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Let (R,⋅,∘) be a braided Hopf brace, i.e. a Hopf brace in the category of Yetter–Drinfeld modules over a Hopf brace (H, ⋅, ∘). In this note, we give the sufficient and necessary conditions which make Radford’s biproduct (R♯H,⋅) and (R♯H∘,∘) become a Hopf brace. [ABSTRACT FROM AUTHOR]

Let (H , ⋅ , ∘) be a Hopf brace. We first show that the category of left compatible modules over (H , ⋅ , ∘) is a tensor category. Next, we show that its Drinfeld centre is equivalent to the category H H ′ Y D of left Yetter–Drinfeld modules over (H , ⋅ , ∘) as a braided tensor category. Finally, we show that Radford's biproduct (R ♯ H , ⋅) and (R ♯ H ∘ , ∘) constitute a Hopf brace, where R is a Hopf algebra in H H ′ Y D . This presents some new method to construct Hopf braces. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the (2+1)‐dimensional compressible Navier–Stokes equation with density‐dependent viscosity coefficients. We introduce a novel power‐type elliptic vortex ansatz and thereby obtain a finite‐dimensional nonlinear dynamical system. The latter is shown to not only have an underlying integrable Ermakov structure of Hamiltonian type, but also admit a Lax pair formulation and associated stationary nonlinear Schrödinger connection. In addition, we construct a class of elliptical vortex solutions termed pulsrodons corresponding to pulsating elliptic warm core eddies and discuss their dynamical behaviors. These solutions have recently found applications in geography, and oceanic and atmospheric dynamics. [ABSTRACT FROM AUTHOR]

Let H be a Hopf algebra with bijective antipode. In this paper we prove that, if A is a Hopf brace with some projection on H, then there exists a compatible braided Hopf brace R such that A is isomorphic to R ♯ H as Hopf braces, where R ♯ H is some Radford's biproduct Hopf algebra. This should be viewed as the brace version of well-known Radford's theorem about Hopf algebras with a projection. This provides a new method to construct Hopf braces by some braided Hopf algebras. [ABSTRACT FROM AUTHOR]

In this paper, we characterize quasi-Yetter–Drinfeld modules over a Hopf algebra H , which was introduced in [Y. Bazlov and A. Berenstein, Braided doubles and rational Cherednik algebras, Adv. Math. 220 (2009), 1466–1530]. We first show that the quasi-Drinfeld center of the category of H -modules is equivalent to the category H H 𝒬 𝒴 𝒟 of quasi-Yetter–Drinfeld modules. Next, we prove that H H 𝒬 𝒴 𝒟 is equivalent to the category of generalized Hopf bimodules. Finally, we show that H H 𝒬 𝒴 𝒟 is also equivalent to the category of quasi-coactions over some Majid's braided group if H is quasi-triangular. [ABSTRACT FROM AUTHOR]

Let H be a weak Hopf algebra with an idempotent element p. Denote by Hp some twisted weak Hopf algebra associated with a generalized Drinfeld twist. Let be a bicovariant differential calculus on H. We give some method to construct a bicovariant differential calculus on Hp by a generalized Drinfeld twist, where is some subspace of Γ determined by. As an application, we present some examples of bicovariant differential calculi on some face algebra and its quantum double. [ABSTRACT FROM AUTHOR]

Let H be a Hopf algebra with bijective antipode. We first define some generalized Hopf bimodules. Next, we show that these Hopf bimodules form a new tensor category with a crossed structure, which is equivalent to the category of some generalized Yetter–Drinfeld modules introduced by Panaite and Staic. Finally, based on this equivalence, we verify that the category of Hopf bimodules admits the structure of a braided G -category in the sense of Turaev. [ABSTRACT FROM AUTHOR]

Let (H,R) be a finite dimensional quasitriangular weak Hopf algebra over a fieldk. We first construct a weak Hopf algebra [Δ(1)(H⊗H)Δ(1)]R, which is based on the subalgebra of the tensor product algebraH⊗H. Next we verify that ifHis factorizable, then the Drinfeld’s quantum double ofHis isomorphic to [Δ(1)(H⊗H)Δ(1)]R. [ABSTRACT FROM AUTHOR]

Let H be a coquasi-triangular Hopf algebra. We first show that the group of braided autoequivalences of the category of H -comodules is isomorphic to the group of braided-commutative bi-Galois objects. Next, by investigating the latter, we obtain that the group of braided autoequivalences of the representation category of Lusztig’s quantum group u q ( s l ( 2 ) ) ′ is isomorphic to the projective special linear group P S L ( 2 ) , where q is a root of unity of odd order N > 1 . [ABSTRACT FROM AUTHOR]

