Nonlinear Lagrangians for Nonlinear Programming Based on Modified Fischer-Burmeister NCP Functions

This paper proposes nonlinear Lagrangians based on modified Fischer-Burmeister NCP functions for solving nonlinear programming problems with inequality constraints. The convergence theorem shows that the sequence of points generated by this nonlinear Lagrange algorithm is locally convergent when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions, and the error bound of solution, depending on the penalty parameter, is also established. It is shown that the condition number of the nonlinear Lagrangian Hessian at the optimal solution is proportional to the controlling penalty parameter. Moreover, the paper develops the dual algorithm associated with the proposed nonlinear Lagrangians. Numerical results reported suggest that the dual algorithm based on proposed nonlinear Lagrangians is effective for solving some nonlinear optimization problems.     

Weakly Nonlinear Time-Adiabatic Theory

We revisit the time-adiabatic theorem of quantum mechanics and show that it can be extended to weakly nonlinear situations, that is to nonlinear Schrödinger equations in which either the nonlinear coupling constant or, equivalently, the solution is asymptotically small. To this end, a notion of criticality is introduced at which the linear bound states stay adiabatically stable, but nonlinear effects start to show up at leading order in the form of a slowly varying nonlinear phase modulation. In addition, we prove that in the same regime a class of nonlinear bound states also stays adiabatically stable, at least in terms of spectral projections.     

Nonlinear diffusion layers

In the practice of block cipher design, there seems to have grown a consensus about the diffusion function that designers choose linear functions with large branch numbers to achieve provable bounds against differential and linear cryptanalysis. In this paper, we propose two types of nonlinear functions as alternative diffusing components. One is based on a nonlinear code with parameters (16,256,6) which is known as a Kerdock code. The other is a general construction of nonlinear functions based on the T-functions, in particular, two automatons with modular addition operations. We show that the nonlinear functions possess good diffusion properties; specifically, the nonlinear function based on a Kerdock code has a better branch number than any linear counterparts, while the automatons achieve the same branch number as a linear near-MDS matrix. The advantage of adopting nonlinear diffusion layers in block ciphers is that, those functions provide extra confusion effect while a comparable performance in the diffusion effect is maintained. As an illustration, we show the application of the nonlinear diffusion functions in two example ciphers, where a 4-round differential characteristic with the optimal number of active Sboxes has a probability significantly lower (\(2^{16}\) and \(2^{10}\) times, respectively) than that of a similar cipher with a linear diffusion layer. As a result, it sheds light upon an alternative strategy of designing lightweight building blocks.     

Nonlinear Feedback Shift Registers via Nonlinear Evolution

Methods for the forward and exact inverse computation of digital data are developed using reversible fully discrete nonlinear evolution equations. Some elementary examples are presented. A special case of the evolution equation presented here reduces to well‐known linear feedback shift registers (LFSRs). LFSRs and various nonlinear versions (NLFSRs) have been studied and used in a variety of applications.     

具有吸收和非线性边界流的非线性扩散方程的爆破估计

本文研究一类具有非线性吸收和非线性边界流的非线性扩散方程，建立了解的爆破速率估计，所得结果依赖于模型中三种非线性机制之间的相互作用。     

Nonlinear Degenerate Parabolic Equation with Nonlinear Boundary Condition

In this paper, we obtain the necessary and sufficient conditions on the global existence of all positive （weak） solutions to a nonlinear degenerate parabolic equation with nonlinear boundary condition.     

Nonlinear Diffusion Processes

We study elliptic systems of strongly nonlinear first-order differential equations in complex form on the plane. For such systems we develop the theory of Hilbert boundary value problems which is very much similar to the well-known theory for a holomorphic vector. Systems of nonlinear elliptic equations describe problems of interaction of several nonlinear stationary processes in the diffusive and convective mass and heat transport by hydrodynamic fluid flows.     

n阶非线性常微分方程的非线性两点边值问题

本文利用打靶法,给出了ｎ阶非线性常微分方程具有非线性两点边界条件的边值问题存在解与存在唯一解的一般性结果,并将所得结果应用于Ｌｉｐｓｃｈｉｔｚ方程的两点过值问题,给出了存在解与存在唯一解的具体的充分条件。     

n阶非线性微分方程的非线性三点边值问题

本文利用广义对接方法,建立了n阶非线性微分方程y(n)=f(x,y,y',…, y(n-1))满足非线性三点边界条件的非线性三点边值问题解的存在性与唯一性定理,所得结果将已有的很多相应结果作为特例.     

Observability of Nonlinear Systems

We consider the observability of systems of the form = Ax +Nx, y = Fx, where A is a linear operator and N and F are nonlinear.We show that if the system is linearized about an equilibriumpoint x_e and the linearized system is continuously initiallyobservable, then the nonlinear system is continuously initiallyobservable in some neighbourhood of x_e. We then look at conditionsunder which solutions of the nonlinear system can be extendedfor all time and consider the problem of stabilizing the systemby feedback controls such that the solutions are eventuallyin the observability neighbourhood of x_e. Finally, we applythese ideas to two systems: a wave equation and a diffusionequation with nonlinear perturbations and nonlinear observations.     

