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Brattka, Vasco ; Greenberg, Noam ; Kalimullin, Iskander ; Soskova, Mariya (Ed.)In her 1990 thesis, Ahmad showed that there is a socalled “Ahmad pair”, i.e., there are incomparable Σ 2 0 enumeration degrees a 0 and a 1 such that every enumeration degree x < a 0 is ⩽ a 1 . At the same time, she also showed that there is no “symmetric Ahmad pair”, i.e., there are no incomparable Σ 2 0 enumeration degrees a 0 and a 1 such that every enumeration degree x 0 < a 0 is ⩽ a 1 and such that every enumeration degree x 1 < a 1 is ⩽ a 0 . In this paper, we first present a direct proof of Ahmad’s second result. We then show that her first result cannot be extended to an “Ahmad triple”, i.e., there are no Σ 2 0 enumeration degrees a 0 , a 1 and a 2 such that both ( a 0 , a 1 ) and ( a 1 , a 2 ) are an Ahmad pair. On the other hand, there is a “weak Ahmad triple”, i.e., there are pairwise incomparable Σ 2 0 enumeration degrees a 0 , a 1 and a 2 such that every enumeration degree x < a 0 is also ⩽ a 1 or ⩽ a 2 ; however neither ( a 0 , a 1 ) nor ( a 0 , a 2 ) is an Ahmad pair.more » « less

Abstract Tropical ascent area (
A _{a}) and high cloud fraction (HCF) are projected to decrease with surface warming in most Coupled Model Intercomparison Project Phase 5 (CMIP5) models. Perturbing deep convective parameters in the Community Atmosphere Model (CAM5) results in a similar spread and correlation between HCF andA _{a}responses to interannual warming compared to the CMIP5 ensemble, with a narrowerA _{a}corresponding to greater HCF reduction. Perturbing cloud physics parameters produces a comparatively smaller range ofA _{a}responses to warming and a dissimilar HCF‐A _{a}relation to that in CMIP5; a narrowerA _{a}corresponds to less HCF reduction, likely due to cloud radiative effects. A narrowing ofA _{a}corresponds to a regime shift toward stronger precipitation in both experiments. We infer that model differences in deep convection parameterization likely play a greater role than differing cloud physics in determining the diverse responses ofA _{a}and HCF to warming in CMIP5. 
An influence diagram is a graphical model of a Bayesian decision problem that is solved by finding a strategy that maximizes expected utility. When an influence diagram is solved by variable elimination or a related dynamic programming algorithm, it is traditional to represent a strategy as a sequence of policies, one for each decision variable, where a policy maps the relevant history for a decision to an action. We propose an alternative representation of a strategy as a graph, called a strategy graph, and show how to modify a variable elimination algorithm so that it constructs a strategy graph. We consider both a classic variable elimination algorithm for influence diagrams and a recent extension of this algorithm that has more relaxed constraints on elimination order that allow improved performance. We consider the advantages of representing a strategy as a graph and, in particular, how to simplify a strategy graph so that it is easier to interpret and analyze.more » « less

null (Ed.)We consider the construction of a polygon P with n vertices whose turning angles at the vertices are given by a sequence A=(α0,…,αn−1) , αi∈(−π,π) , for i∈{0,…,n−1} . The problem of realizing A by a polygon can be seen as that of constructing a straightline drawing of a graph with prescribed angles at vertices, and hence, it is a special case of the well studied problem of constructing an angle graph. In 2D, we characterize sequences A for which every generic polygon P⊂R2 realizing A has at least c crossings, for every c∈N , and describe an efficient algorithm that constructs, for a given sequence A, a generic polygon P⊂R2 that realizes A with the minimum number of crossings. In 3D, we describe an efficient algorithm that tests whether a given sequence A can be realized by a (not necessarily generic) polygon P⊂R3 , and for every realizable sequence the algorithm finds a realization.more » « less

We consider the construction of a polygon P with n vertices whose turning angles at the vertices are given by a sequence A = (α0 , . . . , αn−1 ), αi ∈ (−π,π), for i ∈ {0,...,n − 1}. The problem of realizing A by a polygon can be seen as that of constructing a straightline drawing of a graph with prescribed angles at vertices, and hence, it is a special case of the well studied problem of constructing an angle graph. In 2D, we characterize sequences A for which every generic polygon P ⊂ R2 realizing A has at least c crossings, and describe an efficient algorithm that constructs, for a given sequence A, a generic polygon P ⊂ R2 that realizes A with the minimum number of crossings. In 3D, we describe an efficient algorithm that tests whether a given sequence A can be realized by a (not necessarily generic) polygon P ⊂ R3, and for every realizable sequence finds a realization.more » « less