Search results for: Fokker-Plank stochastic differential equation

Authors: Manlika Rajchakit

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This paper is concerned with robust mean square stability of uncertain stochastic switched discrete time-delay systems. The system to be considered is subject to interval time-varying delays, which allows the delay to be a fast time-varying function and the lower bound is not restricted to zero. Based on the discrete Lyapunov functional, a switching rule for the robust mean square stability for the uncertain stochastic discrete time-delay system is designed via linear matrix inequalities. Finally, some examples are exploited to illustrate the effectiveness of the proposed schemes.

Keywords: robust mean square stability, discrete-time stochastic systems, hybrid systems, interval time-varying delays, lyapunov functional, linear matrix inequalities

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Authors: Joelle Fong

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Older persons are at greatest risk of becoming frail. As survival to the age of 80 and beyond continues to increase, the health and frailty of older Americans has garnered much recent attention among policy makers and healthcare administrators. This paper examines patterns in old-age frailty within a multistate actuarial model that characterizes the stochastic process of biological ageing. Using aggregate population-level U.S. mortality data, we implement a stochastic aging model to examine cohort trends and gender differences in frailty distributions for older Americans born 1865 – 1894. The stochastic ageing model, which draws from the fields of actuarial science and gerontology, is well-established in the literature. The implications for public health insurance programs are also discussed. Our results suggest that, on average, women tend to be frailer than men at older ages and reveal useful insights about the magnitude of the male-female differential at critical age points. Specifically, we note that the frailty statuses of males and females are actually quite comparable from ages 65 to 80. Beyond age 80, however, the frailty levels start to diverge considerably implying that women are moving quicker into worse states of health than men. Tracking average frailty by gender over 30 successive birth cohorts, we also find that frailty levels for both genders follow a distinct peak-and-trough pattern. For instance, frailty among 85-year old American survivors increased in years 1954-1963, decreased in years 1964-1971, and again started to increase in years 1972-1979. A number of factors may have accounted for these cohort differences including differences in cohort life histories, differences in disease prevalence, differences in lifestyle and behavior, differential access to medical advances, as well as changes in environmental risk factors over time. We conclude with a discussion on the implications of our findings on spending for long-term care programs within the broader health insurance system.

Keywords: actuarial modeling, cohort analysis, frail elderly, health

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Authors: Rahul Saini, Roshan Lal

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The present paper deals with the free axisymmetric vibrations of bi-directional functionally graded circular plates using Mindlin’s plate theory and physical neutral surface. The temperature-dependent, as well as temperature-independent mechanical properties of the plate material, varies in radial and transverse directions. Also, temperature profile for one- and two-dimensional temperature variations has been obtained from the heat conduction equation. A simple computational formulation for the governing differential equation of motion for such a plate model has been derived using Hamilton's principle for the clamped and simply supported plates at the periphery. Employing the generalized differential quadrature method, the corresponding frequency equations have been obtained and solved numerically to retain their lowest three roots as the natural frequencies for the first three modes. The effect of various other parameters such as temperature profile, functionally graded indices, and boundary conditions on the vibration characteristics has been presented. In order to validate the accuracy and efficiency of the method, the results have been compared with those available in the literature.

Keywords: bi-directionally FG, GDQM, Mindlin’s circular plate, neutral axis, vibrations

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Authors: Yeliz Mert Kantar, İsmail Yeni̇lmez, Ibrahim Arik

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In this study, the efficiency measurement of the top fifty Turkish Universities has been conducted. The top fifty Turkish Universities are listed by The Scientific and Technological Research Council of Turkey (TÜBITAK) according to the Entrepreneur and Innovative University Index every year. The index is calculated based on four components since 2018. Four components are scientific and technological research competency, intellectual property pool, cooperation and interaction, and economic and social contribution. The four components consist of twenty-three sub-components. The 2021 list announced in January 2022 is discussed in this study. Efficiency analysis have been carried out using the Stochastic Frontier Model. Statistical significance of the sub-components that make up the index with certain weights has been examined in terms of the efficiency measurement calculated through the Stochastic Frontier Model. The relationship between the efficiency ranking estimated based on the Stochastic Frontier Model and the Entrepreneur and Innovative University Index ranking is discussed in detail.

