scholarly journals A polynomial identity implying Schur’s partition theorem

2020 ◽  
Vol 148 (8) ◽  
pp. 3307-3324
Author(s):  
Ali Kemal Uncu
Keyword(s):  
Polynomial Identity ◽  
Partition Theorem

Parallel algorithms for matroid intersection and matroid parity

2015 ◽  
Vol 07 (02) ◽  
pp. 1550019
Author(s):  
Jinyu Huang
Keyword(s):  
Parallel Algorithms ◽  
Polynomial Identity ◽  
Black Box ◽  
Matroid Intersection ◽  
Polynomial Identity Testing ◽  
Identity Testing ◽  
Common Base ◽  
Graphic Matroid ◽  
Matroid Parity ◽  
Nc Algorithms

A maximum linear matroid parity set is called a basic matroid parity set, if its size is the rank of the matroid. We show that determining the existence of a common base (basic matroid parity set) for linear matroid intersection (linear matroid parity) is in NC2, provided that there are polynomial number of common bases (basic matroid parity sets). For graphic matroids, we show that finding a common base for matroid intersection is in NC2, if the number of common bases is polynomial bounded. To our knowledge, these algorithms are the first deterministic NC algorithms for matroid intersection and matroid parity. We also give a new RNC2 algorithm that finds a common base for graphic matroid intersection. We prove that if there is a black-box NC algorithm for Polynomial Identity Testing (PIT), then there is an NC algorithm to determine the existence of a common base (basic matroid parity set) for linear matroid intersection (linear matroid parity).

scholarly journals Euler's partition theorem and the combinatorics of ℓ-sequences

2008 ◽  
Vol 115 (6) ◽  
pp. 967-996 ◽  
Author(s):  
Carla D. Savage ◽  
Ae Ja Yee
Keyword(s):  
Partition Theorem

scholarly journals On commutativity of one-sideds-unital rings

1992 ◽  
Vol 15 (4) ◽  
pp. 813-818
Author(s):  
H. A. S. Abujabal ◽  
M. A. Khan
Keyword(s):  
Polynomial Identity ◽  
Unital Ring ◽  
Unital Rings

The following theorem is proved: Letr=r(y)>1,s, andtbe non-negative integers. IfRis a lefts-unital ring satisfies the polynomial identity[xy−xsyrxt,x]=0for everyx,y∈R, thenRis commutative. The commutativity of a rights-unital ring satisfying the polynomial identity[xy−yrxt,x]=0for allx,y∈R, is also proved.

scholarly journals A unification of two refinements of Euler’s partition theorem

2013 ◽  
pp. 135-147
Author(s):  
William Y. C. Chen ◽  
Henry Y. Gao ◽  
Kathy Q. Ji ◽  
Martin Y. X. Li
Keyword(s):  
Partition Theorem

scholarly journals A partition theorem for α-large sets

1999 ◽  
Vol 160 (1) ◽  
pp. 27-37 ◽  
Author(s):  
Teresa Bigorajska ◽  
Henryk Kotlarski
Keyword(s):  
Partition Theorem ◽  
Large Sets
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