Narrow Results

The study investigates a nonlinear model that describes the dynamics of populations of tumor cells, with a specific emphasis on the phases of cell proliferation and quiescence. The formulation of the model is determined by the age of the cells and time spend by them in each phase. The nonlinearity arises from the transitions between the proliferative and quiescent phases that are influenced by the age of cells and their total count. We thoroughly investigate the stability characteristic, both local and global, of trivial equilibrium points. To discretize the model, we utilize the discontinuous Galerkin scheme for spatial variables and the explicit Runge–Kutta of order four for time-dependent variables. The effectiveness of the discretized scheme is assessed using the convergence rate outcomes. The validity of our results is confirmed through numerical simulations, which also explore the effects of model paraters.

Domain adaptation is an important subfield of transfer learning, it has been successfully applied in many applications of machine learning. Recently, significant theoretical and algorithmic advances have been achieved in domain adaptation. The theoretical analyses for domain adaptation are based on VC dimension and Rademacher complexity. There are also some covering number-based results, but most of these bounds are based on the results of Rademacher complexity, indirectly given by the relationship between covering number and Rademacher complexity. In this paper, we propose a theoretical analysis framework for domain adaptation, thus the error bound can be derived directly by covering number, which is an effective method for analyzing the generalization error in statistical learning theory. We derive generalization error bound for domain adaptation with a class of loss functions satisfying the assumptions. We also propose a mixup contrastive adversarial network for domain adaptation by introducing a mixup module for enhancing the alignment of the source and target domains during domain transfer, and a contrastive learning module for solving class-level alignment after domain transfer. Experimental results demonstrate the effectiveness of the proposed algorithm and the property of the theoretical results.

This paper proposes a new approach to solving three-dimensional linear and nonlinear Volterra integral equations (3D-VIEs) of the second kind using the Taylor collocation method (TCM). We develop an algorithm utilizing Taylor polynomials to approximate solutions of 3D-VIEs within a finite-dimensional space. Moreover, we perform convergence analysis to validate the efficacy of our method. Additionally, we show comparison examples to demonstrate the method’s effectiveness. The comparison highlights the efficiency and reliability of our approach.

This paper presents a new, efficient, accurate, and unconditionally stable second-order time-stepping method for the incompressible thermal micropolar Navier–Stokes equations (TMNSE) using mixed finite elements. The method linearizes the nonlinear convective terms in the momentum equation, microrotation equation, and temperature equation, requiring the solution of a linear problem at each time step. The discrete curvature of the solution is added as a stabilizing term for linear velocity $u$, microrotation velocity $w$, pressure $p$, and temperature $T$ in the equations, respectively. Curvature stabilization ($u^{n+1}-2u^n+u^{n-1}$) is a new concept in computational fluid dynamics (CFD) aimed at improving the commonly used velocity stabilization ($u^{n+1}-u^n$), which only has first-order time accuracy and has adverse effects on important flow quantities such as drag coefficients. We derive a priori error estimates for the fully discrete linear extrapolation curvature stabilization method. The theoretical results and effectiveness of the new method are verified through a series of numerical experiments for $𝜃=\frac{1}{2}$, $\frac{3}{4}$, $\frac{5}{6}$, and $1$ in 2D and 3D, respectively. In particular, this work considers the thermal cavity-driven flow experiment to validate the numerical scheme and obtains good results.

The study has been carried out to analyze the Magnetohydrodynamic (MHD) boundary layer flow, heat and mass transfer of the two-dimensional viscoelastic Oldroyd-B fluid over a vertical stretching sheet in the presence of thermal radiation and chemical reaction with suction/injection in a steady state. The governing equations of the system are partial differential equations, which then give rise to a set of highly nonlinear coupled ordinary differential equations using similarity transformations. The nonlinearity in the differential equations is dealt with quasilinearization technique. The resultant equations are numerically solved using Chebyshev wavelet collocation method. The effects of magnetic field, radiation, chemical reaction and buoyancy parameters are investigated. The numerical values of local skin friction coefficient, local Nusselt number and local Sherwood number are also tabulated and analyzed. We observe that the increase in buoyancy parameters enhances the velocity profiles and larger values of magnetic field decrease the velocity profiles but increases the temperature and concentration profiles. Error analysis has been done to check the convergence of the numerical scheme.

