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Journal articles on the topic 'Geometric and topological aspects of graph theory'
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1
Asif, Muhammad, Hamad Almohamedh, Muhammad Hussain, Khalid M. Alhamed, Abdulrazaq A. Almutairi, and Sultan Almotairi. "An Approach to the Geometric-Arithmetic Index for Graphs under Transformations’ Fact over Pendent Paths." Complexity 2021 (June 24, 2021): 1–13. http://dx.doi.org/10.1155/2021/3745862.
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Graph theory is a dynamic tool for designing and modeling of an interconnection system by a graph. The vertices of such graph are processor nodes and edges are the connections between these processors nodes. The topology of a system decides its best use. Geometric-arithmetic index is one of the most studied graph invariant to characterize the topological aspects of underlying interconnection networks or graphs. Transformation over graph is also an important tool to define new network of their own choice in computer science. In this work, we discuss transformed family of graphs. Let Γ n k , l be the connected graph comprises by k number of pendent path attached with fully connected vertices of the n-vertex connected graph Γ . Let A α Γ n k , l and A α β Γ n k , l be the transformed graphs under the fact of transformations A α and A α β , 0 ≤ α ≤ l , 0 ≤ β ≤ k − 1 , respectively. In this work, we obtained new inequalities for the graph family Γ n k , l and transformed graphs A α Γ n k , l and A α β Γ n k , l which involve GA Γ . The presence of GA Γ makes the inequalities more general than all those which were previously defined for the GA index. Furthermore, we characterize extremal graphs which make the inequalities tight.
2
Islam, Tanweer Ul, Zeeshan Saleem Mufti, Aqsa Ameen, Muhammad Nauman Aslam, and Ali Tabraiz. "On Certain Aspects of Topological Indices." Journal of Mathematics 2021 (May 3, 2021): 1–20. http://dx.doi.org/10.1155/2021/9913529.
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A topological index, also known as connectivity index, is a molecular structure descriptor calculated from a molecular graph of a chemical compound which characterizes its topology. Various topological indices are categorized based on their degree, distance, and spectrum. In this study, we calculated and analyzed the degree-based topological indices such as first general Zagreb index M r G , geometric arithmetic index GA G , harmonic index H G , general version of harmonic index H r G , sum connectivity index λ G , general sum connectivity index λ r G , forgotten topological index F G , and many more for the Robertson apex graph. Additionally, we calculated the newly developed topological indices such as the AG 2 G and Sanskruti index for the Robertson apex graph G.
3
Saeed, Nouman, Kai Long, Tanweer Ul Islam, Zeeshan Saleem Mufti, and Ayesha Abbas. "Topological Study of Zeolite Socony Mobil-5 via Degree-Based Topological Indices." Journal of Chemistry 2021 (June 23, 2021): 1–13. http://dx.doi.org/10.1155/2021/5522800.
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Graph theory is a subdivision of discrete mathematics. In graph theory, a graph is made up of vertices connected through edges. Topological indices are numerical parameters or descriptors of graph. Topological index tells the symmetry of compound and helps us to compare those mathematical values, with boiling point, melting point, density, viscosity, hydrophobic surface area, polarity, etc., of that compound. In the present research paper, degree-based topological indices of Zeolite Socony Mobil-5 are calculated. Names of those topological indices are Randić index, first Zagreb index, general sum connectivity index, hyper-Zagreb index, geometric index, ABC index, etc.
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Bokhary, Syed Ahtsham Ul Haq, Muhammad Imran, and Sadia Manzoor. "On molecular topological properties of dendrimers." Canadian Journal of Chemistry 94, no. 2 (February 2016): 120–25. http://dx.doi.org/10.1139/cjc-2015-0466.
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Topological indices are numerical parameters of a graph that characterize its topology and are usually graph invariant. In a QSAR/QSPR study, physicochemical properties and topological indices such as the Randić, atom–bond connectivity (ABC), and geometric–arithmetic (GA) indices are used to predict the bioactivity of different chemical compounds. Graph theory has found a considerable use in this area of research. In this paper, we study the degree-based molecular topological indices such as ABC4 and GA5 for certain families of dendrimers. We derive the analytical closed formulae for these classes of dendrimers.
5
Manjunath, Muddalapuram, V. Lokesha, Suvarna, and Sushmitha Jain. "Bounds for the Topological Indices of ℘ graph." European Journal of Pure and Applied Mathematics 14, no. 2 (May 18, 2021): 340–50. http://dx.doi.org/10.29020/nybg.ejpam.v14i2.3715.
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Topological indices are mathematical measure which correlates to the chemical structures of any simple finite graph. These are used for Quantitative Structure-Activity Relationship (QSAR) and Quantitative Structure-Property Relationship (QSPR). In this paper, we define operator graph namely, ℘ graph and structured properties. Also, establish the lower and upper bounds for few topological indices namely, Inverse sum indeg index, Geometric-Arithmetic index, Atom-bond connectivity index, first zagreb index and first reformulated Zagreb index of ℘-graph.
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Imran, Muhammad, Muhammad Siddiqui, Amna Abunamous, Dana Adi, Saida Rafique, and Abdul Baig. "Eccentricity Based Topological Indices of an Oxide Network." Mathematics 6, no. 7 (July 18, 2018): 126. http://dx.doi.org/10.3390/math6070126.
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Graph theory has much great advances in the field of mathematical chemistry. Chemical graph theory has become very popular among researchers because of its wide applications in mathematical chemistry. The molecular topological descriptors are the numerical invariants of a molecular graph and are very useful for predicting their bioactivity. A great variety of such indices are studied and used in theoretical chemistry, pharmaceutical researchers, in drugs and in different other fields. In this article, we study the chemical graph of an oxide network and compute the total eccentricity, average eccentricity, eccentricity based Zagreb indices, atom-bond connectivity (ABC) index and geometric arithmetic index of an oxide network. Furthermore, we give analytically closed formulas of these indices which are helpful in studying the underlying topologies.
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Idrees, Nazeran, Muhammad Jawwad Saif, Afshan Sadiq, Asia Rauf, and Fida Hussain. "Topological Indices of H-Naphtalenic Nanosheet." Open Chemistry 16, no. 1 (November 14, 2018): 1184–88. http://dx.doi.org/10.1515/chem-2018-0131.
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AbstractIn chemical graph theory, a single numeric number related to a chemical structure is called a topological descriptor or topological index of a graph. In this paper, we compute analytically certain topological indices for H-Naphtalenic nanosheet like Randic index, first Zagreb index, second Zagreb index, geometric arithmetic index, atom bond connectivity index, sum connectivity index and hyper-Zagreb index using edge partition technique. The first multiple Zagreb index and the second multiple Zagreb index of the nanosheet are also discussed in this paper.
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Bača, Martin, Jarmila Horváthová, Martina Mokrišová, Andrea Semaničová-Feňovčíková, and Alžbeta Suhányiová. "On topological indices of a carbon nanotube network." Canadian Journal of Chemistry 93, no. 10 (October 2015): 1157–60. http://dx.doi.org/10.1139/cjc-2015-0175.
