Search Results (24)

Many applications in finance and biology involve multiplicative noise geometric Brownian motion (GBM)-type stochastic differential equations (SDEs) whose drift carries a single dominant periodic component. Such structures arise in seasonal Black–Scholes option pricing, commodity derivatives with annual price cycles, and stochastic biological oscillators driven by a known frequency; the primary contribution of this paper is a high-precision numerical scheme validated on GBM-type test problems with periodic drift. This paper proposes a Strang operator splitting scheme within the Logarithmic Drift-Diffusion Splitting (LDDS) framework, which splits the SDE in y-space into a deterministic drift ODE sub-step (Step A) and an exactly solvable multiplicative diffusion sub-step (Step B). Step A employs the Periodically Fitted Adams–Bashforth–Moulton fourth-order predictor–corrector method (PABM4), which achieves zero local truncation error for trigonometric forcing terms by introducing additional shift terms and simultaneously imposing polynomial exactness conditions and trigonometric fitting conditions. When the Step A forcing belongs to the PABM4 exact function class ${\mathcal{F}}_{\omega }=\mathrm{span}\{1,t,t^2,\sin \omega t,\cos \omega t\}$, the Strang+PABM4 scheme achieves floating-point precision saturation. We investigate three test problems: the cosine-drift GBM (Test Problem 1), the polynomial–trigonometric mixed drift GBM (Test Problem 2), and a dual-frequency drift applicability test (Test Problem 3). Monte Carlo strong error experiments ($M=1000$ paths) validate that Strang+PABM4 achieves saturation at machine precision (≈${10}^{-15}$) on Test Problems 1 and 2, improving precision by ≈${10}^2\times$ over the best algebraically convergent reference. Test Problem 3 identifies the method’s applicability boundary: when the drift contains a second frequency outside ${\mathcal{F}}_{\omega }$, Strang+PABM4 degrades gracefully to order ≈ 4 without catastrophic failure. The floating-point saturation of Strang+PABM4 is contingent on the drift belonging to ${\widehat{\mathcal{F}}}_{\omega }$; when this condition is violated, the method degrades gracefully to algebraic order ≈ 4, as demonstrated in Test Problem 3. Full article

In a previous paper the authors initiated a study of mutation semigroups, where elementary mutation operations were encoded as total maps on finite sets and analyzed through structural, algebraic, and computational methods. Here we address several of the open problems raised therein. First, we investigate the algebraic characterization of generator sets that force the existence of constant or low-rank maps, linking these conditions to classical results on synchronizing automata. Second, we analyze the computational complexity of contraction-based heuristics, identifying cases where polynomial-time criteria are achievable and others where hardness results emerge. Finally, we discuss connections with quasispecies models in biology and interpret image contractions as mechanisms of error suppression and genomic stability, while noting that rigorous extension to infinite state spaces remains future work. By combining algebraic definitions, structural theorems, and algorithmic analyses, we provide a refined toolkit for understanding mutation collapse and its theoretical implications, with potential applications that require empirical validation beyond the scope of this paper. Full article

Compact finite difference schemes approximate spatial derivatives through implicit relations between neighboring grid points. Despite using compact stencils and relatively simple algebraic structures, these schemes achieve high-order accuracy and spectral-like resolution, reducing dispersion errors while maintaining low numerical dissipation. These properties make them particularly attractive for problems requiring accurate spatial derivatives and computational efficiency, such as wave propagation, aeroacoustics, and turbulent flow simulations. This review presents the main ideas behind compact finite difference schemes, including their derivation from Taylor expansions and Padé approximations, their accuracy properties, and their resolution characteristics through modified wavenumber analysis. The manuscript is intended as a review and practical synthesis, rather than as the proposal of a new numerical scheme, and aims to connect the theoretical construction of compact schemes with their numerical behavior, practical implementation, and representative applications. To support reproducibility, we provide a fully documented open-source Python 3.11 notebook with a reference implementation of the schemes discussed in the paper. The examples include first- and second-order derivative calculations and representative one- and two-dimensional boundary-value problems, including Helmholtz-type equations. Finally, we survey applications across computational fluid dynamics, acoustics, geophysical flows, structural mechanics, biology, electromagnetism, and quantitative finance. Full article