In this article we show that the category of parachain complexes is equivalent to the category of Yetter–Drinfeld modules over the Pareigis's Hopf algebra. [ABSTRACT FROM AUTHOR]

Let ( H , R ) be a quasitriangular weak Hopf algebra over a field k . We show that there is a braided monoidal isomorphism between the Yetter–Drinfeld module category YD H H over H and the category of comodules over some braided Hopf algebra H R in the category M H . Based on this isomorphism, we prove that every braided bi-Galois object A over the braided Hopf algebra H R defines a braided autoequivalence of the category YD H H if and only if A is quantum commutative. In case H is semisimple over an algebraically closed field, i.e. the fusion case, then every braided autoequivalence of YD H H trivializable on M H is determined by such a quantum commutative Galois object. The quantum commutative Galois objects in M H form a group measuring the Brauer group of ( H , R ) as studied in [21] in the Hopf algebra case. [ABSTRACT FROM AUTHOR]

Let H be a quasitriangular weak Hopf algebra. It is proved that the centralizer subalgebra of its source subalgebra in H is a braided group (or Hopf algebra in the category of left H-modules), which is cocommutative and also a left braided Lie algebra in the sense of Majid. [ABSTRACT FROM AUTHOR]

Let B and H be weak Hopf algebras with bijective antipodes SB and SH, respectively. Based on a compatible weak Hopf dual pairing (B, H, σ), we construct a generalized Drinfeld quantum double (B, H) which is a weak T-coalgebra over a twisted semi-direct square of groups. In particular, when B and H are finite dimensional and the above pairing map σ is nondegenerate, (B, H) admits a nontrivial quasitriangular structure. Some explicit examples are given as an application of our theory. [ABSTRACT FROM AUTHOR]

Background: Integrin Subunit Alpha 5 (ITGA5), belongs to the integrin alpha chain family, is vital for promoting cancer cell invasion, metastasis. However, the correlation between ITGA5 expression and immune infiltration in gastrointestinal tumors remain unclear.Methods: The expression level of ITGA5 was detected by Oncomine and Tumor Immune Estimation Resource (TIMER). The association between ITGA5 and prognosis of patients was identified by Kaplan-Meier plotter, Gene Expression Profiling Interactive Analysis 2 (GEPIA2) and PrognoScan. We evaluated the correlation between ITGA5 expression and immune infiltrating level via TIMER. Besides, TIMER, immunohistochemistry (IHC) staining and western blot were used to explore correlations between ITGA5 expression and markers of immune infiltrates cells. Furthermore, we constructed protein-protein interaction (PPI) network and performed functional enrichment by GeneMANIA and Metascape.Results: ITGA5 was generally overexpressed and correlated with worse prognosis in multiple types of gastrointestinal tumors. In addition, ITGA5 expression level was significantly associated with tumor purity and immune infiltration levels of different immune cells in gastrointestinal tumors. Interestingly, immune markers for monocytes, tumor - associated macrophages (TAMs), macrophages 2 (M2) cells and T-helper 2 (Th2) cells were found to be significantly and positively correlated with ITGA5 expression levels in colon and gastric cancer. Results from IHC staining and western blot further proved that markers of Th2 and M2 cell were significantly increased in gastric cancer patients with high ITGA5 expression levels. Lastly, interaction network and function enrichment analysis revealed ITGA5 was mainly involved in "integrin mediated signaling pathway", "leukocyte migration", "cell-substrate adhesion".Conclutions: Our study demonstrated that ITGA5 may act as an essential regulator of tumor immune cell infiltration and a valuable prognostic biomarker in gastrointestinal tumors. Additional work is needed to fully elucidate the underlying mechanisms behind these observations. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a function cascade synchronization method for fractional-order hyperchaotic systems is introduced to achieve the synchronization of two identical fractional-order hyperchaotic systems. It is shown that the method is not only theoretically rigorous, practically feasible, but also a more general one, which contains the complete synchronization, modified projective synchronization and anti-phase synchronization. In order to valid the effectiveness of the proposed method, we give two illustrative examples. Suitable controllers are designed and the function cascade synchronization for fractional-order hyperchaotic systems is achieved. Numerical simulations are performed and shown to fit with our analysis results. Highlights • We propose a function cascade synchronization method for fractional-order hyperchaotic systems, which can be used to achieve the synchronization of the states of two identical fractional-order hyperchaotic systems. • The method is a more general synchronization method, which contains the complete synchronization, modified projective synchronization and anti-phase synchronization etc. • The theoretical results we obtain here may provide a possible way or some guidances to solve problems in encryption and secure communication etc. [ABSTRACT FROM AUTHOR]

