Abstract: The type of AI used to design video game enemies greatly affects the gameplay experience of speed, difficulty, and enjoyment. In most cases, the majority of developers who create 2D platforming games will choose to implement a simple but efficient AI design over an advanced AI model that learns based on experience. One of these simpler AI models that is frequently utilized by 2D platforming game developers is the Finite State Machine (FSM) model. The FSM model creates an organization of the enemy's actions into a limited number of well-defined behaviours, while also indicating how these behaviours relate to one another. We look at how AI uses the FSM method in the 2D platform game "Kirby: Nightmare in Dreamland," which first came out on the Game Boy Advance. The analysis of FSM models enemy AI behaviors and how those behaviors change and when they do so affects how hard the game is. Simulated experiments were conducted on how state time is spread out and how to make things harder by changing the attack cooldown. The results show that FSM-based AI is easy to control, doesn't need a lot of processing power, and has behavior that can be predicted. This makes it a good choice for platform games that are easy to get into. The results show that FSMs are still important in AI research and game design today.

Abstract: Unmanned Aerial Vehicle (UAV), when equipped as communication relays, offer a flexible solution to extend Vehicle-to-Vehicle (V2V) communications beyond fixed infrastructure and Non-Line-of-Sight constraints. In this setting, the allocation of radio resources, across time, frequency and space through beamforming, is challenged by the mobility of Connected and Autonomous Vehicles (CAVs) and their temporal dependencies, as access opportunities depend on prior transmission outcomes such as queue backlog or failed attempts. This paper proposes a Radio Resource Assignment (RRA) framework for UAV-aided V2V networks with beamforming-capable UAV relays. The model discretizes time and space to account for mobility and to track the movement of groups of CAVs across beam segments. The model also incorporates Time Division Multiple Access (TDMA)-based scheduling, beam activation constraints, and realistic traffic generation patterns. Analytical expressions are derived for per-user success probability and system throughput under both, ideal and realistic conditions, and they are validated against simulations, confirming the accuracy of the proposed approximations. Numerical results highlight trade-offs involving UAV altitude and resource allocation interval, while a heuristic beam-activation optimization strategy is shown to further enhance performance, achieving up to 12\% throughput gain over uniform activation.

Abstract: The standard $\varepsilon$--$\delta$ definition of continuity is inherently quantitative, yet the precise dependence of the admissible radius $\delta$ on the accuracy $\varepsilon$ and the base point $x_0$ is rarely treated as an independent mathematical object. In this paper, we introduce the \textit{radius of continuity} through two variants: the radius of pointwise continuity and the radius of uniform continuity, defined as explicit numerical invariants that capture the maximal symmetric neighborhood on which a real-valued function maintains a prescribed tolerance. We establish the fundamental structural properties of these radii, including their behavior under algebraic operations such as sums, products, and compositions, and demonstrate their inverse relationship to the classical modulus of continuity. Furthermore, we prove that the finiteness pattern of these radii characterizes constant versus non-constant functions. To illustrate the utility of this framework, we derive closed-form expressions for the pointwise radius of quadratic polynomials and the uniform radius of the normal probability density function. These examples highlight how the radius of continuity encodes geometric and probabilistic features, such as local curvature and global scale parameters. Ultimately, this perspective bridges the gap between real analysis and quantitative methods in metric geometry, offering a concrete measure of the stability of a function's continuity.

