Can someone give me some trick to be able to formulate an invariant in the formal correction of designs iterative algorithms?

# Tag: analysis

## Real analysis – Integral $ m ( {x in[0,1]: | f (x) |> t }) $ is finite

Suppose $ f in L ^ p[0,1]$ for some $ p $ with $ 1 le p le infty $. Define

$$ h (t): = m ( {x in[0,1]: | f (x) |> t }) $$ where $ m $ It is the measure of Lebesge. Show that $ int_0 ^ infty h (t) ; dt < infty $.

My first thought was to try to use the inequality of Chebyshev for finite $ p $ as follows:

$$ int_0 ^ infty h (t) ; dt = int_0 ^ infty m ( {x in[0,1]: | f (x) |> t }) ; dt le int_0 ^ infty frac1 {t ^ p} int _ { {x in[0,1]: | f (x) |> t }} | f (x) | ^ p ; dmdt $$

And then use the fact that $ f $ is in $ L ^ p $ To join the internal integral by any number $ M_t $ depends on $ t $ that should go to 0 as $ t $ goes to infinity. But this does not seem to be the way forward since it is not clear how to integrate the result $ frac {M_t} {t ^ p} $ And even if I could, it's not obvious to me if the result would be finite. In fact, it seems that it will not be unless $ M_t to0 $ as $ t to0 $.

So, what other methods should I try at this point?

## Vulnerability scanners – Analysis of binaries by ZZUF and PEACH fuzzers

I am a beginner in the software testing area. I installed two different fuzzers ZZUF and PEACH fuzzers. Unfortunately, my lack of knowledge I could not test the C ++ binaries with both fuzzers. The binary test process is not provided in the PEACH and ZZUF tutorials.

For example abc.c

`

int main (void)

{

char session start[16];

char password[16];

```
printf ("Login:");
scanf ("% s", login);
printf ("Password:");
scanf ("% s", password);
if (strcmp (login, "root") == 0) {
if (strcmp (password, "toor") == 0) {
printf ("Success. n");
returns 0;
}
}
printf ("Fail. n");
returns 1;
```

} `

- How can I fuzz binary code or C by ZZUF and PEACH fuzzers?
- Is it possible to test the binary data of LAVA-M (base64, uniq, md5sum, who) or OpenSSL 1.1.0f?

I firmly believe that your help will increase my knowledge.

Thank you

## analysis: prove that $ f $ is a local diffeomorphism of the class $ C ^ 1 $

Leave $ f: V to mathbb {R} ^ {n + 1} $ be defined by $ f (x) = left ( sigma left ( frac {x} {|| x ||} right), || x || – 1 right) $, where $ V = mathbb {R} ^ {n + 1} – { lambda N; lambda in mathbb {R} } $ Y $ sigma $ It is the stereographic projection of the north pole. $ N = (0, …, 0, 1) $.

(i) prove that $ f $ It is a local class diffeomorphism. $ C ^ 1 $

(ii) show that $ f $ it is in fact a class $ C ^ 1 $ global diffeomorphism, finding an expression for $ f ^ {- 1} (y) $.

(iii) Characterize the domain of the inverse. $ f ^ {- 1} $.

We know it by $ S ^ n = {v = (v_1, …, v_n, v_ {n + 1}) in mathbb {R} ^ {n + 1}; || v || = 1 } $, we have $ sigma: S ^ n to mathbb {R} ^ n $ where $ sigma (v) = left ( frac {v_ {1}} {1 – v_ {n + 1}}, …, frac {v_ {n}} {1 – v_ {n + 1} } right) $ It is a homomorphism. We must certainly use the inverse function theorem, but the matrix $ Df (x) $ it's confusing. Was that the way?

## Functional analysis. Ideal for strictly unique operators.

[J. Lindenstrauss and L. Tzafriri. Classical Banach spaces I. Sequence spaces. Springer 1977]. On page 76, after Prop. 2.c.3, it is said that the test in 2.c.3 shows that an operator $ T: ell_p to ell_p $ It is strictly unique if and only if it is compact.

[F. Albiac and N. Kalton. Topics in Banach space theory. Springer 2006] Theorem 5.5.1 says that a weakly compact operator $ T: C (K) to X $ is strictly singular, and Theorem 5.2.3 says that an operator does not weakly compact $ T: C (K) to X $ It is not strictly singular.

