THE Ck-GORDIAN COMPLEX OF KNOTS

2006 ◽  
Vol 15 (01) ◽  
pp. 73-80 ◽  
Author(s):  
YOSHIYUKI OHYAMA
Keyword(s):  
Natural Number ◽  
Simplicial Complex ◽  
The Family

Hirasawa and Uchida defined the Gordian complex of knots which is a simplicial complex whose vertices consist of all knot types in S3by using "a crossing change". In this paper, we define the Ck-Gordian complex of knots which is an extension of the Gordian complex of knots. Let k be a natural number more than 2 and we show that for any knot K0and any given natural number n, there exists a family of knots {K0, K1,…, Kn} such that for any pair (Ki, Kj) of distinct elements of the family, the Ck-distance dCk(Ki, Kj) = 1.

THE GORDIAN COMPLEX OF VIRTUAL KNOTS BY FORBIDDEN MOVES

2013 ◽  
Vol 22 (09) ◽  
pp. 1350051 ◽  
Author(s):  
SUMIKO HORIUCHI ◽  
YOSHIYUKI OHYAMA
Keyword(s):  
Natural Number ◽  
Simplicial Complex ◽  
Virtual Knot ◽  
Virtual Knots ◽  
The Family

Hirasawa and Uchida defined the Gordian complex of knots which is a simplicial complex whose vertices consist of all knot types in S3. In this paper, we define the Gordian complex of virtual knots by using forbidden moves. We show that for any virtual knot K0 and for any given natural number n, there exists a family of virtual knots {K0, K1, …, Kn} such that for any pair (Ki, Kj) of distinct elements of the family, the Gordian distance of virtual knots by forbidden moves dF(Ki, Kj) = 1.

THE GORDIAN COMPLEX OF VIRTUAL KNOTS

2012 ◽  
Vol 21 (14) ◽  
pp. 1250122 ◽  
Author(s):  
SUMIKO HORIUCHI ◽  
KASUMI KOMURA ◽  
YOSHIYUKI OHYAMA ◽  
MASAFUMI SHIMOZAWA
Keyword(s):  
Natural Number ◽  
Simplicial Complex ◽  
Virtual Knot ◽  
Virtual Knots ◽  
The Family

Hirasawa and Uchida defined the Gordian complex of knots which is a simplicial complex whose vertices consist of all knot types in S3. In this paper, we define the Gordian complex of virtual knots which is a simplicial complex whose vertices consist of all virtual knots by using the local move which makes a real crossing a virtual crossing. We show that for any virtual knot K0 and for any given natural number n, there exists a family of virtual knots {K0, K1,…,Kn} such that for any pair (Ki, Kj) of distinct elements of the family, the Gordian distance of virtual knots dv(Ki, Kj) = 1. And we also give a formula of the f-polynomial for the sum of tangles of virtual knots.

scholarly journals DIFFERENTIAL INEQUALITIES AND A MARTY-TYPE CRITERION FOR QUASI-NORMALITY

2018 ◽  
Vol 105 (1) ◽  
pp. 34-45
Author(s):  
JÜRGEN GRAHL ◽  
TOMER MANKET ◽  
SHAHAR NEVO
Keyword(s):  
Natural Number ◽  
Holomorphic Functions ◽  
Differential Inequalities ◽  
The Family ◽  
Type Criterion ◽  
Image Position

We show that the family of all holomorphic functions $f$ in a domain $D$ satisfying $$\begin{eqnarray}\frac{|f^{(k)}|}{1+|f|}(z)\leq C\quad \text{for all }z\in D\end{eqnarray}$$ (where $k$ is a natural number and $C>0$) is quasi-normal. Furthermore, we give a general counterexample to show that for $\unicode[STIX]{x1D6FC}>1$ and $k\geq 2$ the condition $$\begin{eqnarray}\frac{|f^{(k)}|}{1+|f|^{\unicode[STIX]{x1D6FC}}}(z)\leq C\quad \text{for all }z\in D\end{eqnarray}$$ does not imply quasi-normality.

Explicit Constructions of Rödl's Asymptotically Good Packings and Coverings

2000 ◽  
Vol 9 (3) ◽  
pp. 265-276 ◽  
Author(s):  
N. N. KUZJURIN
Keyword(s):  
Natural Number ◽  
Time And Space ◽  
Explicit Constructions ◽  
The Family ◽  
Element Set

For any fixed l < k we present families of asymptotically good packings and coverings of the l-subsets of an n-element set by k-subsets, and an algorithm that, given a natural number i, finds the ith k-subset of the family in time and space polynomial in log n.

scholarly journals Random affine code tree fractals and Falconer–Sloan condition

2014 ◽  
Vol 36 (5) ◽  
pp. 1516-1533 ◽  
Author(s):  
ESA JÄRVENPÄÄ ◽  
MAARIT JÄRVENPÄÄ ◽  
BING LI ◽  
ÖRJAN STENFLO
Keyword(s):  
Natural Number ◽  
General Class ◽  
Satisfy Condition ◽  
The Family ◽  
General Systems ◽  
Real Anal ◽  
Probabilistic Version

