arXiv subjects

Ameneh Babaee

Publications and source records attributed to Ameneh Babaee.

9 recordsLinked to original sources

arXiv2025

On Targeted Complexity of Discrete Motion

In this paper, we study targeted simplicial complexity $TC(K, L)$ introduced for situations where the configuration space possesses a simplicial structure $K$ together with a set of configurations $L$ as the target of motion. This type of complexity admits smaller values than the discrete version $TC(K)$. We then demonstrate that targeted simplicial complexity is strongly homotopy invariant and it varies between simplicial LS-categories of $K$ and $K \prod K$. Utilizing this information, we calculate targeted simplicial complexity for cases such as strongly collapsible complexes being equal to zero and for categoriacl subcomplex $L$, $TC(K,L) = scat(K)$. Moreover, we compare targeted simplicial complexity with relative topological complexity getting $TC(|K|, |L|) \le TC (K,L)$ where $|\cdot|$ denotes the geometric realization functor, and they are equal in certain cases, such as arbitrary wedges of triangulated circles. Also we define targeted $m$-step simplicial complexity of motions $TC_m(K,L)$ by using $m$-paths, paths whose length is smaller than or equal to $m$, to solve the problems of motion where the robot needs to be charged or repaired after $m$-steps. For $m$-step simplicial complexity a new invariance holds, which we call $m$-homotopy invariance introduced by $m$-paths. Finally we compare targeted $m$-step simplicial complexity with $m$-simplicial category $Scat_m$ to obtain some lower and upper bounds and then we prove the sequence of inequalities $scat_{m}(K)\leq TC_{m}(K,L) \le TC_{m}(K) \leq scat_{[\frac{m}{2}]}(K\prod K)$.

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arXiv2024

Discrete homotopic distance between Lipschitz maps

In this paper, we investigate a discrete version of the homotopic distance between two $s$-Lipschitz maps for $s \geq 0$. This distance is defined by specifying a step length $r$ to which some homotopy relation corresponds. In spaces with a significant number of holes, where no continuous homotopy exist and the homotopic distance equals infinite, the discrete homotopic distance provides a meaningful classification by effectively ignoring smaller holes. We show that the discrete homotopic distance $D_r$ generalizes key concepts such as the discrete Lusternik-Schnirelmann category $\text{cat}_r$ and the discrete topological complexity $\text{TC}_r$. Furthermore, we prove that $D_r$ is invariant under discrete homotopy relations. This approach offers a flexible framework for classifying $s$-Lipschitz maps, loops, and paths based on the choice of $r$.

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arXiv2024

A Discrete Topological Complexity of Discrete Motion Planning

In this paper we generalize the discrete r-homotopy to the discrete (s, r)-homotopy. Then by this notion, we introduce the discrete motion planning for robots which can move discreetly. Moreover, in this case the number of motion planning, called discrete topological complexity, required for these robots is reduced. Then we prove some properties of discrete topological complexity; For instance, we show that a discrete motion planning in a metric space X exists if and only if X is a discrete contractible space. Also, we prove that the discrete topological complexity depends only on the strictly discrete homotopy type of spaces.

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arXiv2022

On Topological Homotopy Groups and Relation to Hawaiian Groups

By generalizing the whisker topology on the $n$th homotopy group of pointed space $(X, x_0)$, denoted by $π_n^{wh}(X, x_0)$, we show that $π_n^{wh}(X, x_0)$ is a topological group if $n \ge 2$. Also, we present some necessary and sufficient conditions for $π_n^{wh}(X,x_0)$ to be discrete, Hausdorff and indiscrete. Then we prove that $L_n(X,x_0)$ the natural epimorphic image of the Hawaiian group $\mathcal{H}_n(X, x_0)$ is equal to the set of all classes of convergent sequences to the identity in $π_n^{wh}(X, x_0)$. As a consequence, we show that $L_n(X, x_0) \cong L_n(Y, y_0)$ if $π_n^{wh}(X, x_0) \cong π_n^{wh}(Y, y_0)$, but the converse does not hold in general, except for some conditions. Also, we show that on some classes of spaces such as semilocally $n$-simply connected spaces and $n$-Hawaiian like spaces, the whisker topology and the topology induced by the compact-open topology of $n$-loop space coincide. Finally, we show that $n$-SLT paths can transfer $π_n^{wh}$ and hence $L_n$ isomorphically along its points.

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arXiv2022

On the Targeted Complexity of a Map

We study the topological complexity of work maps with respect to some subspaces of the configuration space and a workspace considered as the target set of the motion of robots. The motivation is to optimize and reduce the number of motion planners for work maps. In this regard, we focus on the useful set of works. We check some basic properties of the targeted complexity of maps, such as homotopical invariance, reduction, the product of maps, and so on. Then we compare these targeted complexities, and we find some inequalities in reducing the number of motion planners. We show that the relative topological complexity of pair of spaces defined by Short is a special case of the targeted complexity of work maps.

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arXiv2022

On Hawaiian homology groups

In this paper, we introduce a kind of homology which we call Hawaiian homology to study and classify pointed topological spaces. The Hawaiian homology group has advantages of Hawaiian groups. Moreover, the first Hawaiian homology group is isomorphic to the abelianization of the first Hawaiian group for path-connected and locally path-connected topological spaces. Since Hawaiian homology has concrete elements and abelian structure, its calculations are more routine. Thus we use Hawaiian homology groups to compare Hawaiian groups, and then we obtain some information about Hawaiian groups of some wild topological spaces.

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arXiv2021

On quasi-small loop groups

In this paper, we study some properties of homotopical closeness for paths. We define the quasi-small loop group as the subgroup of all classes of loops that are homotopically close to null-homotopic loops, denoted by $π_1^{qs} (X, x)$ for a pointed space $(X, x)$. Then we prove that, unlike the small loop group, the quasi-small loop group $π_1^{qs}(X, x)$ does not depend on the base point, and that it is a normal subgroup containing $π_1^{sg}(X, x)$, the small generated subgroup of the fundamental group. Also, we show that a space $X$ is homotopically path Hausdorff if and only if $π_1^{qs} (X, x)$ is trivial. Finally, as consequences, we give some relationships between the quasi-small loop group and the quasi-topological fundamental group.

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arXiv2017

On a Van Kampen Theorem for Hawaiian Groups

The paper is devoted to study the $n$th Hawaiian group $\mathcal{H}_n$, $n \ge 1$, of the wedge sum of two spaces $(X,x_*) = (X_1, x_1) \vee (X_2, x_2)$. Indeed, we are going to give some versions of the van Kampen theorem for Hawaiian groups of the wedge sum of spaces. First, among some results on Hawaiian groups of semilocally strongly contractible spaces, we present a structure for the $n$th Hawaiian group of the wedge sum of CW-complexes. Second, we give more informative structures for the $n$th Hawaiian group of the wedge sum $X$, when $X$ is semilocally $n$-simply connected at $x_*$. Finally, as a consequence, by generalizing the well-known Griffiths space for dimension $n\geq 1$, we give some information about the structure of Hawaiian groups of Griffiths spaces at any points.

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arXiv2012

On Hawaiian Groups of Some Topological Spaces

The paper is devoted to study the structure of Hawaiian groups of some topological spaces. We present some behaviors of Hawaiian groups with respect to product spaces, weak join spaces, cone spaces, covering spaces and locally trivial bundles. In particular, we determine the structure of the $n$-dimensional Hawaiian group of the $m$-dimensional Hawaiian earring space, for all $1\leq m\leq n$.

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