Abstract: We present a breakthrough computational methodology for investi- gating the Riemann Hypothesis, one of the most significant unsolved problems in mathematics. Our approach combines advanced num- ber theory with innovative computational techniques to analyze the distribution of zeros of the Riemann zeta function. We introduce a novel algorithm that identifies previously undetected patterns in zero distributions, providing substantial evidence supporting the Riemann Hypothesis. The computational framework presented allows for veri- fication of the hypothesis to unprecedented heights along the critical line. We demonstrate how our findings have direct applications to cryptography security and primality testing algorithms, potentially transforming computational number theory and its applications

Abstract: Modern computational models tend to become more and more complex, especially in fields like computational biology, physical modelling, social simulation and others. With the increasing complexity of simulations, modern computational architectures demand efficient parallel execution strategies. This paper proposes a novel approach leveraging the reactive streams paradigm as a general-purpose synchronization protocol for parallel simulation. We introduce a method to construct simulation graphs from predefined transition functions, ensuring modularity and reusability. Additionally, we outline strategies for graph optimization and interactive simulation through push and pull patterns. The resulting computational graph, implemented using reactive streams, offers a scalable framework for parallel computation. Through theoretical analysis and practical implementation, we demonstrate the feasibility of this approach, highlighting its advantages over traditional parallel simulation methods. Finally, we discuss future challenges, including automatic graph construction, fault tolerance, and optimization strategies, as key areas for further research.

Abstract: Fractional differential equations (FDEs) have attracted more and more attention in the last years: among them, equations of Caputo type allow for “more natural” initial conditions, when the order is greater than one. As a result, many numerical methods have been devised and investigated for approximating their solution: the Matlab© codes of some of them are also available. The aim of this paper is a systematic comparison of such codes on a selected set of test problems. The obtained results are available on the web.

Abstract: Sets play a foundational role in organizing, understanding, and interacting with the world in our daily lives. They also play a critical role in the functioning and behavior of social robots and artificial intelligence systems, which are designed to interact with humans and their environments in meaningful and socially intelligent ways. A multitude of non-classical set theories emerged during the last half-century aspiring to supplement Cantor’s set theory, allowing sets to be true to the reality of life by supporting for example human imprecision and uncertainty. The aim of this paper is to continue this effort introducing oSets which are sets depending on perception of their observers. In this context, an accessible set is a class of objects for which perception is passive, i.e., it is independent of perception; otherwise, it is said oSet, which cannot be known exactly with respect to its observers, but it can only be approximated by a family of sets representing the diversity of its perception. Thus, the new introduced membership function is a three-place predicate denoted ∈i, where the expression "x∈iX" indicates that "observer i perceives the element x as belonging to the set X". The accessibility notion is related to perception and can be best summarized as follows: "to be accessible is to be perceived" presenting a weaker stance than Berkeley’s idealism, which asserts that "to be is to be perceived".

Abstract: In this study, twelve modified differential evolution algorithms with memory properties and adaptive parameters were proposed to solve the optimization problem. In the experimental process, these modified differential evolution algorithms were applied to 23 continuous test functions. Experiments show that MBDE2 and IHDE-BPSO3 are superior to the original differential evolution algorithm and its extended variants, and the best solutions can be found in most of the problems. It is inducted that the proposed improved differential evolution algorithm can adapt to most problems and obtain better results, and adding the concept of memory property is a great improvement to the capability of the proposed improved differential evolution algorithm.

Abstract: The finite volume method approach in computational fluid dynamics to numerically model a fluid flow problem involves the process of formulating the numerical flux at the faces of the control volume. This process is important in deciding the resolution of the numerical solution, thus its quality. In the current work, the performance of different flux construction methods when solving the one-dimensional Euler equations for an inviscid flow is analyzed through a test problem in the literature having an exact (analytical) solution, which is the Sod problem. The considered flux methods are: exact Riemann solver (Godunov), Roe, Kurganov-Noelle-Petrova, Kurganov and Tadmor, Steger and Warming flux-vector splitting, van Leer flux-vector splitting, AUSM, AUSM+, AUSM+-up, AUFS, five variants of the Harten-Lax-van Leer (HLL) family, and corresponding five variants of the Harten-Lax-van Leer-Contact (HLLC) family, Lax-Friedrichs (Lax), and Rusanov. The methods of exact Riemann solver and van Leer showed excellent performance. The Riemann exact method took the longest runtime, but the spread of runtime among all methods was not large.

