## Legendre Polynomial Identity

I was looking for a method to do the following integral:
$$int^1_{-1}(1-x^2)frac{dP_m(x)}{dx}P_n~dx$$

I know there should be an explicit representation for the result but I am struggling to work it out.

## linear algebra – Problems relates to minimal polynomial and \$AB, BA\$

Let $$A$$ and $$B$$ be complex $$3times 3$$ matrices. The minimal polynomial of $$A$$ is $$x^3-1$$ and the minimal polynomial of $$B$$ is $$(x-1)^3$$. At this time show that $$ABneq BA$$

I know that the characteristic and minimal polynomials of $$AB$$ and $$BA$$ are the same at least one of $$A$$,$$B$$ is invertible. But in general, $$AB$$ and $$BA$$ do not have the same minimal polynomial. Am I right?

How do we answer this question? Any hints would be appreciated.

## complexity theory – Polynomial kernelization for “Set coloring” problem

Given an universum $$U$$ and a system of its subsets $$Fsubseteq mathcal{P}(U)$$, given a $$kin mathbb{Z}_{geq 0}$$. The decision problem is to say whether there exists a $$2$$-coloring of $$U$$ s.t. atleast $$k$$ sets from $$F$$ contain elements of both colors.

I wish to show that there exists kernel with $$2k$$ sets and $$|U|in O(k^2)$$. So far I have these reduction rules:

Rule 1: If there is $$xin U$$ s.t. there’s no $$Win F$$ s.t. $$xin W$$, then we can delete $$x$$ from U and work with a new instance. $$(F,Usetminus{x},k)$$.

Rule 2: If $$Win F$$ is a singleton, then we can remove it and work with $$(Fsetminus {W},U,K)$$ because such $$W$$ will never contain two elements of different colors.

Not really sure how to proceed from here. Any hints appreciated.

## ag.algebraic geometry – Under what conditions is the polynomial of degree \$6\$ irreducible?

Let $$k$$ be a perfect field of characteristic $$p neq 2,3$$ such that $$omega := sqrt(3){1} in k$$, where $$omega neq 1$$. Consider an absolutely irreducible (not necessarily homogenous) quadratic polynomial $$Q in k(s_1, s_2)$$ in two variables $$s_1, s_2$$. Under what conditions is the polynomial $$Q^prime(t_1,t_2) := Q(t_1^3, t_2^3)$$ (of degree $$6$$) absolutely irreducible (or at least irreducible over $$k$$) ?

## complexity theory – Is there a non-deterministic polynomial by time Turing machine such that: \$L(M)in NPC\$ and \$L(overline{M})in P\$

When $$overline{M}$$ is a non-deterministic polynomial by time Turing machine that final states switched: accept to reject and vice versa.
I’m thinking that this equal to $$P=NP$$, but I saw a solution (an example) that I disagree with:
$$M$$ is a non-deterministic polynomial by time Turing machine that decide $$SAT$$, if all that paths are rejected then $$L(overline{M})=Sigma^*in P$$

Is it a valid solution, or as I’m thinking $$L(M)in NPC$$ and $$L(overline{M})in P Leftrightarrow P=NP$$

## algebraic geometry – Reducible polynomial & radicals of ideals..

I was at a seminar about algebraic geometry recently and the first point made was to find out if this object was reducible, thus I want to improve my intuition about reducibility of a polynomial generating an ideal by comparing it against intuition about the capabilities of that ideal having a radical:

If a polynomial is reducible and the factors are equal to each other then a radical of the ideal could exist, such as $$x^{2}+2x+1 = (x+1)(x+1)$$, $$(x+1)^{2}$$ is obviously in that ideal.

If the polynomials reducible and the factors don’t equal eachother then a radical could not be different such as $$(x+1)(x-1) = x^{2} -1$$, neither of the factors raised to a power could at least equal multiplying by the other factor.

If the polynomials irreducible then a radical could not exist the same fashion as a reducible polynomial without equal factors.

So that means in terms of Hilbert Nullstellensatz theorem whichs that $$mathbb{I}(mathbb{V}(I)) = sqrt{I}$$, the ideal of functions vanishing on the zero set of an ideal is the radical of that ideal, that if the ideal is the radical in the first place – it is simply an ideal thats obtained by a reducible polynomial without equal factors or an irreducible polynomial.

