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Bounds on Multiple Self-avoiding Polygons
Hong, Kyungpyo,Oh, Seungsang Canadian Mathematical Society 2018 Canadian mathematical bulletin Vol.61 No.3
<B>Abstract</B><P>A self-avoiding polygon is a lattice polygon consisting of a closed self-avoiding walk on a square lattice. Surprisingly little is known rigorously about the enumeration of self-avoiding polygons, although there are numerous conjectures that are believed to be true and strongly supported by numerical simulations. As an analogous problemto this study, we considermultiple self-avoiding polygons in a confined region as a model for multiple ring polymers in physics. We find rigorous lower and upper bounds for the number pm×n of distinct multiple self-avoiding polygons in the m × n rectangular grid on the square lattice. For m = 2, p2×n = 2<SUP>n−1</SUP> − 1. And for integers m, n ≥ 3,</P><P/>
On 6-Dimensional Nearly Kähler Manifolds
Watanabe, Yoshiyuki,Suh, Young Jin Canadian Mathematical Society 2010 Canadian mathematical bulletin Vol.53 No.3
<B>Abstract</B><P>In this paper we give a sufficient condition for a complete, simply connected, and strict nearly Kähler manifold of dimension 6 to be a homogeneous nearly Kähler manifold. This result was announced in a previous paper by the first author.</P>
Gosset Polytopes in Picard Groups of del Pezzo Surfaces
Canadian Mathematical Society 2012 Canadian journal of mathematics Vol.64 No.1
<B>Abstract</B><P> In this article, we study the correspondence between the geometry of del Pezzo surfaces <I>Sr</I> and the geometry of the <I>r</I>-dimensional Gosset polytopes (<I>r</I> − 4)21. We construct Gosset polytopes (<I>r</I> −4)21 in Pic <I>Sr</I> ꕕ ℚ whose vertices are lines, and we identify divisor classes in Pic <I>Sr</I> corresponding to (<I>a</I> − 1)-simplexes (<I>a</I> ≤ <I>r</I>), (<I>r</I> − 1)-simplexes and (<I>r</I> − 1)-crosspolytopes of the polytope (<I>r</I> − 4)21. Then we explain how these classes correspond to skew <I>a</I>-lines(<I>a</I> ≤ <I>r</I>), exceptional systems, and rulings, respectively.</P><P>As an application, we work on the monoidal transform for lines to study the local geometry of the polytope (<I>r</I>−4)21. And we show that the Gieser transformation and the Bertini transformation induce a symmetry of polytopes 321 and 421, respectively.</P>
Homotopy Classification of Projections in the Corona Algebra of a Non-simple <i>C</i>*-algebra
Brown, Lawrence G.,Lee, Hyun Ho Canadian Mathematical Society 2012 Canadian journal of mathematics Vol.64 No.4
<B>Abstract</B><P>We study projections in the corona algebra of <I>C</I>(<I>X</I>) ⊗ <I>K</I>, where K is the <I>C</I>*-algebra of compact operators on a separable infinite dimensional Hilbert space and <I>X</I> = [0, 1], [0,∞), (−∞,∞), or [0, 1]/﹛0, 1﹜. Using BDF's essential codimension, we determine conditions for a projection in the corona algebra to be liftable to a projection in the multiplier algebra. We also determine the conditions for two projections to be equal in <I>K</I>0, Murray-von Neumann equivalent, unitarily equivalent, or homotopic. In light of these characterizations, we construct examples showing that the equivalence notions above are all distinct.</P>
Real Hypersurfaces in Complex Two-Plane Grassmannians with Reeb Parallel Structure Jacobi Operator
Jeong, Imsoon,Kim, Seonhui,Jin Suh, Young Canadian Mathematical Society 2014 Canadian mathematical bulletin Vol.57 No.4
