Formulas to calculate multivector exponentials in a basis-free representation and orthonormal basis are presented for an arbitrary Clifford geometric algebra , . The formulas are based on the analysis of roots of characteristic polynomial of a multivector. Elaborate examples how to use the formulas in practice are presented. The results are generalised to arbitrary functions of multivector and may be useful in the quantum circuits or in the problems of analysis of evolution of the entangled quantum states.
We show that the gradient of a strongly differentiable function at a point is the limit of a single coordinate-free Clifford quotient between a multi-difference pseudo-vector and a pseudo-scalar, or of a sum of Clifford quotients between scalars (as numerators) and vectors (as denominators), both evaluated at the vertices of a same non-degenerate simplex contracting to that point. Such result allows to fix a issue with a defective definition of pseudo-scalar field in Sobczyck’s Simplicial Calculus. Then, we provide some consequences and conjectures implied by the foregoing results.
We present a framework to model and provide numerical evidence for compartmentalization in the yeast endoplasmic reticulum. Measurement data is collected and an optimal control problem is formulated as a regularized inverse problem. To our knowledge, this is the first attempt in the literature to introduce a PDE-constrained optimization formulation to study the kinetics of fluorescently labeled molecules in budding yeast. Optimality conditions are derived and a gradient descent algorithm allows accurate estimation of unknown key parameters in different cellular compartments. For the first time, the numerical results support the barrier index theory suggesting the presence of a physical diffusion barrier that compartmentalizes the endoplasmic reticulum by limiting protein exchange between the mother and its growing bud. We report several numerical experiments on real data and geometry, with the aim of illustrating the accuracy and efficiency of the method. Furthermore, a relationship between the size ratio of mother and bud compartments and the barrier index ratio is provided.
This paper is concerned with a diffusive predator-prey model with prey-taxis and prey-structure under the homogeneous Neumann boundary condition. The stability of the unique positive constant equilibrium of the predator-prey model is derived. Hopf bifurcation and steady state bifurcation are also concluded.
The Boussinesq equations with partial or fractional dissipation not only naturally generalize the classical Boussinesq equations, but also are physically relevant and mathematically important. Unfortunately, it is not often well understood for many ranges of fractional powers. This paper focuses on a system of the 3D Boussinesq equations with fractional horizontal ( − ∆ h ) α u and ( − ∆ h ) β θ dissipation and proves that if an initial data ( u 0 , θ 0 ) in the Sobolev space H 3 ( R 3 ) close enough to the hydrostatic balance state, respectively, the equations with α , β ∈ ( 1 2 , 1 ] then always lead to a steady solution.
As the COVID-19 continues to mutate, the number of infected people is increasing dramatically, and the vaccine is not enough to fight the mutated strain. In this paper, a SEIR-type fractional model with reinfection and vaccine inefficacy is proposed, which can successfully capture the mutated COVID-19 pandemic. The existence, uniqueness, boundedness and nonnegativeness of the fractional model are derived. Based on the basic reproduction number R 0 , locally stability and globally stability are analyzed. The sensitivity analysis evaluate the influence of each parameter on the R 0 and rank key epidemiological parameters. Finally, the necessary conditions for implementing fractional optimal control are obtained by Pontryagin's Maximum Principle, and the corresponding optimal solutions are derived for mitigation COVID-19 transmission. The numerical results show that humans will coexist with COVID-19 for a long time under the current control strategy. Furthermore, it is particularly important to develop new vaccines with higher protection rates.
This paper is devoted to solve the backward problem for radially symmetric time-fractional diffusion-wave equation under Robin boundary condition. This problem is ill-posed and we apply an iterative regularization method to solve it. The error estimates are obtained under the a priori and a posteriori parameter choice rules. Numerical results show that the proposed method is efficient and stable.
We consider the coupled propagation of an optical field and its second harmonic in a quadratic nonlinear medium governed by a coupled system of Schrodinger equations. We prove the existence of ring-profiled optical vortex solitons appearing as solutions to a constrained minimization problem and as solutions to a min-max problem. In the case of the constrained minimization problem solutions are shown to be positive but the wave propagation constants undetermined, but in the min-max approach the wave propagation constants can be prescribed. The quadratic nonlinearity introduces some interesting properties not commonly observed in other coupled systems in the context of nonlinear optics, such as the system not accepting any semi-trivial solutions, meaning, that optical solitons cannot be observed when, say, one of the beams are off. Additionally, the second harmonic always remains positive.
This paper is concerned with the spreading or vanishing of an epidemic disease which is characterized by a nonlocal diffusion SIR model with nonlocal incidence rate and double free boundaries. We prove that the disease will vanish if the basic reproduction number R 0 < 1 , or the initial area h 0 , the initial datum S 0 , and the expanding ability µ are sufficiently small even that R 0 > 1 , and the disease will spread to the whole area if R 0 > 1 , when h 0 is suitably large or h 0 is small but µ is large enough. Moreover, we also show that the long-time asymptotic limit of the solution when vanishing happens.