## reference request – Interpolation estimate for trigonometric polynomial

Notations:
$$J^s u := sum_{m in mathbb{N}} (1+vert m vert^2)^{s/2}hat u_m e^{imt}$$
$$Vert varphi Vert_{W^{s,p}} := Vert J^s u Vert_{L^p}$$
$$P_n u = sum^{n-1}_{m=-n} hat u_m e^{imt}$$
$$Q_n u = sum^{2n-1}_{k=0} u(t_k) L_k, quad t_k = frac{k}{n} pi, k = 0,1,cdots, 2n-1$$
$$L_k(t) := frac{1}{2n} sum ^{n-1}_{m=-n} e^{im(t – t_k)}, quad k = 0,1,cdots, 2n-1$$
The estimate
$$Vert varphi – P_n varphi Vert_{W^{s,p}} leq C frac{1}{n^{r-s}} Vert varphi Vert_{W^{r,p}}$$
holds for $$p = 2$$ and $$1 with any $$-infty < s < r (see (McLean86)).

Replacing orthogonal projection $$P_n$$ with interpolation operator $$Pi_n$$, the above estimate of case $$p= 2$$ can be found in (Kirsch96) or (Kress14), that is,
$$Vert varphi – Q_n varphi Vert_{W^{s,p}} leq C frac{1}{n^{r-s}} Vert varphi Vert_{W^{r,p}}$$
for any $$0 leq s leq r$$ and $$r > frac{1}{2}$$.

Question: Can the latter estimate also be extended to the case $$1 ?

References

(McLean86) A Spectral Galerkin Method for a Boundary Integral Equation Mathematics of Computation Volume 47, Number 176 October 1986. Pages 597-607.

(Kirsch96) An Introduction to the Mathematical Theory of Inverse Problems. Springer, New York, 1996.

(Kress14) Linear integral equations Third edition. Springer-verlag, New York, 2014.

## analysis – Taylor polynomial of degree \$3\$ around the point \$0\$ using the series expansion

I want to calculate the Taylor polynomial of degree $$3$$ around the point $$0$$ for the function $$f(x)=tan (x)cdot log (1+2x)$$ using the series expansion.

We have that $$tan (x)=x+frac{x^3}{3}+frac{2x^5}{15}+ldots$$ and $$log (1+2x)=2x-2x^2+frac{8x^3}{3}-4x^4+ldots$$.

So is the Taylor polynomial of degree $$3$$ for $$f(x)$$ equal to $$xcdot (2x-2x^2)=2x^2-2x^3$$ ?

## Is there any proof that says "For each problem in NP there is a randomized algorithm that solves that problem in expected polynomial time."

Is it known that "For each problem in NP there is a randomized algorithm that solves it in polynomial time"? If not true then is there any proof of that. Or does it belongs to the unknown domain?

## equation solving – NSolve – Problem with univariate long polynomial

Below I am trying to find the steady state of x1 (x2 is a func of x1) in terms of a variable q1. `x1stst`. Then I wish to compute q1 values giving “trace = 0” in `equq1` (equq1 includes x1 steady states as a function of q1). Unfortunately `NSolve` doesn’t work due to the hideous polynomial of `equq1`

``````(*Find q1*)
q2 = 46.2857;
q3 = 0.111;
q4 = 5.714285714285714`;
q5 = 0.14285714285714288`;
NSolve(q1 q2  (q3 + x1)/(1 + x1) x2 - (q4 x1)/(q5 + x1) == 0 &&
1/(1 + x1^2) - x2 == 0, {x1, x2});
x1stst = NSolve(
q1 q2 (q3 + x1)/(1 + x1) 1/(1 + x1^2) - (q4 x1)/(q5 + x1) == 0, x1);
equq1 = -1 + (5.714285714285714` x1)/(0.14285714285714288` + x1)^2 -
5.714285714285714`/(0.14285714285714288` + x1) - (
46.285714285714285` (0.111` + x1) x2 q1)/(1 + x1)^2 + (
46.285714285714285` x2 q1)/(1 + x1);
NSolve(equq1 == 0 /.
x1stst((1)) /. {x2 -> 1/(1 + x1stst((1))^2)}, q1, Reals)

``````

I tried `Together` `Collect` etc. non of them worked. Looked similar questions but again couldn’t find anything.

Any advice would be appreciated. Thank you.