scholarly journals Splitting numbers and signatures

2016 ◽  
Vol 144 (12) ◽  
pp. 5443-5455
Author(s):  
David Cimasoni ◽  
Anthony Conway ◽  
Kleopatra Zacharova
Keyword(s):  
Splitting Numbers

The reaping and splitting numbers of nice ideals

2014 ◽  
Vol 134 (2) ◽  
pp. 179-192 ◽  
Author(s):  
Rafał Filipów
Keyword(s):  
Splitting Numbers

scholarly journals Splitting numbers of links and the four-genus

2017 ◽  
Vol 146 (1) ◽  
pp. 421-427
Author(s):  
Charles Livingston
Keyword(s):  
Splitting Numbers

scholarly journals The lower semicontinuity of the Frobenius splitting numbers

2010 ◽  
Vol 150 (1) ◽  
pp. 35-46 ◽  
Author(s):  
FLORIAN ENESCU ◽  
YONGWEI YAO
Keyword(s):  
Local Ring ◽  
Lower Semicontinuity ◽  
Lower Semicontinuous ◽  
Prime Characteristic ◽  
Frobenius Splitting ◽  
Mild Conditions ◽  
Splitting Numbers

AbstractWe show that, under mild conditions, the (normalized) Frobenius splitting numbers of a local ring of prime characteristic are lower semicontinuous.

scholarly journals Splitting Numbers of Grids

10.37236/1914 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Dwight Duffus ◽  
Bill Sands
Keyword(s):  
Ordered Set ◽  
Ordered Sets ◽  
Maximal Antichain ◽  
P Elements ◽  
The Family ◽  
Splitting Numbers

For a subset $S$ of a finite ordered set $P$, let $$S\!\uparrow\;=\{x\in P:x\ge s \hbox{ for some }s\in S\}\quad\hbox{and} \quad S\!\downarrow\;=\{x\in P:x\le s \hbox{ for some }s\in S\}.$$ For a maximal antichain $A$ of $P$, let $$s(A)=\max_{A=U\cup D}{|U\!\uparrow|+|D\!\downarrow|\over|P|}\ ,$$ the maximum taken over all partitions $U\cup D$ of $A$, and $$s_k(P)=\min_{A\in {\cal A}(P),|A|=k}s(A)$$ where we assume $P$ contains at least one maximal antichain of $k$ elements. Finally, for a class ${\cal C}$ of finite ordered sets, we define $$s_k({\cal C})=\inf_{P\in {\cal C}}s_k(P).$$ Thus $s_k({\cal C})$ is the greatest proportion $r$ satisfying: every $k$-element maximal antichain of a member $P$ of ${\cal C}$ can be "split" into sets $U$ and $D$ so that $U\!\uparrow\cup\; D\!\downarrow$ contains at least $r|P|$ elements. In this paper we determine $s_k({\cal G}_k)$ for all $k\ge 1$, where ${\cal G}_k=\{{\bf k}\times{\bf n}:n\ge k\}$ is the family of all $k$ by $n$ "grids".

scholarly journals Uniform bounds in F-finite rings and their applications

2017 ◽  
Author(s):  
◽  
Thomas Marion Polstra
Keyword(s):  
Finite Type ◽  
Local Ring ◽  
Lower Semicontinuous ◽  
Multiplicity Function ◽  
Characteristic P ◽  
Finite Rings ◽  
Uniform Bounds ◽  
Frobenius Splitting ◽  
Lower Semi ◽  
Splitting Numbers

This dissertation establishes uniform bounds in characteristic p rings which are either F-finite or essentially of finite type over an excellent local ring. These uniform bounds are then used to show that the Hilbert-Kunz length functions and the normalized Frobenius splitting numbers defined on the spectrum of a ring converge uniformly to their limits, namely the Hilbert-Kunz multiplicity function and the Fsignature function. From this we establish that the F-signature function is lower semicontinuous. Lower semi-continuity of the F-signature of a pair is also established. We also give a new proof of the upper semi-continuity of Hilbert-Kunz multiplicity, a result originally proven by Ilya Smirnov.

scholarly journals Splitting Numbers of Links

2017 ◽  
Vol 60 (3) ◽  
pp. 587-614 ◽  
Author(s):  
Jae Choon Cha ◽  
Stefan Friedl ◽  
Mark Powell
Keyword(s):  
Lower Bounds ◽  
Khovanov Homology ◽  
New Techniques ◽  
Splitting Number ◽  
Splitting Numbers ◽  
Alexander Invariants

AbstractThe splitting number of a link is the minimal number of crossing changes between different components required, on any diagram, to convert it to a split link. We introduce new techniques to compute the splitting number, involving covering links and Alexander invariants. As an application, we completely determine the splitting numbers of links with nine or fewer crossings. Also, with these techniques, we either reprove or improve upon the lower bounds for splitting numbers of links computed by Batson and Seed using Khovanov homology.

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