About positive Dunford-Pettis operators on Banach lattices

We give several characterizations of Banach lattices on which each positive Dunford-Pettis operator is compact. As consequences, we obtain new sufficient and necessary conditions under which a norm of a Banach lattice is order continuous, a positive weakly compact operator is compact and the dual operator of a positive Dunford-Pettis operator is Dunford-Pettis.     

About positive weak Dunford-Pettis operators on Banach lattices

We characterize Banach lattices for which each positive weak Dunford-Pettis operator from a Banach lattice into another dual Banach lattice is almost Dunford-Pettis. Also, we give some sufficient and necessary conditions for which the class of positive weak Dunford-Pettis operators coincides with that of positive Dunford-Pettis operators, and we derive some consequences.     

一类半线性椭圆型方程正整体解的存在性

本文研究了一个半线性椭圆型方程正整体解的存在性问题.利用上下解方法克服了非齐次项出现带来的困难,获得了正整体解存在的必要条件,具体地给出了在此条件下正整体解所处的领域.     

Locally definite normal operators in Krein spaces

We introduce the spectral points of two-sided positive type of bounded normal operators in Krein spaces. It is shown that a normal operator has a local spectral function on sets which are of two-sided positive type. In addition, we prove that the Riesz–Dunford spectral subspace corresponding to a spectral set which is only of positive type is uniformly positive. The restriction of the operator to this subspace is then normal in a Hilbert space.     

Preserving Positive Realness Through Discretization

The importance of positive real transfer functions relies on the fact that they are associated with positive linear systems. Those systems possess the property that their input-output product time-integral, which is a measure of the total enerty, is nonnegative. Such a property can be also formulated in the discrete context. It is shown that a discrete positive real transfer function is obtained from a positive real continuous one of relative order zero being strictly stable poles via discretization by a sampler and zero-order hold device provided that the direct input-output transmission gain is sufficiently large. It is also proved that a discrete positive real transfer function may be obtained from a stable continuous one of relative order zero and high direct input-output gain which posses simple complex conjugate critically stable poles even in the case that this one is not positive real. For that purpose, the use of an appropriate phase-lag or phase lead compensating network for the continuous transfer function may be required to ensure positive realness of the discrete transfer function.     

Perron-frobenius-type theorems for tridiagonal operators

It is proved that in a large class of bounded tridiagonal operators (infinite Jacobi matrices), not necessarily positive or non-negative, positive eigenvalues exist and the eigenvector which corresponds to the greatest of them can be taken strictly positive. It is the unique positive eigenvector up to a constant multiple.     

Completely positive tensors in the complex field

In this paper, we introduce the complex completely positive tensor, which has a symmetric complex decomposition with all real and imaginary parts of the decomposition vectors being non-negative. Some properties of the complex completely positive tensor are given. A semidefinite algorithm is also proposed for checking whether a complex tensor is complex completely positive or not. If a tensor is not complex completely positive, a certificate for it can be obtained; if it is complex completely positive, a complex completely positive decomposition can be obtained.     

Stability of stationary solutions in models of the Calvin cycle

In this paper results are obtained concerning the number of positive stationary solutions in simple models of the Calvin cycle of photosynthesis and the stability of these solutions. It is proved that there are open sets of parameters in the model of Zhu et al. (2009) for which there exist two positive stationary solutions. There are never more than two isolated positive stationary solutions but under certain explicit special conditions on the parameters there is a whole continuum of positive stationary solutions. It is also shown that in the set of parameter values for which two isolated positive stationary solutions exist there is an open subset where one of the solutions is asymptotically stable and the other is unstable. In related models derived from the work of Grimbs et al. (2011), for which it was known that more than one positive stationary solution exists, it is proved that there are parameter values for which one of these solutions is asymptotically stable and the other unstable. A key technical aspect of the proofs is to exploit the fact that there is a bifurcation where the centre manifold is one-dimensional.     

Bi-block positive semidefiniteness of bi-block symmetric tensors

The positive definiteness of elasticity tensors plays an important role in the elasticity theory.In this paper,we consider the bi-block symmetric tensors,which contain elasticity tensors as a subclass.First,we define the bi-block M-eigenvalue of a bi-block symmetric tensor,and show that a bi-block symmetric tensor is bi-block positive(semi)definite if and only if its smallest bi-block M-eigenvalue is(nonnegative)positive.Then,we discuss the distribution of bi-block M-eigenvalues,by which we get a sufficient condition for judging bi-block positive(semi)definiteness of the bi-block symmetric tensor involved.Particularly,we show that several classes of bi-block symmetric tensors are bi-block positive definite or bi-block positive semidefinite,including bi-block(strictly)diagonally dominant symmetric tensors and bi-block symmetric(B)B0-tensors.These give easily checkable sufficient conditions for judging bi-block positive(semi)definiteness of a bi-block symmetric tensor.As a byproduct,we also obtain two easily checkable sufficient conditions for the strong ellipticity of elasticity tensors.     

