scholarly journals Additive structures on f-vector sets of polytopes

2020 ◽  
Vol 20 (2) ◽  
pp. 217-231 ◽  
Author(s):  
Günter M. Ziegler
Keyword(s):  
Additive Structure ◽  
Simple Polytopes ◽  
Simplicial Polytopes ◽  
Affine Semigroup

AbstractWe show that the f-vector sets of d-polytopes have non-trivial additive structure: They span affine lattices and are embedded in monoids that we describe explicitly. Moreover, for many large subclasses, such as the simple polytopes, or the simplicial polytopes, there are monoid structures on the set of f-vectors by themselves: “addition of f-vectors minus the f-vector of the d-simplex” always yields a new f-vector. For general 4-polytopes, we show that the modified addition operation does not always produce an f-vector, but that the result is always close to an f-vector. In this sense, the set of f-vectors of all 4-polytopes forms an “approximate affine semigroup”. The proof relies on the fact for d = 4 every d-polytope, or its dual, has a “small facet”. This fails for d > 4.We also describe a two further modified addition operations on f-vectors that can be geometrically realized by glueing corresponding polytopes. The second one of these may yield a semigroup structure on the f-vector set of all 4-polytopes.

Joint continuity of affine semigroup actions

1980 ◽  
Vol 21 (1) ◽  
pp. 153-165 ◽  
Author(s):  
Dietrich Helmer
Keyword(s):  
Semigroup Actions ◽  
Joint Continuity ◽  
Affine Semigroup

scholarly journals Bigraded Betti numbers of certain simple polytopes

2013 ◽  
Vol 94 (3-4) ◽  
pp. 351-363 ◽  
Author(s):  
I. Yu. Limonchenko
Keyword(s):  
Betti Numbers ◽  
Simple Polytopes

Dualizing Complexes of Affine Semigroup Rings

1990 ◽  
Vol 322 (2) ◽  
pp. 561 ◽  
Author(s):  
Uwe Schafer ◽  
Peter Schenzel
Keyword(s):  
Semigroup Rings ◽  
Affine Semigroup ◽  
Dualizing Complexes

Dagger extension theorem

2011 ◽  
Vol 21 (5) ◽  
pp. 1035-1066 ◽  
Author(s):  
Z. ÉSIK ◽  
T. HAJGATÓ
Keyword(s):  
Unique Solution ◽  
Extension Theorem ◽  
Sufficient Condition ◽  
Iteration Theory ◽  
Additive Structure ◽  
Algebraic Theories ◽  
Constant Morphism

Partial iterative theories are algebraic theories such that for certain morphisms f the equation ξ = f ⋅ 〈ξ, 1p〉 has a unique solution. Iteration theories are algebraic theories satisfying a certain set of identities. We investigate some similarities between partial iterative theories and iteration theories.In our main result, we give a sufficient condition ensuring that the partially defined dagger operation of a partial iterative theory can be extended to a totally defined operation so that the resulting theory becomes an iteration theory. We show that this general extension theorem can be instantiated to prove that every Elgot iterative theory with at least one constant morphism 1 → 0 can be extended to an iteration theory. We also apply our main result to theories equipped with an additive structure.

scholarly journals Divisor class groups of affine semigroup rings associated with distributive lattices

1992 ◽  
Vol 149 (2) ◽  
pp. 352-357 ◽  
Author(s):  
Mitsuyasu Hashimoto ◽  
Takayuki Hibi ◽  
Atsushi Noma
Keyword(s):  
Distributive Lattices ◽  
Semigroup Rings ◽  
Class Groups ◽  
Divisor Class ◽  
Affine Semigroup

scholarly journals Small covers, infra-solvmanifolds and curvature

2015 ◽  
Vol 27 (5) ◽  
Author(s):  
Shintarô Kuroki ◽  
Mikiya Masuda ◽  
Li Yu
Keyword(s):  
Sectional Curvature ◽  
Riemannian Metrics ◽  
Flat Torus ◽  
Combinatorial Structures ◽  
Simple Polytope ◽  
Simple Polytopes ◽  
Small Cover ◽  
Equivalent Conditions

AbstractIt is shown that a small cover (resp. real moment-angle manifold) over a simple polytope is an infra-solvmanifold if and only if it is diffeomorphic to a real Bott manifold (resp. flat torus). Moreover, we obtain several equivalent conditions for a small cover to be homeomorphic to a real Bott manifold. In addition, we study Riemannian metrics on small covers and real moment-angle manifolds with certain conditions on the Ricci or sectional curvature. We will see that these curvature conditions put very strong restrictions on the topology of the corresponding small covers and real moment-angle manifolds and the combinatorial structures of the underlying simple polytopes.

scholarly journals Predictive Constructions Based on Measure-Valued Pólya Urn Processes

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2845
Author(s):  
Sandra Fortini ◽  
Sonia Petrone ◽  
Hristo Sariev
Keyword(s):  
Polish Space ◽  
Asymptotic Properties ◽  
Transition Kernel ◽  
Urn Model ◽  
Additive Structure ◽  
Predictive Distributions ◽  
Pólya Urn ◽  
Random Weights ◽  
Random Transition ◽  
Polya Urn

Measure-valued Pólya urn processes (MVPP) are Markov chains with an additive structure that serve as an extension of the generalized k-color Pólya urn model towards a continuum of possible colors. We prove that, for any MVPP (μn)n≥0 on a Polish space X, the normalized sequence (μn/μn(X))n≥0 agrees with the marginal predictive distributions of some random process (Xn)n≥1. Moreover, μn=μn−1+RXn, n≥1, where x↦Rx is a random transition kernel on X; thus, if μn−1 represents the contents of an urn, then Xn denotes the color of the ball drawn with distribution μn−1/μn−1(X) and RXn—the subsequent reinforcement. In the case RXn=WnδXn, for some non-negative random weights W1,W2,…, the process (Xn)n≥1 is better understood as a randomly reinforced extension of Blackwell and MacQueen’s Pólya sequence. We study the asymptotic properties of the predictive distributions and the empirical frequencies of (Xn)n≥1 under different assumptions on the weights. We also investigate a generalization of the above models via a randomization of the law of the reinforcement.

scholarly journals A sweep-plane algorithm for generating random tuples in simple polytopes

1998 ◽  
Vol 67 (224) ◽  
pp. 1617-1636 ◽  
Author(s):  
Josef Leydold ◽  
Wolfgang Hörmann
Keyword(s):  
Simple Polytopes
Sign in / Sign up

Export Citation Format

Share Document