Stratifying System Research Articles

Stratifying systems and g-vectors

In this paper we study the Cartan matrix associated to the Ext-projective stratifying system induced by a basic and τ -rigid object M in mod ( A ) by means of the g -vectors of the indecomposable direct summands of M . In particular we show that the Cartan group of a stratifying system associated to a τ -rigid module can be calculated directly using these vectors. Moreover we characterise the stratifying systems coming from τ -rigid modules that have a diagonal Cartan matrix.

The size of a stratifying system can be arbitrarily large

In this short note we construct two families of examples of large stratifying systems in module categories of algebras. The first examples consist on stratifying systems of infinite size in the module category of an algebra A. In the second family of examples we show that the size of a finite stratifying system in the module category of a finite dimensional algebra A can be arbitrarily large in comparison to the number of isomorphism classes of simple A-modules. We note that both families of examples are built using well-established results in higher homological algebra.

Immunogenomic Analyses of the Prognostic Predictive Model for Patients With Renal Cancer.

BackgroundRenal cell carcinoma (RCC) is associated with poor prognostic outcomes. The current stratifying system does not predict prognostic outcomes and therapeutic benefits precisely for RCC patients. Here, we aim to construct an immune prognostic predictive model to assist clinician to predict RCC prognosis.MethodsHerein, an immune prognostic signature was developed, and its predictive ability was confirmed in the kidney renal clear cell carcinoma (KIRC) cohorts based on The Cancer Genome Atlas (TCGA) dataset. Several immunogenomic analyses were conducted to investigate the correlations between immune risk scores and immune cell infiltrations, immune checkpoints, cancer genotypes, tumor mutational burden, and responses to chemotherapy and immunotherapy.ResultsThe immune prognostic signature contained 14 immune-associated genes and was found to be an independent prognostic factor for KIRC. Furthermore, the immune risk score was established as a novel marker for predicting the overall survival outcomes for RCC. The risk score was correlated with some significant immunophenotypic factors, including T cell infiltration, antitumor immunity, antitumor response, oncogenic pathways, and immunotherapeutic and chemotherapeutic response.ConclusionsThe immune prognostic, predictive model can be effectively and efficiently used in the prediction of survival outcomes and immunotherapeutic responses of RCC patients.

Immune cell infiltration-based signature for prognosis and immunogenomic analysis in breast cancer.

Breast cancer is one of the most human malignant diseases and the leading cause of cancer-related death in the world. However, the prognostic and therapeutic benefits of breast cancer patients cannot be predicted accurately by the current stratifying system. In this study, an immune-related prognostic score was established in 22 breast cancer cohorts with a total of 6415 samples. An extensive immunogenomic analysis was conducted to explore the relationships between immune score, prognostic significance, infiltrating immune cells, cancer genotypes and potential immune escape mechanisms. Our analysis revealed that this immune score was a promising biomarker for estimating overall survival in breast cancer. This immune score was associated with important immunophenotypic factors, such as immune escape and mutation load. Further analysis revealed that patients with high immune scores exhibited therapeutic benefits from chemotherapy and immunotherapy. Based on these results, we can conclude that this immune score may be a useful tool for overall survival prediction and treatment guidance for patients with breast cancer.

Stratifying Systems through $\tau$-Tilting Theory

In this paper we first show that every non-zero $\tau$-rigid $A$-module induces at least one stratifying system in the module category of $A$. Moreover, we show that each of these stratifying systems can be seen as a signed $\tau$-exceptional sequence.

STRATIFYING SYSTEMS FOR EXACT CATEGORIES

AbstractIn this paper, we develop the theory of stratifying systems in the context of exact categories as a generalisation of the notion of stratifying systems in module categories, which have been studied by different authors. We prove that attached to a stratifying system in an exact category $(\mathcal{A},\mathcal{E})$ there is an standardly stratified algebra B such that the category $\mathscr{F}$F(Θ), of F-filtered objects in the exact category $(\mathcal{A},\mathcal{E})$ is equivalent to the category $\mathscr{F}$(Δ) of Δ-good modules associated to B. The theory we develop in exact categories, give us a way to produce standardly stratified algebras from module categories by just changing the exact structure on it. In this way, we can construct exact categories whose bounded derived category is equivalent to the bounded derived category of an standardly stratified algebra. Finally, applying the relative homological algebra developed by Auslander–Solberg, we can construct examples of stratifying systems that are not a stratifying system in the classical sense, so our approach really produces new stratifying systems.

