelling component to the h-vector is nonnegative. Another consequence is a combinatorial proof that the entries of the h-vector of any ordinary polytope are simple sums of binomial coefficients.

Abstract:
The closed cone of flag vectors of Eulerian partially ordered sets is studied. It is completely determined up through rank seven. Half-Eulerian posets are defined. Certain limit posets of Billera and Hetyei are half-Eulerian; they give rise to extreme rays of the cone for Eulerian posets. A new family of linear inequalities valid for flag vectors of Eulerian posets is given.

Abstract:
A partially ordered set is r-thick if every nonempty open interval contains at least r elements. This paper studies the flag vectors of graded, r-thick posets and shows the smallest convex cone containing them is isomorphic to the cone of flag vectors of all graded posets. It also defines a k-analogue of the Mobius function and k-Eulerian posets, which are 2k-thick. Several characterizations of k-Eulerian posets are given. The generalized Dehn-Sommerville equations are proved for flag vectors of k-Eulerian posets. A new inequality is proved to be valid and sharp for rank 8 Eulerian posets.

Abstract:
Cyclic polytopes are characterized as simplicial polytopes satisfying Gale's evenness condition (a combinatorial condition on facets relative to a fixed ordering of the vertices). Periodically-cyclic polytopes are polytopes for which certain subpolytopes are cyclic. Bisztriczky discovered a class of periodically-cyclic polytopes that also satisfy Gale's evenness condition. The faces of these polytopes are braxtopes, a certain class of nonsimplicial polytopes studied by the authors. In this paper we prove that the periodically-cyclic Gale polytopes of Bisztriczky are exactly the polytopes that satisfy Gale's evenness condition and are braxial (all faces are braxtopes). The existence of other periodically-cyclic Gale polytopes is open.

Abstract:
In a d-simplex every facet is a (d-1)-simplex. We consider as generalized simplices other combinatorial classes of polytopes, all of whose facets are in the class. Cubes and multiplexes are two such classes of generalized simplices. In this paper we study a new class, braxtopes, which arise as the faces of periodically-cyclic Gale polytopes. We give a geometric construction for these polytopes and various combinatorial properties.

Abstract:
Este trabalho apresenta uma análise de séries temporais dos dados de temperatura mínima e temperatura máxima mensal da cidade de Erechim, RS; apresenta-se uma compara o de duas classes de modelos tradicionais de previs o, nomeadamente: modelos da classe ARIMA e modelos de alisamento exponencial. Na classe de modelos ARIMA foram selecionados, utilizando-se critérios de informa o, modelos do tipo SARIMA, que consideram a característica sazonal da temperatura do ar; já para os modelos de alisamento exponencial utilizaram-se os modelos Holt-Winters aditivo, em que as constantes de alisamento s o determinadas de forma a minimizar o erro quadrático médio entre valores previstos e observados; esta análise permitiu a identifica o de componentes como sazonalidade e períodos atípicos. Os modelos de previs o foram comparados para diferentes horizontes de previs o, sendo que os modelos da classe ARIMA se mostraram mais acurados. Os modelos ajustados se mostraram adequados para tra ar previs es das variáveis de temperatura do ar, mostrando-se importantes ferramentas para a climatologia agrícola. This paper presents a time series analysis of the minimum and maximum air temperature of Erechim, RS. A comparison between two traditional classes of the forecasting models, namely: ARIMA models and exponential smoothing models is also presented. In the class of ARIMA models using criteria information, SARIMA type models that consider the seasonal characteristics of air temperature were selected, whereas for exponential smoothing models Holt-Winters additive algorithm were used. Smoothing constants are determined to minimize the mean square error between observed and predicted values. This analysis allowed the identification of components such as seasonality and atypical periods. The model predictions were compared for different forecast horizons. The ARIMA class models proved to be more accurate while the adjusted models were adequate for adjusting forecasts of variables of air temperature, being important tools for agricultural climatology.