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Journal articles on the topic 'Algebraic intersection'

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Published: 4 May 2024

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1

Geske, Christian. "Algebraic intersection spaces." Journal of Topology and Analysis 12, no. 04 (January 9, 2019): 1157–94. http://dx.doi.org/10.1142/s1793525319500778.

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We define a variant of intersection space theory that applies to many compact complex and real analytic spaces [Formula: see text], including all complex projective varieties; this is a significant extension to a theory which has so far only been shown to apply to a particular subclass of spaces with smooth singular sets. We verify existence of these so-called algebraic intersection spaces and show that they are the (reduced) chain complexes of known topological intersection spaces in the case that both exist. We next analyze “local duality obstructions,” which we can choose to vanish, and verify that algebraic intersection spaces satisfy duality in the absence of these obstructions. We conclude by defining an untwisted algebraic intersection space pairing, whose signature is equal to the Novikov signature of the complement in [Formula: see text] of a tubular neighborhood of the singular set.

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2

Patrikalakis, N. M., and P. V. Prakash. "Surface Intersections for Geometric Modeling." Journal of Mechanical Design 112, no. 1 (March 1, 1990): 100–107. http://dx.doi.org/10.1115/1.2912565.

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Evaluation of planar algebraic curves arises in the context of intersections of algebraic surfaces with piecewise continuous rational polynomial parametric surface patches useful in geometric modeling. We address a method of evaluating these curves of intersection that combines the advantageous features of analytic representation of the governing equation of the algebraic curve in the Bernstein basis within a rectangular domain, adaptive subdivision and polyhedral faceting techniques, and the computation of turning and singular points, to provide the basis for a reliable and efficient solution procedure. Using turning and singular points, the intersection problem can be partitioned into subdomains that can be processed independently and which involve intersection segments that can be traced with faceting methods. This partitioning and the tracing of individual segments is carried out using an adaptive subdivision algorithm for Bezier/B-spline surfaces followed by Newton correction of the approximation. The method has been successfully tested in tracing complex algebraic curves and in solving actual intersection problems with diverse features.

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3

Brasselet, J. P. "Intersection of algebraic cycles." Journal of Mathematical Sciences 82, no. 5 (December 1996): 3625–32. http://dx.doi.org/10.1007/bf02362566.

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4

LÓPEZ, HIRAM H., and RAFAEL H. VILLARREAL. "COMPLETE INTERSECTIONS IN BINOMIAL AND LATTICE IDEALS." International Journal of Algebra and Computation 23, no. 06 (September 2013): 1419–29. http://dx.doi.org/10.1142/s0218196713500288.

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For the family of graded lattice ideals of dimension 1, we establish a complete intersection criterion in algebraic and geometric terms. In positive characteristic, it is shown that all ideals of this family are binomial set-theoretic complete intersections. In characteristic zero, we show that an arbitrary lattice ideal which is a binomial set-theoretic complete intersection is a complete intersection.

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5

MANOCHA, DINESH, and JOHN F. CANNY. "A NEW APPROACH FOR SURFACE INTERSECTION." International Journal of Computational Geometry & Applications 01, no. 04 (December 1991): 491–516. http://dx.doi.org/10.1142/s0218195991000311.

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Evaluating the intersection of two rational parametric surfaces is a recurring operation in solid modeling. However, surface intersection is not an easy problem and continues to be an active topic of research. The main reason lies in the fact that any good surface intersection technique has to balance three conflicting goals of accuracy, robustness and efficiency. In this paper, we formulate the problems of curve and surface intersections using algebraic sets in a higher dimensional space. Using results from Elimination theory, we project the algebraic set to a lower dimensional space. The projected set can be expressed as a matrix determinant. The matrix itself, rather than its symbolic determinant, is used as the representation for the algebraic set in the lower dimensional space. This is a much more compact and efficient representation. Given such a representation, we perform matrix operations for evaluation and use results from linear algebra for geometric operations on the intersection curve. Most of the operations involve evaluating numeric determinants and computing the rank, kernel and eigenvalues of matrices. The accuracy of such operations can be improved by pivoting or other numerical techniques. We use this representation for inversion operation, computing the intersection of curves and surfaces and tracing the intersection curve of two surfaces in lower dimension.

