Abstract: Goldbach’s strong conjecture, asserting that every even integer greater than two can be expressed as the sum of two prime numbers, remains one of the oldest unresolved problems in mathematics. Despite overwhelming numerical verification and powerful partial results, a complete analytic proof has remained elusive. At the same time, extensive computations have revealed a striking empirical phenomenon known as Goldbach’s comet: the rapidly growing number of Goldbach representations as a function of the even integer E, forming a characteristic comet-like structure when plotted.This review article provides a comprehensive synthesis of classical analytic number theory, modern distributional results on primes, and recent structural insights in order to explain the existence, shape, and persistence of Goldbach’s comet. We introduce and develop a unified framework based on three complementary quantities: the dominance ratio Ω(E), measuring the growth of available prime density relative to local obstructions; the density field λ, encoding the smooth asymptotic behavior of primes; and the obstruction constant Κ, bounding the maximal effect of local gaps and covariance.We show that Ω(E) diverges, reflecting a fundamental scale separation between global prime density and local irregularities, and that λ-weighted obstructions remain bounded while density grows without bound. This framework explains why no gap-based or covariance-based mechanism can suppress Goldbach representations and why the number of representations necessarily increases. We argue that Goldbach’s conjecture is thereby reduced to a single, well-identified uniform realization problem within existing analytic methods.Rather than claiming a final proof, this article aims to clarify the conceptual structure underlying Goldbach’s conjecture, explain Goldbach’s comet as a necessary consequence of prime density dominance, and position the conjecture within a sharply defined analytic frontier. The result is a coherent, literature-grounded explanation of why Goldbach’s conjecture must be true and what precise technical step remains to complete its proof.

Abstract: An infinite-horizon H_∞ linear-quadratic control problem is considered. This problem has the following features: (i) the quadratic form of the control in the integrand of the cost functional is with a positive small multiplier (small parameter), meaning that the control cost is much smaller than the state cost; (ii) the current cost of the fast state variable in the cost functional is a positive semi-definite (but non-zero) quadratic form. These features require developing a significantly novel approach to asymptotic solution of the matrix Riccati algebraic equation associated with the considered H_∞ problem by solvability conditions. Using this solution, an asymptotic analysis of the H_∞ problem is carried out. This analysis yields parameter-free solvability conditions for this problem and a simplified controller solving this problem. An illustrative example is presented.

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This work develops an analytic framework for Goldbach’s strong conjecture based on symmetry, modular structure, and density constraints of odd integers around the midpoint of an even number. By organizing integers into equidistant pairs about , a tripartite structural law emerges in which every even integer admits representations as composite–composite, prime–composite, or prime–prime sums. This triadic balance acts as a stabilizing mechanism that prevents the systematic elimination of prime–prime representations as the even number grows. The analysis introduces overlapping density windows, DNA-inspired mirror symmetry of primes, and modular residue conservation to show that destructive configurations cannot persist indefinitely. As a result, the classical obstruction known as the covariance barrier is reduced to a narrowly defined analytic condition. The paper demonstrates that Goldbach’s conjecture is structurally enforced for all sufficiently large even integers and that the remaining difficulty is confined to a minimal analytic refinement rather than a combinatorial or probabilistic gap. This places the conjecture within reach of a complete unconditional resolution.

