arXiv subjects

Mohd Salmi Md Noorani

Publications and source records attributed to Mohd Salmi Md Noorani.

5 recordsLinked to original sources

arXiv2025

On the cosine similarity and orthogonality between persistence diagrams

Topological data analysis is an approach to study shape of a data set by means of topology. Its main object of study is the persistence diagram, which represents the topological features of the data set at different spatial resolutions. Multiple data sets can be compared by the similarity of their diagrams to understand their behaviors in relative to each other. The bottleneck and Wasserstein distances are often used as a tool to indicate the similarity. In this paper, we introduce cosine similarity as a new indicator for the similarity between persistence diagrams and investigate its properties. Furthermore, it leads to the new notion of orthogonality between persistence diagrams. It turns out that the orthogonality refers to perfect dissimilarity between persistence diagrams under the cosine similarity. Through data demonstration, the cosine similarity is shown to be more accurate than the standard distances to measure the similarity between persistence diagrams.

math.AT↗

arXiv2022

A high-order predictor-corrector method for initial value problems with fractional derivative involving Mittag-Leffler kernel: epidemic model case study

In this paper, we propose a numerical scheme of the predictor-corrector type for solving nonlinear fractional initial value problems, the chosen fractional derivative is called the Atangana-Baleanu derivative defined in Caputo sense (ABC). This proposed method is based on Lagrangian quadratic polynomials to approximate the nonlinearity implied in the Volterra integral which is obtained by reducing the given fractional differential equation via the properties of the ABC-fractional derivative. Through this technique, we get corrector formula with high accuracy which is implicit as well as predictor formula which is explicit and has the same precision order as the corrective formula. On the other hand, the so-called memory term is computed only once for both prediction and correction phases, which indicates the low cost of the proposed method. Also, the error bound of the proposed numerical scheme is offered. Furthermore, numerical experiments are presented in order to assess the accuracy of the new method on two differential equations. Moreover, a case study is considered where the proposed predictor-corrector scheme is used to obtained approximate solutions of ABC-fractional generalized SI (susceptible-infectious) epidemic model for the purpose of analyzing dynamics of the suggested system as well as demonstrating the effectiveness of the new method to solve systems dealing with real-world problems.

math.NA↗

arXiv2022

A short note on the orbit growth of sofic shifts

A sofic shift is a shift space consisting of bi-infinite labels of paths from a labelled graph. Being a dynamical system, the distribution of its closed orbits may indicate the complexity of the space. For this purpose, prime orbit and Mertens' orbit counting functions are introduced as a way to describe the growth of the closed orbits. The asymptotic behaviours of these counting functions can be implied from the analiticity of the Artin-Mazur zeta function of the space. Despite having a closed-form expression, the zeta function is expressed implicitly in terms of several signed subset matrices, and this makes the study on its analyticity to be seemingly difficult. In this paper, we will prove the asymptotic behaviours of the counting functions for a sofic shift via its zeta function. This involves investigating the properties of the said matrices. Suprisingly, the proof is rather short and only uses well-known facts about a sofic shift, especially on its minimal right-resolving presentation.

math.DS↗

arXiv2016

Renormalization of circle diffeomorphisms with a break-type singularity

Let $f$ be an orientation-preserving circle diffeomorphism with irrational rotation number and with a break point $ξ_{0},$ that is, its derivative $f'$ has a jump discontinuity at this point. Suppose that $f'$ satisfies a certain Zygmund condition dependent on a parameter $γ>0.$ We prove that the renormalizations of $f$ are approximated by Möbius transformations in $C^{1}$-norm if $γ\in (0,1]$ and they are approximated in $C^{2}$-norm if $γ\in (1,+\infty).$ It is also shown, that the coefficients of Möbius transformations get asymptotically linearly dependent.

math.DS↗

arXiv2015

A review of some works in the theory of diskcyclic operators

In this paper, we give a brief review concerning diskcyclic operators and then we provide some further characterizations of diskcyclic operators on separable Hilbert spaces. In particular, we show that if $x\in {\mathcal H}$ has a disk orbit under $T$ that is somewhere dense in ${\mathcal H}$ then the disk orbit of $x$ under $T$ need not be everywhere dense in ${\mathcal H}$. We also show that the inverse and the adjoint of a diskcyclic operator need not be diskcyclic. Moreover, we establish another diskcyclicity criterion and use it to find a necessary and sufficient condition for unilateral backward shifts that are diskcyclic operators. We show that a diskcyclic operator exists on a Hilbert space ${\mathcal H}$ over the field of complex numbers if and only if $\dim({\mathcal H})=1$ or $\dim({\mathcal H})=\infty$ . Finally we give a sufficient condition for the somewhere density disk orbit to be everywhere dense.

math.FA↗