Abstract: All characterizations of Shannon’s entropy include the so-called chain rule, a formula on a hierarchically structured probability distribution, which is based on at least two elementary distributions. We show that the chain rule can be split into two natural components, the well-known additivity of the entropy in case of cross-products and a variant of the chain rule that involves only a single elementary distribution. The latter is given as a proportionality relation and hence allows a vague interpretation as self-similarity, hence intrinsic property of the Shannon entropy. A similar characterization is given also for the Rényi entropy and the min-entropy.

Abstract: With the emergence of Artificial Intelligence (AI) chatbot technologies such as ChatGPT, which utilize AI and Machine Learning (ML) techniques to generate advanced responses to users, the field of education has been transformed drastically. The latest online learning models empowered by AI have proven to have several benefits for students, but these benefits also come with some inherent challenges, which impede students’ learning process and create hurdles for educators. The study aims to identify and analyze the benefits and drawbacks of AI chatbots in educational settings to help overcome existing educational barriers. The paper begins by highlighting the historical evolution of chatbots along with key elements that encompass the architecture of an AI chatbot. The paper also delves into the challenges and limitations associated with the integration of AI chatbots into learning platforms. A systematic literature review methodology has been adopted by dividing the research into several phases, using techniques of inclusion and exclusion criteria to identify appropriate scholarly articles for review. The research findings from this review reveal several benefits of leveraging AI chatbots in learning platforms. AI chatbots like ChatGPT can serve as a virtual tutoring assistant to foster an adaptive learning environment by aiding students with various learning activities, such as learning programming languages, foreign languages, understanding complex concepts, assisting with research activities and providing real-time feedback. Educators can leverage such chatbots to create course content, generate assessments, evaluate students’ performance and use it for data analysis and research. However, this technology possesses significant challenges concerning data security and privacy. In addition, ethical concerns impacting academic integrity and reliance on technology are some of the key concerns. Ultimately, AI chatbots can provide endless opportunities by creating a dynamic interactive learning environment, however, to allow students and teachers to maximize the potential of such a robust technology, it is imperative to first analyze and address the potential risks that AI chatbots pose.

Abstract: The widespread adoption of generative Artificial Intelligence (GenAI) tools in higher education has necessitated the development of appropriate and ethical usage guidelines. This study aims to explore and assess publicly available guidelines covering the use of GenAI tools in universities, following a predefined checklist. We searched and downloaded publicly accessible guidelines on the use of GenAI tools from the websites of the top 50 universities globally, according to the 2025 QS university rankings. From the literature on GenAI use guidelines, we created a 24-item checklist, which was then reviewed by a panel of experts. This checklist was used to assess the characteristics of the retrieved university guidelines. Out of the 50 university websites explored, guidelines were publicly accessible on the sites of 41 institutions. All these guidelines allowed for the use of GenAI tools in academic settings provided that specific instructions detailed in the guidelines were followed. These instructions encompassed securing instructor consent before utilization, identifying appropriate and inappropriate instances for deployment, employing suitable strategies in classroom settings and assessment, appropriately integrating results, acknowledging and crediting GenAI tools, and adhering to data privacy and security measures. However, our study found that only a small number of the retrieved guidelines offered instructions on the AI algorithm (understanding how it works), the documentation of prompts and outputs, AI detection tools, and mechanisms for reporting misconduct. Higher education institutions should develop comprehensive guidelines and policies for the responsible use of GenAI tools. These guidelines must be frequently updated to stay in line with the fast-paced evolution of AI technologies and their applications within the academic sphere.

Abstract: It is well known that the four ancient Greek triangle centers and others have homogeneous barycentric coordinates that are polynomials in the sidelengths a; b; c of a triangle ABC. For example, the circumcenter is represented by the polynomial a(b2 + c2 - a2). It is not so well known that triangle centers having barycentric coordinates such as tanA : tanB : tanC are also representable by polynomials, in this case, by p(a; b; c) : p(b; c; a) : p(c; a; b), where p(a; b; c) = a(a2 + b2 - c2)(a2 + c2 - b2). This paper presents and discusses polynomial representations of triangle centers that have barycentric coordinates of the form f(a; b; c) : f(b; c; a) : f(c; a; b), where f depends on one or more of the functions in the set fcos; sin; tan; sec; csc; cotg. The topics discussed include innite trigonometric orthopoints, the n-Euler line, and symbolic substitution.

