An interested subject in homology theory, is the operators theory. Famous operators are those of ... more An interested subject in homology theory, is the operators theory. Famous operators are those of Adams and Steenrod operators. Some results of such operations are in the works of Gouda & Elhamdadi [1] on Hochschild and cyclic homology. In this article, we are interested in Steenrod operator in Dihedral and Reflexive homology on commutative Banach algebra
We study the relative cohomology theory of Banach algebra and give some important basic theorem o... more We study the relative cohomology theory of Banach algebra and give some important basic theorem of it. More, we give discuss about some of properties which we require it in our investigation. At long last, we ponder the dihedral cohomology aggregate as definitions and hypotheses and we characterize the Banach S-relative dihedral cohomology gathering and some theorems.
Journal of the Egyptian Mathematical Society, Oct 1, 2017
Topological Data Analysis is an emerging field at the intersection of algebraic topology and stat... more Topological Data Analysis is an emerging field at the intersection of algebraic topology and statistical inference aimed at describing the shapes objects represented as point cloud data in the multidimensional space. Since the range of applications of shape analysis is enormous, new tests have given birth to the field of TDA. In this habilitation study three TDA-oriented tests are discussed. A new test based on metric functions is proposed. A small simulation study among the preceding tests has been employed via Monte Carlo simulation. All the mentioned tests in the vignette are activated by real world data within educational field.
In this research, the basic definitions of an operad, graded algebra, and A ∞-module are introduc... more In this research, the basic definitions of an operad, graded algebra, and A ∞-module are introduced. The A ∞-algebras and their (co)homology are studied to obtain the relations between the cyclic and dihedral (co)homology. The L ∞-algebras are discussed, and the relations of the isomorphism between primitive and indecomposable elements in the L ∞-algebras are presented. We demonstrate the relation between cyclic and dihedral (co)homology of L ∞-algebras. Finally, the Mayer-Vietoris sequence of L ∞-algebras is investigated.
European Journal of Pure and Applied Mathematics, 2021
In this paper we study an elementary use of E-infinity modules and E-infinity algebras as togethe... more In this paper we study an elementary use of E-infinity modules and E-infinity algebras as together they have a use in terms of describing triangulated categories. Also, we show an interpretation of E-infinity algebras where the modules are fibrant objects within the categories of differential graded co-algebras and co-modules.
International Journal of Scientific and Research Publications (IJSRP)
In this paper we study the excision property of short sequence to get the long sequence of the di... more In this paper we study the excision property of short sequence to get the long sequence of the dihedral homology and reflexive homology of polynomial algebra. We give a new application of these theorems if we take a new category of graded lie algebras. Another application is relative homology.
An interested subject in homology theory, is the operators theory. Famous operators are those of ... more An interested subject in homology theory, is the operators theory. Famous operators are those of Adams and Steenrod operators. Some results of such operations are in the works of Gouda & Elhamdadi [1] on Hochschild and cyclic homology. In this article, we are interested in Steenrod operator in Dihedral and Reflexive homology on commutative Banach algebra
We study the relative cohomology theory of Banach algebra and give some important basic theorem o... more We study the relative cohomology theory of Banach algebra and give some important basic theorem of it. More, we give discuss about some of properties which we require it in our investigation. At long last, we ponder the dihedral cohomology aggregate as definitions and hypotheses and we characterize the Banach S-relative dihedral cohomology gathering and some theorems.
Journal of the Egyptian Mathematical Society, Oct 1, 2017
Topological Data Analysis is an emerging field at the intersection of algebraic topology and stat... more Topological Data Analysis is an emerging field at the intersection of algebraic topology and statistical inference aimed at describing the shapes objects represented as point cloud data in the multidimensional space. Since the range of applications of shape analysis is enormous, new tests have given birth to the field of TDA. In this habilitation study three TDA-oriented tests are discussed. A new test based on metric functions is proposed. A small simulation study among the preceding tests has been employed via Monte Carlo simulation. All the mentioned tests in the vignette are activated by real world data within educational field.
In this research, the basic definitions of an operad, graded algebra, and A ∞-module are introduc... more In this research, the basic definitions of an operad, graded algebra, and A ∞-module are introduced. The A ∞-algebras and their (co)homology are studied to obtain the relations between the cyclic and dihedral (co)homology. The L ∞-algebras are discussed, and the relations of the isomorphism between primitive and indecomposable elements in the L ∞-algebras are presented. We demonstrate the relation between cyclic and dihedral (co)homology of L ∞-algebras. Finally, the Mayer-Vietoris sequence of L ∞-algebras is investigated.
European Journal of Pure and Applied Mathematics, 2021
In this paper we study an elementary use of E-infinity modules and E-infinity algebras as togethe... more In this paper we study an elementary use of E-infinity modules and E-infinity algebras as together they have a use in terms of describing triangulated categories. Also, we show an interpretation of E-infinity algebras where the modules are fibrant objects within the categories of differential graded co-algebras and co-modules.
International Journal of Scientific and Research Publications (IJSRP)
In this paper we study the excision property of short sequence to get the long sequence of the di... more In this paper we study the excision property of short sequence to get the long sequence of the dihedral homology and reflexive homology of polynomial algebra. We give a new application of these theorems if we take a new category of graded lie algebras. Another application is relative homology.
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Papers by alaa mohamed