Mathstack

Answers are validated before they are marked, so students are not penalised for poor programming skills. Students are given feedback that refers to their specific answer and mistake, mathstack, as if marked by mathstack. STACK can generate random questions mathstack students are shown different variants of questions, and can repeat quizzes with new variants. STACK is free to use and is developed by educators like yourself.

In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory , and to construct fine moduli stacks when fine moduli spaces do not exist. Descent theory is concerned with generalisations of situations where isomorphic , compatible geometrical objects such as vector bundles on topological spaces can be "glued together" within a restriction of the topological basis. In a more general set-up the restrictions are replaced with pullbacks ; fibred categories then make a good framework to discuss the possibility of such gluing. The intuitive meaning of a stack is that it is a fibred category such that "all possible gluings work". The specification of gluings requires a definition of coverings with regard to which the gluings can be considered. It turns out that the general language for describing these coverings is that of a Grothendieck topology.

Mathstack

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They are mathstack higher stacks, mathstack. The Grothendieck topology should be strong enough so that the stack is locally affine in this topology: schemes are locally affine in the Zariski mathstack so this is a good choice for schemes as Serre discovered, algebraic spaces mathstack Deligne—Mumford stacks are locally affine in the etale topology so one usually uses the etale topology for these, while algebraic stacks are locally affine in the smooth topology so one can use the smooth topology in this case, mathstack. In particular, the action sends a tuple, mathstack.

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Mathstack

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Specific feedback Students are given feedback that refers to their specific answer and mistake, as if marked by hand. MR For example, take a projective morphism. There are several inequivalent ways of doing this. Tools Tools. On an algebraic stack one can construct a category of quasi-coherent sheaves similar to the category of quasi-coherent sheaves over a scheme. In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Grothendieck's letter to Serre, Nov 5. Constructions of moduli spaces often proceed by first constructing a larger space parametrizing the objects in question, and then quotienting by group action to account for objects with automorphisms which have been overcounted. Descent theory is concerned with generalisations of situations where isomorphic , compatible geometrical objects such as vector bundles on topological spaces can be "glued together" within a restriction of the topological basis. The descent datum is called effective if the elements x i are essentially the pullbacks of an element x with image V. STACK is used by universities, commercial partners, developers and more, in over 15 countries. The concept of stacks has its origin in the definition of effective descent data in Grothendieck Read Edit View history.

CORDIC and closely related methods known as pseudo-multiplication and pseudo-division or factor combining are commonly used when no hardware multiplier is available e. As such, they all belong to the class of shift-and-add algorithms.

STACK can generate random questions so students are shown different variants of questions, and can repeat quizzes with new variants. The concept of stacks has its origin in the definition of effective descent data in Grothendieck Hidden categories: Articles with short description Short description is different from Wikidata All articles with unsourced statements Articles with unsourced statements from June CS1 maint: location missing publisher. The fiber product of stacks is defined using the usual universal property , and changing the requirement that diagrams commute to the requirement that they 2-commute. An example of a non-affine stack is given by the half-line with two stacky origins. In the same way, moduli spaces of curves, vector bundles, or other geometric objects are often best defined as stacks instead of schemes. On an algebraic stack one can construct a category of quasi-coherent sheaves similar to the category of quasi-coherent sheaves over a scheme. The category c is called a stack over the category C with a Grothendieck topology if it is a prestack over C and every descent datum is effective. Tools Tools. Read Edit View history. These moduli spaces are denoted [3]. There are some minor set theoretical problems with the usual foundation of the theory of stacks, because stacks are often defined as certain functors to the category of sets and are therefore not sets. MR