Keywords: efficiency, entrepreneur and innovative universities, turkish universities, stochastic frontier model, tübi̇tak

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Authors: Brando Vagenende, Marie-Anne Guerry

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For stochastic matrices of any order, the geometric description of the convex set of eigenvalues is completely known. The purpose of this study is to investigate the subset of the monotone matrices. This type of matrix appears in contexts such as intergenerational occupational mobility, equal-input modeling, and credit ratings-based systems. Monotone matrices are stochastic matrices in which each row stochastically dominates the previous row. The monotonicity property of a stochastic matrix can be expressed by a nonnegative lower-order matrix with the same eigenvalues as the original monotone matrix (except for the eigenvalue 1). Specifically, the aim of this research is to focus on the properties of eigenvalues of monotone matrices. For those matrices up to order 3, there already exists a complete description of the convex set of eigenvalues. For monotone matrices of order at least 4, this study gives, through simulations, more insight into the geometric description of their eigenvalues. Furthermore, this research treats in a geometric and algebraic way the properties of eigenvalues of monotone matrices of order at least 4.

Keywords: eigenvalues of matrices, finite Markov chains, monotone matrices, nonnegative matrices, stochastic matrices

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Authors: Zhi Zhang, Liling Cao, Seyedbabak Momenzadeh, Lisa Davey

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Reinforced concrete (RC) flat slab-column systems are commonly used in residential or office buildings, as the flat slab provides efficient clearance resulting in more stories at a given height than regular reinforced concrete beam-slab system. Punching shear of slab-column joints is a critical component of two-way reinforced concrete flat slab design. The unbalanced moment at the joint is transferred via slab moment and shear forces. ACI 318 provides an equation to evaluate the punching shear under the design load. It is important to note that the design code considers gravity and environmental load when considering the design load combinations, while it does not consider the effect from differential foundation settlement, which may be a governing load condition for the slab design. This paper describes how prestressed reinforced concrete slab punching shear is evaluated based on ACI 318 provisions and finite element analysis. A prestressed reinforced concrete slab under differential settlements is studied using the finite element modeling methodology. The punching shear check equation is explained. The methodology to extract data for punching shear check from the finite element model is described and correlated with the corresponding code provisions. The study indicates that the finite element analysis results should be carefully reviewed and processed in order to perform accurate punching shear evaluation. Conclusions are made based on the case studies to help engineers understand the punching shear behavior in prestressed and non-prestressed reinforced concrete slabs.

Keywords: differential settlement, finite element model, prestressed reinforced concrete slab, punching shear

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Authors: Y. Pala, M. O. Ertas

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In this paper, the general Riccati equation is analytically solved by a new transformation. By the method developed, looking at the transformed equation, whether or not an explicit solution can be obtained is readily determined. Since the present method does not require a proper solution for the general solution, it is especially suitable for equations whose proper solutions cannot be seen at first glance. Since the transformed second order linear equation obtained by the present transformation has the simplest form that it can have, it is immediately seen whether or not the original equation can be solved analytically. The present method is exemplified by several examples.

Keywords: Riccati equation, analytical solution, proper solution, nonlinear

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Authors: Krzysztof Pomorski

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In the given work we present universal mathematical framework for modeling of cattle farm system that can set and validate various hypothesis that can be tested against experimental data. The presented work is preliminary but it is expected to be valid tool for future deeper analysis that can result in new class of prediction methods allowing early detection of cow dieseaes as well as cow performance. Therefore the presented work shall have its meaning in agriculture models and in machine learning as well. It also opens the possibilities for incorporation of certain class of biological models necessary in modeling of cow behavior and farm performance that might include the impact of environment on the farm system. Particular attention is paid to the model of coupled oscillators that it the basic building hypothesis that can construct the model showing certain periodic or quasiperiodic behavior.

