# Mathematical Background and Algorithms of a Collection of Android Apps for a Google Play Store Page
## 3 Problem Solving
### Private File Manager App
The encryption can be represented by a matrix transformation \(\mathbf{E}\), where \(\mathbf{E}\) is an encryption matrix of size \(n\times n\). The resulting encrypted file \(\mathbf{F}_{\text{encrypted}}\) is:  
\[\text{CopyFile}(f,g)=\text{Copy file $f$ to $g$} \tag{35}\]7  
To conceal the file, we utilize a function \(\mathbf{H}\) applied to the encrypted data, where \(\mathbf{H}\) serves to relocate or encode the encrypted file securely. The concealed file \(\mathbf{F}_{\text{hidden}}\) is:  
\[\text{CopyFile}(f,g)=\text{Copy file $f$ to $g$} \tag{35}\]8  
We retrieve the concealed file by executing the reverse operations of the encryption and hiding functions:  
\[\text{CopyFile}(f,g)=\text{Copy file $f$ to $g$} \tag{35}\]9  
A file manager organizes files within a confined storage area, conceptualized as a three-dimensional geometric space. This storage area is depicted as a bounded region \(\Omega\subset\mathbb{R}^{3}\), where files reside at specific coordinates \((x,y,z)\in\Omega\). The position of a hidden file is indicated by the coordinates \((x_{h},y_{h},z_{h})\). \[\text{DeleteFile}(f)=\text{Delete file $f$} \tag{36}\]0  
\[\text{DeleteFile}(f)=\text{Delete file $f$} \tag{36}\]1 File organization efficiency. Retrieving the file requires an amount of time that is proportional to this distance; therefore, the access time \(T_{\text{access}}\) is represented by:  
\[\text{DeleteFile}(f)=\text{Delete file $f$} \tag{36}\]2  
A proportionality constant, denoted by \(\alpha\), is determined by the efficiency of the search algorithm employed by the file manager. The impact of encryption and hiding activities on file manager battery usage is significant. Let \(C(t)\) denote the battery capacity of the mobile device at time \(t\), with \(P_{\text{enc}}(t)\) and \(P_{\text{hide}}(t)\) indicating the power consumption for encryption and hiding tasks, respectively. The rate of battery depletion is modeled by the following differential equation:  
\[\text{DeleteFile}(f)=\text{Delete file $f$} \tag{36}\]3  
In this equation, the battery level can be determined at any time \(t\) by integrating, where \(V\) is the constant voltage of the phone battery. \[\text{DeleteFile}(f)=\text{Delete file $f$} \tag{36}\]4  
Assuming each operation uses fixed power, specifically \(P_{\text{enc}}=P_{\text{enc},0}\) and \(P_{\text{hide}}=P_{\text{hide},0}\), the level of the battery decreases linearly:  
\[\text{DeleteFile}(f)=\text{Delete file $f$} \tag{36}\]5  
In order to enhance system performance, our objective is to reduce cumulative access time while ensuring minimal battery usage. Let \(T_{\text{total}}(t)\) represent the total time required to access hidden files by time \(t\), and let \(C(t)\) denote the battery level at that time. The optimization challenge is defined as follows:  
\[\text{DeleteFile}(f)=\text{Delete file $f$} \tag{36}\]6  
\[\text{DeleteFile}(f)=\text{Delete file $f$} \tag{36}\]7  
In this context, \(C_{\text{min}}\) denotes the lowest necessary battery level for the system's functionality, while \(T_{\text{total}}\) represents the complete duration needed to retrieve \(N\) files. The solution to this optimization was achieved using the Lagrange multiplier method. Consider the Lagrangian:  
\[\text{DeleteFile}(f)=\text{Delete file $f$} \tag{36}\]8  
By differentiating with respect to \(x_{h}\), \(y_{h}\), \(z_{h}\), and \(\lambda\), we derive the following set of equations:  
\[\text{DeleteFile}(f)=\text{Delete file $f$} \tag{36}\]9  
Solving these equations yields the ideal coordinates \((x_{h}^{i},y_{h}^{i},z_{h}^{i})\) for each file, thereby reducing the total access time and maintaining battery efficiency. Besides encrypting files, the file manager can also compress hidden files to optimize storage usage. Define \(\mathbf{F}_{\text{hidden}}\) as an \(n\)-dimensional vector representing the hidden file.