<B>Abstract</B><P>In this paper we give a characterization of a real hypersurface of Type (<I>A</I>) in complex two-plane Grassmannians <I>G</I>2(ℂ<SUP>m+2</SUP>), which means a tube over a totally geodesic <I>G</I>2(ℂ<SUP><I>m</I>+1</SUP>) in <I>G</I>2(ℂ<SUP><I>m</I>+2</SUP>), by means of the Reeb parallel structure Jacobi operator ∇<I>ε</I><I>R</I><I>ε</I> = 0.</P>
Real Hypersurfaces in Complex Space Forms with Reeb Flow Symmetric Structure Jacobi Operator
Cho, Jong Taek,Ki, U-Hang Canadian Mathematical Society 2008 Canadian mathematical bulletin Vol.51 No.3
<B>Abstract</B><P>Real hypersurfaces in a complex space form whose structure Jacobi operator is symmetric along the Reeb flow are studied. Among them, homogeneous real hypersurfaces of type (<I>A</I>) in a complex projective or hyperbolic space are characterized as those whose structure Jacobi operator commutes with the shape operator.</P>
Uniform Convexity and the Bishop-Phelps-Bollobás Property
Kwang Kim, Sun,Ju Lee, Han Canadian Mathematical Society 2014 Canadian journal of mathematics Vol.66 No.2
<B>Abstract</B><P>A new characterization of the uniform convexity of Banach space is obtained in the sense of the Bishop-Phelps-Bollobás theorem. It is also proved that the couple of Banach spaces (X;Y) has the Bishop-Phelps-Bollobás property for every Banach space Y when X is uniformly convex. As a corollary, we show that the Bishop-Phelps-Bollobás theorem holds for bilinear forms on ℓp × ℓq (1 < p; q < ∞).</P>
Parabolic Geodesics in Sasakian 3-Manifolds
Cho, Jong Taek,Inoguchi, Jun-ichi,Lee, Ji-Eun Canadian Mathematical Society 2011 Canadian mathematical bulletin Vol.54 No.3
<B>Abstract</B><P>We give explicit parametrizations for all parabolic geodesics in 3-dimensional Sasakian space forms.</P>
Wedge Operations and Torus Symmetries II
Choi, Suyoung,Park, Hanchul Canadian Mathematical Society 2017 Canadian journal of mathematics Vol.69 No.4
<B>Abstract</B><P>A fundamental idea in toric topology is that classes of manifolds with well-behaved torus actions (simply, toric spaces) are classified by pairs of simplicial complexes and (non-singular) characteristic maps. In a previous paper, the authors provided a new way to find all characteristic maps on a simplicial complex K(J) obtainable by a sequence of wedgings from K.The main idea was that characteristic maps on K theoretically determine all possible characteristic maps on a wedge of K.</P><P>We further develop our previous work for classification of toric spaces. For a star-shaped simplicial sphere K of dimension n-1 with m vertices, the Picard number Pic(K) of K is m-n. We call K a seed if K cannot be obtained by wedgings. First, we show that for a fixed positive integer 𝓁, there are at most finitely many seeds of Picard 𝓁 number supporting characteristic maps. As a corollary, the conjecture proposed by V. V. Batyrev in is solved affirmatively.</P><P>Secondly, we investigate a systematicmethod to find all characteristic maps on K(J) using combinatorial objects called (realizable) <I>puzzles</I> that only depend on a seed K. These two facts lead to a practical way to classify the toric spaces of fixed Picard number.</P>
Lie Derivatives and Ricci Tensor on Real Hypersurfaces in Complex Two-plane Grassmannians
Jeong, Imsoon,Dios Pé,rez, Juan de,Suh, Young Jin,Woo, Changhwa Canadian Mathematical Society 2018 Canadian mathematical bulletin Vol.61 No.3
<B>Abstract</B><P>On a real hypersurface <I>M</I> in a complex two-plane Grassmannian <I>G</I>2() we have the Lie derivation and a differential operator of order one associated with the generalized Tanaka-Webster connection . We give a classification of real hypersurfaces <I>M</I> on <I>G</I>2() satisfying , where ξ is the Reeb vector field on <I>M</I> and <I>S</I> the Ricci tensor of <I>M</I>.</P>