Search found 308 matches

by Grigorios Kostakos
Tue Nov 10, 2015 11:39 am
Forum: Real Analysis
Topic: functional equation (01)
Replies: 1
Views: 2188

functional equation (01)

Find all bijective (1-1 & on to) and continuous functions \( h: \mathbb{R}\longrightarrow\mathbb{R}\), such that:
(a) there is \( x_0\in\mathbb{R}\), such that \( h(x_0)=x_0\) and
(b) \(h(x)+h^{-1}(x)=2x\,,\) for all \(x\in\mathbb{R}\) .
by Grigorios Kostakos
Tue Nov 10, 2015 11:35 am
Forum: Real Analysis
Topic: Sequence (01)
Replies: 1
Views: 2228

Sequence (01)

Let \(\{{a_{n}}\}_{n=1}^{\infty}\) a sequence of real numbers such that \[0<a_{1}<a_2\quad {\text{ and}}\quad a_{n+1}=\sqrt{a_{n}\,a_{n-1}}\, , \;n\geqslant2\, .\] a) Prove that the sequence \(\{{a_{n}}\}_{n=1}^{\infty}\) converges. b) Prove that \(\displaystyle\mathop{\lim}\limits_{n\rightarrow{+\i...
by Grigorios Kostakos
Tue Nov 10, 2015 11:30 am
Forum: Real Analysis
Topic: is contractive?
Replies: 0
Views: 1894

is contractive?

A sequence \(\{{\alpha_{n}}\}_{n=1}^{\infty}\) is contractive iff there exists a constant \(c\), with \(0<c<1\), such that, for all \(n\in\mathbb{N}\), holds: \[|a_{n+2}-a_{n+1}|\leqslant c\,|a_{n+1}-a_{n}|\] Examine if the sequence \[a_{n}=({\underbrace{\sin\circ\sin\circ\ldots\circ\sin}_{n-{\rm{ti...
by Grigorios Kostakos
Tue Nov 10, 2015 11:27 am
Forum: Projective Geometry, Solid Geometry
Topic: Inscribed sphere of rhombic triacontahedron
Replies: 1
Views: 4092

Re: Inscribed sphere of rhombic triacontahedron

A rhombic triacontahedron has \(30\) faces, all of which are golden rhombi. A golden rhombus is a rhombus such that the ratio of the long diagonal \(\varDelta\) to the short diagonal \(\delta\) is equal to the golden ratio \(\Phi\), ie \[\frac{\varDelta}{\delta}=\Phi=\frac{1+\sqrt{5}}{2}\quad(1)\,.\...
by Grigorios Kostakos
Mon Nov 09, 2015 4:23 am
Forum: Real Analysis
Topic: Does such function exist?
Replies: 1
Views: 2451

Does such function exist?

Does exist a real function \(f:[a,b]\longrightarrow{\mathbb{R}}\) which is bounded, monotonic and discontinuous in uncountable many points of \([a,b]\) ?
by Grigorios Kostakos
Mon Nov 09, 2015 4:20 am
Forum: Linear Algebra
Topic: Fibonacci numbers as determinants
Replies: 1
Views: 2374

Fibonacci numbers as determinants

Let \(\{{F_{n}}\}_{n=1}^{\infty}\) be the Fibonacci sequence defined as \[F_{n}=F_{n-1}+F_{n-2}\,,\;n\geqslant3\,, \quad F_1=F_2=1\,.\] Prove for every \(n\in\mathbb{N},\,n\geqslant2\) that the \(n\)-th term \(F_n\) of the Fibonacci sequence it is given by the determinant of the \((n-1)\times(n-1)\)...
by Grigorios Kostakos
Mon Nov 09, 2015 1:47 am
Forum: Projective Geometry, Solid Geometry
Topic: Inscribed sphere of rhombic triacontahedron
Replies: 1
Views: 4092

Inscribed sphere of rhombic triacontahedron

Consider a rhombic triacontahedron \(R\) with edge length \(1\) and the inscribed sphere \(S\) of \(R\) (tangent to each of the rhombic triacontahedron's faces). Prove that the radius \(r\) of \(S\) has length \[r=\frac{\Phi^2}{\sqrt{1 + \Phi^2}} =\frac{3 + \sqrt{5}}{\sqrt{10 + 2\sqrt{5}}}\,,\] wher...
by Grigorios Kostakos
Mon Nov 09, 2015 1:40 am
Forum: Differential Geometry
Topic: Circle or line
Replies: 0
Views: 1913

Circle or line

Let \(\overrightarrow{r}(s)\) a natural parametrization of a plane curve. If every tangent line of the curve has the same (constant) distance from a fixed point, prove that the curve must be either a part of a circle, or a part of a line.