Calculation

Geometry of warped product pointwise semi-slant submanifolds in nearly para-Kaehler manifold

2024-11-13T08:59:31+00:00

In this article, firstly we introduce pointwise slant and pointwise semi-slant submanifolds in nearly para-Kaehler manifolds. We demonstrate that there exist pointwise semi-slant non-trivial warped product submanifold $ \mathcal{ M}^T \times_k \mathcal{ M}^{\theta} $ in nearly para-Kaehler manifolds by giving an example. We get a characterization, give certain theorems depending on the casual characters and we reach an optimal inequality.

Triangular numbers and centered square numbers hidden in Pythagorean runs

2024-11-30T16:10:15+00:00

The Pythagorean theorem, which asserts that in a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse and is mathematically expressed as can be generalized to equations with 5, 7, or more variables, resulting in equalities of sums of squares.

If we seek to find t consecutive numbers that satisfy such equations, which can be extended infinitely by increasing the number of variables, and observe the equality of sums of squares for each case, we encounter what are known as Pythagorean runs.

In this study, it was observed that within Pythagorean runs, which can become increasingly complex as we increase the number of variables, there exists a strikingly unique solution set when we restrict ourselves to finding consecutive integers.

By examining the consecutive integers that form these Pythagorean runs, new findings have emerged. Specifically, Pythagorean runs were analyzed using triangular numbers and centered square numbers. A hypothesis was formulated, positing that there is a unique solution involving consecutive integers for Pythagorean runs with figurate numbers. This hypothesis has been proven using both inductive and geometric proof methods.

Almost potent manifolds

2024-11-01T07:25:58+00:00

In this paper we introduce a new manifold, namely potent manifolds. We give examples and investigate the integrability conditions. We also check the curvature relations of Kaehler potent manifolds and show that such manifolds are flat when it has constant sectional curvature. Then we introduce potent sectional curvature and obtain a spacial form of the curvature tensor field when its potent sectional curvature is a constant.

Motion of Galilean particles with curvature and torsion

2024-12-23T07:17:29+00:00

This paper examines the motion of particles governed by an action that depends on the curvature and torsion of their trajectories in the Galilean 3-space $G_{3}.$ We derive the Euler-Lagrange equation corresponding to the action $H(\gamma )=\int\limits_{\gamma}f(\kappa ,\tau )ds$ in $G_{3}$. We present examples to clarify the solutions of the system, clearly explaining their properties and relevance. With examples specifically focusing on the natural Hamiltonian problem derived from the Frenet frame of the curve and a generalization of these natural Hamiltonians, we aim to illustrate their key features and underlying principles.

A study on the topology of graph complexes

2024-12-19T06:08:56+00:00

In this paper, we consider the homotopy types of independence complexes of some graphs. Moreover, we study the homotopy types of graphs which are expanded from a given graph via certain operations. For any graph whose independence complex is contractible, we calculate the homotopy type of clique complex of its central graph. In addition to these, we build a complex from a bipartite graph to calculate homotopy types of some complexes.