Keywords: coupled ordinary differential equations, cattle farm system, numerical methods, stochastic differential equations

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Authors: Sharon-Yasotha Veerayah-Mcgregor, Valipuram Manoranjan

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A backward or inverse problem is known to be an ill-posed problem due to its instability that easily emerges with any slight change within the conditions of the problem. Therefore, only a limited number of numerical approaches are available to solve a backward problem. This paper considers the Nagumo equation, an equation that describes impulse propagation in nerve axons, which also models population growth with the Allee effect. A creative operator splitting numerical scheme is constructed to solve the inverse Nagumo equation. Computational simulations are used to verify that this scheme is stable, accurate, and efficient.

Keywords: inverse/backward equation, operator-splitting, Nagumo equation, ill-posed, finite-difference

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Authors: D. Mehdizadeh, M. Rahimian, M. Eskandari-Ghadi

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This article is concerned with the determination of the static interaction of a vertically loaded rigid circular disc embedded at the interface of a horizontal layer sandwiched in between two different transversely isotropic half-spaces called as tri-material full-space. The axes of symmetry of different regions are assumed to be normal to the horizontal interfaces and parallel to the movement direction. With the use of a potential function method, and by implementing Hankel integral transforms in the radial direction, the government partial differential equation for the solely scalar potential function is transformed to an ordinary 4th order differential equation, and the mixed boundary conditions are transformed into a pair of integral equations called dual integral equations, which can be reduced to a Fredholm integral equation of the second kind, which is solved analytically. Then, the displacements and stresses are given in the form of improper line integrals, which is due to inverse Hankel integral transforms. It is shown that the present solutions are in exact agreement with the existing solutions for a homogeneous full-space with transversely isotropic material. To confirm the accuracy of the numerical evaluation of the integrals involved, the numerical results are compared with the solutions exists for the homogeneous full-space. Then, some different cases with different degrees of material anisotropy are compared to portray the effect of degree of anisotropy.

Keywords: transversely isotropic, rigid disc, elasticity, dual integral equations, tri-material full-space

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Authors: A. B. Disu, M. S. Dada

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This study was conducted to investigate two dimensional heat transfer of a free convective-radiative MHD (Magneto-hydrodynamics) flow with temperature dependent viscosity and heat source of a viscous incompressible fluid in a porous medium between two vertical wavy walls. The fluid viscosity is assumed to vary as an exponential function of temperature. The flow is assumed to consist of a mean part and a perturbed part. The perturbed quantities were expressed in terms of complex exponential series of plane wave equation. The resultant differential equations were solved by Differential Transform Method (DTM). The numerical computations were presented graphically to show the salient features of the fluid flow and heat transfer characteristics. The skin friction and Nusselt number were also analyzed for various governing parameters.

Keywords: differential transform method, MHD free convection, porous medium, two dimensional radiation, two wavy walls

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Authors: Olfa Kaabia, Ilyes Abid, Khaled Guesmi

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This pioneer paper studies whether and how Bitcoin shocks are transmitted to the U.S economy. We employ a new methodology: TVP FAVAR model with stochastic volatility. We use a large dataset of 111 major U.S variables from 1959:m1 to 2016:m12. The results show that Bitcoin shocks significantly impact the U.S. economy. This significant impact is pronounced in a volatile and increasing U.S economy. The Bitcoin has a positive relationship on the U.S real activity, and a negative one on U.S prices and interest rates. Effects on the Monetary Policy exist via the inter-est rates and the Money, Credit and Finance transmission channels.

Keywords: bitcoin, US economy, FAVAR models, stochastic volatility

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Authors: Ramazan Kadiev, Arkadi Ponossov

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Semidiscrete systems contain both continuous and discrete components. This means that the dynamics is mostly continuous, but at certain instants, it is exposed to abrupt influences. Such systems naturally appear in applications, for example, in biological and ecological models as well as in the control theory. Therefore, the study of semidiscrete systems has recently attracted the attention of many specialists. Stochastic effects are an important part of any realistic approach to modeling. For example, stochasticity arises in the population dynamics, demographic and ecological due to a change in time of factors external to the system affecting the survival of the population. In control theory, random coefficients can simulate inaccuracies in measurements. It will be shown in the presentation how to incorporate such effects into semidiscrete systems. Stability analysis is an essential part of modeling real-world problems. In the presentation, it will be explained how sufficient conditions for the moment stability of solutions in terms of the coefficients for linear semidiscrete stochastic equations can be derived using non-Lyapunov technique.

