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.

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.

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.

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.

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}.

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:
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.

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.

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.

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 and？Fuzzy
Rule based PI controller.