Abstract: This study was inspired by Alcantara-Bode’s equivalent to the Riemann Hypothesis published in 1993, the equivalent formulation consisting in the injectivity of an integral operator connected to Riemann Zeta function. Surprisingly, the research on this line has not continued, an explanation would be the lack of criteria for the injectivity of integral operators. This paper aims to fill this gap by proposing a functional-numerical analysis solution exploiting the operator positivity properties on dense sets. The main theorem says that a linear, bounded operator strict positive definite on a dense set of a separable Hilbert space, has its null space containing only the null element, equivalently, it is injective. Having in mind to obtain a generic and useful criterion, we gradually changed the hypothesis of the strict positivity of the operator on a dense set to the involvement at the end, of the associated Hermitian operator that is semi positive on the whole space requesting additional properties related to the positivity of operator approximations on finite dimension subspaces. Then, in order to apply the criterion for Hermitian Hilbert-Schmidt operators, we choose an adequate dense set allowing to obtain operator sparse matrix representations. The criterion applied to the associated Hermitian of the Alcantara-Bode integral operator, showed that the equivalent holds, so the Riemann Hypothesis is true.

Abstract: On the basis of the isomorphic algebraic structures of the field of complex numbers ℂ and the 2-dimensional Euclidean field of real vectors V₂, in terms of identical geometric products of elements, this paper brings integral identities for scalar and vector fields in V₂, which are vector analogues of the well-known integral identities of complex analysis. Consequently, in this paper, Theorem 1., which is a generalized fundamental theorem of integral calculus in the field V₂, is the vector analogue of the Cauchy theorem of complex analysis. Therefore, special attention is paid to the vector analogue of Cauchy's calculus of residues in the field V₂. Finally, at the very end of the paper, the algebraic structure of the 3D field of vectors V₃ is presented, as well as the corresponding fundamental integral identities.

Abstract: Background: In Japan, the number of older adults living alone has been increasing, raising the risk of unnoticed health decline or solitary death. Continuous monitoring using sensors can help detect behavioral changes indicating health issues and has the potential to support both older adults and their families. Methods: We obtained behavior and temperature data, continuously recorded over a long period at 15-min intervals from sensors installed in the homes of nine older adults living alone. After data cleaning, behavioral signals were analyzed using Fourier spectral analysis and multiple regression to extract 13-dimensional behavioral characteristic vectors. We whitened a portion of these behavioral characteristic vectors as benchmark data. We applied the same whitening process to the comparing data using the matrix obtained during this whitening process. By analyzing misclassification rates using boundary variance for benchmark and comparing data, we attempted to detect temporal changes in user behavior and differences between individuals. Results: Spectral analysis revealed 24-hour periodicity in all users’ behavior. By analyzing the misclassification rate using boundary variance for long-term signals, we identified users who maintain consistent behavioral patterns and those exhibiting significant temporal variation. We were also able to detect differences in these behavioral patterns. Conclusions: This study demonstrates that long-term temporal changes in the daily behavior of older adults living alone can be detected using simple continuous sensor data. Our approach is applicable not only for monitoring behavior changes in older adults living alone, but also for observing behavior changes in people with disabilities and children within the home environment.

Abstract: The Green Tech Revolution is humanity’s answer to climate change, featuring game-changing advances in renewable energy, smart infrastructure and circular economies. Successes like Denmark’s wind energy dominance and Norway’s electric vehicle adoption provide proof that scalable, meaningful solutions are attainable. Nevertheless, this revolution in power generation has brought with it major ethical and logistical problems. The environmental impact of mining, global concerns over e-waste, and social risks, including the digital divide and job displacement, reveal a complicated terrain in which innovation can unwittingly exacerbate existing inequities. To manage this, we require a multidimensional Balancing act between Progress and equity. This involves supporting next-generation technologies such as hydrogen fuel and perovskite solar cells, implementing strong policies for sustainable production and recycling, and encouraging individual responsibility via greener consumption. In the end, a truly sustainable future is not going to be delivered by technology; it has to be a imperative and equitable partnership between governments, corporations and citizens. It is only through such concerted efforts that the gains of the green transition can be shared universally, so that planetary health and social justice march forward hand in hand.

