Minimum Entropy Orientations

We study graph orientations that minimize the entropy of the in-degree sequence. The problem of finding such an orientation is an interesting special case of the minimum entropy set cover problem previously studied by Halperin and Karp [Theoret. Comput. Sci., 2005] and by the current authors [Algorithmica, to appear]. We prove that the minimum entropy orientation problem is NP-hard even if the graph is planar, and that there exists a simple linear-time algorithm that returns an approximate solution with an additive error guarantee of 1 bit. This improves on the only previously known algorithm which has an additive error guarantee of log_2 e bits (approx. 1.4427 bits).Comment: Referees' comments incorporate

Spin, statistics, orientations, unitarity

A topological quantum field theory is Hermitian if it is both oriented and complex-valued, and orientation-reversal agrees with complex-conjugation. A field theory satisfies spin-statistics if it is both spin and super, and $360^\circ$-rotation of the spin structure agrees with the operation of flipping the signs of all fermions. We set up a framework in which these two notions are precisely analogous. In this framework, field theories are defined over $\mathrm{Vect}_{\mathbb R}$, but rather than being defined in terms of a single tangential structure, they are defined in terms of a bundle of tangential structures over $\mathrm{Spec}(\mathbb R)$. Bundles of tangential structures may be etale-locally equivalent without being equivalent, and Hermitian field theories are nothing but the field theories controlled by the unique nontrivial bundle of tangential structures that is etale-locally equivalent to Orientations. This bundle owes its existence to the fact that $\pi_1^{et}(\mathrm{Spec}(\mathbb R)) = \pi_1{BO}$. We interpret Deligne's "existence of super fiber functors" theorem as implying that in a categorification of algebraic geometry in which symmetric monoidal categories replace commutative rings, $\pi_2^{et}(\mathrm{Spec}(\mathbb R)) = \pi_2{BO}$. There are eight bundles etale-locally equivalent to Spins, one of which is distinguished; upon unpacking the meaning of that distinguished tangential structure, one arrives at field theories that are both Hermitian and satisfy spin-statistics. Finally, we formulate a notion of "reflection-positivity" and prove that if an etale-locally-oriented field theory is reflection-positive then it is necessarily Hermitian, and if an etale-locally-spin field theory is reflection-positive then it necessarily both satisfies spin-statistics and is Hermitian. The latter result is a topological version of the famous Spin-Statistics Theorem.Comment: To appear in Algebraic and Geometric Topolog

Review on learning orientations

The need has arises towards the consideration of individual difference to let learners engage in and responsible for their own learning, retain information longer, apply the knowledge more effectively, have positive attitudes towards the subject, have more interest in learning materials, score higher and have high intrinsic motivation level. As regard to the importance of individual differences, Martinez (2000) has grounded a new theory, which is Intentional Learning Theory that covered individual aspects of cognitive, intention, social and emotion. This theory hypothesizes that the fundamental of understanding how individual learns, interact with an environment, performs, engages in learning, experiences learning, and assimilate and accommodate the new knowledge is by understanding individual’s fundamental emotions and intentions about how to use learning, why it is important, when the suitable time, and how it can accomplish personal goals and change. The intent of this theory is to focus on emotions and intentions of an individual regarding why, when and how learning goals are organized, processed, and achieved. In conclusion, Learning Orientations introduced by this theory describes the disposition of an individual in approaching, managing and achieving their learning intentionally and differently from others

Counting Planar Eulerian Orientations

Inspired by the paper of Bonichon, Bousquet-M\'elou, Dorbec and Pennarun, we give a system of functional equations which characterise the ordinary generating function, $U(x),$ for the number of planar Eulerian orientations counted by edges. We also characterise the ogf $A(x)$, for 4-valent planar Eulerian orientations counted by vertices in a similar way. The latter problem is equivalent to the 6-vertex problem on a random lattice, widely studied in mathematical physics. While unable to solve these functional equations, they immediately provide polynomial-time algorithms for computing the coefficients of the generating function. From these algorithms we have obtained 100 terms for $U(x)$ and 90 terms for $A(x).$ Analysis of these series suggests that they both behave as $const\cdot (1 - \mu x)/\log(1 - \mu x),$ where we conjecture that $\mu = 4\pi$ for Eulerian orientations counted by edges and $\mu=4\sqrt{3}\pi$ for 4-valent Eulerian orientations counted by vertices.Comment: 26 pages, 20 figure

Achievement goals and motivational responses in tennis: Does the context matter?

Objectives: This study examined: (a) whether athletes’ goal orientations differ across training and competition; (b) whether goal orientations predict effort, enjoyment, and psychological skill use differently in training and competition; and (c) whether goal orientations predict perceived improvement in training and perceived performance in competition. Method: Participants were 116 competitive tennis players (mean age = 19.99, SD = 5.82), who completed questionnaires measuring goal orientations, effort, enjoyment, and psychological skill use in training and competition, perceived improvement in training, and perceived performance in competition. Results: Dependent t-tests revealed that athletes reported higher task orientation in training than in competition and higher ego orientation in competition than in training, while Pearson product-moment correlations revealed a high cross-contextual consistency for both task and ego goal orientations between training and competition. Regression analyses indicated that task orientation predicted positively effort, enjoyment, self-talk, and goal setting in both contexts, perceived improvement in training, and perceived performance in competition. An interaction effect also emerged whereby ego orientation predicted positively effort in competition only when task orientation was low or average. Conclusions: The findings suggest that goal orientations may differ between training and competition; task orientation is the goal that should be promoted in both contexts; and the context may affect the relationship between goal orientations and effort, enjoyment, and goal setting

Even Orientations and Pfaffian graphs

We give a characterization of Pfaffian graphs in terms of even orientations, extending the characterization of near bipartite non--pfaffian graphs by Fischer and Little \cite{FL}. Our graph theoretical characterization is equivalent to the one proved by Little in \cite{L73} (cf. \cite{LR}) using linear algebra arguments