By introducing an elliptic vortex ansatz, the 2+1-dimensional two-layer fluid system is reduced to a finite-dimensional nonlinear dynamical system. Time-modulated variables are then introduced and multicomponent Ermakov systems are isolated. The latter is shown to be also Hamiltonian, thereby admitting general solutions in terms of an elliptic integral representation. In particular, a subclass of vortex solutions is obtained and their behaviors are simulated. Such solutions have recently found applications in oceanic and atmospheric dynamics. Moreover, it is proved that the Hamiltonian system is equivalent to the stationary nonlinear cubic Schrödinger equations coupled with a Steen-Ermakov-Pinney equation. [ABSTRACT FROM AUTHOR]

Let H be a coquasitriangular quantum groupoid. In this paper, using a suitable idempotent element e in H, we prove that eH is a braided group (or a braided Hopf algebra in the category of right H-comodules), which generalizes Majid's transmutation theory from a coquasitriangular Hopf algebra to a coquasitriangular weak Hopf algebra. [ABSTRACT FROM AUTHOR]

Reinforcing the visible photon absorption and charge separation are the key issues to maximize the photocatalytic performance of graphitic carbon nitride. Herein, holey bridging-nitrogen-defected graphitic carbon nitride nanosheets were prepared through solid-state copolymerization and subsequently thermal annealing with melamine and hexamethylenetetramine as the precursors. Numerous pores and bridging nitrogen defects that embedded into the thin-layer framework were evidenced through comprehensive characterization. The synthesized textural and electronic structure enables the significant improvement of photocatalytic hydrogen production, with the optimized sample of D-CNNS(0.3) representing a hydrogen evolution rate of 2497.1 μmol∙g−1∙h−1 under visible light irradiation (λ > 420 nm). This is about 10.4 and 41.1 folds improvement compared with pristine nanosheets and bulk carbon nitride, respectively. Both experimental and theoretical results demonstrate the bridging nitrogen defects are beneficial to enhance photoabsorption, promote charge separation and transfer. Together with the enlarged surface area, the optimized nanosheet sample shows a dramatically improved quantum yield in visible region. [Display omitted] • Bridging-nitrogen defects modified g-C 3 N 4 nanosheets were facilely prepared. • Textural and electronic structures were simultaneously regulated. • Optimized H 2 production reached 2495.1 μmol g−1 h−1 (λ > 420 nm). • Apparent quantum yield was 3.24% at 420 ± 10 nm. • Photoabsorption and charge separation could be induced by bridging-nitrogen defects. [ABSTRACT FROM AUTHOR]

Sulfur-doped holey carbon nitride nanosheets were facilely prepared through subtly controlling of thiocyanuric acid precursor, resulting into an apparent quantum yield of 10 % at 420 nm for hydrogen production from water. • Ultrathin sulfur-doped holey carbon nitride nanosheets were successfully prepared via self-templating approach. • Optimized S-CN(0.1) performed superior hydrogen evolution rate of 6225.4 μmol g−1 h−1 (λ> 420 nm), almost 45 times higher than the pristine bulk one. • An apparent quantum yield of 10 % at 420 nm was achieved for hydrogen production. • A reliable and universal method was developed to realize morphological evolution of graphitic carbon nitride with increasing reaction sites. Surface engineering is an efficient way to enhance photoabsorption, promote charge separation and boost photocatalysis. Herein, sulfur-doped holey g-C 3 N 4 nanosheets have been prepared through a universal self-templating approach with thiocyanuric acid as the single-precursor. By subtly controlling the feeding amount of precursor, the synthesized sulfur-doped holey g-C 3 N 4 nanosheets exhibit excellent visible-light driven photocatalytic hydrogen production activity. The optimized catalyst presents a hydrogen evolution rate of 6225.4 μmol g−1h−1, with an apparent quantum yield of 10 % at 420 nm. Comprehensive characterizations and theoretical calculations suggest that the enhanced photocatalysis is attributed to the synergy of the enlarged surface area, the negatively-shifted conduction band, and the narrowed bandgap due to sulfur-doping and ultra-thin two-dimensional topology. This work highlights the importance of controlling the precursor dosage and inducing sulfur doping into the polymer, providing a promising and reliable strategy to simultaneously regulate the nanostructural and electronic structure of g-C 3 N 4 for highly efficient photocatalysis. [ABSTRACT FROM AUTHOR]