arXiv subjects
Publications and source records attributed to J. F. Peters.
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.
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.
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.
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.
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.
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.
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.
This article introduces strongly near smooth manifolds. The main results are (i) second countability of the strongly hit and far-miss topology on a family $\mathcal{B}$ of subsets on the Lodato proximity space of regular open sets to which singletons are added, (ii) manifold strong proximity, (iii) strong proximity of charts in manifold atlases implies that the charts have nonempty intersection. The application of these results is given in terms of the nearness of atlases and charts of proximal manifolds and what are known as Vorono\" i manifolds.
This article introduces strongly proximal continuous (s.p.c.) functions, strong proximal equivalence (s.p.e.) and strong connectedness. A main result is that if topological spaces $X,Y$ are endowed with compatible strong proximities and $f:X\longrightarrow Y$ is a bijective s.p.e., then its extension on the hyperspaces $\CL(X)$ and $\CL(Y)$, endowed with the related strongly hit and miss hypertopologies, is a homeomorphism. For a topological space endowed with a strongly near proximity, strongly proximal connectedness implies connectedness but not conversely. Conditions required for strongly proximal connectedness are given. Applications of s.p.c. and strongly proximal connectedness are given in terms of strongly proximal descriptive proximity.
This article introduces strongly near proximity, which represents a new kind of proximity called \emph{almost proximity}. A main result in this paper is the introduction of a hit-and-miss topology on ${CL}(X)$, the hyperspace of nonempty closed subsets of $X$, based on the strongly near proximity.
This article introduces the {\it strongly hit and far-miss as well as hit and strongly far miss hypertopologies on $\textrm{CL}(X)$ associated with} ${\mathscr{B}}$, a nonempty family of subsets on the topological space $X$. They result from the strong farness and strong nearness proximities. The main results in this paper stem from the Hausdorffness of $(\textrm{CL}(X), τ_{\doublevee, \mathscr{B}})$ and $(\textrm{RCL}(X), τ^\doublewedge_\mathscr{B} ) $, where $\textrm{RCL}(X)$ is the space of regular closed subsets of $X$. To obtain the results, special local families are introduced.
This article introduces strongly far proximity, which is associated with Lodato proximity $δ$. A main result in this paper is the introduction of a hit-and-miss topology on $\mbox{CL}(X)$, the hyperspace of nonempty closed subsets of $X$, based on the strongly far proximity.
A main result in this paper is the proof that proximal Delaunay triangulation regions are convex polygons. In addition, it is proved that every Delaunay triangulation region has a local Leader uniform topology.
A main result in this paper is the proof that proximal Voronoï regions are convex polygons. In addition, it is proved that every collection of proximal Voronoï regions has a Leader uniform topology.
We prove that a pair of proximal Klee-Phelps convex groupoids $A(\circ)$,$B(\circ)$ in a finite-dimensional normed linear space $E$ are normed proximal, {\em i.e.}, $A(\circ)\ δ B(\circ)$ if and only if the groupoids are normed proximal. In addition, we prove that the groupoid neighbourhood $N_z(\circ)\subseteq S_z(\circ)$ is convex in $E$ if and only if $N_z(\circ) = S_z(\circ)$.
Significant temperature effects on the electrokinetic transport in a nanochannel with a slab geometry are demonstrated using a molecular dynamics (MD) model. A system consisting of Na+ and Cl- ions dissolved in water and confined between fixed crystalline silicon walls with negatively charged inner surfaces in an external electric field was investigated. Lennard-Jones (LJ) force fields and Coulomb electrostatic interactions with Simple Point Charge Extended (SPC/E) model were used to represent the interactions between ions, water molecules, and channel wall atoms. Dependence of the flow of water and ions on the temperature was examined. The magnitude of the water flux and even its direction are shown to be significantly affected by temperature. In particular, the previously reported flow reversal phenomenon does not occur at higher temperature. Temperature dependence of the flux was attributed to the charge redistribution and to the changes in viscosity of water.