Some properties of minimal solutions for a fuzzy relation equation

Some properties of the solution set and minimal solutions of a fuzzy relation equation are considered. In this paper, we show the necessary and sufficient condition for existence of a minimal solution of a finite fuzzy relation equation defined on infinite index sets.     

On the Kahler-Ricci Flow on Projective Manifolds of General Type

This note concerns the global existence and convergence of the solution for Kahler-Ricci flow equation when the canonical class, Kx, is numerically effective and big. We clarify some known results regarding this flow on projective manifolds of general type and also show some new observations and refined results.     

On the K？hler-Ricci Flow on Projective Manifolds of General Type

Abstract This note concerns the global existence and convergence of the solution for K?hler-Ricci flow equation when the canonical class, K_X, is numerically effective and big. We clarify some known results regarding this flow on projective manifolds of general type and also show some new observations and refined results. (Dedicated to the memory of Shiing-Shen Chern) * Partially supported by NSF grants and a Simons fund.     

Bifurcation for minimal surface equation in hyperbolic 3-manifolds

Initiated by the work of Uhlenbeck in late 1970s, we study existence, multiplicity and asymptotic behavior for minimal immersions of a closed surface in some hyperbolic three-manifold, with prescribed conformal structure on the surface and second fundamental form of the immersion. We prove several results in these directions, by analyzing the Gauss equation governing the immersion. We determine when existence holds, and obtain unique stable solutions for area minimizing immersions. Furthermore, we find exactly when other (unstable) solutions exist and study how they blow-up. We prove our class of unstable solutions exhibit different blow-up behaviors when the surface is of genus two or greater. We establish similar results for the blow-up behavior of any general family of unstable solutions. This information allows us to consider similar minimal immersion problems when the total extrinsic curvature is also prescribed.     

On the Minimal Nonnegative Solution of Nonsymmetric Algebraic Riccati Equation

We study perturbation bound and structured condition number about the minimal nonnegative solution of nonsymmetric algebraic Riccati equation, obtaining a sharp perturbation bound and an accurate condition number. By using the matrix sign function method we present a new method for finding the minimal nonnegative solution of this algebraic Riccati equation. Based on this new method, we show how to compute the desired M-matrix solution of the quadratic matrix equation X^2 - EX - F = 0 by connecting it with the nonsymmetric algebraic Riccati equation, where E is a diagonal matrix and F is an M-matrix.     

Existence of solutions for anti-periodic boundary value problems of nonlinear impulsive functional integro-differential equations of mixed type

We discuss the existence of minimal and maximal solutions for a class of first order nonlinear impulsive functional integro-differential equations of mixed type with anti-periodic boundary conditions. The main tool of study is the monotone iterative technique.     

Asymptotic speeds of spread and traveling wave solutions of a second order integrodifference equation without monotonicity

This paper is concerned with the propagation modes of a second order integrodifference equation without monotonicity. The equation cannot generate monotone semifolws. By constructing auxiliary functions/equations and applying some known results, the minimal wave speed of traveling wave solutions and asymptotic speeds of spread are established.     

Global existence of classical solutions to the minimal surface equation with slow decay initial value

In this paper, we prove that the global existence of classical solutions to the Cauchy problem for the minimal surface equation with slow decay initial value in Minkowskian space time.     

On the Dynamics of the Classical Transverse Field Ising Chain

We investigate stationary and travelling wave solutions of a special lattice differential equation in one space dimension. Depending on a parameter λ, results are given on the existence, shape and stability for these kind of solutions. The analysis of travelling wave solutions leads us to a functional differential equation with both forward and backward shifts. The existence of solutions of this equation will be proved by use of the implicit function theorem. In particular, we consider kink solutions and periodic solutions.     

On differential equations characterized by their Lie point symmetries

We study the geometry of differential equations determined uniquely by their point symmetries, that we call Lie remarkable. We determine necessary and sufficient conditions for a differential equation to be Lie remarkable. Furthermore, we see how, in some cases, Lie remarkability is related to the existence of invariant solutions. We apply our results to minimal submanifold equations and to Monge-Ampère equations in two independent variables of various orders.     

The area minimizing problem in conformal cones,I

In this paper we study the area minimizing problem in some kinds of conformal cones. This concept is a generalization of the cones in Euclidean spaces and the cylinders in product manifolds. We define a non-closed-minimal (NCM) condition for bounded domains. Under this assumption and other necessary conditions we establish the existence of bounded minimal graphs in mean convex conformal cones. Moreover those minimal graphs are the solutions to corresponding area minimizing problems. We can solve the area minimizing problem in non-mean convex translating conformal cones if these cones are contained in a larger mean convex conformal cones with the NCM assumption. We give examples to illustrate that this assumption can not be removed for our main results.     

