Manifold Theory Research Articles

Bifurcations in a General Delay Sel’kov–Schnakenberg Reaction–Diffusion System

The dynamics of a delay Sel’kov–Schnakenberg reaction–diffusion system are explored. The existence and the occurrence conditions of the Turing and the Hopf bifurcations of the system are found by taking the diffusion coefficient and the time delay as the bifurcation parameters. Based on that, the existence of codimension-2 bifurcations including Turing–Turing, Hopf–Hopf and Turing–Hopf bifurcations are given. Using the center manifold theory and the normal form method, the universal unfolding of the Turing–Hopf bifurcation at the positive constant steady-state is demonstrated. According to the universal unfolding, a Turing–Hopf bifurcation diagram is shown under a set of specific parameters. Furthermore, in different parameter regions, we find the existence of the spatially inhomogeneous steady-state, the spatially homogeneous and inhomogeneous periodic solutions. Discretization of time and space visualizes these spatio-temporal solutions. In particular, near the critical point of Hopf–Hopf bifurcation, the spatially homogeneous periodic and inhomogeneous quasi-periodic solutions are found numerically.

Mathematical analysis and simulation of Ebola virus disease spread incorporating mitigation measures

In order to understand the dynamics of the Ebola virus disease (EVD), this research developed a mathematical model that includes quarantine and public education campaigns as control measures. The model’s equilibrium points are displayed and the effective basic reproduction number Reff is estimated. Bifurcation theory is used for the stability analysis of endemic equilibrium state and general bifurcation theory is used to prove the existence of endemic equilibrium state. We evaluated the nature of the endemic equilibrium state of the model’s equation near the disease-free equilibrium, Reff=1 and introduce a bifurcation parameter. The results of the centre manifold theory are used to demonstrate that there is nontrivial endemic equilibrium near the disease-free equilibrium. We demonstrated the characteristics of the endemic equilibrium state that is close to the disease-free equilibrium as well as the fact that there is a potentially unstable endemic equilibrium condition. In other words, the sickness is slowly declining and will eventually disappear, as in the case of West Africa. Finally, we simulated the model developed to study the dynamics of the diseases with varying parameters using Homotopy Perturbation Method (HPM) to validate the qualitative analysis of the model. The result confirmed the hypothesis of our research that if quarantine and public enlightenment is properly used for the mitigation of the disease, the disease outcomes will drastically reduced. The results presented in this research will be useful for public health experts to contain the Ebola disease spread most especially in Africa.

Mathematical Analysis of COVID-19 model with Vaccination and Partial Immunity to Reinfection

COVID-19 is an infectious respiratory disease caused by a new virus, called SARS-CoV-2. Since itsinception, it has been a major cause of deaths and illnesses in the general population across the globe. Inthis paper, we have formulated and theoretically analyzed a non-linear deterministic model for COVID-19transmission dynamics by incorporating vaccination of the susceptible population. The system properties,such as the boundedness of solutions, the basic reproduction number R0, the local stability of disease-freeequilibrium(DFE), and endemic equilibrium (EE) points, are explored. Besides, the Lyapunov function isutilized to prove the global stability of both DFE and EE. The bifurcation analysis was carried out by utilizingthe center manifold theory. Then, the model is fitted with real COVID-19 cumulative data of infected casesin Kenya as from March 30, 2020, to March 30, 2022. Furthermore, sensitivity analysis was performed forthe proposed model to ascertain the relative significance of model parameters to COVID-19 transmissiondynamics. The simulations revealed that the spread of COVID-19 can be curtailed not only via vaccinationof susceptible populations but also increased administration of COVID-19 booster vaccine to the vaccinatedpersons and early detection and treatment of asymptomatic individuals.

Various topological complexities of small covers and real Bott manifolds

In this paper, we compute the LS-category and equivariant LS-category of a small cover and its real moment angle manifold. We calculate a tight lower bound for the topological complexity of many small covers over a product of simplices. Then we compute symmetric topological complexity of several small covers over a product of simplices. We calculate the LS one-category of real Bott manifolds and infinitely many small covers.

