Binary Differential Operators Research Articles

Classification of 𝔞𝔣𝔣(3|1)-invariant binary differential operators on weighted densities and cohomology

In this paper, we consider the [Formula: see text]-module structure on the spaces of differential bilinear operators acting on the superspaces of weighted densities. We classify [Formula: see text]-invariant binary differential operators acting on the spaces of weighted densities. This result allows us to compute the first [Formula: see text]-relative differential cohomology of [Formula: see text] with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. Finally, we deduce the classification of the [Formula: see text]-relative extension of [Formula: see text] modules [Formula: see text]

Invariant bilinear operators and the second 𝔰𝔩(2)-relative cohomology of the Lie algebra of vector fields on ℝ

Let [Formula: see text] be the Lie algebra of smooth vector fields on [Formula: see text] and [Formula: see text] be the space of bilinear differential operators acting on weighted densities. In this paper, we classify [Formula: see text]-invariant skew-symmetric binary differential operators from [Formula: see text] to [Formula: see text] vanishing on [Formula: see text]. This result allows us to compute the second differential [Formula: see text]-relative cohomology of [Formula: see text] with coefficients in [Formula: see text].

An Empirical Analysis of Green Technology Innovation and Ecological Efficiency Based on a Greenhouse Evolutionary Ventilation Algorithm Fuzzy-Model

The combination and convergence of energy-intensive industries developed by ecological factors based on energy clusters is discussed in this paper. Here, a few models for the prediction of greenhouse effects are used as a single type of modeling. In this model, the solar panel system is included as a measure of the greenhouse effect; Commitment Unit (CU) formulations are changed with flouted logic because solar integrations and other unknown variables are intermittent. In general, the greenhouse model with natural ventilation temperature prediction is incomplete, in which the resulting fluid logical CU problem can be solved with an evolutionary algorithm based on the definition and the theory of quantum calculation. This paper proposes a Fuzzy Model-Based Quantum Greenhouse Evolutionary Ventilation Algorithm (FM-BSQGEVA) which helps to minimize the CU problem. The QGEVA is updated to include a hierarchy-group-oriented scheme to tackle the non-linear nature of the issue and its multifaceted nature. The QGEVA is further developed to support a new binary differential operator and several genetic algorithm operators with a redefined rotational angle look-up. The chances that such operators are used on separate solutions are affected by stating the membership function based on their related fitness. The fitness function is calculated through a combination of the penalty function, objective function and the added fluid function. The models built can be used to regulate and control natural ventilation in greenhouse effects. This finding shows that an energy-intensive industrial cluster’s environmental chain of the industry has improved eco-efficiency.

On osp(2|2)-relative cohomology of the Lie superalgebra of contact vector fields and deformations

Abstract We consider the 1 | 2 -dimensional real superspace R 1 | 2 endowed with its standard contact structure defined by the 1-form α . The conformal Lie superalgebra K ( 2 ) acts on R 1 | 2 as the Lie superalgebra of contact vector fields; it contains the Mobius superalgebra o s p ( 2 | 2 ) . We classify o s p ( 2 | 2 ) -invariant superskew-symmetric binary differential operators from K ( 2 ) ∧ K ( 2 ) to D λ , μ vanishing on o s p ( 2 | 2 ) , where D λ , μ is the superspace of linear differential operators between the superspaces of weighted densities. This result allows us to compute the second differential o s p ( 2 | 2 ) -relative cohomology of K ( 2 ) with coefficients in D λ , μ . We study generic formal o s p ( 2 | 2 ) -trivial deformations of the K ( 2 ) -module structure on the direct sum of the superspaces of weighted densities. This work is the simplest superization of a result by Bouarroudj (2007).

The binary aff(n|1)-invariant differential operators on weighted densities on the superspace R1|n and aff(n|1)-relative cohomology

We consider the $\mathfrak{aff}(n|1)-$module structure on the spaces of differential bilinear operators acting on the superspaces of weighted densities. We classify $\mathfrak{aff}(n|1)-$invariant binary differential operators acting on the spaces of weighted densities. This result allows us to compute the first $\mathfrak{aff}(n|1)-$relative differential cohomology of $\mathcal{K}(n)$ with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities.

