Ken’ichi Ohshika, Athanase Papadopoulos · arXiv (Cornell University) · 2018
We prove two rigidity results for automorphism groups of the spaces ML(S) of\nmeasured laminations on a closed hyperbolic surface S and PML(S) of projective\nmeasured laminations on this surface. The results concern the homeomorphisms of\nML(S) that preserve the geometric intersection between laminations and the\nhomeomorphisms of PML(S) that preserve the zero sets of these intersection\nfunctions.\n
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Kenrô Furutani, Nobukazu Ôtsuki · Kyoto journal of mathematics · 1993
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Bohdan Zelinka · Časopis pro pěstování matematiky · 1981
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Koerner, J, Consiglio Nazionale delle Ricerche, Rome (Italy). Istituto di Analisi dei Sistemi ed Informatica · OpenGrey (Institut de l'Information Scientifique et Technique) · 1991
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T. Winiarski · Annales Polonici Mathematici · 1986
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Pawlik, Arkadiusz · Jagiellonian University Repository (Jagiellonian University)
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Sotiris Nikoletseas, Christoforos Raptopoulos, Paul G. Spirakis · Lecture notes in computer science · 2012
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Paul R. Thie · Proceedings of the American Mathematical Society · 1970
Let A be an analytic set of pure dimension p in a neighborhood of the origin in Cn, and assume that 0 is a point of A. The Lelong Number of A at 0, denoted by n(0, A), is defined as the limit as r goes to zero of the volume of A in a ball of radius r, center 0, divided by the volume of a p-dimensional linear subspace in such a ball (see [5]). Suppose now that A is locally a complete intersection at 0, that is, that there exists a neighborhood U of 0 and q= n-p functions fl, * **, fq holomorphic on U such that
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Fan Yunzheng · SHILAP Revista de lepidopterología · 2004
Let G be a 2-connected simple graph of order n with the independence number α. We show here that ∀u, v ∈ V (G) and any z ∈ {u, v}, w ∈ V (G)\\{u, v} with d(w, z) = 2, if |N(u) ∩ N(w) | ≥ α − 1 or |N(v) ∩ N(w) | ≥ α − 1, then G is Hamiltonian, unless G belongs to a kind of special graphs.
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SUNGMUN, CHO · Open Access System for Information Sharing (Pohang University of Science and Technology) · 2019
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Iwona Krzyżanowska · Topology and its Applications · 2010
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Bautista Serrano, Humberto · Scholar Works at UT Tyler (The University of Texas at Tyler) · 2019
Let G be a finite, nontrivial group. In a paper in 1994, Cohn defined the covering number of a finite group as the minimum number of nontrivial proper subgroups whose union is equal to the whole group. This concept has received considerable attention lately, mainly due to the importance of recent discoveries. In this thesis we study a dual concept to the covering number. We define the intersection number of a finite group as the minimum number of maximal subgroups whose intersection is equal to the Frattini subgroup. Similarly we define the inconjugate intersection number of a finite group as the minimum number of inconjugate maximal subgroups whose intersection is equal to the Frattini subgroup. We study basic properties for these intersection numbers and determine its values for certain types of finite groups. Then we single out some similarities and differences between the covering number and the intersection number in specific types of groups. Finally we point to some directions of future developments and research.
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Vincent Despré, Francis Lazarus · HAL (Le Centre pour la Communication Scientifique Directe) · 2015
The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e. of any curve obtained by continuous deformation. Given a curve $c$ represented by a closed walk of length at most $\ell$ on a combinatorial surface of complexity $n$ we describe simple algorithms to (1) compute the geometric intersection number of $c$ in $O(n+ \ell^2)$ time, (2) construct a curve homotopic to $c$ that realizes this geometric intersection number in $O(n+\ell^4)$ time, (3) decide if the geometric intersection number of $c$ is zero, i.e. if $c$ is homotopic to a simple curve, in $O(n+\ell\log\ell)$ time. The algorithms for (2) and (3) are restricted to orientable surfaces, but the algorithm for (1) is also valid on non-orientable surfaces. To our knowledge, no exact complexity analysis had yet appeared on those problems. An optimistic analysis of the complexity of the published algorithms for problems (1) and (3) gives at best a $O(n+g^2\ell^2)$ time complexity on a genus $g$ surface without boundary. No polynomial time algorithm was known for problem (2) for surfaces without boundary. Interestingly, our solution to problem (3) provides a quasi-linear algorithm to a problem raised by Poincaré more than a century ago. Finally, we note that our algorithm for problem (1) extends to computing the geometric intersection number of two curves of length at most $\ell$ in $O(n+ \ell^2)$ time.