Note that $ ell_ infty $ is a $ C (K) $ space with $ K $ The Stone-Cech compaction of the set of positive integers.

## Analysis: Is it possible to find the expression of $ c $ in MVT for a specific function?

Yes $ f (x) $ It is a differentiable function in the interval. $[x_0,x_1]$

The MVT implies the equation. $ frac {f (x_0) -f (x_1)} {x_0-x_1} = f & # 39; (c) $ where, $ c in[x_0 ,x_1]$.

Is it possible to find the exact expression of $ c $ using $ x_0 $ Y $ x_1 $

?

Let's make the case much simpler.

How to find the equation of a tangent to the function. $ f (x) = e ^ x $ which parallel

to the intersection between $ (1, e) $ Y $ (m, e ^ m) $.

## real analysis – Show that $ g (x) = x + f (x) $ is the projective

Leave $ f: { mathbb {R}} ^ n rightarrow { mathbb {R}} ^ n $ be continuously differentiable and $ C in (0.1) $ a constant, so that $ {|| Df (x) ||} _ {op} leq C $ $ forall x in { mathbb {R}} n $ with $ op $ Being an operator standard.

Show that $ g: { mathbb {R}} ^ n rightarrow { mathbb {R}} ^ n $, $ g (x) = x + f (x) $ It is overjective.

I tried to follow:

$ g (x) = x + f (x) $

$ Leftrightarrow Dg (x) = Dx + Df (x) $

$ Leftrightarrow {|| Df (x) ||} _ {op} = {|| Dg (x) -Dx ||} _ {op} leq C $

I'm not really sure how to move forward from here. Is it possible to use the sub-additive of the matrix rules even if there is a minus sign in the equation?

I'm grateful for every clue.

## real analysis – inverse triangle inequality in $ L ^ p $

They require some inequalities of generalization. I do not know if they are true or not. Can you help me ?

Here we are talking about $ L ^ p $ spaces with $ p> 1 $.

I know that in the real line:

$$ || x | – | and || <| x-y | <| x | + | and | $$ equivalently

$$ || x | – | and || <| x + y | <| x | + | and | $$

Now I am trying to find similar inequalities in the spaces of Lebesgues.

I already found that:

$$ (| x + y |) ^ p leq 2 ^ {p-1} (| x | ^ p + | y | ^ p) $$ Thanks to Jensen's inequality.

I also know that Minkowski's inequality tells me:

$$ || f + g || _ {L ^ p} leq || f || _ {L ^ p} + || g || _ {L ^ p} $$

Now I'm looking for something in the other limit. I mean, like my

Friends told me that it should be true:$$ | || f || _ {L ^ p} – || g || _ {L ^ p} | leq || f-g || _ {L ^ p} $$ Y

equivalently$$ | || f || _ {L ^ p} – || g || _ {L ^ p} | leq || f + g || _ {L ^ p} $$

I would also like to find something like this:

$$ lambda | (| x | ^ p – | and | ^ p) | leq (| x + y |) ^ p $$

Do you know if there is something like those 2 inequalities and, if so, how the tests?

Thank you !

## Functional analysis: the optimal asymptotic behavior of the coefficient in Hardy-Littlewood's maximum inequality

It is well known that $ f in L ^ 1 ( mathbb {R ^ n}) $,$ mu (x in mathbb {R ^ n} | Mf (x)> lambda) le frac {C_n} { lambda} int _ { mathbb {R ^ n}} | f | mathrm {d mu} $, where $ C_n $ It is a constant only depends on $ n $.

It is easy to see $ C_n le 2 ^ n $, but how to determine its optimal asymptotic behavior? For example, it makes $ C_n $ bounded in $ n $? Is $ C_n $ bounded by polynomials in $ n $?

## Real analysis – Identification of a determining condition.

Has the following condition been studied and, if so, is there some known class of functions that satisfies it?

Condition.For a fixed $ n> 0 $, everyone $ 2 times 2 $ under the matrix

$$

begin {bmatrix}

1 & x & dotsm & x ^ n \

1 & f & dotsm & f ^ n

end {bmatrix}

$$

are linearly independent about $ Bbb {Z} $, where $ f: Bbb {R} to Bbb {R} $ Y $ f neq 0, x $.

In other words, I would like to characterize the functions. $ f $ for which $ x ^ si ^ j – x ^ jf ^ i $, with $ 0 leq i <j leq n $are linearly independent about $ Bbb {Z} $.