Reputation
158,645
Next tag badge:
100/100 score
20/20 answers
Badges
5 98 258
Impact
~1.9m people reached

1d
comment What is known about optimization of spectral properties of matrices over finite fields?
Sturm's theorem is a theorem. It is exact! With its help, we can count roots in an interval without actually finding them. For a very simple case, we can easily tell that the polynomial $x^7 + x - 1$ has exactly one positive root, without having to compute it.
1d
comment Linear space obtained from another one factoring out the constant.
Factoring out what constants?
1d
comment How to design the analyticity of a function to find when the variable becomes zero
A better solution to what? What exactly is the problem? In what way does $|x_0| |x_1| \ldots |x_n|$ not work?
1d
comment proof that length of difference of projections implies equality of length of normals
"Projection on a convex set" meaning the closest point in the (presumably closed) convex set.
2d
comment What is known about optimization of spectral properties of matrices over finite fields?
en.wikipedia.org/wiki/Sturm%27s_theorem maplesoft.com/support/help/Maple/view.aspx?path=sturm
2d
comment Exact conditions under which the arithmetic progression $\{bk + r\}_{\{k\in\mathbb{N}\}}$ contains 0,1, or 2 primes
You just said the prime has to be $p$. That's one, not two. The real question is whether it is $0$ primes or $1$ prime.
2d
comment What is known about optimization of spectral properties of matrices over finite fields?
Well, you could use something like Maple's "RootOf" notation. In principle it is possible to decide questions such as " is $\lambda \le 2 \sqrt{d-1}$" by using Sturm's theorem to count roots of polynomials in intervals.
Apr
16
comment What is known about optimization of spectral properties of matrices over finite fields?
By the way, there are very good ways to get numerical approximations to the eigenvalues of a matrix. These generally do not involve working with the characteristic polynomial.
Apr
16
comment What is known about optimization of spectral properties of matrices over finite fields?
If by "exactly" you mean as expressions in rational numbers and radicals, then that is correct, in general the spectral gap of a graph can not be expressed in that way.
Apr
16
comment What is known about optimization of spectral properties of matrices over finite fields?
If you're looking for real eigenvalues, what does the "over finite fields" in the title mean?
Apr
16
comment How would multiplying money work?
They might buy you a square meal.
Apr
16
comment What subsets of $\mathbb R$ have exactly two limit points?
No: an uncountable set has uncountably many limit points. In fact there is a set of limit points homeomorphic to the Cantor set.
Apr
15
comment Product of two infinite sequences
Just out of curiosity, what was your "probabilistic proof"?
Apr
15
comment How to find every 10 digit number starting from 4?
Alternate hint: every integer from $\ldots$ to $\ldots$.
Apr
15
comment How to find every 10 digit number starting from 4?
By "starting from number 4", do you mean the first digit is 4? How many other digits are there? How many possible values are there for each?
Apr
15
comment Explain $(\|Mx\|_2)^2 = (M^Tx)^T(M^Tx) $ (positive definite, positive semi definite)
Perhaps that first $M$ should be $M^T$? $(\|Mx\|_2)^2 = (M^T x)^T (M^T x)$ is false.
Apr
15
comment How to integrate $(e^x + 2x)^2$?
Expand and do it term-by-term.
Apr
15
comment If $A$ is idempotent and $B=(I-A)$, then $BA'=I$
The question is wrong. If $A$ is an idempotent matrix other than $0$, $I - A$ has less than full rank, so you certainly can't have $B A' = I$.
Apr
15
comment seeking for Newton's like inequalities as sufficient condition for polynomial to have only real zeros
Yes, but isn't the Huchinson and Kurtz result what you're asking for in question (1)?
Apr
15
comment Real function such that preimage of every constant is measurable
Yes, this is a good counterexample.