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Search Results: 1 - 10 of 200519 matches for " P. P. Ntumba "
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On the endomorphisms of some sheaves of functions
Patrice P. Ntumba
Mathematics , 2013,
Abstract: Given a $C^\infty$ real manifold $X$ and $\mathcal{C}^m_X$ its sheaf of $m$-times differentiable real-valued functions, we prove that the sheaf $\mathcal{D}^{m, r}_X$ of differential operators of order $\leq m$ with coefficient functions of class $C^r$ can be obtained in terms of the sheaf $\mathcal{H}om_{\mathbb{R}_X}(\mathcal{C}^m_X, \mathcal{C}^r_X)$ of morphisms of $\mathcal{C}^m_X$ into $\mathcal{C}^r_X$. The superscripts $m$ and $r$ are integers.
On $\mathcal A$-Transvections and Symplectic $\mathcal A$-Modules
Patrice P. Ntumba
Mathematics , 2009,
Abstract: In this paper, building on prior joint work by Mallios and Ntumba, we show that $\mathcal A$-\textit{transvections} and \textit{singular symplectic }$\mathcal A$-\textit{automorphisms} of symplectic $\mathcal A$-modules of finite rank have properties similar to the ones enjoyed by their classical counterparts. The characterization of singular symplectic $\mathcal A$-automorphisms of symplectic $\mathcal A$-modules of finite rank is grounded on a newly introduced class of pairings of $\mathcal A$-modules: the \textit{orthogonally convenient pairings.} We also show that, given a symplectic $\mathcal A$-module $\mathcal E$ of finite rank, with $\mathcal A$ a \textit{PID-algebra sheaf}, any injective $\mathcal A$-morphism of a \textit{Lagrangian sub-$\mathcal A$-module} $\mathcal F$ of $\mathcal E$ into $\mathcal E$ may be extended to an $\mathcal A$-symplectomorphism of $\mathcal E$ such that its restriction on $\mathcal F$ equals the identity of $\mathcal F$. This result also holds in the more general case whereby the underlying free $\mathcal A$-module $\mathcal E$ is equipped with two symplectic $\mathcal A$-structures $\omega_0$ and $\omega_1$, but with $\mathcal F$ being Lagrangian with respect to both $\omega_0$ and $\omega_1$. The latter is the analog of the classical \textit{Witt's theorem} for symplectic $\mathcal A$-modules of finite rank.
Symplectic Reduction of Sheaves of $\mathcal{A}$-modules
A. Mallios,P. P. Ntumba
Mathematics , 2008,
Abstract: Given an arbitrary sheaf $\mathcal{E}$ of $\mathcal{A}$-modules (or $\mathcal{A}$-module in short) on a topological space $X$, we define \textit{annihilator sheaves} of sub-$\mathcal{A}$-modules of $\mathcal{E}$ in a way similar to the classical case, and obtain thereafter the analog of the \textit{main theorem}, regarding classical annihilators in module theory, see Curtis[\cite{curtis}, pp. 240-242]. The familiar classical properties, satisfied by annihilator sheaves, allow us to set clearly the \textit{sheaf-theoretic version} of \textit{symplectic reduction}, which is the main goal in this paper.
Fundamentals for Symplectic $\mathcal{A}$-modules
Anastasios Mallios,Patrice P. Ntumba
Mathematics , 2007,
Abstract: Sheaf theoretically based Abstract Differential Geometry incorporates and generalizes all the classical differential geometry. Here, we undertake to partially explore the implications of Abstract Differential Geometry to classical symplectic geometry. The full investigation will be presented elsewhere.
Biorthogonality in $\mathcal A$-Pairings and Hyperbolic Decomposition Theorem for $\mathcal A$-Modules
Patrice P. Ntumba,Adaeze C. Orioha
Mathematics , 2009,
Abstract: In this paper, as part of a project initiated by A. Mallios consisting of exploring new horizons for \textit{Abstract Differential Geometry} ($\grave{a}$ la Mallios), \cite{mallios1997, mallios, malliosvolume2, modern}, such as those related to the \textit{classical symplectic geometry}, we show that results pertaining to biorthogonality in pairings of vector spaces do hold for biorthogonality in pairings of $\mathcal A$-modules. However, for the \textit{dimension formula} the algebra sheaf $\mathcal A$ is assumed to be a PID. The dimension formula relates the rank of an $\mathcal A$-morphism and the dimension of the kernel (sheaf) of the same $\mathcal A$-morphism with the dimension of the source free $\mathcal A$-module of the $\mathcal A$-morphism concerned. Also, in order to obtain an analog of the Witt's hyperbolic decomposition theorem, $\mathcal A$ is assumed to be a PID while topological spaces on which $\mathcal A$-modules are defined are assumed \textit{connected}.