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6

Yokura, Shoji. "Algebraic cycles and intersection homology." Proceedings of the American Mathematical Society 103, no. 1 (January 1, 1988): 41. http://dx.doi.org/10.1090/s0002-9939-1988-0938641-x.

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7

Abhyankar, Shreeram S., Srinivasan Chandrasekar, and Vijaya Chandru. "Improper intersection of algebraic curves." ACM Transactions on Graphics 9, no. 2 (April 1990): 147–59. http://dx.doi.org/10.1145/78956.78957.

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8

Mandal, Satya, and Yong Yang. "Intersection theory of algebraic obstructions." Journal of Pure and Applied Algebra 214, no. 12 (December 2010): 2279–93. http://dx.doi.org/10.1016/j.jpaa.2010.02.027.

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9

Levine, Marc. "Intersection theory in algebraic cobordism." Journal of Pure and Applied Algebra 221, no. 7 (July 2017): 1645–90. http://dx.doi.org/10.1016/j.jpaa.2016.12.022.

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10

Sederberg, T. W. "Algorithm for algebraic curve intersection." Computer-Aided Design 21, no. 9 (November 1989): 547–54. http://dx.doi.org/10.1016/0010-4485(89)90015-8.

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11

Abhyankar, Shreeram S., Srinivasan Chandrasekar, and Vijaya Chandru. "Intersection of algebraic space curves." Discrete Applied Mathematics 31, no. 2 (April 1991): 81–96. http://dx.doi.org/10.1016/0166-218x(91)90062-2.

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12

HOHMEYER, MICHAEL E. "A SURFACE INTERSECTION ALGORITHM BASED ON LOOP DETECTION." International Journal of Computational Geometry & Applications 01, no. 04 (December 1991): 473–90. http://dx.doi.org/10.1142/s021819599100030x.

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A robust and efficient surface intersection algorithm that is implementable in floating point arithmetic, accepts surfaces algebraic or otherwise and which operates without human supervision is critical to boundary representation solid modeling. To the author's knowledge, no such algorithms has been developed. All tolerance-based subdivision algorithms will fail on surfaces with sufficiently small intersections. Algebraic techniques, while promising robustness, are presently too slow to be practical and do not accept non-algebraic surfaces. Algorithms based on loop detection hold promise. They do not require tolerances except those associated with machine associated with machine arithmetic, and can handle any surface for which there is a method to construct bounds on the surface and its Gauss map. Published loop detection algorithms are, however, still too slow and do not deal with singularities. We present a new loop detection criterion and discuss its use in a surface intersection algorithms. The algorithm, like other loop detection based intersection algorithms, subdivides the surfaces into pairs of sub-patches which do not intersect in any closed loops. This paper presents new strategies for subdividing surfaces in a way that causes the algorithms to run quickly even when the intersection curve(s) contain(s) singularities.

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13

Lai, Kailing, Fanning Meng, and Huanqi He. "The Intersection Multiplicity of Intersection Points over Algebraic Curves." International Journal of Mathematics and Mathematical Sciences 2023 (May 5, 2023): 1–13. http://dx.doi.org/10.1155/2023/6346685.

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In analytic geometry, Bézout’s theorem stated the number of intersection points of two algebraic curves and Fulton introduced the intersection multiplicity of two curves at some point in local case. It is meaningful to give the exact expression of the intersection multiplicity of two curves at some point. In this paper, we mainly express the intersection multiplicity of two curves at some point in R 2 and A K 2 under fold point, where char K = 0 . First, we give a sufficient and necessary condition for the coincidence of the intersection multiplicity of two curves at some point and the smallest degree of the terms of these two curves in R 2 . Furthermore, we show that two different definitions of intersection multiplicity of two curves at a point in A K 2 are equivalent and then give the exact expression of the intersection multiplicity of two curves at some point in A K 2 under fold point.

