A binary preference relation on a real vector space V with zero element 0 satisfying four axioms induces a utility function U = f ◦ u : V → R where u:V → R is linear as a map of vector spaces and f : R → R is weakly monotone. The fourth axiom is key: X − Y ( ) 0 ⇒ X ( ) Y (X, Y ∈ V ). The utility function is linear if and only if also satisfies the converse of the fourth axiom. A consequence is...

As the capacity to summarize of the preference map diminishes with the number of consumers (the number of axes to interpret increases), a non-metric PCA is sometimes used to reduce the number of necessary axes. The non-metric PCA consists of a monotone transform of the data so that the variability explained by the k (k=2 or 3) first axes is maximized. This transform implies that we consider the...

We provide sufficient conditions for a monotone function with finite set of outcomes to be cyclically monotone. Using these conditions, we show that any defined on the domain gross substitutes is The result also extends generalized and complements.

In the paper, we develop the notation of lattice-wise categories as concrete categories (see [10]) of lattices. Namely, the categories based on [24] with lattices as objects and at least monotone maps between them as morphisms. As examples, we introduce the categories UPS, CONT, and ALG with complete, continuous, and algebraic lattices, respectively, as objects and directed suprema preserving m...

In this paper we deal with the regularity of planar optimal transport maps, in a “critical” case not covered presently in the literature. Following a suggestion by J.Maly, and the ideas in [1], we relate the regularity problem to a “rigidity” problem for partial differential inclusions which might be interesting in its own right. Let us start with the first problem. We give a formulation in ter...

This paper provides an expository introduction to monotone and near-monotone biochemical network structures. Monotone systems respond in a predictable fashion to perturbations, and have very robust dynamical characteristics. This makes them reliable components of more complex networks, and suggests that natural biological systems may have evolved to be, if not monotone, at least close to monoto...

Abstract We study a nonlinear evolutionary quasi–variational–hemivariational inequality (in short, (QVHVI)) involving set-valued pseudo-monotone map. The central idea of our approach consists introducing parametric variational problem that defines selection associated with (QVHVI). prove the solvability by employing surjectivity theorem for sum operators, combined Minty’s formulation and techni...

We study a single-period optimal transport problem on R2 with covariance-type cost function c(x,y)=(x1−y1)(x2−y2) and backward martingale constraint. show that plan γ is if only there maximal monotone set G supports the x-marginal of such c(x,y)=minz∈Gc(z,y) for every (x,y)∈suppγ. obtain sharp regularity conditions uniqueness an its representation in terms map. Our motivated by variant classica...

We present shortest path maps that support min-link queries from any source point on a line segment to destination points in either a simple polygon or a polygonal domain (a polygon with holes). These structures are more versatile than the traditional shortest path map from a fixed source point, and we use them to compute the min-link Fréchet distance between two polygonal curves. In addition t...