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Journal articles on the topic 'BOUNDARY VERTICES'

Author: Grafiati

Published: 11 September 2023

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1

Chartrand, Gary, David Erwin, Garry L. Johns, and Ping Zhang. "Boundary vertices in graphs." Discrete Mathematics 263, no. 1-3 (February 2003): 25–34. http://dx.doi.org/10.1016/s0012-365x(02)00567-8.

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2

Mariumuthu, G., and M. S. Saraswathy. "Dynamics of Boundary Graphs." Journal of Scientific Research 5, no. 3 (August 29, 2013): 447–55. http://dx.doi.org/10.3329/jsr.v5i3.14866.

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In a graph G, the distance d(u,v) between a pair of vertices u and v is the length of a shortest path joining them. A vertex v is a boundary vertex of a vertex u if for all The boundary graph B(G) based on a connected graph G is a simple graph which has the vertex set as in G. Two vertices u and v are adjacent in B(G) if either u is a boundary of v or v is a boundary of u. If G is disconnected, then each vertex in a component is adjacent to all other vertices in the other components and is adjacent to all of its boundary vertices within the component. Given a positive integer m, the mth iterated boundary graph of G is defined as A graph G is periodic if for some m. A graph G is said to be an eventually periodic graph if there exist positive integers m and k >0 such that We give the necessary and sufficient condition for a graph to be eventually periodic. Keywords: Boundary graph; Periodic graph. © 2013 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v5i3.14866 J. Sci. Res. 5 (3), xxx-xxx (2013)

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3

Cáceres, José, Carmen Hernando, Mercè Mora, Ignacio M. Pelayo, María L. Puertas, and Carlos Seara. "On geodetic sets formed by boundary vertices." Discrete Mathematics 306, no. 2 (February 2006): 188–98. http://dx.doi.org/10.1016/j.disc.2005.12.012.

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4

Li, Tao, and Wen Liang Chen. "A Robust Clipping Algorithm for Overlap Surface Patches." Applied Mechanics and Materials 701-702 (December 2014): 136–40. http://dx.doi.org/10.4028/www.scientific.net/amm.701-702.136.

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By revising geometry and topology information of boundaries, a robust clipping method is proposed for overlapped surface patches. All the boundary vertices of one patch are marked on the basis of their relationship with the boundary loops of the other patch. Then the overlap regions are recognized according to the vertices flags. If the overlap regions are slim and only one boundary curve in each patch is intersected, the corresponding parts of the curves are subdivided and their vertices are repositioned, and the interfered curves are sewed together. Otherwise, the boundaries are reorganized according to vertex flags. There are two candidate clipping schemes for each overlapped patches and the one with fewer boundary curves is the final result. Examples verify robustness of the algorithms.

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5

Boyarsky, Alexey, Bogdan Kulik, and Oleg Ruchayskiy. "String field theory vertices, integrability and boundary states." Journal of High Energy Physics 2003, no. 11 (November 21, 2003): 045. http://dx.doi.org/10.1088/1126-6708/2003/11/045.

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6

Avdonin, Sergei, and Julian Edward. "An Inverse Problem for Quantum Trees with Delta-Prime Vertex Conditions." Vibration 3, no. 4 (November 17, 2020): 448–63. http://dx.doi.org/10.3390/vibration3040028.

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In this paper, we consider a non-standard dynamical inverse problem for the wave equation on a metric tree graph. We assume that the so-called delta-prime matching conditions are satisfied at the internal vertices of the graph. Another specific feature of our investigation is that we use only one boundary actuator and one boundary sensor, all other observations being internal. Using the Neumann-to-Dirichlet map (acting from one boundary vertex to one boundary and all internal vertices) we recover the topology and geometry of the graph together with the coefficients of the equations.

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Alexandrov, Stepan Andreevich, Nikolay Vladimirovich Bogachev, Andrei Yurievich Vesnin, and Andrei Aleksandrovich Egorov. "On volumes of hyperbolic right-angled polyhedra." Sbornik: Mathematics 214, no. 2 (2023): 148–65. http://dx.doi.org/10.4213/sm9740e.

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New upper bounds for the volumes of right-angled polyhedra in hyperbolic space $\mathbb{H}^3$ are obtained in the following three cases: for ideal polyhedra with all vertices on the ideal hyperbolic boundary; for compact polyhedra with only finite vertices; and for finite-volume polyhedra with vertices of both types. Bibliography: 23 titles.

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8

Shastry, Aditya, and Nidhi Khandelwal. "Antibandwidth of a Graph." Mapana - Journal of Sciences 11, no. 4 (August 6, 2012): 59–64. http://dx.doi.org/10.12723/mjs.23.4.

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The antibandwidth problem consists of placing the vertices of a graph on a line in consecutive integer points in such a way that the minimum difference of adjacent vertices is maximized. This problem is NP- hard. In this paper, we find some bounds for antibandwidth using some invariants of graphs. We prove that considerating the interior boundary and the exterior boundary when estimating the antibandwidth of connected graphs gives the same results.