Abstract: The position paper serves scientists from all scientific disciplines to get a quick, concise, and easy-to-understand primer that describes the basic principles of design and applications of massively-parallel computations and models. The thesis of the position paper is, “What is the estimated direction of development of massively-parallel computing techniques utilizing emergents as observed in all scientific disciplines?” The birth of massively-parallel computations (MPCs) in the 1940s is closely related to the development of both early computers and simulations of nuclear processes that are operating within matter. It would be demonstrated that MPCs are becoming front and center in many research areas after a long delay that was forced by the previous inaccessibility of MPC computers. Another impetus to the development of MPCs came in the form of generalization of mathematical descriptions of observed natural phenomena using differential equations. More specifically, discretization schemes of differential equations got implemented in MPC simulations. Those achievements opened doors to the development of advanced descriptions of natural phenomena. Currently, there exist a number of MPC techniques that are gaining an increasing influence on the description of self-organization, emergence, replication, self-replication, and error-resilience within living and nonliving systems. A promising class of cellular automata, which is capable of describing a wide range of processes observed within natural phenomena, is called emergent information processing (EIP). The EIP approach is opening doorstowards future descriptions of many observed biological phenomena that are notoriously resisting mathematical and computational descriptions. Fixed and ad hoc networks, which are observed in living and non-living systems, could be interpreted using EIP methodology; this offers an opportunity for computational studies of emergent processes operating within them. Demonstrated approaches represent a toolkit that could be applied in all areas of research that are dealing with living systems. The usefulness of the EIP approach is demonstrated on a special case of emergent synchronization simulation, which could be applied in the design of artificial, biocompatible pacemaker implants and computers utilizing emergent computations.

Abstract: Finding all possibly efficient solutions of an interval multiple objective linear programming (IMOLP) problem with interval coefficients in the objective functions, the constraint matrix and the right-hand side vector is dealt with. Up to now, there are no known methods that can find all possibly efficient solutions of an IMOLP problem with interval coefficients in the objective functions and the right-hand side vector. In this paper, we propose a method to find all possibly efficient solutions of an IMOLP problem with interval coefficients in the objective functions and the right-hand side vector. Some sufficient conditions to obtain all possibly efficient solutions of an IMOLP problem in a general case are also given.

Abstract: Detecting anomalous events in high-dimensional behavioral data is a fundamental challenge in modern cybersecurity, particularly in scenarios involving stealthy advanced persistent threats (APTs). Traditional anomaly-detection techniques rely on heuristic notions of distance or density, yet rarely offer a mathematically coherent description of how sparse events can be formally separated from the dominant behavioral structure. This study introduces a density-metric manifold framework that unifies geometric, topological, and density-based perspectives into a single analytical model. Behavioral events are embedded in a five-dimensional manifold equipped with a Euclidean metric and a neighborhood-based density operator. Anomalies are formally defined as points whose local density falls below a fixed threshold, and we prove that such points occupy geometrically separable regions of the manifold. The theoretical foundations are supported by experiments conducted on openly available cybersecurity datasets, including ADFA-LD and UNSW-NB15, where we demonstrate that low-density behavioral patterns correspond to structurally rare attack configurations. The proposed framework provides a mathematically rigorous explanation for why APT-like behaviors naturally emerge as sparse, isolated regions in high-dimensional space. These results offer a principled basis for high-dimensional anomaly detection and open new directions for leveraging geometric learning in cybersecurity.

Abstract: This paper presents a framework for synthesizing bee bioacoustic signals associated with hive events. While existing approaches like WaveGAN have shown promise in audio generation, they often fail to preserve the subtle temporal and spectral features of bioacoustic signals critical for event-specific classification. The proposed method, MCWaveGAN, extends WaveGAN with a Markov Chain refinement stage, producing synthetic signals that more closely match the distribution of real bioacoustic data. Experimental results show that this method captures signal characteristics more effectively than WaveGAN alone. Furthermore, when integrated into a classifier, synthesized signals improved hive status prediction accuracy. These results highlight the potential of the proposed method to alleviate data scarcity in bioacoustics and support intelligent monitoring in smart beekeeping, with broader applicability to other ecological and agricultural domains.

