Search

This paper introduces an axiomatic basis for measuring the energy characteristic of vibrating dynamical systems. The basic approach is to compare non-modulated vs. modulated waveforms in measuring energy during the vibratory motion $m(t)$ at time $t$ of moving object such as off-road vehicle oscillating movements recorded in an infrared (IR) video. Modulation of $m(t)$ is achieved either physically by adjusting the load on a spring system or geometrically by adjusting the frequency $\omega$ of the Euler exponential in $m(t)e^{\pm j \omega t}dt$. Expenditure of energy $E_{m(t)}$ by a system is measured in terms of the area bounded by the motion $m(t)$ waveform at time $t$., Comment: 16 pages, 16 figures

This paper introduces an axiomatic approach in the theory of energy dissipation in Hilbert envelopes on waveforms emanating from various vibrating systems. A Hilbert envelope is a curve tangent to peak points on a motion waveform. The basic approach is to compare non-modulated vs. modulated waveforms in measuring energy loss during the vibratory motion $m(t)$ at time $t$ of moving object such as a walker, runner or biker recorded in a video. Modulation of $m(t)$ is achieved by using Mersenne primes to adjust the frequency $\omega$ of the Euler exponential in $m(t)e^{\pm j2\pi \omega t}{\rm d}t$. Expediture of energy $E_{m(t)}$ by a system is measured in terms of the area bounded by the motion $m(t)$ waveform at time $t$., Comment: 12 pages, 6 figures

This paper introduces advances in the geometry of the transforms for cross ratio of four points in a line in the Desargues affine plane. The results given here have a clean, based Desargues affine plan axiomatic's and definitions of addition and multiplication of points on a line in this plane, and for skew field properties. In this paper are studied, properties and results related to the some transforms for cross ratio for 4-points, in a line, which we divide into two categories, \emph{Invariant} and \emph{Preserving} transforms for cross ratio. The results in this paper are (1) the cross-ratio of four points is \emph{Invariant} under transforms: Inversion, Natural Translation, Natural Dilation, Mobi\"us Transform, in a line of Desargues affine plane. (2) the cross-ratio of four points is \emph{Preserved} under transforms: parallel projection, translations and dilation's in the Desargues affine plane., Comment: 16 pages, 8 figures, 24 References. arXiv admin note: substantial text overlap with arXiv:2209.02636, arXiv:2209.14241, arXiv:2208.12745

This paper introduces advances in the geometry of the cross ratio of four co-linear points in in the Desargues affine plane. The cross-ratio of co-linear points of a skew field in the Desargues affine plane. The results given here have a clean rendition, based on Desargues affine plane axiomatics, skew field properties and the addition and multiplication of planar co-linear points., Comment: 16 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:2208.12745; text overlap with arXiv:2209.02636

This paper introduces invariant transforms that preserve the ratio of either two or three co-linear points in the Desargues affine plane skew field. The results given here have a clean, geometric presentation based based Desargues affine plan axiomatic and definitions with skew field properties. The main results in this paper, are (1) ratio of two and three points is \emph{Invariant} under transforms: Inversion, Natural Translation, Natural dilatation, Mobi\"us Transform, in a line of Desargues affine plane. (2) parallel projection of a pair of lines in the Desargues affine plane preserves the ratio of two and three points, (3) translations in the Desargues affine plane preserve the ratio of 2 and 3 points and (4) dilatation in the Desargues affine plane preserve the ratio of 2 and 3 points., Comment: 20 pages, 12 figures. arXiv admin note: text overlap with arXiv:2208.12745

This paper introduces advances in the geometry of the ratio of either two or three points in a line in the Desargues affine plane, and we see this as a ratio of elements of skew field which are constructed over a line in Desargues affine plane. The results given here have a clean, geometric presentation based Desargues affine plan axiomatics and definitions of addition and multiplication of points on a line in this plane, and for skew field properties. The results in this paper are: (1) study of properties for ratio of two and three points, in a line on Desargues affine plane. Also, we discuss the cases related to the "line-skew field" characteristic, when it is two and when it is different from two. (2) we have construct the maps for ratio points-set, for two and three points, and have prove that, this maps are bijections of the lines. (3) set of ratio points (for two and for three points) with addition and multiplication of points, forms a skew fields, for more, this skew fields are sub-skew fields of the 'line-skew field' on Desargues affine plane. (4) Every Dyck polygon containing co-linear ratio vertices in the Desargues affine plane has a free group presentation., Comment: 25 pages, 5 figures