Abstract:

This paper offers an in-depth analysis of the role of the Internet of Things (IoT) in fire safety systems, emphasizing fire detection, localization, and evac- uation. Through a bibliometric analysis, we identify pivotal research trends and advancements in IoT-based sensors and devices. We discuss the integration of emerging technologies to enhance fire safety system performance and delve into the primary network architectures and communication protocols vital for efficient IoT-based fire safety systems. The paper concludes by highlighting challenges, research gaps, and prospective directions for IoT in fire safety.

Abstract: IIn this paper we derive a Plancherel theorem specific for a WidderLambert type integral transform by employing the corresponding Plancherel theorem for the classical Mellin transform. These findings lead us to explore a class of functions in connection with Salem’s equivalence to the Riemann hypothesis.

Abstract: This study investigates the application of modified Wasserstein Generative Adversarial Networks with Gradient Penalty (WGAN-GP) to generate synthetic RGB and infrared (IR) datasets for precision agriculture, particularly targeting the detection of Raphanus Raphanistrum (wild radish). Traditional WGAN models face challenges such as vanishing gradients and poor convergence, which hinder their effectiveness in generating high-quality synthetic data. To address these issues, this work proposes modifications that include replacing fully connected layers with convolutional and transposed convolutional layers, combined with batch normalization, to improve the fidelity of generated images and training stability. The experimental results demonstrate that the modified WGAN-GP produces superior synthetic images compared to other GAN variants, especially in maintaining structural similarity for RGB datasets. However, generating high-quality IR images remains challenging due to inherent spectral complexities, with consistently lower SSIM (Structural Similarity Index) scores across models. This study highlights the importance of architectural modifications and auxiliary learning strategies in enhancing GAN performance, especially for complex agricultural datasets. The findings suggest that future work should focus on integrating attention mechanisms and advanced loss functions to further improve model stability and the quality of generated synthetic data. These advancements are vital for the broader adoption of GANs in precision agriculture, enabling enhanced data augmentation for machine learning applications that support effective crop monitoring and management.

Abstract: In the article, we define a sequence F: F(0) = 2, F(1) = 10, F(2) = 16, F(3) = 22, F(4) = 26, F(4) = 28, · · · , F(i) = F(i − 1) + l, where the number l ≤ 8 is defined by using prime numbers and matrix algebra. The sequence includes only even numbers and can be used to define a series of numbers F(i) + 1, the non-Goldbach sequence, that contains all numbers that cannot be written as the sum of two primes. The result is related to Goldbach’s conjecture. The famous Goldbach conjecture states that every even integer greater than 2 can be expressed as the sum of two primes.

Abstract: This study presents a comprehensive characterization of iterative steganography, a distinct class of information-hiding techniques, and proposes a formal mathematical model for their description. We introduce a novel quantitative measure, the Incremental Information Function (IIF), designed to evaluate information gain in iterative steganographic methods. The IIF provides a comprehensive framework for analyzing the step-by-step process of embedding information into a cover medium, focusing on the cumulative effects of each iteration in the encoding and decoding cycles. The practical application and efficacy of the proposed method are demonstrated through detailed case studies in video steganography. These examples illustrate the utility of the IIF in delineating the properties and characteristics of iterative steganographic techniques. Our analysis reveals that the IIF effectively captures the incremental nature of information embedding and serves as a valuable tool for assessing the robustness and capacity of steganographic systems. This research offers significant insights into the field of information hiding, particularly in the development and evaluation of advanced steganographic methods. The IIF emerges as a promising analytical tool for researchers, providing a quantitative approach to understanding and optimizing iterative steganographic techniques.

Abstract: The P versus NP problem is a cornerstone of theoretical computer science, asking whether problems that are easy to check are also easy to solve. "Easy" here means solvable in polynomial time, where the computation time grows proportionally to the input size. While this problem's origins can be traced to John Nash's 1955 letter, its formalization is credited to Stephen Cook and Leonid Levin. Despite decades of research, a definitive answer remains elusive. Central to this question is the concept of NP-completeness. If even one NP-complete problem could be solved efficiently, it would imply that all problems in NP could be solved efficiently, proving P equals NP. This research proposes that a notoriously difficult NP-complete problem can be solved efficiently, thereby potentially establishing the equivalence of P and NP. This work is an expansion and refinement of the article "Note for the P versus NP problem", published in IPI Letters.