## To remove the invariant in an iterative analysis?

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

## 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];

``````printf ("Login:");

if (strcmp (login, "root") == 0) {
if (strcmp (password, "toor") == 0) {
printf ("Success.  n");
returns 0;
}
}
printf ("Fail.  n");
returns 1;
``````

} `

1. How can I fuzz binary code or C by ZZUF and PEACH fuzzers?
2. 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.

$$(| x + y |) ^ p leq 2 ^ {p-1} (| x | ^ p + | y ​​| ^ p)$$ 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 $$0 leq i are linearly independent about $$Bbb {Z}$$.