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Path planning is a crucial algorithmic approach for designing robot behaviors. Sampling-based approaches, like rapidly exploring random trees (RRTs) or probabilistic roadmaps, are prominent algorithmic solutions for path planning problems. Despite its exponential convergence rate, RRT can only find suboptimal paths. On the other hand, $\textrm{RRT}^*$, a widely-used extension to RRT, guarantees probabilistic completeness for finding optimal paths but suffers in practice from slow convergence in complex environments. Furthermore, real-world robotic environments are often partially observable or with poorly described dynamics, casting the application of $\textrm{RRT}^*$ in complex tasks suboptimal. This paper studies a novel algorithmic formulation of the popular Monte-Carlo tree search (MCTS) algorithm for robot path planning. Notably, we study Monte-Carlo Path Planning (MCPP) by analyzing and proving, on the one part, its exponential convergence rate to the optimal path in fully observable Markov decision processes (MDPs), and on the other part, its probabilistic completeness for finding feasible paths in partially observable MDPs (POMDPs) assuming limited distance observability (proof sketch). Our algorithmic contribution allows us to employ recently proposed variants of MCTS with different exploration strategies for robot path planning. Our experimental evaluations in simulated 2D and 3D environments with a 7 degrees of freedom (DOF) manipulator, as well as in a real-world robot path planning task, demonstrate the superiority of MCPP in POMDP tasks., Comment: Accepted: RA-L & IROS 2022

This paper introduces proximal homotopic cycles, which lead to the main results in this paper, namely, extensions of the Mitsuishi-Yamaguchi Good Coverning Theorem with different forms of Tanaka good cover of an Alexandrov space equipped with a proximity relation and extensions of the Jordan curve theorem. An application of these results is also given., Comment: 19 pages, 7 figures. arXiv admin note: substantial text overlap with arXiv:2104.05601

This paper introduces homotopic nerve complexes in a planar Whitehead CW space and their Rotman free group presentations. Nerve complexes were introduced by P.S. Alexandrov during the 1930s and recently given a formal structure from a computational topology perspective by H. Edelsbrunner and J.L. Harer in 2010. A homotopic nerve results from the nonvoid intersection of a collection of homotopic 1-cycles. Briefly, a 1-cycle is a finite sequence of path-connected vertexes with no end vertex and with a nonvoid interior. A homotopic 1-cycle has the structure of a 1-cycle in a CW space in which cycle edges are replaced by homotopic maps. A group $G(V,+)$ containing a basis $\mathcal{B}$ is {\em free}, provided every member of $V$ can be written as a linear combination of elements (generators) of the basis $\mathcal{B}\subset V$. Let $\bigtriangleup$ be the members $v$ of $V$, each written as a linear combination of the basis elements of $\mathcal{B}$. A presentation of $G(V,+)$ is a mapping $\mathcal{B}\times \bigtriangleup\to G(\left\{v\in V:=sum_{k\in \mathbb{Z}\atop g\in \mathcal{B}}{kg}\right\},+)$. The main results in this paper are (1) Every homotopic vortex nerve has a free group presentation and (2) For a vortex nerve that consists of a finite collection of closed, convex sets in Euclidean space, the nerve and union of sets in the nerve have the same homotopy type., Comment: 15 pages, 8 figures, Dedicated to Yuriy Trokhymchuk & Saroja V. Banavar

This paper introduces proximal path cycles, which lead to the main results in this paper, namely, extensions of the Mitsuishi-Yamaguchi Good Coverning Theorem with different forms of Tanaka good cover of an Alexandrov space equipped with a proximity relation as well as extension of the Jordan curve theorem. In this work, a {\bf path cycle} is a sequence of maps $h_1,\dots,h_i,\dots,h_{n-1}\mbox{mod}\ n$ in which $h_i:[0,1]\to X$ and $h_i(1) = h_{i+1}(0)$ provide the structure of a path-connected cycle that has no end path. An application of these results is also given for the persistence of proximal video frame shapes that appear in path cycles., Comment: 22 pages, 7 figures