We calculate the almost sure dimension for a general class of random affine code tree fractals in $\mathbb{R}^{d}$. The result is based on a probabilistic version of the Falconer–Sloan condition $C(s)$ introduced in Falconer and Sloan [Continuity of subadditive pressure for self-affine sets. Real Anal. Exchange 34 (2009), 413–427]. We verify that, in general, systems having a small number of maps do not satisfy condition $C(s)$. However, there exists a natural number $n$ such that for typical systems the family of all iterates up to level $n$ satisfies condition $C(s)$.

scholarly journals Gordian complexes of knots and virtual knots given by region crossing changes and arc shift moves

2020 ◽  
Vol 29 (10) ◽  
pp. 2042008
Author(s):  
Amrendra Gill ◽  
Madeti Prabhakar ◽  
Andrei Vesnin
Keyword(s):  
Simplicial Complex ◽  
Infinite Family ◽  
High Dimensional ◽  
Dimensional Simplex ◽  
Virtual Knot ◽  
Virtual Knots ◽  
The Family

Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in [Formula: see text]. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using [Formula: see text]-move and forbidden moves. In this paper, we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in the virtual knot diagram which results in reversing orientation locally between two consecutive crossings. We show the existence of an arbitrarily high-dimensional simplex in both the Gordian complexes, i.e. by region crossing change and by the arc shift move. For any given knot (respectively, virtual knot) diagram we construct an infinite family of knots (respectively, virtual knots) such that any two distinct members of the family have distance one by region crossing change (respectively, arc shift move). We show that the constructed virtual knots have the same affine index polynomial.

THE GORDIAN COMPLEX OF KNOTS

2002 ◽  
Vol 11 (03) ◽  
pp. 363-368 ◽  
Author(s):  
MIKAMI HIRASAWA ◽  
YOSHIAKI UCHIDA
Keyword(s):  
Simplicial Complex ◽  
Infinite Family ◽  
The Family

In this paper, we define the Gordian complex of knots, which is a simplicial complex whose vertices consist of all oriented knot types in the 3-sphere. We show that for any knot K, there exists an infinite family of distinct knots containing K such that any pair (Ki, Kj) of the member of the family, the Gordian distance dG(Ki, Kj) = 1.

scholarly journals Bucshbaum simplicial posets

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Jonathan Browder ◽  
Steven Klee
Keyword(s):  
Simplicial Complex ◽  
Necessary Conditions ◽  
Betti Numbers ◽  
Geometric Realization ◽  
The Family ◽  
International Audience ◽  
Simplicial Poset ◽  
Simplicial Cell ◽  
Topological Data

International audience The family of Buchsbaum simplicial posets generalizes the family of simplicial cell manifolds. The $h'-$vector of a simplicial complex or simplicial poset encodes the combinatorial and topological data of its face numbers and the reduced Betti numbers of its geometric realization. Novik and Swartz showed that the $h'-$vector of a Buchsbaum simplicial poset satisfies certain simple inequalities. In this paper we show that these necessary conditions are in fact sufficient to characterize the h'-vectors of Buchsbaum simplicial posets with prescribed Betti numbers.

On the paradox of grounded classes

1955 ◽  
Vol 20 (2) ◽  
pp. 140-140 ◽  
Author(s):  
Richard Montague
Keyword(s):  
Natural Number ◽  
Set Theory ◽  
Infinite Sequence ◽  
Axiom Of Choice ◽  
The Other ◽  
Other Hand ◽  
The Family ◽  
The One ◽  
The Axiom Of Choice ◽  
Regular Classes

Mr. Shen Yuting, in this Journal, vol. 18, no. 2 (June, 1953), stated a new paradox of intuitive set-theory. This paradox involves what Mr. Yuting calls the class of all grounded classes, that is, the family of all classes a for which there is no infinite sequence b such that … ϵ bn ϵ … ϵ b2ϵb1 ϵ a.Now it is possible to state this paradox without employing any complex set-theoretical notions (like those of a natural number or an infinite sequence). For let a class x be called regular if and only if (k)(x ϵ k ⊃ (∃y)(y ϵ k · ~(∃z)(z ϵ k · z ϵ y))). Let Reg be the class of all regular classes. I shall show that Reg is neither regular nor non-regular.Suppose, on the one hand, that Reg is regular. Then Reg ϵ Reg. Now Reg ϵ ẑ(z = Reg). Therefore, since Reg is regular, there is a y such that y ϵ ẑ(z = Reg) · ~(∃z)(z ϵ z(z = Reg) · z ϵ y). Hence ~(∃z)(z ϵ ẑ(z = Reg) · z ϵ Reg). But there is a z (namely Reg) such that z ϵ ẑ(z = Reg) · z ϵ Reg.On the other hand, suppose that Reg is not regular. Then, for some k, Reg ϵ k · [1] (y)(y ϵ k ⊃ (∃z)(z ϵ k · z ϵ y)). It follows that, for some z, z ϵ k · z ϵ Reg. But this implies that (ϵy)(y ϵ k · ~(ϵw)(w ϵ k · w ϵ y)), which contradicts [1].It can easily be shown, with the aid of the axiom of choice, that the regular classes are just Mr. Yuting's grounded classes.

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