Abstract: The solution of the incompressible Navier-Stokes equations with pressure Poisson equation has been an area of intensive research and extensive applications. Nevertheless, the proper formulation of pressure boundary conditions for solid walls has remained theoretically unestablished, since the existing specifications are unable to generate a truly solenoidal velocity field (i.e., a velocity field with zero divergence). Some prior researches have claimed divergence-free boundary conditions but lack rigorous validity. In this study, we clarify a longtime misconception and point out that the regular pressure boundary condition for solid walls will make the system ill-posed and non-divergence-free, because the system doesn’t need any pressure boundary condition in theory. However, solving the pressure Poisson equation still requires a boundary closure. To resolve this contradiction, we propose a new pressure Poisson equation and associated Neumann boundary condition based on the corrective pressure. They are derived in detail, examined for the pressure compatibility, analyzed for numerical stability with the normal mode method, and validated successfully with test problems. The new formulation produces strictly divergence-free velocity solutions with enhanced accuracy and stability due to its theoretical soundness.

Abstract: Nanoimprint lithography (NIL) master fidelity is governed by coupled variations beginning with resist spin-coating, proceeding through electron-beam exposure, and culminating in anisotropic etch transfer. We present an integrated, physics-based simulation chain. First, it includes a spin-coating thickness model that combines Emslie–Meyerhofer scaling with a Bornside edge correction. The simulated wafer-scale map at 4000 rpm exhibits the canonical center-rise and edge-bead profile with a 0.190–0.206 μm thickness range, while the locally selected 600 nm × 600 nm tile shows <0.1 nm variation, confirming an effectively uniform region for downstream analysis. Second, it couples an e-beam lithography (EBL) module in which column electrostatics and trajectory-derived spot size feed a hybrid Gaussian–Lorentzian proximity kernel; under typical operating conditions ( ≈ 2–5 nm), the model yields low CD bias (ΔCD = 2.38/2.73 nm), controlled LER (2.18/4.90 nm), and stable NMSE (1.02/1.05) for isolated versus dense patterns. Finally, the exposure result is passed to a level-set reactive ion etching (RIE) model with angular anisotropy and aspect-ratio-dependent etching (ARDE), which reproduces density-dependent CD shrinkage trends (4.42% versus 7.03%) consistent with transport-limited profiles in narrow features. Collectively, the simulation chain accounts for stage-to-stage propagation—from spin-coating thickness variation and EBL proximity to ARDE-driven etch behavior—while reporting OPC-aligned metrics such as NMSE, ΔCD, and LER. In practice, mask process correction (MPC) is necessary rather than optional: the simulator provides the predictive model, metrology supplies updates, and constrained optimization sets dose, focus, and etch set-points under CD/LER requirements.

Abstract: Due to their complexity and non-linearity, metaheuristic algorithms have become the gold standard in problem solving for those problems that cannot be solved by standard computational solutions. However, the global performance of these algorithms is strongly linked to the population structuring and the mechanism of replacing the worst solutions within the population. In this paper, an Adaptive Artificial Hummingbird Algorithm (AAHA), a new version of the basic AHA, is introduced and designed to enhance performance by studying the impacts of different population initialization methods within a broad and continual migration form. For the initialization phase, four methods: the Gaussian chaotic map, the Sinus chaotic map, the Opposite-based learning (OBL), and the Diagonal Uniform Distribution (DUD) are proposed as an alternative to the random population initialization method. A new strategy is proposed for replacing the worst solution in the migration phase. The new strategy used the best solution as an alternative to the worst solution with simple and effective local search. The proposed strategy stimulates exploitation and exploration when using the best solution and local search, respectively. The proposed AAHA algorithm is tested through various benchmark functions with different characteristics under many statistical indices and tests. Additionally, the AAHA results are benchmarked against other optimization algorithms to assess their effectiveness. The proposed AAHA algorithm outperformed alternatives in both speed and reliability. DUD-based initialization enabled the fastest convergence and optimal solutions. These findings underscore the significance of initialization in metaheuristics and highlight the efficacy of the AAHA algorithm for complex continuous optimization problems.