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14

Girsh, A. "Intersection Operation on a Complex Plane." Geometry & Graphics 9, no. 1 (July 22, 2021): 20–28. http://dx.doi.org/10.12737/2308-4898-2021-9-1-20-28.

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Two plane algebraic curves intersect at the actual intersection points of these curves’ graphs. In addition to real intersection points, algebraic curves can also have imaginary intersection points. The total number of curves intersection points is equal to the product of their orders mn. The number of imaginary intersection points can be equal to or part of mn. The position of the actual intersection points is determined by the graphs of the curves, but the imaginary intersection points do not lie on the graphs of these curves, and their position on the plane remains unclear. This work aims to determine the geometry of imaginary intersection points, introduces into consideration the concept of imaginary complement for these algebraic curves in the intersection operation, determines the form of imaginary complements, which intersect at imaginary points. The visualization of imaginary complements clarifies the curves intersection picture, and the position of the imaginary intersection points becomes expected.

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15

Lin, Xiaojin. "Intersection complex via residue." JUSTC 52, no. 7 (2022): 3. http://dx.doi.org/10.52396/justc-2021-0263.

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16

Catanzaro, Michael J., Vladimir Y. Chernyak, and John R. Klein. "Exciton scattering via algebraic topology." Journal of Topology and Analysis 11, no. 02 (June 2019): 251–72. http://dx.doi.org/10.1142/s1793525319500110.

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This paper introduces an intersection theory problem for maps into a smooth manifold equipped with a stratification. We investigate the problem in the special case when the target is the unitary group [Formula: see text] and the domain is a circle. The first main result is an index theorem that equates a global intersection index with a finite sum of locally defined intersection indices. The local indices are integers arising from the geometry of the stratification. The result is used to study a well-known problem in chemical physics, namely, the problem of enumerating the electronic excitations (excitons) of a molecule equipped with scattering data.

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17

ATAGÜN, Akın Osman, and Hüseyin KAMACI. "Set-generated soft subrings of rings." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 71, no. 4 (December 30, 2022): 993–1006. http://dx.doi.org/10.31801/cfsuasmas.1013172.

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This paper focuses on the set-oriented operations and set-oriented algebraic structures of soft sets. Relatedly, in this paper, firstly some essential properties of $\alpha$-intersection of soft set are investigated, where $\alpha$ is a non-empty subset of the universal set. Later, by using $\alpha$-intersection of soft set, the notion of set-generated soft subring of a ring is introduced. The generators of soft intersections and products of soft subrings are given. Some related properties about generators of soft subrings are investigated and illustrated by several examples.

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18

NAZIR, SHAHEEN. "ON THE INTERSECTION OF RATIONAL TRANSVERSAL SUBTORI." Journal of the Australian Mathematical Society 86, no. 2 (April 2009): 221–31. http://dx.doi.org/10.1017/s1446788708000372.

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AbstractWe show that under a suitable transversality condition, the intersection of two rational subtori in an algebraic torus (ℂ*)n is a finite group which can be determined using the torsion part of some associated lattice. We also give applications to the study of characteristic varieties of smooth complex algebraic varieties. As an example we discuss A. Suciu’s line arrangement, the so-called deleted B3-arrangement.

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19

DICKENSTEIN, ALICIA, and ENRIQUE A. TOBIS. "INDEPENDENT SETS FROM AN ALGEBRAIC PERSPECTIVE." International Journal of Algebra and Computation 22, no. 02 (March 2012): 1250014. http://dx.doi.org/10.1142/s0218196711006819.

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In this paper, we study the basic problem of counting independent sets in a graph and, in particular, the problem of counting antichains in a finite poset, from an algebraic perspective. We show that neither independence polynomials of bipartite Cohen–Macaulay graphs nor Hilbert series of initial ideals of radical zero-dimensional complete intersections ideals, can be evaluated in polynomial time, unless #P = P. Moreover, we present a family of radical zero-dimensional complete intersection ideals JP associated to a finite poset P, for which we describe a universal Gröbner basis. This implies that the bottleneck in computing the dimension of the quotient by JP (that is, the number of zeros of JP) using Gröbner methods lies in the description of the standard monomials.