While the quantum emergence of spacetime is becoming a major research topic in physics, the quantum emergence of intelligence has not been widely researched in quantum information science (QIS). Following causal-logical quantum gravity theory, bipolar entropy vs. entropy and negative entropy (or negentropy) are reviewed and distinguished for quantum emergence/submergence of quantum agent (QA) and quantum intelligence (QI) in algebraic terms. This work refers to QA as an entangled bipolar string/superstring in bipolar dynamic equilibrium (BDE) and QI being centered on logically definable causality in regularity, mind-light-matter unity, and brain-universe similarity. ER = EPR is extended to ER ≥≥ EPR for the mathematical scalability of bipolar strings and their collective entanglement. The extension leads to a number of conjectures, testable predictions, and theorems. The term “equilibraton” is proposed as a type of EPR or bipolar generic string to serve as an entropic stitch to collectively hold the universe together as a quantum entanglement in BDE with ubiquitous, regulated local emergence and submergence of QA&QI. Equilibraton leads to the concept of bipolar entropy square—a complete entropic solution to the background issue in quantum gravity. With complete background independence, energy/information conservational bipolar entropy, energy/information invariance, bipolar entropy non-additivity, and equilibrium-based plateau concavity are introduced. The nature of the one-dimensional arrow of time is conjectured. As a unification of order and disorder for equilibrium-based regulation, bipolar entropy bridges QA&QI to agentic AI, where quantum-bio-economics can be viewed as a topological intervention of a natural dynamic equilibrium in a social or natural world. Use cases are reviewed to illustrate the practical and theoretical aspects of bipolar entropy in business management, quantum-bio-economics, quantum cryptography, physics, and biology. Eddington–Einstein’s comments on entropy are revisited. It is expected that bipolar entropy will bring quantum emergence/submergence to agentic AI&QI for entangled machine thinking and imagination as a naturally scalable and testable foundation of real-world quantum gravity, quantum information science (QIS), quantum cognition and quantum biology (QCQB) to enhance Large Language AI Models (LLMs) and machine intelligence. Full article

Genetic sequences play a central role in biological and medical research, and mathematics provides powerful means for their representation and analysis. Conventional approaches, such as the fuzzy polynucleotide space ${[0, 1]}^{12}$, model codons as 12-dimensional vectors, but this comes at the cost of high dimensionality. In this study, we introduce two new models, Vector-Fuzzy-I and Vector-Fuzzy-II, that map codons and genetic sequences into the 4-dimensional Euclidean space ℝ⁴ using vector algebra and fuzzy set theory. In the first model, sequence structure is represented by successive vector addition, while in the second, it is represented by positional frequencies normalized by nucleotide locations. These low-dimensional representations are unique, preserve sequence order, and allow effective measurement of similarity and difference via Euclidean metrics. Compared with the fuzzy polynucleotide space, the proposed models achieve dimensionality reduction while enhancing the resolution of sequence differentiation. Our approach offers new mathematical perspectives for sequence analysis in theoretical biology. Full article

Persistent homology is a powerful tool in topological data analysis that captures the multi-scale topological features of data. In this work, we provide a mathematical introduction to persistent homology and demonstrate its application to protein–protein interaction networks. We combine persistent homology with algebraic connectivity, a graph-theoretic measure of network robustness, to analyze the topology and stability of PPI networks. An example is provided to illustrate the methodology and its potential applications in systems biology. Full article

The tumor microenvironment is a highly dynamic and complex system where cellular interactions evolve over time, influencing tumor growth, immune response, and treatment resistance. In this study, we develop a graph-theoretic framework to model the tumor microenvironment, where nodes represent different cell types, and edges denote their interactions. The temporal evolution of the tumor microenvironment is governed by fundamental biological processes, including proliferation, apoptosis, migration, and angiogenesis, which we model using differential equations with stochastic effects. Specifically, we describe tumor cell population dynamics using a logistic growth model incorporating both apoptosis and random fluctuations. Additionally, we construct a dynamic network to represent cellular interactions, allowing for an analysis of structural changes over time. Through numerical simulations, we investigate how key parameters such as proliferation rates, apoptosis thresholds, and stochastic fluctuations influence tumor progression and network topology. Our findings demonstrate that graph theory provides a powerful mathematical tool to analyze the spatiotemporal evolution of tumors, offering insights into potential therapeutic strategies. This approach has implications for optimizing cancer treatments by targeting critical network structures within the tumor microenvironment. Full article