Towards a Materialist Reading of Thorstein Veblen's Notion of (Economic) Institution

Thorstein Veblen once considered his work on instincts to be his only important contribution to economic theory. Instincts are the conditions and causes behind the formation of habits of thought, while the latter are the sine qua non elements of institutions. The article poses the question: If Veblenʼs relation instincts-habits of thought-institutions were to be thought of as a formal system, what role would they conceptually occupy? It interprets habits of thought as pure ideas in a Platonist fashion (eidos)—multiplicities thought as Ideas—conceived by Georg Cantorʼs theory of manifolds and philosophically assessed by Alain Badiou positing that “mathematics is ontology.” The article aims: (1) to show how habits of thought, as institutions, abstracted from all content can be thought of as set-theoretic multiplicities; (2) to relate Veblenʼs prime instinct, the instinct of workmanship, in equivalence with set theory’s axiom of the empty set—pointing towards Veblenʼs materialist orientation.

A Novel, Efficient Approach to Simulate the 3D Resolved Spatial and Temporal Development of Space-Charge Layers in Solid Electrolytes

The role of space-charge layers (SCLs) on the overall battery performance of solid-state batteries is heavily discussed in the scientific community ranging from dominant to negligible [1]. Reasons for this are that direct measurements of SCLs in experiments are difficult and so far, simulation models that can efficiently resolve the transient development of the shape of SCLs while being thermodynamically consistent are missing. The different length scales of SCLs on the one hand and of the microstructure of a battery cell on the other hand make a spatial discretization of realistic microstructures with resolved SCLs infeasible. To date, this was circumvented by using phenomenological models that do not require a spatial discretization of the SCL, by applying thermodynamically consistent models to simplified one-dimensional geometries, or by performing simulations on the sub-continuum scale, e.g. DFT simulations or kinetic Monte Carlo simulations. As all models, the mentioned models have limitations. In this case, they are for example either thermodynamically not consistent or limited to small, not representative geometries.In this work [2], we propose a model that enables the prediction of the spatial and temporal development of SCLs within geometrically resolved microstructures while being thermodynamically consistent [3]. We exploit that effects in SCLs are predominantly one-dimensional. Furthermore, it can be shown, that SCLs develop orthogonal to the interface between the solid electrolyte and the electrode in the steady state. This allows to project the equations in the regions of the solid electrolyte where SCLs are expected in one dimension. In the remaining region of the solid electrolyte, the equations remain three-dimensional. A consistent coupling strategy between the three-dimensional and the one-dimensional region is derived and is irrevocably connected to the fulfilment of conservation properties. This modelling approach allows to drastically reduce the size of the numerical discretization required to resolve SCLs in space, as a fine discretization is only required in one direction embedded into a coarse three-dimensional discretization which represents the geometric microstructure. Thus, the required computational effort significantly reduces.The continuous equations are discretized in space using the finite element method. The obtained nonlinear system of equations is solved with the Newton-Raphson scheme using one monolithic system, i.e. the concurrent solution of the differently discretized regions to ensure robustness and efficiency of the solver. Tailoring an iterative solver by utilizing physical information of the system (similar to [4]), enables to apply the model to realistic microstructures with more than 10 million unknowns.With the proposed approach it is possible for the first time to quantify the influence of realistic microstructures on the transient and spatial formation process of SCLs. This work is the first to show that the formation process of SCLs at the interface between solid electrolyte and electrode is strongly influenced by the geometry of the microstructure (see Fig. 1): The dominating trend is a propagation of the SCL through the composite electrode, i.e. from the separator side to the current collector side of each electrode. Simultaneously, an inhomogeneity in the plane being orthogonal to the dominant direction is observed. This can be attributed to different conduction paths within this two-dimensional manifold due to the irregularity of the investigated microstructures and their geometric complexity. Furthermore, characteristic points in the microstructure can be identified where more charge is stored in the SCL during the transient development compared to the steady state. Again, this is attributed to the complex microstructures in interplay with the impedance of the bulk electrolyte and of the SCL. For these observations a modelling approach that resolves SCLs in realistic microstructures is mandatory.