Keywords: abrupt changes, exponential stability, regularization, stochastic noises

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Authors: Hassen M. Ouakad

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In this paper, the influence of van der Waals, as well as electrostatic forces on the structural behavior of MEMS and NEMS actuators, has been investigated using of a Euler-Bernoulli beam continuous model. In the proposed nonlinear model, the electrostatic fringing-fields and the mid-plane stretching (geometric nonlinearity) effects have been considered. The nonlinear integro-differential equation governing the static structural behavior of the actuator has been derived. An original Galerkin-based reduced-order model has been developed to avoid problems arising from the nonlinearities in the differential equation. The obtained reduced-order model equations have been solved numerically using the Newton-Raphson method. The basic design parameters such as the pull-in parameters (voltage and deflection at pull-in), as well as the detachment length due to the van der Waals force of some investigated micro- and nano-actuators have been calculated. The obtained numerical results have been compared with some other existing methods (finite-elements method and finite-difference method) and the comparison showed good agreement among all assumed numerical techniques.

Keywords: MEMS, NEMS, fringing-fields, mid-plane stretching, Galerkin

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Authors: Seyed Amin Vakili, Sahar Sadat Vakili, Seyed Ehsan Vakili, Nader Abdoli Yazdi

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Differential equations are of fundamental importance in engineering and applied mathematics, since many physical laws and relations appear mathematically in the form of such equations. The equilibrium state of structures consisting of one-dimensional elements can be described by an ordinary differential equation. The response of these kinds of structures under the loading, namely relationship between the displacement field and loading field, can be predicted by the solution of these differential equations and on satisfying the given boundary conditions. When the effect of change of geometry under loading is taken into account in modeling of equilibrium state, then these differential equations are partially integrable in quartered. They also exhibit instability characteristics when the structures are loaded compressively. The purpose of this paper is to represent the ability of the Modified Newmark Method in analyzing flexural-torsional instability of struts for both bifurcation and non-bifurcation structural systems. The results are shown to be very accurate with only a small number of iterations. The method is easily programmed, and has the advantages of simplicity and speeds of convergence and easily is extended to treat material and geometric nonlinearity including no prismatic members and linear and nonlinear spring restraints that would be encountered in frames. In this paper, these abilities of the method will be extended to the system of linear differential equations that govern strut flexural torsional stability.

Keywords: instability, torsion, flexural, buckling, modified newmark method stability

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Authors: Deva Kanta Phukan

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An attempt is made where an electrically conducting fluid is injected from a fixed outer cylindrical casing onto an inner moving cylindrical rod. A magnetic field is applied parallel to the axis of the cylindrical rod. The basic governing set of partial differential equations for conservation of mass and momentum are reduced to a set of non-linear ordinary differential equation by introducing similarity transformation, which are integrated numerically. A perturbation solution for the case of large magnetic parameter is derived for constant Reynolds number.

Keywords: annular axi-symmetric stagnation flow, conducting fluid, magnetic field, moving cylinder

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Authors: Shangerganesh Lingeshwaran

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In this presentation, we have performed numerical simulations for a reaction-diffusion equation with various nonlinear density-dependent diffusion operators and proliferation functions. The mathematical model represented by parabolic partial differential equation is considered to study the invasion of gliomas (the most common type of brain tumors) and to describe the growth of cancer cells and response to their treatment. The unknown quantity of the given reaction-diffusion equation is the density of cancer cells and the mathematical model based on the proliferation and migration of glioma cells. A standard Galerkin finite element method is used to perform the numerical simulations of the given model. Finally, important observations on the each of nonlinear diffusion functions and proliferation functions are presented with the help of computational results.

