‎In this paper‎, ‎we first study the boundedness of generalized‎ weighted composition operators from minimal Möbius invariant‎ ‎subspace to the $n$th weighted type space‎. ‎Then‎, ‎we obtain‎ ‎equivalence conditions for its boundedness by using the Bell‎ ‎polynomials‎. ‎We also find some estimates for the essential‎ ‎norm of this operator and use them to present equivalence‎ ‎conditions for the compactness of such an operator‎.

For a given subspace as a solution space of a linear ODE, we defi ne a special linear parametric group action and prolong it to the jet bundle. We determine these group parameters by moving frame method and prove that these group parameters are the fi rst integrals of the given ODE. These fi rst integrals are used to construct the general form of operators which preserve given subspace invariant.

In this paper, option pricing is given via stochastic analysis and invariant subspace method. Finally numerical solutions is driven and shown via diagram. The considered model is one of the most well known non-linear time series model in which the switching mechanism is controlled by an unobservable state variable that follows a first-order Markov chain. Some analytical solutions for option pricing are given under our considered model. Then numerical solutions are presented via finite difference method.

Let $\mathcal{P}_q(n)$ be the set of all subspaces in the vector space $\mathbb{F}_q^n$. There is a subspace distance $d_S(U,V)$ between any two subspaces $U$ and $V$. A subspace code is also a subset of $\mathcal{P}_q(n)$. It is known that $d_S(U,V)\geq d_H(\nu(\pi U),\nu(\pi V))$, where $\pi\in S_n$, $\nu(U)$ denotes the pivot vector of $E(U)$ and $E(U)$ is the reduced row echelon form of the generator matrix of $U$. In this paper, we show that if $E(U)$ and $E(V)$ have at most one non-zero entry in each rows and each columns then the equality holds. Moreover, we introduce the sets $\mathcal{G}_{U,V}=\{\pi\in S_n\mid d_S(U,V)=d_H(\nu(\pi U),\nu(\pi V))\}$ for any $U,V\in\mathcal{P}_q(n)$ and examine them in the spaces $\mathcal{P}_2(4)$, $\mathcal{P}_2(5)$, $\mathcal{P}_2(6)$ and $\mathcal{P}_3(4)$. It is shown that the groups $1$, $\mathbb{Z}_2$, $\mathbb{Z}_2\times \mathbb{Z}_2$, $S_3$, $S_4$ and $1$, $\mathbb{Z}_2$, $\mathbb{Z}_2\times \mathbb{Z}_2$, $S_3$, $D_8$, $S_3\times \mathbb{Z}_2$, $S_4$, $S_5$ appears between these sets in $\mathcal{P}_2(4)$ and $\mathcal{P}_2(5)$, respectively. Moreover, the groups $1$, $\mathbb{Z}_2$, $\mathbb{Z}_2\times \mathbb{Z}_2$, $S_3$, $D_8$, $\mathbb{Z}_2\times \mathbb{Z}_2 \times \mathbb{Z}_2$, $S_3\times \mathbb{Z}_2$, $D_8\times \mathbb{Z}_2$, $S_4$, $S_3\times S_3$, $S_4\times \mathbb{Z}_2$, $(S_3\times S_3)$:$2$, $S_5$, $S_6$ and $1$, $\mathbb{Z}_2$, $\mathbb{Z}_2\times \mathbb{Z}_2$, $S_3$, $D_8$, $S_4$ appears between these sets in $\mathcal{P}_2(6)$ and $\mathcal{P}_3(4)$, respectively.

Mobius Syndrome is rare, heterogeneous and nonprogressive congentital disorder that is mostly seen sporadically and has been reported few cases. The principle clinical features of the syndrome are bilateral abducens paresis with different degree, incomplete and bilateral facal nerve paresis, orofacial and limbs anomalies. The main cause of clinical syndrome is agenesis of cranial nerve nucleus in the ponse. Our patient is an 18-Year old man with history of poor sucking, lack of impressions during crawing and bilateral club foot at birth time. His parents noticed disorder of eye movements in the infancy period. these findings had stable, non progressive course. Sign that have been found in the clinical assessment are: Bilateral, peripheral Type facial never paresis more on the left, bilateral club foot, complete inability of conjugated gaze to the right convegence of eyes ant left gaze, divergence of eyes at upward gaze and Goiter of grade two. Paraclinical studies such as brain MRI, Echocardiography, ECG, Liver and kideny fuction tests, thyroid hormones and audiometery were normal. NCV Showed axonal ytpe neuopathy of both facial nerves.