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A numerical quantity that characterizes the whole structure of a graph is called a topological index. The concept of Randić (Rα), atom−bond connectivity (ABC), and geometric−arithmetic (GA) topological indices was established in chemical graph theory based on vertex degrees. In this paper, we study a carbon nanotube network that is motivated by the molecular structure of a regular hexagonal lattice and determine Rα, ABC, and GA indices for this important class of networks.
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Bojarski, B., and G. Khimshiashvili. "Global Geometric Aspects of Riemann–Hilbert Problems." gmj 8, no. 4 (December 2001): 713–26. http://dx.doi.org/10.1515/gmj.2001.713.
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Abstract We discuss some global properties of an abstract geometric model for Riemann–Hilbert problems introduced by the first author. In particular, we compute the homotopy groups of elliptic Riemann–Hilbert problems and describe some connections with the theory of Fredholm structures which enable one to introduce more subtle geometrical and topological invariants for families of such problems.
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Liu, Jia-Bao, Imran Khalid, Mohammad Tariq Rahim, Masood Ur Rehman, Faisal Ali, and Muhammad Salman. "Eccentric topological properties of a graph associated to a finite dimensional vector space." Main Group Metal Chemistry 43, no. 1 (October 6, 2020): 164–76. http://dx.doi.org/10.1515/mgmc-2020-0020.
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AbstractA topological index is actually designed by transforming a chemical structure into a number. Topological index is a graph invariant which characterizes the topology of the graph and remains invariant under graph automorphism. Eccentricity based topological indices are of great importance and play a vital role in chemical graph theory. In this article, we consider a graph (non-zero component graph) associated to a finite dimensional vector space over a finite filed in the context of the following eleven eccentricity based topological indices: total eccentricity index; average eccentricity index; eccentric connectivity index; eccentric distance sum index; adjacent distance sum index; connective eccentricity index; geometric arithmetic index; atom bond connectivity index; and three versions of Zagreb indices. Relationship of the investigated indices and their dependency with respect to the involved parameters are also visualized by evaluating them numerically and by plotting their results.
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Mathews, Daniel V. "Twisty itsy bitsy topological field theory." International Journal of Mathematics 25, no. 10 (September 2014): 1450097. http://dx.doi.org/10.1142/s0129167x14500979.
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We extend the topological field theory (itsy bitsy topological field theory) of our previous work from mod 2 to twisted coefficients. This topological field theory is derived from sutured Floer homology (SFH) but described purely in terms of surfaces with signed points on their boundary (occupied surfaces) and curves on those surfaces respecting signs (sutures). It has information-theoretic (itsy) and quantum-field-theoretic (bitsy) aspects. In the process we extend some results of SFH, consider associated ribbon graph structures, and construct explicit admissible Heegaard decompositions.
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Ul Haq Bokhary, Syed Ahtsham, Muhammad Imran, Shehnaz Akhter, and Sadia Manzoor. "Molecular topological invariants of certain chemical networks." Main Group Metal Chemistry 44, no. 1 (January 1, 2021): 141–49. http://dx.doi.org/10.1515/mgmc-2021-0010.
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Abstract Topological descriptors are the graph invariants that are used to explore the molecular topology of the molecular/chemical graphs. In QSAR/QSPR research, physico-chemical characteristics and topological invariants including Randić, atom-bond connectivity, and geometric arithmetic invariants are utilized to corelate and estimate the structure relationship and bioactivity of certain chemical compounds. Graph theory and discrete mathematics have discovered an impressive utilization in the area of research. In this article, we investigate the valency-depended invariants for certain chemical networks like generalized Aztec diamonds and tetrahedral diamond lattice. Moreover, the exact values of invariants for these categories of chemical networks are derived.
13
Zhang, Xiujun, Xinling Wu, Shehnaz Akhter, Muhammad Jamil, Jia-Bao Liu, and Mohammad Farahani. "Edge-Version Atom-Bond Connectivity and Geometric Arithmetic Indices of Generalized Bridge Molecular Graphs." Symmetry 10, no. 12 (December 14, 2018): 751. http://dx.doi.org/10.3390/sym10120751.
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Topological indices are graph invariants computed by the distance or degree of vertices of the molecular graph. In chemical graph theory, topological indices have been successfully used in describing the structures and predicting certain physicochemical properties of chemical compounds. In this paper, we propose a definition of generalized bridge molecular graphs that can model more kinds of long chain polymerization products than the bridge molecular graphs, and provide some results of the edge versions of atom-bond connectivity ( A B C e ) and geometric arithmetic ( G A e ) indices for some generalized bridge molecular graphs, which have regular, periodic and symmetrical structures. The results of this paper offer promising prospects in the applications for chemical and material engineering, especially in chemical industry research.
14
Liu, Jia-Bao, Muhammad Kashif Shafiq, Haidar Ali, Asim Naseem, Nayab Maryam, and Syed Sheraz Asghar. "Topological Indices of mth Chain Silicate Graphs." Mathematics 7, no. 1 (January 4, 2019): 42. http://dx.doi.org/10.3390/math7010042.
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A topological index is a numerical representation of a chemical structure, while a topological descriptor correlates certain physico-chemical characteristics of underlying chemical compounds besides its numerical representation. A large number of properties like physico-chemical properties, thermodynamic properties, chemical activity, and biological activity are determined by the chemical applications of graph theory. The biological activity of chemical compounds can be constructed by the help of topological indices such as atom-bond connectivity (ABC), Randić, and geometric arithmetic (GA). In this paper, Randić, atom bond connectivity (ABC), Zagreb, geometric arithmetic (GA), ABC4, and GA5 indices of the mth chain silicate S L ( m , n ) network are determined.
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Baig, Abdul Qudair, Muhammad Imran, Waqas Khalid, and Muhammad Naeem. "Molecular description of carbon graphite and crystal cubic carbon structures." Canadian Journal of Chemistry 95, no. 6 (June 2017): 674–86. http://dx.doi.org/10.1139/cjc-2017-0083.
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Graph theory plays a vital role in modeling and designing any chemical structure or chemical network. Chemical graph theory helps in understanding the molecular structural properties of a molecular graph. The molecular graph consists of atoms called vertices and chemical bonds between atoms called edges. In this article, we study the chemical graphs of carbon graphite and crystal structure of cubic carbon. Moreover, we compute and give closed formulas of degree-based additive topological indices, mainly the first and second Zagreb indexes, general Randić index, atom bond connectivity index, geometric arithmetic index, fourth atom bond connectivity index, and fifth geometric arithmetic index of carbon graphite denoted by CG(m, n) for t levels, and crystal structure cubic carbon denoted for n levels.
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Yang, Hong, Muhammad Aamer Rashid, Sarfraz Ahmad, Saima Sami Khan, and Muhammad Kamran Siddiqui. "On Molecular Descriptors of Face-Centered Cubic Lattice." Processes 7, no. 5 (May 13, 2019): 280. http://dx.doi.org/10.3390/pr7050280.