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9

Yahya Mohamed, S., and N. Subashini. "Bipolar fuzzy graphs based on boundary and interior vertices." Malaya Journal of Matematik S, no. 1 (2020): 506–10. http://dx.doi.org/10.26637/mjm0s20/0096.

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10

Chen, Lingyun, and Weigen Yan. "Spanning Trees of the Generalised Union Jack Lattice." Zeitschrift für Naturforschung A 71, no. 4 (April 1, 2016): 331–35. http://dx.doi.org/10.1515/zna-2015-0415.

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AbstractThe Union Jack lattice UJL(n, m) with toroidal boundary condition can be obtained from an n×m square lattice with toroidal boundary condition by inserting a new vertex vf to each face f and adding four edges (vf, ui(f)), where u1(f), u2(f), u3(f), and u4(f) are four vertices on the boundary of f. The Union Jack lattice has been studied extensively by statistical physicists. In this article, we consider the problem of enumeration of spanning trees of the so-called generalised Union Jack lattice UDn, which is obtained from the Aztec diamond $AD_n^t$ of order n with toroidal boundary condition by inserting a new vertex vf to each face f and adding four edges (vf, ui(f)), where u1(f), u2(f), u3(f) and u4(f) are four vertices on the boundary of f.

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11

Liu, Hou-lin, Cui Dai, Liang Dong, and Ming-gao Tan. "A Novel Mesh Quality Improvement Method for Boundary Elements." Journal of Applied Mathematics 2012 (2012): 1–13. http://dx.doi.org/10.1155/2012/109542.

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In order to improve the boundary mesh quality while maintaining the essential characteristics of discrete surfaces, a new approach combining optimization-based smoothing and topology optimization is developed. The smoothing objective function is modified, in which two functions denoting boundary and interior quality, respectively, and a weight coefficient controlling boundary quality are taken into account. In addition, the existing smoothing algorithm can improve the mesh quality only by repositioning vertices of the interior mesh. Without destroying boundary conformity, bad elements with all their vertices on the boundary cannot be eliminated. Then, topology optimization is employed, and those elements are converted into other types of elements whose quality can be improved by smoothing. The practical application shows that the worst elements can be eliminated and, with the increase of weight coefficient, the average quality of boundary mesh can also be improved. Results obtained with the combined approach are compared with some common approach. It is clearly shown that it performs better than the existing approach.

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12

Wang, Zhihua, and Hongmei Kang. "A Polynomial Splines Identification Method Based on Control Nets." Complexity 2020 (August 19, 2020): 1–12. http://dx.doi.org/10.1155/2020/7103963.

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In this study, based on Polynomial Splines with control nets, an identification method is investigated. We introduce polynomial splines with control nets defined over T-mesh. The basic idea is to extend T-vertices such that those T-vertices become interior cross vertices or boundary vertices. To this end, we introduce the design-suitable T-mesh for constructing polynomial splines with control net. In design-suitable T-meshes, there are no extra basis vertices produced by an appropriate extension of T-vertices. The basis functions are defined over each vertex in a design-suitable T-mesh by the means of constructing PHT-splines basis functions.

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13

Aslan, Ersin, and Alpay Kirlangic. "Lexicographic Product and Isoperimetric Number." ISRN Combinatorics 2013 (September 2, 2013): 1–6. http://dx.doi.org/10.1155/2013/243981.

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The isoperimetric number of a graph , denoted by , was introduced by Mohar (1987). A graph and a subset of its vertices are given, and let denote the edge boundary of , the set of edges which connects vertices in to vertices not in . The isoperimetric number of is defined as . In this paper, some results about the isoperimetric number of graphs obtained by graph operations are given.

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14

Mansour, Toufik, and Reza Rastegar. "Convex polyominoes revisited: enumeration of outer site perimeter, interior vertices, and boundary vertices of certain degrees." Journal of Difference Equations and Applications 26, no. 7 (July 2, 2020): 1013–41. http://dx.doi.org/10.1080/10236198.2020.1813730.

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15

Gong, Dian Xuan, and Feng Gong Lang. "On the Dimension of Bivariate C¹ Cubic Spline Space with Homogeneous Boundary Conditions over a CT Triangulation." Applied Mechanics and Materials 50-51 (February 2011): 488–92. http://dx.doi.org/10.4028/www.scientific.net/amm.50-51.488.

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A bivariate spline is a piecewise polynomial with some smoothness de ned on a parti- tion. In this paper, we mainly study the dimensions of bivariate C1 cubic spline spaces S1;0 3 (CT ) and S1;1 3 (CT ) with homogeneous boundary conditions over CT by using interpolating technique, where CT stands for a CT triangulation. The dimensions are related with the numbers of the inter vertices and the singular boundary vertices. The results of this paper can be applied in many elds such as the nite element method for partial di erential equation, computer aided design, numerical approximation, and so on.

