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An admissible self-financing portfolio superhedges a contingent claim when its terminal wealth is at least the claim payoff almost surely. The least permitted initial cost is the superhedging price. Exact claim replication requires equality.
A nonsingular holomorphic function of chiral superfields remains chiral because . Dependence on conjugate superfields generally destroys the constraint. This explains the holomorphic form of a superpotential.
A chiral spinor superfield carries a Spinor representation of the Lorentz group index and satisfies . The chiral field-strength superfield is a fermionic example: its lowest component is the Weyl spinor gaugino. Its spinor index distinguishes it from a scalar chiral superfield, even though both obey the same chirality constraint.
For and , apply the Chernoff bound to using . Optimization at proves the displayed estimate. It remains useful when the additive lower threshold in the chi-squared concentration inequality is nonpositive. For , choosing gives a tail at most because .
For a vertical eddy diffusivity , mixing over a length takes . The condition that transport outrun a chemical relaxation time therefore gives . Taking instead of one atmospheric scale height changes this estimate by a factor of one hundred. The atmospheric scale height itself depends on both temperature and gravity.
For a nondecreasing real function on an interval and nonnegative measurable function ,The pushforward of is dominated by Lebesgue measure: on each interval, the integral of the derivative is at most the increase of . This handles jumps and singular parts without a smoothness assumption.
A hyperbolic change of integration variable uses identities such as to simplify square roots. For an open matter-dominated Friedmann solution, makes the cosmic-time integral elementary.
For a ring homomorphism , an -module becomes an -module by the displayed action. Its underlying abelian group stays the same.
For a ring homomorphism , tensoring an -module with the -bimodule gives an -module. This functor is left adjoint to restriction of scalars.
For a ring homomorphism and a left -module , the left -action on is . This construction is right adjoint to restriction of scalars.
For a Radon-Nikodym derivative with , its conditional-expectation process is a nonnegative martingale. If almost surely, it defines an equivalent probability measure. A stochastic exponential satisfying the Novikov condition is a common positive density process.
A chain map inducing isomorphisms on all homology groups. The same definition applies to a map of cochain complexes using cohomology groups. A quasi-isomorphism need not be an isomorphism in each degree.
For a CW complex, applying to its integral cellular chain complex gives a cochain complex. The differential is precomposition with the cellular boundary. For finitely many cells it is represented by the transpose of the boundary matrix, reduced in the coefficient group when appropriate.
For a CW complex, the homology of its cellular chain complex naturally equals its singular homology. The relative groups of successive skeleton pairs, free on the cells in the corresponding dimension and zero in other dimensions, yield this identification through their connecting maps.
The inverse autoregressive polynomial is an analytic function on a disk larger than the unit disk. The Cauchy estimate gives exponentially decaying coefficients of its infinite moving-average representation. Summing their products gives for some .
Mass inflation is the growth of the interior effective mass and curvature near an inner Cauchy horizon under suitably perturbed black-hole data. It can obstruct regular spacetime extensions that exist for an exact Reissner-Nordstrom spacetime. The extension regularity matters: curvature blowup alone does not rule out every continuous-metric extension.
Near a nondegenerate outgoing inner Cauchy horizon, a regular null coordinate behaves as , where is the magnitude of the inner-horizon surface gravity. Transforming stress components gives . Even a small power-law tail can therefore cause an unbounded locally measured flux, motivating mass inflation.
For , , differentiation gives the minimum at the displayed diameter. Equal strength and gravity contributions occur instead at . These two transition conventions coincide only for .
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