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## real analysis – Periodical functions with \$f^{(n)} = f\$, but \$f^{(k)} neq f\$ for \$kin {1,ldots,n-1}\$

For any infinitely differentiable function $$f: mathbb{R}to mathbb{R}$$ and positive integer $$kinmathbb{N}$$, let $$f^{(k)}$$ denote the $$k$$-th derivative of $$f$$.

For which $$ninmathbb{N}$$, $$n>1$$, is there a periodical function $$f:mathbb{R}to mathbb{R}$$ with the property that $$f^{(n)} = f$$, but $$f^{(k)} neq f$$ for $$kin {1,ldots,n-1}$$?

## real analysis – Finiteness of a bilinear combination

For $$jinmathbb{N}$$, consider continuous functions $$f_j:(0,1)tomathbb{mathbb{R}^+}$$ such that
$$sup_{tin(0,1)}sum_jf_j(t)<+infty,$$
namely $$f_j(t)in L_t^{infty}((0,1),l_j^1(mathbb{N}))$$. I would like to understand whether the quantity
$$S_f:=sum_{j,kinmathbb{N}}int_0^1f_j(t)f_k(t)dt$$
is finite. Some observations:

• If for every $$j$$ the function $$f_j$$ is increasing in $$t$$, then $$S_f$$ is finite. Indeed,

$$S_fleq sum_{jinmathbb{N}}f_j(T)sum_{kinmathbb{N}}int_0^1 f_k(t)dtleq sum_{jinmathbb{N}}f_j(T)int_0^1sup_{tin(0,1)}sum_{jinmathbb{N}}f_k(t)dt<+infty$$

Similarly, if the functions $$f_j$$ attains their maxima on a finite set of points in $$(0,1)$$, then $$S_f$$ is finite.

• More generally, $$S_f$$ is finite if $$f_j(t)in L_j^1L_t^{infty}$$, but of course this is not always the case. For example, one can take (a smooth modification of) $$f_j(t):=delta_{t=j^{-1}}$$. Nevertheless, also in this case $$S_f$$ is finite.

• I tried to consider the case when the $$f_j$$ have plenty of oscillations, but I was unable to find a counterexample.

So my question is the following: is it true that $$S_f$$ is always finite?

In the context I’m interested, there is an additional (weak) control of the derivatives of $$f_j$$, of the form $$2^{-j}f_j’in L^1(0,1)$$, so one could also try to prove the finiteness of $$S_f$$ under this additional assumption.

Thank you for any suggestion.

## complexity classes – What is the computational class of a pushdown automaton with real values?

Say there is a push-down automata, in this example I’ll use a deadfish-like set:

`+: increase x by 1`

`0: set x to 0`

`ln: set x to ln(x) <-- real valued result`

With x being an infinite precision real-valued variable, does this allow said machine to have more power than if it was operating on integers? Or am I misunderstanding something?

## real analysis – Homeomorphism between two metric spaces via identity

Suppose $$(X,d)$$ is a complete metric space with $$U_1,U_2,…$$ nonempty open subsets, with none equal to $$X.$$ Let $$U= bigcap_{n=1}^{infty } U_n neq emptyset$$ and define $$d_n$$ on
$$U_n$$ as $$d_{n}(x,y) =text{min} (D_{n} (x,y),1)$$ where $$D_{n}(x,y) =d(x,y)+lvert frac{1}{d(x,U_n^c) } – frac{1}{d(y,U_n^c)} rvert.$$ Define $$D(x,y)=sum_{n=1}^{infty } frac{1}{2^n} d_{n} (x,y).$$

If I want to show that $$(U,d)$$ is homeomorphic to $$(U,D)$$ by the identity function, then I can show that $$(U,d)$$ and $$(U,D)$$ have the same open sets.

$$implies$$
Let $$V$$ be an open set in $$(U,d)$$. Let $$x in V$$ and $$r<1.$$ If
$$d(x,y), then $$y in V.$$ $$f(x)=x$$ so $$x in f ^{-1} (V).$$ If
$$D(x,y) then $$f(y)=y in V$$ so $$y in f ^{-1} (V).$$ Thefore $$f ^{-1} (V)$$ is an open set.

$$impliedby$$
Let $$x in f ^{-1} (V)$$ open set. Let $$r<1.$$ $$D(x,y) $$f(x)=x in V.$$ If $$d(x,y) then $$y in f ^{-1} (V)$$
so $$f(y)=y in V.$$ Thus $$V$$ is an open set so $$(U,d)$$ is homeomorphic
to $$(U,D).$$

Is it correct to use a radius of less than 1? I did this because $$D(x,y)$$ is less than or equal to 1.

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## real analysis – Proving the limit exists when point tends to the cluster point of a set

Let c be a cluster point of A $$subset mathbb{R}$$ and $$f: A rightarrow mathbb{R}$$ be a function. Suppose that for every sequence in {$$x_{n}$$} in A, such that $$lim x_{n} = c$$, the sequence $${f(x_{n})}_{0}^{infty}$$ is Cauchy. Prove that $$lim _{xrightarrow c}f(x)$$ exists.

I did not know how to approach this question, but I found a skeleton answer here:

Why do I need to show uniqueness?

I still don’t understand how we show that the limit exists though.

## real analysis – Definition of the flow of an ODE and its inverse

At lesson, the teacher considers a flow $$Phi$$ given by the solutions of the ode system for $$tin(0, T)$$ and $$xinmathbb R^d$$,
$$begin{cases} y'(s)=b(y(s), s),&sleq T\ y(t)=x end{cases},qquad(star)$$
that is $$Phi(x, t, s)=y(s)$$ solving $$(star)$$. He said that we will be mostly concerned with $$Phi(cdot, 0, cdot)$$. The field $$b$$ is assumed to be Lipschitz continuous in both variables and bounded.

Then, he intoduces the inverse $$Psi$$ of the above flow as follows: $$Psi(x, 0, s)=y(s)$$ satisfying
$$begin{cases} y'(s)=-b(y(s), t-s),&s
and he said that $$Psi$$ is such that
$$Phi(Psi(x, 0, s), 0, s)=x,quad Psi(Phi(x, 0, s), 0, s)=x.qquad (starstar)$$
I do not understand $$(starstar)$$. Can someone help me? Maybe is the definition of the inverse wrong?

Thank you