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

High lubricity meets load capacity: cartilage mimicking bilayer structure by brushing up stiff hydrogels from subsurface

Natural articular cartilage has ultralow friction even at high squeezing pressure. Biomimicking cartilage with soft materials has been and remains a grand challenge in the fields of materials science and engineering. Inspired by the unique structural features of the articular cartilage, as well as by its remarkable lubrication mechanisms dictated by the properties of the superficial layers, a novel archetype of cartilage‐mimicking bilayer material by robustly entangling thick hydrophilic polyelectrolyte brushes into the subsurface of a stiff hydrogel substrate is developed. The topmost soft polymer layer provides effective aqueous lubrication, whereas the stiffer hydrogel layer used as a substrate delivers the load‐bearing capacity. Their synergy is capable of attaining low friction coefficients (order 0.010) under heavily loaded conditions (order 10 MPa contact pressure) in water environment, a performance incredibly close to that of natural articular cartilage. The bioinspired material can maintain low friction even when subjected to 50k reciprocating cycles under high contact pressure, with almost no wear observed on the sliding track. These findings are theoretically explained and compounded by multiscale simulations used to shed light on the mechanisms responsible for this remarkable performance. This work opens innovative technology routes for developing cartilage‐mimicking ultralow friction soft materials

Rota–Baxter (Co)algebra Equation Systems and Rota–Baxter Hopf Algebras

We introduce and discuss the notions of Rota–Baxter bialgebra equation systems and Rota–Baxter Hopf algebras. Then we construct a lot of examples based on Hopf quasigroups

Rota–Baxter (Co)algebra Equation Systems and Rota–Baxter Hopf Algebras

We introduce and discuss the notions of Rota–Baxter bialgebra equation systems and Rota–Baxter Hopf algebras. Then we construct a lot of examples based on Hopf quasigroups

A METHOD OF CONSTRUCTING BRAIDED HOPF ALGEBRAS

Let A and B be two Hopf algebras and R ∈ Hom(B ⊗ A, A ⊗ B), the twisted tensor product Hopf algebra A#RB was introduced by S. Caenepeel et al in [3] and further studied in our recent work [6]. In this paper we give the necessary and sufficient conditions for A#RB to be a Hopf algebra with a projection. Furthermore, a braided Hopf algebra A is constructed by twisting the multiplication of A through a (γ, R)-pair (A, B). Finally we give a method to construct Radford’s biproduct directly by defining the module action and comodule action from the twisted tensor biproduct.

Double crossed biproducts and related structures

Let $H$ be a bialgebra. Let $\sigma: H\otimes H\to A$ be a linear map, where $A$ is a left $H$-comodule coalgebra, and an algebra with a left $H$-weak action $\triangleright$. Let $\tau: H\otimes H\to B$ be a linear map, where $B$ is a right $H$-comodule coalgebra, and an algebra with a right $H$-weak action $\triangleleft$. In this paper, we improve the necessary conditions for the two-sided crossed product algebra $A\#^{\sigma} H~{^{\tau}\#} B$ and the two-sided smash coproduct coalgebra $A\times H\times B$ to form a bialgebra (called double crossed biproduct) such that the condition $b_{[1]}\triangleright a_0\otimes b_{[0]}\triangleleft a_{-1}=a\otimes b$ in Majid's double biproduct (or double-bosonization) is one of the necessary conditions. On the other hand, we provide a more general two-sided crossed product algebra structure via Brzez\'nski's crossed product and give some applications.Comment: Communications in Algebra,202

Robust Super‐Lubricity for Novel Cartilage Prototype Inspired by Scallion Leaf Architecture

The simultaneous achievement -under physiologically high contact pressures- of ultra-low friction, nearly zero surface wear, and long lifetime in the development of human cartilage prosthetics is still a big challenge. In this work, inspired by the unique lubrication mechanism of scallion leaves resulting from the synergy of oriented surface micro-topography and mucus hydration, a novel layered soft hydrogel as cartilage prototype is developed by chemically embedding thick hydrophilic polyelectrolyte brush chains into the sub-surface of a high strength anisotropic hydrogel bulk. It exhibits an anisotropic polymer network with unique mechanical properties (tensile strength: 8.3 to 23.7 MPa; elastic modulus 20.0 to 30.0 MPa), anisotropic hydrated surface texture, super-lubricity, and excellent wear resistance. Thydrogel architecture can exhibit low coefficient of friction (COF) less than approximate to 0.01 under a wide range of contact stresses (0.2 to 2.4 MPa) and maintain cartilage-like long-lasting (50k sliding cycles) robust super-lubricity (COF approximate to 0.006) and nearly-zero wear under high contact pressure (approximate to 2.4 MPa) condition. Theoretical underpinning reveals how multiscale surface anisotropy, mechanics, and hydration regulate super-low friction generation. This work provides a novel design paradigm for the fabrication of robust soft materials with extraordinary lubricity as implantable prototypes and coatings.A novel anisotropic layered lubrication hydrogel (ALLH) is engineered by mimicking the architecture and lubrication mechanism of natural scallion leaf (NSL). The ALLH exhibits low coefficient of friction (COF) less than approximate to 0.01 under a wide range of contact stresses and demonstrates long-lasting (50k sliding cycles) robust super-lubricity (COF approximate to 0.006) along with nearly-zero wear under high contact pressure (approximate to 2.4 MPa).imag

3D Printing of an Oil/Water Mixture Separator with In Situ Demulsification and Separation

Currently, many meshes, membranes, and fabrics with extreme wettability of superhydrophobicity/superoleophilicity, or superhydrophilicity and underwater superoleophobicity are promising candidates for oil/water mixture separation. Nevertheless, a facile yet effective way to design and fabricate porous mesh still remains challenging. In this work, fused deposition modeling (FDM) 3D printing of Fe/polylactic acid (PLA) composites was employed to fabricate superhydrophilic and underwater superoleophobic mesh (S-USM) with hydrogel coatings via the surface polymerization of Fe(II)-mediated redox reaction. In addition, salt of aluminum chloride was incorporated within the hydrogel coating, which was attributed to strengthening the demulsification of oil-in-water emulsions, resulting in efficient separation of oil-in-water mixtures. The S-USM was efficient for a wide range of oil-in-water mixtures, such as dodecane, diesel, vegetable oil, and even crude oil, with a separation efficiency of up to 85%. In this study, the flexible design and fabrication of 3D printing were used for the facile creation of spherical oil skimmers with hydrogel coatings that were capable of removing the floating oil. Most importantly, this work is expected to promote post-treatment processes using 3D printing as a new manufacturing technology and, in this way, a series of devices of specific shape and function will be expanded to satisfy desired requirements and bring great convenience to personal life