# Mathematical Background and Algorithms of a Collection of Android Apps for a Google Play Store Page
## 3 Problem Solving
### Novel Torch App Which Can Be Switched by Double-Tapping Anywhere on the Screen  
The methodology of the novel torch app is to first study mathematical aspects such as the power consumption of the mobile phone, while the torch LED is turned on, and then implement all the findings in the mobile phone app. The scientific impact of the novel torch app is that it improves research in fields such as energy-efficient mobile technology, adaptive brightness management, human-computer interaction, and technologies for emergency response. Moreover, it contributes to progress in optimizing low-power LEDs and improving accessibility features. Figure 17 shows the flowchart of the novel torch app. Algorithm 3 shows the most important steps of the novel torch app. ```
1:procedureToggleFlashlight
2:ifFlashlightStatus = OFF then
3:TurnOnFlashlight()
4:FlashlightStatus \(\leftarrow\) ON
5:Return \"Flashlight Turned ON\"
6:else
7:TurnOffFlashlight()
8:FlashlightStatus \(\leftarrow\) OFF
9:Return \"Flashlight Turned OFF\"
10:endif
11:endprocedure
```  
**Algorithm 3** Toggle flashlight in novel torch app. There is a need for a torch app that can be toggled really easily by tapping anywhere on the phone screen; a solution would be this novel torch app. The creation method and mathematical background will be presented next. With this application, the user can turn the flashlight of the mobile device on/off by double-tapping anywhere on the screen. The problem of the app starting accidentally was solved by adding a double-tap to turn the flashlight on and a double-tap to turn  
Figure 16: The pop-up with the information/help message shown after tapping the information button. the flashlight off. The application was designed to be extremely simple and easy to use. The application also has a notification that turns the flashlight on/off by tapping on the notification. To make the application usable, the user must allow camera access for the application, as the application uses the flashlight near the camera of the mobile device. Next, the implementation of the application will be shown. Equation (63) shows a mathematical expression of the flashlight toggling app. \[\text{Flashlight(state)}= \begin{cases}\text{ON},&\text{if state}=\text{TRUE}\\ \text{OFF},&\text{if state}=\text{FALSE}\end{cases}  \tag{63}\]  
The flashlight function is represented by a flashlight. The current state of the flashlight is represented by the TRUE or FALSE state. If the state is TRUE, the function will return ON, showing that the flashlight is turned ON, and OFF if the state is FALSE, showing that the flashlight is turned OFF. The illumination \(E\) (measured in lux) at a distance \(d\) from the flashlight is determined by Equation (64). \[E=\frac{I}{d^{2}} \tag{64}\]  
In this context, \(I\) represents the luminous intensity in candelas (cd). Battery life \(t\) can be determined using Equation (65). \[t=\frac{C\times\eta}{P} \tag{65}\]  
Figure 17: Flowchart of the novel torch app. Here:  
* \(C\) represents the battery capacity in milliampere-hours (mAh). * \(\eta\) denotes the LED driver's efficiency (dimensionless, usually ranging from 0 to 1). * \(P\) corresponds to the power usage in watts (W). The thermal resistance, denoted as \(R_{\theta}\) and measured in \({}^{\circ}\)C/W, can be determined using Equation (66). \[R_{\theta}=\frac{T_{j}-T_{a}}{P} \tag{66}\]  
Where:  
* \(T_{j}\) is the junction temperature of the LED (\({}^{\circ}\)C). * \(T_{a}\) is the ambient temperature (\({}^{\circ}\)C). * \(P\) is the power dissipation in watts (W). As demonstrated in Equation (67), the total luminous flux \(\Phi\) (measured in lumens) can be determined using the luminous efficacy \(\eta_{l}\) (measured in lumens per watt) and the power consumption \(P\) (measured in watts). \[\Phi=\eta_{l}\times P \tag{67}\]  
The flashlight app's power usage is described by  
\[P=V\times I \tag{68}\]  
Where:  
* \(V\) represents the voltage level of the cell phone battery. * \(I\) is the amount of current consumed by the LED.