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Apr
14
comment Simple groups and words
At least, you should add that the word does not contain $n$th power of a word, where $n$ is the period of $S$; otherwise you may have words like $a^kb^na^{n-k}$ etc.
Apr
1
comment Square root of a complex matrix commuting with a given one
@GeoffRobinson: Oh yes, ypu are right. Sorry. I have added some words about this case; hopefully now they are correct;)...
Apr
1
revised Square root of a complex matrix commuting with a given one
Correction of the last (wrong) statement
Apr
1
revised Square root of a complex matrix commuting with a given one
deleted 1 characters in body
Apr
1
answered Square root of a complex matrix commuting with a given one
Mar
27
answered Symmetry on a sphere
Mar
26
answered What is the name for the type of matrices?
Mar
26
comment What is the name for the type of matrices?
You may regard $M_{a_1,\dots,a_{2^n}}$ as the `addition table' for $n$-dimensional vector space over $\mathbb F_2$.
Mar
3
comment About generalized Minkowski inequality
Recall that the Minkowski inequality turns into equality when $(x_1,\dots,x_n)$ and $(y_1,\dots,y_n)$ are proportional. Thus, if you perturb the function $x^p$ a bit in a neighborhood of some point $a$ (keeping monotonicity, convexity, whatnot) then you may set $a=x_n$ or $a=x_n+y_n$ refuting your inequality for the perturbed function.
Mar
2
comment Triangle with largest perimeter in a convex region
Sorry, my previous comment was a result of a miscomputation. The circle is indeed the best among ellipses.
Mar
2
comment Triangle with largest perimeter in a convex region
Oh, sorry; that was me who miscalculated...
Feb
27
comment The maximal discrete parallelepiped in a convex body
In fact, we may even set $D=d$...
Feb
27
comment The maximal discrete parallelepiped in a convex body
But $c_d$ may depend on $d$...
Feb
27
answered The maximal discrete parallelepiped in a convex body
Feb
27
comment The maximal discrete parallelepiped in a convex body
Surely, your parallelepiped may be degenerate?
Feb
25
comment Consecutive square values of cubic polynomials
Just to bound $c$ from below: the polynomial $24x^3-135x^2+192x$ has square values at $0,1,2,3,4$.
Jan
28
answered Union of Permutations
Jan
28
comment Union of Permutations
For $n=6$, you may achieve more permutations keeping $s=12$. Set $U_1=U_2=\{1,2\}$, $U_3=U_4=\{3,4\}$, $U_5=U_6=\{5,6\}$. This pattern clearly fits for 8 permutations. So the optimal configuration should be a different one. (To make it more evident, compare $5!=120$ permutations obtained by permutations of the first 5 numbers with $2^7=128$ permutations obtained by switchings of 7 pairs of numbers.)
Jan
28
comment Union of Permutations
Just a reformulation which may look more familiar to someone. Given that the permanent of a 0-1 matrix $A$ is at least $k$, find the minimal sum of its elements.
Jan
25
comment Extending a line-arrangement so that the bounded components of its complement are triangles
It would be also interesting to find whether the answer to the question is affirmative if we replace lines by pseudolines (that is --- unbounded curves or olylines satisfying the property that every two of them intersect at exactly one point; here we omit parallel lines).