Keywords: glioma invasion, nonlinear diffusion, reaction-diffusion, finite eleament method

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Authors: Maryam Khazaei Pool, Lori Lewis

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This is a summary of articles based on higher order B-splines methods and the variation of B-spline methods such as Quadratic B-spline Finite Elements Method, Exponential Cubic B-Spline Method, Septic B-spline Technique, Quintic B-spline Galerkin Method, and B-spline Galerkin Method based on the Quadratic B-spline Galerkin method (QBGM) and Cubic B-spline Galerkin method (CBGM). In this paper, we study the B-spline methods and variations of B-spline techniques to find a numerical solution to the Burgers’ equation. A set of fundamental definitions, including Burgers equation, spline functions, and B-spline functions, are provided. For each method, the main technique is discussed as well as the discretization and stability analysis. A summary of the numerical results is provided, and the efficiency of each method presented is discussed. A general conclusion is provided where we look at a comparison between the computational results of all the presented schemes. We describe the effectiveness and advantages of these methods.

Keywords: Burgers’ equation, Septic B-spline, modified cubic B-spline differential quadrature method, exponential cubic B-spline technique, B-spline Galerkin method, quintic B-spline Galerkin method

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Authors: Yutae Lee

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In this paper, we consider the performance and availability of a redundancy model. The redundancy model is a form of resilience that ensures service availability in the event of component failure. This paper considers a 2N redundancy model. In the model there are at most one active service unit and at most one standby service unit. The active one is providing the service while the standby is prepared to take over the active role when the active fails. We design our analysis model using Stochastic Reward Nets, and then evaluate the performance and availability of 2N redundancy model using Stochastic Petri Net Package (SPNP).

Keywords: availability, performance, stochastic reward net, 2N redundancy

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Authors: Homa Ghave, Parmis Shahmaleki

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This paper presents a new multi-criteria stochastic mathematical model for a single machine scheduling with outsourcing allowed. There are multiple jobs processing in batch. For each batch, all of job or a quantity of it can be outsourced. The jobs have stochastic processing time and lead time and deterministic due dates arrive randomly. Because of the stochastic inherent of processing time and lead time, we use the chance constrained programming for modeling the problem. First, the problem is formulated in form of stochastic programming and then prepared in a form of deterministic mixed integer linear programming. The objectives are considered in the model to minimize the maximum tardiness and outsourcing cost simultaneously. Several procedures have been developed to deal with the multi-criteria problem. In this paper, we utilize the concept of satisfaction functions to increases the manager’s preference. The proposed approach is tested on instances where the random variables are normally distributed.

Keywords: single machine scheduling, multi-criteria mathematical model, outsourcing strategy, uncertain lead times and processing times, chance constrained programming, satisfaction function

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Authors: Tadas Telksnys, Zenonas Navickas, Minvydas Ragulskis

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Necessary and sufficient conditions for the existence of second order solitary solutions to the Hodgkin-Huxley equation are derived in this paper. The generalized multiplicative operator of differentiation helps not only to construct closed-form solitary solutions but also automatically generates conditions of their existence in the space of the equation's parameters and initial conditions. It is demonstrated that bright, kink-type solitons and solitary solutions with singularities can exist in the Hodgkin-Huxley equation.

Keywords: Hodgkin-Huxley equation, solitary solution, existence condition, operator method

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Authors: Raphael Naryongo, Philip Ngare, Anthony Waititu

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This paper deals with numerical simulation of Wishart processes for a single asset risky pricing model whose volatility is described by Wishart affine diffusion processes. The multi-factor specification of volatility will make the model more flexible enough to fit the stock market data for short or long maturities for better returns. The Wishart process is a stochastic process which is a positive semi-definite matrix-valued generalization of the square root process. The aim of the study is to model the log asset stock returns under the double Wishart stochastic volatility model. The solution of the log-asset return dynamics for Bi-Wishart processes will be obtained through Euler-Maruyama discretization schemes. The numerical results on the asset returns are compared to the existing models returns such as Heston stochastic volatility model and double Heston stochastic volatility model

