Unifying cycles

Two-literal clauses of the form L leftarrow R occur quite frequently in logic programs, deductive databases, and - disguised as an equation - in term rewriting systems. These clauses define a cycle if the atoms L and R are weakly unifiable, i.e., if L unifies with a new variant of R. The obvious problem with cycles is to control the number of iterations through the cycle. In this paper we consider the cycle unification problem of unifying two literals G and F modulo a cycle. We review the state of the art of cycle unification and give new results for a special type of cycles called unifying cycles, i.e., cycles L leftarrow R for which there exists a substitution sigma such that sigmaL = sigmaR. Altogether, these results show how the deductive process can be efficiently controlled for special classes of cycles without losing completeness

Revealed Preference and the Number of Commodities

This work consists of two parts: First, it is shown that for a two-dimensional commodity space any homothetic utility function that rationalizes each pair of observations in a set of consumption data also rationalizes the entire set of observations. The result is stated as a pairwise version of Varian’s Homothetic Axiom of Revealed Preference and is used to provide a simplified nonparametric test of homotheticity. In the second part a unifying proof technique is presented to show that the Weak Axiom of Revealed Preference (WARP) implies the Strong Axiom of Revealed Preference (SARP) for two commodities yet not for more commodities. It also shows that preference cycles can be of arbitrary length.While these results are already known, the proof here generalizes and unifies the existing ones insofar as it gives necessary and sufficient conditions for preference cycles to exist. It is then shown that in two dimensions the necessary condition cannot be fulfilled, whereas in more than two dimensions the sufficient conditions can always be met. The proof admits an intuitive understanding of the reason by giving a geometric interpretion of preference cycles as paths on indifference surfaces.Homotheticity, nonparametric tests, preference cycles, revealed preference, SARP,WARP

Unifying Markov Properties for Graphical Models

Several types of graphs with different conditional independence interpretations --- also known as Markov properties --- have been proposed and used in graphical models. In this paper we unify these Markov properties by introducing a class of graphs with four types of edges --- lines, arrows, arcs, and dotted lines --- and a single separation criterion. We show that independence structures defined by this class specialize to each of the previously defined cases, when suitable subclasses of graphs are considered. In addition, we define a pairwise Markov property for the subclass of chain mixed graphs which includes chain graphs with the LWF interpretation, as well as summary graphs (and consequently ancestral graphs). We prove the equivalence of this pairwise Markov property to the global Markov property for compositional graphoid independence models.Comment: 31 Pages, 6 figures, 1 tabl

(WP 2018-05) Specialization, Fragmentation, and Pluralism in Economics

This paper investigates whether specialization in research is causing economics to become an increasingly fragmented and diverse discipline with a continually rising number of niche-based research programs and a declining role for dominant cross-science research programs. It opens by framing the issue in terms of centrifugal and centripetal forces operating on research in economics, and then distinguishes descriptive from normative pluralism. It reviews recent research regarding the JEL code and the economics’ J. B. Clark Award that points towards rising specialization and fragmentation of research in economics. It then reviews five related arguments that might explain increasing specialization and fragmentation in economics: (i) Smith’s early division of labor view, (ii) Kuhn’s later thinking about the importance of specialization, (iii) Heiner’s behavioral burden of knowledge argument, (iv) Ross innovation-diffusion analysis and Arthur’s theory of technological change as determinants of specialization in science, and (v) the effects of space and culture or internationalization on innovation appropriation. The paper then discusses what descriptive pluralism implies about normative pluralism, and makes a case for multidisciplinarity over interdisciplinarity as a basis for arguments promoting pluralism. The paper closes with brief comments on the issue of specialization and pluralism in the wider world outside economics and science

