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This study presents some explicit exact solutions for nonlinear geophysical ocean waves in the β-plane approximation near the equator. The solutions are provided in Lagrangian coordinates by describing the path of each particle. The unidirectional equatorially trapped waves are symmetric about the equator and propagate eastward above the thermocline and beneath the near-surface layer to which wind effects are confined. At each latitude the flow pattern represents a traveling wave.

Analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed: within a period each particle experiences a backward-forward motion with a slight forward drift. [PUBLICATION ABSTRACT]

In recent years two nonlinear dispersive partial differential equations have attracted much attention due to their integrable structure. We prove that both equations arise in the modeling of the propagation of shallow water waves over a flat bed. The equations capture stronger nonlinear effects than the classical nonlinear dispersive Benjamin–Bona–Mahoney and Korteweg–de Vries equations. In particular, they accommodate wave breaking phenomena.

This paper develops a new approach in the analysis of the Camassa-Holm equation. By introducing a new set of independent and dependent variables, the equation is transformed into a semilinear system, whose solutions are obtained as fixed points of a contractive transformation. These new variables resolve all singularities due to possible wave breaking. Returning to the original variables, we obtain a semigroup of global solutions, depending continuously on the initial data. Our solutions are conservative, in the sense that the total energy equals a constant, for almost every time.[PUBLICATION ABSTRACT]

The Camassa-Holm equation has a number of constants of motion arising as eigenvalues of an associated spectral problem. We give a description of the spectral picture and discuss the scattering problem.

The evaluation of large language models (LLMs) increasingly relies on other LLMs acting as automated judges. While this approach offers scalability and efficiency, it raises serious concerns regarding evaluator reliability, positional bias, and ranking stability. This paper presents a scalable framework for diagnosing positional bias and instability in LLM-based evaluation by using controlled pairwise comparisons judged by multiple independent language models. The system supports mirrored comparisons with reversed response order, prompt injection, and surface-level perturbations (e.g., paraphrasing, lexical noise), enabling fine-grained analysis of evaluator consistency and verdict robustness. Over 3600 pairwise comparisons were conducted across five instruction-tuned open-weight models using ten open-ended prompts. The top-performing model (gemma:7b-instruct) achieved a 66.5% win rate. Evaluator agreement was uniformly high, with 100% consistency across judges, yet 48.4% of verdicts reversed under mirrored response order, indicating strong positional bias. Kendall’s Tau analysis further showed that local model rankings varied substantially across prompts, suggesting that semantic context influences evaluator judgment. All evaluation traces were stored in a graph database (Neo4j), enabling structured querying and longitudinal analysis. The proposed framework provides not only a diagnostic lens for benchmarking models but also a blueprint for fairer and more interpretable LLM-based evaluation. These findings underscore the need for structure-aware, perturbation-resilient evaluation pipelines when benchmarking LLMs. The proposed framework offers a reproducible path for diagnosing evaluator bias and ranking instability in open-ended language tasks. Future work will apply this methodology to educational assessment tasks, using rubric-based scoring and graph-based traceability to evaluate student responses in technical domains.

Lately, the logistics sector has seen accelerated development, which has led to general economic growth, but, at the same time, it has caused considerable environmental damage due to the excessive consumption and emissions that are currently affecting society at large. Since logistics activities are considered some of the most polluting economic activities, this present article aims to present the advantages of implementing the green logistics concept. To this purpose, the activity of a logistics centre in Romania was analysed, with a focus on the greenhouse gases (GHGs) produced as a consequence of this economic activity, and its carbon footprint was calculated according to the GHG Protocol. Although this global standard is based on an integrated approach to how GHG emissions are calculated, there is limited evidence about its degree of implementation by companies. The results of the analysis revealed that the consumption of energy and fuel by the logistics sector has a significant impact on the environment. This impact is maintained, albeit at a smaller scale, even if the technology is replaced and the equipment used by companies to carry out their activities is increasingly performant.

We present an exact solution of the nonlinear governing equations for geophysical water waves in the β‐plane approximation near the Equator. The solution describes in the Lagrangian framework equatorially trapped waves propagating eastward in a stratified inviscid fluid. Key Points We present an explicit solution describing equatorially trapped waves The solution is obtained in Lagrangian coordinates The closed form of the solution offers a clear picture of the flow structure

The author presents an explicit exact solution to the governing equations for geophysical equatorial waves in the β-plane setting. The solution describes equatorially trapped waves propagating eastward above the thermocline and beneath the near-surface layer where wind effects are confined. At great depths the water is still, while the transition toward the large-amplitude oscillation of the thermocline is accommodated by an eastward-flowing current. Above the thermocline a flow reversal occurs, with the underlying current flowing westward close to the layer where wind effects are confined.

Industrial development has implicitly led to the development of new systems that increase the ability to provide services and products in real time. Autonomous mobile robots are considered some of the most important tools that can help both industry and society. These robots offer a certain autonomy that makes them indispensable in industrial activities. However, some elements of these robots are not yet very well outlined, such as their construction, their lifetime and energy consumption, and the environmental impact of their activity. Within the context of European regulations (here, we focus on the Green Deal and the growth in greenhouse gas emissions), any industrial activity must be analyzed and optimized so that it is efficient and does not significantly impact the environment. The added value of this paper is its examination of the activities carried out by mobile robots and the impact of their electronic components on the environment. The proposed analysis employs, as a central point, an analysis of mobile robots from the point of view of their electronic components and the impact of their activity on the environment in terms of energy consumption, as evaluated by calculating the emission of greenhouse gases (GHGs). The way in which the activity of a robot impacts the environment was established throughout the economic flow, as well as by providing possible methods of reducing this impact by optimizing the robot’s activity. The environmental impact of a mobile robot, in regard to its electronic components, will also be analyzed when the period of operation is completed.