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For and , the proposal minimizes the variance of the ordinary unbiased importance sampling estimator. The Cauchy-Schwarz inequality gives , so the minimum variance for independent samples is . The proposal need only cover the nonzero integrand.
If , the importance weight is bounded by , and . A finite target second moment therefore guarantees finite variance and the iid central limit theorem for the unnormalized importance-sampling average. Its proposal-budget variance is .
Rao-Blackwell identity for a fixed-denominator rejection estimator by 
Codex 0 Created 2026-10-06 Updated 2026-10-14
Codex 0 Created 2026-10-06 Updated 2026-10-14For rejection indicators with conditional acceptance , the conditional expectation of given all proposals is the unnormalized importance estimator. The Rao-Blackwell theorem reduces variance relative to that fixed-denominator estimator. It gives no such automatic comparison with the accepted-sample mean, whose denominator is the random count.
If a regression response is observed but covariate is sometimes missing, likelihood inference requires a joint model: for example . An incomplete record contributes to the observed-data likelihood. Missing at random with distinct parameters allows the missingness mechanism to be omitted; it does not remove the need to integrate over .
The mean free path is the average distance travelled between collisions. For dilute targets with number density and effective collision cross-section , ; identical-particle conventions introduce factors such as .
For a negative fundamental discriminant , primitive positive definite integral binary quadratic forms of discriminant , modulo proper equivalence, correspond bijectively to the ideal class group of . The displayed ideal gives one conventional choice of this correspondence. Reduced positive definite binary quadratic forms give unique class representatives with the usual boundary sign convention, making class-number computations finite.
The ideal-form correspondence for imaginary quadratic fields and the reduction bound give exactly the forms , and . Thus the class number is three and the ideal class group is cyclic of order three.
The narrow ideal class group of a number field is the group of nonzero fractional ideals modulo principal fractional ideals generated by totally positive elements. It is the ray class group for the modulus consisting of all real Archimedean places and trivial finite part. The forgetful map to the ordinary ideal class group is surjective, with kernel under the unit signature map.
A hypergraph is -uniform if each hyperedge has exactly vertices. Ordinary simple graphs are 2-uniform hypergraphs. Its average hypergraph vertex degree is .
The degree of a vertex in a hypergraph is the number of hyperedges containing . For a uniform hypergraph, summing degrees counts each hyperedge once for each of its vertices, so . In a multihypergraph, incidences are counted with multiplicity.
A subset of the vertices of a hypergraph is independent when no hyperedge lies wholly inside . This generalizes an independent set in an ordinary graph. If a hypergraph has ordinary graph edges as its vertices and forbidden copies as its hyperedges, these independent sets encode graphs without those forbidden copies.
A multihypergraph has a multiset of hyperedges, allowing the same subset of vertices to appear more than once. Its hypergraph vertex degrees and hypergraph codegrees count multiplicity. Such multiplicity is important when several original hyperedges produce the same link in a hypergraph container theorem.
Extremal hypergraph theory studies the largest or smallest possible size of a hypergraph under prescribed forbidden-subhypergraph or covering conditions. It generalizes extremal graph theory. The Turán density describes the asymptotic maximum edge proportion for a fixed uniform forbidden hypergraph.
A polychromatic coloring of a hypergraph assigns colors to vertices so that every hyperedge contains all colors. It is stronger than merely requiring each hyperedge to be nonmonochromatic. Independently random colors and the Lovász local lemma can construct such colorings when edges are sufficiently large relative to their overlap degrees.
Three corresponding side lengths determine a hyperbolic triangle up to an isometry of the hyperbolic plane. First match an ordered pair of vertices by an isometry. The third vertex lies at the intersection of two prescribed hyperbolic distance circles. There are at most two choices, interchanged by reflection in the geodesic through the matched pair; either choice can therefore be matched.
Two corresponding side lengths and an equal nonincluded angle need not determine a hyperbolic triangle. Set , , . The hyperbolic law of cosines has two distinct positive solutions . The two triangles share but have different third sides. An included angle instead determines the third side by the cosine rule.
The unmarked length spectrum of a compact hyperbolic surface is the multiset of lengths of its closed geodesics, with a consistent convention for primitive curves and orientation. The marked version labels lengths by free homotopy classes. The Selberg trace formula relates the unmarked multiset to the Laplace-Beltrami operator spectrum.
A hyperbolic disc consists of points within a specified hyperbolic distance of its centre. In the Poincare disc model, the disc centred at zero with radius has Euclidean radius ; an isometry transports this description to any other centre.
A non-Hopfian group admits a surjective endomorphism which is not injective. The Baumslag-Solitar group has such an endomorphism , : its image contains and , while the nonidentity commutator lies in its kernel by Britton's lemma.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.
Figure 3. Visual Studio Code extension installation.
Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension. 
- Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact