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For one internal species, the grand canonical partition function of a classical ideal gas satisfies , with quantum gas fugacity and thermal de Broglie wavelength . Differentiating the grand potential gives , hence . An internal partition factor must be included consistently if internal states are available.
The chemical potential assigned to a specified standard state. With dimensionless thermodynamic activity , the per-particle convention is ; a molar convention uses . The standard state changes both and the activity definition without changing observable equilibrium.
Although the energy in velocities is , the Hamiltonian in canonical variables isThe field therefore enters Hamilton's equations through the relation between canonical and mechanical momentum, even though a magnetic field does no work.
For negligible natural radius and release from rest at , . The effective radial potential is a sum of and terms, whose two turning radii have squared product . HenceThese formulas assume . At exactly zero field the ideal ring collapses to zero radius instead of making a regular positive-radius turning point.
For a real random variable with characteristic function , the Gil-Pelaez inversion formula recovers the cumulative distribution function at every continuity point:At a jump the same expression gives . The integral is interpreted as the indicated improper limit, not assumed absolutely convergent. The sign follows from the convention: its imaginary part is . This is useful when a characteristic function is available algebraically but a density is not.[ref]
Note on the inversion theorem by J. Gil-Pelaez. pp. 481–482
In a chair conformation of cyclohexane, axial bonds point approximately parallel to the ring's threefold axis, alternating up and down around the ring. Equatorial bonds point outward around its perimeter, also alternating in their up/down relationship to the ring.
A ring flip exchanges axial and equatorial positions but preserves a substituent's up or down orientation. Thus axial does not mean up, and equatorial does not mean down. In methylcyclohexane the equatorial methyl conformation is favored because an axial methyl group has unfavorable close contacts with axial substituents on carbons three and five.[ref]
The flow and head at which a pump's characteristic curve intersects the hydraulic system's required-head curve. Increasing system resistance at unchanged speed normally reduces flow on a conventional falling pump curve.[ref]
Thermodynamics, Heat Transfer, and Fluid Flow, Volume 3 by the United States Department of Energy. Module HT-03, pp.51–53, System Operating Point
For a uniform square plate of side hinged along an edge, and . An impulse normal to the plate, applied a distance below the centre, gives and signed hinge impulse . Therefore the hinge impulse vanishes at below the centre, a distance from the hinge. The sign of identifies the reaction direction; its magnitude is .
An element of a group is central when it commutes with every element of that group. The set of all such elements is the center of a group. Equivalently, its conjugacy class is a singleton: for every is exactly the commutation condition. This lets centrality identify one-element conjugacy classes without enumerating every conjugation.
A cellulose microfibril is an organized assembly of cellulose chains. Surface accessibility and chain packing help determine which glycosidic bonds enzymes can attack.
Specification driven by information inherited within a cell, such as asymmetrically partitioned maternal material. It can coexist with external inputs later in development.
Specification that depends on a cell's interactions with its surroundings. Neighbour exchange and controlled signal perturbation can test it causally.
The ability of cells to move among epithelial, mesenchymal and intermediate regulatory states. The benefit of a state can vary between dissemination, persistence and metastatic outgrowth.
A supplied precursor that must be chemically converted into the catalytically active form. Activation stoichiometry need not equal the reagent consumption of each subsequent catalytic turnover.
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