CONSTRUCTION OF FUNDAMENTAL THEOREMS OF FRACTIONAL CALCULUS

This paper discusses the theory of derivatives and integrals in the form of fractions with a particular order initiated by Lioville. Specifically, regarding the correlation between fractional derivatives and integrals, by examining definitions, determining the kernel function, and applying them to several examples, so a general formula will be obtained regarding the relationship between the two. This formula is the product of the fractional derivative of an order of a polynomial function of m-degree which is equal to the (n+1) th derivative of the related order fractional integral of a polynomial function of -degree that the truth is proved by using Mathematical Induction

OPTIMAL CONTROL OF MATHEMATICAL MODELS IN BIOENERGY SYSTEMS AS EMPOWERMENT OF SUSTAINABLE ENERGY SOURCES

Energy has a very important role in everyday life. Dependence on non-renewable energy increases its vulnerability to supply instability, making it important to seek alternative energy sources to overcome this dependence. Bioenergy is an alternative energy produced from organic materials such as biomass. Control of renewable energy is needed to increase production and empowerment. In this research, a mathematical model of biogas production growth in the form of differential equations formed with optimal control modifications is proposed. Completion of the model is carried out by forming an objective function, as well as determining the Hamilton function and Lagrange function. Numerical simulations in the model show that providing control can increase biogas production as a sustainable energy source

THE CHEMICAL TOPOLOGICAL GRAPH ASSOCIATED WITH THE NILPOTENT GRAPH OF A MODULO RING OF PRIME POWER ORDER

Chemical topological graph theory constitutes a subdomain within mathematical chemistry that leverages graph theory to model chemical molecules.  In this context, a chemical graph serves as a graphical representation of molecular structures. Specifically, a chemical molecule is portrayed as a graph wherein atoms are denoted as vertices, and the interatomic bonds are represented as edges within the graph. Various molecular properties are intricately linked to the topological indices of these molecular graphs. Notably, commonly employed indices encompass the Wiener Index, the Gutman Index, and the Zagreb Index.  This study is directed towards elucidating the numerical invariance and topological indices inherent to a nilpotent graph originating from a modulo integer ring with prime order. Consequently, the investigation seeks to discern how the Wiener Index, the Zagreb Index, and other characteristics of the nilpotent graph manifest within a ring of integers modulo prime order powers

SOMBOR INDEX AND ITS GENERALIZATION OF POWER GRAPH OF SOME GROUP WITH PRIME POWER ORDER

Graphs are an intriguing topic of discussion due to their numerous applications, particularly in chemistry. Topological indices derived from graph representations of molecules enable us to predict various properties of these compounds, including their physical characteristics, chemical reactivity, biological activity, toxicity, and atom-to-atom interactions. More recently, graphs have also been utilized to depict abstract mathematical objects such as groups. A notable example of graph representation in group theory is seen in power graphs. This research explores new graph topological indices based on vertex degrees, inspired by the Euclidean metric, particularly the Sombor index, and its application to the power graph of the integer modulo group and the dihedral group. The primary outcome of this study is the derivation of a general formula for the Sombor index and its generalization

Recognizing the Spatial Distribution and Voronoi Patterns of the Recorded Earthquake Epicenters in Sunda Strait, Indonesia

Currently, Sunda Strait is one of the most active transportation hubs. However, this region also bears a notable history of geohazards associated with the dynamics of tectonic activity of the Eurasian and Indo-Australian tectonic plates, such as the super-eruption of Krakatoa volcano in 1883, the Sunda Strait tsunami in 2018, and decades of frequent earthquakes. To address these challenges, this study conducted a statistical analysis of the frequency and distribution of seismic activities in the Sunda Strait region based on recorded epicenter data in the United States Geological Survey's (USGS) Earthquake catalog. We assembled 440 multivariate earthquake data points between 1990 and 2023 (over three decades). The results of this study indicate that the machine learning approach precisely identifies four relevant parameters for -means clustering, followed by an analysis of silhouette values to recognize Voronoi patterns. These statistical patterns also have significant implications for the number of epicenter clusters and recognizing their spatial distribution. It provides a new understanding of the spatial-temporal characteristics and locates the list of frequent earthquake regions. Having all the necessary information would help to comprehensively evaluate geohazard risks in Sunda Strait region

Learning With Error for Digital Image Encryption

Learning With Error (LWE) is one of the development of a system linear equation that add some noise or error. These problems have good potential for cryptography, especially for the development of Key Exchange Mechanism (KEM). Moreover, the question is whether LWE can be applied for digital image security or not. The digital image consists of hundreds of pixels that can be interpreted as a matrix. Each Pixel is encrypted with LWE so that the image becomes unidentified or cipher

PRIME LABELING OF SOME WEB GRAPHS WITHOUT CENTER

The prime labeling of a graph  $G$ of order $n$ is a bijection function from the set of vertices in $G$ to the set of the first $n$ positive integers, such that any two adjacent points in $G$ have labels that are coprime to each other. In this paper  we discuss the primality of the graph $W_0(2,n)$ along with its combinations with similar graphs and various types of edges subdivisions in the graph $W_0(2,n)$. Moreover, it is also presented the necessary and sufficient conditions for the graph to be prime

ROBUST PREDICTION INTERVALS FOR INDONESIAN INFLATION: A BIAS-CORRECTED BOOTSTRAP APPROACH

Inflation is important to be analyzed due to its impact is felt across various aspects of the economy and individuals' lives. This research aimed to develop robust and reliable predictions concerning Indonesian inflation using the bias-corrected bootstrap method for an AR model. The data utilized spanned from January 2020 to September 2023 and was obtained from Bank Indonesia's website. The analysis provided the optimal order in the AR model, which resulted in p=2 as the best order (AIC=-1.858, BIC=-1.698, and HQ=-1.798). The number of bootstrap replications used was B=100, 250, 500, and 1000. The analysis was conducted using R Studio. The analysis results indicated that the model employed for prediction analysis was highly stable, with all point forecasts indicating result consistency. The prediction results suggested that inflation in Indonesia was expected to decrease in the upcoming 5 months. The results also revealed that the bias-corrected bootstrap approach could provide forecasting results with a higher level of accuracy. This research contributed to the understanding and forecasting of Indonesian inflation, emphasizing model stability and consistent results

ANALYSIS OF THE EFFECT OF STUART NUMBER AND RADIATION ON VISCOUS FLUID FLOW

Computational fluid dynamics (CFD) is a numerical solution of fluid flow problems built from applied mathematical modeling. The problem of the flow of a viscous fluid which is influenced by a magnetic field gives rise to a boundary layer, from this boundary layer a dimensional building equation is formed. The governing equation is obtained from the continuity equation, momentum equation, and energy equation, then transformed into a non-dimensional equation by substituting non-dimensional variables and transformed into a similarity equation. The numerical solution to this problem uses the Keller Box method. The numerical simulation results show that the Stuart Number increases the velocity profile, while the temperature profile decreases. The effect of radiation parameters on the velocity profile did not change significantly, but the temperature profile decreased