Abstract: Bounds for positive definite sets such as attractors of dynamic systems are typically characterized by Lyapunov-like functions. These Lyapunov functions and their time derivatives must satisfy certain definiteness conditions, whose verification usually requires considerable experience. If the system and a Lyapunov-like candidate function are polynomial, the definiteness conditions lead to Boolean combinations of polynomial equations and inequalities with quantifieres that can be formally solved using quantifier elimination. Unfortunately, the known algorithms for quantifier elimination require considerable computing power, meaning that many problems cannot be solved within a reasonable amount of time. In this context, it is particularly important to find a suitable mathematical formulation of the problem. This article develops a method that reduces the expected computational effort required for the necessary verification of definiteness conditions. The approach is illustrated using the example of the Chua system with cubic nonlinearity.

Abstract: This paper presents an analytic and computational framework to study noise in the distribution of prime numbers through layers based on multi-step prime gaps. Given the n-th prime pn, we consider the differences g(k,n) := p(n+k) − p(n), group these distances for different values of k and normalize them by an appropriate logarithmic scale. This produces, for each k, a discrete-time signal that can be analysed with tools from signal processing and control theory (Fourier analysis, resonators, RMS sweeps, and linear combinations). The main conceptual point is that all these layers should inherit the same underlying “noise” coming from the nontrivial zeros of the Riemann zeta function, up to deterministic filtering factors depending on k. We make this idea precise in a heuristic spectral inheritance hypothesis, motivated by the explicit formula and the change of variables t = log x. We then implement the construction computationally for hundreds of thousands of primes and several layers k, and we measure the response of bandpass resonators as a function of their central frequency. The numerical results show three phenomena: (i) the magnitude spectra of the layers share prominent peaks at the same frequencies, (ii) the RMS responses of resonators for different layers display aligned “hills” in the frequency domain, and (iii) a nearly perfect flattening is obtained by taking an appropriate linear combination of layers obtained via singular value decomposition. These findings support the picture that the same spectral content is present across layers and that the differences lie mainly in deterministic gain factors. The framework is deliberately modest from the point of view of number theory: we do not attempt to prove new theorems about primes, but rather to provide a coherent signal- processing view in which standard tools from electrical engineering can be used to explore the noisy side of the prime distribution in a structured way.

Abstract: Boundary value problems for systems of integrodifferential equations appear in all branches of science and engineering. Accuracy in modeling complex processes requires the specification of nonlocal boundary conditions, including multipoint and integral conditions. These kinds of problems are even harder to solve. In this paper, we present solvability criteria and a direct operator method for constructing the exact solution to systems of linear ordinary integrodifferential equations with general nonlocal boundary conditions. Several examples are solved to demonstrate the effectiveness of the method. The results equally apply to nonlocal boundary value problems for systems of ordinary differential, loaded differential, and loaded integrodifferential equations.

Abstract: This paper proposes a novel reversible arithmetic framework that addresses the irreversible information loss problem in traditional arith metic systems caused by specific operations such as multiplication by zero and division by zero. The system extends each numerical value into a mathematical object containing both the current value and com plete computational history, redefining the four fundamental arith metic operations to ensure traceability of computational processes. We establish the axiomatic foundation of this system, prove its key properties, and demonstrate its practical application value through er ror source tracing in differential equation solving, financial modeling, and machine learning. Theoretical analysis shows that while maintain ing computational reversibility, the system controls space complexity within acceptable limits through optimization techniques.

Abstract: The first (main) aim of the article is to define and practically apply the parameterized Kolmogorov-Smirnov goodness-of-fit test for normality. These modifications consist in varying a formula for calculating the empirical distribution function (EDF). The second contribution is to expand the EDF family with four new proposals. The third contribution is to create a family of alternative distributions, consisting of both older and newer distributions that, thanks to their flexibility, belong to all groups of skewness and kurtosis signs. Critical values are obtained using 106 order statistics for sample sizes n = 10, 20 and at a significance level a = 0.05. The fourth contribution is to calculate the power of the analyzed tests for alternative distributions based on 105 values of test statistics, with parameters selected so that they are similar to the normal distribution in various ways. The effectiveness of the analyzed tests is illustrated by analyzing real datasets.