Modern radio telescopes are routinely reaching depths where normal starforming galaxies are the dominant observed population. Realising the potential of radio as a tracer of star formation and black hole activity over cosmic time involves achieving such depths over representative volumes, with radio forming part of a larger multiwavelength campaign. In pursuit of this we used the Karl G. Jansky Very Large Array (VLA) to image $\sim$5 deg$^{2}$ of the VIDEO/XMM-LSS extragalactic deep field at 1--2 GHz. We achieve a median depth of 16 $\mu$Jy beam$^{-1}$ with an angular resolution of 4.5\arcsec. Comparisons with existing radio observations of XMM-LSS showcase the improved survey speed of the upgraded VLA: we cover 2.5 times the area and increase the depth by $\sim$20\% in 40\% of the time. Direction-dependent calibration and wide-field imaging were required to suppress the error patterns from off-axis sources of even modest brightness. We derive a catalogue containing 5,762 sources from the final mosaic. Sub-band imaging provides in-band spectral indices for 3,458 (60\%) sources, with the average spectrum becoming flatter than the canonical synchrotron slope below 1 mJy. Positional and flux-density accuracy of the observations, and the differential source counts are in excellent agreement with those of existing measurements. A public release of the images and catalogue accompanies this article., Comment: 13 pages, 8 figures, 3 tables, accepted for publication in MNRAS

When approaching the atomically thin limit, defects and disorder play an increasingly important role in the properties of two-dimensional materials. Superconductivity is generally thought to be vulnerable to these effects, but here we demonstrate the contrary in the case of oxygenation of ultrathin tantalum disulfide (TaS$_2$). Our first-principles calculations show that incorporation of oxygen into the TaS$_2$ crystal lattice is energetically favourable and effectively heals sulfur vacancies typically present in these crystals, thus restoring the carrier density to the intrinsic value of TaS$_2$. Strikingly, this leads to a strong enhancement of the electron-phonon coupling, by up to 80% in the highly-oxygenated limit. Using transport measurements on fresh and aged (oxygenated) few-layer TaS$_2$, we found a marked increase of the superconducting critical temperature ($T_{\mathrm{c}}$) upon aging, in agreement with our theory, while concurrent electron microscopy and electron-energy loss spectroscopy confirmed the presence of sulfur vacancies in freshly prepared TaS$_2$ and incorporation of oxygen into the crystal lattice with time. Our work thus reveals the mechanism by which certain atomic-scale defects can be beneficial to superconductivity and opens a new route to engineer $T_{\mathrm{c}}$ in ultrathin materials.

In this paper, we have established boundaries of cardinal numbers of nonempty sets in finite non-$T_1$ topological spaces using interval analysis. For a finite set with known cardinality, we give interval estimations based on the closure and interior of the set. In this paper, we give new results for the cardinalities of non-empty semi-open sets in non-$T_1$ topological spaces as well as in extremely disconnected and hyperconnected topological spaces., Comment: 6 pages

This paper introduces an order proximity on a collection of objects induced by a partial order using the Smirnov closeness measure on a Sz\'{a}z relator space. A Sz\'{a}z relator is a nonempty family of relations defined on a nonvoid set $K$. The Smirnov closeness measure provides a straightforward means of assembling partial ordered of pairwise close sets. In its original form, Ju. M. Smirnov closeness measure $\delta(A,B) = 0$ for a pair of nonempty sets $A,B$ with nonvoid intersection and $\delta(A,B) = 1$ for non-close sets. A main result in this paper is that the graph obtained by the proximity is equivalent to the Hasse diagram of the order relation that induces it. This paper also includes an application of order proximity in detecting sequences of video frames that have order proximity., Comment: 21 pages, 4 figures, dedicated to Ju.M. Smirnov and S.A. Naimpally

This article introduces proximal cell complexes in a hyperconnected space. Hyperconnectedness encodes how collections of path-connected sub-complexes in a Alexandroff-Hopf-Whitehead CW space are near to or far from each other. Several main results are given, namely, a hyper-connectedness form of CW (Closure Finite Weak topology) complex, the existence of continuous functions that are paths in hyperconnected relator spaces and hyperconnected chains with overlapping interiors that are path graphs in a relator space. An application of these results is given in terms of the definition of cycles using the centroids of triangles., Comment: 15 pages, 5 figures

This paper introduces an extension of descriptive intersection and provides a framework for descriptive unions of nonempty sets. Fibre bundles provide structures that characterize spatially near as well as descriptively near sets, their descriptive intersection and their unions. The properties of four different forms of descriptive unions are given. A main result given in this paper is the equivalence between ordinary set intersection and a descriptive union. Applications of descriptive unions are given with respect to Jeffs-Novik convex unions and descriptive unions in digital images., Comment: 19 pages, 6 figures

