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: 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: 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: 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 feature vectors. We attempted to detect temporal changes and behavioral characteristics by whitening these data and performing correspondence analysis. Results: Spectral analysis revealed 24-hour periodicity in all users’ behavior. Based on changes in the maximum component value and adjusted R2, individuals were classified into a stable group (SG) and a fluctuating group (FG). Boundary variance and false error analyses confirmed that behavioral temporal changes and individual characteristics could be detected objectively. Conclusions: The findings showed that temporal changes in daily behavior among older adults living alone can be detected using simple continuous sensor data, suggesting potential for early detection of health-related changes and preventive support in home.

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.

Abstract: The sequence space of all real-valued sequences, denoted Seq(R), is typically investigated through the lens of infinite-dimensional vector spaces, utilizing Banach space norms or Schauder bases. This work proposes a complementary, constructive classification based instead on the asymptotic limit profile encoded by the pair lim infa_n, lim supa_n. We demonstrate that this perspective naturally partitions Seq(R) into seven mutually disjoint macroscale blocks, covering behaviors from finite convergence to bounded and unbounded oscillation. For each block, we provide explicit closed-form representative sequences and establish that every constituent class possesses the cardinality of the continuum. Furthermore, we investigate the structural relationships between these blocks at two distinct levels of granularity. At the macroscale, we employ injective mappings to define an idealized connectivity graph, while at the microscale, we introduce a connection relation governed by the Hadamard (pointwise) product. This dual analysis reveals a rich directed graph structure where the block of finite convergent sequences functions as a global attractor with no outgoing connections. Statistical comparisons between the idealized and realized adjacency matrices indicate that the pointwise product structure realizes approximately two-thirds of the theoretically possible macroscale relations. Ultimately, this partition-based framework endows the seemingly chaotic space Seq(R) with a transparent, geometrically interpretable internal structure.

Abstract: Given Hilbert space operators A,B and X, let △_A,B and δ_A,B denote, respectively, the elementary operators △_A,B(X) = I − AXB and the generalised derivation δ_A,B(X) = AX − XB. This paper considers the structure of operators D^m _d1,d2 (I) = 0 and Dm d1,d2 compact, where m is a positive integer, D =△ or δ, d1 =△_A∗,B∗ or δA_∗,B∗ and d2 = △_A,B or δ_A,B. This is a continuation of the work done by C. Gu for the case △^m _δA∗,B∗, _δA,B (I) = 0, and the author with I.H. Kim for the cases △^m _{δA∗,B∗,δA,B} (I) = 0 or △^m _{δA∗,B∗,δA,B} is compact, and δ^m _{△A∗,B∗,△A,B} (I) = 0 or δ^m _{△A∗,B∗,δA,B} is compact. Operators D^m _d1,d2 (I) = 0 are examples of operators with finite spectrum, indeed the operators A,B have at most a two point spectrum, and if D^m _d1,d2 is compact, then (the non-nilpotent operators) A, B are algebraic. D^m _d1,d2 (I) = 0 implies Dⁿ _d1,d2 (I) = 0 for integers n ≥ m: the reverse implication, however, fails. It is proved that D^m _d1,d2 (I) = 0 implies Dd1,d2 (I) = 0 if and only if of A and B (are normal, hence) satisfy a Putnam-Fuglede commutativity property.

Abstract: The challenge of integrating external knowledge into visual reasoning frameworks has motivated a growing interest in models capable of bridging perceptual understanding with abstract, non-visual information. Unlike conventional visual question answering settings, knowledge-driven VQA demands a joint interpretation of visible cues and facts that are absent from the image itself. This paper introduces a new perspective on this task and proposes \textsc{KV-Trace}, a unified semantic tracing framework that emphasizes iterative knowledge refinement and structured visual interpretation. Instead of treating visual and knowledge modalities as homogeneous sources, our framework explicitly distinguishes their representational roles and organizes them into a progressive reasoning pipeline. Through a dynamic knowledge memory space and a query-sensitive semantic propagation mechanism, \textsc{KV-Trace} composes multi-stage reasoning steps that evolve according to the underlying question. Extensive experiments conducted on the KRVQR and FVQA benchmarks demonstrate that our model achieves improved reasoning depth and generalization capacity. Additional ablation studies further verify the contribution of each reasoning component and highlight the interpretability benefits gained from explicit knowledge structuring.