Fuzzy logic‐based thermal generation scheduling strategy with solar‐battery system using advanced quantum evolutionary method

This study presents a fuzzy logic-based strategy to solve thermal generation scheduling [namely unit commitment (UC)] problem integrated with an equivalent solar-battery system using quantum inspired evolutionary algorithm. Solar-battery system is included with this model as a measure of green-house effect. The crisp formulations of UC are modified utilising fuzzy logics, because of the inherent intermittency involved in solar energy integration and other uncertain variables. An evolutionary algorithm based on the concept and principle of quantum computation is applied to solve the resultant fuzzy logic-based UC problem. Conventional quantum evolutionary algorithm (QEA) is modified by incorporating a hierarchy-group oriented scheme to deal with the non-linear and multi-peak nature of the problem. QEA is further advanced facilitating some genetic algorithm operators and a new binary differential operator along with rotation operator with a redefined rotational angle look-up table. The probabilities of using such operators on individual solution(s) are fuzzified by defining membership function based on associated fitness of that individual. The fitness function is then determined by combining the objective function, penalty function and the aggregated fuzzy membership function. The proposed fuzzy logic-based QEA (FLQEA) is applied to UC problem in two different scaled power systems. Provided simulation results will show the effectiveness of FLQEA.

Supertransvectants, cohomology, and deformations

Over the (1, N)-dimensional real superspace, N = 2, 3, we classify \documentclass[12pt]{minimal}\begin{document}$\mathfrak {osp}(N|2)$\end{document}osp(N|2)-invariant binary differential operators acting on the superspaces of weighted densities, where \documentclass[12pt]{minimal}\begin{document}$\mathfrak {osp}(N|2)$\end{document}osp(N|2) is the orthosymplectic Lie superalgebra. This result allows us to compute the first differential \documentclass[12pt]{minimal}\begin{document}$\mathfrak {osp}(N|2)$\end{document}osp(N|2)-relative cohomology of the Lie superalgebra \documentclass[12pt]{minimal}\begin{document}$\mathcal {K}(N)$\end{document}K(N) of contact vector fields with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. We classify generic formal \documentclass[12pt]{minimal}\begin{document}$\mathfrak {osp}(3|2)$\end{document}osp(3|2)-trivial deformations of the \documentclass[12pt]{minimal}\begin{document}$\mathcal {K}(3)$\end{document}K(3)-module structure on the superspaces of symbols of differential operators. We prove that any generic formal \documentclass[12pt]{minimal}\begin{document}$\mathfrak {osp}(3|2)$\end{document}osp(3|2)-trivial deformation of this \documentclass[12pt]{minimal}\begin{document}$\mathcal {K}(3)$\end{document}K(3)-module is equivalent to its infinitesimal part. This work is the simplest generalization of a result by the first author et al. [Basdouri, I., Ben Ammar, M., Ben Fraj, N., Boujelbene, M., and Kammoun, K., “Cohomology of the Lie superalgebra of contact vector fields on \documentclass[12pt]{minimal}\begin{document}$\mathbb {K}^{1|1}$\end{document}K1|1 and deformations of the superspace of symbols,” J. Nonlinear Math. Phys. 16, 373 (2009)10.1142/S1402925109000431].

Localized Spatial Soliton Excitations in (2 + 1)-Dimensional Nonlinear Schrödinger Equation with Variable Nonlinearity and an External Potential

We report on the localized spatial soliton excitations in the multidimensional nonlinear Schrödinger equation with radially variable nonlinearity coefficient and an external potential. By using Hirota's binary differential operators, we determine a variety of external potentials and nonlinearity coefficients that can support nonlinear localized solutions of different but desired forms. For some specific external potentials and nonlinearity coefficients, we discuss features of the corresponding (2+1)-dimensional multisolitonic solutions, including ring solitons, lump solitons, and soliton clusters.

The binary invariant differential operators on weighted densities on the superspace R1∣n and cohomology

Over the (1,n)-dimensional real superspace, n>1, we classify K(n)-invariant binary differential operators acting on the superspaces of weighted densities, where K(n) is the Lie superalgebra of contact vector fields. This result allows us to compute the first differential cohomology of K(n) with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities—a superization of a result by Feigin and Fuchs [“Homology of the Lie algebras of vector fields on the line,” Funct. Anal. Appl. 14, 201 (1980)]. We explicitly give 1-cocycles spanning these cohomology spaces.