In this paper, we study the conjecture of Kühnel and Lutz, who state that a combinatorial triangulation of the product of two spheres
S
i
×
S
j
with
j
≥
i
is tight if and only if it has exactly
i
+
2
...j
+
4
vertices. To approach this conjecture, we use graded Betti numbers of Stanley–Reisner rings. By using recent results on graded Betti numbers, we prove that the only if part of the conjecture holds when
j
>
2
i
and that the if part of the conjecture holds for triangulations all whose vertex links are simplicial polytopes. We also apply this algebraic approach to obtain lower bounds on the numbers of vertices and edges of triangulations of manifolds and pseudomanifolds.
The Specht ideal of shape λ, where λ is a partition, is the ideal generated by all Specht polynomials of shape λ. Haiman and Woo proved that these ideals are reduced and found their universal Gröbner ...bases. In this short note, we give a short proof for these results.
Intensive care unit (ICU) survivors after traumatic brain injury (TBI) frequently have serious disabilities with subsequent difficulty in reintegration into society. We aimed to investigate outcomes ...for ICU survivors after moderate to severe TBI (msTBI) and to identify predictive factors of return home (RH) and return to work (RTW). This single-center retrospective cohort study was conducted on all trauma patients admitted to the emergency ICU of our hospital between 2013 and 2017. Of these patients, adult (age ≥ 18 years) msTBI patients with head Abbreviated Injury Scale ≥ 3 were extracted. We performed univariate/multivariate logistic regression analyses to explore the predictive factors of RH and RTW. Among a total of 146 ICU survivors after msTBI, 107 were included (median follow-up period: 26 months). The RH and RTW rates were 78% and 35%, respectively. Multivariate analyses revealed that the predictive factors of RH were age < 65 years (P < 0.001), HR < 76 bpm (P = 0.015), platelet count ≥ 19× 104/μL (P = 0.0037), D-dimer < 26 μg/mL (P = 0.034), and Glasgow Coma Scale (GCS) score > 8 (P = 0.0015). Similarly, the predictive factors of RTW were age < 65 years (P < 0.001) and GCS score > 8 (P = 0.0039). This study revealed that "age" and "GCS score on admission" affected RH and RTW for ICU survivors after msTBI.
Recently, it was proved by Anari–Oveis Gharan–Vinzant, Anari–Liu–Oveis Gharan–Vinzant and Brändén–Huh that, for any matroid M, its basis generating polynomial and its independent set generating ...polynomial are log-concave on the positive orthant. Using these, they obtain some combinatorial inequalities on matroids including a solution of strong Mason's conjecture. In this paper, we study the strictness of the log-concavity of these polynomials and determine when equality holds in these combinatorial inequalities. We also consider a generalization of our result to morphisms of matroids.
Balanced subdivisions and flips on surfaces MURAI, SATOSHI; SUZUKI, YUSUKE
Proceedings of the American Mathematical Society,
03/2018, Letnik:
146, Številka:
3
Journal Article
Recenzirano
Odprti dostop
In this paper, we show that two balanced triangulations of a closed surface are not necessarily connected by a sequence of balanced stellar subdivisions and welds. This answers a question posed by ...Izmestiev, Klee and Novik. We also show that two balanced triangulations of a closed surface are connected by a sequence of three local operations, which we call the pentagon contraction, the balanced edge subdivision and the balanced edge weld. In addition, we prove that two balanced triangulations of the 2-sphere are connected by a sequence of pentagon contractions and their inverses if none of them are the octahedral sphere.
Let
n
be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in
G
L
(
n
,
C
)
/
B
such that its ...associated graded ring has graded pieces (i.e., homogeneous components) isomorphic to rings which are related to the cohomology rings of Hessenberg varieties in
G
L
(
n
-
1
,
C
)
/
B
, showing the inductive nature of these rings. In previous work, the first two authors, together with Abe and Masuda, gave an explicit presentation of these cohomology rings in terms of generators and relations. We introduce a new set of polynomials which are closely related to the relations in the above presentation and obtain a sequence of equivalence relations they satisfy; this allows us to derive our filtration. In addition, we obtain the following three corollaries. First, we give an inductive formula for the Poincaré polynomial of these varieties. Second, we give an explicit monomial basis for the cohomology rings of regular nilpotent Hessenberg varieties with respect to the presentation mentioned above. Third, we derive a basis of the set of linear relations satisfied by the images of the Schubert classes in the cohomology rings of regular nilpotent Hessenberg varieties. Finally, our methods and results suggest many directions for future work; in particular, we propose a definition of “Hessenberg Schubert polynomials” in the context of regular nilpotent Hessenberg varieties, which generalize the classical Schubert polynomials. We also outline several open questions pertaining to them.
In this paper, we study simplicial complexes whose Stanley–Reisner rings are almost Gorenstein and have a-invariant zero. We call such a simplicial complex an almost Gorenstein* simplicial complex. ...To study the almost Gorenstein* property, we introduce a new class of simplicial complexes which we call uniformly Cohen–Macaulay simplicial complexes. A d-dimensional simplicial complex Δ is said to be uniformly Cohen–Macaulay if it is Cohen–Macaulay and, for any facet F of Δ, the simplicial complex Δ∖{F} is Cohen–Macaulay of dimension d. We investigate fundamental algebraic, combinatorial and topological properties of these simplicial complexes, and show that almost Gorenstein* simplicial complexes must be uniformly Cohen–Macaulay. By using this fact, we show that every almost Gorenstein* simplicial complex can be decomposed into those of having one dimensional top homology. Also, we give a combinatorial criterion of the almost Gorenstein* property for simplicial complexes of dimension ≤2.
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