formal vector
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2012 ◽  
Vol 257 (2) ◽  
pp. 471-489 ◽  
Author(s):  
Li Ren ◽  
Qiang Mu ◽  
Yongzheng Zhang
Keyword(s):  

2012 ◽  
Vol 53 (3) ◽  
pp. 033506
Author(s):  
Li Ren ◽  
Qiang Mu ◽  
Yongzheng Zhang
Keyword(s):  

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Adam Towsley ◽  
Jonathan Pakianathan ◽  
David H. Douglass

Covariance is used as an inner product on a formal vector space built on random variables to define measures of correlation across a set of vectors in a -dimensional space. For , one has the diameter; for , one has an area. These concepts are directly applied to correlation studies in climate science.


10.37236/1999 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Mahir Bilen Can ◽  
Bruce E. Sagan

Let $\Pi_n$ denote the set of all set partitions of $\{1,2,\ldots,n\}$. We consider two subsets of $\Pi_n$, one connected to rook theory and one associated with symmetric functions in noncommuting variables. Let ${\cal E}_n\subseteq\Pi_n$ be the subset of all partitions corresponding to an extendable rook (placement) on the upper-triangular board, ${\cal T}_{n-1}$. Given $\pi\in\Pi_m$ and $\sigma\in\Pi_n$, define their slash product to be $\pi|\sigma=\pi\cup(\sigma+m)\in\Pi_{m+n}$ where $\sigma+m$ is the partition obtained by adding $m$ to every element of every block of $\sigma$. Call $\tau$ atomic if it can not be written as a nontrivial slash product and let ${\cal A}_n\subseteq\Pi_n$ denote the subset of atomic partitions. Atomic partitions were first defined by Bergeron, Hohlweg, Rosas, and Zabrocki during their study of $NCSym$, the symmetric functions in noncommuting variables. We show that, despite their very different definitions, ${\cal E}_n={\cal A}_n$ for all $n\ge0$. Furthermore, we put an algebra structure on the formal vector space generated by all rook placements on upper triangular boards which makes it isomorphic to $NCSym$. We end with some remarks.


2008 ◽  
Vol 15 (1) ◽  
pp. 129-148
Author(s):  
Ilya Shapiro ◽  
Xiang Tang
Keyword(s):  

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