Sublinear Systems Research Articles

Almost minimizers for a sublinear system with free boundary

We study vector-valued almost minimizers of the energy functional ∫D|∇u|2+21+qλ+(x)|u+|q+1+λ-(x)|u-|q+1dx,0<q<1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\int _D\\left( |\ abla \ extbf{u}|^2+\\frac{2}{1+q}\\left( \\lambda _+(x)|\ extbf{u}^+|^{q+1}+\\lambda _-(x)|\ extbf{u}^-|^{q+1}\\right) \\right) dx,\\quad 0<q<1. \\end{aligned}$$\\end{document}For Hölder continuous coefficients lambda _pm (x)>0, we take the epiperimetric inequality approach and prove the regularity for both almost minimizers and the set of “regular" free boundary points.

Log Canonical Thresholds on Burniat Surfaces with $K^2 = 6$ via Pluricanonical Divisors

Let $S$ be a Burniat surface with $K_S^2 = 6$ and $\varphi$ be the bicanonical map of $S$. In this paper we show optimal lower bounds of log canonical thresholds of members of pluricanonical sublinear systems of $S$ via Klein group $G$ induced by $\varphi$. Indeed, for a positive even integer $m$, the log canonical threshold of members of an invariant (resp. anti-invariant) part of $|mK_S|$ is greater than or equal to $1/(2m)$ (resp. $1/(2m-2)$). For a positive odd integer $m$, the log canonical threshold of members of an invariant (resp. anti-invariant) part of $|mK_S|$ is greater than or equal to $1/(2m-5)$ (resp. $1/(2m)$). The inequalities are all optimal.

On Enriques-Fano threefolds and a conjecture of Castelnuovo

Let Wsubset mathbb {P}^{13} be the image of the rational map defined by the linear system of the sextic surfaces of mathbb {P}^3 having double points along the edges of a tetrahedron. Let mathcal {L} be the linear system of the hyperplane sections of W. It is known that a general Sin mathcal {L} is an Enriques surface. The aim of this paper is to study the sublinear system mathcal {L}_{bullet }subset mathcal {L} of the hyperplane sections of W having a triple point at a general point w in W. We will show that a general element of mathcal {L}_{bullet } is birational to an elliptic ruled surface and that the image of W via the rational map defined by mathcal {L}_{bullet } is a cubic Del Pezzo surface Delta subset mathbb {P}^3 with 4 nodes. Interestingly, this fact appears to be related to a conjecture of Castelnuovo.

Rough layer scattering filled by elliptical cylinders from the method of moments combined with the characteristic basis function method and the Kirchoff approximation.

In this paper, the electromagnetic field scattered by several 2D scatterers of any shape is calculated rigorously from the boundary integral equations discretized by the method of moments with the point matching method and pulse basis functions. In addition, the resulting linear system is efficiently solved from the domain decomposition method named the characteristic basis function method. To accelerate the computation of the primary basis functions, which requires solving sublinear systems, the Kirchoff approximation is applied for metallic and dielectric objects. The efficiency of the method is tested on several applications met in practice: stack of rough interfaces separating homogeneous media, collection of metallic and dielectric elliptical cylinders, collection of coated elliptical cylinders, and a combination of the previous scenarios.

Homoclinic orbits of sub-linear Hamiltonian systems with perturbed terms

By using variational methods, we obtain the existence of homoclinic orbits for perturbed Hamiltonian systems with sub-linear terms. To the best of our knowledge, there is no published result focusing on the perturbed and sub-linear Hamiltonian systems.

Synthetic Adaptive Fuzzy Disturbance Observer and Sliding-Mode Control for Chaos-Based Secure Communication Systems

This paper aims to present secure communication based on master and slave Lorenz chaotic systems. To apply a form of the linear synchronization control method, the mathematical models of master and slave systems were reformed into Takagi-Sugeno (T-S) fuzzy systems. First, the Lorenz chaotic system was completely changed to the form of the T-S fuzzy model with two sublinear systems and two boundary fuzzy membership functions. Second, a newly adaptive disturbance observer (ADOB) was proposed with a high convergent speed for the synchronization system, which was based on the basic nonlinear disturbance observer. Third, adaptive sliding-mode control (ASMC) has been constructed to synchronize the master and slave systems of a secure communication system. The stability of the proposed control algorithms was shown by solving the Lyapunov condition with the support of Young's inequality. The synchronization of two nonidentical chaotic Lorenz systems was utilized to encrypt and decrypt the data. Transmitted and decrypted signals are used to show that the proposed algorithms are adequate for the secure communication system. To confirm the originality and power of the proposed algorithms, secure communication between two computers was implemented perfectly through an internet router and electronic circuit communication scenarios.

Multiple periodic solutions for one-sided sublinear systems: A refinement of the Poincaré-Birkhoff approach

In this paper we prove the existence of multiple periodic (harmonic and subharmonic) solutions for a class of planar Hamiltonian systems which includes the case of the second order scalar ODE $x'' + a(t)g(x) = 0$ with $g$ satisfying a one-sided condition of sublinear type. We consider the classical approach based on the Poincare-Birkhoff fixed point theorem as well as some refinements on the side of the theory of topological horseshoes. A Duffing-type equation and an exponential nonlinearity case are studied as applications.