Keywords: euler schemes, log-asset return, infinitesimal generator, wishart diffusion affine processes

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Authors: A. I. Ma’ali, R. B. Adeniyi, A. Y. Badeggi, U. Mohammed

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An error estimation of the integrated formulation of the Lanczos tau method for some class of ordinary differential equations was reported. This paper is concern with the generalization of tau approximants and their corresponding error estimates for some class of ordinary differential equations (ODEs) characterized by m + s =3 (i.e for m =1, s=2; m=2, s=1; and m=3, s=0) where m and s are the order of differential equations and number of overdetermination, respectively. The general result obtained were validated with some numerical examples.

Keywords: approximant, error estimate, tau method, overdetermination

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Authors: Nicolò Vaiana, Giorgio Serino

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The nonlinear time history analysis of seismically base-isolated structures can require a significant computational effort when the behavior of each seismic isolator is predicted by adopting the widely used differential equation Bouc-Wen model. In this paper, a nonlinear exponential model, able to simulate the response of seismic isolation bearings within a relatively large displacements range, is described and adopted in order to reduce the numerical computations and speed up the nonlinear dynamic analysis. Compared to the Bouc-Wen model, the proposed one does not require the numerical solution of a nonlinear differential equation for each time step of the analysis. The seismic response of a 3d base-isolated structure with a lead rubber bearing system subjected to harmonic earthquake excitation is simulated by modeling each isolator using the proposed analytical model. The comparison of the numerical results and computational time with those obtained by modeling the lead rubber bearings using the Bouc-Wen model demonstrates the good accuracy of the proposed model and its capability to reduce significantly the computational effort of the analysis.

Keywords: base isolation, computational efficiency, nonlinear exponential model, nonlinear time history analysis

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Authors: Lina Wu, Ye Li

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An equivalent relation between a harmonic form and a closed co-closed form is established on a complete non-compact manifold. This equivalence has been generalized for a differential k-form ω from Lq spaces to non-Lq spaces when q=2 in the context of p-balanced growth where p=2. Especially for a simple differential k-form on a complete non-compact manifold, the equivalent relation has been verified with the extended scope of q for from finite q-energy in Lq spaces to infinite q-energy in non-Lq spaces when with 2-balanced growth. Generalized Hadamard Theorem, Cauchy-Schwarz Inequality, and Calculus skills including Integration by Parts as well as Convergent Series have been applied as estimation techniques to evaluate growth rates for a differential form. In particular, energy growth rates as indicated by an appropriate power range in a selected test function lead to a balance between a harmonic differential form and a closed co-closed differential form. Research ideas and computational methods in this paper could provide an innovative way in the study of broadening Lq spaces to non-Lq spaces with a wide variety of infinite energy growth for a differential form.

Keywords: closed forms, co-closed forms, harmonic forms, L^q spaces, p-balanced growth, simple differential k-forms

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Authors: Nara Guimarães, Marcelo Aquino Martorano, Douglas Gouvêa

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An investigation into Cahn-Hilliard equation was carried out through numerical simulation to identify a possible phase separation for one and two dimensional domains. It was observed that this equation can reproduce important mass fluxes necessary for phase separation within the miscibility gap and for coalescence of particles.

Keywords: Cahn-Hilliard equation, miscibility gap, phase separation, dimensional domains

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Authors: Ishit Sheth, Chandrasekhar Jinendran, Chinmaya Ranjan Sahu

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A fourteen degree of freedom (DOF) ride and handling control mathematical model is developed for a car using generalized boltzmann hamel equation which will create a basis for design of ride and handling controller. Mathematical model developed yield equations of motion for non-holonomic constrained systems in quasi-coordinates. The governing differential equation developed integrates ride and handling control of car. Model-based systems engineering approach is implemented for simulation using matlab/simulink, vehicle’s response in different DOF is examined and later validated using commercial software (ADAMS). This manuscript involves detailed derivation of full car vehicle model which provides response in longitudinal, lateral and yaw motion to demonstrate the advantages of the developed model over the existing dynamic model. The dynamic behaviour of the developed ride and handling model is simulated for different road conditions.

