For converting light energy into electricity, an optical pendulum generator was designed by combining photomechanical movement of liquid-crystalline actuator (LCA) with Faraday's law of electromagnetic induction. Bilayer cantilever actuators were first fabricated with LDPE and LCA. Their photomechanical movement drove the attached copper coils to cut magnetic line of force generating electricity. The output electricity was proportional to the changing rate of the magnetic flux, which was greatly influenced by light intensity, film thickness, and sample size. Continuous electrical output was also achieved. This simple strategy may expand applications of photoactive materials in the capture and storage of light energy.

We determine the $$L_\infty $$ L ∞ -algebra that controls deformations of a relative Rota–Baxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying $$\mathsf {Lie}\mathsf {Rep}$$ Lie Rep  pair by the dg Lie algebra controlling deformations of the relative Rota–Baxter operator. Consequently, we define the cohomology of relative Rota–Baxter Lie algebras and relate it to their infinitesimal deformations. A large class of relative Rota–Baxter Lie algebras is obtained from triangular Lie bialgebras and we construct a map between the corresponding deformation complexes. Next, the notion of a homotopy relative Rota–Baxter Lie algebra is introduced. We show that a class of homotopy relative Rota–Baxter Lie algebras is intimately related to pre-Lie$$_\infty $$ ∞ -algebras.

Bilayer photomechanical films are fabricated by depositing one layer of molecular azobenzene chromophores onto flexible low-density polyethylene substrates. The photoinduced bending and unbending behavior of five azobenzene derivatives including azobenzene, 4-hydroxy-azobenzene, 4-((4-hydroxyphenyl)diazenyl)bezoitrile, 4-((4-methoxyph-enyl)diazenyl)phenol, and 4-(phenyldiazenyl)phenol is systematically studied by considering the incident light intensity and the thickness of the coated chromophore layers. Precise control of photoinduced curling of the bilayer film is successfully achieved upon irradiation with two beams of UV light, and the curled films can be recovered by thermal relaxation in the dark. The easily fabricated bilayer films show fast photomechanical response, strong photoinduced stress, and stability similar to crosslinked polymeric films.

The main object of study of this paper is the notion of a LieDer pair, i.e. a Lie algebra with a derivation. We introduce the concept of a representation of a LieDer pair and study the corresponding cohomologies. We show that a LieDer pair is rigid if the second cohomology group is trivial, and a deformation of order n is extensible if its obstruction class, which is defined to be an element is the third cohomology group, is trivial. We classify central extensions of LieDer pairs using the second cohomology group with the coefficient in the trivial representation. For a pair of derivations, we define its obstruction class and show that it is extensible if and only if the obstruction class is trivial. Finally, we classify Lie2Der pairs using the third cohomology group of a LieDer pair.

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In this paper, we first construct the controlling algebras of embedding tensors and Lie–Leibniz triples, which turn out to be a graded Lie algebra and an $$L_\infty $$ L ∞ -algebra respectively. Then we introduce representations and cohomologies of embedding tensors and Lie–Leibniz triples, and show that there is a long exact sequence connecting various cohomologies. As applications, we classify infinitesimal deformations and central extensions using the second cohomology groups. Finally, we introduce the notion of a homotopy embedding tensor which will induce a Leibniz$$_\infty $$ ∞ -algebra. We realize Kotov and Strobl’s construction of an $$L_\infty $$ L ∞ -algebra from an embedding tensor, as a functor from the category of homotopy embedding tensors to that of Leibniz$$_\infty $$ ∞ -algebras, and a functor further to that of $$L_\infty $$ L ∞ -algebras.

AC) stimuli. Xiao et al. [30] fabricated a graphene/poly(vinyldene fluoride) (PVDF) bilayer actuator by coating of PVDF solution onto a porous graphene paper. The hybrid film exhibited electrodriven vibration having rapid response rate, large displacement, and durable stability. In those electrothermal actuators, Joule heating was generated when electric current passed through the graphene film. Then the thermal expansion resulted in the large amount of deflection of the graphene/ polymer bilayer film. Polymer actuator is advantageous because of its high sensitivity, low energy input, and larger expansion rate. In this work, reduced graphene oxide (RGO)-based actuator was fabricated by spin-coating a reduced graphene oxide solution onto a polymer substrate obtaining a bilayer structure actuator. Poly dimethylsiloxane (PDMS) or polyethylene (PE) polymer substrate was used to support the electric-heated RGO layer and enlarge the thermal expansion deformation. [31] The bilayer film showed responsive bending motion under DC or AC voltage driving. Several actuate modes were proposed according to the bilayer structure design and driving current control. Experimental Section Preparation of RGOGraphite powder (2 g) was added into 46 mL of concentrated H 2 SO 4 , followed by adding 1 g NaNO 3 into above mixture under stirring and cooling in an ice bath condition. The mixture was continuous stirred while 6 g KMnO 4 was added slowly to keep the temperature of mixture below 5 °C. Then the mixture was kept at 35 °C for 30 min, followed by adding 90 mL deionized water while stirring, and the temperature would rise up to 95 °C. The mixture was kept stirring for a further 30 min, and 100 mL deionized water and 10 mL 30% H 2 O 2 were added in sequence. The oxidized material was then washed with 1:10 (v:v) HCl solution one time and deionized water three times to remove metal ions, followed by centrifugation. The collected product was dried in a vacuum drying oven at 45 °C.To obtain RGO, the as-synthesized GO powder (10 mg) was suspended in 20 mL of distilled water by ultrasonication until a yellowish-brown colloid was obtained. Subsequently, few drops of ammonia solution (35%) were added to increase the pH up to 8 allowing stability of the sheets. 15 µL hydrazine hydrate (0.1 m) solution was added to the above solution and refluxed at 98 °C for 100 min in a water bath, affording the formation Electrothermal Actuator Electrodriven bilayer actuator is designed and fabricated by spin-coating a reduced graphene oxide solution onto a polymer substrate. The bonded interface properties, electrical and thermal conductivities are characterized through scanning electron microscopy and X-ray diffraction. The bilayer actuator exhibits fast and large bending response when a direct current voltage is applied to the graphene layer. Whereas it exhibits oscillation when an alternating current voltage is applied to the bilayer actuator. The effects of the layer structure and the electro-operation methods on the bending motions are studied. Two new ...

In this paper, we introduce the cohomology theory of relative Rota–Baxter operators on Leibniz algebras. We use the cohomological approach to study linear and formal deformations of relative Rota–Baxter operators. In particular, the notion of Nijenhuis elements is introduced to characterize trivial linear deformations. Formal deformations and extendibility of order [Formula: see text] deformations of a relative Rota–Baxter operator are also characterized in terms of the cohomology theory.