The finite frame theory is a very important part offrame theory due to its significant relevance in various branchesof mathematical applications. Studying controlled inite frames isthe goal of the work. To this end, we introduce controlled framesin a inite-dimensional Hilbert space and study some properties ofthem. The main class of inite frames in frame applied problemsis Parseval frames. By viewpoint to this, a brief discussion aboutParseval frames is presented and also Parseval controlled framesare investigated. Afterward, the paper characterizes all operatorsthat construct controlled inite frames. Furthermore, controlledinite frames are also considered as a proper subset of dual framesby the equivalency relation between frames.

<abstract><p>One of the most famous equations that are widely used in various branches of physics, mathematics, financial markets, etc. is the Langevin equation. In this work, we investigate the existence of the solution for two generalized fractional hybrid Langevin equations under different boundary conditions. For this purpose, the problem of the existence of a solution will become the problem of finding a fixed point for an operator defined in the Banach space. To achieve the result, one of the recent fixed point techniques, namely the $ \alpha $-$ \psi $-contraction technique, will be used. We provide sufficient conditions to use this type of contraction in our main theorems. In the calculations of the auxiliary lemmas that we present, the Mittag-Leffler function plays a fundamental role. The fractional derivative operators used are of the Caputo type. Two examples are provided to demonstrate the validity of the obtained theorems. Also, some figures and a table are presented to illustrate the results.</p></abstract>

In this paper, we prove the Hyers-Ulam stability of functional equation

In this paper, the linear operators on M nm which preserve or strongly preserve chain majorization are characterized.