Nonlinear Separation Approach to Constrained Extremum Problems

In this paper, by virtue of a nonlinear scalarization function, two nonlinear weak separation functions, a nonlinear regular weak separation function, and a nonlinear strong separation function are first introduced, respectively. Then, by the image space analysis, a global saddle-point condition for a nonlinear function is investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Finally, some necessary and sufficient optimality conditions are obtained for constrained extremum problems.     

Doubly Nonlinear Degenerate Parabolic Systems with Coupled Nonlinear Boundary Conditions

In this paper, we study the global existence and the global nonexistence of doubly nonlinear degenerate parabolic systems with nonlinear boundary conditions. We first prove a local existence result by the regularization method. Next, we construct a weak comparison principle. Then we discuss the large time behavior of solutions by using a modified upper and lower solution methods and constructing various upper and lower solutions. Necessary and sufficient conditions on the global existence of all positive (weak) solutions are obtained.     

Homogeneous Nonlinear Chemical Reactions

The following contribution is designed to illuminate the interface between nonlinear differential equations and their applications to nonlinear chemical reactions. We shall see that most of the generic phenomena known in nonlinear dynamics have actually been observed in chemical experiments.     

Nonlinear Waves of Vorticity

This is a review of solutions of the vorticity equation for two-dimensional flow of an inviscid incompressible fluid that represent nonlinear waves. Geophysical applications are emphasized. Some of the solutions are valid in the beta-plane of Rossby. Some are related to weakly nonlinear perturbations of basic parallel flows and axisymmetric flows, to initial-value problems of hydrodynamic instability and to variational principles of minimal enstrophy or maximal entropy. Some have been found by exploiting well-known ideas of the theory of solitons. In addition to listing known solutions and presenting a synthesis of their relationship to other fluid dynamic results, we report a few new ideas and new solutions for strongly nonlinear waves.     

A New Jacobi Elliptic Function Expansion Method for Solvinga Nonlinear PDE Describing Pulse Narrowing Nonlinear Transmission Lines

In this article, we apply the first elliptic function equation to find a new kind of solutions of nonlinear partial differential equations (PDEs) based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method. New exact solutions to the Jacobi elliptic functions of a nonlinear PDE describing pulse narrowing nonlinear transmission lines are given with the aid of computer program, e.g. Maple or Mathematica. Based on Kirchhoff's current law and Kirchhoff's voltage law, the given nonlinear PDE has been derived and can be reduced to a nonlinear ordinary differential equation (ODE) using a simple transformation. The given method in this article is straightforward and concise, and can be applied to other nonlinear PDEs in mathematical physics. Further results may be obtained.     

Nonlinear separation approach to inverse variational inequalities

In this paper, we employ the image space analysis to investigate an inverse variational inequality (for short, IVI) with a cone constraint. By virtue of the nonlinear scalarization function commonly known as the Gerstewitz function, three nonlinear weak separation functions, two nonlinear regular weak separation functions and a nonlinear strong separation function are first introduced. Then, by these nonlinear separation functions, theorems of the weak and strong alternative and some optimality conditions for IVI with a cone constraint are derived without any convexity. In particular, a global saddle-point condition for a nonlinear function is investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Finally, two gap functions and an error bound for IVI with a cone constraint are obtained.     

Non-Exponential Stabilization of Nonlinear Discrete-Time Systems

In this paper, we investigate the stabilization problem for a class of nonlinear discrete-time control systems. We consider semilinear systems with a nonlinear perturbation satisfying a Hölder-type condition. Based on a new discrete inequality of Gronwall type, we establish sufficient conditions for the global and local weak stabilizability of semilinear and nonlinear discrete-time systems, respectively.     

带非线性边界条件的非线性抛物型方程组

本文讨论带非线性边界条件的抛物型方程组ｕｔ＝Δｕｍ，ｖｔ＝Δｖｍ，ｘ∈Ω，ｔ＞０，ｕｎ＝ｖｐ，ｖｎ＝ｕｑ，ｘ∈Ω，ｔ＞０，ｕ（ｘ，０）＝ｕ０（ｘ）δ＞０，ｖ（ｘ，０）＝ｖ０（ｘ）δ＞０，ｘ∈Ω（Ｉ）解的整体存在性和在有限时刻爆破问题．其中ｍ，ｐ，ｑ＞０，ΩＩＲＮ是有界光滑区域，δ＞０可以充分小．     

一类求解非线性等式和不等式约束优化问题的区间算法

在Moore二分法的基础上,通过构造的区间列L中标志矢量R的分量取值来删除部分不满足约束条件的区域,将非线性约束优化问题转化为初始域子域上的无约束优化问题,该算法可利用极大熵方法求解多目标优化问题,理论分析和数值结果均表明,这种算法是稳定且可靠的.