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16

Vitkauskas, Jonas, and Artūras Štikonas. "Relations between Spectrum Curves of Discrete Sturm-Liouville Problem with Nonlocal Boundary Conditions and Graph Theory. II." Lietuvos matematikos rinkinys 62 (December 15, 2021): 1–8. http://dx.doi.org/10.15388/lmr.2021.25128.

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In this paper, relations between discrete Sturm--Liouville problem with nonlocal integral boundary condition characteristics (poles, critical points, spectrum curves) and graphs characteristics (vertices, edges and faces) were found. The previous article was devoted to the Sturm--Liouville problem in the case two-points nonlocal boundary conditions.

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17

Jang, Chang Doo, and Yun Keun Han. "An Approach to Efficient Nesting and Cutting Path Optimization of Irregular Shapes." Journal of Ship Production 15, no. 03 (August 1, 1999): 129–35. http://dx.doi.org/10.5957/jsp.1999.15.3.129.

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A new nesting and cutting path generation algorithm is proposed and a PC-based GUI system is established for user convenience. The proposed algorithm can reduce the calculation time by updating the boundary of raw material after each part is placed and by selecting the vertices of the boundary as initial allocation points. Grouping the parts by their areas and applying the column nesting can improve the nesting efficiency, e.g., waste ratio. Also, optimal cutting paths are found based on an algorithm by allowing all the convex vertices of the parts to be piercing points compared with existing work that used only fixed piercing points.

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18

Kaliuzhnyi-Verbovetskyi, D., and V. Pivovarchik. "RECOVERING THE SHAPE OF A QUANTUM CATERPILLAR TREE BY TWO SPECTRA." Mechanics And Mathematical Methods 5, no. 1 (June 30, 2023): 14–24. http://dx.doi.org/10.31650/2618-0650-2023-5-1-14-24.

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existence of co-spectral (iso-spectral) graphs is a well-known problem of the classical graph theory. However, co-spectral graphs exist in the theory of quantum graphs also. In other words, the spectrum of the Sturm-Liouville problem on a metric graph does not determine alone the shape of the graph. Сo-spectral trees also exist if the number of vertices exceeds eight. We consider two Sturm-Liouville spectral problems on an equilateral metric caterpillar tree with real L2 (0,l) potentials on the edges. In the first (Neumann) problem we impose standard conditions at all vertices: Neumann boundary conditions at the pendant vertices and continuity and Kirchhoff’s conditions at the interior vertices. The second (Dirichlet) problem differs from the first in that in the second problem we set the Dirichlet condition at the root (one of the pendant vertices of the stalk of the caterpillar tree, i.e. the central path of it). Using the asymptotics of the eigenvalues of these two spectra we find the determinant of the normalized Laplacian of the tree and the determinant of the prime submatrix of the normalized laplacian obtained by deleting the row and the column corresponding to the root. Expanding the fraction of these determinants into continued fraction we receive full information on the shape of the tree. In general case this continued fraction is branched. We prove that in the case of a caterpillar tree the continued fraction does not branch and the spectra of the Neumann and Dirichlet problems uniquely determine the shape of the tree. A concrete example is shown. The known pair of co-spectral trees with minimal number (eight) of vertices belongs to the class of caterpillar trees. Keywords: metric graph, tree, pendant vertex, interior vertex, edge, caterpillar tree, Sturm-Liouville equation, potential, eigenvalues, spectrum, Dirichlet boundary condition, Neumann boundary condition, root, continued fraction, adjacency matrix, prime submatrix, normalized Laplacian

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19

Orthuber, E., and J. Avbelj. "3D BUILDING RECONSTRUCTION FROM LIDAR POINT CLOUDS BY ADAPTIVE DUAL CONTOURING." ISPRS Annals of Photogrammetry, Remote Sensing and Spatial Information Sciences II-3/W4 (March 11, 2015): 157–64. http://dx.doi.org/10.5194/isprsannals-ii-3-w4-157-2015.

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This paper presents a novel workflow for data-driven building reconstruction from Light Detection and Ranging (LiDAR) point clouds. The method comprises building extraction, a detailed roof segmentation using region growing with adaptive thresholds, segment boundary creation, and a structural 3D building reconstruction approach using adaptive 2.5D Dual Contouring. First, a 2D-grid is overlain on the segmented point cloud. Second, in each grid cell 3D vertices of the building model are estimated from the corresponding LiDAR points. Then, the number of 3D vertices is reduced in a quad-tree collapsing procedure, and the remaining vertices are connected according to their adjacency in the grid. Roof segments are represented by a Triangular Irregular Network (TIN) and are connected to each other by common vertices or - at height discrepancies - by vertical walls. Resulting 3D building models show a very high accuracy and level of detail, including roof superstructures such as dormers. The workflow is tested and evaluated for two data sets, using the evaluation method and test data of the “ISPRS Test Project on Urban Classification and 3D Building Reconstruction” (Rottensteiner et al., 2012). Results show that the proposed method is comparable with the state of the art approaches, and outperforms them regarding undersegmentation and completeness of the scene reconstruction.

