functional programming – My implementation of Clojure’s assoc-in

This is my implementation of Clojure’s assoc-in function. I am looking for tips on making it more idiomatic and in general, better.

(defn my-assoc-in
  (m (& ks) v)
  (let (sorted-keys (reverse ks))
    (loop (curr-key (first sorted-keys)
           rem-keys (rest sorted-keys)
           curr-map (get-in m (reverse rem-keys))
           acc-val  (assoc curr-map curr-key v))
      (if (empty? rem-keys)
        acc-val
        (let (next-key (first rem-keys)
              next-rem (rest rem-keys)
              next-map (get-in m next-rem)
              next-val (assoc next-map next-key acc-val))
          (recur next-key next-rem next-map next-val))))))

Finding two nontrivial functional dependencies that follow from F (but are not in F)

Given below is the set F of functional dependencies for the relational schema:

R = {A,B,C,D,E,F,G}.

F = {AB → C,BC → D,G → F,AE → FG}

Show two nontrivial functional dependencies that follow from F:

Which is :

AB → CD

ABC → D

I don’t know how those two derived from? Could someone walk me through? thanks

functional analysis – Relation between fractional and integer Sobolev norms

I encountered a situation where I have to add two norms defined on the boundary:

$$C_1 ||u||^2_{L_2(partialOmega)} + C_2 ||u||^2_{H^{3/2}(partial Omega)},$$

but do not really know how to manipulate this expression. Is there a relation between the norms?

Functional inequalities involving the condition $left(int_0^t f(x)dxright)^2 ge int_0^t f(x)^3dx$

I was reading the solution to a functional inequality in an article when the author made the following remark without giving any proof: let $f(x): (0, infty)to(0, infty)$ be locally integrable and such that
$$left(int_0^t f(x)dxright)^2 ge int_0^t f(x)^3dx$$
for all $t>0$. Then, the following statement is true:

$int_0^t f(x)^gamma dx le frac{1}{gamma +1}left(2int_0^t f(x)dxright)^{(gamma + 1)/2}$ for all positive $t$ and $gamma in (1,3)$.

Again, there is no proof in the article, so I don’t know if this is fairly easy to prove or very involved. One thing that might be worth mentioning is that the inequalities above become exact when $f(x)=x$. I am wondering if anyone has an idea or have seen these before.

functional analysis – On a subspace that is isomorphic to a dense subspace

Let $X$ be an infinite dimensional Banach space and let $M$ be a dense subspace of $X$, i.e., $overline{M}=X$. Let $N$ be another subspace of $X$ such that $N$ is topologically isomorphic to $M$.
Then, is it true that $N$ is necessarily dense in $X$, i.e., $overline{N}=X$ ? I think (intuitively) this should be true. But, how to proceed to prove this fact? Could anybody provide a hint?

functional analysis – Biconjugate formula in non-separated locally convex spaces

I have a locally convex space $X$ with topological dual $X^*$ and coupling $langle x,x^* rangle:=x^*(x), xin X, x^*in X^*$.

For $f:Xtooverline{mathbb{R}}$ one defines its convex conjugate by $f^*(x^*):=sup{langle x,x^* rangle-f(x)mid xin X}$ and similarly for $g:X^*tooverline{mathbb{R}}$ one defines its convex conjugate by $g^*(x):=sup{langle x,x^* rangle-g(x^*)mid x^*in X^*}$.

The biconjugate formula states that for a proper convex lower semicontinuous $f:Xtooverline{mathbb{R}}$ one has $f^{**}=f$.

The proof we went over in class is based on $X$ being Hausdorff separated and relies on separation theorems for the epigraph.

My question is whether the biconjugate formula holds when the space $X$ is NOT Hausdorff separated?

functional analysis – Show that a sequence is eventually null

Let H be a separable Hilbert space and ${e_n}$ and ${f_n}$ be orthonormal basis in H. Define the linear operator:

begin{equation*}
Sx = sum_{n=1}^{infty} mu_n (x,e_n)f_n.
end{equation*}

I have to prove that if the dimension of Im S is finite, then ${mu_n}$ is eventually null.

This is my attempt.