Sinceits introduction in the 19th century to address geometric problems, topology as a methodology has undergone a series of evolutions, encompassing branches of geometric topology, point-set topology (analytic topology), algebraic topology, and differential topology, gradually permeating into various interdisciplinary applied fields. Starting from disciplines with typical geometric characteristics such as geography, physics, biology, and computer science, topology has found its way to economic fields in the 20th century. Given that the introduction of topology to economics is relatively new and presents features of being fragmented and non-systematic, this review aimed to provide scholars with a systematic evolution map to refine the characteristics of topology as a methodology applied in economics and finance, thereby aiding future potential interdisciplinary developments in these fields. By collecting abundant literature indexed in SCOPUS/WoS and other famous databases, with a qualitative analysis to classify and summarize it, we found that topological methods were introduced to modern economics when dealing with dynamic optimization, functional analysis, and convex programming problems, including famous applications such as uncovering equilibrium with fixed-point theorems in Walrasian economics. Topology can help uncover and refine the topological properties of these function space transformations, thus finding unchangeable features. Meanwhile, in contemporary economics, topology is being used for high-dimension reduction, complex network construction, and structural data mining, combined with techniques of machine learning, and applied to high-dimensional time series and structure analysis in financial markets. The most famous practical applications include the use of topological data analysis (TDA) and topological machine learning (TML) for different applied problems. Full article

Our paper reviews some key concepts in chemical reaction network theory and mathematical epidemiology, and examines their intersection, with three goals. The first is to make the case that mathematical epidemiology (ME), and also related sciences like population dynamics, virology, ecology, etc., could benefit by adopting the universal language of essentially non-negative kinetic systems as developed by chemical reaction network (CRN) researchers. In this direction, our investigation of the relations between CRN and ME lead us to propose for the first time a definition of ME models, stated in Open Problem 1. Our second goal is to inform researchers outside ME of the convenient next generation matrix (NGM) approach for studying the stability of boundary points, which do not seem sufficiently well known. Last but not least, we want to help students and researchers who know nothing about either ME or CRN to learn them quickly, by offering them a Mathematica package “bootcamp”, including illustrating notebooks (and certain sections below will contain associated suggested notebooks; however, readers with experience may safely skip the bootcamp). We hope that the files indicated in the titles of various sections will be helpful, though of course improvement is always possible, and we ask the help of the readers for that. Full article

Diffusion equations play a crucial role in various scientific and technological domains, including mathematical biology, physics, electrical engineering, and mathematics. This article presents a new formulation of the diffusion equation in the context of electrical engineering. Specifically, the behaviour of the physical quantity of charge carriers (such as concentration) is examined within semiconductor materials. The primary focus of this work is to solve the two-dimensional, time-fractional, nonlinear drift reaction–diffusion equation by applying an appropriate numerical scheme. In recent years, researchers working on nonlinear diffusion equations have proposed several numerical methods, with the shifted airfoil collocation method being one such efficient technique for solving nonlinear partial differential equations. This collocation approach effectively reduces the considered two-dimensional, time-fractional, nonlinear drift reaction–diffusion equation to a system of algebraic equations. The efficiency and effectiveness of the proposed method are validated through an error analysis, comparing the exact solution and the proposed numerical solution for a specific form of the considered mathematical model. The variations in the concentration of charge carriers, driven by the effects of drift and reaction terms, are displayed graphically as the system transitions from a fractional order to an integer order. Full article

In this article, we study the fractional form of a well-known dynamical system from mathematical biology, namely, the Lotka–Volterra model. This mathematical model describes the dynamics of a predator and prey, and we consider here the fractional form using the Rabotnov fractional-exponential (RFE) kernel. In this work, we derive an approximate formula of the fractional derivative of a power function ${\zeta }^p$ in terms of the RFE kernel. Next, by using the spectral collocation method (SCM) based on the shifted Vieta–Lucas polynomials (VLPs), the fractional differential system is reduced to a set of algebraic equations. We provide a theoretical convergence analysis for the numerical approach, and the accuracy is verified by evaluating the residual error function through some concrete examples. The results are then contrasted with those derived using the fourth-order Runge-Kutta (RK4) method. Full article