Alternative technologies of composite of highly loaded aircraft structures: a qualitative method for making multicriteria decisions: Part III. Research of the methodological basis in decision-making: technological constructions in in the assessment toolki

The third part of the article proceeds from the starting points in the decision-making problem (DMP) specified in the first stages of research [1, 2]. Here, we continue to compare the predominance (first of all, quality) of autoclave and non-autoclave alternative technological processes (ATP) as part of the stages of TP(e) with a linear algorithm of end-to-end action in the manufacture of carbon fiber (CF) aircraft structures (AC) such as highly loaded wing stringer panels (HLS) of B787, A350, MC-21, CSeries mainline aircraft as a component of hierarchical systems. To describe and study them, the following were involved: initial technological and verbal models, technological and mathematical model of an autonomous dynamic system (ADS) and a number of two-dimensional manifolds (topology) to it in the form of technological and geometric models. The article continues the study of development in this direction by approximating manifolds by polyhedra. Including co-cellular and double-formed structures - pentagonal pyramids and bipyramids, as well as bodies of revolution around them - cone and bicon. Examples of schematic and technological interpretive modeling are presented. The methodological basis is the main provisions of decision-making theory, factor analysis and system-process approach with the involvement of practice results (expert analysis), first of all, the definition of technological factors of processes with criterion assessments of the advantages, components of alternatives in their competitive strategies.

Exploring the interplay between climate change and schistosomiasis transmission dynamics

Schistosomiasis, a neglected tropical disease caused by parasitic worms, poses a major public health challenge in economically disadvantaged regions, especially in Sub-Saharan Africa. Climate factors, such as temperature and rainfall patterns, play a crucial role in the transmission dynamics of the disease. This study presents a deterministic model that aims to evaluate the temporal and seasonal transmission dynamics of schistosomiasis by examining the influence of temperature and rainfall over time. Equilibrium states are examined to ascertain their existence and stability employing the center manifold theory, while the basic reproduction number is calculated using the next-generation technique. To validate the model's applicability, demographic and climatological data from Uganda, Kenya, and Tanzania, which are endemic East African countries situated in the tropical region, are utilized as a case study region. The findings of this study provide evidence that the transmission of schistosomiasis in human populations is significantly influenced by seasonal and monthly variations, with incidence rates varying across countries depending on the frequency of temperature and rainfall. Consequently, the region is marked by both schistosomiasis emergencies and re-emergences. Specifically, it is observed that monthly mean temperatures within the range of 22–27 °C create favorable conditions for the development of schistosomiasis and have a positive impact on the reproduction numbers. On the other hand, monthly maximum temperatures ranging from 27 to 33 °C have an adverse effect on transmission. Furthermore, through sensitivity analysis, it is projected that by the year 2050, factors such as the recruitment rate of snails, the presence of parasite egg-containing stools, and the rate of miracidia shedding per parasite egg will contribute significantly to the occurrence and control of schistosomiasis infections. This study highlights the significant influence of seasonal and monthly variations, driven by temperature and rainfall patterns, on the transmission dynamics of schistosomiasis. These findings underscore the importance of considering climate factors in the control and prevention strategies of schistosomiasis. Additionally, the projected impact of various factors on schistosomiasis infections by 2050 emphasizes the need for proactive measures to mitigate the disease's impact on vulnerable populations. Overall, this research provides valuable insights to anticipate future challenges and devise adaptive measures to address schistosomiasis transmission patterns.

Experimental and Theoretical Investigations of Shunt Currents between Alkaline Water Electrolyzers