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20

Khan, W. A., and B. Davvaz. "Soft Intersection Nearsemirings and Its Algebraic Applications." Lobachevskii Journal of Mathematics 41, no. 3 (March 2020): 362–72. http://dx.doi.org/10.1134/s1995080220030105.

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21

Joshua, Roy. "Higher Intersection Theory on Algebraic Stacks: I." K-Theory 27, no. 2 (October 2002): 133–95. http://dx.doi.org/10.1023/a:1021116524762.

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22

Joshua, Roy. "Higher Intersection Theory on Algebraic Stacks: II." K-Theory 27, no. 3 (November 2002): 197–244. http://dx.doi.org/10.1023/a:1021648227488.

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23

Elber, Gershon, Tom Grandine, and Myung-Soo Kim. "Surface self-intersection computation via algebraic decomposition." Computer-Aided Design 41, no. 12 (December 2009): 1060–66. http://dx.doi.org/10.1016/j.cad.2009.07.008.

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24

Wu, Jinming. "Real intersection points of piecewise algebraic curves." Applied Mathematics Letters 25, no. 10 (October 2012): 1299–303. http://dx.doi.org/10.1016/j.aml.2011.11.032.

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25

Gonzales, Richard P. "Algebraic rational cells and equivariant intersection theory." Mathematische Zeitschrift 282, no. 1-2 (September 22, 2015): 79–97. http://dx.doi.org/10.1007/s00209-015-1533-5.

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26

Parkin, Spencer T. "The Intersection of Rays with Algebraic Surfaces." Advances in Applied Clifford Algebras 24, no. 3 (April 12, 2014): 809–15. http://dx.doi.org/10.1007/s00006-014-0460-6.

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27

Cheng, Jin-San, Bingwei Zhang, Yikun Xiao, and Ming Li. "Topology driven approximation to rational surface-surface intersection via interval algebraic topology analysis." ACM Transactions on Graphics 42, no. 4 (July 26, 2023): 1–16. http://dx.doi.org/10.1145/3592452.

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Computing the intersection between two parametric surfaces (SSI) is one of the most fundamental problems in geometric and solid modeling. Maintaining the SSI topology is critical to its computation robustness. We propose a topology-driven hybrid symbolic-numeric framework to approximate rational parametric surface-surface intersection (SSI) based on a concept of interval algebraic topology analysis (IATA) , which configures within a 4D interval box the SSI topology. We map the SSI topology to an algebraic system's solutions within the framework, classify and enumerate all topological cases as a mixture of four fundamental cases (or their specific sub-cases). Various complicated topological situations are covered, such as cusp points or curves, tangent points (isolated or not) or curves, tiny loops, self-intersections, or their mixtures. The theoretical formulation is also implemented numerically using advanced real solution isolation techniques, and computed within a topology-driven framework which maximally utilizes the advantages of the topology maintenance of algebraic analysis, the robustness of iterative subdivision, and the efficiency of forward marching. The approach demonstrates improved robustness under benchmark topological cases when compared with available open-source and commercial solutions, including IRIT, SISL, and Parasolid.

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28

Coltoiu, Mihnea. "On Barth’s conjecture concerining." Nagoya Mathematical Journal 145 (March 1997): 99–123. http://dx.doi.org/10.1017/s0027763000006127.

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A classical, still unsolved problem, is the following: is every connected curve A ⊂ P3 a set-theoretic complete intersection? It is clear that if A is a set-theoretic complete intersection then:a) The algebraic cohomology groups vanish for every coherent algebraic sheaf on P3.b) The analytic cohomology groups vanish for every coherent analytic sheaf on P3\A.

