Takuro Abe
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View article: Addition theorems for Ziegler pairs of hyperplane arrangements
Addition theorems for Ziegler pairs of hyperplane arrangements Open
Inspired by Terao's freeness conjecture, we examine Ziegler pairs, which are pairs of hyperplane arrangements that share the same underlying matroid but have different modules of logarithmic derivations. In this paper, we present a general…
View article: Solomon-Terao polynomials and Castelnouvo-Mumford regularity of hyperplane arrangements
Solomon-Terao polynomials and Castelnouvo-Mumford regularity of hyperplane arrangements Open
The Solomon-Terao bi-polynomial was introduced by Solomon and Terao which degenerates to the characteristic polynomial of hyperplane arrangements. Also, it was proved recently that the other specialization of the Solomon-Terao bi-polynomia…
View article: Projective dimension of weakly chordal graphic arrangements
Projective dimension of weakly chordal graphic arrangements Open
A graphic arrangement is a subarrangement of the braid arrangement whose set of hyperplanes is determined by an undirected graph. A classical result due to Stanley, Edelman and Reiner states that a graphic arrangement is free if and only i…
View article: Arrangements, Matroids and Logarithmic Vector Fields
Arrangements, Matroids and Logarithmic Vector Fields Open
The focus of this workshop was on the ongoing interaction between geometric aspects of matroid theory with various directions in the study of hyperplane arrangements. A hyperplane arrangement is exactly a linear realization of a (loop-free…
View article: Baseline inflammatory status affects the prognostic impact of statins in patients with peripheral arterial disease
Baseline inflammatory status affects the prognostic impact of statins in patients with peripheral arterial disease Open
View article: Free Reflection Multiarrangements and Quasi-Invariants
Free Reflection Multiarrangements and Quasi-Invariants Open
To a complex reflection arrangement with an invariant multiplicity function one can relate the space of logarithmic vector fields and the space of quasi-invariants, which are both modules over invariant polynomials. We establish a close re…
View article: Sacubitril/Valsartan: A Game-changer for Hypertensive Patients Beyond Blood Pressure Control?
Sacubitril/Valsartan: A Game-changer for Hypertensive Patients Beyond Blood Pressure Control? Open
View article: Prognostically Optimal Heart Rate at Discharge in Hospitalized Patients With Heart Failure and Atrial Fibrillation
Prognostically Optimal Heart Rate at Discharge in Hospitalized Patients With Heart Failure and Atrial Fibrillation Open
The findings suggest that controlling HR to less than 80 beats/min at discharge may lead to better clinical outcomes in patients with HF and persistent AF.
View article: A Hodge filtration of logarithmic vector fields for well-generated complex reflection groups
A Hodge filtration of logarithmic vector fields for well-generated complex reflection groups Open
Given an irreducible well-generated complex reflection group, we construct an explicit basis for the module of vector fields with logarithmic poles along its reflection arrangement. This construction yields in particular a Hodge filtration…
View article: Cokernels of the Euler restriction map of logarithmic derivation modules
Cokernels of the Euler restriction map of logarithmic derivation modules Open
There are two restriction maps of the logarithmic modules of plane arrangements in a three dimensional vector space. One is the Euler restriction and the other is the Ziegler restriction. The dimension of the cokernel of the Ziegler restri…
View article: Generalization of the addition and restriction theorems from free arrangements to the class of projective dimension one
Generalization of the addition and restriction theorems from free arrangements to the class of projective dimension one Open
We study a generalized version of Terao’s addition theorem for free arrangements to the category of those with projective dimension one. Namely, we give a formulation to determine the algebraic structure of the logarithmic derivation modul…
View article: Worpitzky-compatible sets and the freeness of arrangements between Shi and Catalan
Worpitzky-compatible sets and the freeness of arrangements between Shi and Catalan Open
Given an irreducible root system, the Worpitzky-compatible subsets are defined by a geometric property of the alcoves inside the fundamental parallelepiped of the root system. This concept is motivated and mainly understood through a latti…
View article: Analyzing drilling noise in rotational atherectomy: Improving safety and effectiveness through visualization and anomaly detection using autoencoder—A preclinical study
Analyzing drilling noise in rotational atherectomy: Improving safety and effectiveness through visualization and anomaly detection using autoencoder—A preclinical study Open
Background and Aims As the population of aging societies continues to grow, the prevalence of complex coronary artery diseases, including calcification, is expected to increase. Rotational atherectomy (RA) is an essential technique for tre…
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View article: On Ziegler's conjectures for logarithmic derivations of arrangements
On Ziegler's conjectures for logarithmic derivations of arrangements Open
In his paper and thesis in 1989, Ziegler posed several conjectures regarding commutative algebra related to hyperplane arrangements. In this article, we revisit two of them. One is on generic cuts of free arrangements, and the other has to…