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20

Farooq, Omer, Michał Ławniczak, Afshin Akhshani, Szymon Bauch, and Leszek Sirko. "The Generalized Euler Characteristics of the Graphs Split at Vertices." Entropy 24, no. 3 (March 9, 2022): 387. http://dx.doi.org/10.3390/e24030387.

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We show that there is a relationship between the generalized Euler characteristic Eo(|VDo|) of the original graph that was split at vertices into two disconnected subgraphs i=1,2 and their generalized Euler characteristics Ei(|VDi|). Here, |VDo| and |VDi| denote the numbers of vertices with the Dirichlet boundary conditions in the graphs. The theoretical results are experimentally verified using microwave networks that simulate quantum graphs. We demonstrate that the evaluation of the generalized Euler characteristics Eo(|VDo|) and Ei(|VDi|) allow us to determine the number of vertices where the two subgraphs were initially connected.

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21

Wang, Biao, Guoping Wu, Qiang Zhao, Yaozhu Li, Yiyuan Gao, and Jiangfeng She. "A Topology-Preserving Simplification Method for 3D Building Models." ISPRS International Journal of Geo-Information 10, no. 6 (June 20, 2021): 422. http://dx.doi.org/10.3390/ijgi10060422.

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Simplification of 3D building models is an important way to improve rendering efficiency. When existing algorithms are directly applied to simplify multi-component models, generally composed of independent components with strong topological dependence, each component is simplified independently. The consequent destruction of topological dependence can cause unreasonable separation of components and even result in inconsistent conclusions of spatial analysis among different levels of details (LODs). To solve these problems, a novel simplification method, which considers the topological dependence among components as constraints, is proposed. The vertices of building models are divided into boundary vertices, hole vertices, and other ordinary vertices. For the boundary vertex, the angle between the edge and component (E–C angle), denoting the degree of component separation, is introduced to derive an error metric to limit the collapse of the edge located at adjacent areas of neighboring components. An improvement to the quadratic error metric (QEM) algorithm was developed for the hole vertex to address the unexpected error caused by the QEM’s defect. A series of experiments confirmed that the proposed method could effectively maintain the overall appearance features of building models. Compared with the traditional method, the consistency of visibility analysis among different LODs is much better.

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22

Schütt, Carsten, and Elisabeth Werner. "Random polytopes with vertices on the boundary of a convex body." Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 331, no. 9 (November 2000): 697–701. http://dx.doi.org/10.1016/s0764-4442(00)01685-2.

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23

Jendrol', Stanislav. "Triangles with restricted degrees of their boundary vertices in plane triangulations." Discrete Mathematics 196, no. 1-3 (February 1999): 177–96. http://dx.doi.org/10.1016/s0012-365x(98)00172-1.

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24

Konyrkulzhayeva, М. N. "THE DIRICHLET PROBLEM ON THE ORIENTED GRAPHS." BULLETIN Series of Physics & Mathematical Sciences 70, no. 2 (June 30, 2020): 84–90. http://dx.doi.org/10.51889/2020-2.1728-7901.12.

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Differential operators on graphs often arise in mathematics and different fields of science such as mechanics, physics, organic chemistry, nanotechnology, etc. In this paper the solutions of the Dirichlet problem for a differential operator on a star-shaped graph are deduced. And the differential operator with standard matching conditions in the internal vertices and the Dirichlet boundary conditions at boundary vertices are studied. Task is a model the oscillation of a simple system of several rods with an adjacent end. In work the formula of the Green function of the Dirichlet problem for the second order equation on directed graph is showed. Spectral analysis of differential operators on geometric graphs is the basic mathematical apparatus in solving modern problems of quantum mechanics.

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25

NURGES, ÜLO, and ENNU RÜSTERN. "ON THE ROBUST STABILITY OF DISCRETE-TIME SYSTEMS." Journal of Circuits, Systems and Computers 09, no. 01n02 (February 1999): 37–50. http://dx.doi.org/10.1142/s0218126699000050.

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A sufficient stability condition for monic Schur polynomials is obtained via the so-called reflection coefficients of polynomials and the discrete version of Kharitonov's weak theorem. The discrete Kharitonov theorem defines only (n - 1)-dimensional stable hyperrectangle for n-degree monic polynomials. By the use of a linear Schur invariant transformation we put stable line segments through vertices of this hyperrectangle and find an n-dimensional stable polytope with all vertices on the stability boundary.

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26

Chávez, María-José, Seiya Negami, Antonio Quintero, and María Trinidad Villar-Liñán. "A generating theorem of punctured surface triangulations with inner degree at least 4." Mathematica Slovaca 69, no. 5 (October 25, 2019): 969–78. http://dx.doi.org/10.1515/ms-2017-0281.

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Abstract Given any punctured surface F2, we present a method for generating all of F2 triangulations with inner vertices of degree ≥ 4 and boundary vertices of degree ≥ 3. The method is based on a set of expansive operations which includes the well-known vertex splitting and octahedron addition. By reversing this method we get a procedure to obtain minimal triangulations by a sequence of intermediate triangulations, all of them within the given family.

