368 research outputs found

    On Some Special Property of The Farey Sequence

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    In this paper, some special property of the Farey sequence is discussed. We prove in each term of the Farey sequence, the sum of elements in the denominator is two times of the sum of elements in the numerator. We also prove that the Farey sequence contains a palindrome structure

    Transmission of electromagnetic power through a biological medium

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    Primary goal of this work is to study transmission of EM power through a multilayered biological medium. For a particular case study, EM power transmission from an external transmitter to a coupled receiver implanted inside a biological medium simulating a human body is studied to find solutions for factors such as optimum transmission frequency and excitation current. Different aspects of interaction of EM waves with biological bodies and tissues are discussed. Two major factors that may affect transmission of EM power through a biological body are absorption and reflection of EM waves. A simulation in which exact Maxwell\u27s equations are solved to find E field distribution in cross-sectional planes of a human body with the implanted receiver takes into account both absorption and reflection accurately. A simplified model for a human body with an implanted receiver and an external transmitter is developed here. Main motivation is to find E field distribution throughout the model and find energy density coupling between the transmitter and the receiver regions. Edge based finite element simulations are carried out on the model for a number of frequencies between 1 kHz and 9 GHz and frequency dependent values for EM properties such as relative permittivity and conductivity of biological tissues are used for all the simulations. Energy density coupling, E field coupling and S parameters showing reflection at the excitation port are obtained from the simulated results. Energy coupling is found to be almost constant with values near 0.01 between 1 kHz and 500 MHz. Current densities are below the thermal safe current density level even for an excitation current density of 3x106 A.m-2 in the transmitter. Although model used for simulation is simplistic, the results obtained are useful to study EM power losses in a multilayered biological medium. Results can be applied to find safe limits of excitation current density for transmitting EM power through a biological medium such as a human body without causing any damage due to heating

    On compatible Leibniz algebras

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    In this paper, we study compatible Leibniz algebras. We characterize compatible Leibniz algebras in terms of Maurer-Cartan elements of a suitable differential graded Lie algebra. We define a cohomology theory of compatible Leibniz algebras which in particular controls a one-parameter formal deformation theory of this algebraic structure. Motivated by a classical application of cohomology, we moreover study the abelian extension of compatible Leibniz algebras

    Cohomology, deformations and extensions of Rota-Baxter Leibniz algebras

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    A Rota-Baxter Leibniz algebra is a Leibniz algebra (g,[ , ]g)(\mathfrak{g},[~,~]_{\mathfrak{g}}) equipped with a Rota-Baxter operator T:g→gT : \mathfrak{g} \rightarrow \mathfrak{g}. We define representation and dual representation of Rota-Baxter Leibniz algebras. Next, we define a cohomology theory of Rota-Baxter Leibniz algebras. We also study the infinitesimal and formal deformation theory of Rota-Baxter Leibniz algebras and show that our cohomology is deformation cohomology. Moreover, We define an abelian extension of Rota-Baxter Leibniz algebras and show that equivalence classes of such extensions are related to the cohomology groups.Comment: 25 Page

    Involutive and oriented dendriform algebras

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    Dendriform algebras are certain splitting of associative algebras and arise naturally from Rota-Baxter operators, shuffle algebras and planar binary trees. In this paper, we first consider involutive dendriform algebras, their cohomology and homotopy analogs. The cohomology of an involutive dendriform algebra splits the Hochschild cohomology of an involutive associative algebra. In the next, we introduce a more general notion of oriented dendriform algebras. We develop a cohomology theory for oriented dendriform algebras that closely related to extensions and governs the simultaneous deformations of dendriform structures and the orientation.Comment: 22 pages; Subsection 3.4 is newly added; comments are welcom
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