On the size dependence of the King stratification index

The study reported in this paper addresses an aspect of the modelling of particle stratification as it applies to the processing of minerals. It focuses on the phenomenological model developed by King and the extent to which the single parameter in that model, i.e. the stratification index, is dependent on the size of particles in the stratifying system. Based on a careful experimental investigation in a batch jig, the study found that, contrary to expectation and current theory, the size dependence of the index is weak or absent over the range of conditions investigated. The implications of the findings for both theory and modelling practice and for Rao’s proposed modification of King’s model are discussed. The study also presents a number of findings related to experimental procedures for obtaining precise measurements of parameters in stratification models.

Assessing the value of risk indices of postoperative nausea and vomiting in ambulatory surgical patients

Postoperative and postdischarge nausea and vomiting have profound impact on the efficient delivery of quality healthcare. In addition to patient dissatisfaction, physical morbidities as well as unplanned hospital admissions may result. It is important to risk stratify and intervene on patients at risk. The aim of this review is to explore the benefits and shortcomings of the scoring systems commonly used today. Two widely-used risk stratification systems identify and score patient-, anesthesia-, and surgery-related risk factors for postoperative/postdischarge nausea and vomiting in the adult population. These systems have much overlap, with previous history of nausea and vomiting being the biggest risk factor. There is also a risk stratifying system for pediatric patients. Based on the preoperative score, a patient may be deemed low, intermediate, or high risk. Once stratified, appropriate medications and/or other interventions may be planned to prevent postoperative/postdischarge nausea and vomiting. Risk-stratifying scoring systems seek to identify patients at risk for postoperative/postdischarge nausea and vomiting. A protocol-based approach is recommended. A number of risk stratification systems have been validated and have been simplified for widespread adoption. Some institutions report a decrease in postoperative nausea and vomiting rates when an algorithm is followed based on one of the three scoring systems. By identifying at risk patients, an anesthetic can be better planned. A reduction in the incidence of postoperative/postdischarge nausea and vomiting will have a direct effect upon patient satisfaction, morbidity, and healthcare costs.

Stratifying systems over the hereditary path algebra with quiver $$\mathbb {A}_{p,q}$$ A p , q

The authors have proved in [J. Algebra Appl. 14 (2015), no. 6] that the size of a stratifying system over a finite-dimensional hereditary path algebra $A$ is at most $n$, where $n$ is the number of isomorphism classes of simple $A$-modules. Moreover, if $A$ is of Euclidean type a stratifying system over $A$ has at most $n-2$ regular modules. In this work, we construct a family of stratifying systems of size $n$ with a maximal number of regular elements, over the hereditary path algebra with quiver $\widetilde{\mathbb {A}}_{p,q} $, canonically oriented.

Stratifying systems over hereditary algebras

This paper deals with stratifying systems over hereditary algebras. In the case of tame hereditary algebras we obtain a bound for the size of the stratifying systems composed only by regular modules and we conclude that stratifying systems cannot be complete. For wild hereditary algebras, with more than two vertices, we show that there exists a complete stratifying system whose elements are regular modules. In any other case, we conclude that there are no stratifying system consisting of regular modules.

Split-By-Nilpotent Extensions Algebras and Stratifying Systems

Let Γ and Λ be artin algebras such that Γ is a split-by-nilpotent extension of Λ by a two sided ideal I of Γ. Consider the change of rings functors G: =ΓΓΛ ⊗Λ − and F: =ΛΛΓ ⊗Γ −. In this article, by assuming that I Λ is projective, we find the necessary and sufficient conditions under which a stratifying system (Θ, ≤) in modΛ can be lifted to a stratifying system (GΘ, ≤) in mod(Γ). Furthermore, by using the functors F and G, we study the relationship between their filtered categories of modules; and some connections with their corresponding standardly stratified algebras are stated (see Theorem 5.12, Theorem 5.15 and Theorem 5.18). Finally, a sufficient condition is given for stratifying systems in mod(Γ) in such a way that they can be restricted, through the functor F, to stratifying systems in mod(Λ).

Audism

More than 30 years ago, Tom Humphries coined the term “audism” to describe audiocentric (based on hearing and speaking) assumptions and attitudes of supremacy. Only a handful of scholarly articles mention the concept of audism and not one of those is published outside of Deaf Cultural Studies. In this article, audism is broadly defined in the ideological contexts of individual, institutional, metaphysical, and laissez-faire prejudices. Audism is further explained in the context of overt, covert, and aversive practices of discrimination. Examples of the intersections of the theory and practice of audiocentric privilege are explored. Based on critical observations of audism as a stratifying system of oppression, four recommendations are made: increasing public awareness of Deaf American contributions to society (multiculturalism), infusing Deaf-centric curriculum content in education (equity), advocating intergroup dialogues as a transformative pedagogy that further exposes audism as a social injustice (intercultural responsibility), and promoting community service opportunities (ethical citizenship) for students to do volunteer work in the Deaf American Community.

