An exchange matrix @math dominates an exchange matrix @math if the signs of corresponding entries weakly agree, with the entry of @math always having weakly greater absolute value. When @math dominates @math , interesting things happen in many cases (but not always): the identity map between the associated mutation-linear structures is often mutation-linear; the mutation fan for @math often refines the mutation fan for @math ; the scattering (diagram) fan for @math often refines the scattering fan for @math ; and there is often an injective homomorphism from the principal-coefficients cluster algebra for @math to the principal-coefficients cluster algebra for @math , preserving @math -vectors and sending the set of cluster variables for @math (or an analogous larger set) into the set of cluster variables for @math (or an analogous larger set). The scope of the description "often" is not the same in all four contexts and is not settled in any of them. In this paper, we prove theorems that provide examples of these dominance phenomena.
Several other lines of research touch tangentially on the ideas presented in this paper. The notion of dominance appears in @cite_14 @cite_10 as an integral part of the definition of . In @cite_14 @cite_10 , the condition is broader in the sense that it allows passing to submatrices and or making a global sign change. Also, the condition in @cite_14 @cite_10 is applied to extended exchange matrices, not just to exchange matrices. Some surface-cutting constructions similar to resection have appeared in @cite_19 @cite_10 , but resection is different because it aims to do something different, namely to construct dominance relations without restricting to submatrices and without freezing'' variables.
@cite_19 @cite_14 @cite_10
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Top-k dominating queries combine the natural idea of selecting the k best items with a comprehensive “goodness” criterion based on dominance. A point p1 dominates p2 if p1 is as good as p2 in all attributes and is strictly better in at least one. Existing works address the problem in settings where data objects are multidimensional points. However, there are domains where we only have access to the distance between two objects. In cases like these, attributes reflect distances from a set of input objects and are dynamically generated as the input objects change. Consequently, prior works from the literature cannot be applied, despite the fact that the dominance relation is still meaningful and valid. For this reason, in this work, we present the first study for processing top-k dominating queries over distance-based dynamic attribute vectors, defined over a metric space. We propose four progressive algorithms that utilize the properties of the underlying metric space to efficiently solve the problem and present an extensive, comparative evaluation on both synthetic and real-world datasets. In this paper, we study the dominance relation under a stochastic setting. Let @math be a set of @math colored stochastic points in @math , each of which is associated with an existence probability. We investigate the problem of computing the probability that a realization of @math contains inter-color dominances, which we call the (CSD) problem. We propose the first algorithm to solve the CSD problem for @math in @math time. On the other hand, we prove that, for @math , even the CSD problem with a restricted color pattern is #P-hard. In addition, even if the existence probabilities are restricted to be @math , the problem remains #P-hard for @math . A simple FPRAS is then provided to approximate the desired probability in any dimension. We also study a variant of the CSD problem in which the dominance relation is considered with respect to not only the standard basis but any orthogonal basis of @math . Specifically, this variant, which we call the free-basis colored stochastic dominance (FBCSD) problem, considers the probability that a realization of @math contains inter-color dominances with respect to any orthogonal basis of @math . We show that the CSD problem is polynomial-time reducible to the FBCSD problem in the same dimension, which proves the #P-hardness of the latter for @math . Conversely, we reduce the FBCSD problem in @math to the CSD problem in @math , by which an @math time algorithm for the former is obtained. Our objective is to investigate the implications of the “replacement principle” for the fair allocation of an infinitely divisible commodity among agents with single-peaked preferences. The principle says that when one of the components of the data entering the description of the problem to be solved changes, all of the relevant agents should be affected in the same direction: they all gain or they all lose. We apply it to situations in which the preferences of one of the agents may change, under the name ofwelfare-domination under preference-replacement. We show that there is no selection from the no-envy and Pareto solution satisfying it. Then, we weaken it by limiting its application to situations in which the change is not so disruptive that it turns the economy from one in which there is “too little” of the commodity to one in which there is “too much,” or conversely. We show that there is only one selection from the no-envy and Pareto solution satisfying this property andreplication-invariance. It is the “uniform rule,” a solution that has played a central role in previous analyses of the problem.Journal of Economic LiteratureClassification Numbers: D51, D63, D70. Let G be a graph and @t:V(G)->N be an assignment of thresholds to the vertices of G. A subset of vertices D is said to be dynamic monopoly (or simply dynamo) if the vertices of G can be partitioned into subsets D"0,D"1,...,D"k such that D"0=D and for any i=1,...,k-1 each vertex v in D"i"+"1 has at least t(v) neighbors in D"[email protected][email protected]?D"i. Dynamic monopolies are in fact modeling the irreversible spread of influence such as disease or belief in social networks. We denote the smallest size of any dynamic monopoly of G, with a given threshold assignment, by dyn(G). In this paper, we first define the concept of a resistant subgraph and show its relationship with dynamic monopolies. Then we obtain some lower and upper bounds for the smallest size of dynamic monopolies in graphs with different types of thresholds. Next we introduce dynamo-unbounded families of graphs and prove some related results. We also define the concept of a homogeneous society that is a graph with probabilistic thresholds satisfying some conditions and obtain a bound for the smallest size of its dynamos. Finally, we consider dynamic monopoly of line graphs and obtain some bounds for their sizes and determine the exact values in some special cases.