What would be the Newton's method in the form $x_{k+1}=g(x_k)$ to solve the equation $$f(x)=x^bx+b^2-d^2=0$$ in which both $b>0,d>0$ are parameters? Additionally, I need to show that $|g'(x)|\le 1/2$ whenever $|x-b|\ge d/\sqrt{2}$ and also that $## Chain rule for variational derivatives and differentiation of an integral February 10 Assume that I have the following functional: $$F[t]=\int_{\Omega} f( u_1(x,t),u_2(x,t),...,u_N(x,t) ) dV$$ where$\Omega \in \mathbb{R}^k$. Is it true that: $$\frac{dF}{dt}= \int_{\Omega} \Sigma_{i=1}^N \frac{\delta F}{\delta u_i } \frac{\partial ## Continuity of v: \mathcal{B} (M;N) \times M \rightarrow N, v(f,x) = f(x). February 10 1 Let (M,d_M), (N,d_N) be metric spaces and v: \mathcal{B} (M;N) \times M \rightarrow N, v(f,x) = f(x). Then v is continuous at (f_0,x_0)\in \mathcal{B}(M;N) \times M \iff f_0 : M \rightarrow N is continuous at x_0 \in M$$\mathcal{B}(M ## Cauchy continuous implies standard continuity February 10 Let$f$be Cauchy continuous.$f$is Cauchy continuous if for any Cauchy sequence$\{x_{n}\}$in$(X,d_{X})$,$\{f(x_{n})\}$is a Cauchy sequence in$(Y,d_{Y})$. Show that Cauchy continuous$\implies$continuous.Since$f$is Cauchy continuous, we hav ## complete the proof for this statement February 10 3 $$\forall x \in \mathbb{R}, x \neq 0 \implies \frac{1}{x^2\:+3}\:<\:\frac{4}{5}\:$$I thought of doing the contrapositive but not sure what to do next. $$\frac{1}{x^{2\:}+3}\:\ge \frac{4}{5}$$$$\implies 0\ge 4x^2\:+7$$Because$x^2+3 > 0$for every ## Proving that a statement about$ February 10    3

I need to do assignment for my homework, in which I need to prove that the following statement is valid. $$(s<t \text{ and } t<u)\implies(s<u)$$I need to do this assignment using the laws and definitions of inequality.The problem is that I don'

## How to find the limit of a matrix $P^n = UD^nU^{-1}$ where $D$ is a diagonal matrix of eigenvalues and $U$ a matrix of eigenvectors February 10

If we have a matrix where $P = UDU^{-1}$, where $D$ is a diagonal matrix of real eigenvalues that are less than or equal to 1, and $U$ is the corresponding matrix of eigenvectors, how can we show that the limit of $P^n = UD^nU^{-1} \to u_1v_1'$, wher

## Explicit formula for $e_k = 4e_{k-1} + 5$ February 10

The sequence looks like this:$e_0 = 2$$e_1 = 4(e_{}) + 5 = 13$$e_2 = 4(e_{}) + 5 = 57$$e_3 = 4(e_{}) + 5 = 233$$e_4 = 4(e_{}) + 5 = 937$How would I go about finding the explicit formula for this? For something a little simpler it's fairly

The "normal" definition of a sufficient statistics is via independence of the pdf (conditional on the statistic) of the parameter $\theta$. The Fisher-Neyman theorem gives a nice characterization:The statistic $T$ is sufficient iff $f(x;\theta) ## Frechet differentiability, asymptotic normality February 10 I try to prove the asymptotic normality from the Frechet differentiability. Consider$$T(G)-T(F)=L_{F}(G-F)+o\left(d_{\star}(G,F)\right)$$and$$L_{F}\left(F_{n}-F\right)=L_{F}\left(\frac{1}{n}\sum_{i=1}^{n}\left(\delta_{x_{i}}-x\right)\right)=\frac{1} ## Mathematical definitions of infill asymptotics February 10 1 I am writing a paper that uses infill asymptotics and one of my reviewers has asked me to please provide a rigorous mathematical definition of what infill asymptotics is (i.e., with math symbols and notation).I can't seem to find any in the literatur ## Derive the asymptotic distribution of$\frac{2}{n(n-1)}\sum\sum_{i February 10

Derive the asymptotic distribution of Gini's mean diference, which is defined as $\frac{2}{n(n-1)}\sum\sum_{i<j}|X_{i}-X_{j}|$.This is an exercise of Asyptotic Statistics by A.W. van der Vaart. I know it's needed to calculate the mean of kernel funct

## On the computation of the Hessian matrix. February 10

I'm trying to compute the Hessian matrix of a data fit of an ODE model to some data. Below is a cut out of the instructions I'm following (which can also be found at Gutenkunst, R.N., Waterf ...

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