Remarks on the Box Problem
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Authors
Nets Hawk Katz
Elliot Krop
Mauro Maggioni
Issue Date
Type
Journal Article, Academic Journal
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Abstract
Let Iijk be an N ×N ×N
tensor of 1’s and 0’s. We say I satisfies the (E) condition provided it does not
contain 1’s at all the corners of any nontrivial box. That is, I satisfies the (E)
condition, if for any i1
= i2, j1
= j2, k1
= k2, we have that
Π2l
=1Π2
m=1Π2
n=1Iiljmkn = 0.
We prove the following Theorem. Let p be any prime. There is a p3 − 1 × p3 − 1 × p3 − 1 tensor I
satisfying the (E) condition containing p2(p3 − 1)2 many 1’s.