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Face-centered cubic lattice F C C ( n ) has received extensive consideration as of late, inferable from its recognized properties and non-poisonous nature, minimal effort, plenitude, and basic creation process. The graph of a face-centered cubic cross-section contains cube points and face centres. A topological index of a molecular graph G is a numeric amount identified with G, which depicts its topological properties. In this paper, using graph theory tools, we computed the molecular descriptors (topological indices)—to be specific, Zagreb-type indices, a forgotten index, a Balaban index, the fourth version of an atom–bond connectivity index, and the fifth version of a geometric arithmetic index for face-centered cubic lattice F C C ( n ) .
17
Zhang, Zhiqiang, Zeshan Saleem Mufti, Muhammad Faisal Nadeem, Zaheer Ahmad, Muhammad Kamran Siddiqui, and Muhammad Reza Farahani. "Computing Topological Indices for Para-Line Graphs of Anthracene." Open Chemistry 17, no. 1 (November 13, 2019): 955–62. http://dx.doi.org/10.1515/chem-2019-0093.
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AbstractAtoms displayed as vertices and bonds can be shown by edges on a molecular graph. For such graphs we can find the indices showing their bioactivity as well as their physio-chemical properties such as the molar refraction, molar volume, chromatographic behavior, heat of atomization, heat of vaporization, magnetic susceptibility, and the partition coefficient. Today, industry is flourishing because of the interdisciplinary study of different disciplines. This provides a way to understand the application of different disciplines. Chemical graph theory is a mixture of chemistry and mathematics, which plays an important role in chemical graph theory. Chemistry provides a chemical compound, and graph theory transforms this chemical compound into a molecular graphwhich further is studied by different aspects such as topological indices.We will investigate some indices of the line graph of the subdivided graph (para-line graph) of linear-[s] Anthracene and multiple Anthracene.
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Imran, Muhammad, Abdul Qudair Baig, and Haidar Ali. "On topological properties of dominating David derived networks." Canadian Journal of Chemistry 94, no. 2 (February 2016): 137–48. http://dx.doi.org/10.1139/cjc-2015-0185.
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Topological indices are numerical parameters of a graph that characterize its molecular topology and are usually graph invariant. In a QSAR/QSPR study, the physico-chemical properties and topological indices such as the Randić, atom–bond connectivity (ABC), and geometric–arithmetic (GA) indices are used to predict the bioactivity of chemical compounds. Graph theory has found a considerable use in this important area of research. All of the studied interconnection networks in this paper are constructed by the Star of David network. In this paper, we study the general Randić, first Zagreb, ABC, GA, ABC4 and GA5, indices for the first, second, and third types of dominating David derived networks and give closed formulas of these indices for these networks. These results are useful in network science to understand the underlying topologies of these networks.
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Imran, Muhammad, Abdul Qudair Baig, Shafiq Ur Rehman, Haidar Ali, and Roslan Hasni. "Computing topological polynomials of mesh-derived networks." Discrete Mathematics, Algorithms and Applications 10, no. 06 (December 2018): 1850077. http://dx.doi.org/10.1142/s1793830918500775.
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Topological descriptors are numerical parameters of a molecular graph which characterize its molecular topology and are usually graph invariant. In QSAR/QSPR study, physico-chemical properties and topological indices such as Randić, atom-bond connectivity [Formula: see text] and geometric-arithmetic (GA) index are used to predict the bioactivity of chemical compounds. Graph theory has found a considerable use in this area of research. The counting polynomials are those polynomials having at exponent the extent of a property partition and coefficients the multiplicity/occurrence of the corresponding partition. All of the studied interconnection mesh networks in this paper are motivated by the molecular structure of a Sodium chloride NaCl. In this paper, Omega, Sadhana and PI polynomials are computed for mesh-derived networks. These polynomials were proposed on the ground of quasi-orthogonal cut edge strips in polycyclic graphs. These polynomials count equidistant and non-equidistant edges in graphs. Moreover, the analytical closed formulas of these polynomials for mesh-derived networks are computed for the first time.
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Liu, Jia-Bao, Haidar Ali, Muhammad Shafiq, and Usman Munir. "On Degree-Based Topological Indices of Symmetric Chemical Structures." Symmetry 10, no. 11 (November 9, 2018): 619. http://dx.doi.org/10.3390/sym10110619.
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A Topological index also known as connectivity index is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound. Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariant. In QSAR/QSPR study, physico-chemical properties and topological indices such as Randić, atom-bond connectivity (ABC) and geometric-arithmetic (GA) index are used to predict the bioactivity of chemical compounds. Graph theory has found a considerable use in this area of research. In this paper, we study HDCN1(m,n) and HDCN2(m,n) of dimension m , n and derive analytical closed results of general Randić index R α ( G ) for different values of α . We also compute the general first Zagreb, ABC, GA, A B C 4 and G A 5 indices for these Hex derived cage networks for the first time and give closed formulas of these degree-based indices.
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Imran, Muhammad, and Shehnaz Akhter. "Degree-based topological indices of double graphs and strong double graphs." Discrete Mathematics, Algorithms and Applications 09, no. 05 (October 2017): 1750066. http://dx.doi.org/10.1142/s1793830917500665.
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The topological indices are useful tools to the theoretical chemists that are provided by the graph theory. They correlate certain physicochemical properties such as boiling point, strain energy, stability, etc. of chemical compounds. For a graph [Formula: see text], the double graph [Formula: see text] is a graph obtained by taking two copies of graph [Formula: see text] and joining each vertex in one copy with the neighbors of corresponding vertex in another copy and strong double graph SD[Formula: see text] of the graph [Formula: see text] is the graph obtained by taking two copies of the graph [Formula: see text] and joining each vertex [Formula: see text] in one copy with the closed neighborhood of the corresponding vertex in another copy. In this paper, we compute the general sum-connectivity index, general Randi[Formula: see text] index, geometric–arithmetic index, general first Zagreb index, first and second multiplicative Zagreb indices for double graphs and strong double graphs and derive the exact expressions for these degree-base topological indices for double graphs and strong double graphs in terms of corresponding index of original graph [Formula: see text].
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Rao, Yongsheng, Ammarah Kanwal, Riffat Abbas, Saima Noureen, Asfand Fahad, and Muhammad Imran Qureshi. "Some degree-based topological indices of caboxy-terminated dendritic macromolecule." Main Group Metal Chemistry 44, no. 1 (January 1, 2021): 165–72. http://dx.doi.org/10.1515/mgmc-2021-0016.
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Abstract In the modern era of the chemical science, the chemical graph theory has contributed significantly to exploring the properties of the chemical compounds. Currently, the computation of the topological indices is one of the most active directions of the research in the area of the chemical graph theory. The main feature of the study of the topological indices is its its ability of predicting the various physio-chemical properties. In this article, we compute several degree-based topological indices for the caboxy-terminated dendritic macromolecule. We compute Harmonic index, atom-bond connectivity index, geometric arithmetic index, sum connectivity index, inverse sum index, symmetric division degree, and Zagreb indices for caboxy-terminated dendritic macromolecule. The obtained results have potential to predict biochemical properties such as viscosity, entropy, and boiling point.