Abstract: The semantic interpretation of actions is deeply intertwined with how change unfolds over time, space, and interaction. Prior theoretical and computational work has suggested that explicitly modeling three-dimensional motion---including object positions and orientations evolving through time---should offer a privileged pathway for encoding fine-grained verb meaning, especially for distinctions such as \textit{roll} versus \textit{slide}. At the same time, the vast majority of multimodal language models rely almost exclusively on two-dimensional visual inputs, implicitly assuming that such projections suffice to ground linguistic meaning. In this work, we revisit this assumption through a systematic and tightly controlled comparison of visual and motion-based modalities. We construct self-supervised encoders over both 2D video observations and 3D trajectory data, and probe the resulting representations for their capacity to discriminate verb-level semantic categories. Contrary to prevailing intuition, our empirical analysis reveals that representations learned from 2D visual streams are competitive with, and in some cases indistinguishable from, those derived from explicit 3D trajectories. These findings complicate the widely held belief that richer environmental encodings automatically lead to superior semantic representations, and suggest that the relationship between perceptual fidelity and linguistic abstraction is more nuanced than often assumed. Our study offers early evidence that effective verb representation may emerge from multiple perceptual pathways, motivating a rethinking of how embodiment and modality interact in multimodal language learning.

Abstract: We will prove the generalized stability of an additive-quadratic-cubic functional equation in the sprit of Găvruţa.

Abstract: Classical real analysis rigorously defines convergence via ε − N criteria, yet it frequently regards the specific entry index N as a mere artifact of proof rather than an intrinsic property. This paper fills this quantitative void by developing a radius of convergence framework for the sequence space Seq(R). We define an index-based radius ρ_a(ε) alongside a rescaled geometric radius ρ∗_a (ε); the latter maps the unbounded index domain to a finite interval, establishing a structural analogy with spatial radii familiar in analytic function theory. We systematically analyze these radii within a seven-block partition of the sequence space, linking them to liminf-limsup profiles and establishing their stability under algebraic operations like sums, products, and finite modifications. The framework’s practical power is illustrated through explicit asymptotic inversions for sequences such as Fibonacci ratios, prime number distributions, and factorial growth. By transforming the speed of convergence into a geometric descriptor, this approach bridges the gap between asymptotic limit theory and constructive analysis, offering a unified, fine-grained measure for both convergent and divergent behaviors.

Abstract: The first aim of this study is to point out new aspects of approximation theory applied to a few classes of holomorphic functions, via Vitali’s theorem. The approximation is made with the aid of the complex moments of the involved functions, that are defined similarly to the moments of a real valued continuous function. Applying uniform approximation of continuous functions on compact intervals via Korovkin’s theorem, the hard part concerning uniform approximation on compact subsets of the complex plane follows according to Vitali’s theorem. The theorem on the set of zeros of a holomorphic function is also applied. In the end, existence and uniqueness of solution for a mul-tidimensional moment problem is characterized in terms of limits of sums of quadratic expressions. This is an application appearing at the end of the title. Consequences resulting from the first part of the paper are pointed out with the aid of functional calculus for self-adjoint operators.

Abstract: Accurate evaluation of pressure distribution at the socket–limb interface is essential for improving prosthetic fit and comfort in transfemoral amputees. This study aimed to develop a data-driven framework that integrates machine learning–based segmentation with finite element method (FEM) to quantitatively assess interface pressure during socket application. MRI data from a transfemoral amputee were processed using a custom image segmentation algorithm to extract adipose tissue, femur, and ischium, achieving high F-measure scores. The segmented tissues were reconstructed into 3D models, refined through outlier removal and surface smoothing, and used for FEM simulations in LS-DYNA. Pressure values were extracted at nine sensor locations and compared with experimental measurements. The results showed consistent polarity between measured and simulated values across all points. Furthermore, at the eight locations excluding the ischial tuberosity (IS) region, a statistically significant and moderately strong positive correlation was observed between measured and simulated pressures (r = 0.7485, p < 0.05). Notably, positive pressure regions demonstrated close agreement between experimental and simulated values, whereas the discrepancy observed at the IS region was likely influenced by the medial boundary conditions introduced to prevent unrealistic tissue displacement. This difference highlights a limitation of the current simulation setup. Overall, the proposed framework demonstrated reliable pressure estimation and offers a promising approach for personalized prosthetic socket design through automated anatomical modeling and simulation.

