## Why $\mathcal{O}(n^2)$ multiplication of coefficient required for canonical form of polynomial?

I was going to a textbook and am not able to understand the following:

Let $$F(x)$$ is given as a product $$F(x) = \sum_{i=0}^{n} (x – a_i)$$. Transforming $$F(x)$$ to its canonical form by consecutively multiplying the $$i$$th monomial with the product of the first $$i-1$$ monomials requires $$\mathcal{O}(n^2)$$ multiplications of coefficients.

A canonical form of polynomial is $$\sum_{i=0}^{n} c_i x_i$$.

Why $$\mathcal{O}(n^2)$$?

## Degree bounds and coefficient size in elimination theory?

Suppose we have polynomials is of form $$h_1(x_1,\dots,x_n)-c_1\in\mathbb Z[x_1,\dots,x_n]$$ $$\vdots$$ $$h_n(x_1,\dots,x_n)-c_n\in\mathbb Z[x_1,\dots,x_n]$$ where $$h_1(x_1,\dots,x_n),\dots,h_n(x_1,\dots,x_n)\in\mathbb Z[x_1,\dots,x_n]$$ are homogeneous polynoials of degree $$d$$ with each $$x_i$$ degree $$1$$ (that is only monomials of form $$x_{i_1},\dots,x_{i_d}$$ on the condition that each coefficient is random in $$[-B,B]$$ with $$c_1,\dots,c_n$$ being of absolute value at most $$B^{1+\frac d{2^d-1}}$$ then if you use elimination theory we can successively eliminate variables and reduce to an univariate polynomial.

Let the absolute value of the maximum coefficient size $$B’$$ of final univariate polynomial in reduced form (no common gcd of coefficients) and the let the degree be $$d’$$.

Then can $$B’^{\frac1{d’}+\epsilon}>B^{\frac1{d(2^d-1)}}$$ hold at arbitrarily small $$\epsilon>0$$ as $$B$$ increases and $$d$$ is fixed?

## O(1) algorithm to get approximate number that is larger as a binomial coefficient

Ideally I need a to calculate the binomial coefficient $${p \choose n}$$. But since the fastest algorithm to do this is an $$\mathcal{O}(n)$$ algorithm I would look to do something different. I don’t really care for the exact number as long as it is reasonably close and is at least equal or larger then the actual number. My number should be within a factor of 2 of actual binomial coefficient. Is there an $$\mathcal{O}(1)$$ way to compute such a number?

## Correlation coefficient of a WSS process

Could someone please tell me why the correlation coefficient of a WSS process is :

$$\rho_{XX}(\tau) = \frac{\operatorname{K}_{XX}(\tau)}{\sigma^2} = \frac{\operatorname{E}[(X_t – \mu)(X_{t+\tau} – \mu)]}{\sigma^2}$$

$$\rho_{XX}(\tau) = \frac{\operatorname{K}_{XX}(\tau)}{\sigma_{t}\sigma_{t+\tau}} = \frac{\operatorname{E}[(X_t – \mu)(X_{t+\tau} – \mu)]}{\sigma_{t}\sigma_{t+\tau}}$$

?

Where :$$~~~\sigma_{t}^2=\sigma^2=\operatorname{E}[X^2(t)]-\operatorname{E}^2[X(t)]$$ .

Stated in another way , does a WSS process imply that :

$$\operatorname{E}[X^2(t+\tau)]=\operatorname{E}[X^2(t)]$$ ?

(https://en.wikipedia.org/wiki/Autocovariance)

## Well posedness of wave equations whith time depending coefficient

Let us consider the folooiwng wave equation $$\begin{array}{rrr} y_{tt}=y_{xx}+a(t,x)y_{t} & \text{in} & (0,T)\times (0,1), \ y(0,t)=0\text{ , }y(t,1)=0 & \text{in} & (0,T), \ y(0,x)=y_{0}(x)\text{ , }y_{t}(0,x)=y_{1}(x) & \text{in} & (0,1). \end{array}$$ Assume that for example that $$\left( {{y_0},{y_1}} \right) \in H_0^1(0,1) \times {L^2}(0,1)$$ (resp. $$\left( {{y_0},{y_1}} \right) \in {L^2}(0,1) \times {H^{ – 1}}(0,1)$$). In the case of time depending coefficients semigroups theory fails, we have to deal with this problem otherwise. I saw some books and I found that $$a$$ must be $$C^1$$ in time. Is this the optimal assumption? Because I could solve this problem by characterestics and I didn’t need to this assumption, it was just $$a \in {L^2}((0,T) \times (0,1))$$. Any suggestions?. Thank you.

## Is the following variant of the Universal Coefficient Theorem valid?

A version of the Universal Coefficient Theorem that relates the integer cohomology of a group $$G$$ to its cohomology with coefficients in an abelian group $$M$$ is as follows:

$$H^n(G,M) = H^n(G,\mathbb Z) \otimes M \quad \times \quad\text{Tor}_1^{\mathbb Z}(H^{n+1}(G,\mathbb Z), M)$$

It is assumed in this expression that $$G$$ acts trivially on coefficients. Now suppose $$G$$ is an extension of the group $$\mathbb Z_2$$ by some group $$G_0$$ and $$M = \mathbb Z_2$$. I am interested in finding the cohomology of $$G$$ with coefficients in the module $$\mathbb Z^{sgn}$$, whose coefficients are twisted by the $$\mathbb Z_2$$ factor, while $$G_0$$ has trivial action. I wanted to know if the following statement, which looks like the statement of the UCT, is actually valid:

$$H^n(G,\mathbb Z_2) = H^n(G,\mathbb Z^{sgn}) \otimes \mathbb Z_2 \quad \times \quad\text{Tor}_1^{\mathbb Z}(H^{n+1}(G,\mathbb Z^{sgn}), \mathbb Z_2)$$

(So just to be clear, the l.h.s. has trivial action on $$\mathbb Z_2$$, while the r.h.s. has nontrivial action on $$\mathbb Z^{sgn}$$. I have checked this by hand for some simple examples and it works out, but I don’t have any proof. )

## how to get coefficient list from a polynomial with negative powers

Say a polynomial x^2-x^(-2), I need to extract its coefficient. I tried the command CoefficientList[x^2-x^(-2)], but no result comes out.

I am confused why it happens. How can I get the desired result {-1,0,0,0,1}?

It should be an easy question, but I do not know the suitable command. Would you please give me some tips?

## determining the coefficient matrix

We have a set of equations as

a x + b y = c z + d w e x + f y = g z + h w 

We wish to have a matrix as below :

How can we reach the m (matrix)