A generalization of Rader's utility representation theorem

Rader's utility representation theorem guarantees the existence of an upper semicontinuous utility function for any upper semicontinuous total preorder on a second countable topological space. In this paper we present a generalization of Rader's theorem to not necessarily total preorders that are weakly upper semicontinuous.Weakly upper semicontinuous preorder; utility function

Sublinear and continuous order-preserving functions for noncomplete preorders

We characterize the existence of a nonnegative, sublinear and continuous order-preserving function for a not necessarily complete preorder on a real convex cone in an arbitrary topological real vector space. As a corollary of the main result, we present necessary and sufficient conditions for the existence of such an order-preserving function for a complete preorder.Comment: 8 page

Weak continuity of preferences with nontransitive indifference

We characterize weak continuity of an interval order on a topological space by using the concept of a scale in a topological space.Weakly continuous interval order; continuous numerical representation

A selection of maximal elements under non-transitive indifferences

In this work we are concerned with maximality issues under intransitivity of the indifference. Our approach relies on the analysis of "undominated maximals" (cf., Peris and Subiza, J Math Psychology 2002). Provided that an agent's binary relation is acyclic, this is a selection of its maximal elements that can always be done when the set of alternatives is finite. In the case of semiorders, proceeding in this way is the same as using Luce's selected maximals. We put forward a sufficient condition for the existence of undominated maximals for interval orders without any cardinality restriction. Its application to certain type of continuous semiorders is very intuitive and accommodates the well-known "sugar example" by Luce

A note on maximal elements for acyclic binary relations on compact topological spaces

I introduce the concept of a weakly tc-upper semicontinuous acyclic binary relation on a topological space (X,t), which appears as slightly more general than other concepts of continuity which have been introduced in the literature in connection with the problem concerning the existence of maximal elements. By using such a notion, I show that if an acyclic binary relation on a compact topological space is weakly tc-upper semicontinuous, then there exists a maximal element relative to such a binary relation. In this way I generalize existing results concerning the existence of maximal elements on compact topological spaces

A selection of maximal elements under non-transitive indifferences

In this work we are concerned with maximality issues under intransitivity of the indifference. Our approach relies on the analysis of "undominated maximals" (cf., Peris and Subiza, J Math Psychology 2002). Provided that an agent's binary relation is acyclic, this is a selection of its maximal elements that can always be done when the set of alternatives is finite. In the case of semiorders, proceeding in this way is the same as using Luce's selected maximals. We put forward a sufficient condition for the existence of undominated maximals for interval orders without any cardinality restriction. Its application to certain type of continuous semiorders is very intuitive and accommodates the well-known "sugar example" by Luce.Maximal element; Selection of maximals; Acyclicity; Interval order; Semiorder

Equilibrium a la Cournot and Supply FunctionEquilibrium in Electricity Markets underLinear Price/Demand and Quadratic Costs

We present a comparison between the Cournot and the Supply Function Equilibrium models under linear price/demand, linear supply functions and quadratic costs

Characterization of Useful Topologies in Mathematical Utility Theory by Countable Chain Conditions

Under the additional assumption of complete regularity, we furnish a simple characterization of all the topologies such that every continuous total preorder is representable by a continuous utility function. In particular, we prove that a completely regular topology satisfies such property if, and only if, it is separable and every linearly ordered collection of clopen sets is countable. Since it is not restrictive to refer to completely regular topologies when dealing with this kind of problem, this is, as far as we are concerned, the simplest characterization of this sort available in the literature. All the famous utility representation theorems are corollaries of our result

Free meals on long-distance cruisers: the vampire fish rides giant catfishes in the Amazon

