Muhammad Ahsan

The current work is dedicated to simulating the natural convection of Al2O3-Cu/Water hybrid nanofluid within the tilted quarter-elliptical chamber being affected by a magnetic field at different angles. The ellipse equation is considered for modeling the chamber in such a way that the ratio of height to length takes a constant value of 2. The temperature inside the chamber's curving wall has been kept low or cold, while the small smooth wall of the chamber is presumed adiabatic. The chamber’s larger wall is subjected to three different types of heating. All chamber walls have no velocity. The effect of the Hartmann number (Ha=0, 15, 30, 45), angle of exerted magnetic field (λ=0, 45◦, 90◦), chamber wall heating type (sinusoidal heating, linear heating, and constant temperature heating), chamber inclination angle (Г=-90◦, 0, +90◦), and the Rayleigh number (Ra=103, 104, 105) on the values related to the properties of heat and flow transport have undergone extensive study. The correctness of the results was guaranteed by comparing the findings of prior investigations with the current work.

Assume that G = (V(G), E(G)) is a connected graph. For a set of vertices W_E⊆V(G), two edges g₁, g₂ ∈ E(G) are distinguished by a vertex x₁ ∈ W_E, if d(x₁, g₁) ≠ d(x₁, g₂). W_E is termed edge metric generator for G if any vertex of W_E distinguishes every two arbitrarily distinct edges of graph G. Furthermore, the edge metric dimension of G, indicated by edim(G), is the cardinality of the smallest W_E for G. The edge metric dimensions of the dragon, kayak paddle, cycle with chord, generalized prism, and necklace graphs are calculated in this article.

Recent advances in graph theory, linear algebra, and commutative algebra render us to tackle problems in one bough of mathematics with assistance and guidance from others. We will elaborate foremost and conceptually fathomless homological invariants inextricably linked with circulant matrices and cycles through various path lengths in this article, as well as a class of Koszul algebra, which portrays combinatorial correlation, in the end.

The most productive application of graph theory in chemistry is the representation of molecules by the graphs, where vertices and edges of graphs are the atoms and valence bonds between a pair of atoms, respectively. For a vertex and an edge of a connected graph , the minimum number from distances of with and is called the distance between and . If for every two distinct edges , there always exists such that , then is named as an edge metric generator. The minimum number of vertices in is known as the edge metric dimension of . In this paper, we calculate the edge metric dimension of ortho-polyphenyl chain graph , meta-polyphenyl chain graph , and the linear [n]-tetracene graph and also find the edge metric dimension of para-polyphenyl chain graph . It has been proved that the edge metric dimension of , , and is bounded, while <svg...

Toeplitz networks are used as interconnection networks due to their smaller diameter, symmetry, simpler routing, high connectivity, and reliability. The edge metric dimension of a network is recently introduced, and its applications can be seen in several areas including robot navigation, intelligent systems, network designing, and image processing. For a vertex s and an edge g = s 1 s 2 of a connected graph G , the minimum number from distances of s with s 1 and s 2 is called the distance between s and g . If for every two distinct edges s 1 , s 2 ∈ E G , there always exists w 1 ɛ W E ⊆ V G , such that d s 1 , w 1 ≠ d s 2 , w 1 ; then, W E is named as an edge metric generator. The minimum number of vertices in W E is known as the edge metric dimension of G . In this study, we consider four families of Toeplitz networks T n 1,2 , T n 1,3 , T n 1,4 , and T n 1,2,3 and studied their edge metric dimension. We prove that for all n ≥ 4 , e dim T n 1,2 = 4 , for n ≥ 5 , e dim T n 1,3 = 3 , and for n ≥ 6 , e dim T n 1,4 = 3 . We further prove that for all n ≥ 5 , e dim T n 1,2,3 ≤ 6 , and hence, it is bounded.

A method to use for mapping VR-Box joystick with shooting game input controls that include key mapping of every joystick key from the joystick with the game itself. Key mapping was observed and then the source code keys we're mapped with the joystick keys. The joystick holds major 4 buttons with a 360-degree moveable joystick with 2 supporting keys making them total of 6 keys. In this paper we will discuss mapping of Google Virtual Reality plugin with First Person Shooting plugin. The integration of joystick with VR-Box helped the gameplay even more controllable. The joystick keys are responsible for invoking key features of game as it can be connected via Bluetooth with the mobile headset to ensure long range wireless connectivity. The device converts the keys triggered from joystick to enable the features inside the game.