Mathematics and Computer Science: Research Updates Vol. 12

6G-Enabled NB-IoT Framework for Satellite-Integrated Industrial IoT

C. Rajabhushanam — 2026-06-20

This study develops a 6G-enabled NB-IoT framework for satellite-integrated Industrial Internet of Things (IIoT) applications, with emphasis on signal generation, routing, and cloud service enablement for device-centric industrial and smart-city environments. The proposed framework is aligned with the ISRO-RESPOND Basket 2024 requirements for NB-IoT in satellite communication and considers IEEE 802.15.4-based wireless sensor networking, 3GPP-oriented cellular access, and edge-cloud platform services. The manuscript presents a conceptual architecture in which wireless sensor nodes, relay links, base station interfaces, and cloud resources support producer-consumer data flows between device, machine, and application layers. At the physical layer, the study discusses SC-FDMA uplink and OFDMA downlink operation, pulse-shaping, cyclic-prefix use, sub-carrier mapping, and RF power-amplifier design to minimise peak-to-average power ratio and intersymbol interference. The proposed use of a Zener-diode-assisted RF power-amplifier circuit is positioned as a mechanism for maintaining voltage stability and supporting energy-aware transmission in narrowband channels. At the networking layer, the framework incorporates TCP/IP encapsulation, link-state routing, BGP, OSPF, RIP, ZRP, VLAN/WLAN/PAN connectivity, and packet retransmission logic to support end-to-end data transport. The service layer maps NB-IoT and wireless-sensor data flows to cloud and edge orchestration for fire services, waste management, transport, street lighting, emergency medical services, and smart-meter applications. It also organises assumptions on topology, data acquisition, quality of experience, and routing resilience for later technical evaluation. The chapter therefore provides an integrated design-oriented description of a satellite-assisted NB-IoT/IIoT framework that combines waveform considerations, routing procedures, data acquisition, and cloud-based service access. The work is primarily architectural and is intended to inform subsequent simulation, implementation, and validation.

Contextualised Pedagogy in Teaching Integrals: Evidence from Goma Secondary Schools

Paul Twatahamahoro Bihame — 2026-06-20

Integral calculus is a fundamental area of mathematics that supports the calculation of areas, volumes and averages, as well as the modelling of phenomena in several scientific and technical fields. In secondary schools in Goma, Democratic Republic of the Congo (DRC), the teaching and learning of integrals take place in a socio-educational context marked by limited resources, insufficient continuing teacher training and weak contextualisation of textbooks and classroom activities. This study analysed the conditions under which integrals are taught and learned in Goma secondary schools, with particular attention to contextualised pedagogy and educational innovation. A descriptive mixed-method design was used. Data were collected from 16 mathematics teachers and 295 students drawn from 16 schools through textbook analysis, questionnaires and diagnostic tests. Quantitative data were processed using descriptive statistics, while open-ended responses and textbook content were examined through thematic content analysis. The findings show that the textbooks used in the schools introduce integrals mainly after derivatives and present them predominantly as inverse operations, with exercises that are largely procedural and weakly connected to local situations. Teachers reported limited access to continuing training and teaching resources, and classroom practice remained mainly lecture-based. Students perceived integral calculus as difficult and abstract, frequently confused the integral with the primitive, and encountered substantial difficulty when solving contextualised problems. However, many students indicated that practical examples, guided exercises and visual or digital supports could improve their understanding. The study concludes that improving the teaching and learning of integrals in Goma requires stronger didactic transposition, locally contextualised resources and sustained support for mathematics teachers.

Mapping the Memory Landscape: Understanding RAM

N K Kaphungkui — 2026-06-20

This chapter presents a practical, hardware-oriented explanation of random-access memory (RAM) and its relationship to low-level programming concepts. It is designed for learners who encounter difficulty with pointers, dynamic memory, stack behaviour, heap allocation, and the apparent abstraction of program storage in C and C++. The discussion begins by describing RAM as a linear sequence of byte-addressable cells, each identified by a unique address commonly represented in hexadecimal form. This model is then used to explain how variables, arrays, strings, objects, compiled instructions, and pointers occupy specific regions within memory. The chapter further examines the distinction between stack and heap memory, emphasising their different lifetimes, management models, growth directions, and roles in function execution and dynamic allocation. It also connects these software-level ideas with hardware-level organisation by comparing stack and heap concepts with the internal RAM structure of the 8051 microcontroller and the segmented memory architecture of the 8086 microprocessor. In addition, the chapter discusses processor registers, cache hierarchy, RAM, and secondary storage as layers of a memory-speed hierarchy, explaining why access time varies across these storage locations and why global variables may incur higher access costs than register-resident local values. By linking programming abstractions to the physical and architectural organisation of memory, the chapter offers a structured conceptual map for understanding pointer arithmetic, stack frames, heap management, register use, cache effects, embedded-system memory constraints, and low-level program behaviour. The overall aim is to support clearer reasoning about memory management and performance without treating memory-related errors as mysterious or disconnected from hardware reality.

Approximation of Lipschitz Functions via Product Summability Methods

Prabir Jena — 2026-06-20

Product summability methods provide a useful framework for studying the approximation of Fourier series when ordinary convergence does not fully capture the behaviour of functions with limited smoothness. This study examines the degree of approximation of periodic functions belonging to the Lipschitz class by applying the product mean obtained from the Cesàro and Euler summability methods to the associated Fourier series. The work is placed within summability theory and harmonic analysis, where the control of approximation error is central to understanding the convergence of transformed trigonometric series. The manuscript first reviews the relevant notions of degree of approximation, Lipschitz continuity, Cesàro summability, Euler summability and product means, and then establishes the principal theorem for functions that are Lebesgue integrable on the stated interval and periodic with period 2π. The theorem gives an estimate for the approximation error in relation to the smoothness parameter of the Lipschitz class and the order of the Fourier approximation. The analysis indicates that the combined Cesàro–Euler product mean provides a structured procedure for obtaining approximation estimates for functions whose regularity is controlled but not necessarily differentiable. The results contribute to the study of summability-based Fourier approximation by clarifying how a product method can be used in the Lipschitz setting. The discussion further notes the relevance of such approximation techniques to areas in which Fourier representations are used, including signal analysis, numerical methods and computational modelling. Overall, the study presents a mathematically focused treatment of product summability as a tool for estimating the approximation of Lipschitz functions through Fourier series, without extending the conclusion beyond the established theorem.