This study introduces a novel fractional-order model to investigate the interplay between cancer and obesity and their treatment. Initially, we examine the solution’s existence and uniqueness for the proposed model. Additionally, we establish the boundedness of these solutions. Subsequently, we identify some potential equilibrium points of the cancer–obesity model and investigate their stability. To address the considered model, we propose fractional Euler’s and Adam’s methods. Theoretical and numerical analyses are conducted to assess the error estimates and performance of both methods with varying fractional-order derivatives. Moreover, we formulate an optimal control problem concerning cancer density and drug concentration. We delve into the existence of control and explore the first-order optimality conditions. We validate the analytical findings through numerical computations, demonstrating that administering drugs with control variables enhance immunity levels and reduce the burden of cancer.

Minimal deterministic finite automata (DFAs) can be reduced further at the expense of a finite number of errors. Recently, such minimization algorithms have been improved to run in time O(n log n), where n is the number of states of the input DFA, by [GAWRYCHOWSKI and JEŻ: Hyper-minimisation made efficient. Proc. MFCS, LNCS 5734, 2009] and [HOLZER and MALETTI: An n log n algorithm for hyper-minimizing a (minimized) deterministic automaton. Theor. Comput. Sci. 411, 2010]. Both algorithms return a DFA that is as small as possible, while only committing a finite number of errors. These algorithms are further improved to return a DFA that commits the least number of errors at the expense of an increased (quadratic) run-time. This solves an open problem of [BADR, GEFFERT, and SHIPMAN: Hyper-minimizing minimized deterministic finite state automata. RAIRO Theor. Inf. Appl. 43, 2009]. In addition, an experimental study on random automata is performed and the effects of the existing algorithms and the new algorithm are reported.

Two exponentially fitted two-derivative Runge–Kutta (EFTDRK) methods of algebraic order four are derived. The asymptotic expressions of the local errors for large energies are obtained. The numerical results in the integration of the radial Schrödinger equation with the Woods–Saxon potential show the high efficiency of our new methods compared to some famous optimized codes in the literature.

A family of exponentially fitted two-step hybrid methods for the numerical integration of the Schrödinger equation is investigated. The asymptotic expressions of the local errors for large energies are presented. The stability of the new methods is analyzed. Numerical results are reported for the widely used Woods–Saxon potential to show the efficiency of the new methods. The error analysis is clearly tested by the resonance problem.

In this paper, we analyze equal width wave equation using multi-quadric quasi-interpolation (MQQI) scheme. The traveling wave solutions called solitary waves: Single solitary wave motion, interaction of two and three solitary waves and evolution of solitons have been investigated computationally. Several comparisons are made to demonstrate the effectiveness of the proposed method. Error norms are used to observe the accuracy and efficiency of this scheme.

Modified three-derivative Runge–Kutta (MTHDRK) methods for the numerical solution of the resonant state for the Schrödinger equation are investigated. Order conditions are presented and oscillation-fitting conditions are derived. Two practical fifth-order explicit MTHDRK methods are constructed and the error analysis is carried out for large energy. The numerical results are presented for the numerical solution of the Schrödinger equation to show the robustness of our new methods.

The aim of this paper is to use the second derivative of Chebyshev polynomials (SDCHPs) as basis functions for solving linear and nonlinear boundary value problems (BVPs). Then, the operational matrix for the derivative was established by using SDCHPs. The established matrix via mixing between two spectral methods, collocation and Galerkin, has been applied to solve BVPs. Consequently, an error analysis is investigated to ensure the convergence of the technique used. Finally, we solved some problems involving real-life applications and compared their solutions with exact and other solutions from different methods to verify the accuracy and efficiency of this method.

For a periodic-review inventory system with stochastic and continuous demands and without setup costs, it is known that there exists a base-stock policy to be optimal. For a finite-horizon system with non-stationary parameters, it is essential to find approximate optimal policies because of the computational intractability of finding the optimal policy. This paper provides an approach, by which an analytical bound of the error between the costs of the approximate optimal policy and the optimal policy is given. It is shown that the approximate optimal policy converges to the optimal policy as the size of the computational grid decreases. Numerical results are presented to illustrate the methodology.

This work provides a technical applied description of the residual power series method (RPSM) to develop a fast and accurate algorithm for mixed hyperbolic–elliptic systems of conservation laws with Riemann initial datum. The RPSM does not require discretization, reduces the system to an explicit system of algebraic equations and consequently of massive and complex computations, and provides the solution in a form of Taylor power series expansion of closed-form exact solution (if exists). Theoretically, convergence hypotheses are discussed, and error bounds of exponential rates are derived. Numerically, the convergence and stability of approximate solutions are achieved for systems of mixed type. The reported results, with application to general Cauchy problems, which rise in diverse branches of physics and engineering, reveal the reliability, efficiency, and economical implementation of the proposed algorithm for handling nonlinear partial differential equations in applied mathematics.