Strong solutions to a parabolic equation with linear growth with respect to the gradient variable

In this paper we prove existence and uniqueness of strong solutions to the homogeneous Neumann problem associated to a parabolic equation with linear growth with respect to the gradient variable. This equation is a generalization of the time-dependent minimal surface equation. Existence and regularity in time of the solution is proved by means of a suitable pseudoparabolic relaxed approximation of the equation and a passage to the limit.     

Stochastic Cahn-Hilliard equations driven by Poisson random measures

We study a stochastic Cahn-Hilliard equation driven by a Poisson random measure with Neumann boundary conditions. The global weak solution is established for the equation. Moreover, the existence of a Lyapunov function for the equation and an invariant measure associated with the transition semigroup are proved.     

ON THE MINIMAL NONNEGATIVE SOLUTION OF ONSYMMETRIC ALGEBRAIC RICCATI EQUATION

We study perturbation bound and structured condition number about the minimalnonnegative solution of nonsymmetric algebraic Riccati equation,obtaining a sharp per-turbation bound and an accurate condition number.By using the matrix sign functionmethod we present a new method for finding the minimal nonnegative solution of this al-gebraic Riccati equation.Based on this new method,we show how to compute the desiredM-matrix solution of the quadratic matrix equation X~2-EX-F=0 by connecting itwith the nonsymmetric algebraic Riccati equation,where E is a diagonal matrix and F isan M-matrix.     

极大代数上线性系统的3维最小实现

研究了极大代数上线性系统的单输入单输出的最小实现问题.根据无穷序列{gi}∞0的特征方程将3-阶周期序列{gi}∞0分为4类,分别讨论了序列{gi}∞0存在3维最小实现的充要条件.     

完备强对偶原子分配格上的不可约极小并分解及其应用

在完备强对偶原子分配格上引入了不可约极小并分解的概念,给出了元素存在不可约极小并分解的一些充要条件.证明了当元素恰有一个下邻时,该元索就足完全并既约元;有两个下邻时,元素的不可约极小并分解与不可约完全并既分解是等价的;下邻多于两个时,元素的不可约极小并分解不一定足不可约完全并既分解.最后证明了模糊关系方程有极小解的充要条件是方程左边有大于等于右手项的系数或右手项系数有不可约极小并分解.     

Existence and uniqueness of pseudo almost periodic solutions to some abstract partial neutral functional-differential equations and applications

The paper considers the existence and uniqueness of pseudo almost periodic (mild) solutions to some classes of first-order partial neutral functional-differential equations. Upon making some suitable assumptions, existence and uniqueness results are obtained. Applications include both a partial integro-differential equation arising in control systems and the scalar reaction-diffusion equation with delay.     

Minimisers of the Allen–Cahn equation and the asymptotic Plateau problem on hyperbolic groups

We investigate the existence of non-constant uniformly-bounded minimal solutions of the Allen–Cahn equation on a Gromov-hyperbolic group. We show that whenever the Laplace term in the Allen–Cahn equation is small enough, there exist minimal solutions satisfying a large class of prescribed asymptotic behaviours. For a phase field model on a hyperbolic group, such solutions describe phase transitions that asymptotically converge towards prescribed phases, given by asymptotic directions. In the spirit of de Giorgi's conjecture, we then fix an asymptotic behaviour and let the Laplace term go to zero. In the limit we obtain a solution to a corresponding asymptotic Plateau problem by Γ-convergence.     

Uniform attractors for non-autonomous motion equations of viscoelastic medium

Sufficient conditions for existence of minimal uniform trajectory attractors and uniform global attractors of non-autonomous evolution equations in Banach spaces are obtained. It is not assumed that the symbol space of an equation is a compact metric space and that the family of trajectory spaces corresponding to this symbol space is translation-coordinated or closed in any sense. Using these results, existence of minimal uniform trajectory attractors and uniform global attractors for weak solutions of the boundary value problem for motion equations of an incompressible viscoelastic medium with the Jeffreys constitutive law is shown.     

Minimal graphs in {M \times \mathbb{R}}

We study minimal graphs in . First, we establish some relations between the geometry of the domain and the existence of certain minimal graphs. We then discuss the problem of finding the maximal number of disjoint domains Ω ⊂ M that admit a minimal graph that vanishes on ∂Ω. When M is two-dimensional and has non-negative sectional curvature, we prove that this number is 3. This was proved by Tkachev in . Maria Fernanda Elbert was partially supported by CNPq and Faperj.