A pivotal component of national climate strategies is the transition of the energy sector from fossil fuels to green energy. In this context, electrochemical processes will play a significant role in the future, requiring electrochemical systems with high efficiency and maximum system life cycles.1,2 For industrial electrochemical systems capital and operating costs are often reduced by a system design with a shared and circulating electrolyte supply providing ionically conductive cell-to-cell pathways. Under such conditions, parasitic ion migration occurs between adjacent cells, known as shunt currents.3,4 Shunt currents can have severe implications, such as decrease in faraday efficiency, material corrosion or interferences with instrumentation 3,5,6.To ensure high efficiency and maximum system life cycle, shunt current estimation is of high importance for the design of electrochemical multi-cell systems. A widely used approach in the literature is the theoretical, model-based approach using equivalent circuit models for shunt current determination7–9. Only a few publications demonstrate experimental determination methods 10,11. Furthermore, most shunt current studies in the field of electrochemistry focus on redox flow batteries.This work shows an innovative experimental approach for direct shunt current measurements between two alkaline water electrolysis cells with shared and circulating electrolyte feed under various conditions. The flow field design enables the insertion of reference electrodes into the flow cells for direct measurement of the potential differences between adjacent electrodes. Combined with the experimental data of the ionic tube resistances, the cell-to-cell shunt currents were accurately determined. Furthermore, an equivalent circuit model was created, fed with experimental data and validated with measured results. After successful validation the model was extended to electrolysis systems with more than two cells.Experimental data and simulations are in good agreement. The conducted experiments show the impact of temperature, cell voltage and tube manifold geometry on shunt current formation between alkaline water electrolyzers. Simulations performed are carried out to calculate shunt currents as a function of these parameters in large multi-cell systems. Furthermore, efficiency losses and corrosion processes as a result of shunt currents are estimated based on the results of this work.Literature Baños R, Manzano-Agugliaro F, Montoya FG, Gil C, Alcayde A, Gómez J. Optimization methods applied to renewable and sustainable energy: A review. Renew Sustain Energy Rev. 2011;15(4):1753-1766. doi:10.1016/j.rser.2010.12.008Yan Z, Hitt JL, Turner JA, Mallouk TE. Renewable electricity storage using electrolysis. Proc Natl Acad Sci U S A. 2020;117(23):12558-12563. doi:10.1073/pnas.1821686116Delgado NM, Monteiro R, Cruz J, Bentien A, Mendes A. Shunt currents in vanadium redox flow batteries – a parametric and optimization study. Electrochim Acta. 2022;403:139667. doi:10.1016/j.electacta.2021.139667Kaminski EA, Savinell RF. A Technique for Calculating Shunt Leakage and Cell Currents in Bipolar Stacks Having Divided or Undivided Cells. J Electrochem Soc. 1983;130(5):1103-1107. doi:10.1149/1.2119891Yin C, Guo S, Fang H, Liu J, Li Y, Tang H. Numerical and experimental studies of stack shunt current for vanadium redox flow battery. Appl Energy. 2015;151:237-248. doi:10.1016/j.apenergy.2015.04.080Pletcher D, Walsh FC. Industrial Electrochemistry. Springer Science & Business Media; 2012.Schaeffer JA, Chen L Der, Seaba JP. Shunt current calculation of fuel cell stack using Simulink®. J Power Sources. 2008;182(2):599-602. doi:10.1016/j.jpowsour.2008.04.014Wandschneider FT, Röhm S, Fischer P, Pinkwart K, Tübke J, Nirschl H. A multi-stack simulation of shunt currents in vanadium redox flow batteries. J Power Sources. 2014;261:64-74. doi:10.1016/j.jpowsour.2014.03.054Ye Q, Hu J, Cheng P, Ma Z. Design trade-offs among shunt current, pumping loss and compactness in the piping system of a multi-stack vanadium flow battery. J Power Sources. 2015;296:352-364. doi:10.1016/j.jpowsour.2015.06.138Rous̆ar I, Cezner V. Experimental Determination and Calculation of Parasitic Currents in Bipolar Electrolyzers with Application to Chlorate Electrolyzer. J Electrochem Soc. 1974;121(5):648. doi:10.1149/1.2401878Fink H, Remy M. Shunt currents in vanadium flow batteries: Measurement, modelling and implications for efficiency. J Power Sources. 2015;284:547-553. doi:10.1016/j.jpowsour.2015.03.057

Revealing hidden patterns in deep neural network feature space continuum via manifold learning

Deep neural networks (DNNs) extract thousands to millions of task-specific features during model training for inference and decision-making. While visualizing these features is critical for comprehending the learning process and improving the performance of the DNNs, existing visualization techniques work only for classification tasks. For regressions, the feature points lie on a high dimensional continuum having an inherently complex shape, making a meaningful visualization of the features intractable. Given that the majority of deep learning applications are regression-oriented, developing a conceptual framework and computational method to reliably visualize the regression features is of great significance. Here, we introduce a manifold discovery and analysis (MDA) method for DNN feature visualization, which involves learning the manifold topology associated with the output and target labels of a DNN. MDA leverages the acquired topological information to preserve the local geometry of the feature space manifold and provides insightful visualizations of the DNN features, highlighting the appropriateness, generalizability, and adversarial robustness of a DNN. The performance and advantages of the MDA approach compared to the existing methods are demonstrated in different deep learning applications.