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29

Gao, Xiao-Shan, Wei Li, and Chun-Ming Yuan. "Intersection theory in differential algebraic geometry: Generic intersections and the differential Chow form." Transactions of the American Mathematical Society 365, no. 9 (February 12, 2013): 4575–632. http://dx.doi.org/10.1090/s0002-9947-2013-05633-4.

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30

Klos, Basia. "Kernels of Algebraic Curvature Tensors of Symmetric and Skew-Symmetric Builds." PUMP Journal of Undergraduate Research 6 (August 26, 2023): 301–16. http://dx.doi.org/10.46787/pump.v6i0.3456.

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The kernel of an algebraic curvature tensor is a fundamental subspace that can be used to distinguish between different algebraic curvature tensors. Kernels of algebraic curvature tensors built only of canonical algebraic curvature tensors of a single build have been studied in detail. We consider the kernel of an algebraic curvature tensor R that is a sum of canonical algebraic curvature tensors of symmetric and skew-symmetric build. An obvious way to ensure that the kernel of R is nontrivial is to choose the involved bilinear forms such that the intersection of their kernels is nontrivial. We present a construction wherein this intersection is trivial but the kernel of R is nontrivial. We also show how many bilinear forms satisfying certain conditions are needed in order for R to have a kernel of any allowable dimension.

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31

Jana, Chiranjibe, Madhumangal Pal, Faruk Karaaslan, and Aslihan Sezgi̇n. "(α,β)-Soft Intersectional Rings and Ideals with their Applications." New Mathematics and Natural Computation 15, no. 02 (June 20, 2019): 333–50. http://dx.doi.org/10.1142/s1793005719500182.

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Molodtsov initiated the soft set theory, providing a general mathematical framework for handling uncertainties that we encounter in various real-life problems. The main object of this paper is to lay a foundation for providing a new soft algebraic tool for considering many problems that contain uncertainties. In this paper, we introduce a new kind of soft ring structure called [Formula: see text]-soft-intersectional ring based on some results of soft sets and intersection operations on sets. We also define [Formula: see text]-soft-intersectional ideal and [Formula: see text]-soft-intersectional subring, and investigate some of their properties using these new concepts. We obtain some results in ring theory based on [Formula: see text]-soft intersection sense and its application in ring structures. Furthermore, we provide relationships between soft-intersectional ring and [Formula: see text]-soft-intersectional ring, soft-intersectional ideal and [Formula: see text]-soft-intersectional ideal.

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32

Kowalski, Piotr. "AX–SCHANUEL CONDITION IN ARBITRARY CHARACTERISTIC." Journal of the Institute of Mathematics of Jussieu 18, no. 06 (November 8, 2017): 1157–213. http://dx.doi.org/10.1017/s1474748017000378.

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We prove a positive characteristic version of Ax’s theorem on the intersection of an algebraic subvariety and an analytic subgroup of an algebraic group [Ax, Some topics in differential algebraic geometry. I. Analytic subgroups of algebraic groups, Amer. J. Math. 94 (1972), 1195–1204]. Our result is stated in a more general context of a formal map between an algebraic variety and an algebraic group. We derive transcendence results of Ax–Schanuel type.

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33

DEVELİ, Ülkü, and Filiz ÇITAK. "A New Perspective on $k$-Ideals of a Semiring via Soft Intersection Ideals." Journal of New Theory, no. 41 (December 31, 2022): 18–34. http://dx.doi.org/10.53570/jnt.1145507.

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In recent years, soft sets have become popular in various fields. For this reason, many studies have been carried out in the field of algebra. In this study, soft intersection k-ideals are defined with the help of a semiring, and some algebraic structures are examined. Moreover, the quotient rings are defined by k-semiring. Isomorphism theorems are examined by quotient rings. Finally, some algebraic properties are investigated by defining soft intersection maximal k-ideals.

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34

Ma, Xueling, and Hee Sik Kim. "(M,N)-Soft Intersection BL-Algebras and Their Congruences." Scientific World Journal 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/461060.

