Infinitesimal (BiHom-)bialgebras of any weight (I): Basic definitions and properties

The purpose of this paper is to introduce and study $\lambda$-infinitesimal BiHom-bialgebras (abbr. \l-infBH-bialgebra) and some related structures. They can be seen as an extension of \l-infinitesimal bialgebras considered by Ebrahimi-Fard, including Joni and Rota's infinitesimal bialgebras as well as Loday and Ronco's infinitesimal bialgebras, and including also infinitesimal BiHom-bialgebras introduced by Liu, Makhlouf, Menini, Panaite. In this paper, we provide various relevant constructions and new concepts. Two ways are provided for a unitary (resp. counitary) algebra (coalgebra) to be a \l-infBH-bialgebra and the notion of \l-infBH-Hopf module is introduced and discussed. It is proved, in connexion with nonhomogeneous (co)associative BiHom-Yang-Baxter equation, that every (left BiHom-)module (resp. comodule) over a (anti-)quasitriangular (resp. (anti-)coquasitriangular) \l-infBH-bialgebra carries a structure of \l-infBH-Hopf module. Moreover, two approaches to construct BiHom-pre-Lie (co)algebras from \l-infBH-bialgebras are presented

When Leibniz Bialgebras are Nijenhuis?

Leibniz algebras can be seen as a "non-commutative" analogue of Lie algebras. Nijenhuis operators on Leibniz algebras introduced by Cari\~{n}ena, Grabowski, and Marmo in [J. Phys. A: Math. Gen. 37(2004)] are (1, 1)-tensors with vanishing Nijenhuis torsion. Recently triangular Leibniz bialgebras were introduced by Tang and Sheng in [J. Noncommut. Geom. 16(2022)] via the twisting theory of twilled Leibniz algebras. In this paper we find that Leibniz algebras are very closely related to Nijenhuis operators, and prove that a triangular symplectic Leibniz bialgebra together with a dual triangular structure must possess Nijenhuis operators, which makes it possible to study Nijehhuis geometry from the perspective of Leibniz algebras. At the same time, we regain the classical Leibniz Yang-Baxter equation by using the tensor form of classical $r$-matrics. At last we give the classification of triangular Leibniz bialgebras of low dimensions

Generalized Composition Operators from ℬ

Let 0<p<∞, let -2<q<∞, and let φ be an analytic self-map of and g∈H(). The boundedness and compactness of generalized composition operators (Cφgf)(z)=∫0z‍f'(φ(ξ))g(ξ)dξ, z∈, f∈H(), from ℬμ (ℬμ,0) spaces to QK,ω(p,q) spaces are investigated

Construction of an immunogenic cell death-based risk score prognosis model in breast cancer

Immunogenic cell death (ICD) is a form of regulated cell death that elicits immune response. Common inducers of ICD include cancer chemotherapy and radiation therapy. A better understanding of ICD might contribute to modify the current regimens of anti-cancer therapy, especially immunotherapy. This study aimed to identify ICD-related prognostic gene signatures in breast cancer (BC). An ICD-based gene prognostic signature was developed using Lasso-cox regression and Kaplan-Meier survival analysis based on datasets acquired from the Cancer Genome Atlas and Gene Expression Omnibus. A nomogram model was developed to predict the prognosis of BC patients. Gene Set Enrichment Analysis (GESA) and Gene Set Variation Analysis (GSVA) were used to explore the differentially expressed signaling pathways in high and low-risk groups. CIBERSORT and ESTIMATE algorithms were performed to investigate the difference of immune status in tumor microenvironment of different risk groups. Six genes (CALR, CLEC9A, BAX, TLR4, CXCR3, and PIK3CA) were selected for construction and validation of the prognosis model of BC based on public data. GSEA and GSVA analysis found that immune-related gene sets were enriched in low-risk group. Moreover, immune cell infiltration analysis showed that the immune features of the high-risk group were characterized by higher infiltration of tumor-associated macrophages and a lower proportion of CD8+ T cells, suggesting an immune evasive tumor microenvironment. We constructed and validated an ICD-based gene signature for predicting prognosis of breast cancer patients. Our model provides a tool with good discrimination and calibration abilities to predict the prognosis of BC, especially triple-negative breast cancer (TNBC)