Mathstack

Answers are validated before they are marked, mathstack students are not penalised for poor programming skills. Students are given feedback that refers to their specific answer and mistake, as buprenorphine pronunciation marked by hand, mathstack. STACK can generate random questions so students are shown different variants of questions, mathstack, and can repeat quizzes with new variants.

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.

Mathstack

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In mathematics a stack or 2-sheaf is, roughly speaking, mathstack, a sheaf that takes values in categories mathstack than sets. Annals of Mathematics.

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The art of speaking to AI chatbots is continuing to frustrate and baffle people. A study attempting to fine-tune prompts fed into a chatbot model found that, in one instance, asking it to speak as if it were on Star Trek dramatically improved its ability to solve grade-school-level math problems. The study, first reported by New Scientist , was published on February 9 on arXiv , a server where scientists can share preliminary findings before they have been validated by careful scrutiny from peers. Instead, they were trying to figure out if they could capitalize on the "positive thinking" trend. People attempting to get the best results out of chatbots have noticed the output quality depends on what you ask them to do , and it's really not clear why. This would suggest it's not only what you ask the AI model to do, but how you ask it to act while doing it that influences the quality of the output.

Mathstack

Stuck on a tricky math problem? Google's newest app will use AI to help you solve it. Two years ago, Google announced the purchase of a math problem-solving app called Photomath. And earlier this week, that app was officially brought under the company's app umbrella.

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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. Contributions are welcomed and encouraged. 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. This usually seems to lead to an equivalent category of quasi-coherent sheaves, but is easier to use: for example it is easier to compare with the etale topology on algebraic spaces. In this paper they also introduced Deligne—Mumford stacks , which they called algebraic stacks, though the term "algebraic stack" now usually refers to the more general Artin stacks introduced by Artin Some authors require this as a property of stacks, rather than of prestacks. The Grothendieck topology should be strong enough so that the stack is locally affine in this topology: schemes are locally affine in the Zariski topology so this is a good choice for schemes as Serre discovered, algebraic spaces and 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. Roughly speaking, Deligne—Mumford stacks can be thought of as algebraic stacks whose objects have no infinitesimal automorphisms. For example, if a few points have non-trivial stabilisers, then the categorical quotient will not exist among schemes, but it will exist as a stack. Instead of using the smooth topology for algebraic stacks one often uses a modification of it called the Lis-Et topology short for Lisse-Etale: lisse is the French term for smooth , which has the same open sets as the smooth topology but the open covers are given by etale rather than smooth maps.

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.

More generally one can define the notion of an n -sheaf or n —1 stack, which is roughly a sort of sheaf taking values in n —1 categories. Differentiable stacks and topological stacks are defined in a way similar to algebraic stacks, except that the underlying category of affine schemes is replaced by the category of smooth manifolds or topological spaces. Annals of Mathematics. This is because these are the points where the cover ramifies. For general algebraic stacks the etale topology does not have enough open sets: for example, if G is a smooth connected group then the only etale covers of the classifying stack BG are unions of copies of BG, which are not enough to give the right theory of quasicoherent sheaves. There are several ways to deal with this problem:. The descent datum is called effective if the elements x i are essentially the pullbacks of an element x with image V. Toggle limited content width. Main article: Quasi-coherent sheaf on an algebraic stack. Instead of using the smooth topology for algebraic stacks one often uses a modification of it called the Lis-Et topology short for Lisse-Etale: lisse is the French term for smooth , which has the same open sets as the smooth topology but the open covers are given by etale rather than smooth maps. 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 Lis-Et topology has a subtle technical problem: a morphism between stacks does not in general give a morphism between the corresponding topoi. In the same way, moduli spaces of curves, vector bundles, or other geometric objects are often best defined as stacks instead of schemes. Training and Events. 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.