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For example, in detecting a dissimilar coin in three weighings ( = ), the maximum number of coins that can be analyzed is = .Note that with weighings and coins, it is not always possible to determine the nature of the last coin (whether it is heavier or lighter than the rest), but only that the other coins are all the same, implying that the last coin is the ...
A stack is called a stack in groupoids or a (2,1)-sheaf if it is also fibered in groupoids, meaning that its fibers (the inverse images of objects of C) are groupoids. Some authors use the word "stack" to refer to the more restrictive notion of a stack in groupoids.
The outer coin makes two rotations rolling once around the inner coin. The path of a single point on the edge of the moving coin is a cardioid.. The coin rotation paradox is the counter-intuitive math problem that, when one coin is rolled around the rim of another coin of equal size, the moving coin completes not one but two full rotations after going all the way around the stationary coin ...
Conversely, given a commutative diagram, it defines a poset category, where: the objects are the nodes, there is a morphism between any two objects if and only if there is a (directed) path between the nodes, with the relation that this morphism is unique (any composition of maps is defined by its domain and target: this is the commutativity ...
In Euclidean geometry, two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation , rotation and reflection .
The probability of drawing another gold coin from the same box is 0 in (a), and 1 in (b) and (c). Thus, the overall probability of drawing a gold coin in the second draw is 0 / 3 + 1 / 3 + 1 / 3 = 2 / 3 . The problem can be reframed by describing the boxes as each having one drawer on each of two sides. Each ...
By contrast, the category with a single object and a single morphism is not equivalent to the category with two objects and only two identity morphisms. The two objects in are not isomorphic in that there are no morphisms between them. Thus any functor from to will not be essentially surjective. Consider a category with one object , and two ...
A gerbe on a topological space [1]: 318 is a stack of groupoids over that is locally non-empty (each point has an open neighbourhood over which the section category of the gerbe is not empty) and transitive (for any two objects and of () for any open set , there is an open covering = {} of such that the restrictions of and to each are connected by at least one morphism).