MORSE AND CODIMENSION 1 FUNCTIONS ON THE PROJECTIVE PLANE

The topological properties of functions on two-dimensional spaces are described using an invariant called the Reeb graph. In the case of Morse functions on closed oriented two-dimensional manifolds, this serves as a complete topological invariant for homotopy equivalence. For non-oriented two-dimensional manifolds and manifolds with boundaries, additional information is required to construct a complete topological invariant. If a linear order is imposed on the set of vertices, it becomes a complete topological invariant with respect to topological equivalence. The research explores the topological classification of Morse functions and functions of corank 1 on the projective plane. In these cases, the Reeb graph is an oriented rooted tree. An analytical expression is derived for the number of oriented Reeb graphs with a root. For functions of corank 1 and a fixed number of critical points, all possible Reeb graphs are constructed.

L-INTEGRAL FOR ISOPERIMETRIC EXTREMALS OF ROTATION

In this paper, the geodetic flux on a spherical tangent bundle of a two-dimensional Riemannian manifold with the Sasaki metric is considered and it is shown that, if the basis manifold is locally isometric of the surface of rotation, then the Hamiltonian system corresponding to the flow is fully integrated according to Liouville. Hence, as a consequence, the flow trajectories are in quadratures.

Optimal control model for the infectiology of staphylococcus aureus with dual transmission pathways

Staphylococcus aureus has been increasing globally, which negatively affects food safety, human and veterinary medicine. It colonizes about 30% of the world population, causing superficial skin and soft tissue infections. The dynamics of direct (human-to-human) and indirect (environment-to-human) transmission, shedding, and treatment failure of the disease remains an open epidemiological challenge. Therefore, this study introduces a mathematical model that incorporates both direct and indirect transmission cycle, effect of shedding, treatment failure and time-dependent control parameters. We derive an expression for the basic reproduction number, R0s, under different scenarios. Using comparison, Lyapunov and center manifold theories, we establish that the non-optimal control model is globally asymptotically stable whenever R0s<1 with a forward bifurcation diagram. The shedding of pathogens into the environment and dual transmission routes are the key determinants in the model that increased the number of colonized individuals. Optimal control model was exploring to unveil the impact of time-dependent hygiene practice, treatment and hydrogen peroxide for good human health and clean environment. We show also the control efficacies of the proposed intervention strategies. The key finding is that having triple controls (as in strategy 7) is better than having two controls (as in strategies 4–6) or a single control (as strategies 1–3), but economically very expensive. In a situation where two controls are the only available option, then the use of either improved hygiene practice and treatment (strategy 4) or treatment and hydrogen peroxide (strategy 6) be implemented for controlling staph infection in humans and environment. Whereas, in a case only one control is affordable, then optimized treatment is better than either improved hygiene practice or hydrogen peroxide. We hope that the outcomes of this study have presented policy-makers and decision-makers of low-, middle-and high-income countries with appropriate alternative interventions to implement while dealing with staph infections.

Arnold diffusion in Hamiltonian systems on infinite lattices

AbstractWe consider a system of infinitely many penduli on an ‐dimensional lattice with a weak coupling. For any prescribed path in the lattice, for suitable couplings, we construct orbits for this Hamiltonian system of infinite degrees of freedom which transfer energy between nearby penduli along the path. We allow the weak coupling to be next‐to‐nearest neighbor or long range as long as it is strongly decaying. The transfer of energy is given by an Arnold diffusion mechanism which relies on the original V. I Arnold approach: to construct a sequence of hyperbolic invariant quasi‐periodic tori with transverse heteroclinic orbits. We implement this approach in an infinite dimensional setting, both in the space of bounded ‐sequences and in spaces of decaying ‐sequences. Key steps in the proof are an invariant manifold theory for hyperbolic tori and a Lambda Lemma for infinite dimensional coupled map lattices with decaying interaction.

The stochastic stability and H∞‐fuzzy control of stochastic bifurcation of a doubly‐fed induction generator

AbstractConsidering the dynamic behavior of doubly‐fed induction generators (DFIGs) under the influence of random factors, this paper not only analyzes the phenomenon of stochastic instability and bifurcation of a DFIG dynamic variable in its random space when they are affected by environmental noise, but also proposes a method based on the Tkagi–Sugneo (T–S) fuzzy control strategy to control its stochastic bifurcation. First, a four‐dimensional stochastic dynamic DFIG model is established by using multiplicative white noise to simulate the influence of environmental noise on electrical variables, and stochastic central manifold theory is used to reduce the dimensionality of a planar model in the bifurcation neighborhood. Then, the stochastic stability of the model is investigated based on singular boundary theory, while the steady‐state probability density of the stochastic DFIG is determined using the Fokker–Plank–Kolmogorov equation to obtain the location and probability density of its stochastic P‐bifurcation. Finally, the influence of stochastic bifurcation behavior is eliminated by H∞‐fuzzy output feedback control. The numerical simulation results indicate that the location and probability of stochastic bifurcation in a DFIG will vary with the change in noise intensity, and the bifurcation parameter values and stochastic stability domain are obtained. The harm caused by random factors can be solved based on H∞‐fuzzy output feedback, which provides a theoretical basis for the stable operation of the DFIG system.