Precision in measurement of glucose level in artificial pancreas is a challenging task and mandatory requirement for the proper functioning of artificial pancreas. A suitable machine learning technique for the measurement of glucose level in artificial pancreas may play crucial role in the management of diabetes. Therefore in the present work, a comparison has been made among few machine learning (ML) techniques for measurement of glucose levels in artificial pancreas because the machine learning is an astounding technology of artificial intelligence, and widely applicable in various fields such as medical science, robotics, environmental science, etc. The models namely decision tree (DT), random forest (RF), support vector machine (SVM), and K-nearest neighbours (KNN), based on supervised learning, are proposed for the dataset of Pima Indian to predict and classify the diabetes mellitus. Ensuring the predictions and accuracy up to the level of DMT2, the comparative behavior of all four models has been discussed. The machine learning models developed here stratifies and predicts whether an individual is diabetic or not based on the features available in the data set. Dataset passes through pre-processing and machine learning algorithms are fitted to train the dataset, and then the performance of the test results has been discussed. Error matrix (EM) has been generated to measure the accuracy score of the models. The accuracies in prediction and classification of DMT2 models are 71%, 77%, 78%, and 80% for DT, SVM, RF and KNN algorithms respectively. The KNN model has shown a more precise result in comparison to other models. The proposed methods have shown astounding behaviour in terms of accuracy in the prediction of diabetes mellitus as compared to previously developed methods.
In this work we propose the construction of discrete-time systems with two time scales in which infectious diseases dynamics are involved. We deal with two general situations. In the first, we consider that individuals affected by the disease move between generalized sites on a faster time scale than the dynamics of the disease itself. The second situation includes the dynamics of the disease acting faster together with another slower general process. Once the models have been built, conditions are established so that the analysis of the asymptotic behavior of their solutions can be carried out through reduced models. This is done using known reduction results for discrete-time systems with two time scales. These results are applied in the analysis of two new models. The first of them illustrates the first proposed situation, being the local dynamics of the SIS-type disease. Conditions are found for the eradication or global endemicity of the disease. In the second model, a case of co-infection with a primary disease and an opportunistic disease is treated, the latter acting faster than the former. Conditions for eradication and endemicity of co-infection are proposed.
In this paper, we consider switched systems that allow discontinuous jumps in the state when switching occurs. In particular we analyze the case where these jumps are dictated by a state reset rule. We show that it is possible to study the state trajectories of a switched system with state reset by means of some switched systems without state resets, i.e., with continuous state trajectories. We establish some sufficient conditions for stability of switched systems with state reset using the associated systems without reset. The obtained results are used in numerical examples to show the effectiveness of the proposed approach.
We consider the system of linear equations arisen from the finite element discretization of the distributed optimal control problem with time-periodic parabolic equations. A block alternating splitting iteration (BASI) method is presented for solving the obtained system. We prove that the BASI method is unconditionally convergent. We derive the BASI preconditioner and present an estimation formula for the parameter of the BASI preconditioner. Numerical results are presented to verify the efficiency of both the BASI method and the BASI preconditioner. Comparison with some existing methods are also given.
From decomposition method for operators, we consider Newton-like iterative processes for approximating solutions of nonlinear operators in Banach spaces. These iterative processes maintain the quadratic convergence of Newton's method. Since the operator decomposition method has its highest degree of application in non-differentiable situations, we construct Newton-type methods using symmetric divided differences, which allow us to improve the accessibility of the methods. Experimentally, by studying the basins of attraction of these methods, observe an improvement in the accessibility of derivative-free iterative processes that are normally used in these non-differentiable situations, such as the classic Steffensen's method. In addition, we study both the local and semi-local convergence of the Newton-type methods considered.
We propose a modification of a method based on Fourier analysis to obtain the Floquet characteristic exponents for periodic homogeneous linear systems, which shows a high precision. This modification uses a variational principle to find the correct Floquet exponents among the solutions of an algebraic equation. Once we have these Floquet exponents, we determine explicit approximated solutions. We test our results on systems for which exact solutions are known to verify the accuracy of our method including one dimensional periodic potentials of interest in quantum physics. Using the equivalent linear system, we also study approximate solutions for homogeneous linear equations with periodic coefficients.
This paper firstly finishes off the exact solutions of Riemann-Liouville fractional differential time-delay oscillatory system of order ρ∈(1 ,2) by using two newly defined delayed perturbations of Mittag-Leffler matrix functions and constant variation method. In the light of the exact solutions, we explore the finite time stability of the nonhomogeneous fractional oscillatory differential equations with pure delay. Ultimately, an example is cited to verify the rationality of the results. Through our method, the public problems left by Mahmudov in 2022 were partially solved.
In this paper, we consider the initial boundary value problem for a pseudo-parabolic Kirchhoff equation with logarithmic nonlinearity. Using the potential well method, we obtain a threshold result of global existence and finite-time blow-up for the weak solutions with initial energy J ( u 0 ) ≤ d . When the initial energy J ( u 0 ) > d , we find another criterion for the vanishing solution and blow-up solution. We also establish the decay rate of the global solution and estimate the life span of the blow-up solution. Meanwhile, we study the existence of the ground state solution to the corresponding stationary problem.