Revealing the time structure of physical or biological objects is usually performed thanks to the tools of signal processing such as the fast Fourier transform, Ramanujan sum signal processing, and many other techniques. For space-time topological objects in physics and biology, we propose a type of algebraic processing based on schemes in which the discrimination of singularities within objects is based on the space-time-spin group $SL(2,\mathbb{C})$. Such topological objects possess an homotopy structure encoded in their fundamental group, and the related $SL(2,\mathbb{C})$ multivariate polynomial character variety contains a plethora of singularities somehow analogous to the frequency spectrum in time structures. Our approach is applied to a model of quantum computing based on an Akbulut cork in exotic $R_4$, to an hyperbolic model of topological quantum computing based on magic states and to microRNAs in genetics. Such diverse topics reveal the manifold of possibilities of using the concept of a scheme spectrum. Full article

By nature, growth regulatory networks in biology are dynamic and stochastic, and feedback regulates their growth function at different ages. In this study, we carried out a stochastic modeling of growth networks and demonstrated this method using three mixed effect four-parameter Gompertz-type diffusion processes and a combination thereof using the conditional normal copula function. Using the conditional normal copula, newly derived univariate distributions can be combined into trivariate and bivariate distributions, and their corresponding conditional bivariate and univariate distributions. The link between the predictor variable and the remaining one or two explanatory variables can be formalized using copula-type densities and a numerical integration procedure. In this study, for parameter estimation, we used a semiparametric maximum pseudo-likelihood estimator procedure, which was characterized by a two-step technique, namely, separately estimating the parameters of the marginal distributions and the parameters of the copula. The results were illustrated using two observed longitudinal datasets, the first of which included the age, diameter, and potentially available area of 39,437 trees (48 stands), while the second included the age, diameter, potentially available area, and height of 8604 trees (47 stands) covering uneven mixed-species (pine, spruce, and birch) stands. All results were implemented using the MAPLE symbolic algebra system. Full article

The absence of a conventional association between the cell–cell cohabitation and its emergent dynamics into cliques during development has hindered our understanding of how cell populations proliferate, differentiate, and compete (i.e., the cell ecology). With the recent advancement of single-cell RNA sequencing (RNA-seq), we can potentially describe such a link by constructing network graphs that characterize the similarity of the gene expression profiles of the cell-specific transcriptional programs and analyze these graphs systematically using the summary statistics given by the algebraic topology. We propose single-cell topological simplicial analysis (scTSA). Applying this approach to the single-cell gene expression profiles from local networks of cells in different developmental stages with different outcomes reveals a previously unseen topology of cellular ecology. These networks contain an abundance of cliques of single-cell profiles bound into cavities that guide the emergence of more complicated habitation forms. We visualize these ecological patterns with topological simplicial architectures of these networks, compared with the null models. Benchmarked on the single-cell RNA-seq data of zebrafish embryogenesis spanning 38,731 cells, 25 cell types, and 12 time steps, our approach highlights gastrulation as the most critical stage, consistent with the consensus in developmental biology. As a nonlinear, model-independent, and unsupervised framework, our approach can also be applied to tracing multi-scale cell lineage, identifying critical stages, or creating pseudo-time series. Full article

Protein self-assembly is a common feature in biology and is often required for a myriad of fundamental processes, such as enzyme activity, signal transduction, and transport of solutes across membranes, among others. There are several techniques to find and assess homo-oligomer formation in proteins. Naturally, all these methods have their limitations, meaning that at least two or more different approaches are needed to characterize a case study. Herein, we present a new method to study protein associations using intrinsic fluorescence lifetime with phasors. In this case, the method is applied to determine the equilibrium dissociation constant (K_D) of human peroxiredoxin 1 (hPrx1), an efficient cysteine-dependent peroxidase, that has a quaternary structure comprised of five head-to-tail homodimers non-covalently arranged in a decamer. The hPrx1 oligomeric state not only affects its activity but also its association with other proteins. The excited state lifetime of hPrx1 has distinct values at high and low concentrations, suggesting the presence of two different species. Phasor analysis of hPrx1 emission lifetime allowed for the identification and quantification of hPrx1 decamers, dimers, and their mixture at diverse protein concentrations. Using phasor algebra, we calculated the fraction of hPrx1 decamers at different concentrations and obtained K_D (1.1 × 10⁻²⁴ M⁴) and C_0.5 (1.36 μM) values for the decamer–dimer equilibrium. The results were validated and compared with size exclusion chromatography. In addition, spectral phasors provided similar results despite the small differences in emission spectra as a function of hPrx1 concentration. The phasor approach was shown to be a highly sensitive and quantitative method to assess protein oligomerization and an attractive addition to the biophysicist’s toolkit. Full article