complex analysis – Express $f(t) = cos(4t)+sin(6t)$ as trigonometric polynomial.

I’m asked to express the function $f(t) = cos(4t)+sin(6t)$ as trigonometric polynomial of the form $sum_{n=-N}^N c_ne^{inwt}$. My first approach was to use Eulers formula to arrive at $f(t)= frac{1}{2}e^{i4t}+frac{1}{2}e^{-i4t}+frac{1}{2i}e^{i6t}-frac{1}{2i}e^{-i6t}$, but I’m not sure if this is correct and how I would turn this into a series.

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real analysis – derivative of determinant

I am struggling to understand how to use of the definition of a derivative and definition of a limit to compute the derivative of the determinant function of a 2×2 matrix. It is a subset of $R^4$ under Euclidean norm.
Im very new to math analysis.
I don’t know where to start, I know that from the derivative def that the lim as h goes to 0 is $|((f(x+h)-f(x))/h|$ = a
for f:R->R

lexical analysis – What makes a nested function in C so complicated?

Disclaimer: I’m not a compiler expert, but some of my best friends are! 🙂

I do a lot of embedded systems programming where stack (and other memory) is limited. Most of my code is written in gcc, so I thought nested functions could be useful for reducing stack usage. Here’s an example of what I had in mind:

(Though the algorithm isn’t important for this question, assume that list_traverse takes a pointer to a list head and a function. The function gets called with each element in turn, returning false when it wants the traversal to stop.)

bool list_contains(list_t *head, list_t *item) {
  bool found = false;
  bool contains_aux(list_t *list) {
    if (list == item) {
      found = true;
      return false;
    }
    return true;
  }
  list_traverse(head, contains_aux);
  return found;
}

I assumed that the nested contains_aux() function would simply be compiled “inside” the list_contains() and be able to reference the lexically closed variables (found and item). But I was wrong.

Instead, looking at the generated code, I see that it installs some sort of trampoline function on the stack and calls that before getting to the contains_aux() function.

I guess that makes sense: contains_aux() can’t share the same stack frame as list_contains() because we’re another function deeper in the stack when its called (via list_traverse()).

So is there some way to accomplish this sort of lexical closure efficiently? Or should I just resign myself using un-nested functions and passing a “context” object to contains_aux()?

calculus and analysis – question on correct use of Limit for multivariable function

V 12.1 on windows.

This limit $lim_{(x rightarrow 0,yrightarrow 0)} frac{x^2-y^2}{x^2+y^2}$ depends on the direction. So the limit does not exist, or could be written as Maple does it, which is $-1dots1$, here is the help from Maple on this:

enter image description here

How can one get Mathematica to give this result? Now Mathematica says the limit is $1$. I tried the Direction option but not able to make it change its mind.

f = (x^2 - y^2)/(x^2 + y^2);
Limit(f, {x -> 0, y -> 0})
(* 1 *)

But we see the limit depends on the direction

 Limit(Limit(f, x -> 0), y -> 0)
 (* -1 *)

 Limit(Limit(f, y -> 0), x -> 0)
 (*  1 *)

Here is also Maple to confirm

restart;
f:=(x^2-y^2)/(x^2+y^2);
limit(f, (x=0,y=0));

enter image description here

Btw, this is not the only one I found, here is another

f = (x^2*y^2)/(x^4 + y^4);
Limit(f, {x -> 0, y -> 0})
(* 0 *)

Maple gives

restart;
f:=x^2*y^2/(x^4+y^4);
limit(f,(y=0,x=0))
  (* 0 .. 1/2 *)

And another one (this one is from youtube actually, so you can see they also say there the limit does not exist)

f = (x^4 - 4 y^2)/(x^2 + 2 y^2);
Limit(f, {x -> 0, y -> 0})
(* 0 *)

restart;
f:=(x^4-4*y^2)/(x^2+2*y^2);
limit(f, (x=0,y=0));
(* -2 .. 0 *)

So I have feeling I am not using Limit in Mathematica correctly, or missing something about its correct use, but do not now know how to correct it. As I said, I tried different Direction option.

functional analysis – Can we use the duality notation such that the second variable is an element of the measure space?

I read article New Sequential Compactness Results for Spaces of Scalarly Integrable Functions by Erik J. Balder, on page 8 : Author defined the function $a: Ttimes Eto mathbb{R}$ by the usual duality between $E$ and $E^*=F$ $( ⟨.,.⟩)$ such that:
$$
a(t,x)=langle x, trangle
$$

But normally the second variable $(tin T)$ must be an element of dual of $E£, right? Why does the author make this notation?

An idea please
enter image description here

fa.functional analysis – Reference request: sufficiently smooth functions on the plane belong to the projective tensor square of $L^2$ of the line

Let $newcommand{ptp}{widehat{otimes}}ptp$ denote the projective tensor product of Banach spaces. Some back of the envelope calcuations, using the Fourier transform and Plancherel/Parseval, suggest to me that the following should be true:
$newcommand{Real}{{bf R}}$

for some $k > 1$, every compactly supported $C^k$-function $Real^2toReal$ belongs to $L^2(Real)ptp L^2(Real)$.

(Recall that in contrast, there are continuous functions on $(0,1)^2$ that do not belong to $C(0,1)ptp C(0,1)$.)

If the claim above is true, I would like to know if there are standard references, perhaps from the world of integral kernel operators or Sobolev spaces, which I could cite, rather than reinventing the wheel (and probably getting suboptimal values of $k$).

In a slightly different direction, I would also be interested to know of references which prove analogous resuts for $C^k$ functions (suitably interpreted) on compact connected Lie groups.

real analysis – we want to prove that $(n,n+1) cap mathbb{N}$ is empty

Indeed, let $S(n)$ be the statement ${ n : (n,n+1) cap mathbb{N} = varnothing }$. Clearly, $(1,2) cap mathbb{N} = varnothing$, thus $1 in S(n)$. Assume now that $S(n)$ is true. Then,

$$ (n+1,n+2) cap mathbb{N} = (n,n+1) cap (mathbb{N} setminus (n+1,n+2) ) cap mathbb{N} = (n,n+1) cap mathbb{N} cap ( mathbb{N} setminus (n+1,n+2) ) = varnothing $$

and the claim follows by the principle of mathematical induction.

Is this correct proof?