Journal of the Indonesian Mathematical Society

Modelling The US Dollar Index Using Continuous Hidden Markov

David Vijanarco Martal — 2026-09-01

This study employs the continuous hidden Markov model (HMM) to model the index data of the US Dollar index from 2018 to 2024. HMM is used to predict and analyze hidden patterns that generated the data. The data is modelled using a continuous HMM with 11 hidden states and lognormal distributions with different parameters for each hidden state. The accuracy of the continuous HMM is measured by MAPE. The MAPE value for both training and testing data is very low, less than 4%. This means that the continuous HMM can be used to model the data accurately. The plot shows accurate predictions between simulated data and real data, and furthermore, the model can capture the fluctuations of the data.

Homomorphisms of Modules over Two Skew Generalized Power Series Rings

Ahmad Faisol — 2026-09-01

Let R and R' be rings with identity and let S be a strictly ordered monoid equipped with twisting homomorphisms into the endomorphism rings of R and R'. Using these data, we consider two skew generalized power series rings associated with R and R'. Starting from an (R,R')-module M, we construct an induced module consisting of generalized power series with coefficients in M and show that it naturally becomes a module over the two skew generalized power series rings through a suitable trilinear action. We then study homomorphisms in this framework. In particular, we prove that every (R,R')-module homomorphism between two modules induces a corresponding homomorphism between their associated generalized power series modules. Furthermore, we provide sufficient conditions describing when a generalized power series belongs to the kernel of the induced homomorphism in terms of the coefficientwise behavior of the original map. These results extend the theory of skew generalized power series modules from the classical single-ring setting to a two-ring context and provide a foundation for further developments, including generalized isomorphism results and related algebraic applications.

An Existence and Uniqueness of Solutions for a Stochastic Differential Equations with \psi-Caputo Fractional Derivative

R. Hariharan — 2026-09-01

The present work investigates the existence and uniqueness of solutions for a fractional stochastic differential equations with $\psi$-Caputo fractional derivatives. We establish important theoretical conclusion and examine the energy growth bounds and asymptotic behavior of the stochastic solution using the Banach's fixed point theorem. Additionally, we consider our analysis into single-valued function derived from multivalued mappings. We provide examples to validate in our methodology.

A More Precise Elbow Method For Optimum K-means Clustering

Indra Herdiana — 2026-09-01

K-means clustering is an unsupervised clustering method that requires an initial decision of number of clusters. One method to determine the number of clusters is the elbow method, a heuristic method that relies on a graphical plot of the within-cluster sum of squares against the number of clusters. The method uses the number based on the elbow point, the point closest to $90^\circ$ that indicates the most optimum number of clusters. This research improves the elbow method such that the selection of cluster numbers is unbiased towards visual interpretation. We use the analytical geometric formula to calculate an angle between lines and real analysis principle of derivative to simplify the elbow point determination. We also consider every possibility of the elbow method graph behaviour such that the algorithm is universally applicable. The result is that the elbow point can be measured precisely with a simple algorithm that does not involve complex functions or calculations. This improved method gives an alternative of more reliable cluster determination method that contributes to more optimum k-means clustering, obtained by the fewest clusters with the lowest error.

A New Slope Averaging Technique in Heun's Method: Combining Harmonic and Root Mean Square Means

Syamsudhuha Syamsudhuha — 2026-09-01

This paper introduces a modified version of Heun's method for solving initial value problems (IVPs) in ordinary differential equations (ODEs). The proposed method replaces the standard arithmetic mean used for the average slope in Heun's method with a novel combination of harmonic mean and root mean square (RMS) for the slopes of the tangent lines. This alteration is designed to enhance the accuracy and performance of the method. Through theoretical analysis, we establish that the modified method retains stability and consistency properties, crucial for reliable numerical solutions. Extensive numerical experiments demonstrate that the method performs well across a range of step sizes, showing notable improvements in accuracy for both small and large step sizes when compared to the standard Heun's method. These results suggest that the new approach offers a robust alternative for solving IVPs, particularly in scenarios where adaptive step sizing or high precision is required.

Explicit Formulas for the Anti-Trace of 3-by-3 Matrix Powers via Generating Functions

Jeriko Gormantara — 2026-09-01

We study the anti-trace—the sum of anti-diagonal entries—of powers of 3-by-3 matrices. Unlike the ordinary trace, the anti-trace has no spectral characterization, so closed forms for the anti-trace of matrix powers are valuable. Starting from the Cayley--Hamilton recurrence for matrix powers and solving a trivariate generating function, we derive explicit closed-form expressions whose coefficients are signed multinomial combinations of the sums of principal minors and their anti-principal counterparts. The formulas are purely algebraic and remain valid over any commutative ring. As a structural application, we analyze block anti-diagonal matrices of size 3m-by-3m.

The Non-Normal Cyclic Subgroup Graph Associated with Quasidihedral Groups of Order 16

Nur Nabilah Abdul Razak — 2026-09-01

Let \textsl{G} be a group and \textsl{H} be a non-normal cyclic subgroup of \textsl{G}. The non-normal cyclic subgroup graph, denoted as $\Gamma^{NN}_{H}(G)$, is defined as a directed graph with vertex set elements of \textsl{G} such that for two distinct elements \textsl{x} and \textsl{y} in \textsl{G}, \textsl{x} is the initial vertex and \textsl{y} is the terminal vertex of an edge if their product is in \textsl{H}. In 2020, the idea of the non-normal subgroup graph, which is the extension of the subgroup graph was developed. This research identifies the non-normal cyclic subgroup graphs connected to order 16 quasidihedral groups. Firstly, the Groups, Algorithms and Programming (GAP) software is used to identify the non-normal cyclic subgroups. The definition of $\Gamma^{NN}_{H}(G)$ is then used to calculate the adjacency and direction of the vertices. Lastly, Maple 2016 software will be used to visualize these graphs.