ADAM: Analysis of Discrete Models of Biological Systems Using Computer Algebra

Background: Many biological systems are modeled qualitatively with discrete models, such as probabilistic Boolean networks, logical models, Petri nets, and agent-based models, with the goal to gain a better understanding of the system. The computational complexity to analyze the complete dynamics of these models grows exponentially in the number of variables, which impedes working with complex models. Although there exist sophisticated algorithms to determine the dynamics of discrete models, their implementations usually require labor-intensive formatting of the model formulation, and they are oftentimes not accessible to users without programming skills. Efficient analysis methods are needed that are accessible to modelers and easy to use. Method: By converting discrete models into algebraic models, tools from computational algebra can be used to analyze their dynamics. Specifically, we propose a method to identify attractors of a discrete model that is equivalent to solving a system of polynomial equations, a long-studied problem in computer algebra. Results: A method for efficiently identifying attractors, and the web-based tool Analysis of Dynamic Algebraic Models (ADAM), which provides this and other analysis methods for discrete models. ADAM converts several discrete model types automatically into polynomial dynamical systems and analyzes their dynamics using tools from computer algebra. Based on extensive experimentation with both discrete models arising in systems biology and randomly generated networks, we found that the algebraic algorithms presented in this manuscript are fast for systems with the structure maintained by most biological systems, namely sparseness, i.e., while the number of nodes in a biological network may be quite large, each node is affected only by a small number of other nodes, and robustness, i.e., small number of attractors

Microgravity: A Teacher's Guide With Activities in Science, Mathematics, and Technology

The purpose of this curriculum supplement guide is to define and explain microgravity and show how microgravity can help us learn about the phenomena of our world. The front section of the guide is designed to provide teachers of science, mathematics, and technology at many levels with a foundation in microgravity science and applications. It begins with background information for the teacher on what microgravity is and how it is created. This is followed with information on the domains of microgravity science research; biotechnology, combustion science, fluid physics, fundamental physics, materials science, and microgravity research geared toward exploration. The background section concludes with a history of microgravity research and the expectations microgravity scientists have for research on the International Space Station. Finally, the guide concludes with a suggested reading list, NASA educational resources including electronic resources, and an evaluation questionnaire

Edge-Orders

Canonical orderings and their relatives such as st-numberings have been used as a key tool in algorithmic graph theory for the last decades. Recently, a unifying concept behind all these orders has been shown: they can be described by a graph decomposition into parts that have a prescribed vertex-connectivity. Despite extensive interest in canonical orderings, no analogue of this unifying concept is known for edge-connectivity. In this paper, we establish such a concept named edge-orders and show how to compute (1,1)-edge-orders of 2-edge-connected graphs as well as (2,1)-edge-orders of 3-edge-connected graphs in linear time, respectively. While the former can be seen as the edge-variants of st-numberings, the latter are the edge-variants of Mondshein sequences and non-separating ear decompositions. The methods that we use for obtaining such edge-orders differ considerably in almost all details from the ones used for their vertex-counterparts, as different graph-theoretic constructions are used in the inductive proof and standard reductions from edge- to vertex-connectivity are bound to fail. As a first application, we consider the famous Edge-Independent Spanning Tree Conjecture, which asserts that every k-edge-connected graph contains k rooted spanning trees that are pairwise edge-independent. We illustrate the impact of the above edge-orders by deducing algorithms that construct 2- and 3-edge independent spanning trees of 2- and 3-edge-connected graphs, the latter of which improves the best known running time from O(n^2) to linear time

Transit functions on graphs (and posets)

The notion of transit function is introduced to present a unifying approachfor results and ideas on intervals, convexities and betweenness in graphs andposets. Prime examples of such transit functions are the interval function I andthe induced path function J of a connected graph. Another transit function isthe all-paths function. New transit functions are introduced, such as the cutvertextransit function and the longest path function. The main idea of transitfunctions is that of ‘transferring’ problems and ideas of one transit functionto the other. For instance, a result on the interval function I might suggestsimilar problems for the induced path function J. Examples are given of howfruitful this transfer can be. A list of Prototype Problems and Questions forthis transferring process is given, which suggests many new questions and openproblems.graph theory;betweenness;block graph;convexity;distance in graphs;interval function;path function;induced path;paths and cycles;transit function;types of graphs