Abstract: This paper presents a comprehensive stochastic framework for solving Fredholm integral equations of the second kind, combining biased and unbiased Monte Carlo approaches with modern variance-reduction techniques. Starting from the Liouville-Neumann series representation, several biased stochastic algorithms are formulated and analyzed, including the classical Markov Chain Monte Carlo (MCM), Crude Monte Carlo (CMCM), and quasi-Monte Carlo methods based on modified Sobol sequences (MCA-MSS-1, MCA-MSS-2, and MCA-MSS-2-S). Although these algorithms achieve substantial accuracy improvements through stratification and symmetrisation, they remain limited by the systematic truncation bias inherent in finite-series approximations. To overcome this limitation, two unbiased stochastic algorithms are developed and investigated: the classical Unbiased Stochastic Algorithm (USA) and a new enhanced version, the Novel Unbiased Stochastic Algorithm (NUSA). The NUSA method introduces adaptive absorption control and kernel-weight normalization, yielding significant variance reduction while preserving exact unbiasedness. Theoretical analysis and numerical experiments confirm that NUSA provides superior stability and accuracy, achieving uniform sub-10−3 relative errors in both one- and multi-dimensional problems, even for strongly coupled kernels with ∥K∥L2≈1. Comparative results show that the proposed NUSA algorithm combines the robustness of unbiased estimation with the efficiency of quasi-Monte Carlo variance reduction. It offers a scalable, general-purpose stochastic solver for high-dimensional Fredholm integral equations, making it well-suited for advanced applications in computational physics, engineering analysis, and stochastic modeling.

Abstract: We present a framework to implement Boolean logic using only smooth functions (sums, products, and low-cost nonlinearities) without hard thresholds or explicit comparators, while exactly preserving truth tables on the Boolean vertices {0,1}. The core is a family of polynomial gate primitives (AND, OR, XOR, NAND, etc.) that are exact at the vertices and are interleaved with a smooth booleanizer B_n(x) = xⁿ / (xⁿ + (1−x)ⁿ) that stabilizes trajectories toward the attractors 0 and 1 without discretization. We prove (i) exactness on the Boolean vertices; (ii) existence of attractors with local contraction for B_n; (iii) a block-contraction condition that yields stability in deep cascades; and (iv) a phase/harmonic variant via the encoding s = 2b − 1∈ {−1, +1} (equivalently s = cos(π(1 − b))). We synthesize a continuous half-adder and a continuous SR latch (smooth ODEs), and we outline an analog-hardware route using standard op-amp adders/scalers, analog multipliers, and exponential (log-antilog) cells. We discuss sensitivity, cost, speed, and robustness under variation, and we identify research directions linking continuous logic to neuromorphic and AI-centric analog-digital co-design.

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Reconstructing the continuous, three-dimensional (3D) distribution of intracranial electric potential from sparse, non-invasive scalp electroencephalography (EEG) is a central inverse problem in computational neuroimaging. This work introduces the Electro- Diffusion Physics-Informed Neural Network (ED-PINN), a coordinate-based neural representation that enforces the governing quasi-static Maxwellian electro-diffusion equation, ∇ · (σ(x)∇ϕ(x)) = −I(x), as a soft constraint during training. By parameterizing the potential field ϕ(x) as a continuous function, ED-PINN integrates sparse electrode measurements, Dirichlet/Neumann boundary conditions, and collocation-based PDE residuals into a single, unified objective function. This mesh-free approach enables the reconstruction of physically consistent, differentiable volumetric fields without the need for explicit domain meshing required by traditional methods like FEM or BEM. We demonstrate the efficacy of this approach on a canonical three-layer spherical head model with realistic tissue conductivities and synthetic Gaussian sources. We present a quantitative and qualitative evaluation, analyze primary error sources, and outline a clear roadmap for extensions to anatomically realistic geometries and anisotropic conductivity tensors derived from diffusion MRI. The experiments show that ED-PINN produces smooth, differentiable potential fields and localizes sources to sub-centimeter accuracy under the studied conditions. The paper includes detailed implementation notes, training recipes suitable for Colab/CPU environments, and a curated bibliography to ensure reproducibility.

