Calculation

Type-1 interpolating sesqui-f-harmonic maps between Riemannian manifolds

2026-03-31T12:19:41+00:00

In this paper, first, we introduce and study a new map called a type-1 interpolating sesqui-$f$-harmonic map. Then, we provide necessary and sufficient conditions for a differentiable curve in a Riemannian space form to be a type-1 interpolating sesqui-$f$-harmonic. These conditions are presented in a main theorem and investigated in several subcases. Moreover, we analyze type-1 interpolating sesqui-$f$-harmonic curves on $S^{n}(1)$ and $H^{n}(-1)$.

Clairaut conformal hemi-slant submersions from Kahler manifolds

2026-05-05T17:34:22+00:00

In this paper, we introduce and study Clairaut conformal hemi-slant submersions from K\"ahler manifolds onto Riemannian manifolds. This class of maps combines the geometry of Clairaut conformal submersions with the hemi-slant decomposition of the vertical distribution in the almost Hermitian manifolds. We first establish a characterization theorem for Clairaut conformal hemi-slant submersions in terms of the geodesic behavior on the total manifold, the mean curvature of the fibers, and the behavior of the dilation along the fibers. We then derive equivalent formulations of the Clairaut condition adapted to the slant and anti-invariant components of the vertical distribution and obtain refined decompositions of the Clairaut relation and the harmonicity condition with respect to the hemi-slant splitting. Furthermore, we investigate the stability of the Clairaut conformal hemi-slant structure under conformal deformations of the total metric. We also study curvature properties of such submersions and obtain vertical sectional, scalar and Ricci curvature decomposition formulas compatible with the hemi-slant structure. In particular, the vertical curvature is decomposed into its slant, anti-invariant and mixed components, revealing the geometric influence of the hemi-slant splitting on the Clairaut and harmonic structures. Finally, we provide an explicit nontrivial example illustrating the theory.

Second variation of $F$-Einstein-Hilbert functional

2026-04-07T08:41:30+00:00

This article describes a formula for second variation of generalized Einstein-Hilbert functional on Riemannian manifolds.
This work extends the definition of stable Einstein manifolds, and we present some properties.

Some properties of geodesics and F-geodesics on tangent bundle with gradient Sasaki metric

2026-05-30T10:17:40+00:00

In this paper, we investigate the properties of geodesics and $F$-geodesics on the tangent bundle equipped with the gradient Sasaki metric. First, we establish necessary and sufficient conditions for a curve to be a geodesic. We then study the behavior of $F$-geodesics and $F$-planar curves on the tangent bundle with respect to the induced Levi-Civita connection. Our theoretical results are supported by explicit examples that illustrate the behavior of these curves.

The cosmological Barker equation: An extended analytical framework for local group dynamics and collision timing mechanisms

2026-05-02T07:35:04+00:00

This paper extends the classical Barker equation, a historical cornerstone of celestial mechanics, to encompass complex cosmological and galactic perturbation effects. The orbital dynamics of the Local Group, overwhelmingly dominated by the impending convergence of the Milky Way and Andromeda galaxies, are typically analyzed via computationally expensive N-body simulations or the highly idealized Timing Argument. In this study, we introduce a modified effective potential that simultaneously accounts for the universal expansion driven by the Hubble flow and the extended mass distributions of dark matter halos. The resulting Cosmological Barker Equation yields the ultimate collision timescale of the binary system in a closed-form analytic expression that depends exclusively on initial boundary conditions. The analytically derived expansion terms mathematically demonstrate precisely how the underlying cosmic flow delays gravitational collapse, manifesting as a higher-order perturbation. Concurrently, the inclusion of a dedicated dark matter parameter allows the theoretical architecture to be seamlessly calibrated against modern numerical simulations. Ultimately, this expanded analytical framework provides a transparent and intuitive mathematical alternative for investigating the orbital mechanics of macroscopic galactic systems, profoundly enhancing physical insight while entirely circumventing the traditional computational burden.

Comparison between analytical and MATLAB solution of wave equation using finite difference methods

2026-06-03T09:48:02+00:00

This paper presents a comparative study of the analytical solution and the numerical solution of the one-dimensional wave equation using Finite Difference Methods (FDM) in MATLAB. The wave equation models various physical phenomena, such as vibrations in strings and sound waves. The numerical solutions were obtained using Finite Difference Methods FDM approaches and compared with the analytical solution to evaluate accuracy and stability. The results show that the numerical solution should match the exact solution closely since r=1 ensures stability and accuracy. The maximum error should be very small. The finite difference method is fast for this problem due to its simple time-stepping formula. The elapsed time per run should be minimal.