The lattice Boltzmann method, which can simulate the macroscopic behavior of fluid flow, is a relatively new approach. The error analysis of the lattice Boltzmann BGK model is proposed based on extrapolation technique. According to the Filippova procedure, the error estimation formula is derived. The theory analysis shows that the error of lattice Boltzmann method is proportional to the lattice size and the relaxation parameter. The non-equilibrium part of the distribution function leads to the main error. Finally, the proposed error analysis method is validated by numerical simulation of the lid-driven flows in a square cavity. The numerical results agree well with the theory analysis results.

Paper chip is an efficient and inexpensive device for pesticide residues detection. However, the reasons of detection error are not clear, which is the main problem to hinder the development of pesticide residues detection. This paper focuses on error analysis for pesticide detection performed on paper-based microfluidic chip devices, which test every possible factor to build the mathematical models for detection error. In the result, double-channel structure is selected as the optimal chip structure to reduce detection error effectively. The wavelength of 599.753 nm is chosen since it is the most sensitive detection wavelength to the variation of pesticide concentration. At last, the mathematical models of detection error for detection temperature and prepared time are concluded. This research lays a theory foundation on accurate pesticide residues detection based on paper-based microfluidic chip devices.

A novel method is proposed to deal with the problem of building recognition in forward-looking infrared (FLIR) image sequences. Two computational models are deduced in this paper: one is the perspective transformation model, through which an image is transformed perspectively from downward-looking state to forward-looking state; the other is the indirect location model, through which the position of a building is computed in an FLIR image. In addition, in order to illustrate the application scope of our method, the error analysis is presented. The proposed approach is validated by extensive experiments, with images taken in different weather conditions, seasons and at different times. Superior recognition results were obtained on three FLIR image sequences we collected.

The laser ranging has extensive detection range and high measurement accuracy and so it is widely applied in navigation systems of the autonomous robot. But different environmental conditions have different influences on the transmission of the laser, producing errors of various degrees to measurement. The proposed range of influence in research on laser ranging LMS200 which is used in robot navigation under dissimilar conditions, by analyzing the primary factor that affects LMS200 ranging in a room and outside, contributing to the experimental contrast, obtaining experimental effects under environmentally influencing factors, discovering reflectivity, incidence, mixed pixel and visibility and so on that are environmental factors truly affecting practical application, builds the foundation for carrying out the ranging application of robot navigation. Mobile robots need enough sensor information of movements for carrying out autonomous navigation. The information establishment environment model in the completely unknown environment, depends upon itself to provide sensors to the robot that carries out autonomous localization and navigation. This paper also systematically introduces the applied research of laser ranging in mobile robot autonomous navigation, carrying out the scanning of the environment through the two-dimensional range finder sensor, proposed in autonomous navigation of map building and localization that uses two-dimensional scanning to obtain three-dimensional data stream, the experiment confirms that this algorithm allows the robot to obtain a very good three-dimensional visual structure drawing.

Image misalignment during image stitching is a common issue for the mechanically stitched CCDs in the aerial cameras due to the relative image motion between CCDs. In this paper, we analyze the error in imaging stitching for the mechanically stitched CCDs and propose a compensation method based on position and orientation system (POS). First, the imaging relationship of overlapping pixels in mechanical stitching is analyzed. The imaging model of adjacent CCD is constructed according to the collinear equation. The effects of stitching staggered distance, carrier attitude change and flight speed on the image mosaic of overlapping areas are given. Monte Carlo algorithm is used to analyze the statistical value of image mosaic error in overlapping area under typical working conditions. Then, a geometric correction method based on POS recording of external orientation elements of adjacent CCD imaging is proposed. The overlapping area stitching error is reduced from 14.9 pixels to 0.4 pixels which meets the engineering requirements. The mechanical stitching mathematical model, analysis and correction method established in this paper have strong engineering and application significance for mechanical stitching of aerial cameras.

This paper presents linear and bilinear shape transformations including basic transformations, analyzes their geometric properties, and provides computer algorithms. The shape transformations can be used to simplify the recognition of Roman letters, Chinese characters and other pictorial patterns by normalizing their shapes to the standard forms.

Important theoretical analyses have been performed to illustrate that the linear and bilinear transformations are applicable to computer recognition of digitized patterns. A number of pictorial examples have been computed to confirm the analyses and conclusions made.