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The purpose of this paper is to give a foundation for providing a new soft algebraic tool in considering many problems containing uncertainties. In order to provide these new soft algebraic structures, we discuss a new soft set-(M, N)-soft intersection set, which is a generalization of soft intersection sets. We introduce the concepts of (M, N)-SI filters of BL-algebras and establish some characterizations. Especially, (M, N)-soft congruences in BL-algebras are concerned.

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35

LAFACE, ROBERTO, and PIOTR POKORA. "LOCAL NEGATIVITY OF SURFACES WITH NON-NEGATIVE KODAIRA DIMENSION AND TRANSVERSAL CONFIGURATIONS OF CURVES." Glasgow Mathematical Journal 62, no. 1 (January 18, 2019): 123–35. http://dx.doi.org/10.1017/s0017089518000575.

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AbstractWe give a bound on the H-constants of configurations of smooth curves having transversal intersection points only on an algebraic surface of non-negative Kodaira dimension. We also study in detail configurations of lines on smooth complete intersections $X \subset \mathbb{P}_{\mathbb{C}}^{n + 2}$ of multi-degree d = (d1, …, dn), and we provide a sharp and uniform bound on their H-constants, which only depends on d.

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36

Trenkler, Götz, and Dietrich Trenkler. "Intersection of three planes revisited – an algebraic approach." International Journal of Mathematical Education in Science and Technology 48, no. 2 (September 12, 2016): 285–89. http://dx.doi.org/10.1080/0020739x.2016.1228016.

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37

Weber, Andrzej. "Formality of equivariant intersection cohomology of algebraic varieties." Proceedings of the American Mathematical Society 131, no. 9 (April 23, 2003): 2633–38. http://dx.doi.org/10.1090/s0002-9939-03-07138-7.

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38

Gadgil, Siddhartha, and Suhas Pandit. "Algebraic and geometric intersection numbers for free groups." Topology and its Applications 156, no. 9 (May 2009): 1615–19. http://dx.doi.org/10.1016/j.topol.2008.12.039.

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39

Hübl, R., and E. Kunz. "On the intersection of algebraic curves and hypersurfaces." Mathematische Zeitschrift 227, no. 2 (February 1998): 263–78. http://dx.doi.org/10.1007/pl00004375.

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40

M. Srividya, D. Vidhya, and G. Jayalalitha. "A STUDY ON QUASI UNIFORM DYNAMICAL SYSTEM." International Journal of Scientific Research in Modern Science and Technology 2, no. 12 (December 29, 2023): 31–37. http://dx.doi.org/10.59828/ijsrmst.v2i12.167.

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This paper introduces a new concept called fuzzy rough algebraic quasi uniformity on a fuzzy rough TM dynamical system. It is highlighting the properties of fuzzy rough quasi uniform dynamical system space. This paper proves that any finite intersection of fuzzy rough quasi uniform open algebraic is a fuzzy rough quasi uniform open algebraic and any finite union of fuzzy rough quasi uniform closed algebraic is a fuzzy rough quasi uniform closed algebraic.

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41

Ma, Xueling, Jianming Zhan, and Bijan Davvaz. "Applications of soft intersection sets to hemirings via SI-h-bi-ideals and SI-h-quasi-ideals." Filomat 30, no. 8 (2016): 2295–313. http://dx.doi.org/10.2298/fil1608295m.

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The aim of this paper is to lay a foundation for providing a soft algebraic tool in considering many problems that contains uncertainties. In order to provide these soft algebraic structures, we introduce the concepts of SI-h-bi-ideals and SI-h-quasi-ideals of hemirings. The relationships between these kinds of soft intersection h-ideals are established. Finally, some characterizations of h-hemiregular, h-intra-hemiregular and h-quasi-hemiregular hemirings are investigated by these kinds of soft intersection h-ideals.

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42

HUMPHRIES, STEPHEN P. "INTERSECTION THEORIES AND CHEBYSHEV POLYNOMIALS III." Journal of Algebra and Its Applications 08, no. 01 (February 2009): 53–81. http://dx.doi.org/10.1142/s0219498809003187.