Abstract: Alzheimer’s Disease (AD) is a progressive neurodegenerative disorder that often begins decades before clinical symptoms manifest. Early detection remains critical for effective intervention, particularly in younger adults where biomarker deviations may signal pre-symptomatic risk. This research presents a computational modeling framework to predict cognitive impairment progression and stratify individuals into risk zones based on age-specific biomarker thresholds. The model integrates sigmoid-based data generation to simulate non-linear biomarker trajectories reflective of real-world disease progression. Core biomarkers—including CSF Aβ42, Amyloid PET, CSF Tau, and MRI FDG-PET—were analyzed simultaneously to compute a Cognitive Impairment (CI) score, dynamically adjusted for age. Higher CSF Aβ42 levels consistently demonstrated a protective effect, while elevated Amyloid PET and Tau levels increased cognitive risk. Age-specific CI thresholds prevented the overestimation of risk in younger individuals and the underestimation in older cohorts. The study highlights the model’s potential to identify individuals in risk zones, enabling targeted early interventions. Furthermore, the framework supports retrospective disease trajectory analysis, offering clinicians insights into optimal intervention windows even after symptom onset. Future work aims to validate the model using longitudinal real-world datasets and expand its predictive capacity through machine learning techniques and the integration of genetic and lifestyle factors. Ultimately, this research contributes to advancing precision medicine approaches in Alzheimer’s Disease by providing a scalable computational tool for early risk assessment and intervention planning.

Abstract:

In this paper, we demonstrate that the Maxwell eigenvalue problem can be solved by a nonconforming finite element and multigrid method. By using an appropriate operator, the eigenvalue problem can be viewed as a curl-curl problem. We obtain the approximate optimal error estimates on graded mesh. We also prove the convergence of the W-cycle and full multigrid algorithms for the corresponding discrete problem. The performance of these algorithms is illustrated by numerical experiments.

Abstract: It is known that the input-output model is based on the balance of inter-industry linkages, however, these conditions do not ensure equilibrium transactions among its agents, both producers and consumers. To address this issue, this work focuses on creating, justifying, and computational models for determine equilibrium transactions within the platform industry and ecosystems of Kazakhstan's financial and insurance activities industry. The research employs a conceptual design for computational models, integrating input (produce/selling, payments, imports) and output (purchase/buying, final demand, exports) transactions to analyze equilibrium growth in platform industries and ecosystems using OECD data from Kazakhstan's statistics (1995–2018). The main results of this study include computational models for determine equilibrium within the platform industry and its ecosystems of Kazakhstan's financial and insurance activities. Their application involves identifying time phases of undervalued and overvalued transactions, classifying time phases by their alignment with equilibrium transactions within the domain of feasible solutions, and analyzing the dynamics of equilibrium growth, enabling more efficient resource allocation and promoting sustainable growth for the industry and its ecosystem.

Abstract:

Wolfram’s Elementary Cellular Automata (ECA) serve as fundamental models for studying discrete dynamical systems, yet their classification remains challenging under traditional statistical and heuristic methods. By leveraging tools from algebraic topology, homotopy theory and differential geometry, we establish a formal connection between topological invariants and ECA’s structural properties and evolution. We analyse the role of Betti numbers, Euler characteristics, edge complexity and persistent homology in achieving robust separation of the four ECA classes. Additionally, we apply coarse proximity theory and assessed the applicability of Poincaré duality, Nash embedding and Seifert–van Kampen theorems to quantify large-scale connectivity patterns. We find that Class 1 automata exhibit simple, contractible topological spaces, indicating minimal structural complexity, while Class 2 automata exhibit periodic fluctuations in their topological features, reflecting their cyclic structure and repeating patterns. Class 3 automata exhibit a higher variance in their structural properties with persistent topological features forming and dissolving across scales, a signature of chaotic evolution. Class 4 automata exhibit statistically significant increases in higher-dimensional topological voids, suggesting the appearance of stable formations. Edge complexity and fractal dimension emergd as the strongest predictors of increasing computational and topological complexity, confirming that self-similarity and structural complexity play a crucial role in distinguishing cellular automata classes. Further, we address the critical distinction between Class 3 and Class 4 automata, which holds paramount importance in practical applications. Our approach establishes a mathematical framework for automaton classification by identifying emergent structures, with potential applications in computational physics, artificial intelligence and theoretical biology.

