Koszul Differential Graded Algebras and BGG Correspondence II

The concept of Koszul differential graded (DG for short) algebra is introduced in [8].Let A be a Koszul DG algebra.If the Ext-algebra of A is finite-dimensional,i.e.,the trivial module A k is a compact object in the derived category of DG A-modules,then it is shown in [8] that A has many nice properties.However,if the Ext-algebra is infinitedimensional,little is known about A.As shown in [15] (see also Proposition 2.2),A k is not compact if H(A) is finite-dimensional.In this paper,it is proved that the Koszul duality theorem also holds when H(A) is finite-dimensional by using Foxby duality.A DG version of the BGG correspondence is deduced from the Koszul duality theorem.

Derived Equivalences and Recollements of Differential Graded Algebras and Schemes

In this paper, we first prove for two differential graded algebras （DGAs） A, B which are derived equivalent to k-algebras A, F, respectively, that ：D（Ak B） ≈D（Ak Г）. In particular, Hp^b（Ak B） ≈ Hb（proj-A k Г）. Secondly, for two quasi-compact and sepa- rated schemes X, Y and two algebras A, B over k which satisfy ：D（Qcoh（X）） ≈：D（A） and ：D（Qcoh（Y）） ≈D（B）, we show that ：D（Qcoh（X × Y）） ≈ 79（AB） and ：Db（Coh（X × Y）） ≈Db（mod-（A B））. Finally, we prove that if X is a quasi-compact and separated scheme over k, then ：D（Qcoh（X ~ pl）） admits a recollement relative to D（Qcoh（X））, and we de- scribe the functors in the recollement explicitly. This recollement induces a recollement of bounded derived categories of coherent sheaves and a recollement of singularity categories. When the scheme X is derived equivalent to a DGA or algebra, then the recollement which we get corresponds to the recollement of DGAs in [14] or the recollement of upper triangular algebras in [7].

Differential Graded Structures on Path Categories

In this paper, we consider the graded path category associated to a quiver. We investigate all n-differentials on such a category, and also study the associated graded Lie algebra. Moreover, a necessary and sufficient condition is given for the graded path categorv to admit a DG category structure.

T2-weighted imaging-based radiomic-clinical machine learning model for predicting the differentiation of colorectal adenocarcinoma

BACKGROUND The study on predicting the differentiation grade of colorectal cancer(CRC)based on magnetic resonance imaging(MRI)has not been reported yet.Developing a non-invasive model to predict the differentiation grade of CRC is of great value.AIM To develop and validate machine learning-based models for predicting the differ-entiation grade of CRC based on T2-weighted images(T2WI).METHODS We retrospectively collected the preoperative imaging and clinical data of 315 patients with CRC who underwent surgery from March 2018 to July 2023.Patients were randomly assigned to a training cohort(n=220)or a validation cohort(n=95)at a 7:3 ratio.Lesions were delineated layer by layer on high-resolution T2WI.Least absolute shrinkage and selection operator regression was applied to screen for radiomic features.Radiomics and clinical models were constructed using the multilayer perceptron(MLP)algorithm.These radiomic features and clinically relevant variables(selected based on a significance level of P<0.05 in the training set)were used to construct radiomics-clinical models.The performance of the three models(clinical,radiomic,and radiomic-clinical model)were evaluated using the area under the curve(AUC),calibration curve and decision curve analysis(DCA).RESULTS After feature selection,eight radiomic features were retained from the initial 1781 features to construct the radiomic model.Eight different classifiers,including logistic regression,support vector machine,k-nearest neighbours,random forest,extreme trees,extreme gradient boosting,light gradient boosting machine,and MLP,were used to construct the model,with MLP demonstrating the best diagnostic performance.The AUC of the radiomic-clinical model was 0.862(95%CI:0.796-0.927)in the training cohort and 0.761(95%CI:0.635-0.887)in the validation cohort.The AUC for the radiomic model was 0.796(95%CI:0.723-0.869)in the training cohort and 0.735(95%CI:0.604-0.866)in the validation cohort.The clinical model achieved an AUC of 0.751(95%CI:0.661-0.842)in the training cohort and 0.676(95%CI:0.525-0.827)in the validation cohort.All three models demonstrated good accuracy.In the training cohort,the AUC of the radiomic-clinical model was significantly greater than that of the clinical model(P=0.005)and the radiomic model(P=0.016).DCA confirmed the clinical practicality of incorporating radiomic features into the diagnostic process.CONCLUSION In this study,we successfully developed and validated a T2WI-based machine learning model as an auxiliary tool for the preoperative differentiation between well/moderately and poorly differentiated CRC.This novel approach may assist clinicians in personalizing treatment strategies for patients and improving treatment efficacy.

