arXiv subjects

Alamgir Hossain

Publications and source records attributed to Alamgir Hossain.

3 recordsLinked to original sources

arXiv2026

Study of attractors and fractal functions on the product spaces and Dimensional aspects

In this paper, the product of the Hausdorff metric on the product space is defined and the equivalency between the product Hausdorff metric and the Hausdorff metric on the product space is established. The finite product of the iterated function systems (IFS) on the product space is considered and the relation between the attractor of the product IFS and the attractors of the co-ordinate IFSs is studied. Dimension bounds of the homogeneous and inhomogeneous attractors on the product space is established. Also, the product fractal interpolation function on the higher dimensional space is constructed.

math.DS↗

arXiv2026

Quantization Dimension of $1$-variable Random Self-Similar Measures

The quantization problem for random fractals presents unique challenges due to the lack of uniform geometric scaling inherent in deterministic systems. In this article, we establish the almost sure quantization dimension for a class of $1$-variable (homogeneously) random self-similar measures. Unlike the deterministic setting, where the dimension is derived from a fixed pressure function, we prove that in the random case, the quantization dimension $κ_{r}$ is the unique zero of the expectation of the topological pressure. We rigorously justify this by exploiting the ergodicity of the shift map on the symbolic space to control distortion errors across non-uniform scales. Our results highlight the thermodynamic formalism underlying the quantization of random dynamical systems.

math.DS↗

arXiv2020

A computational methodology for two-dimensional fluid flows

A weighted residual collocation methodology for simulating two-dimensional shear-driven and natural convection flows has been presented. Using a dyadic mesh refinement, the methodology generates a basis and a multiresolution scheme to approximate a fluid flow. To extend the benefits of the dyadic mesh refinement approach to the field of computational fluid dynamics, this article has studied an iterative interpolation scheme for the construction and differentiation of a basis function in a two-dimensional mesh that is a finite collection of rectangular elements. We have verified that, on a given mesh, the discretization error is controlled by the order of the basis function. The potential of this novel technique has been demonstrated with some representative examples of the Poisson equation. We have also verified the technique with a dynamical core of two-dimensional flow in primitive variables. An excellent result has been observed-on resolving a shear layer and on the conservation of the potential and the kinetic energies with respect to previously reported benchmark simulations. In particular, the shear-driven simulation at CFL = 2.5 (Courant-Friedrichs-Lewy) and $\mathcal Re = 1\,000$ (Reynolds number) exhibits a linear speedup of CPU time with an increase of the time step, $Delta t$. For the natural convection flow, the conversion of the potential energy to the kinetic energy and the conservation of total energy is resolved by the proposed method. The computed streamlines and the velocity fields have been demonstrated.

physics.flu-dyn↗