arXiv subjects

Nikolaos E. Sofronidis

Publications and source records attributed to Nikolaos E. Sofronidis.

6 recordsLinked to original sources

arXiv2018

On homeomorphisms and $C^{1}$ maps

Our purpose in this article is first, following [8], to prove that if $α$, $β$ are any points of the open unit disc $D(0;1)$ in the complex plane ${\bf C}$ and $r$, $s$ are any positive real numbers such that ${\overline{D}}( α;r) \subseteq D(0;1)$ and ${\overline{D}}( β;s) \subseteq D(0;1)$, then there exist $t \in (0,1)$ and a homeomorphism $h : {\overline{D}}(0;1) \rightarrow {\overline{D}}(0;1)$ such that ${\overline{D}}( α;r) \subseteq D(0;t)$, ${\overline{D}}( β;s) \subseteq D(0;t)$, $h \left[ {\overline{D}}( α;r) \right] = {\overline{D}}( β;s)$ and $h = id$ on ${\overline{D}}(0;1) \setminus D(0;t)$, and second, following [9], to prove that if $q \in {\bf N} \setminus \{ 0, 1 \} $ and ${\bf B}({\bf 0};1)$ is the open unit ball in ${\bf R}^{q}$, while for any $t>0$, we set $f^{(t)}( {\bf x} ) = \frac{ t {\bf x} }{ 1 + (t-1) \Vert {\bf x} \Vert }$, whenever ${\bf x} \in {\overline{\bf B}}({\bf 0};1)$, then $f^{(t)} \rightarrow id$ in $C^{1} \left( {\overline{\bf B}}({\bf 0};1) , {\bf R}^{q} \right) $ as $t \rightarrow 1^{+}$.

math.GM↗

arXiv2017

On geometry and mechanics

Our purpose in this article is first, following [14], to find the topological upper limits of projections of secant planes to $C^{1}$ surfaces and the topological upper limits of projections of secant hyperplanes to $C^{1}$ hypersurfaces and second to prove that every $C^{2}$ space curve can be the solution of the principle of stationary action.

math.GM↗

arXiv2017

Diffeomorphisms of the closed unit disc converging to the identity

If $\mathcal{G}$ is the group (under composition) of diffeomorphisms $f : {\bar{D}}(0;1) \rightarrow {\bar{D}}(0;1)$ of the closed unit disc ${\bar{D}}(0;1)$ which are the identity map $id : {\bar{D}}(0;1) \rightarrow {\bar{D}}(0;1)$ on the closed unit circle and satisfy the condition $det(J(f)) > 0$, where $J(f)$ is the Jacobian matrix of $f$ or (equivalently) the Fréchet derivative of $f$, then $\mathcal{G}$ equipped with the metric $d_{\mathcal{G}}(f,g) = \Vert f-g \Vert_{\infty } + \Vert J(f) - J(g) \Vert_{\infty }$, where $f$, $g$ range over $\mathcal{G}$, is a metric space in which $d_{\mathcal{G}} \left( f_{t} , id \right) \rightarrow 0$ as $t \rightarrow 1^{+}$, where $f_{t}(z) = \frac{ tz }{ 1 + (t-1) \vert z \vert }$, whenever $z \in {\bar{D}}(0;1)$ and $t \geq 1$.

math.GM↗

arXiv2016

Fixed point free homeomorphisms of the complex plane

Our purpose in this article is to prove that the group $H({\bf C})$ of homeomorphisms of the complex plane ${\bf C}$ is a metric group equipped with the metric induced by uniform convergence of homeomorphisms and their inverses on compacts and the set $$\left\{ h \in H({\bf C}) : ( \forall z \in {\bf C} )( h(z) \neq z ) \right\} $$ of fixed point free homeomorphisms of the complex plane is a conjugacy invariant dense $G_{δ}$ subset of $H({\bf C})$.

math.GN↗

arXiv2015

On continuous Polish group actions and equivalence relations

Let $X = \left\{P \in [0,1]^{\bf N} : \left(\forall ν\in {\bf N} \right) \left(P \left(\{ν\} \right) > 0 \right) \wedge \sum\limits_{ν= 0}^{\infty} P \left(\{ν\} \right) = 1 \right\} $ be the Polish space of probability measures on ${\bf N}$, each of which assigns positive probability to every elementary event, while for any $P \in X$, let $Γ_{P} = \left\{ξ\in L^{1}({\bf N}, P) : \left(\forall ν\in {\bf N} \right) \left(ξ(ν) > 0 \right) \wedge \sum\limits_{ν= 0}^{\infty} ξ(ν) P \left(\{ν\} \right) = 1 \right\} $ and let $Φ_{P} : Γ_{P} \ni ξ\mapsto Φ_{P}(ξ) \in X$ be defined by the relation $\left(Φ_{P}(ξ) \right) \left(\{ν\} \right) = ξ(ν) P \left(\{ν\} \right) $, whenever $ν\in {\bf N}$. If we consider the equivalence relation $E = \left\{(P,Q) \in X^{2} : \left(\exists ξ\in Γ_{P} \right) \left(Q = Φ_{P}(ξ) \right) \right\} $, the Polish space ${\bf P} = \left\{{\bf x} \in {\ell}^{1} \left({\bf R} \right) : \left(\forall n \in {\bf N} \right) \left({\bf x}(n) > 0 \right) \right\} $ and the commutative Polish group ${\bf G} = \left\{{\bf g} \in (0, \infty)^{\bf N} : \lim\limits_{n \rightarrow \infty}{\bf g}(n) = 1 \right\} $, while we set $\left({\bf g} \cdot {\bf x} \right) (n) = {\bf g}(n){\bf x}(n)$, whenever ${\bf g} \in {\bf G}$, ${\bf x} \in {\bf P}$ and $n \in {\bf N}$, then $E$ is definable and it admits a strong approximation by the turbulent Polish group action of ${\bf G}$ on ${\bf P}$.

math.LO↗

arXiv2015

Variational inequalities

If $- \infty < α< β< \infty $ and $f \in C^{3} \left( [ α, β] \times {\bf R}^{2} , {\bf R} \right) $ is bounded, while $y \in C^{2} \left( [ α, β] , {\bf R} \right) $ solves the typical one-dimensional problem of the calculus of variations to minimize the function $$F \left( y \right) = \int_{ α}^{ β}f \left( x, y(x), y'(x) \right) dx,$$ then for any ${ϕ} \in C^{2} \left( [ α, β] , {\bf R} \right) $ for which ${ϕ}^{(k)}( α) = {ϕ}^{(k)}( β) = 0$ for every $k \in \{ 0, 1, 2 \} $, we prove that $\int_{α}^{β} \left( \frac{ {\partial }^{2}f }{ \partial y^{2} } {ϕ}^{2} - \frac{ {\partial }^{3}f }{ \partial y^{2} \partial y' } 2 {ϕ}^{3} \right) dx$ $\geq \int_{α}^{β} \left( \frac{ {\partial }^{2}f }{ \partial y \partial y' } 2 ϕϕ' + \frac{ {\partial }^{3}f }{ \partial y {\partial y'}^{2} } 2 {ϕ}^{2} ϕ' + \frac{ {\partial }^{2}f }{ {\partial y'}^{2} } ϕϕ" + \frac{ {\partial }^{3}f }{ \partial y {\partial y'}^{2} } ϕ' {ϕ}^{2} + \frac{ {\partial }^{3}f }{ {\partial y'}^{3} } ϕ{ϕ'}^{2} \right) dx$, so either the above are variational inequalities of motion or the Lagrangian of motion is not $C^{3}$.

math.OC↗