Let’s suppose that $rank; T=k$.

$$Sx= sum_{n=1}^infty alpha_n(x|e_n)f_n= sum_{n=1}^k alpha_n(x|e_n)f_n+ sum_{n=k+1}^infty alpha_n(x|e_n)f_n= sum_{n=1}^k alpha_n(x|e_n)f_n$$

So we have:

begin{equation} label{0}
sum_{n=k+1}^infty alpha_n(x|e_n)f_n=0
end{equation}

As ${f_n}$ is a basis:
$$ alpha_n(x|e_n) =0 quad forall ngeq k+1 $$

I don’t know how to conclude rigorously that $alpha_n=0$.

I’d appreciate if someone could revise this.

Thank you.

javascript: conditioning the start of an animation to a button (functional component of React + JS)

Hello Goodnight. I have a course to do. It is a Tinder style. One of the additional challenges is animating when you click the buttons. I've already made the animation, but I don't know how to get it started by simply clicking a certain button.
I am using React and the languages ​​I learned are just CSS and JS, ah ## Headers ## and it has to be in a functional component, because I am using Hooks.

Here is an example: https://astro-match.surge.sh/

Object to animate:


{perfil.name}, {perfil.age}

{perfil.bio}

Thank you!

Functional analysis: how to find a sequence that converges with the fixed random variable

Suppose $ mathscr {A} $ is a convex set $ L ^ infty (P) $, $ mathscr {B} $ is the closing of $ mathscr {A} $ under the topology $ σ (L ^ infty (P), L ^ 1 (P)) $. For each $ X in mathscr {B} $, we can find a sequence $ {X_n } _ {n geq 1} subseteq mathscr {A} $ S t. $ X_n xrightarrow {L_1} X $?

I thought about the problem a lot but still had no answer, so I ask for help here. Can you give me proof of existence or a counterexample? Thank you.

Functional analysis: example of a non-reflective Banach space and two sequences.

Leave $ (E, mathcal {A}, mu) $ be a space of finite measure and $ X $ To be a Banach space. The set of all Bochner integrable functions from $ E $ inside $ X $ is denoted by $ mathcal {L} _X ^ 1 $.

Yes $ X $ is thoughtful, we have the following theorem

Theorem 1

Leave $ (f_n) _ {n geq 1} subset mathcal {L} _ {X} ^ 1 $ it is a sequence with: $$ sup_n int_ {E} { | f_n | d mu} < infty. $$
Then there is $ h _ { infty} in mathcal {L} _ { mathbb {R}} ^ 1 $ and a subsequence $ (g_k) _k $ of $ (f_n) _n $ such that for each subsequence $ (h_m) _m $ of $ (g_k) _k $ : $$ frac {1} {i} sum_ {j = 1} ^ {i} {h_j (t)} to h _ { infty} (t) text {weakly in} X text {a.e. } $$

Proof of this result exists in the article "Infinite dimension extension of a Komlos theorem"by Erik J. Balder (theorem A)

Yes $ X $ not thoughtful, we have the following theorem

Theorem 2

Leave $ (f_n) _ {n geq 1} subset mathcal {L} _ {X} ^ 1 $ it is a sequence with:
$$
begin {cases}
bullet ~~ {f_n (t) } text {is relatively weakly compact a.e.,} \
bullet ~~ sup_n int_ {E} { | f_n | d mu} < infty. \
end {cases}
$$

Then there is $ h _ { infty} in mathcal {L} _ { mathbb {R}} ^ 1 $ and a subsequence $ (g_k) _k $ of $ (f_n) _n $ such that for each subsequence $ (h_m) _m $ of $ (g_k) _k $ : $$ frac {1} {i} sum_ {j = 1} ^ {i} {h_j (t)} to h _ { infty} (t) text {weakly in} X text {a.e. } $$

Proof of this result exists in the article "Infinite dimension extension of a Komlos theorem"by Erik J. Balder (theorem B)

My problem:

I want an example of a non-reflective space of Banach and two sequences, so that:

  • The first sequence is limited in $ mathcal {L} _ {X} ^ 1 $ but it does not verify the consequence of theorem 1.

  • The second sequence that verifies the hypotheses of Theorem 2 and its consequences.