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1 - A note on the new basis in the mod 2 Steenrod algebraJournal of Linear and Topological Algebra , Issue 2 , Year , Spring 2018The Mod $2$ Steenrod algebra is a Hopf algebra that consists of the primary cohomology operations, denoted by $Sq^n$, between the cohomology groups with $\mathbb{Z}_2$ coefficients of any topological space. Regarding to its vector spac MoreThe Mod $2$ Steenrod algebra is a Hopf algebra that consists of the primary cohomology operations, denoted by $Sq^n$, between the cohomology groups with $\mathbb{Z}_2$ coefficients of any topological space. Regarding to its vector space structure over $\mathbb{Z}_2$, it has many base systems and some of the base systems can also be restricted to its sub algebras. On the contrary, in addition to the work of Wood, in this paper we define a new base system for the Hopf subalgebras $\mathcal{A}(n)$ of the mod $2$ Steenrod algebra which can be extended to the entire algebra. The new base system is obtained by defining a new linear ordering on the pairs $(s+t,s)$ of exponents of the atomic squares $Sq^{2^s(2^t-1)}$ for the integers $s\geq 0$ and $t\geq 1$. Manuscript profile -
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2 - Invariant elements in the dual Steenrod algebraJournal of Linear and Topological Algebra , Issue 4 , Year , Summer 2019In this paper, we investigate the invariant elements of the dual mod $p$ Steenrod subalgebra ${\mathcal{A}_p}^*$ under the conjugation map $\chi$ and give bounds on the dimensions of $(\chi-1)({\mathcal{A}_p}^*)_d$, where $({\mathcal{A}_p}^*)_d$ MoreIn this paper, we investigate the invariant elements of the dual mod $p$ Steenrod subalgebra ${\mathcal{A}_p}^*$ under the conjugation map $\chi$ and give bounds on the dimensions of $(\chi-1)({\mathcal{A}_p}^*)_d$, where $({\mathcal{A}_p}^*)_d$ is the dimension of ${\mathcal{A}_p}^*$ in degree $d$. Manuscript profile -
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3 - Conjectures on the anti-automorphism of Z-basis of the Steenrod algebraJournal of Linear and Topological Algebra , Issue 1 , Year , Winter 2021In this paper, we compute the images of some of the $Z$-basis elements under the anti-automorphism map $\chi$ of the mod 2 Steenrod algebra $\mathcal{A}_2$ and propose some conjectures based on our computations.In this paper, we compute the images of some of the $Z$-basis elements under the anti-automorphism map $\chi$ of the mod 2 Steenrod algebra $\mathcal{A}_2$ and propose some conjectures based on our computations. Manuscript profile -
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4 - Topological spaces induced by homotopic distanceJournal of Linear and Topological Algebra , Issue 2 , Year , Spring 2022Topological complexity which plays an important role in motion planning problem can be generalized to homotopic distance $\mathrm{D}$ as introduced in \cite{MVML}. In this paper, we study the homotopic distance and mention that it can be realize MoreTopological complexity which plays an important role in motion planning problem can be generalized to homotopic distance $\mathrm{D}$ as introduced in \cite{MVML}. In this paper, we study the homotopic distance and mention that it can be realized as a pseudometric on $\mathrm{Map}(X,Y)$. Moreover we study the topology induced by the pseudometric $\mathrm{D}$. In particular, we consider the space $\mathrm{Map}(S^1,S^1)$ and use the non-compactness of it to talk about the non-compactness of $\mathrm{Map}(X,Y)$. Manuscript profile