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27

Abello, James, Vladimir Estivill-Castro, Thomas Shermer, and Jorge Urrutia. "Illumination of Orthogonal Polygons with Orthogonal Floodlights." International Journal of Computational Geometry & Applications 08, no. 01 (February 1998): 25–38. http://dx.doi.org/10.1142/s0218195998000035.

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We provide the first tight bound for covering an orthogonal polygon with n vertices and h holes with vertex floodlights (guards with restricted angle of vision). In particular, we provide tight bounds for the number of orthogonal floodlights, placed at vertices or on the boundary, sufficient to illuminate the interior or the exterior of an orthogonal polygon with holes. Our results lead directly to very simple linear, and thus optimal, algorithms for computing a covering of an orthogonal polygon.

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28

Vitkauskas, Jonas, and Artūras Štikonas. "Relations between spectrum curves of discrete Sturm-Liouville problem with nonlocal boundary conditions and graph theory." Lietuvos matematikos rinkinys 61 (February 18, 2021): 1–6. http://dx.doi.org/10.15388/lmr.2020.22474.

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Sturm-Liouville problem with nonlocal boundary conditions arises in many scientific fields such as chemistry, physics, or biology. There could be found some references to graph theory in a discrete Sturm-Liouville problem, especially in investigation of spectrum curves. In this paper, relations between discrete Sturm-Liouville problem with nonlocal boundary conditions characteristics (poles, critical points, spectrum curves) and graphs characteristics (vertices, edges and faces) were found.

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29

Soner, Seren, and Can Ozturan. "Generating Multibillion Element Unstructured Meshes on Distributed Memory Parallel Machines." Scientific Programming 2015 (2015): 1–10. http://dx.doi.org/10.1155/2015/437480.

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We present a parallel mesh generator called PMSH that is developed as a wrapper code around the open source sequential Netgen mesh generator. Parallelization of the mesh generator is carried out in five stages: (i) generation of a coarse volume mesh; (ii) partitioning of the coarse mesh; (iii) refinement of coarse surface mesh to produce fine surface submeshes; (iv) remeshing of each fine surface submesh to get a final fine mesh; (v) matching of partition boundary vertices followed by global vertex numbering. A new integer based barycentric coordinate method is developed for matching distributed partition boundary vertices. This method does not have precision related problems of floating point coordinate based vertex matching. Test results obtained on an SGI Altix ICE X system with 8192 cores confirm that our approach does indeed enable us to generate multibillion element meshes in a scalable way.

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30

Bendito, E., A. M. Encinas, and A. Carmona. "Eigenvalues, eigenfunctions and Green's functions on a path via Chebyshev polynomials." Applicable Analysis and Discrete Mathematics 3, no. 2 (2009): 282–302. http://dx.doi.org/10.2298/aadm0902282b.

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In this work we analyze the boundary value problems on a path associated with Schr?dinger operators with constant ground state. These problems include the cases in which the boundary has two, one or none vertices. In addition, we study the periodic boundary value problem that corresponds to the Poisson equation in a cycle. Moreover, we obtain the Green's function for each regular problem and the eigenvalues and their corresponding eigenfunctions otherwise. In each case, the Green's functions, the eigenvalues and the eigenfunctions are given in terms of first, second and third kind Chebyshev polynomials.

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31

HENDERSON, MICHAEL E. "MULTIPLE PARAMETER CONTINUATION: COMPUTING IMPLICITLY DEFINED k-MANIFOLDS." International Journal of Bifurcation and Chaos 12, no. 03 (March 2002): 451–76. http://dx.doi.org/10.1142/s0218127402004498.

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We present a new continuation method for computing implicitly defined manifolds. The manifold is represented as a set of overlapping neighborhoods, and extended by an added neighborhood of a boundary point. The boundary point is found using an expression for the boundary in terms of the vertices of a set of finite, convex polyhedra. The resulting algorithm is quite simple, allows adaptive spacing of the computed points, and deals with the problems of local and global overlap in a natural way. The algorithm is robust (the new points need only be near the boundary), and is well suited to problems with large embedding dimension, and small to moderate dimension.

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32

Borodin, Oleg V. "Triangles with restricted degree sum of their boundary vertices in plane graphs." Discrete Mathematics 137, no. 1-3 (January 1995): 45–51. http://dx.doi.org/10.1016/0012-365x(94)e0144-7.

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33

Pushpam, P. Roushini Leely, and D. Yokesh. "Differentials in certain classes of graphs." Tamkang Journal of Mathematics 41, no. 2 (June 30, 2010): 129–38. http://dx.doi.org/10.5556/j.tkjm.41.2010.664.