Abstract: To improve the classification accuracy about K-means model for Coal-Gangue-Image, this paper proposed an Coal-Gangue-Image Classification Method (AF-WPOA-Kmeans-CCM) based on K-means and an improved wolf-pack-optimization using Adaptive Adjustment Factor.Firstly, this paper proposed an improved wolf-pack-optimization algorithm(AF-WPOA) using Adaptive Adjustment Factor Strategy(AF) aimed to dynamically adjust the update amount of the population for maintaining the genetic diversity of the population while accelerating convergence speed as well as ASGS devoted to enhance the algorithm's global exploration capability in hunting mechanism and local exploitation power in siege mechanism.Secondly,different weights for each feature dimension were adopted to more accurately reflect the varying importance of different feature dimensions in the classification results while Hilbert space was used to eliminate the drawback of low feature dimensions of the images from this paper resulted to achieving higher classification accuracy in high-dimensional space.Finally,AF-WPOA-Kmeans-CCM was formed by adopting the AF-WOPA to optimize K-means clustering model dedicated to accurate classification of Coal-Gangue-Image. Experimental results show that AF-WPOA outperforms GA, PSO and LWCA, while AF-WPOA-Kmeans-CCM performs better than Kmeans-CCM, GA-Kmeans-CCM, PSO-Kmeans-CCM, and LWCA-Kmeans-CCM—achieving 93.03% and 91.55% classification ac-curacy on the training and testing sets, respectively. Unfortunately, Coal-Gangue-Image often have diverse sources and significant feature differences. This makes it hard for fixed cluster centers to fully represent the numerous, multi-source, and highly variable unknown samples to be predicted.

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In this article the proof of the binary Goldbach conjecture via Chen’s weak conjecture are established (any integer greater than three is the sum and the difference of two positive primes). To this end, a "localised" algorithm is developed for the construction of two recurrent sequences of extreme Goldbach decomponents (U_2n and (V_2n), ((U_2n) dependent of (V_2n)) verifying: for any integer \( n \ge 2 \)(U_2n) and (V_2n) are positive primes and U_2n + V_2n = 2n. To form them, a third sequence of primes (W_2n) is defined for any integer \( n \ge 3 \) by W_2n = Sup \( (p \in P : p \le 2n - 3) \), \( P \)denoting the set of positive primes. The Goldbach conjecture has been proved for all even integers 2n between 4 and 4.10¹⁸ and in the neighbourhoodof 10¹⁰⁰,10²⁰⁰and 10³⁰⁰ for intervals of amplitude 10⁹. The table of extreme Goldbach decomponents, compiled using the programs in Appendix 15 and written with the Maxima and Maple scientific computing software, as well as files from ResearchGate, Internet Archive, and the OEIS, reaches values of the order of 2n = 10⁵⁰⁰⁰. Algorithms for locating Goldbach's decomponentss for very large values of 2n are also proposed. In addition, a global proof by strong recurrence "finite ascent and descent method" on all the Goldbach decomponents is provided by using sequences of primes (Wq_2n) defined by: Wq_2n = Sup \( (p \in P : p \le 2n - q) \) for any odd positive prime q, and a further proof by Euclidean divisions of 2n by its two assumed extreme Goldbach decomponents is announced by identifying uniqueness, coincidence and consistency of the two operations. Next, a majorization of U_2n by n^0.525, 0.7 ln^2.2(n) with probability one and 5 ln^1.3(n) on average for any integer n large enough is justified. Finally, the Lagrange-Lemoine-Levy (3L) conjecture and its generalization called "Bachet-Bézout-Goldbach"(BBG) conjecture are proven by the same type of method. In Aditional notes, we provide heuristic estimates for Goldbach's comet and presented a graphical synthesis using a reversible Goldbach tree (parallel algorithm).