Abstract:

To understand phase transition processes like solidification, phase field models are frequently employed. These models couple the energy (heat) equation for temperature with a nonlinear parabolic partial differential equation (p.d.e.) that includes a second unknown, the phase, which takes characteristic values, such as zero in the solid phase and one in the liquid phase. We consider a simplified phase field model described by a system of parabolic p.d.e’s, q(ϕ)ϕt = ∇ · (A(ϕ)∇ϕ) + f (ϕ, u), ut = Δu + [p(ϕ)]t, where ϕ(x, y, t) represents the phase indicator function and u(x, y, t) denotes the temperature. The functions q, p, and f are given scalars, and A is a 2×2 diagonal matrix dependent on ϕ. This system is posed for t ≥ 0 on a rectangle in the x, y plane with appropriate boundary and initial conditions. We solve the system using a finite difference method that uses for both equations the Crank-Nicolson-ADI scheme. We prove a convergence result for the method and show results of numerical experiments verifying its order of accuracy. The isotropic system is numerically solved using Crank-Nicolson-ADI finite difference discretization for both equations. The initial-boundary-value problem is considered with homogeneous Dirichlet boundary conditions for ϕ and u. The paper presents preliminary results on finite difference approximations, establishes the main result, showing that finite difference approximations to u and ϕ converge in the discrete L2 and H1 norms with bounds of order Δt2 + h2, given a stability condition of Δt h ≤ σ. Finally, numerical experiments confirm the convergence orders.

Abstract: The identification of influential nodes in complex networks is fundamental for assessing their importance, particularly when simultaneously considering topological structure and nodal attributes. In this paper, we introduce SL-WLEN (Semi-local Centrality with Weighted and Lexicographic Extended Neighborhood), a novel centrality metric designed to identify the most influential nodes in complex networks. SL-WLEN integrates topological structure and nodal attributes by combining local components (degree and nodal values) with semi-local components (Local Relative Average Shortest Path LRASP and lexicographic ordering), thereby overcoming limitations of existing methods that treat these aspects independently. The incorporation of lexicographic ordering preserves the relative importance of nodes at each neighborhood level, ensuring that those with high values maintain their influence in the final metric without distortions from statistical aggregations. The metric was validated on a chip manufacturing quality control network comprising 1,555 nodes, where each node represents a critical process characteristic. The weighted connections between nodes reflect correlations among characteristics, enabling the evaluation of how changes propagate through the system and affect final product quality. Robustness testing demonstrates that SL-WLEN maintains high stability under various perturbations: preserving Top-1 rankings (98%) and correlations (R²>0.92) even with 50% link removal, while maintaining robustness above 80% under moderate network modifications. These findings evidence its effectiveness for complex network analysis in dynamic environments.

Abstract: We write large numbers by grouping them in exponentially increasing groups:1,2,4,8,.... , 1, 10, 100, 1000, ..., or any other arbitrary power base. Is there another way? Let's explore incremental, rather than exponential grouping, but pile these groups on top of one another. Group 1 holds one number 1, Group 2 holds the next two numbers: 2,3, group 3 holds the next three numbers: 4,5,6,.... etc. Every natural number will find its place in a group in which it will have its in-group count. E.g. number 5 is count 2 in group 3. Applying iteratively, we have before us a natural way to express integers. History of math has taught us that advanced representation leads to profound insight (e.g algebra v. arithmetic). It is worthy, therefore, to explore what this grouping, let's call it numerization can offer us.

Abstract: In this paper, we focus on develop high-order and structure-preserving numerical algorithm for the two-dimensional nonlinear space fractional Schr\"{o}dinger equations. By using the scalar auxiliary variable method with the composite Simpson's formula, and a fourth-order numerical differential formula for Riesz derivative, an effective difference scheme is constructed. Meanwhile, the conserving properties of the numerical solution is also studied.

Abstract: Many engineering problems are characterized by points of discontinuity in the first derivative of the phase variables. Such complicated discrete-continuous problems are called hybrid or event-continuous systems. The specification and computer analysis of the designated class of systems are illustrated by the common problem of a servo drive with the PWM controller. A mathematical model from the class of Cauchy problems with non-linear control logic is presented to resolve the task discussed. The designed model is performed in two stages: the mathematical model specification (modelling) and implementation (simulation). Two approaches to make a model specification are considered, the first one is the traditional structural technique, and the advanced method is based on the finite state machine paradigm. Structural and finite state software models are built in progressive simulation tools, such as Matlab, SimInTech, Ptolemy II and ISMA. Computer implementation of the software models of hybrid systems is performed using platform built-in libraries with numerical methods and algorithms for correct event detection. The results of computational experiments for transient processes of phase variables are qualitatively identical, that proves the adequacy of the models discussed. Quantitative errors, caused by using different integration techniques and event detection algorithms, are acceptable. The algorithm for asymptotic event detection, suggested by the Russian platform ISMA developers, is considered in detail. This technique demonstrates the best results by the one-sidedness criterion, which is explained by using the author’s algorithm for the integration step determining and the event function dynamics monitoring in the vicinity of the event limits.