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Vincent Despré, Francis Lazarus · Journal of the ACM · 2019
The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e., of any curve obtained by continuous deformation. Given a curve c represented by a closed walk of length at most ℓ on a combinatorial surface of complexity n , we describe simple algorithms to (1) compute the geometric intersection number of c in O ( n + ℓ 2 ) time, (2) construct a curve homotopic to c that realizes this geometric intersection number in O ( n +ℓ 4 ) time, and (3) decide if the geometric intersection number of c is zero, i.e., if c is homotopic to a simple curve, in O ( n +ℓ log ℓ) time. The algorithms for (2) and (3) are restricted to orientable surfaces, but the algorithm for (1) is also valid on non-orientable surfaces. To our knowledge, no exact complexity analysis had yet appeared on those problems. An optimistic analysis of the complexity of the published algorithms for problems (1) and (3) gives at best a O ( n + g 2 ℓ 2 ) time complexity on a genus g surface without boundary. No polynomial time algorithm was known for problem (2) for surfaces without boundary. Interestingly, our solution to problem (3) provides a quasi-linear algorithm to a problem raised by Poincaré more than a century ago. Finally, we note that our algorithm for problem (1) extends to computing the geometric intersection number of two curves of length at most ℓ in O ( n + ℓ 2 ) time.
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Paul R. Thie · Proceedings of the American Mathematical Society · 1970
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Diljit Singh · arXiv (Cornell University) · 2017
For any simple digraph $D$ we offer a new proof for the intersection number of its middle digraph, $M(D)$; while doing so we also solve for the intersection number when $D$ has loops. In addition, a new transformation, the union of $D$ and its subdivision digraph, is introduced and its intersection number calculated in full generality. For the total digraph, we extend previous arguments letting us solve for the intersection number of $T(D)$ with $D$ possibly having loops, but under the restriction loops of the digraph only touch (are to or from) themselves, sinks, and sources.
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Fan Yunzheng · Boletim da Sociedade Paranaense de Matemática · 2009
Let G be a 2-connected simple graph of order n with the independence number\alpha. We show here that \forall u; v \in V (G)\backslash\{u,v\} and any z \in \{u,v\}; w \in V (G)\backslash \{u,v\}; with d(w; z) = 2, if |N(u) \ cap N(w)| \geq \alpha - 1 or |N(v) \cap N(w)| \geq \alpha - 1, then G is Hamiltonian, unless G belongs to a kind of special graphs.
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Pinaki Mondal · arXiv (Cornell University) · 2016
We explicitly characterize when the Milnor number at the origin of a polynomial or power series (over an algebraically closed field k of arbitrary characteristic) is the minimum of all polynomials with the same Newton diagram, which completes works of Kushnirenko (Invent. Math., 1976) and Wall (J. Reine Angew. Math., 1999). Given a fixed collection of n convex integral polytopes in R^n, we also give an explicit characterization of systems of n polynomials supported at these polytopes which have the maximum number (counted with multiplicity) of isolated zeroes on k^n, or more generally, on a union of torus orbits on k^n; this completes the program (undertaken by many authors including Khovanskii (Funkcional. Anal. i Prilozen, 1978), Huber and Sturmfels (Discrete Comput. Geom., 1997), Rojas (J. Pure Appl. Algebra, 1999)) of the extension to k^n of Bernstein's theorem (Funkcional. Anal. i Prilozen, 1975) on number of solutions of n polynomials on (k^*)^n. Our solutions to these two problems are connected by the computation of the intersection multiplicity at the origin of n hypersurfaces determined by n generic polynomials.
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N. S. Bolshakova · Discrete Mathematics and Applications · 2008
Latin squares C , D of order n are called pseudo-orthogonal if any two rows of the matrices C and D have exactly one common element. We give conditions for existence of families consisting of t pseudo-orthogonal Latin squares of order n . It is proved that the intersection number of a complete r -partite graph equals n 2 if and only if there exists a family consisting of r – 2 pairwise pseudo-orthogonal Latin squares of order n . It is proved that if , where prols( n ) is the maximum t such that there exists a set of t pseudo-orthogonal Latin squares of order n , then the intersection number of the graph is equal to n 2 . Applications of the obtained results to calculating the intersection number of some graphs are given.
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Alexandr Buryak, Sergey Shadrin · arXiv (Cornell University) · 2009
We give a new proof of Faber's intersection number conjecture concerning the top intersections in the tautological ring of the moduli space of curves $\M_g$. The proof is based on a very straightforward geometric and combinatorial computation with double ramification cycles.
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