Abstract: In this paper we are studying the meromorphic integrability of a two-dimensional Hamiltonian system with a homogeneous potential of degree 6. The approach used in this work is the theory of the Ziglin-Moralez-Ruiz-Ramis-Simo. Within the scope of this theory, the study of such systems is reduced to determining the differential Galois group of a linear differential equation, obtained as a projection onto the tangent bundle of the phase curve of its non-equilibrium solution - Variation Equations (VE). In the case of Hamiltonian systems with homogeneous potentials, the variation equations are hypergeometric. If a standard approach is used to study such a system, it is necessary to calculate a Darboux point, which is not always easy. In this paper we can skip this difficulty by reducing VE to a Legendre equation We use the results for solvability of the Galois group of the associated Legendre equation for study a Hamiltonian system with a homogeneous polynomial potential of degree 6. For the full study, the second variations are used and conditions for a non-zero logarithmic term in their solutions are found.

Abstract: Descriptive statistics are a cluster of statistical techniques used to summarise, organise, and communicate general features of data already gathered. JASP (Jeffrey’s Amazing Statistics Program) is an open-source, cross-platform software package designed to make statistical analyses accessible, transparent, and reproducible. JASP integrates both frequentist and Bayesian methods within an intuitive interface that emphasises ease of use and high-quality output in the APA format. Unlike traditional statistical software, JASP reduces the technical burden of analysis through drag-and-drop functionality, automated data handling, and direct export options for results, figures and syntax. Its integrated data library and educational resources render it especially advantageous for students, while its sophisticated features, such as regression, factor analysis, and Bayesian modelling, provide robust tools for researchers. This review offers a comprehensive overview of the JASP environment, encompassing file management, data handling, analysis menus, and visualisation tools. Furthermore, it emphasises fundamental statistical principles, including measures of central tendency, dispersion, and data integrity. Researchers can learn data behaviour and enhance their skills using this tutorial, without needing extensive statistical programming knowledge.

Abstract: In this paper, we investigate the Cauchy problem for the coupled generalized Korteweg-de Vries system driven by white noise. We prove local well-posedness for data in $ H^{s} \times H^{s},$ with $ s>1/2$. The key ingredients that we used in this paper are multilinear estimates in Bourgain spaces, the Itô formula and a fixed point argument. Our result improves the local well-posedness result of Gomes and Pastor \cite{gomes2021solitary}.

Abstract: The ϵ-δ definition of continuity, foundational to real analysis, continues to attract sustained interest across the mathematics community. This paper responds to this interest by providing a comprehensive, systematic survey of direct ϵ-δ proofs for a broad spectrum of functions. We meticulously analyze 54 prominent real-valued functions, categorizing them into eight distinct clusters to highlight recurring proof structures and methodologies. For each function, we present a step-by-step proof alongside an explicit formula for δ in terms of ϵ and the point of continuity. Beyond serving as a robust pedagogical resource for students, instructors, and independent learners, this collection demystifies the proof-writing process by showcasing the elegant, unifying logic underlying a seemingly diverse set of problems. The work’s organized structure and detailed examples offer clarity where confusion often resides, ultimately fostering a deeper intuition for the core principles of continuity. By transforming a collection of challenging proofs into an accessible and navigable reference, this atlas opens up new avenues for further research in the field for mathematicians.

Abstract: Large Language Models (LLMs) play a key role in social simulations but most existing virtual agents retain little if any dynamic adaptability and do not convincingly express different emotional states, limiting the fidelity of the simulation despite advancements in generation population-aligned persona. We present EvoPersona, a design to enhance population-aligned persona with both dynamic contextual awareness and emotional dynamics. EvoPersona has two components: a Contextual Awareness Module, built on an instruction-tuned small LLM, that allows the agent (persona) to adapt its behavior and language style in response to situational cues in real-time; and an Emotional Dynamics Module based on an evolving internal emotional state that is driven by real-time emotional input analysis and subject to Reinforcement Learning supervised feedback to ensure that the emotional state is naturalistic and, importantly, consistent with the context. We report results from five extensive experiments showing continued strong population-level psychological alignment along with significantly better contextual coherence and emotional realism than baseline state-of-the-art models.