Monge gave a number of examples to illustrate the difficulties of the problem. Plainly if arbitrary measures are allowed it may not be possible to transport at all. If μ is a single point mass and ν consists of two point masses, each with weight 1/2, then no map T can “split μ in half”. Where transportation maps do exist, there may be many different optimal ones, even for measures on the line. ...

we introduce a new concept of general $g$-$eta$-monotone operator generalizing the general $(h,eta)$-monotone operator cite{arvar2, arvar1}, general $h-$ monotone operator cite{xiahuang} in banach spaces, and also generalizing $g$-$eta$-monotone operator cite{zhang}, $(a, eta)$-monotone operator cite{verma2}, $a$-monotone operator cite{verma0}, $(h, eta)$-monotone operator cite{fanghuang}...

We prove generalizations of Löwner’s results on matrix monotone functions to several variables. We give a characterization of when a function of d variables is locally monotone on d-tuples of commuting self-adjoint n-by-n matrices. We prove a generalization to several variables of Nevanlinna’s theorem describing analytic functions that map the upper half-plane to itself and satisfy a growth con...

Results are obtained about the existence and behavior of hereditarily weakly confluent maps of continua onto the unit circle S1. A simple and useful necessary and sufficient condition is given for a map of a continuum, X, onto S1 to be hereditarily weakly confluent (HWC). It is shown that when X is arcwise connected, an HWC map of X onto S1 is monotone with arcwise connected fibers. A number of...

we introduce a new concept of general $g$-$eta$-monotone operator generalizing the general $(h,eta)$-monotone operator cite{arvar2, arvar1}, general $h-$ monotone operator cite{xiahuang} in banach spaces, and also generalizing $g$-$eta$-monotone operator cite{zhang}, $(a, eta)$-monotone operator cite{verma2}, $a$-monotone operator cite{verma0}, $(h, eta)$-monotone operator cite{fanghuang}, $h$-...

In this note we consider a collection C of one parameter families of unimodal maps of [0, 1]. Each family in the collection has the form {μf} where μ ∈ [0, 1]. Denoting the kneading sequence of μf by K(μf), we will prove that for each member of C, the map μ 7→ K(μf) is monotone. For interest, μf(x) = 4μx(1− x) and μf(x) = μ sin(πx) are shown to belong to C. This extends the work of Masato Tsuji...

0. Introduction. Over the past thirty years, a powerful theory of monotone dynamical systems has been developed by many authors. A partial list of contributors would include N. Alikakos, E. N. Dancer, M. Hirsch, P. Hess, M.A. Krasnoselskii, U. Krause, H. Matano, P. Polacik, H.L. Smith, P. Takac and H. Thieme. If one understands the subject more generally as a chapter in the study of linear and ...

For two classes of monotone maps on the n-dimensional positive orthant we show that for a discrete dynamical system induced by a map the origin of Rn+ is globally asymptotically stable, if and only if the map Γ is such that for any point in s ∈ R+, s 6= 0, the image-vector Γ(s) is such that at least one component is strictly less than the corresponding component of s. One class is the set of n ...

We study monotonicity properties for minimizers of transport problems. In the one-dimensional case, we present an algorithm to construct minimizing monotone transport plans by “monotonizing” a given minimizing transport plan. This method applies in particular to the case of the L1-Wasserstein metric where we prove the existence of monotone minimizers for arbitrary marginals. We find that monoto...

We study countably piecewise continuous, piecewise monotone interval maps. We establish a necessary and sufficient criterion for the existence of a nondecreasing semiconjugacy to a map of constant slope in terms of the existence of an eigenvector of an operator acting on a space of measures. Then we give sufficient conditions under which this criterion is not satisfied. Finally, we give example...