This paper introduces path triangulation of points in a bounded, simply connected surface region, replacing ordinary triangles in a Delaunay triangulation with path triangles from homotopy theory. A {\bf path triangle} has a border that is a sequence of paths $h:I\to X, I=[0,1]$. The main results in this paper are that (1) a cone $D\times I$ collapses to a path triangle $h\bigtriangleup K$, extending E.C. Zeeman's collapsible dunce hat cone theorem, (2) an ordinary path triangle with geometrically realized straight edges generalizes Veech's billiard triangle, (3) a billiard ball $K\times I$ collapses to a round path triangle geometrically realized as a triangle with curviliear edges, (4) a geometrically realized homotopy system defined in terms of free group presentations of path triangulations of finite cell complexes extends J.H.C. Whitehead's homotopy system geometric realization theorem and (5) every path triangulation of a cell complex is a good cover., Comment: 16 pages, 6 figures, 23 references, dedicated to J.H.C. Whitehead and S.V. Banavar

This article introduces free group representations of planar vortexes in a CW space that are a natural outcome of results for amenable groups and fixed points found by M.M. Day during the 1960s and a fundamental result for fixed points given by L.E.J. Brouwer., Comment: 11 pages, 1 Figure. arXiv admin note: substantial text overlap with arXiv:2007.04394

This paper introduces shape boundary regions in descriptive proximity forms of CW (Closure-finite Weak) spaces as a source of amiable fixed subsets as well as almost amiable fixed subsets of descriptive proximally continuous (dpc) maps. A dpc map is an extension of an Efremovi\v{c}-Smirnov proximally continuous (pc) map introduced during the early-1950s by V.A. Efremovi\v{c} and Yu. M. Smirnov. Amiable fixed sets and the Betti numbers of their free Abelian group representations are derived from dpc's relative to the description of the boundary region of the sets. Almost amiable fixed sets are derived from dpc's by relaxing the matching description requirement for the descriptive closeness of the sets. This relaxed form of amiable fixed sets works well for applications in which closeness of fixed sets is approximate rather than exact. A number of examples of amiable fixed sets are given in terms of wide ribbons. A bi-product of this work is a variation of the Jordan Curve Theorem and a Fixed Cell Complex Theorem, which is an extension of the Brouwer Fixed Point Theorem., Comment: 17 pages, 6 figures, Dedicated to Sanjo Zlobec, honouring his 80th birthday

This article introduces descriptive fixed sets and their properties in descriptive proximity spaces viewed in the context of planar ribbon complexes. These fixed sets are a byproduct of descriptive proximally continuous maps that spawn fixed subsets, eventual fixed subsets and almost fixed subsets of the maps. For descriptive continuous map $f$ on a descriptive proximity space $X$, a subset $A$ of $X$ is fixed, provided the description of $f(A)$ matches the desription of $A$. In terms ribbon complexes in a CW space, an Abelian group representation of a ribbon is Day-amenable and each amenable ribbon has a fixed point. A main result in this paper is that if $h$ is a proximal descriptive conjugacy between maps $f,g$, then if $A$ is an [ordinary, eventual, almost] descriptively fixed subset of $f$, then $h(A)$ is a descriptively fixed subset of $g$., Comment: 15 pages, 5 figures

This article introduces planar ribbons, Vergili ribbon complexes and ribbon nerves in Alexandroff-Hopf-Whitehead CW (Closure finite Weak) topological spaces. A {\em planar ribbon} (briefly, {ribbon}) in a CW space is the closure of a pair of nesting, non-concentric filled cycles that includes the boundary but does not include the interior of the inner cycle. Each planar ribbon has its own distinctive shape determined by its outer and inner boundaries and the interior within its boundaries. A Vergili ribbon complex (briefly, ribbon complex) in a CW space is a non-void collection of countable planar ribbons. A ribbon nerve is a nonvoid collection of planar ribbons (members of a ribbon complex) that have nonempty intersection. A planar CW space is a non-void collection of cells (vertexes, edges and filled triangles) that may or may not be attached to other and which satisfy Alexandroff-Hopf-Whitehead containment and intersection conditions. In the context of CW spaces, planar ribbons, ribbon complexes and ribbon nerves are characterized by Betti numbers derived from standard Betti numbers $\mathcal{B}_0$ (cell count), $\mathcal{B}_1$ (cycle count) and $\mathcal{B}_2$ (hole count), namely, $\mathcal{B}_{rb}$ and $\mathcal{B}_{rbNrv}$ introduced in this paper. Results are given for collections of ribbons and ribbon nerves in planar CW spaces equipped with an approximate descriptive proximity, division of the plane into three bounded regions by a ribbon and Brouwer fixed points on ribbons. In addition, the homotopy types of ribbons and ribbon nerves are introduced., Comment: 14 pages, 5 figures, dedicated to E. Betti and S.A. Naimpally

