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

Data-driven system modeling and optimal control for nonlinear dynamical systems

With the increasing complexity of modern industry processes, robotics, transportation, aerospace, power grids, an exact model of the physical systems are extremely hard to obtain whereas abundant of time-series data can be captured from these systems. This makes it a important and demanding research area to investigate feasibility of using data to learn behaviours of systems and design controllers where the end goal generally evolves around stabilization. Transfer operators i.e. Perron-Frobenius and Koopman operators play an undeniable role in advanced research of nonlinear dynamical system stabilization. These operators have been a alternate direction of how we generally approach dynamical systems, providing linear representations for even strongly nonlinear dynamics. There is tremendous benet of acquiring a linear model of a system using these models but, there remains a challenge of innite dimension for such models. To deal with it, we can approximate a nite dimensional matrix of these operators e.g. Koopman matrix using Extended Dynamic Mode Decomposition (EDMD) or Naturally Structured Dynamic Mode Decomposition (NSDMD). Using duality property of Koopman and P-F operators we can derive formulation for P-F matrix from Koopman matrix. Once we have a linear approximation of the system, Lyapunov measure approach can be used along with a linear programming based computational framework for stability analysis and design of almost everywhere stabilizing controller. In this work, we propose a complete structure to stabilize a system that does not have an explicit model and only requires black box input output time-series data. On a separate work, we show a set-oriented approach can be used to control and stabilize systems with known dynamics model however having stochastic parameters. Essentially, this work proposes two approaches to stabilize a nonlinear system using both of known system model with inherent uncertainty and stabilize a black box system entirely using input-output data

Non-abelian extensions of Rota-Baxter Lie algebras and inducibility of automorphisms

A Rota-Baxter Lie algebra $\mathfrak{g}_T$ is a Lie algebra $\mathfrak{g}$ equipped with a Rota-Baxter operator $T : \mathfrak{g} \rightarrow \mathfrak{g}$. In this paper, we consider non-abelian extensions of a Rota-Baxter Lie algebra $\mathfrak{g}_T$ by another Rota-Baxter Lie algebra $\mathfrak{h}_S.$ We define the non-abelian cohomology $H^2_{nab} (\mathfrak{g}_T, \mathfrak{h}_S)$ which classifies {equivalence classes of} such extensions. Given a non-abelian extension $$0 \rightarrow \mathfrak{h}_S \xrightarrow{i} \mathfrak{e}_U \xrightarrow{p} \mathfrak{g}_T \rightarrow 0$$ of Rota-Baxter Lie algebras, we also show that the obstruction for a pair of Rota-Baxter automorphisms in $\mathrm{Aut}(\mathfrak{h}_S ) \times \mathrm{Aut}(\mathfrak{g}_T)$ to be induced by an automorphism in $\mathrm{Aut}(\mathfrak{e}_U)$ lies in the cohomology group $H^2_{{nab}} (\mathfrak{g}_T, \mathfrak{h}_S)$. As a byproduct, we obtain the Wells short-exact sequence in the context of Rota-Baxter Lie algebras.Comment: Any comments/suggestions are welcom

Study of cyclic bursting loading on needle-punched nonwovens: Part II – Change in air permeability and compression behavior

Efforts have been made to investigate the physical properties of needle-punched nonwovens, such as compression and air permeability, before and after the application of cyclic bursting pressure. It is observed that the structural parameters, such as areal density, needling density and fibre fineness, have significant effect on the aforementioned fabric characteristics. More compression is observed in a sample with low mass density. Sample made of coarser fibre shows lower compression as compared to finer fibre samples. The air permeability decreases with increase in mass density. On conducting cyclic bursting test on a sample, the values of air permeability and compression change significantly. The cyclic parameters, such as pressure magnitude and rest time at cyclic peak pressure, cause significant structural changes and reorientation of the fibre during deformation. Increase in pressure and rest time causes increase in compression parameter (α) and air permeability. Nevertheless, at low cyclic pressure level (10% of bursting pressure), the network becomes compact due to fibrous reorientation, and therefore both the values of compression parameter (α) and air permeability show initial drop. On examining the samples made of coarser and finer fibres, the variation is observed in the extent of change in the air permeability