This article introduces proximal planar vortex 1-cycles, resembling the structure of vortex atoms introduced by William Thomson (Lord Kelvin) in 1867 and recent work on the proximity of sets that overlap either spatially or descriptively. Vortex cycles resemble Thomson's model of a vortex atom, inspired by P.G. Tait's smoke rings. A vortex cycle is a collection of non-concentric, nesting 1-cycles with nonempty interiors (i.e., a collection of 1-cycles that share a nonempty set of interior points and which may or may not overlap). Overlapping 1-cycles in a vortex yield an Edelsbrunner-Harer nerve within the vortex. Overlapping vortex cycles constitute a vortex nerve complex. Several main results are given in this paper, namely, a Whitehead CW topology and a Leader uniform topology are outcomes of having a collection of vortex cycles (or nerves) equipped with a connectedness proximity and the case where each cluster of closed, convex vortex cycles and the union of the vortex cycles in the cluster have the same homotopy type., Comment: 10 figures, 25 pages

This article introduces descriptive cellular homology on cell complexes, which is an extension of J.H.C. Whitehead's CW topology. A main result is that a descriptive cellular complex is a topology on fibres in a fibre bundle. An application of two forms of cellular homology is given in terms of the persistence of shapes in CW spaces., Comment: 6 pages, 4 figures

The MIGHTEE large survey project will survey four of the most well-studied extragalactic deep fields, totalling 20 square degrees to $\mu$Jy sensitivity at Giga-Hertz frequencies, as well as an ultra-deep image of a single ~1 square degree MeerKAT pointing. The observations will provide radio continuum, spectral line and polarisation information. As such, MIGHTEE, along with the excellent multi-wavelength data already available in these deep fields, will allow a range of science to be achieved. Specifically, MIGHTEE is designed to significantly enhance our understanding of, (i) the evolution of AGN and star-formation activity over cosmic time, as a function of stellar mass and environment, free of dust obscuration; (ii) the evolution of neutral hydrogen in the Universe and how this neutral gas eventually turns into stars after moving through the molecular phase, and how efficiently this can fuel AGN activity; (iii) the properties of cosmic magnetic fields and how they evolve in clusters, filaments and galaxies. MIGHTEE will reach similar depth to the planned SKA all-sky survey, and thus will provide a pilot to the cosmology experiments that will be carried out by the SKA over a much larger survey volume., Comment: Proceedings of Science, "MeerKAT Science: On the Pathway to the SKA", Stellenbosch, 25-27 May 2016

This paper introduces the geodesics of triangulated image object shapes. Both rectilinear and curvilinear triangulations of shapes are considered. The triangulation of image object shapes leads to collections of what are known as nerve complexes that provide a workable basis for the study of shape geometry.A nerve complex is a collection of filled triangles with a common vertex. Each nerve complex triangle has an extension called a spoke, which provides an effective means of covering shape interiors. This leads to a geodesic-based metric for shape approximation which offers a straightforward means of assessing, comparing and classifying the shapes of image objects with high acuity., Comment: 32 pages, 11 figures

In a computational topology of digital images, simplexes are replaced by Delta sets in approximating image object shapes. For simplicity, simplexes and Delta sets are restricted to the Euclidean plane. A planar simplex is either a vertex, a line segment or a filled triangle. In this study of image shapes, a planar Delta set is a sequence of ordered simplicial complexes. The basic approach is to approximate an image shape by decomposing an image region containing the shape into combinations of Delta sets called Delta complexes. This approach to image shapes is motivated by the ease with which shapes covered by Delta complexes can be measured and compared. A number of basic results directly related to shape analysis are also given in the context of Delta complex proximities., Comment: 20 pages, 12 figures

This article introduces proximal Cech nerves and Cech complexes, restricted to finite, bounded regions $K$ of the Euclidean plane. A Cech nerve is a collection of intersecting balls. A Cech complex is a collection of nerves that cover $K$. Cech nerves are proximal, provided the nerves are close to each other, either spatially or descriptively. A Cech nerve has an advantage over the usual Alexandroff nerve, since we need only identify the center and fixed radius of each ball in a Cech nerve instead of identifying the three vertices of intersecting filled triangles (2-simplexes) in an Alexandroff nerve. As a result, Cech nerves more easily cover $K$ and facilitate approximation of the shapes of irregular finite, bounded planar regions. A main result of this article is an extension of the Edelsbrunner-Harer Nerve Theorem for descriptive and non-descriptive Cech nerves and Cech complexes, covering $K$., Comment: 11 pages, 2 figures, keywords: Ball, Cech Complex, Cech Nerve, Cover, Homotopic Equivalence, Proximity

