Weighted Rota-Baxter operators on associative algebras are closely related to modified Yang-Baxter equations, splitting of algebras, weighted infinitesimal bialgebras, and play an important role in mathematical physics. For any λ ∈ k, we construct a differential graded Lie algebra whose Maurer-Cartan elements are given by λ-weighted relative Rota-Baxter operators. Using such characterization, we define the cohomology of a λ-weighted relative Rota-Baxter operator T , and interpret this as the Hochschild cohomology of a suitable algebra with coefficients in an appropriate bimodule. We study linear, formal and finite order deformations of T from cohomological points of view. Among others, we introduce Nijenhuis elements that generate trivial linear deformations and define a second cohomology class to any finite order deformation which is the obstruction to extend the deformation. In the end, we also consider the cohomology of λ-weighted relative Rota-Baxter operators in the Lie case and find a connection with the case of associative algebras.

Abstract. -When a soft elastic cylinder is bent beyond a critical radius of curvature, a sharp fold in the form of a kink appears at its inner side while the outer side remains smooth. The critical radius increases linearly with the diameter of the cylinder while remaining independent of its elastic modulus, although, its maximum deflection at the location of the kink depends on both the diameter and the modulus of the cylinders. Experiments are done also with annular cylinders of varying wall thickness which exhibits both the kinking and the ovalization of the cross-section. The kinking phenomenon appears to occur by extreme localization of curvature at the inner side of a post-buckled cylinder.Highly deformable, soft elastic materials occur in many different applications e.g. soft tissues, artificial organs, therapeutic patches, shock absorbers, dampeners, platforms for microfluidic devices etc. In these variety of applications the material is exposed to many different form of mechanical loads, which, due to the large deformability of these materials, can generate such responses which are different from that commonly observed with the liner elastic systems. An example is the surface wrinkling which is the most common form of mechanical response in elastic objects subjected to compressive stresses e.g. engendered by bending of an elastic block [1], or by sliding of a rubber block over a hard surface [2] or by the transverse poisson contraction of a uniaxially stretched rubber sheet [3]; however, here we report a different kind of behavior observed with soft hydrogel under similar circumstances. Our experiments with hydrogel cylinders bent beyond a critical curvature show that in stead of wrinkling, the material here responds by the appearance of one single sharp fold in the form of a kink at the inner side of the cylinder. As the cylinder is progressively bent, at a critical radius of curvature, the kink appears with a abrupt jump accompanied by the reversal of the curvature in the region close proximity to the kink; at the kink the curvature shoots up to infinity. While kinking phenomenon has been observed with slender biological filaments like DNA [4] or bacterial flagella [5] and inorganic fibers like multi-walled nano-tubes [6], where kinks appear rather intrinsically, mediated by the dual effects of local defects and intermolecular interactions, the kinking phenomenon observed in our system appears to be akin to the classical Euler's buckling instability [7], albeit localized at the inner side of a post-buckled cylinder. In this letter we have characterized this instability by using solid and annular cylinders of varying inner and outer diameters and by varying the modulus of the gel material.c EDP Sciences

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A relative Rota–Baxter algebra is a triple ( A, M, T) consisting of an algebra A, an A-bimodule M, and a relative Rota–Baxter operator T. Using Voronov’s derived bracket and a recent work of Lazarev, Sheng, and Tang, we construct an L∞[1]-algebra whose Maurer–Cartan elements are precisely relative Rota–Baxter algebras. By a standard twisting, we define a new L∞[1]-algebra that controls Maurer–Cartan deformations of a relative Rota–Baxter algebra ( A, M, T). We introduce the cohomology of a relative Rota–Baxter algebra ( A, M, T) and study infinitesimal deformations in terms of this cohomology (in low dimensions). As an application, we deduce cohomology of triangular skew-symmetric infinitesimal bialgebras and discuss their infinitesimal deformations. Finally, we define homotopy relative Rota–Baxter operators and find their relationship with homotopy dendriform algebras and homotopy pre-Lie algebras.

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