Water filtration using nonwoven cartridge filter system

A cartridge based fibrous filtration system has been designed and investigated. For the fibrous medium, needle-punch nonwoven structure has been selected and different nonwoven samples are analysed for the water purification. A series of different needle-punched nonwovens made of polypropylene have been produced by changing mass per unit area, needling density and fibre linear density. A chemical oxygen demand test has been employed to obtain the filtration efficiency (FE). The FE obtained for these samples ranges from 8.84% to 78.04% in purifying the reference water (mud water). It has been found that the FE increases with increase in mass per unit area and needling density (p < 0.01). Also, the filter media made of finer fibres displays higher FE than coarser fibre (p < 0.01). On examining the filter performance in multiple filtration cycles, it is found that the FE increases initially and finally reaches to a saturation value. A good correlation (r2 > 0.95) has been found for the FE of each cycle with the air permeability, thickness and weight density of the loaded filter. The FE of the bare nonwoven (maximum FE) further improves (95%) by incorporating activated particles

Effect of enzymatic process on characteristics of cottonized industrial hemp fibre

This paper reports an investigation on the bacteria-based enzyme's cottonization of industrial hemp fibre. The industrial hemp fibres are enzyme processed to eliminate massive non-cellulosic portions from the fibre to enhance their fineness and softness. Box- Behnken response surface methodology is applied to optimize the effect of different concentrations, temperature and time by using enzymes on chemical and physical properties, like weight loss, average strength, length, chemical composition, and surface modification to simulate cotton feel touch. The results show that the pectin, lignin, hemicellulose, and other impurities are removed under the enzyme and alkali refining processes. The effect of the enzyme concentration, treatment time, and treatment temperature is found significant on weight loss, fibre diameter, crystallinity, cellulose, hemicelluloses content, and tenacity. The effluent load of the enzyme process by measuring COD is also found less than that of the alkali processing for industrial hemp fibre. Industrial hemp fibre can be a perfect model of the sustainability of fibre from the plant's inception into the entire life cycle, leading to the biodegradable product reducing less effluent load in an environment.

Cyclic bursting loading on needle-punched nonwovens: Part I – Distention behavior

The present study aims at examining different needle-punched polypropylene nonwovens under different cyclic bursting pressure. Various fabric parameters including mass density, punch density and fibre fineness have been investigated. For the cyclic test, the distension is measured at different cyclic bursting pressures proportionate to the bursting strength of the fabric. Other parameters including the cyclic pressure magnitude, the number of cycles, and the rest time at peak pressure have also been investigated. It has been found that the bursting strength and distension of fabric increase with an increase in mass density, while they show opposite trend with punch density and fibre denier (p < 0.01). The distension value of each sample increases with an increase in the cyclic parameters i.e. number of loading cycle, rest time and pressure peak

Water filtration using nonwoven cartridge filter system

72-79A cartridge based fibrous filtration system has been designed and investigated. For the fibrous medium, needle-punch nonwoven structure has been selected and different nonwoven samples are analysed for the water purification. A series of different needle-punched nonwovens made of polypropylene have been produced by changing mass per unit area, needling density and fibre linear density. A chemical oxygen demand test has been employed to obtain the filtration efficiency (FE). The FE obtained for these samples ranges from 8.84% to 78.04% in purifying the reference water (mud water). It has been found that the FE increases with increase in mass per unit area and needling density (p r2 > 0.95) has been found for the FE of each cycle with the air permeability, thickness and weight density of the loaded filter. The FE of the bare nonwoven (maximum FE) further improves (95%) by incorporating activated particles