View article: Projective dimension of weakly chordal graphic arrangements
Projective dimension of weakly chordal graphic arrangements Open
A graphic arrangement is a subarrangement of the braid arrangement whose set of hyperplanes is determined by an undirected graph. A classical result due to Stanley, Edelman and Reiner states that a graphic arrangement is free if and only i…
View article: Free paths of arrangements of hyperplanes
Free paths of arrangements of hyperplanes Open
We study the free path problem, i.e., if we are given two free arrangements of hyperplanes, then we can connect them by free arrangements or not. We prove that if an arrangement $\mathcal{A}$ and $\mathcal{A} \setminus \{H,L\}$ are free, t…
View article: Addition-deletion theorems for the Solomon-Terao polynomials and $B$-sequences of hyperplane arrangements
Addition-deletion theorems for the Solomon-Terao polynomials and $B$-sequences of hyperplane arrangements Open
We prove the addition-deletion theorems for the Solomon-Terao polynomials, which have two important specializations. Namely, one is to the characteristic polynomials of hyperplane arangements, and the other to the Poincarè polynomials of t…
View article: Baseline inflammatory status affects the prognostic impact of statins in patients with peripheral arterial disease
Baseline inflammatory status affects the prognostic impact of statins in patients with peripheral arterial disease Open
Background Statins bring favorable effects on the clinical prognosis of patients with atherosclerotic disease partly through their anti-inflammatory properties. However, this effect has not been fully verified in patients with peripheral a…
View article: Direct Oral Anticoagulants Affect Activated Clotting Time During and Bleeding Events After Percutaneous Coronary Intervention
Direct Oral Anticoagulants Affect Activated Clotting Time During and Bleeding Events After Percutaneous Coronary Intervention Open
Background Inappropriately high activated clotting time (ACT) during percutaneous coronary intervention (PCI) is associated with an increased risk of bleeding events. However, whether the prescription of direct oral anticoagulants (DOAC) a…
View article: Generalization of the addition and restriction theorems from free arrangements to the class of projective dimension one
Generalization of the addition and restriction theorems from free arrangements to the class of projective dimension one Open
We study a generalized version of Terao's famous addition theorem for free arrangements to the category of those with projective dimension one. Namely, we give a criterion to determine the algebraic structure of logarithmic derivation modu…
View article: Myocardial bridging may shed light on one aspect of unexplained myocardial infarction with non-obstructive coronary arteries
Myocardial bridging may shed light on one aspect of unexplained myocardial infarction with non-obstructive coronary arteries Open
View article: Deletion-Restriction for Logarithmic Forms on Multiarrangements
Deletion-Restriction for Logarithmic Forms on Multiarrangements Open
We consider the behaviour of logarithmic differential forms on arrangements and multiarrangements of hyperplanes under the operations of deletion and restriction, extending early work of Günter Ziegler. The restriction of logarithmic forms…
View article: Free reflection multiarrangements and quasi-invariants
Free reflection multiarrangements and quasi-invariants Open
To a complex reflection arrangement with an invariant multiplicity function one can relate the space of logarithmic vector fields and the space of quasi-invariants, which are both modules over invariant polynomials. We establish a close re…
View article: Vertex-weighted Digraphs and Freeness of Arrangements Between Shi and Ish
Vertex-weighted Digraphs and Freeness of Arrangements Between Shi and Ish Open
We introduce and study a digraph analogue of Stanley's $ψ$-graphical arrangements from the perspectives of combinatorics and freeness. Our arrangements form a common generalization of various classes of arrangements in literature including…
View article: Roots of the characteristic polynomials of hyperplane arrangements and their restrictions and localizations
Roots of the characteristic polynomials of hyperplane arrangements and their restrictions and localizations Open
Terao's factorization theorem shows that if an arrangement is free, then its characteristic polynomial factors into the product of linear polynomials over the integer ring. This is not a necessary condition, but there are not so many non-f…
View article: On some freeness-type properties for line arrangements
On some freeness-type properties for line arrangements Open
By way of Ziegler restrictions we study the relation between nearly free plane arrangements and combinatorics and we give a Yoshinaga-type criterion for plus-one generated plane arrangements.
View article: Addition–deletion results for the minimal degree of logarithmic derivations of hyperplane arrangements and maximal Tjurina line arrangements
Addition–deletion results for the minimal degree of logarithmic derivations of hyperplane arrangements and maximal Tjurina line arrangements Open
View article: Projective dimensions of hyperplane arrangements
Projective dimensions of hyperplane arrangements Open
We establish a general theory for projective dimensions of the logarithmic derivation modules of hyperplane arrangements. That includes the addition-deletion and restriction theorem, Yoshinaga-type result, and the division theorem for proj…
View article: On complex supersolvable line arrangements
On complex supersolvable line arrangements Open