53 research outputs found
Bimodules over relative Rota-Baxter algebras and cohomologies
A relative Rota-Baxter algebra is a generalization of a Rota-Baxter algebra.
Relative Rota-Baxter algebras are closely related to dendriform algebras. In
this paper, we introduce bimodules over a relative Rota-Baxter algebra that
fits with the representations of dendriform algebras. We define the cohomology
of a relative Rota-Baxter algebra with coefficients in a bimodule and then
study abelian extensions of relative Rota-Baxter algebras in terms of the
second cohomology group. Finally, we consider homotopy relative Rota-Baxter
algebras and classify skeletal homotopy relative Rota-Baxter algebras in terms
of the above-defined cohomology.Comment: Comments and suggestions are welcom
Non-abelian extensions of Rota-Baxter Lie algebras and inducibility of automorphisms
A Rota-Baxter Lie algebra is a Lie algebra
equipped with a Rota-Baxter operator . In this paper, we consider non-abelian extensions of a
Rota-Baxter Lie algebra by another Rota-Baxter Lie algebra
We define the non-abelian cohomology which classifies {equivalence classes of}
such extensions. Given a non-abelian extension
of Rota-Baxter Lie algebras, we also show that the obstruction for a pair of
Rota-Baxter automorphisms in to be induced by an automorphism in
lies in the cohomology group . As a byproduct, we obtain the Wells
short-exact sequence in the context of Rota-Baxter Lie algebras.Comment: Any comments/suggestions are welcom
Recent Advances in Lossy Mode Resonance-Based Fiber Optic Sensors: A Review
Fiber optic sensors (FOSs) based on the lossy mode resonance (LMR) technique have gained substantial attention from the scientific community. The LMR technique displays several important features over the conventional surface plasmon resonance (SPR) phenomenon, for planning extremely sensitive FOSs. Unlike SPR, which mainly utilizes the thin film of metals, a wide range of materials such as conducting metal oxides and polymers support LMR. The past several years have witnessed a remarkable development in the field of LMR-based fiber optic sensors; through this review, we have tried to summarize the overall development of LMR-based fiber optic sensors. This review article not only provides the fundamental understanding and detailed explanation of LMR generation but also sheds light on the setup/configuration required to excite the lossy modes. Several geometries explored in the literature so far have also been addressed. In addition, this review includes a survey of the different materials capable of supporting lossy modes and explores new possible LMR supporting materials and their potential applications in sensing
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