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Ahmadini, Abdullah Ali H., Ali N. A. Koam, Ali Ahmad, Martin Bača, and Andrea Semaničová–Feňovčíková. "Computing Vertex-Based Eccentric Topological Descriptors of Zero-Divisor Graph Associated with Commutative Rings." Mathematical Problems in Engineering 2020 (August 24, 2020): 1–6. http://dx.doi.org/10.1155/2020/2056902.
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The applications of finite commutative ring are useful substances in robotics and programmed geometric, communication theory, and cryptography. In this paper, we study the vertex-based eccentric topological indices of a zero-divisor graphs of commutative ring ℤp2×ℤq, where p and q are primes.
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Ding, Lei, Syed Ahtsham Ul Haq Bokhary, Masood Ur Rehman, Usman Ali, Hirra Mubeen, Quaid Iqbal, and Jia-Bao Liu. "Degree-Based Indices of Some Complex Networks." Journal of Mathematics 2021 (March 17, 2021): 1–16. http://dx.doi.org/10.1155/2021/5531357.
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A topological index is a numeric quantity assigned to a graph that characterizes the structure of a graph. Topological indices and physico-chemical properties such as atom-bond connectivity ABC , Randić, and geometric-arithmetic index GA are of great importance in the QSAR/QSPR analysis and are used to estimate the networks. In this area of research, graph theory has been found of considerable use. In this paper, the distinct degrees and degree sums of enhanced Mesh network, triangular Mesh network, star of silicate network, and rhenium trioxide lattice are listed. The edge partitions of these families of networks are tabled which depend on the sum of degrees of end vertices and the sum of the degree-based edges. Utilizing these edge partitions, the closed formulae for some degree-based topological indices of the networks are deduced.
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Afzal, Deeba, Farkhanda Afzal, Mohammad Reza Farahani, and Samia Ali. "On Computation of Recently Defined Degree-Based Topological Indices of Some Families of Convex Polytopes via M-Polynomial." Complexity 2021 (July 7, 2021): 1–11. http://dx.doi.org/10.1155/2021/5881476.
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Topological indices are of incredible significance in the field of graph theory. Convex polytopes play a significant role both in various branches of mathematics and also in applied areas, most notably in linear programming. We have calculated some topological indices such as atom-bond connectivity index, geometric arithmetic index, K-Banhatti indices, and K-hyper-Banhatti indices and modified K-Banhatti indices from some families of convex polytopes through M-polynomials. The M-polynomials of the graphs provide us with a great help to calculate the topological indices of different structures.
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Cancan, Murat, Kerem Yamaç, Ziyattin Taş, and Mehmet Şerif Aldemir. "On Some Ve-Degree and Harmonic Molecular Topological Properties of Carborundum." ARO-THE SCIENTIFIC JOURNAL OF KOYA UNIVERSITY 8, no. 1 (May 15, 2020): 65. http://dx.doi.org/10.14500/aro.10560.
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Carborundum, also known as silicon carbide which containing carbon and silicon, is a semiconductor. Molecular topological properties of physical substances are important tools to investigate the underlying topology of these substances. Ev-degree and ve-degree based on the molecular topological indices have been defined as parallel to their corresponding classical degree based topological indices in chemical graph theory. Classical degree based topological properties of carborundum have been investigated recently. As a continuation of these studies, in this study, we compute novel ve-degree harmonic, ve-degree sum-connectivity, ve-degree geometric-arithmetic, and ve-degree atom-bond connectivity, the first and the fifth harmonic molecular topological indices of two carborundum structures.
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Baig, Abdul Qudair, Muhammad Imran, and Haidar Ali. "On topological indices of poly oxide, poly silicate, DOX, and DSL networks." Canadian Journal of Chemistry 93, no. 7 (July 2015): 730–39. http://dx.doi.org/10.1139/cjc-2014-0490.
Full textAbstract:
Topological indices are numerical parameters of a graph that characterize its topology and are usually graph invariant. In a QSAR/QSPR study, physicochemical properties and topological indices such as Randić, atom–bond connectivity (ABC), and geometric–arithmetic (GA) indices are used to predict the bioactivity of chemical compounds. Graph theory has found a considerable use in this area of research. In this paper, we study different interconnection networks and derive analytical closed results of the general Randić index (Rα(G)) for α = 1, [Formula: see text], –1, [Formula: see text] only, for dominating oxide network (DOX), dominating silicate network (DSL), and regular triangulene oxide network (RTOX). All of the studied interconnection networks in this paper are motivated by the molecular structure of a chemical compound, SiO4. We also compute the general first Zagreb, ABC, GA, ABC4, and GA5 indices and give closed formulae of these indices for these interconnection networks.
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Li, Zhi Jie, Chang Hua Li, Xin Liu, and Pu Liang Zheng. "Graph Embedding Method Based on Space Syntax and Improved K-Means Clustering." Advanced Materials Research 1044-1045 (October 2014): 1163–68. http://dx.doi.org/10.4028/www.scientific.net/amr.1044-1045.1163.
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The main drawbacks of structural pattern recognition compared to statistical pattern recognition are the high computation complexity and fewer processing tools that are available in the domain. To bridge the gap between the structural and statistical pattern recognition, a new graph embedding method based on space syntax and improved K-means clustering is proposed. The present paper uses the space syntax theory to build quantitative description of the nodes’ topological features, and then combines the proposed topological features with non-topological features in other aspects of the domain to construct node feature set using an improved K-means clustering algorithm, and then maps the graph into vector space explicitly by a statistical approach. Thus SVM can be applied to achieve graph classification. The experimental results show that such an embedding method can achieve higher classification accuracy in different graph datasets.
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Kwun, Young, Abaid Virk, Waqas Nazeer, M. Rehman, and Shin Kang. "On the Multiplicative Degree-Based Topological Indices of Silicon-Carbon Si2C3-I[p,q] and Si2C3-II[p,q]." Symmetry 10, no. 8 (August 3, 2018): 320. http://dx.doi.org/10.3390/sym10080320.
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The application of graph theory in chemical and molecular structure research has far exceeded people’s expectations, and it has recently grown exponentially. In the molecular graph, atoms are represented by vertices and bonds by edges. Topological indices help us to predict many physico-chemical properties of the concerned molecular compound. In this article, we compute Generalized first and multiplicative Zagreb indices, the multiplicative version of the atomic bond connectivity index, and the Generalized multiplicative Geometric Arithmetic index for silicon-carbon Si2C3−I[p,q] and Si2C3−II[p,q] second.
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ASOREY, MANUEL, and FERNANDO FALCETO. "GEOMETRIC REGULARIZATION AND GAUGE INVARIANCE IN CHERN–SIMONS THEORIES." International Journal of Modern Physics A 07, no. 02 (January 20, 1992): 235–56. http://dx.doi.org/10.1142/s0217751x92000156.