Pairings of Sheaves of $\mathcal{A}$-Modules through Bilinear $\mathcal{A}$-Morphisms
A. Mallios,PP Ntumba
Mathematics , 2008,
Abstract: It is proved that for any free $\mathcal{A}$-modules $\mathcal{F}$ and $\mathcal{E}$ of finite rank on some $\mathbb{C}$-algebraized space $(X, \mathcal{A})$ a \textit{degenerate} bilinear $\mathcal{A}$-morphism $\Phi: \mathcal{F}\times \mathcal{E}\longrightarrow \mathcal{A}$ induces a \textit{non-degenerate} bilinear $\mathcal{A}$-morphism $\bar{\Phi}: \mathcal{F}/\mathcal{E}^\perp\times \mathcal{E}/\mathcal{F}^\perp\longrightarrow \mathcal{A}$, where $\mathcal{E}^\perp$ and $\mathcal{F}^\perp$ are the \textit{orthogonal} sub-$\mathcal{A}$-modules associated with $\mathcal{E}$ and $\mathcal{F}$, respectively. This result generalizes the finite case of the classical result, which states that given two vector spaces $W$ and $V$, paired into a field $k$, the induced vector spaces $W/V^\perp$ and $V/W^\perp$ have the same dimension. Some related results are discussed as well.
Abstract Geometric Algebra. Orthogonal and Symplectic Geometries
PP Ntumba,Ac Orioha
Mathematics , 2008,
Abstract: Our main interest in this paper is chiefly concerned with the conditions characterizing \textit{orthogonal and symplectic abstract differential geometries}. A detailed account about the sheaf-theoretic version of the \textit{symplectic Gram-Schmidt theorem} and of the \textit{Witt's theorem} is also given.
Cofibrations in the Category of Frolicher Spaces. Part I
B. Dugmore,PP. Ntumba
Mathematics , 2007,
Abstract: Cofibrations are defined in the category of Fr\"olicher spaces by weakening the analog of the classical definition to enable smooth homotopy extensions to be more easily constructed, using flattened unit intervals. We later relate smooth cofibrations to smooth neighborhood deformation retracts. The notion of smooth neighborhood deformation retract gives rise to an analogous result that a closed Fr\"olicher subspace $A$ of the Fr\"olicher space $X$ is a smooth neighborhood deformation retract of $X$ if and only if the inclusion $i: A\hookrightarrow X$ comes from a certain subclass of cofibrations. As an application we construct the right Puppe sequence.
Hybrid Fuzzy Controller Based Frequency Regulation in Restructured Power System  [PDF]
P. Anitha, P. Subburaj
Circuits and Systems (CS) , 2016, DOI: 10.4236/cs.2016.76065
Abstract: This paper discusses the implementation of Load Frequency Control (LFC) in restructured power system using Hybrid Fuzzy controller. The formulation of LFC in open energy market is much more challenging; hence it needs an intelligent controller to adapt the changes imposed by the dynamics of restructured bilateral contracts. Fuzzy Logic Control deals well with uncertainty and indistinctness while Particle Swarm Optimization (PSO) is a well-known optimization tool. Abovementioned techniques are combined and called as Hybrid Fuzzy to improve the dynamic performance of the system. Frequency control of restructured system has been achieved by automatic Membership Function (MF) tuned fuzzy logic controller. The parameters defining membership function has been tuned and updated from time to time using Particle Swarm Optimization (PSO). The robustness of the proposed hybrid fuzzy controller has been compared with conventional fuzzy logic controller using performance measures like overshoot and settling time following a step load perturbation. The motivation for using membership function tuning using PSO is to show the behavior of the controller for a wide range of system parameters and load changes. Error based analysis with parametric uncertainties and load changes is tested on a two-area restructured power system.
Integral Performance Criteria Based Analysis of Load Frequency Control in Bilateral Based Market  [PDF]
P. Anitha, P. Subburaj
Circuits and Systems (CS) , 2016, DOI: 10.4236/cs.2016.76086
Abstract: Performance index based analysis is made to examine and highlight the effective application of Particle Swarm Optimization (PSO) to optimize the Proportional Integral gains for Load Frequency Control (LFC) in a restructured power system that operates under Bilateral based policy scheme. Various Integral Performance Criteria measures are taken as fitness function in PSO and are compared using overshoot, settling time and frequency and tie-line power deviation following a step load perturbation (SLP). The motivation for using different fitness technique in PSO is to show the behavior of the controller for a wide range of system parameters and load changes. Error based analysis with parametric uncertainties and load changes are tested on a two-area restructured power system. The results of the proposed PSO based controller show the better performance compared to the classical Ziegler-Nichols (Z-N) tuned PI andFuzzy Rule based PI controller.
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