Abstract: The Whale Optimization Algorithm (WOA) is a bio-inspired metaheuristic algorithm known for its simple structure and ease of implementation. However, WOA suffers from issues such as premature convergence, low population diversity in the later stages of iteration, slow convergence rate, low convergence accuracy, and an imbalance between exploration and exploitation. In this paper, we proposed an enhanced whale optimization algorithm with multi-strategy (LSEWOA). LSEWOA employs Good Nodes Set Initialization to generate uniformly distributed whale individuals, a newly designed Leader-Followers Search-for-Prey Strategy, a Spiral-based Encircling Prey strategy inspired by the concept of Spiral flight, and an Enhanced Spiral Updating Strategy. Additionally, we redesigned the update mechanism for parameter a to better balance exploration and exploitation. The effectiveness of the proposed LSEWOA was evaluated using CEC2005, and the impact of each improvement strategy was analyzed. We also performed a quantitative analysis of LSEWOA and compare it with other state-of-art metaheuristic algorithms. Finally, we applied LSEWOA to nine engineering design optimization problems to verify its capability in solving real-world optimization challenges. Experimental results demonstrated that LSEWOA outperformed better than other algorithms and successfully addressed the shortcomings of the classic WOA.

Abstract: In this paper, we present a new explicit formula for the sum of the chordal distance between the generalized singular values of Grassmann matrix pairs, based on Riemannian optimization models. The new formulas involve small-size unitary matrices and real orthogonal matrices derived from Riemannian optimization models. We then apply Newton’s method on Riemannian manifolds to efficiently solve the variable matrices involved. The new explicit formulas provide significant improvements over the existing theoretical and computational upper bounds.

Abstract: This article provides a survey about polyhedral embeddings of triangular regular maps of genus g, $2 \leq g \leq 14$, and of neighborly spatial polyhedra. An old conjecture of Gr\"unbaum from 1967, although disproved in 2000, lies behind this investigation. We discuss all duals of these polyhedra as well, whereby we accept, e.g., the Szilassi torus with its non-convex faces to be a dual of the M\"obius torus. A numerical optimization approach of the second author for finding such embeddings, was first applied to finding (unsuccessfully) a dual polyhedron of one of the 59 closed oriented surfaces with the complete graph of 12 vertices as their edge graph. The same method has been successfully applied for finding polyhedral embeddings of triangular regular maps of genus g, $2 \leq g \leq 14$. The effectiveness of the new method has led to ten additional new polyhedral embeddings of triangular regular maps and their duals. There does exist symmetrical polyhedral embeddings of all triangular regular maps with genus g, $2 \leq g \leq 14$, except a single undecided case of genus 13.

Abstract:

In this paper, we analyze the weather dynamics in Vancouver, Canada from 2009-2019 using quantum information theory. The novel approach taken in this work demonstrates that applying quantum information principles to classical problems, such as weather analysis, can yield new features and valuable insights that would otherwise be overlooked. Historical data was examined using entropy, and coherence measures, revealing connections between quantumlevel phenomena and macro-scale weather patterns. Key findings include the role of quantum coherence in weather shifts and evidence of quantum entanglement producing nonlinear weather dynamics. The results demonstrate the value of quantum information theory for enhancing weather forecasting and climate modeling.

Abstract: This study investigates the use of quantum computing, particularly Grover's algorithm, to improve genetic diagnostics for DiGeorge syndrome compared to conventional computational techniques. We employed the IBM Qiskit framework to simulate Grover's algorithm for the rapid and precise identification of pathogenic gene sequences. Background: Conventional genetic diagnostic methods are laborious, delaying essential treatment decisions. Quantum computing, capable of swiftly processing large datasets, offers substantial improvements in diagnostic speed and precision. Materials and Methods: We executed Grover's algorithm using Qiskit, evaluating its performance relative to classical algorithms based on diagnostic time and accuracy. We visualized results using R Studio with the ggplot2 and dplyr libraries. Results: The quantum methodology significantly reduced diagnostic duration from 300 seconds to 45 seconds and improved accuracy from 85% to 98%, surpassing traditional techniques. Conclusions: Our findings indicate that quantum computing can transform genetic diagnostics by enabling faster and more accurate identification of genetic disorders, thus promoting earlier and more personalized treatments. Future research should focus on improving the scalability of quantum computers and incorporating effective quantum algorithms into clinical workflows.