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Some perturbative aspects of Chern–Simons theories are analyzed in a geometric-regularization framework. In particular, we show that the independence from the gauge condition of the regularized theory, which insures its global meaning, does impose a new constraint on the parameters of the regularization. The condition turns out to be the one that arises in pure or topologically massive Yang–Mills theories in three-dimensional space–times. One-loop calculations show the existence of nonvanishing finite renormalizations of gauge fields and coupling constant which preserve the topological meaning of Chern–Simons theory. The existence of a (finite) gauge-field renormalization at one-loop level is compensated by the renormalization of gauge transformations in such a way that the one-loop effective action remains gauge-invariant with respect to renormalized gauge transformations. The independence of both renormalizations from the space–time volume indicates the topological nature of the theory.
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Nnaji, Bartholomew O., and Tzong-Shyan Kang. "Interpretation of CAD models through neutral geometric knowledge." Artificial Intelligence for Engineering Design, Analysis and Manufacturing 4, no. 1 (February 1990): 15–45. http://dx.doi.org/10.1017/s0890060400002225.
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A generalized approach to fast interpretation of objects and their features has so far eluded researchers. In manufacturing, this interpretation can be approached from the vision point of view or from the CAD data perspective. Presently, CAD systems are widely used in several aspects of manufacturing production. It is therefore more efficient to use CAD data for object reasoning in manufacturing, especially when systems will eventually be data driven. Components can be modelled on a CAD system using various modelling techniques and the representation of their geometric information is still CAD system dependent. However, the advent of the Initial Graphics Exchange Specification (IGES) now makes it possible to represent CAD data in a neutral and standard manner.This paper describes a scheme for recognizing and representing features for CAD data extracted using the IGES interface. The concepts developed are based on graph-based feature representation, where features are represented by a set of faces as well as their topological adjacency.Strategies for classifying features and methods of decomposing a complicated feature into several simpler features for recognition purposes are discussed.
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SATI, HISHAM. "ON THE GEOMETRY OF THE SUPERMULTIPLET IN M-THEORY." International Journal of Geometric Methods in Modern Physics 08, no. 07 (November 2011): 1519–51. http://dx.doi.org/10.1142/s0219887811005774.
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The massless supermultiplet of 11-dimensional supergravity can be generated from the decomposition of certain representation of the exceptional Lie group F4 into those of its maximal compact subgroup Spin(9). In an earlier paper, a dynamical Kaluza–Klein origin of this observation is proposed with internal space the Cayley plane, 𝕆P2, and topological aspects are explored. In this paper we consider the geometric aspects and characterize the corresponding forms which contribute to the action as well as cohomology classes, including torsion, which contribute to the partition function. This involves constructions with bilinear forms. The compatibility with various string theories are discussed, including reduction to loop bundles in ten dimensions.
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Hajij, Mustafa, and Paul Rosen. "An Efficient Data Retrieval Parallel Reeb Graph Algorithm." Algorithms 13, no. 10 (October 12, 2020): 258. http://dx.doi.org/10.3390/a13100258.
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The Reeb graph of a scalar function that is defined on a domain gives a topologically meaningful summary of that domain. Reeb graphs have been shown in the past decade to be of great importance in geometric processing, image processing, computer graphics, and computational topology. The demand for analyzing large data sets has increased in the last decade. Hence, the parallelization of topological computations needs to be more fully considered. We propose a parallel augmented Reeb graph algorithm on triangulated meshes with and without a boundary. That is, in addition to our parallel algorithm for computing a Reeb graph, we describe a method for extracting the original manifold data from the Reeb graph structure. We demonstrate the running time of our algorithm on standard datasets. As an application, we show how our algorithm can be utilized in mesh segmentation algorithms.
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Cowan, Richard. "A mosaic of triangular cells formed with sequential splitting rules." Journal of Applied Probability 41, A (2004): 3–15. http://dx.doi.org/10.1017/s0021900200112161.
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The iterative division of a triangle by chords which join a randomly-selected vertex of a triangle to the opposite side is investigated. Results on the limiting random graph which eventuates are given. Aspects studied are: the order of vertices; the fragmentation of chords; age distributions for elements of the graph; various topological characterisations of the triangles. Different sampling protocols are explored. Extensive use is made of the theory of branching processes.
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Cowan, Richard. "A mosaic of triangular cells formed with sequential splitting rules." Journal of Applied Probability 41, A (2004): 3–15. http://dx.doi.org/10.1239/jap/1082552186.
Full textAbstract:
The iterative division of a triangle by chords which join a randomly-selected vertex of a triangle to the opposite side is investigated. Results on the limiting random graph which eventuates are given. Aspects studied are: the order of vertices; the fragmentation of chords; age distributions for elements of the graph; various topological characterisations of the triangles. Different sampling protocols are explored. Extensive use is made of the theory of branching processes.
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Roushon, S. K. "The Farrell-Jones isomorphism conjecture for 3-manifold groups." Journal of K-Theory 1, no. 1 (November 30, 2007): 49–82. http://dx.doi.org/10.1017/is007011012jkt005.
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AbstractWe show that the Fibered Isomorphism Conjecture (FIC) of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for fundamental groups of a large class of 3-manifolds. We also prove that if the FIC is true for irreducible 3-manifold groups then it is true for all 3-manifold groups. In fact, this follows from a more general result we prove, namely we show that if the FIC is true for each vertex group of a graph of groups with trivial edge groups then the FIC is true for the fundamental group of the graph of groups. This result is part of a program to prove the FIC for the fundamental group of a graph of groups where all the vertex and edge groups satisfy the FIC. A consequence of the first result gives a partial solution to a problem in the problem list of R. Kirby. We also deduce that the FIC is true for a class of virtually P D3-groups.Another main aspect of this article is to prove the FIC for all Haken 3-manifold groups assuming that the FIC is true for B-groups. By definition a B-group contains a finite index subgroup isomorphic to the fundamental group of a compact irreducible 3-manifold with incompressible nonempty boundary so that each boundary component is of genus ≥ 2. We also prove the FIC for a large class of B-groups and moreover, using a recent result of L.E. Jones we show that the surjective part of the FIC is true for any B-group.
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Li, Qiongling, Xinwei Li, Xuetong Wang, Yuxia Li, Kuncheng Li, Yang Yu, Changhao Yin, Shuyu Li, and Ying Han. "Topological Properties of Large-Scale Cortical Networks Based on Multiple Morphological Features in Amnestic Mild Cognitive Impairment." Neural Plasticity 2016 (2016): 1–14. http://dx.doi.org/10.1155/2016/3462309.