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Let $X subset V$ be a set of vertices in a graph $G = (V, E)$. The boundary $B(X)$ of $X$ is defined to be the set of vertices in $V-X$ dominated by vertices in $X$, that is, $B(X) = (V-X) cap N(X)$. The differential $ partial(X)$ of $X$ equals the value $ partial(X) = |B(X)| - |X|$. The differential of a graph $G$ is defined as $ partial(G) = max { partial(X) | X subset V }$. It is easy to see that for any graph $G$ having vertices of maximum degree $ Delta(G)$, $ partial(G) geq Delta (G) -1$. In this paper we characterize the classes of unicyclic graphs, split graphs, grid graphs, $k$-regular graphs, for $k leq 4$, and bipartite graphs for which $ partial(G) = Delta(G)-1$. We also determine the value of $ partial(T)$ for any complete binary tree $T$.

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34

Okoth, Isaac Owino. "Refined enumeration of 2-noncrossing trees." Notes on Number Theory and Discrete Mathematics 27, no. 2 (June 2021): 201–10. http://dx.doi.org/10.7546/nntdm.2021.27.2.201-210.

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A 2-noncrossing tree is a connected graph without cycles that can be drawn in the plane with its vertices on the boundary of circle such that the edges are straight line segments that do not cross and all the vertices are coloured black and white with no ascent (i, j), where i and j are black vertices, in a path from the root. In this paper, we use generating functions to prove a formula that counts 2-noncrossing trees with a black root to take into account the number of white vertices of indegree greater than zero and black vertices. Here, the edges of the 2-noncrossing trees are oriented from a vertex of lower label towards a vertex of higher label. The formula is a refinement of the formula for the number of 2-noncrossing trees that was obtained by Yan and Liu and later on generalized by Pang and Lv. As a consequence of the refinement, we find an equivalent refinement for 2-noncrossing trees with a white root, among other results.

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35

Greenblatt, Rafael L. "Discrete and zeta-regularized determinants of the Laplacian on polygonal domains with Dirichlet boundary conditions." Journal of Mathematical Physics 64, no. 4 (April 1, 2023): 043301. http://dx.doi.org/10.1063/5.0062138.

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For [Formula: see text], a connected, open, bounded set whose boundary is a finite union of disjoint polygons whose vertices have integer coordinates, the logarithm of the discrete Laplacian on [Formula: see text] with Dirichlet boundary conditions has an asymptotic expression for large L involving the zeta-regularized determinant of the associated continuum Laplacian. When Π is not simply connected, this result extends to Laplacians acting on two-valued functions with a specified monodromy class.

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36

PAPADOPOULOU, EVANTHIA. "k-PAIRS NON-CROSSING SHORTEST PATHS IN A SIMPLE POLYGON." International Journal of Computational Geometry & Applications 09, no. 06 (December 1999): 533–52. http://dx.doi.org/10.1142/s0218195999000315.

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This paper presents a simple O(n+k) time algorithm to compute the set of knon-crossing shortest paths between k source-destination pairs of points on the boundary of a simple polygon of n vertices. Paths are allowed to overlap but are not allowed to cross in the plane. A byproduct of this result is an O(n) time algorithm to compute a balanced geodesic triangulation which is easy to implement. The algorithm extends to a simple polygon with one hole where source-destination pairs may appear on both the inner and outer boundary of the polygon. In the latter case, the goal is to compute a collection of non-crossing paths of minimum total cost. The case of a rectangular polygonal domain where source-destination pairs appear on the outer and one inner boundary12 is briefly discussed.

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Zhang, Xiao Qing, Miao Le Hou, Guang Zhu, and Yun Gang Hu. "Calculation of Areas of Cultural Relics Surface Defects Based on the Triangular Mesh Model." Advanced Materials Research 446-449 (January 2012): 3452–56. http://dx.doi.org/10.4028/www.scientific.net/amr.446-449.3452.

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In order to solve the problem that need exact and scientific data in checking and restoring cultural relics, this paper presents a novel algorithm that statistics defect areas of cultural relics by calculating holes area in the in triangular mesh models.First, build the topological relationship between triangles, vertices and edges and extract boundary using boundary property of triangular mesh. Next, the holes bounding edges are linked in sequence into holes polygon. Finally, distinguish holes boundary and model exterior boundary by means of triangular mesh topological characteristics and the areas of three-dimensional holes polygon are calculated to statistics defect areas of cultural relics through the method of coordinate. Through experiments, it proved that this algorithm was correctly and reasonable.

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Turchi, N., and F. Wespi. "Limit theorems for random polytopes with vertices on convex surfaces." Advances in Applied Probability 50, no. 4 (November 29, 2018): 1227–45. http://dx.doi.org/10.1017/apr.2018.58.

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Abstract We consider the random polytope Kn, defined as the convex hull of n points chosen independently and uniformly at random on the boundary of a smooth convex body in ℝd. We present both lower and upper variance bounds, a strong law of large numbers, and a central limit theorem for the intrinsic volumes of Kn. A normal approximation bound from Stein's method and estimates for surface bodies are among the tools involved.