This article introduces a theory of proximal nerve complexes and nerve spokes, restricted to the triangulation of finite regions in the Euclidean plane. A nerve complex is a collection of filled triangles with a common vertex, covering a finite region of the plane. Structures called $k$-spokes, $k\geq 1$, are a natural extension of nerve complexes. A $k$-spoke is the union of a collection of filled triangles that pairwise either have a common edge or a common vertex. A consideration of the closeness of nerve complexes leads to a proximal view of simplicial complexes. A practical application of proximal nerve complexes is given, briefly, in terms of object shape geometry in digital images., Comment: 16 pages, 9 figures

In this study, we provide mathematical and practice-driven justification for using $[0,1]$ normalization of inconsistency indicators in pairwise comparisons. The need for normalization, as well as problems with the lack of normalization, are presented. A new type of paradox of infinity is described., Comment: 15 pages, 3 figures

The theory of descriptive nearness is usually adopted when dealing with sets that share some common properties even when the sets are not spatially close, i.e., the sets have no members in common. Set description results from the use of probe functions to define feature vectors that describe a set and the nearness of sets is given by their proximities. A probe on a non-empty set $X$ is a real-valued function $\Phi: X \rightarrow \mathbb{R}^n$, where $\Phi(x)= (\phi_1(x),.., \phi_n(x))$. We establish a connection between relations on an object space $X$ and relations on the feature space $\Phi(X).$ Having as starting point the Peters proximity, two sets are \emph{descriptively near}, if and only if their descriptions intersect. In this paper, we construct a theoretical approach to a more visual form of proximity, namely, descriptive proximity, which has a broad spectrum of applications. We organize descriptive proximities on two different levels: weaker or stronger than the Peters proximity. We analyze the properties and interplay between descriptions on one side and classical proximities and overlap relations on the other side., Comment: 12 pages, 6 figures

This paper investigates problems in the characterization of the proximity of digital discs. Based on the $\mathcal{L}_1$-metric structure for the 2D digital plane and using a Jaccard-like metric, we determine numerical characters for intersecting digital discs., Comment: 6 pages, 4 figures

This paper introduces two proximal forms of Lenzen physical geometry, namely, an \emph{axiomatized strongly proximal physical geometry} that is built on simplicial complexes with the dualities and sewing operations derived from string geometry and an \emph{axiomatized descriptive proximal physical geometry} in which spatial regions are described based on their features and the descriptive proximities between regions. This is a computational proximity approach to a Lenzen geometry of physical space. In both forms of physical geometry, region is a primitive. Intuitively, a region is a set of connected subregions. The primitive in this geometry is \emph{region}, instead of \emph{point}. Each description of a region with $n$ features is a mapping from the region to a feature vector in $\mathbb{R}^n$. In the feature space, proximal physical geometry has the look and feel of either Euclidean, Riemannian, or non-Euclidean geometry, since we freely work with the relations between points in the feature space. The focus in these new forms of geometry is the relation between individual regions with their own distinctive features such as shape, area, perimeter and diameter and the relation between nonempty sets of regions. The axioms for physical geometry as well as the axioms for proximal physical geometry are given and illustrated. Results for parallel classes of regions, descriptive fibre bundles and BreMiller-Sloyer sheaves are given. In addition, a region-based Borsuk-Ulam Theorem as well as a Wired Friend Theorem are given in the context of both forms of physical geometry., Comment: 27 pages, 15 figures

This paper introduces a string-based extension of the Borsuk-Ulam Theorem (denoted by strBUT). A string is a region with zero width and either bounded or unbounded length on the surface of an $n$-sphere or a region of a normed linear space. In this work, an $n$-sphere surface is covered by a collection of strings. For a strongly proximal continuous function on an $n$-sphere into $n$-dimensional Euclidean space, there exists a pair of antipodal $n$-sphere strings with matching descriptions that map into Euclidean space $\mathbb{R}^n$. Each region $M$ of a string-covered $n$-sphere is a worldsheet (denoted by $\mbox{wsh}M$). For a strongly proximal continuous mapping from a worldsheet-covered $n$-sphere to $\mathbb{R}^n$, strongly near antipodal worldsheets map into the same region in $\mathbb{R}^n$. An application of strBUT is given in terms of the evaluation of Electroencephalography (EEG) patterns., Comment: 14 pages, 8 figures. arXiv admin note: substantial text overlap with arXiv:1605.02987