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Previous studies have demonstrated that amnestic mild cognitive impairment (aMCI) has disrupted properties of large-scale cortical networks based on cortical thickness and gray matter volume. However, it is largely unknown whether the topological properties of cortical networks based on geometric measures (i.e., sulcal depth, curvature, and metric distortion) change in aMCI patients compared with normal controls because these geometric features of cerebral cortex may be related to its intrinsic connectivity. Here, we compare properties in cortical networks constructed by six different morphological features in 36 aMCI participants and 36 normal controls. Six cortical features (3 volumetric and 3 geometric features) were extracted for each participant, and brain abnormities in aMCI were identified by cortical network based on graph theory method. All the cortical networks showed small-world properties. Regions showing significant differences mainly located in the medial temporal lobe and supramarginal and right inferior parietal lobe. In addition, we also found that the cortical networks constructed by cortical thickness and sulcal depth showed significant differences between the two groups. Our results indicated that geometric measure (i.e., sulcal depth) can be used to construct network to discriminate individuals with aMCI from controls besides volumetric measures.
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Zahra, Nida, and Muhammad Ibrahim. "On topological properties of hierarchical hypercube network based on Ve and Ev degree." Main Group Metal Chemistry 44, no. 1 (January 1, 2021): 185–93. http://dx.doi.org/10.1515/mgmc-2021-0022.
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Abstract Grid implementation is a principal unit in electrical and electronic engineering but it depends on the domain of these projects. For example, depending on the grid and the signal processing in that fields of electronic and electrical engineering, such as more abstract mathematics in signal conversion and e-transmission theory griding, etc. Provides transmission through grid nodes. Graph theory is very useful in research fields. As topological indices, there are more actual numbers associated with chemical composition complaints connected to the chemical grid with physical and chemical properties and reactions. In this paper, we expand the work to interconnected grid and examine the first Zagreb, the second Zagreb, Randic, sum-connectivity, harmonic, geometric, and atom bond connectivity exponents of hierarchical hypercube network based on vertex-edge and edge-vertex degree.
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Maksimov, Anatolii G., Arsenii D. Zavalishin, Maxim V. Abramov, and Alexander L. Tulupyev. "Chemoinformatics: Applications of Computer Science in the Analysis of Chemical Structures (on the Example of Cadmium Sulfide)." Computer tools in education, no. 4 (December 28, 2019): 44–54. http://dx.doi.org/10.32603/2071-2340-2019-4-44-54.
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The article is aimed at demonstrating the practical application of graph theory as a subsection of the theoretical foundations of computer science in solving one of the interdisciplinary problems — describing the structure of the cadmium sulfide molecule using methods and indices of chemoinformatics. The article presents the results of calculations of the atom-bond connectivity index (ABC), of the geometric and arithmetic index GA, of the generalized Randic index, GA5 and ABC4, of the Zagreb indices for the chemical graph of cadmium sulfide (CdS). Topological indices for cadmium sulfide are considered for the first time, although the task of calculating these indices is not new in itself. The relevance of the results is emphasized by the fact that cadmium sulfide is widely used in various fields, such as optoelectronics, photodetectors, photoresistors, etc.
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Jian, Feng, and Shi Dandan. "Complex Network Theory and Its Application Research on P2P Networks." Applied Mathematics and Nonlinear Sciences 1, no. 1 (January 1, 2016): 45–52. http://dx.doi.org/10.21042/amns.2016.1.00004.
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AbstractAdvances in complex networks of Peer-to-Peer (P2P) networks were reviewed and summarized. The paper outlines some important topological properties such as degree, average path length and clustering coefficient at first, and then three kinds of most important network mechanism models are introduced, including random graph model, small world model and scale-free model. A simple description about research status for P2P networks based on complex networks is made from three aspects: positive research, network mechanism model, network broadcast and control. Some developing prospects of complex networks of P2P are pointed out finally. Complex network provides new ideas and methods to deal with many complex problems including P2P networks.
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Chatterjee, Subhojyoti, and Feng Wang. "Electronic structures of hexane isomers studied using quantum mechanics and graph theory." Journal of Theoretical and Computational Chemistry 14, no. 02 (March 2015): 1550014. http://dx.doi.org/10.1142/s0219633615500145.
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Electronic and topological properties of hexane ( C 6 H 14) and its five geometric isomers are systematically studied quantum mechanically using several techniques such as positron–electron annihilation gamma-ray spectra, C1s binding energy spectra and carbon nuclear magnetic resonance (NMR) spectra, as well as information derived from graph theory. It is revealed that the Doppler-shift in the gamma-ray spectra of the hexane isomers is in the vicinity of the n-hexane molecule with small structural dependency, in agreement with the fact that the measured Doppler-shifts of other linear alkanes are in the vicinity of hexane. The present study further reveals the electronic structures of hexane isomers, which are deeply rooted into the carbon core electrons, more than mere properties in the valence space. The calculations show that the highest occupied molecular orbitals (HOMOs) of the isomers exhibit less important roles in gamma-ray spectra; whereas the electron–positron annihilation is dominated by the electrons of the lowest occupied valence orbitals (LOVOs) and other valence electrons underneath the HOMO electrons, in agreement with previous findings. The present study further reveals that the C1s binding energies of the isomers exhibit association with the nodes of the isomers using graph theory. That is, more branched carbons likely engage with larger chemical shift, which is indicated by the largest eigenvalues (LEVs) of the adjacency matrix (AM) from graph theory. The chemical shift of the carbon NMR spectra is revealed by the LEVs of the Laplacian matrix (LM) obtained from chemical graph theory.
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Gustafsson, Björn. "Vortex motion and geometric function theory: the role of connections." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 377, no. 2158 (September 30, 2019): 20180341. http://dx.doi.org/10.1098/rsta.2018.0341.
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We formulate the equations for point vortex dynamics on a closed two-dimensional Riemannian manifold in the language of affine and other kinds of connections. This can be viewed as a relaxation of standard approaches, using the Riemannian metric directly, to an approach based more on local coordinates provided with a minimal amount of extra structure. The speed of a vortex is then expressed in terms of the difference between an affine connection derived from the coordinate Robin function and the Levi–Civita connection associated with the Riemannian metric. A Hamiltonian formulation of the same dynamics is also given. The relevant Hamiltonian function consists of two main terms. One of the terms is the well-known quadratic form based on a matrix whose entries are Green and Robin functions, while the other term describes the energy contribution from those circulating flows which are not implicit in the Green functions. One main issue of the paper is a detailed analysis of the somewhat intricate exchanges of energy between these two terms of the Hamiltonian. This analysis confirms the mentioned dynamical equations formulated in terms of connections. This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.
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Koptev, A. N., and A. Yu Myasnikov. "Development of ordinary harness for aircraft onboard cable networks." VESTNIK of Samara University. Aerospace and Mechanical Engineering 18, no. 4 (January 21, 2020): 76–86. http://dx.doi.org/10.18287/2541-7533-2019-18-4-76-86.