Liquid-vapor phase transition in a stratifying lead-zinc system upon a reduction in pressure

The boundaries of the liquid-vapor phase transition at 101325, 100, and 10 Pa were calculated on the basis of the values of the vapor pressure of the components in a lead-zinc system as determined by the boiling point method. It was shown that at low pressures the region of the coexistence of the liquid and vapor is superimposed on the stratification region of the state diagram.

On the Relative Socle for Stratifying Systems

Let A be a finite dimensional k-algebra, (Θ, ≤) be a stratifying system in mod(A) and ℱ(Θ) be the class of Θ-filtered A-modules. In this article, we give the definition and also study some of the properties of the relative socle in ℱ(Θ). We approach the relative socle in three ways. Namely, we view it as (1) a Θ-semisimple subobject of M having the largest Θ-length, (2) a maximal Θ-semisimple subobject of M, and (3) a minimal Θ-essential subobject of M.

HOMOLOGICAL SYSTEMS IN MODULE CATEGORIES OVER PRE-ORDERED SETS

We introduce the so-called homological systems in a module category over a pre-ordered set, which generalize the notion of a stratifying system over a linearly ordered set, and study both the corresponding modules filtrated by the systems and algebras stratified by the systems. In particular, we recover the tilting theory for pre-standardly stratified algebras, and get a general formula for computing the Cartan determinants of pre-standardly stratified algebras in terms of standard modules and simple modules. Also, the finitistic dimension of a given algebra, and the relative homological dimensions of full subcategories of the modules related to a homological system, are discussed. As an application, we get a new bound for the finitistic dimension of a pre-standardly stratified algebra.

On Young modules of general linear groups

We study ℓ-permutation modules of finite general linear groups GL n ( q ) acting on partial flags in the natural module, where the coefficient field of the modules has characteristic ℓ, for ℓ ∤ q . We call the indecomposable summands of these permutation modules linear Young modules. We determine their vertices and Green correspondents, by methods relying only on the representation theory of GL n ( q ) . Furthermore, we show that when the multiplicative order of q modulo ℓ is strictly greater than 1, the Specht modules for GL n ( q ) in characteristic ℓ form a stratifying system. This implies in particular, that for GL n ( q ) -modules with Specht filtration, the filtration multiplicities are independent of the filtration. This is an analogue of a recent theorem by Hemmer and Nakano.

Quadratic forms associated to stratifying systems

Let R be an algebra, and let ( θ , ≼ ) be a stratifying system of R-modules. If the category F ( θ ) is θ-directing, then we prove that ind F ( θ ) is finite. In order to do that, we introduce a quadratic form q θ which depends on θ. Moreover, we also give sufficient conditions to get the correspondence X ↦ dim ̲ θ X from ind F ( θ ) to the set of positive roots of q θ .

Applications of stratifying systems to the finitistic dimension

Given an Ext-injective stratifying system of Λ -modules ( θ , Y ̲ , ≼ ) satisfying that the projective dimension of Y is finite, we prove that the finitistic dimension of the algebra Λ is equal to the finitistic dimension of the category I ( θ ) = { X ∈ mod Λ : Ext Λ 1 ( -, X ) | F ( θ ) = 0 } . Moreover, using the theory of stratifying systems we obtain bounds for the finitistic dimension of Λ . In particular, we get the optimal bound 2 n - 2 for the finitistic dimension of a standardly stratified algebra with n simples.

Stratifying Systems via Relative Projective Modules

ABSTRACT In this paper we point out that the “Process of standardization”, given in Dlab and Ringel (1992), and also the “Comparison method” given in Platzeck and Reiten (2001) can be generalized. To do so, we introduce the concept of relative projective stratifying system and prove a result from which the Theorem 2 in Dlab and Ringel (1992) and Proposition 2.1 in Ringel (1991) follows. #Communicated by A. Happel. ‡Dedicated to Raymundo Bautista on his 60th birthday.

On Standardly Stratified Algebras

Let A be a finite dimensional algebra over an algebraically closed field k. For any fixed partial ordering of an index set,Λ say,labelling the simple A-modules L(i),there are standard modules,denoted by Δ(i),i ∈ Λ. By definition,Δ(i) is the largest quotient of the projective cover of L(i) having composition factors L(j) with j ≤ i. Denote by ℱ(Δ) the category of A-modules which have a filtration whose quotients are isomorphic to standard modules. The algebra A is said to be standardly stratified if all projective A-modules belong to ℱ(Δ). In this paper we define a “stratifying system” and we show that this produces a module Y,whose endomorphism ring A is standardly stratified. In particular,we construct stratifying systems for special biserial self-injective algebras.