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Uspenskii, Alexandr Alexandrovich, and Pavel Dmitrievich Lebedev. "EUCLIDEAN DISTANCE TO A CLOSED SET AS A MINIMAX SOLUTION OF THE DIRICHLET PROBLEM FOR THE HAMILTON–JACOBI EQUATION." Tambov University Reports. Series: Natural and Technical Sciences, no. 124 (2018): 797–804. http://dx.doi.org/10.20310/1810-0198-2018-23-124-797-804.

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A combined (jointing analytical methods and computational procedures) approach to the construction of solutions in a class of boundary-value problems for a Hamiltonian-type equation is proposed. In the class of problems under consideration, the minimax (generalized) solution coincides with the Euclidean distance to the boundary set. The properties of this function are studied depending on the geometry of the boundary set and the differential properties of its boundary. Methods are developed for detecting pseudo-vertices of a boundary set and for constructing singular solution sets with their help. The methods are based on the properties of local diffeomorphisms and use partial one-sided limits. The effectiveness of the research approaches developed is illustrated by the example of solving a planar timecontrol problem for the case of a nonconvex target set with boundary of variable smoothness.

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Pébay, Philippe P., and David Thompson. "Communication-Free Streaming Mesh Refinement." Journal of Computing and Information Science in Engineering 5, no. 4 (March 22, 2005): 309–16. http://dx.doi.org/10.1115/1.2052806.

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This article presents a technique for the adaptive refinement of tetrahedral meshes. What makes this method new is that no neighbor information is required for the refined mesh to be compatible everywhere. Refinement consists of inserting new vertices at edge midpoints until some tolerance (geometric or otherwise) is met. For a tetrahedron, the six edges present 26=64 possible subdivision combinations. The challenge is to triangulate the new vertices (i.e., the original vertices plus some subset of the edge midpoints) in a way that neighboring tetrahedra always generate the same triangles on their shared boundary. A geometric solution based on edge lengths was developed previously, but did not account for geometric degeneracies (edges of equal length). This article provides a solution that works in all cases, while remaining entirely communication-free.

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Hernández, R., J. A. García, and C. Mañoso. "Interval Continuous Plant Identification from Value Sets." Journal of Applied Mathematics 2012 (2012): 1–32. http://dx.doi.org/10.1155/2012/840603.

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This paper shows how to obtain the values of the numerator and denominator Kharitonov polynomials of an interval plant from its value set at a given frequency. Moreover, it is proven that given a value set, all the assigned polynomials of the vertices can be determined if and only if there is a complete edge or a complete arc lying on a quadrant. This algorithm is nonconservative in the sense that if the value-set boundary of an interval plant is exactly known, and particularly its vertices, then the Kharitonov rectangles are exactly those used to obtain these value sets.

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BARLETTI, LUIGI. "LINEAR TRANSPORT OF PARTICLES ON NETWORKS." Mathematical Models and Methods in Applied Sciences 06, no. 02 (March 1996): 279–94. http://dx.doi.org/10.1142/s0218202596000596.

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We introduce a model of transport of particles in a network, which is represented by a connected graph with m vertices and n edges. Each edge represents a one-dimensional conductor of particles, whose behavior is described by means of a linear Boltzmann-like equation. In graph vertices, a system of linear boundary conditions is given which takes into account the exchanges of particles between the edges. The well-posedness of the initial value problem is studied into an abstract L1-like setting and the structure of the solution is given for simplest case of pure streaming transport.

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Chen, Jie, Cheng Hui Gao, and Bing Wei He. "A New Hole Filling Algorithm in Space for Triangular Model." Advanced Materials Research 279 (July 2011): 200–206. http://dx.doi.org/10.4028/www.scientific.net/amr.279.200.

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Aiming at filling the holes which were generated from uncompleted point cloud data in reverse engineering, a new hole filling algorithm in space is presented. Firstly, the holes boundary was identified and pretreated, and the hole boundary feature datum was established and the boundary was projected on it to form a projection polygon. Secondly, the smallest angle of the projection polygon was found out to determine the corresponding boundary point as the mesh growing point. The original hole was covered by the new meshes covering and then filling algorithm was completed. Finally, the neighborhood points of hole boundary vertex were selected as the sampling points for the least squares fitting adjustment of new filled vertices position, which aims at the preparation hole filled result. Examples are given to prove that the method has good accuracy and stability of the hole filling.

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Buterin, Sergey. "On Recovering Sturm–Liouville-Type Operators with Global Delay on Graphs from Two Spectra." Mathematics 11, no. 12 (June 13, 2023): 2688. http://dx.doi.org/10.3390/math11122688.

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We suggest a new formulation of the inverse spectral problem for second-order functional-differential operators on star-shaped graphs with global delay. The latter means that the delay, which is measured in the direction of a specific boundary vertex, called the root, propagates through the internal vertex to other edges. Now, we intend to recover the potentials from the spectra of two boundary value problems on the graph with a common set of boundary conditions at all boundary vertices except the root. For simplicity, we focus on star graphs with equal edges when the delay parameter is not less than their length. Under the assumption that the common boundary conditions are of the Robin type and they are known and pairwise linearly independent, the uniqueness theorem is proven and a constructive procedure for solving the proposed inverse problem is obtained.