具有阶段结构和周期投放的非自治单种系统的周期解

研究一类带周期投放的非自治阶段结构单种群模型,利用泛函微分方程的单调和凹算子理论,得到系统存在一个全局吸引正周期的充分条件。     

Stability of discrete-time positive evolution operators on ordered Banach spaces and applications

In this paper we discuss stability problems for a class of discrete-time evolution operators generated by linear positive operators acting on certain ordered Banach spaces. Our approach is based upon a new representation result that links a positive operator with the adjoint operator of its restriction to a Hilbert subspace formed by sequences of Hilbert–Schmidt operators. This class includes the evolution operators involved in stability and optimal control problems for linear discrete-time stochastic systems. The inclusion is strict because, following the results of Choi, we have proved that there are positive operators on spaces of linear, bounded and self-adjoint operators which have not the representation that characterize the completely positive operators. As applications, we introduce a new concept of weak-detectability for pairs of positive operators, which we use to derive sufficient conditions for the existence of global and stabilizing solutions for a class of generalized discrete-time Riccati equations. Finally, assuming weak-detectability conditions and using the method of Lyapunov equations we derive a new stability criterion for positive evolution operators.     

Geometry of the copositive and completely positive cones

The copositive cone, and its dual the completely positive cone, have useful applications in optimisation, however telling if a general matrix is in the copositive cone is a co-NP-complete problem. In this paper we analyse some of the geometry of these cones. We discuss a way of representing all the maximal faces of the copositive cone along with a simple equation for the dimension of each one. In doing this we show that the copositive cone has faces which are isomorphic to positive semidefinite cones. We also look at some maximal faces of the completely positive cone and find their dimensions. Additionally we consider extreme rays of the copositive and completely positive cones and show that every extreme ray of the completely positive cone is also an exposed ray, but the copositive cone has extreme rays which are not exposed rays.     

Complete controllability and contractibility in multimodal systems

The concept of a strictly positive definite set of Hermitian matrices is introduced. It is shown that a strictly positive definite set is always a positive definite set, and conditions are found under which a positive definite set is strictly positive definite. We also show that a set of Hermitian matrices is strictly positive definite if and only if some nonnegative linear combination of these matrices is a positive definite matrix. For state dimension two, we use this concept to find necessary and sufficient conditions for a two-mode completely controllable irreducible multimodal system to be contractible relative to an elliptic norm. For general state dimensions, we give necessary and sufficient conditions for a special-type two-mode completely controllable irreducible system to be contractible relative to a weakly monotone norm. Applying the above results, we show that, for state dimension two, there exists a completely controllable two-mode system which is not contractible relative to either an elliptic or a weakly monotone norm. We leave open the question whether or not complete controllability implies contractibility, relative to some norm, for multimodal systems of two or more modes.     

The residual finiteness of positive one-relator groups

It is proven that every positive one-relator group which satisfies the condition has a finite index subgroup which splits as a free product of two free groups amalgamating a finitely generated malnormal subgroup. As a consequence, it is shown that every positive one-relator group is residually finite. It is shown that positive one-relator groups are generically and hence generically residually finite. A new method is given for recognizing malnormal subgroups of free groups. This method employs a 'small cancellation theory' for maps between graphs. Received: August 4, 2000     

A semidefinite algorithm for completely positive tensor decomposition

A symmetric tensor, which has a symmetric nonnegative decomposition, is called a completely positive tensor. In this paper, we characterize the completely positive tensor as a truncated moment sequence, and transform the problem of checking whether a tensor is completely positive to checking whether its corresponding truncated moment sequence admits a representing measure, then present a semidefinite algorithm to solve it. If a tensor is not completely positive, a certificate for it can be obtained; if it is completely positive, a nonnegative decomposition can be obtained.     

Steady spatial asymptotics for the Vlasov–Poisson system

A collisionless plasma is modelled by the Vlasov–Poisson system in three space dimensions. A fixed background of positive charge, which is independent of time and space, is assumed. The situation in which mobile negative ions balance the positive charge as ∣x∣ tends to infinity is considered. Hence, the total positive charge and the total negative charge are infinite. Smooth solutions with appropriate asymptotic behaviour were shown to exist locally in time in a previous work. This paper studies the time behaviour of the net charge and a natural quantity related to energy, and shows that neither is constant in time in general. Also, neither quantity is positive definite. When the background density is a decreasing function of ∣v∣, a positive definite quantity is constructed which remains bounded. A priori bounds are obtained from this. Copyright © 2003 John Wiley & Sons, Ltd.     

Compact Manifolds with Positive Einstein Curvature

In this paper we study positive Einstein curvature which is a condition on the Riemann curvature tensor intermediate between positive scalar curvature and positive sectional curvature. We prove some constructions and obstructions for positive Einstein curvature on compact manifolds generalizing similar well known results for the scalar curvature. Finally, because our problem is relatively new, many open questions are included.     

Invariant sublattices for positive operators

There are, by now, many results which guarantee that positive operators on Banach lattices have non-trivial closed invariant sublattices. In particular, this is true for every positive compact operator. Apart from some results of a general nature, in this paper we present several examples of positive operators on Banach lattices which do not have non-trivial closed invariant sublattices. These examples include both AM-spaces and Banach lattices with an order continuous norm and which are and are not atomic. In all these cases we can ensure that the operators do possess non-trivial closed invariant subspaces.     

Positive and Nontrivial Solutions for the Urysohn Integral Equation

We establish new criteria for the existence of either positive or nonzero solutions of the Urysohn integral equation. We also discuss the existence of an interval of positive eigenvalues and sufficient conditions for the existence of at least a positive eigenvalue with a nonzero or positive eigenfunction for the Urysohn integral operator. Among others, we employ techniques based on fixed point index theory for compact maps, which are new for this type of equation.     

On the inertia and range of transformations of positive definite matrices

Results are established which relate the range and inertia of general transformations on positive definite matrices. Included are a bound theorem for certain eigenvalues of these transformations, a characterization of positive definite preserving, completely positive transformations, and generalizations of the theorems of Stein [8] and of Stein and Pfeffer [9].