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The article is devoted to one of the most important components of the onboard complex of aircraft equipment - the onboard cable network. The contents of the design documentation for the aircraft onboard cable network are disclosed. The statement of the problem of designing harnesses is defined in general terms. The main stages of designing aircraft onboard cable networks are described on the verbal level, as well as in the form of logical algorithms and graph-algorithms. Some theoretical aspects of designing aircraft onboard cable networks are presented. The concepts of topological space, topological structure, and continuous mapping of the harness structure into the aircraft structure are introduced. Geometric research of an ordinary cable harness of the onboard cable network led to the need to consider the harnesses as a geometric complex in the framework of combinatorial topology. An example of compiling a table of connections of ordinary harnesses for the aircraft onboard system of ultra-short wave communication is given. The rules and requirements for the information content of the table of connections of an ordinary harness to the aircraft on-board system are emphasized. Mention is made of the need to integrate ordinary harnesses into a complex one consisting of tens or even hundreds of ordinary harnesses to simplify the process of installation of the onboard cable network in the aircraft.
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Pincak, Richard, Alexander Pigazzini, Saeid Jafari, and Cenap Ozel. "The “Emerging” Reality from “Hidden” Spaces." Universe 7, no. 3 (March 23, 2021): 75. http://dx.doi.org/10.3390/universe7030075.
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The main purpose of this paper is to show and introduce some new interpretative aspects of the concept of “emergent space” as geometric/topological approach in the cosmological field. We will present some possible applications of this theory, among which the possibility of considering a non-orientable wormhole, but mainly we provide a topological interpretation, using this new approach, to M-Theory and String Theory in 10 dimensions. Further, we present some conclusions which this new interpretation suggests, and also some remarks considering a unifying approach between strings and dark matter. The approach shown in the paper considers that reality, as it appears to us, can be the “emerging” part of a more complex hidden structure. Pacs numbers: 11.25.Yb; 11.25.-w; 02.40.Ky; 02.40.-k; 04.50.-h; 95.35.+d.
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Chen, Yufei, Xiaodong Yue, Caiming Zhong, and Gang Wang. "Functional Region Annotation of Liver CT Image Based on Vascular Tree." BioMed Research International 2016 (2016): 1–13. http://dx.doi.org/10.1155/2016/5428737.
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Anatomical analysis of liver region is critical in diagnosis and treatment of liver diseases. The reports of liver region annotation are helpful for doctors to precisely evaluate liver system. One of the challenging issues is to annotate the functional regions of liver through analyzing Computed Tomography (CT) images. In this paper, we propose a vessel-tree-based liver annotation method for CT images. The first step of the proposed annotation method is to extract the liver region including vessels and tumors from the CT scans. And then a 3-dimensional thinning algorithm is applied to obtain the spatial skeleton and geometric structure of liver vessels. With the vessel skeleton, the topology of portal veins is further formulated by a directed acyclic graph with geometrical attributes. Finally, based on the topological graph, a hierarchical vascular tree is constructed to divide the liver into eight segments according to Couinaud classification theory and thereby annotate the functional regions. Abundant experimental results demonstrate that the proposed method is effective for precise liver annotation and helpful to support liver disease diagnosis.
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Udrişte, Constantin, Mădălina Constantinescu, Ionel Ţevy, and Oltin Dogaru. "Dualities in Nonholonomic Optimization." Annals of West University of Timisoara - Mathematics and Computer Science 54, no. 2 (December 1, 2016): 149–66. http://dx.doi.org/10.1515/awutm-2016-0020.
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Abstract This article deals with optimizing problems whose restrictions are nonholonomic. The central issue relates to dual nonholonomic programs (what they mean and how they are solved?) when the nonholonomic constraints are given by Pfaff equations. We emphasize that nonholonomic critical points are not the classical ones and that the nonholonomic Lagrange multipliers are not the classical (holonomic) Lagrange multipliers. Topological significance of Lagrange multipliers and dual function theory introduced by EDO and EDP are key results. Also new Riemannian geometries attached to a given nonholonomic constrained optimization problem are introduced. The original results are surprising and include: (i) aspects derived from the Vranceanu theory of nonholonomic manifolds, and from the geometric distributions theory, (ii) optimal problems in Darboux canonical coordinates.
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BOI, LUCIANO. "IDEAS OF GEOMETRIZATION, GEOMETRIC INVARIANTS OF LOW-DIMENSIONAL MANIFOLDS, AND TOPOLOGICAL QUANTUM FIELD THEORIES." International Journal of Geometric Methods in Modern Physics 06, no. 05 (August 2009): 701–57. http://dx.doi.org/10.1142/s0219887809003783.
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The aim of the first part of this paper is to make some reflections on the role of geometrical and topological concepts in the developments of theoretical physics, especially in gauge theory and string theory, and we show the great significance of these concepts for a better understanding of the dynamics of physics. We will claim that physical phenomena essentially emerge from the geometrical and topological structure of space–time. The attempts to solve one of the central problems in 20th theoretical physics, i.e. how to combine gravity and the other forces into an unitary theoretical explanation of the physical world, essentially depends on the possibility of building a new geometrical framework conceptually richer than Riemannian geometry. In fact, it still plays a fundamental role in non-Abelian gauge theories and in superstring theory, thanks to which a great variety of new mathematical structures has emerged. The scope of this presentation is to highlight the importance of these mathematical structures for theoretical physics. A very interesting hypothesis is that the global topological properties of the manifold's model of space–time play a major role in quantum field theory (QFT) and that, consequently, several physical quantum effects arise from the nonlocal changing metrical and topological structure of these manifold. Thus the unification of general relativity and quantum theory require some fundamental breakthrough in our understanding of the relationship between space–time and quantum process. In particular the superstring theories lead to the guess that the usual structure of space–time at the quantum scale must be dropped out from physical thought. Non-Abelian gauge theories satisfy the basic physical requirements pertaining to the symmetries of particle physics because they are geometric in character. They profoundly elucidate the fundamental role played by bundles, connections, and curvature in explaining the essential laws of nature. Kaluza–Klein theories and more remarkably superstring theory showed that space–time symmetries and internal (quantum) symmetries might be unified through the introduction of new structures of space with a different topology. This essentially means, in our view, that "hidden" symmetries of fundamental physics can be related to the phenomenon of topological change of certain class of (presumably) nonsmooth manifolds. In the second part of this paper, we address the subject of topological quantum field theories (TQFTs), which constitute a remarkably important meeting ground for physicists and mathematicians. TQFTs can be used as a powerful tool to probe geometry and topology in low dimensions. Chern–Simons theories, which are examples of such field theories, provide a field theoretic framework for the study of knots and links in three dimensions. These are rare examples of QFTs which can be exactly (nonperturbatively) and explicitly solved. Abelian Chern–Simons theory provides a field theoretic interpretation of the linking and self-linking numbers of a link (i.e. the union of a finite number of disjoint knots). In non-Abelian theories, vacuum expectation values of Wilson link operators yield a class of polynomial link invariants; the simplest of them is the well-known Jones polynomial. Powerful methods for complete analytical and nonperturbative computation of these knot and link invariants have been developed. From these invariants for unoriented and framed links in