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Search Results: 1 - 10 of 201300 matches for " Patrice 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.
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}.
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.
Situating Cornerstone Propositions  [PDF]
Patrice Philie
Open Journal of Philosophy (OJPP) , 2012, DOI: 10.4236/ojpp.2012.24038
Abstract: Ostensibly, Wittgenstein’s last remarks published in 1969 under the title On Certainty are about epistemology, more precisely about the problem of scepticism. This is the standard interpretation of On Certainty. But I contend, in this paper, that we will get closer to Wittgenstein’s intentions and perhaps find new and illuminating ways to interpret his late contribution if we keep in mind that his primary goal was not to provide an answer to scepticism. In fact, I think that the standard reading (independently of its fruitfulness with dealing with scepticism) leads to a distorted view of Wittgenstein’s contribution in On Certainty. In order to see that, scepticism will first be briefly characterised, and then I will attempt to circumscribe more precisely the standard reading of On Certainty. In section 4, three exegetical arguments against the standard reading are offered – the hope being that the weight of these three arguments, taken together, instils doubt in the reader’s mind about the correctness of the standard reading. The paper concludes with an attempt to gesture at the philosophical significance of On Certainty once we set aside the standard reading.
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.
Assessment of Polyamines and Trehalose in Wheat Microspores Culture for Embryogenesis and Green Regenerated Plants  [PDF]
Amina Redha, Patrice Suleman
American Journal of Plant Sciences (AJPS) , 2013, DOI: 10.4236/ajps.2013.411275

Most aspects of microspore culture protocol have the capacity to cause stress to microspores, hence, less stressful treatments might be required to avoid deleterious effects. In stressed plants, polyamines and trehalose can act as compatible solutes or osmoprotectants by stabilizing proteins and biological membranes. To improve green plant regeneration in wheat microspore culture, this study assessed the effects of polyamines (putrecine, spermidine, spermine) and trehalose on androgenic response namely embryogenesis, green plant regeneration and ploidy of green plants regenerated in three spring wheat genotypes. Microspores of the genotypes produced significant numbers of embryos and green plants among polyamine treatments but trehalose had no effect (P ≤ 0.05). Polyamine treatments for 30 min generally produced more green plants per 100 microspores than the 60 min treatments in all three genotypes. At least three out of twelve polyamine treatments in each genotype improved the production of double haploid plants and seed setting in regenerants. Wheat genotype, concentration and duration of polyamine treatment had significant impact on embryogenesis and regeneration of green plants in this study. The study also showed that polyamines could be used to accelerate cultivar development in wheat breeding.

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