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.

Abstract: The ability to understand implicit relationships between events plays an important role in higher-level natural language processing, though the current methods for event, or relationship extraction, struggle with multi-event chains where the logical relationship is unstated and spans multiple sentences or paragraphs. Current approaches generally rely on explicit indicators or preset relation types to identify event relationships and do not account for reasoning and common sense knowledge. In response to this gap we introduce Context-Aware Implicit Relation Discovery (CAIRD) - a framework for detecting and extracting unstated relationships that carry semantic importance across event sequences such as causal, temporal and conditional relationships. CAIRD comprises an Event Chain Context Encoder for sequential understanding, a Common Sense Knowledge Augmenter to incorporate outside knowledge, and an Implicit Relation Detector that learns a representation of relations within a continuous space, along with a Relation Fusion and Output Module. We also introduce the Implicit Narrative Graph dataset for annotating implicit relations, and the Eventual Common Sense knowledge graph for outside augmentation. Experiments show that CAIRD performs better than strong baselines for text relations and knowledge-enhanced approaches suggesting the utility of external knowledge, and the overall utility of the proposed framework architecture to capture both rich implicit logical properties of more complex, longer event sequences.

Abstract: In this paper, an analytic Fourier–Feynman transform(FFT) and a convolution product(CP) associated with bounded linear operator(BLOP)s on abstract Wiener space(AWS) B are defined. The existences of the FFT and the CP of certain bounded functionals on B are also provided. Additionally, three kinds of relationships between the FFT and the CP are investigated. It turned out in this paper that the relations between them as well as the concepts of the transform and the convolution involve previous researches performed with Gaussian processes on classical Wiener space C₀[0,T]. That is, the Gaussian processes used in previous researches are Banach space BLOPs on C₀[0,T].

Abstract: The Collatz conjecture, also known as the 3n + 1 problem, remains one of the most famous unsolved problems in mathematics. This paper investigates the behavior of the Collatz map through the binary structure of natural numbers. We establish quantitative connections between the fractional part of log₂n, the density of zeros and ones in binary expansions, and the 2-adic valuation v₂(3n + 1). For an explicit infinite subclass of integers with zero density at least 1/2 in their binary expansion (approximately 2n−1 numbers of binary length n), we rigorously verify the conjecture, proving that trajectories reach the cycle {4, 2, 1} in at most O((log₂n)²) steps. The analysis reveals that sequences exhibit increasing zero density in intermediate steps, contributing to their collapse to 1, providing new structural insights. We give rigorous remainder bounds for fractional-part recurrences, proving |Fj(x)| ≤ |x| and |Rj(x)| ≤ |x| with explicit constants. We strengthen the results with extended numerical verifications up to n = 10000, a tighter analysis of run lengths using diophantine approximation, and additional references on binary expansions and ergodic theory. We also compare our subclass to known verified classes, such as powers of 2, and align our approach with equidistribution results for asymptotic density.

Abstract:

Predicting periods of heightened stock-price volatility helps investors and policy makers manage risk during geopolitical and macroeconomic shocks. This study models the short-term volatility of seven influential U.S. technology companies—Apple, Microsoft, Alphabet, Amazon, Nvidia, Tesla and Meta—collectively known as the “Magnificent Seven.” We build classification models to distinguish between high- and low-volatility regimes using daily stock prices, technical indicators and sentiment signals derived from tariff news between 1 January 2018 and 30 April 2025. The United States Trade Representative announced in May 2024 that tariffs on semiconductors will rise from 25% to 50% and tariffs on electric vehicles will increase from 25% to 100% these actions highlight the importance of trade policy for tech stocks. Our methodology computes a rolling 14-day standard deviation to label volatility regimes and applies logistic regression, decision trees and random forest classifiers. The random forest model tuned with Optuna outperforms other methods, achieving 0.69 accuracy, 0.64 precision, 0.65 recall, 0.64 F1 and a ROC–AUC of 0.72 on out-of-sample data. Feature importance analysis shows that tariff sentiment, average true range and Bollinger band width are the strongest predictors of volatility. The models and visualizations, along with a reproducible code appendix, offer investors and policy makers a transparent framework for assessing the impact of tariff announcements on market turbulence.

Abstract: The COVID-19 pandemic caused people worldwide to have continuous mental distress over the years due to ongoing uncertainty, economic instability, and changes in the living environment. However, the perceived stress of individuals is not uniform due to many factors such as their biological differences, gendered social roles, living conditions, and experiences. This study investigates the gendered impact of marital status on mental health, specifically, feelings of nervousness or stress, one year after the COVID-19 pandemic, while controlling sociodemographic variables: age, education, employment, family size, and national-level happiness scores using responses from over 24 countries from COVIDiSTRESS Global Survey II data and explainable machine learning models. Our results revealed that age, gender, and national happiness were the strongest predictors of stress, with relationship status also playing a moderate but meaningful role. Notably, women in dating or cohabitating relationships indicated significantly higher model-inferred stress levels, while marriage appeared more protective, especially for women. These findings highlight the complex interplay between relational, personal, and national factors in shaping mental health outcomes and call for gender-sensitive mental health policies in the post-pandemic period or public health crisis.