The trichomycterid catfishes known as candirus are renowned for their blood feeding, but information on their habits under natural conditions is very fragmentary and generally restricted to hosts or habitats. We recorded an undescribed species of the vandelliine genus Paracanthopoma riding the giant jau catfish, Zungaro zungaro (Pimelodidae), in the upper Amazon. The candirus were found on the host's caudal and pectoral fins, as well as the base of the dorsal fin, with their snouts buried up to the eyes in the tough skin of the catfish host. All of them had small amounts of partly digested blood in the distal part of the gut. Along the host's dorsal fin base we found a few additional tiny holes, most of them healed. We suggest that Paracanthopoma feeds on the gill chamber of its hosts, and that the individuals we found were taking a ride partly buried into the host's skin. Our assumption seems supported by the widespread behaviour of vandelliine candirus taking blood from the gill region of their hosts, and by a report of Paracanthopoma parva found on the gills of another species of giant catfish, Brachyplatystoma vaillanti. Additionally, the Paracanthopoma sp. individuals we examined were not gorged with blood as usual for several vandelliines. Species within the genus Paracanthopoma have the longest and most robust snout, and the longest and strongest dentary teeth among blood-feeding candirus, which fit their drilling needs. Taking a ride on a giant host would be advantageous for Paracanthopoma candirus for several reasons: 1) dispersal; 2) no need to search for hosts to feed; and 3) protection from predators. The alternative explanation that Paracanthopoma takes blood from the tiny holes it drills in the skin seems unlikely, due to the recent finding that species of the genus Vandellia are unable to take blood from their hosts actively and cut open a major branchial artery to gorge themselves with blood due to the host's arterial pressure instead. The body parts of the host the Paracanthopoma sp. individuals were attached on have no large vessels that would supply them with plenty of blood. Thus, drilling a hole on a giant host skin seems to serve mostly to anchor the Paracanthopoma candirus to their long-distance cruising catfish host. If our assumption holds true, then species of this genus exemplify an instance of phoresis (hitch-hiking) among the blood-feeding candirus.Os bagres tricomicterídeos conhecidos como candirus são famosos por se alimentarem de sangue, mas as informações sobre seus hábitos, em condições naturais, são fragmentárias e restritas aos seus hospedeiros ou ambientes. Registramos uma espécie não descrita de candiru do gênero Paracanthopoma (Vandelliinae) sobre um jaú, Zungaro zungaro (Pimelodidae), no alto Rio Amazonas. Os candirus estavam sobre as nadadeiras caudal e peitoral e junto à base da dorsal, com seus focinhos enterrados até a altura dos olhos, no tegumento espesso do bagre hospedeiro. Os candirus continham pequenas quantidades de sangue parcialmente digerido na porção distal de seus tubos digestórios. Havia diversos orifícios rasos próximos à base da nadadeira dorsal do hospedeiro, a maioria cicatrizada. Sugerimos que Paracanthopoma se alimente na câmara branquial dos seus hospedeiros e que os candirus estejam viajando parcialmente enterrados na pele do jaú. Nossa suposição está apoiada no hábito de tomar sangue na região branquial dos hospedeiros, predominante entre os Vandelliinae, bem como por um registro de Paracanthopoma parva sobre as brânquias de uma outra espécie de grande bagre (Brachyplatystoma vaillanti). Além disso, os indivíduos de Paracanthopoma sp. não estavam empanturrados com sangue, como é usual para Vandelliinae. As espécies de Paracanthopoma têm o focinho mais longo e robusto entre os candirus hematófagos, além de dentes mandibulares longos e muito fortes, características adequadas ao hábito de perfurar a pele do hospedeiro. Viajar no corpo do hospedeiro seria vantajoso por diversos motivos: 1) dispersão; 2) não haver necessidade de procurar hospedeiros para se alimentar; 3) proteção contra predadores. A explicação alternativa, de que Paracanthopoma toma sangue nos pequenos furos que escava, não parece plausível, devido à recente descoberta de que espécies de Vandellia são incapazes de tomar sangue ativamente, pois fazem uma incisão numa das artérias branquiais e valem-se da pressão arterial do hospedeiro para bombear sangue dentro do seu tubo digestório. As partes do hospedeiro, em que os candirus estavam fixados, não têm vasos sangüíneos de calibre adequado para este tipo de alimentação. Portanto, escavar um furo na pele de um hospedeiro deve servir principalmente para ancorar os candirus durante os longos percursos do seu hospedeiro. Caso a nossa sugestão seja plausível, as espécies de Paracanthopoma representam um exemplo de forese em candirus hematófagos.109114Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq

Lifting Theorems for Continuous Order-Preserving Functions and Continuous Multi-Utility

We present some lifting theorems for continuous order-preserving functions on locally and σ -compact Hausdorff preordered topological spaces. In particular, we show that a preorder on a locally and σ -compact Hausdorff topological space has a continuous multi-utility representation if, and only if, for every compact subspace, every continuous order-preserving function can be lifted to the entire space. Such a characterization is also presented by introducing a lifting property of ≾-C-compatible continuous order-preserving functions on closed subspaces. The assumption of paracompactness is also used in connection to lifting conditions