Non Naturally Reductive Einstein Metrics on the Orthogonal Group via Real Flag Manifolds

We obtain new invariant Einstein metrics on the compact Lie groups SO(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{SO}\\,}}(n)$$\\end{document} which are not naturally reductive. This is achieved by using the real flag manifolds SO(k1+⋯+kp)/SO(k1)×⋯×SO(kp)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{SO}\\,}}(k_1+\\cdots +k_p)/{{\\,\ extrm{SO}\\,}}(k_1)\ imes \\cdots \ imes {{\\,\ extrm{SO}\\,}}(k_p)$$\\end{document} and by imposing certain symmetry assumptions in the set of all left-invariant metrics on SO(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{SO}\\,}}(n)$$\\end{document}.

Regioisomers of Aromatic Thioethers with Aggregation‐Induced Phosphorescence

AbstractIn this contribution, we describe a library of nine different molecular asterisks, all based on aromatic thioethers. Here, the substitution patterns of the central core unit bearing two nitriles and the rotors featuring methoxy groups were permuted. In addition, the photophysical properties in the solid and aggregated state were investigated. X‐ray diffractometry confirmed the molecular structures. In this frame, the molecular packing in the crystal lattice of the two polymorphs of mNoM was extensively studied and a correlation between photoluminescence and secondary interactions was evaluated. In the solid state, all compounds showed emission in the yellow to red part of the electromagnetic spectrum (555–644 nm) with lifetimes of up to 25 μs and photoluminescence quantum yields of up to 47 %, depending on the substitution pattern. As all the evaluated compounds perform as good emitters with aggregation‐induced phosphorescence (AIP), these compounds can be regarded as multivalent platforms for future applications in supramolecular assemblies or as biomedical labels due to the versatile substitution patterns, manifold geometries as well as appealing photophysical properties.

Bimeromorphic geometry of LCK manifolds

A locally conformally Kähler (LCK) manifold is a complex manifold M M which has a Kähler structure on its cover, such that the deck transform group acts on it by homotheties. Assume that the Kähler form is exact on the minimal Kähler cover of M M . We prove that any bimeromorphic map M ′ → M M’\rightarrow M is in fact holomorphic; in other words, M M has a unique minimal model. This can be applied to a wide class of LCK manifolds, such as the Hopf manifolds, their complex submanifolds and to OT manifolds.

Generalisation of affine Lie algebras on compact real manifolds

We report on recent work concerning a new type of generalised Kac-Moody algebras based on the spaces of differentiable mappings from compact manifolds or homogeneous spaces onto compact Lie groups.

Stability and optimal control of two products innovation diffusion system

The current study is aimed at analysing a four-compartmental system to examine the interaction and market dissemination of two product categories. In the first category, the product is low-priced and preferred by economically backward people. In the second category, the product is high-priced and liked by people in high-income groups. The non-user population is the potential buyer of products in the market, and frustrated people are fed up with both products. The nonnegativity and boundedness of the solution to the system help in ensuring the well-defined nature of the model. The viability of the model with UFE (user-free equilibrium), an equilibrium point in the absence of users of low-priced products, and interior equilibria is analysed. The analysis of the local and global stability of the model’s equilibria is conducted with reference to ℛ0. We examine the system’s transcritical bifurcation using the Centre Manifold Theory at ℛ0=1. The Hopf bifurcation’s occurrence condition has been identified. The frustration class population gets reduced using the optimal control theory. The extended optimum control model is used to build the Hamiltonian function, which is then solved using Pontryagin’s maximum principle to obtain the lowest cost. To realize which parameters impact the basic influence numbers most, we also run a sensitivity analysis on them. Finally, simulations are conducted to support the theoretical model.