Disturbance Observer-Based Linear Matrix Inequality for the Synchronization of Takagi-Sugeno Fuzzy Chaotic Systems

This article presents a synchronization control method based on poles’ placement, disturbances, and uncertainty estimation (DUE) for a pair of Takagi-Sugeno fuzzy systems. First, a 3-D chaotic system was completely converted into a Takagi-Sugeno (T-S) fuzzy model by applying the nonlinearity sector method, which consists of if-then rules and sub-linear systems. Second, two identical T-S fuzzy systems with different initial conditions were synchronized by applying the linear matrix inequality (LMI) to place the eigenvalues of the state error equations in the stable region. Third, the sum of the time-varying disturbances and uncertainties of two nonidentical T-S fuzzy systems were deleted by a disturbance and uncertainty estimation. The given output signals confirmed that the proposed method is suitable and ideal for synchronizing T-S fuzzy systems. The ideas of control theory were implemented by using two experimental scenarios in MATLAB Simulink for two computers connected via an internet router and an electronics circuit’s communication.

State augmented feedback controller design approach for T-S fuzzy system with complex actuator saturations

The control problem of T-S fuzzy system with actuator amplitude, rate and acceleration saturations is addressed in this paper, where state augmented feedback controller with LMIs (Liner Matrix Inequalities) constraint conditions are proposed. Dynamic decoupling method is applied to fuzzify the input magnitude saturation nonlinearity into several sub-linear systems with fuzzy rules, thus a new T-S fuzzy system with input rate and acceleration saturations can be obtained. Then PDC (parallel distributed compensation) and NPDC (non-PDC) controller are both designed for the new T-S fuzzy system. The first and second order derivatives of input variable are given to obtain an augmented fuzzy system. As the augmented system output variable, the input rate is represented by first order derivative of the input term, the rate saturation constraint is described through norm bounded method. Moreover, the polytopic approach is used to replace the second order derivative of the input term, and a state augmented feedback NPDC controller is designed, the domain of attraction optimization process is also given. Finally, two practical examples are presented to show the effectiveness of proposed method.

Existence of infinitely many solutions of Dirac equations with sublinear nonlinearity

In this note we use an recent improvement of Clark’s theorem by Liu and Wang (Nonlinear Anal 32:1015–1037, 2015) to prove two existence results of infinitely many solutions for sublinear Dirac equations and systems on compact spin manifolds.

Existence of periodic solutions of sublinear Hamiltonian systems

The variational method is used to obtain some existence theorems of periodic solutions of sublinear systems with or not with impacts under suitable growth conditions. Compared with normal systems, impact systems need additional conditions to ensure the existence of periodic bouncing solutions.

Positive solutions for a class of sublinear elliptic systems

In this paper, we are concerned with the existence of positive solutions of the semilinear elliptic system

Existence, uniqueness and stability of positive solutions to a general sublinear elliptic systems

In this paper, we make use of a new stability result and bifurcation theory to study the existence and uniqueness of positive solutions to semilinear elliptic systems with some general sublinear conditions. Moreover, we obtain the precise global bifurcation diagrams of the system in a single monotone solution curve. MSC:35J55, 35B32.

Coherent Price Systems and Uncertainty-Neutral Valuation

We consider fundamental questions of arbitrage pricing arising when the uncertainty model incorporates volatility uncertainty. The resulting ambiguity motivates a new principle of preference-free valuation. By establishing a microeconomic foundation of sublinear price systems, the principle of ambiguity-neutral valuation imposes the novel concept of equivalent symmetric martingale measures. Such measures exist when the asset price with uncertain volatility is driven by Peng's G-Brownian motion.

New examples of curves with a one-dimensional family of pencils of minimal degree

We give a geometric construction of sub-linear systems on a K3 surface consisting of smooth curves C with infinitely many $${g^1_{{\rm gon}(C)}}$$ ’s.

Existence of solutions for a class of second-order sublinear and linear Hamiltonian systems with impulsive effects

Existence of solutions for a class of second-order sublinear and linear Hamiltonian systems with impulsive effects

Periodic and homoclinic solutions generated by impulses for asymptotically linear and sublinear Hamiltonian system

In this paper, we get the existence of periodic and homoclinic solutions for a class of asymptotically linear or sublinear Hamiltonian systems with impulsive conditions via variational methods. However, without impulses, there is no homoclinic or periodic solution for the system considered in this paper. Moreover, our results can be used to study the existence of periodic and homoclinic solutions of difference equations.

Periodic solutions for sublinear systems via variational approach

The variational method is used to obtain some existence theorems for periodic solutions of second order systems with or without impulsive under weak sublinear growth conditions.

Farkas’ lemma for separable sublinear inequalities without qualifications

We show that the celebrated Farkas lemma for linear inequality systems continues to hold for separable sublinear inequality systems. As a consequence, we establish a qualification-free characterization of optimality for separable sublinear programming problems which include classes of robust linear programming problems. We also deduce that the Lagrangian duality always holds for these programming problems without qualifications.

Characteristic 2 approach to bivariate interpolation problems

We investigate bivariate Hermite interpolation problems in characteristic 2. Given a nonnegative integer t, we describe all the sub-linear systems generated by monomials, in which there is no curve passing through a general point with multiplicity at least 2t. As an application, we show that a certain linear system of plane curves with ten base points is non-special.