Intereting Posts

Function $\mathbb{R}\to\mathbb{R}$ that is continuous and bounded, but not uniformly continuous
Are complex determinants for matrices possible and if so, how can they be interpreted?
How to find a random axis or unit vector in 3D?
Lebesgue integral of $\chi_{\mathbb{Q}}: \mathbb{R} \rightarrow \mathbb{R}$
Adjacency matrix for n-dimensional hypercube graph
Problem on Galois theory and irreducible polynomial
The Monster PolyLog Integral $\int_0^\infty \frac{Li_n(-\sigma x)Li_m(-\omega x^2)}{x^3}dx$
$13\mid4^{2n+1}+3^{n+2}$
Does the special Pell equation $X^2-dY^2=Z^2$ have a simple general parameterization?
How can I solve this triple integral $\iiint_{B} y\;dxdydz$ on a defined set?
Is this an integrable function?
Recovering connection from parallel transport
Equation that can easily be changed to output the digit in 1's, 10's ,100's etc?
if $Q$ and $P$ are distinct $p$-Sylow subgroups then $Q\not\subseteq N_G(P)$.
Proof that $a^b>b^a$ if $a<b$ are integers larger or equal to two and $(a,b)\neq (2,3),(2,4)$

Let $x\in\mathbb R$ be a periodic point with lenght 2 of the recursion $x_{n+1}=f(x_n)$

My book about Dynamic systems says that this recursion has a fixed point now, because a periodic point with length 2 exists.

May you could help me to prove this statement?

- Dense set in the unit circle- reference needed
- Can we construct a Koch curve with similarity dimension $s\in$?
- Find and classify the bifurcations that occur as $\mu$ varies for the system
- Picard iteration (general)
- A rational orbit that's provably dense in the reals?
- Michaelis-Menten steady state hypothesis

- Dulac's criterion and global stability connection
- A differentiable manifold of class $\mathcal{C}^{r}$ tangent to $E^{\pm}$ and representable as graph
- Properties of the space of $ T $-invariant probability measures over a compact topological space.
- In this case, does $\{x_n\}$ converge given that $\{x_{2m}\}$ and $\{x_{2m+1}\}$ converge?
- Solve a system of second order differential equations
- What can Lefschetz fixed point theorem tell us about the group of automorphisms of a compact Riemann surface?
- Proof Strategy for a Dynamical System of Points on the Plane
- How does rounding affect Fibonacci-ish sequences?
- If $f(x)=x^2-x-1$ and $f^n(x)=f(f(\cdots f(x)\cdots))$, find all $x$ for which $f^{3n}(x)$ converges.
- How to obtain a possible state space representation of this 2nd order transfer function?

In general if $f$ is continuous, and there exists an integer k>1 and a real number $a$ such that $f^k(a)=a$ then there exists a real number $r$ such that $f(r)=r$.

**Proof:** Let $g(x)=f(x)-x$. Since:

$g(a)+g(f(a))+g(f^2(a))+…+g(f^{k-1}(a))=f^k(a)-a=0$, therefore either one of the numbers $g(a),g(f(a)),g(f^2(a)),…,g(f^{k-1}(a))$ is zero (in this case we are done) or one of these numbers is positive and another number is negative. Assume WLOG that the second case holds. Let $g(f^i(a))g(f^j(a))<0$ (it means that they have different signs). Since, $g$ has a sign change in the interval $[\min(f^i(a),f^j(a)),\max(f^i(a),f^j(a))]$, thus $g$ has a root.

Also, if you like hitting flies with sledgehammers, this directly follows from Sharkovskii’s theorem.

- How do I integrate the following? $\int{\frac{(1+x^{2})\mathrm dx}{(1-x^{2})\sqrt{1+x^{4}}}}$
- Characteristic of a product ring?
- What is the probability that the center of the circle is contained within the triangle?
- The number of paths on a graph of a fixed length w/o repeatings
- Prove composition is a measurable function
- Insertion sort proof
- Suppose that $f(x)$ is continuous on $(0, \infty)$ such that for all $x > 0$,$f(x^2) = f(x)$. Prove that $f$ is a constant function.
- Lebesgue outer measure of $\cap\mathbb{Q}$
- $t$ – th graded piece of the coordinate ring of $Y \times Z$
- Intuition behind proof of bounded convergence theorem in Stein-Shakarchi
- Convergence of the series $\sqrtn-1$
- $\operatorname{MaxSpec}(A)$ closed
- Examples of non-obvious isomorphisms following from the first isomorphism theorem
- A game on a graph
- $ z^n = a_n + b_ni $ Show that $ b_{n+2} – 2b_{n+1} + 5b_n = 0 $ (complex numbers)