Abstract: Social media platforms have become critical spaces where consumers and investors publicly react to major corporate events. These online reactions provide rich text data for analyzing brand sentiment and evaluating marketing campaigns. This study examines how sentiment toward Apple changed the company’s 2020 product launch within Reddit finance communities. Using a dataset of 297,533 Reddit comments mentioning Apple’s ticker (“AAPL”) posted between November 2016 and October 2021 in finance-related subreddits, comments were labeled as occurring before or after the September 11, 2020, launch. Sentiment was measured using VADER, a lexicon‐ and rule‐based sentiment analyzer optimized for social media text (Hutto & Gilbert, 2014). Descriptive statistics, correlation analyses, and independent‐samples t tests compared sentiment and engagement (upvotes) across periods and explored relationships among sentiment, text length, and upvotes. Overall sentiment was slightly positive (M = 0.13), with a small but statistically significant increase after the launch (Before: M = 0.12; After: M = 0.14). Upvotes did not differ meaningfully by period. Correlations showed that stronger sentiment was associated with longer comments but was essentially unrelated to upvotes. As an exploratory extension, a small labeled subset of comments was used to pilot fine‐tuning a transformer-based model with the Unsloth framework, building on evidence that domain-specific transformers such as FinBERT outperform lexicon-based methods on financial text (Araci, 2019). The findings suggest that Apple’s 2020 launch modestly improved conversational tone in Reddit finance discussions without changing engagement, and they highlight the value of combining fast lexicon methods with modern transformers for campaign evaluation.

Abstract: In this paper we present a method to get the prime counting function p(x) and other arithmetical functions than can be generated by a Dirichlet series, first we use the general variational method to derive the solution for a Fredholm Integral equation of first kind with symmetric Kernel K(x,y)=K(y,x), after that we find another integral equations with Kernels K(s,t)=K(t,s) for the Prime counting function and other arithmetical functions generated by Dirichlet series, then we could find a solution for π(x) and \( \sum_{n \le x} a(n) = A(x) \), solving δJ[ϕ]=0 for a given functional J, so the problem of finding a formula for the density of primes on the interval [2,x], or the calculation of the coefficients for a given arithmetical function a(n), can be viewed as some “Optimization” problems that can be attacked by either iterative or Numerical methods (as an example we introduce Rayleigh-Ritz and Newton methods with a brief description) Also we have introduced some conjectures about the asymptotic behavior of the series \( \Xi_n(x) = \sum_{p \le x} p^{\,n} = S_n(x)\ \) for n>0 , and a new expression for the Prime counting function in terms of the Non-trivial zeros of Riemann Zeta and its connection to Riemman Hypothesis and operator theory.

Abstract: This study analyzes New Zealand-sourced greenhouse gas data to identify trends and factors of emission. These findings are expected to guide and enable the creation of efficient policies and sustainable practices supportive of New Zealand's goals on climate change for a greener and sustainable future. The study tries to highlight the rising greenhouse gas levels which have been impacting agriculture, transportation, and public health. New Zealand's industrial landscape is so diverse that it provides a very special opportunity to study the trends in emissions and identification of effective mitigation strategies. The review covers major trends by sector, region, and household activities from 2007 to 2022. Based on the findings, agriculture is the largest contributor to emissions, accounting for 18.6% of total GHG outputs. Primary industries, forestry, and fishing are also big contributors, while urban transportation shows a flat trend. Regional disparities are reflected, with Waikato, Canterbury, and Auckland having the highest emissions.

