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The network simplex algorithm specializes the simplex algorithm to network flows. A basis is represented by a spanning tree with nonbasic flows at their bounds; potentials give reduced costs, and entering arcs produce cycle pivots. Degeneracy in linear programming is particularly common in assignment instances.
An uncapacitated minimum-cost flow has nonnegative edge flows without finite upper capacities. The flow balances prescribe net supply at each vertex. For the oriented incidence matrix convention with tail and head , vertex network dual potentials have network reduced costs . Nonnegative network reduced costs and complementary slackness certify a feasible optimum. A feasible negative-cost directed cycle makes the problem unbounded below.
A flow balance prescribes net outgoing minus incoming flow at a vertex. Positive is supply and negative is demand; zero is ordinary flow conservation. Summing all balances requires total supply to equal total demand. In a directed graph, these equations are represented by the oriented incidence matrix.
Sufficient statistic for a symmetric uniform sample by 
Codex 0 Created 2026-10-06 Updated 2026-11-05
Codex 0 Created 2026-10-06 Updated 2026-11-05For independent samples uniform on with , the absolute maximum is a minimal sufficient statistic, since the likelihood is . The pair of sample minimum and maximum is a sufficient statistic but is not minimal: simultaneous sign reversal preserves and generally changes that pair.
For a metric space or pseudometric space , this is the minimum number of balls of radius , with centers in , needed to cover . The displayed grid bound uses closed balls. For open balls, replacing the radius by a fixed constant factor gives the same entropy estimates. For and , a uniform grid gives . These polynomial growth bounds make the Dudley entropy integral finite.
An -net of a metric space is a subset such that every has for some . Its least possible finite size is the metric covering number with closed balls. A maximal set with pairwise distances greater than is an -net.
In a metric space , the closed ball centred at of radius is . It is a closed set, since the distance to is a continuous function. Its boundary need not be the whole corresponding distance sphere in an arbitrary metric space.
For with , the saddle point descent is the real axis and ascent is the imaginary axis. Let the endpoint have . Its outward steepest descent contour satisfieswith the continuous square-root branch from . The imaginary part of stays fixed and its real part increases, so the exponential decays. The path tends to positive or negative real infinity according to the sign of .
Since , the endpoint Laplace integral isReversing the contour changes the sign. A purely imaginary nonzero is a special case: reaches zero, so the path passes through the saddle and cannot be treated as one regular endpoint path without specifying how it continues.
For an integrand , a contour along which is constant and decreases away from the contributing endpoint or saddle. This suppresses oscillation while localizing the large-parameter integral.
A common-effect synthesis treats study estimates as estimates of one shared effect. For independent approximately unbiased estimates with variances , use with . This model does not accommodate additional between-study heterogeneity.
Systematic displacement of an individual study estimate caused by design, conduct, analysis or selective reporting. Combining estimates can reduce random error without reducing this bias. Bias-adjusted meta-analysis and analyses restricted or stratified by relevant bias domains can assess its impact.
For independent subgroup estimates with variances , the null contrast has Wald statistic . If evidence overlaps, include minus twice the covariance in the contrast variance. A nonsignificant comparison is not proof of equivalence.
The reflection-symmetric Newtonian gravitational potential is harmonic off the plane. Its normal-derivative jump is , so the Poisson equation for Newtonian gravity gives surface density of a disk and volume mass density . The midplane circular speed is constant. The scale-free model has infinite total mass and does not use a finite-mass boundary condition at infinity.
The planar Mestel disk has , giving independent centered Gaussian distributions for with velocity dispersions . Its streaming velocity is zero. A maximally prograde stellar distribution instead has mean and azimuthal variance , while all second raw moments remain unchanged.
Social choice theory studies how individual preferences determine a collective decision. A social choice function selects one alternative from a preference profile; strategyproofness asks whether agents can benefit by misrepresenting their preferences.
A mechanism or social choice function is strategyproof if truthful reporting is a weakly dominant strategy: for every true type and every collection of other reports, no misreport yields strictly higher utility or a strictly preferred outcome. This is stronger than merely requiring truthful reports to form a Nash equilibrium.
An auction allocates goods using submitted messages and a specified payment rule. Private-value auctions model each bidder's valuation as its own private information. Equilibrium bids depend on the allocation and payment rules, not only on valuations.
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