Keywords: Full Vehicle Model, MBSE, Non Holonomic Constraints, Boltzmann Hamel Equation

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Authors: Mustafa Bayram Gücen, Coşkun Yakar

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In this study, we have investigated the strict stability of fuzzy differential systems and we compare the classical notion of strict stability criteria of ordinary differential equations and the notion of strict stability of fuzzy differential systems. In addition that, we present definitions of stability and strict stability of fuzzy differential equations and also we have some theorems and comparison results. Strict Stability is a different stability definition and this stability type can give us an information about the rate of decay of the solutions. Lyapunov’s second method is a standard technique used in the study of the qualitative behavior of fuzzy differential systems along with a comparison result that allows the prediction of behavior of a fuzzy differential system when the behavior of the null solution of a fuzzy comparison system is known. This method is a usefull for investigating strict stability of fuzzy systems. First of all, we present definitions and necessary background material. Secondly, we discuss and compare the differences between the classical notion of stability and the recent notion of strict stability. And then, we have a comparison result in which the stability properties of the null solution of the comparison system imply the corresponding stability properties of the fuzzy differential system. Consequently, we give the strict stability results and a comparison theorem. We have used Lyapunov second method and we have proved a comparison result with scalar differential equations.

Keywords: fuzzy systems, fuzzy differential equations, fuzzy stability, strict stability

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Authors: Beata Jackowska-Zduniak

Abstract:

We formulate and analyze a mathematical model describing dynamics of the hypothalamus-pituitary-thyroid homoeostatic mechanism in endocrine system. We introduce to this system two types of couplings and delay. In our model, feedback controls the secretion of thyroid hormones and delay reflects time lags required for transportation of the hormones. The influence of delayed feedback on the stability behaviour of the system is discussed. Analytical results are illustrated by numerical examples of the model dynamics. This system of equations describes normal activity of the thyroid and also a couple of types of malfunctions (e.g. hyperthyroidism).

Keywords: mathematical modeling, ordinary differential equations, endocrine system, delay differential equation

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Authors: Fang Liu, JunShuai Xue, JiaJia Yao, XueYan Yang, ZuMao Li, GuanLin Wu, HePeng Zhang, ZhiPeng Sun

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III-Nitride resonant tunneling diode (RTD) is one of the most promising candidates for multiple-valued logic (MVL) elements. Here, we report a monolithic integration of GaN resonant tunneling diodes to realize multiple negative differential resistance (NDR) regions for MVL application. GaN RTDs, composed of a 2 nm quantum well embedded in two 1 nm quantum barriers, are grown by plasma-assisted molecular beam epitaxy on free-standing c-plane GaN substrates. Negative differential resistance characteristic with a peak current density of 178 kA/cm² in conjunction with a peak-to-valley current ratio (PVCR) of 2.07 is observed. Statistical properties exhibit high consistency showing a peak current density standard deviation of almost 1%, laying the foundation for the monolithic integration. After complete electrical isolation, two diodes of the designed same area are connected in series. By solving the Poisson equation and Schrodinger equation in one dimension, the energy band structure is calculated to explain the transport mechanism of the differential negative resistance phenomenon. Resonant tunneling events in a sequence of the series-connected RTD pair (SCRTD) form multiple NDR regions with nearly equal peak current, obtaining three stable operating states corresponding to ternary logic. A frequency multiplier circuit achieved using this integration is demonstrated, attesting to the robustness of this multiple peaks feature. This article presents a monolithic integration of SCRTD with multiple NDR regions driven by the resonant tunneling mechanism, which can be applied to a multiple-valued logic field, promising a fast operation speed and a great reduction of circuit complexity and demonstrating a new solution for nitride devices to break through the limitations of binary logic.

Keywords: GaN resonant tunneling diode, multiple-valued logic system, frequency multiplier, negative differential resistance, peak-to-valley current ratio

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