Graceful Vit Labeling: A New Approach and Its Applications to Graphs

Felicia Servina Djuang — 2026-09-01

Consider an undirected, simple graph $ G = (V(G), E(G)) $. A graceful labeling of graph $ G $ is an injective function $f: V(G) \to \{0, 1, 2, \dots, |E(G)|\} $ such that the induced edge labels, defined by $f^*(uv)=|f(u)-f(v)|$ for every edge $uv \in E(G)$, are all distinct. In this paper, we introduce a new type of graceful labeling, called \textbf{graceful vit labeling}. A graceful labeling $f$ of graph $G$ is called a graceful vit labeling if the vertex weight function $w_f: V(G) \to \mathbb{N}$, defined by $w_f(v)=f(v)+\sum_{uv \in E(G)}f^*(uv)$ for every $v \in V(G)$, assigns pairwise distinct weights to all vertices in graph $G$. In other words, no two vertices have the same sum of their own label and the labels of all edges incident to them. In this paper we present various examples of graphs that admit graceful vit labeling and explores structural properties and necessary conditions for their existence.

Analysis of Primary Resonance in Damped Duffing-Helmholtz Oscillator Excited by an Amplitude-Modulated Signal

Chaudhary Anunay Kumar — 2026-09-01

The effect of amplitude modulated signal on the primary resonance on a Duffing-Helmholtz oscillator has been investigated using ($i$) a multiple time scale perturbation method involving two different time scales i.e., regular time ($T_0$) and a slow time, ($T_1$), to analyze the system response and ($ii$) its modified version which is valid for larger values of expansion parameter viz., asymptotic ordering parameter, incorporating Lindstedt-Poincare transformation method that allows the interaction of the amplitude and frequency of the response. The modification of the amplitude- frequency plots, exhibiting bi-stable behavior, caused by different values of modulation amplitude and damping has been obtained both analytically and numerically for various parameters of the system near the primary resonance. The peak amplitude of the response is shown to get lowered in strongly nonlinear Duffing-type system. Phase plane trajectories of stable and unstable manifolds are presented in different instances.

Antimagic Labeling of Graph Unions of Trees and 4-Cycles

Poh Hwa Ong — 2026-09-01

Let $G(V,E)$ be a graph with $V(G)$ as the set of vertices and $E(G)$ as the set of edges. A labeling is a bijection $f:E(G)\to\{1,2,\cdots, |E(G)|\}$. For each vertex $v$, let $\phi(v)$ be the sum of the labels on the edges incident to $v$. If all values $\phi(v)$ are distinct, the labeling is antimagic, and the graph is antimagic if such a labeling exists. This paper explores the concept of antimagic labeling in graph theory, with a particular focus on the union of various tree structures, including stars, single brooms, and double brooms. We apply extended Skolem sequences to prove that for a wide range of parameters, unions of multiple 3-paths with appropriately many 4-cycles yield antimagic graphs. Additionally, we analyze combinations of 4-cycles paired with different tree structures.

Atom of the Lattice of All N-Radicals

Puguh Wahyu Prasetyo — 2026-09-01

In this paper, we study atomic elements in the lattices of radical classes. A minimal element of the lattice of all $N$-radicals (respectively, supernilpotent radicals and special radicals) is called an $N$-atom (respectively, a supernilpotent atom and a special atom). We construct a class of $N$-atoms generated by prime essential rings and investigate their structural properties. We show that these $N$-atoms are closely related to supernilpotent atoms and special atoms, and we clarify the precise relationships among these three types of atoms. In particular, we prove that for a prime essential ring $R$, the associated radicals $\widetilde{\mathit{l}}_{R}$ and $\overline{\mathit{l}}_{R}$ generate, respectively, a special atom and a supernilpotent atom. This result provides a positive answer to an open question concerning which prime essential rings give rise to atomic elements in the lattices of all $N-$ radical, special radicals and supernilpotent radicals.

Coprime Degree of a Finite Group

Abdul Gazir Syarifudin — 2026-09-01

The coprime probability of a finite group has been defined, and the coprime probability of a p-group for any prime number p has also been obtained. In this paper, we refine the concept by introducing the coprime degree of a finite group. We calculate the coprime degrees of cyclic groups, nilpotent groups, dihedral groups, and general quaternion groups.

On Graceful Labeling of Some Power Graphs

Gurvinder Singh — 2026-09-01

Power graphs of groups and semigroups constitute a significant class of graphs in algebraic graph theory, bridging algebraic structures and graph theoretic concepts. Further, labeling of graphs, particularly those endowed with an algebraic structure as their vertex set, is a vibrant area of research. This article contributes to this burgeoning field by exploring new classes of finite groups whose power graphs admit or forbid an important type of labeling called graceful labeling.

SG-Lightlike Submanifolds of a Locally Bronze Semi-Riemannian Manifold Equipped With (l,m)-Type Connection

Rajinder Kaur — 2026-09-01

This research work introduces the geometry of the SG (Screen Generic)-lightlike submanifolds of a locally bronze semi-Riemannian manifold endowed with an (l,m)-type connection. The characterization theorems on geodesicity of such submanifolds with respect to the integrability and parallelism of the distributions are established. It is shown that there exists no coisotropic, isotropic or totally proper SG-lightlike submanifold of a locally bronze semi-Riemannian manifold. Assertions for the smooth transversal vector fields in totally umbilical proper SG-lightlike submanifold are obtained. Furthermore, the structure of a minimal SG-lightlike submanifold of a locally bronze semi-Riemannian manifold is detailed with an example.