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Panasenko, G., K. Pileckas, and B. Vernescu. "Steady state non-Newtonian flow in a thin tube structure: equation on the graph." St. Petersburg Mathematical Journal 33, no. 2 (March 4, 2022): 327–40. http://dx.doi.org/10.1090/spmj/1702.

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The dimension reduction for the viscous flows in thin tube structures leads to equations on the graph for the macroscopic pressure with Kirchhoff type junction conditions at the vertices. Nonlinear equations on the graph generated by the non-Newtonian rheology are treated here. The existence and uniqueness of a solution of this problem is proved. This solution describes the leading term of an asymptotic analysis of the stationary non-Newtonian fluid motion in a thin tube structure with no-slip boundary condition on the lateral boundary.

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Gashnikov, M. V. "Adaptive interpolation based on optimization of the decision rule in a multidimensional feature space." Computer Optics 44, no. 1 (February 2020): 101–8. http://dx.doi.org/10.18287/2412-6179-co-661.

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An adaptive multidimensional signal interpolator is proposed, which selects an interpolating function at each signal point by means of the decision rule optimized in a multidimensional feature space using a decision tree. The search for the dividing boundary when splitting the decision tree vertices is carried out by a recurrence procedure that allows, in addition to the search for the boundary, selecting the best pair of interpolating functions from a predetermined set of functions of an arbitrary form. Results of computational experiments in nature multidimensional signals are presented, confirming the effectiveness of the adaptive interpolator.

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DINITZ, YEFIM, SHIMON EVEN, and MARIA ZAPOLOTSKY. "A COMPACT LAYOUT OF THE BUTTERFLY." Journal of Interconnection Networks 04, no. 01 (March 2003): 53–75. http://dx.doi.org/10.1142/s0219265903000738.

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For the Butterfly of N input/output vertices we present a layout on the rectilinear (square) grid of area ½N2 + o(N2). A lower bound of the same order is proved. The encompassing rectangle which defines the area is 45° slanted w.r.t. the grid axes and the input/output vertices are not on the boundary of this rectangle. The layout is component-scalable, i.e., if one allocates for each switch a square of a × a area, the layout remains of area ½N2 + o(N2); that is, the value of a affects only the o(N2) term. The layout is also free of knock-knees.

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MAZZEO, RAFE, and JULIE ROWLETT. "A heat trace anomaly on polygons." Mathematical Proceedings of the Cambridge Philosophical Society 159, no. 2 (June 19, 2015): 303–19. http://dx.doi.org/10.1017/s0305004115000365.

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AbstractLet Ω0be a polygon in$\mathbb{R}$2, or more generally a compact surface with piecewise smooth boundary and corners. Suppose that Ωεis a family of surfaces with${\mathcal C}$∞boundary which converges to Ω0smoothly away from the corners, and in a precise way at the vertices to be described in the paper. Fedosov [6], Kac [8] and McKean–Singer [13] recognised that certain heat trace coefficients, in particular the coefficient oft0, are not continuous as ε ↘ 0. We describe this anomaly using renormalized heat invariants of an auxiliary smooth domainZwhich models the corner formation. The result applies to both Dirichlet and Neumann boundary conditions. We also include a discussion of what one might expect in higher dimensions.

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Popa, Sorin. "Some ergodic properties for infinite graphs associated with subfactors." Ergodic Theory and Dynamical Systems 15, no. 5 (October 1995): 993–1003. http://dx.doi.org/10.1017/s0143385700009731.

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AbstractWe prove that the restriction of the graph of a subfactor, ΓN,M, to an infinite subset of vertices with finite boundary has the same norm as ΓN,W. In particular, if N φ M is extremal with [M : N] > 4 and ΓN,M has an A∞, tail then ΓN, M = A∞.

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Sintunavarat, Wutiphol, and Ali Turab. "A unified fixed point approach to study the existence of solutions for a class of fractional boundary value problems arising in a chemical graph theory." PLOS ONE 17, no. 8 (August 12, 2022): e0270148. http://dx.doi.org/10.1371/journal.pone.0270148.

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A theory of chemical graphs is a part of mathematical chemistry concerned with the effects of connectedness in chemical graphs. Several researchers have studied the solutions of fractional differential equations using the concept of star graphs. They employed star graphs because their technique requires a central node with links to adjacent vertices but no edges between nodes. The purpose of this paper is to extend the method’s range by introducing the concept of an octane graph, which is an essential organic compound having the formula C8H18. In this manner, we analyze a graph with vertices annotated by 0 or 1, which is influenced by the structure of the chemical substance octane, and formulate a fractional boundary value problem on each of the graph’s edges. We use the Schaefer and Krasnoselskii fixed point theorems to investigate the existence of solutions to the presented boundary value problems in the framework of the Caputo fractional derivative. Finally, two examples are provided to highlight the importance of our results in this area of study.

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