S3, an invariant for any three-manifold can be easily constructed by exploiting the Lickorish–Wallace surgery presentation of three-manifolds. This invariant up to a normalization is the partition function of the Chern–Simons field theory. Even perturbative analysis of Chern–Simons theories are rich in their mathematical structure; these provide a field theoretic interpretation of Vassiliev knot invariants. In Donaldson–Witten theory perturbative methods have proved their relations to Donaldson invariants. Nonperturbative methods have been applied after the work by Seiberg and Witten on N = 2 supersymmetric Yang–Mills theory. The outcome of this application is a totally unexpected relation between Donaldson invariants and a new set of topological invariants called Seiberg–Witten invariants. Not only in mathematics, Chern–Simons theories find important applications in three- and four-dimensional quantum gravity also. Work on TQFT suggests that a quantum gravity theory can be formulated in three-dimensional space–time. Attempts have been made in the last years to formulate a theory of quantum gravity in four-dimensional space–time using "spin networks" and "spin foams". More generally, the developments of TQFTs represent a sort of renaissance in the relation between geometry and physics. The most important (new) feature of present developments is that links are being established between quantum physics and topology. Maybe this link essentially rests on the fact that both quantum theory and topology are characterized by discrete phenomena emerging from a continuous background. One very interesting example is the super-symmetric quantum mechanics theory, which has a deep geometric meaning. In the Witten super-symmetric quantum mechanics theory, where the Hamiltonian is just the Hodge–Laplacian (whereas the quantum Hamiltonian corresponding to a classical particle moving on a Riemannian manifold is just the Laplace–Beltrami differential operator), differential forms are bosons or fermions depending on the parity of their degrees. Witten went to introduce a modified Hodge–Laplacian, depending on a real-valued function f. He was then able to derive the Morse theory (relating critical points of f to the Betti numbers of the manifold) by using the standard limiting procedures relating the quantum and classical theories. Super-symmetric QFTs essentially should be viewed as the differential geometry of certain infinite-dimensional manifolds, including the associated analysis (e.g. Hodge theory) and topology (e.g. Betti numbers). A further comment is that the QFTs of interest are inherently nonlinear, but the nonlinearities have a natural origin, e.g. coming from non-Abelian Lie groups. Moreover there is usually some scaling or coupling parameter in the theory which in the limit relates to the classical theory. Fundamental topological aspects of such a quantum theory should be independent of the parameters and it is therefore reasonable to expect them to be computable (in some sense) by examining the classical limit. This means that such topological information is essentially robust and should be independent of the fine analytical details (and difficulties) of the full quantum theory. In the last decade much effort has been done to use these QFTs as a conceptual tool to suggest new mathematical results. In particular, they have led to spectacular progress in our understanding of geometry in low dimensions. It is most likely no accident that the usual QFTs can only be renormalized in (space–time) dimensions ≤4, and this is precisely the range in which difficult phenomena arise leading to deep and beautiful theories (e.g. the work of Thurston in three dimensions and Donaldson in four dimensions). It now seems clear that the way to investigate the subtleties of low-dimensional manifolds is to associate to them suitable infinite-dimensional manifolds (e.g. spaces of connections) and to study these by standard linear methods (homology, etc.). In other words we use QFT as a refined tool to study low-dimensional manifolds.
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He, Zeyuan, and Simon D. Guest. "On rigid origami I: piecewise-planar paper with straight-line creases." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, no. 2232 (December 2019): 20190215. http://dx.doi.org/10.1098/rspa.2019.0215.
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Origami (paper folding) is an effective tool for transforming two-dimensional materials into three-dimensional structures, and has been widely applied to robots, deployable structures, metamaterials, etc. Rigid origami is an important branch of origami where the facets are rigid, focusing on the kinematics of a panel-hinge model. Here, we develop a theoretical framework for rigid origami, and show how this framework can be used to connect rigid origami and its cognate areas, such as the rigidity theory, graph theory, linkage folding and computer science. First, we give definitions regarding fundamental aspects of rigid origami, then focus on how to describe the configuration space of a creased paper. The shape and 0-connectedness of the configuration space are analysed using algebraic, geometric and numeric methods. In the algebraic part, we study the tangent space and generic rigid-foldability based on the polynomial nature of constraints for a panel-hinge system. In the geometric part, we analyse corresponding spherical linkage folding and discuss the special case when there is no cycle in the interior of a crease pattern. In the numeric part, we review methods to trace folding motion and avoid self-intersection. Our results will be instructive for the mathematical and engineering design of origami structures.
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Nespolo, Massimo. "Does mathematical crystallography still have a role in the XXI century?" Acta Crystallographica Section A Foundations of Crystallography 64, no. 1 (December 21, 2007): 96–111. http://dx.doi.org/10.1107/s0108767307044625.
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Mathematical crystallography is the branch of crystallography dealing specifically with the fundamental properties of symmetry and periodicity of crystals, topological properties of crystal structures, twins, modular and modulated structures, polytypes and OD structures, as well as the symmetry aspects of phase transitions and physical properties of crystals. Mathematical crystallography has had its most evident success with the development of the theory of space groups at the end of the XIX century; since then, it has greatly enlarged its applications, but crystallographers are not always familiar with the developments that followed, partly because the applications sometimes require some additional background that the structural crystallographer does not always possess (as is the case, for example, in graph theory). The knowledge offered by mathematical crystallography is at present only partly mirrored inInternational Tables for Crystallographyand is sometimes still enshrined in more specialist texts and publications. To cover this communication gap is one of the tasks of the IUCr Commission on Mathematical and Theoretical Crystallography (MaThCryst).
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Tang, Jinfu, Suyu Zhong, Yaojing Chen, Kewei Chen, Junying Zhang, Gaolang Gong, Adam S. Fleisher, Yong He, and Zhanjun Zhang. "Aberrant White Matter Networks Mediate Cognitive Impairment in Patients with Silent Lacunar Infarcts in Basal Ganglia Territory." Journal of Cerebral Blood Flow & Metabolism 35, no. 9 (April 15, 2015): 1426–34. http://dx.doi.org/10.1038/jcbfm.2015.67.
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Silent lacunar infarcts, which are present in over 20% of healthy elderly individuals, are associated with subtle deficits in cognitive functions. However, it remains largely unclear how these silent brain infarcts lead to cognitive deficits and even dementia. Here, we used diffusion tensor imaging tractography and graph theory to examine the topological organization of white matter networks in 27 patients with silent lacunar infarcts in the basal ganglia territory and 30 healthy controls. A whole-brain white matter network was constructed for each subject, where the graph nodes represented brain regions and the edges represented interregional white matter tracts. Compared with the controls, the patients exhibited a significant reduction in local efficiency and global efficiency. In addition, a total of eighteen brain regions showed significantly reduced nodal efficiency in patients. Intriguingly, nodal efficiency–behavior associations were significantly different between the two groups. The present findings provide new aspects into our understanding of silent infarcts that even small lesions in subcortical brain regions may affect large-scale cortical white matter network, as such may be the link between subcortical silent infarcts and the associated cognitive impairments. Our findings highlight the need for network-level neuroimaging assessment and more medical care for individuals with silent subcortical infarcts.