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We investigate the properties of certain operators that were used in previous papers to create geometric and algebraic intersection number functions. Using these operators we show how to construct a general intersection theory and then we investigate properties of certain polynomials that arise naturally in this context. We indicate connections to twin primes.

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43

ABTAHI, F., H. G. AMINI, H. A. LOTFI, and A. REJALI. "AN ARBITRARY INTERSECTION OF Lp-SPACES." Bulletin of the Australian Mathematical Society 85, no. 3 (February 16, 2012): 433–45. http://dx.doi.org/10.1017/s0004972711003510.

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AbstractFor a locally compact group G and an arbitrary subset J of [1,∞], we introduce ILJ(G) as a subspace of ⋂ p∈JLp(G) with some norm to make it a Banach space. Then, for some special choice of J, we investigate some topological and algebraic properties of ILJ(G) as a Banach algebra under a convolution product.

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44

Liu, Yin, and Paul Zsombor-Murray. "INTERSECTION CURVES BETWEEN QUADRIC SURFACES OF REVOLUTION." Transactions of the Canadian Society for Mechanical Engineering 19, no. 4 (December 1995): 435–53. http://dx.doi.org/10.1139/tcsme-1995-0023.

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A novel algebraic method is used to determine the intersection between two biquadratic surfaces. The underlying idea is to deal with functional decomposition by reducing the order of high-degree functions and thereby overcome the problems common to other methods.

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45

Laface, Roberto, and Piotr Pokora. "Towards the weighted Bounded Negativity Conjecture for blow-ups of algebraic surfaces." manuscripta mathematica 163, no. 3-4 (December 11, 2019): 361–73. http://dx.doi.org/10.1007/s00229-019-01157-2.

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AbstractIn the present paper we focus on a weighted version of the Bounded Negativity Conjecture, which predicts that for every smooth projective surface in characteristic zero the self-intersection numbers of reduced and irreducible curves are bounded from below by a function depending on the intesection of curve with an arbitrary big and nef line bundle that is positive on the curve. We gather evidence for this conjecture by showing various bounds on the self-intersection number of curves in an algebraic surface. We focus our attention on blow-ups of algebraic surfaces, which have so far been neglected.

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46

Guo, Hongfeng, Bing Xing, Ziwei Ming, and Jun-E. Feng. "Algebraic Representation of Topologies on a Finite Set." Mathematics 10, no. 7 (April 2, 2022): 1143. http://dx.doi.org/10.3390/math10071143.

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Since the 1930s, topological counting on finite sets has been an interesting work so as to enumerate the number of corresponding order relations on the sets. Starting from the semi-tensor product (STP), we give the expression of the relationship between subsets of finite sets from the perspective of algebra. Firstly, using the STP of matrices, we present the algebraic representation of the subset and complement of finite sets and corresponding structure matrices. Then, we investigate respectively the relationship between the intersection and union and intersection and minus of structure matrices. Finally, we provide an algorithm to enumerate the numbers of topologies on a finite set based on the above theorems.

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47

Boratynski, M. "Locally Complete Intersection Multiple Structures on Smooth Algebraic Curves." Proceedings of the American Mathematical Society 115, no. 4 (August 1992): 877. http://dx.doi.org/10.2307/2159329.

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Hakopian, H., and D. Voskanyan. "On the Intersection Points of Two Plane Algebraic Curves." Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) 54, no. 2 (March 2019): 90–97. http://dx.doi.org/10.3103/s1068362319020055.

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Boratyński, M. "Locally complete intersection multiple structures on smooth algebraic curves." Proceedings of the American Mathematical Society 115, no. 4 (April 1, 1992): 877. http://dx.doi.org/10.1090/s0002-9939-1992-1120504-1.

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Barbanera, Franco, and Maribel Fernández. "Intersection type assignment systems with higher-order algebraic rewriting." Theoretical Computer Science 170, no. 1-2 (December 1996): 173–207. http://dx.doi.org/10.1016/s0304-3975(96)80706-7.

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