Multi-object optimization design for differential and grading toothed roll crusher using a genetic algorithm

Our differential and grading toothed roll crusher blends the advantages of a toothed roll crusher and a jaw crusher and possesses characteristics of great crushing,high breaking efficiency,multi-sieving and has,for the moment,made up for the short- comings of the toothed roll crusher.The moving jaw of the crusher is a crank-rocker mechanism.For optimizing the dynamic per- formance and improving the cracking capability of the crusher,a mathematical model was established to optimize the transmission angleγand to minimize the travel characteristic value m of the moving jaw.Genetic algorithm is used to optimize the crusher crank-rocker mechanism for multi-object design and an optimum result is obtained.According to the implementation,it is shown that the performance of the crusher and the cracking capability of the moving jaw have been improved.

一类Gerstenhaber代数的构造

For any K-algebra A,based on Hochschild complex and Hochschild coho-mology of A,we construct a new Gerstenhaber algebra,and give Gerstenhaber algebra epimorphism from the new Gerstenhaber algebra to the Gerstenhaber algebra of the Hochschild cohomology of A.

Compact DG modules and Gorenstein DG algebras

When the base connected cochain DG algebra is cohomologically bounded, it is proved that the difference between the amplitude of a compact DG module and that of the DG algebra is just the projective dimension of that module. This yields the unboundedness of the cohomology of non-trivial regular DG algebras.When A is a regular DG algebra such that H(A) is a Koszul graded algebra, H(A) is proved to have the finite global dimension. And we give an example to illustrate that the global dimension of H(A) may be infinite, if the condition that H(A) is Koszul is weakened to the condition that A is a Koszul DG algebra. For a general regular DG algebra A, we give some equivalent conditions for the Gorensteiness.For a finite connected DG algebra A, we prove that Dc(A) and Dc(A op) admit Auslander-Reiten triangles if and only if A and A op are Gorenstein DG algebras. When A is a non-trivial regular DG algebra such that H(A) is locally finite, Dc(A) does not admit Auslander-Reiten triangles. We turn to study the existence of Auslander-Reiten triangles in D lf b (A) and D lf b (A op) instead, when A is a regular DG algebra.

DG algebra structures on AS-regular algebras of dimension 2

Let A be a connected cochain DG algebra, whose underlying graded algebra is an Artin-Schelter regular algebra of global dimension 2 generated in degree 1. We give a description of all possible differential of A and compute H（A）. Such kind of DG algebras are proved to be strongly Gorenstein. Some of them serve as examples to indicate that a connected DG algebra with Koszul underlying graded algebra may not be a Koszul DG algebra defined in He and We （J Algebra, 2008, 320： 2934-2962）. Unlike positively graded chain DG algebras, we give counterexamples to show that a bounded below DC A-module with a free underlying graded A^#-module may not be semi-projective.

DG Poisson algebra and its universal enveloping algebra

We introduce the notions of differential graded(DG) Poisson algebra and DG Poisson module. Let A be any DG Poisson algebra. We construct the universal enveloping algebra of A explicitly, which is denoted by A^(ue). We show that A^(ue) has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over A is isomorphic to the category of DG modules over A^(ue). Furthermore, we prove that the notion of universal enveloping algebra A^(ue) is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra.