In this article, we prove a finiteness result on the number of log minimal models for 3-folds in
$\operatorname{char}p>5$
. We then use this result to prove a version of Batyrev’s conjecture on the ...structure of nef cone of curves on 3-folds in characteristic
$p>5$
. We also give a proof of the same conjecture in full generality in characteristic 0. We further verify that the duality of movable curves and pseudo-effective divisors hold in arbitrary characteristic. We then give a criterion for the pseudo-effectiveness of the canonical divisor
$K_{X}$
of a smooth projective variety in arbitrary characteristic in terms of the existence of a family of rational curves on
$X$
.
In this paper, we will compute Fourth atom-bond connectivity index ABC_4(G) and Fifth geometric-arithmetic connectivity index GA_5(G), by considering G as para-line graph of some convex polytopes.
It is known that every lattice polytope is unimodularly equivalent to a face of some reflexive polytope. A stronger question is to ask whether every (0,1)-polytope is unimodularly equivalent to a ...facet of some reflexive polytope. A large family of (0,1)-polytopes are the edge polytopes of finite simple graphs. In the present paper, it is shown that, by giving a new class of reflexive polytopes, each edge polytope is unimodularly equivalent to a facet of some reflexive polytope. Furthermore, we extend the characterization of normal edge polytopes to a characterization of normality for these new reflexive polytopes.
A lattice polytope
P
⊂
R
d
is called a locally anti-blocking polytope if for any closed orthant
R
ε
d
in
R
d
,
P
∩
R
ε
d
is unimodularly equivalent to an anti-blocking polytope by reflections of ...coordinate hyperplanes. We give a formula for the
h
∗
-polynomials of locally anti-blocking lattice polytopes. In particular, we discuss the
γ
-positivity of
h
∗
-polynomials of locally anti-blocking reflexive polytopes.
The paper is devoted to perfect and almost perfect homogeneous polytopes in Euclidean spaces. We have classified perfect and almost perfect polytopes among all regular polytopes and all semiregular ...polytopes except Archimedean solids and two four-dimensional Gosset polytopes. Also we have constructed some non-regular homogeneous polytopes that are (or are not) perfect and posed some unsolved questions.
Abstract
A
d
-dimensional lattice polytope
P
is Gorenstein if it has a multiple
rP
that is a reflexive polytope up to translation by a lattice vector. The difference
$$d+1-r$$
d
+
1
-
r
is called the ...degree of
P
. We show that a Gorenstein polytope is a lattice pyramid if its dimension is at least three times its degree. This was previously conjectured by Batyrev and Juny. We also present a refined conjecture and prove it for IDP Gorenstein polytopes.
Motivated by understanding the quality of tractable convex relaxations of intractable polytopes, Ko et al. gave a closed-form expression for the volume of a standard relaxation Q(G) of the Boolean ...quadric polytope (also known as the (full) correlation polytope) P(G) of the complete graph G=Kn. We extend this work to structured sparse graphs. In particular, we (i) demonstrate the existence of an efficient algorithm for vol(Q(G)) when G has bounded treewidth, (ii) give closed-form expressions (and asymptotic behaviors) for vol(Q(G)) for all stars, paths, and cycles, and (iii) give a closed-form expression for vol(P(G)) for all cycles. Further, we demonstrate that when G is a cycle, the simple relaxation Q(G) is a very close model for the much more complicated P(G). Additionally, we give some computational results demonstrating that this behavior of the cycle seems to extend to more complicated graphs. Finally, we speculate on the possibility of extending some of our results to cactii or even series–parallel graphs.
Abstract
The polyhedra with
A
3
,
B
3
/
C
3
,
H
3
reflection symmetry group
G
in the real 3
D
space are considered. The recursive rules for finding orbits with smaller radii, which provide the ...structures of nested polytopes, are demonstrated.
Expansion of random 0/1 polytopes Leroux, Brett; Rademacher, Luis
Random structures & algorithms,
March 2024, 2024-03-00, 20240301, Letnik:
64, Številka:
2
Journal Article
Recenzirano
Odprti dostop
A conjecture of Milena Mihail and Umesh Vazirani (Proc. 24th Annu. ACM Symp. Theory Comput., ACM, Victoria, BC, 1992, pp. 26–38.) states that the edge expansion of the graph of every 0/1$$ 0/1 $$ ...polytope is at least one. Any lower bound on the edge expansion gives an upper bound for the mixing time of a random walk on the graph of the polytope. Such random walks are important because they can be used to generate an element from a set of combinatorial objects uniformly at random. A weaker form of the conjecture of Mihail and Vazirani says that the edge expansion of the graph of a 0/1$$ 0/1 $$ polytope in ℝd$$ {\mathrm{\mathbb{R}}}^d $$ is greater than one over some polynomial function of d$$ d $$. This weaker version of the conjecture would suffice for all applications. Our main result is that the edge expansion of the graph of a random 0/1$$ 0/1 $$ polytope in ℝd$$ {\mathrm{\mathbb{R}}}^d $$ is at least 112d$$ \frac{1}{12d} $$ with high probability.
Open problems on k-orbit polytopes Cunningham, Gabe; Pellicer, Daniel
Discrete mathematics,
June 2018, 2018-06-00, Letnik:
341, Številka:
6
Journal Article
Recenzirano
Odprti dostop
We present 35 open problems on combinatorial, geometric and algebraic aspects of k-orbit abstract polytopes. We also present a theory of rooted polytopes that has appeared implicitly in previous work ...but has not been formalized before.
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