This paper introduces a string-based extension of the Borsuk-Ulam Theorem (denoted by strBUT). A string is a region with zero width and either bounded or unbounded length on the surface of an $n$-sphere or a region of a normed linear space. In this work, an $n$-sphere surface is covered by a collection of strings. For a strongly proximal continuous function on an $n$-sphere into $n$-dimensional Euclidean space, there exists a pair of antipodal $n$-sphere strings with matching descriptions that map into Euclidean space $\mathbb{R}^n$. Each region $M$ of a string-covered $n$-sphere is a worldsheet. For a strongly proximal continuous mapping from a worldsheet-covered $n$-sphere to $\mathbb{R}^n$, strongly near antipodal worldsheets map into the same region in $\mathbb{R}^n$. This leads to a wired friend theorem in descriptive string theory. An application of strBUT is given in terms of the evaluation of Electroencephalography (EEG) patterns., Comment: 17 pages, 8 figures

This paper introduces Edelsbrunner-Harer nerve in collections of Voronoi regions (called nucleus clusters) endowed with one or more proximity relations. The main results in this paper are that a maximal nucleus cluster (MNC) in a Voronoi Tessellation is a strongly proximal Edelsbrunner-Harer nerve, each MNC nerve and the union of the sets in the MNC have the same homotopy type., Comment: 10 pages, 4 figures

Color image segmentation is a crucial step in many computer vision and pattern recognition applications. This article introduces an adaptive and unsupervised clustering approach based on Voronoi regions, which can be applied to solve the color image segmentation problem. The proposed method performs region splitting and merging within Voronoi regions of the Dirichlet Tessellated image (also called a Voronoi diagram) , which improves the efficiency and the accuracy of the number of clusters and cluster centroids estimation process. Furthermore, the proposed method uses cluster centroid proximity to merge proximal clusters in order to find the final number of clusters and cluster centroids. In contrast to the existing adaptive unsupervised cluster-based image segmentation algorithms, the proposed method uses K-means clustering algorithm in place of the Fuzzy C-means algorithm to find the final segmented image. The proposed method was evaluated on three different unsupervised image segmentation evaluation benchmarks and its results were compared with two other adaptive unsupervised cluster-based image segmentation algorithms. The experimental results reported in this article confirm that the proposed method outperforms the existing algorithms in terms of the quality of image segmentation results. Also, the proposed method results in the lowest average execution time per image compared to the existing methods reported in this article., Comment: 21 pages, 5 figures

This paper introduces nucleus clustering in Voronoi tessellations of plane surfaces with applications in the geometry of digital images. A \emph{nucleus cluster} is a collection of Voronoi regions that are adjacent to a Voronoi region called the cluster nucleus. Nucleus clustering is a carried out in a strong proximity space. Of particular interest is the presence of maximal nucleus clusters in a tessellation. Among all of the possible nucleus clusters in a Voronoi tessellation, clusters with the highest number of adjacent polygons are called \emph{maximal nucleus clusters}. The main results in this paper are that strongly near nucleus clusters are strongly descriptively near and every collection of Voronoi regions in a tessellation of a plane surface is a Zelins'kyi-Soltan-Kay-Womble convexity structure., Comment: 7 pages, 2 figures

Current advances in neurosciences deal with the functional architecture of the central nervous system, paving the way for holistic theories that improve our understanding of brain activity. From topology, a strong concept comes into play in the understanding of brain signals, namely, continuous mapping of the signals onto a hyper-sphere, a 4-dimensional space equipped with a doughnut-like shape that is not detected by observers living in a 3-dimensional world. We evaluate the features of the imperceptible 4th dimension based on resting state functional magnetic resonance imaging series. In particular, we looked for simultaneous activation of antipodal signals on the surface of a cortical hyper-sphere. In this article, we demonstrate that spontaneous brain activity displays the typical features which reveal the existence of a functional hyper-sphere. The suggestion here is that the brain is embedded in a hyper-sphere, which helps solve long-standing mysteries concerning our psychological activities such mind-wandering and memory retrieval or the ability to connect, pasts and future events., Comment: 12 pages, 9 figures