This article introduces vortex nerve complexes in CW (Closure finite Weak) topological spaces, which first appeared in works by P. Alexandroff, H. Hopf and J.H.C. Whitehead during the 1930s. A vortex nerve is a CW complex containing one or more intersecting path-connected cycles. Each vortex nerve has its own distinctive shape. Both vortex nerve shapes (bounded planar surfaces with nonempty interior) and holes (bounded planar surfaces with empty interior that live inside and define shapes) have boundaries that are path-connected cycles. In the context of CW complexes, the usual Betti numbers $\mathcal{B}_0$ (cell count), $\mathcal{B}_1$ (cycle count) and $\mathcal{B}_2$ (hole count) provide a basis for the introduction of several new Betti numbers, namely, vortex $\mathcal{B}_{vtex}$, vortex nerve $\mathcal{B}_{vNrv}$ and shape $\mathcal{B}_{sh}$ introduced in this paper. In addition, results are given for CW complexes equipped with a descriptive proximity as well as for the homotopy types of vortex nerves and the complexes and cycles contained in the nerves., Comment: 9 pages, 1 figure, dedicated to Enrico Betti and Som Naimpally

This paper introduces ordered skew fields that result from the construction of a skew field over an ordered line in a Desargues affine plane. A special case of a finite ordered skew field in the construction of a skew field over an ordered line in a Desargues affine plane in Euclidean space, is also considered. Two main results are given in this paper: (1) every skew field constructed over a skew field over an ordered line in a Desargues affine plane is an ordered skew field and (2) every finite skew field constructed over a skew field over an ordered line in a Desargues affine plane in $\mathbb{R}^2$ is a finite ordered skew field., Comment: 14 pages, 9 figures

This article introduces an application of Ghrist barcodes in the study of persistent Betti numbers derived from vortex nerve complexes found in triangulations of video frames. A Ghrist barcode is a topology of data pictograph useful in representing the persistence of the features of changing shapes. The basic approach is to introduce a free Abelian group representation of intersecting filled polygons on the barycenters of the triangles of Alexandroff nerves. An Alexandroff nerve is a maximal collection of triangles with a common vertex in the triangulation of a finite, bounded planar region. In our case, the planar region is a video frame. A Betti number is a count of the number of generators in a finite Abelian group. The focus here is on the persistent Betti numbers across sequences of triangulated video frames. Each Betti number is mapped to an entry in a Ghrist barcode. Two main results are given, namely, vortex nerves are Edelsbrunner-Harer nerve complexes and the Betti number of a vortex nerve equals $k+2$ for a vortex nerve containing $k$ edges attached between a pair of vortex cycles in the nerve., Comment: 14 pages, 9 figures

This paper considers dilations and translations of lines in the Desargues affine plane. A dilation of a line transforms each line into a parallel line whose length is a multiple of the length of the original line. In addition to the usual Playfair axiom for parallel lines in an affine plane, further conditions are given for distinct lines to be parallel in the Desargues affine plane. This paper introduces the dilation of parallel lines in a finite Desargues affine plane that is a bijection of the lines. Two main results are given in this paper, namely, each dilation in a finite Desarguesian plane is an isomorphism between skew fields constructed over isomorphic lines and each dilation in a finite Desarguesian plane occurs in a Pappian space., Comment: 15 pages, 6 figures

This article introduces planar shape signatures derived from homology nerves, which are intersecting 1-cycles in a collection of homology groups endowed with a proximal relator (set of nearness relations) that includes a descriptive proximity. A 1-cycle is a closed, connected path with a zero boundary in a simplicial complex covering a finite, bounded planar shape. The signature of a shape sh A (denoted by sig(sh A)) is a feature vector that describes sh A. A signature sig(sh A) is derived from the geometry, homology nerves, Betti number, and descriptive CW topology on the shape sh A. Several main results are given, namely, (a) every finite, bounded planar shape has a signature derived from the homology group on the shape, (b) a homology group equipped with a proximal relator defines a descriptive Leader uniform topology and (c) a description of a homology nerve and union of the descriptions of the 1-cycles in the nerve have same homotopy type., Comment: 15 pages; 4 figures

This article considers proximal planar shapes in terms of the proximity of shape nerves and shape nerve complexes. A shape nerve is collection of 2-simplexes with nonempty intersection on a triangulated shape space. A planar shape is a shape nerve complex, which is a collection of shape nerves that have nonempty intersection. A main result in this paper is the homotopy equivalence of a planar shape nerve complex and the union of its nerve sub-complexes., Comment: 13 pages, 5 figures. arXiv admin note: text overlap with arXiv:1704.05909

This article reports on simulations that show how, starting with a form of neural lattice structure, it is possible to reversibly generate many alternative isomers with a lower structural symmetry, which results from twisting two hexagons around a central bond., Comment: 11 pages, 5 figures

The focus of this article is on the detection and classification of patterns based on groupoids. The approach hinges on descriptive proximity of points in a set based on the neighborliness property. This approach lends support to image analysis and understanding and in studying nearness of image segments. A practical application of the approach is in terms of the analysis of natural images for pattern identification and classification., Comment: 9 pages, 6 figures

This paper introduces a visibility relation $v$ (and the strong visibility relation $\mathop{v}\limits^{\doublewedge}$) on proximal Delaunay meshes. A main result in this paper is that the visibility relation $v$ is equivalent to Wallman proximity. In addition, a Delaunay triangulation region endowed with the visibility relation $v$ has a local Leader uniform topology., Comment: 6 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:1411.6260