Abstract: This study applies semantic and sentiment analysis to explain why large language model (LLM)–predicted hashtags differ from hashtags chosen by human content creators for YouTube long-form video descriptions. Using the Public Health Advocacy Dataset (PHAD), which contains social-media videos related to tobacco products (University of Arkansas CVIU Lab, n.d.), the project examines whether the sentiment expressed in each description particularly emotional tone or motivational language, helps explain why some LLM predictions match human labels and others do not. An LLM (Qwen-3) predicts hashtags based solely on video descriptions, and mismatches between predicted and human-assigned hashtags are then analyzed. In this study, two approaches are used to measure sentimental features: LIWC categories capturing tones, and curiosity-related wording, and VADER polarity scores catching fine-grained emotional tone. Both sentiment models are applied to the validation dataset to compare matched and mismatched cases. The LLM reached an accuracy of 55.19%. Results show no significant sentiment differences between correct and incorrect predictions, suggesting that mismatches are not driven by emotional or motivational cues and that the LLM’s errors are more likely related to semantic ambiguity or category complexity rather than sentiment.

Abstract: The SISSI/SGCI framework (Spectral Information Similarity System Interface / Spectral Generalized Coherence Index) provides a unified harmonic–geometric model for quantifying vibrational information flow across molecular systems. While Version 1 introduced the mathematical formulation of the harmonic operator and its coherence functional, this Version 2 presents the first real-world experimental validation using isotopic vibrational spectra. Benzene vs. benzene-d6 and water H2O vs. heavy water (D2O) serve as benchmark systems to test sensitivity, robustness, and harmonic alignment performance. We introduce a fully reproducible end-to-end pipeline including JCAMP-DX parsing, baseline correction, normalization, uniform resampling, local harmonic curvature mapping, sliding-window coherence tracking, zero-matching distance (ZMD), and Monte Carlo null-model comparison. Results show that SISSI/SGCI identifies isotopic vibrational shifts with significantly higher precision than classical spectral similarity measures (correlation, cosine similarity, RMS error), and remains stable under synthetic noise conditions. This experimental validation demonstrates that SISSI/SGCI is not only a mathematically rigorous formalism but also a practical high-resolution tool for analyzing vibrational information, with potential applications in spectroscopy, materials science, computational chemistry, and information-theoretic descriptions of molecular dynamics. All datasets, figures, and the complete reproducible demonstration package are openly released.

Abstract: Fractional transforms have emerged as powerful analytical tools that bridge the time, frequency, and scale domains by introducing a fractional-order parameter into the kernel of classical transforms. This survey provides an overview of the mathematical foundations and distributional frameworks of several key fractional transforms, with emphasis on their formulation within appropriate spaces of generalized functions. Particular attention is devoted to the quasiasymptotic behavior of distributions in relation to the asymptotic properties of their corresponding fractional transforms. Moreover, we apply the considered transforms to the same sample signal and perform a comparative analysis.

Abstract: We study Geraghty-type non-self mappings within the framework of best proximity point theory. By introducing auxiliary functions with subsequential convergence, we establish general conditions ensuring the existence and uniqueness of best proximity points. Our results extend and unify earlier work on proximal and Kannan-type contractions under a Geraghty setting, and we provide counterexamples showing that the auxiliary assumptions are essential. As an illustration, we construct an explicit non-self alignment mapping on subsets of \( \mathbb{R} \)² for which all hypotheses can be verified and the unique best proximity point, as well as the convergence of the associated proximal iteration, can be computed in closed form.

Abstract: In this paper, we study the behavior of the m − th derivatives of general algebraic polynomials in weighted Bergman spaces defined in domains of the complex plane bounded by piecewise smooth curves with nonzero exterior angles and zero interior angles. Our approach involves establishing upper bounds on the growth of these derivatives not only interior of the unbounded domain but also on the closure of given domain. Through this analysis, we reveal detailed patterns in the growth of the m − th derivatives of algebraic polynomials throughout the complex plane, depending on the properties of the weighted function and the domain.