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CRISPR (/ˈkrɪspər/) (an acronym for clustered regularly interspaced short palindromic repeats) is a family of DNA sequences found in the genomes of prokaryotic organisms such as bacteria and archaea.[2] These sequences are derived from DNA fragments of bacteriophages that had previously infected the prokaryote. They are used to detect and ...
- CRISPR
- Escherichia coli
the free encyclopedia that anyone can edit. William T. Stearn (16 April 1911 – 9 May 2001) was a British botanist. Born in Cambridge, he was largely self-educated. He was head librarian at the Royal Horticultural Society 's Lindley Library in London from 1933 to 1952, and then moved to the Natural History Museum where he was a scientific ...
Wikipedia[note 3] is a free content online encyclopedia written and maintained by a community of volunteers, known as Wikipedians, through open collaboration and the use of the wiki-based editing system MediaWiki. Wikipedia is the largest and most-read reference work in history.[3][4] It is consistently ranked as one of the ten most popular ...
- Characterizing Properties
- Useful Relations
- Second Derivatives
- Standard Integrals
- Taylor Series Expressions
- Infinite Products and Continued Fractions
- Comparison with Circular Functions
- Relationship to The Exponential Function
- Hyperbolic Functions For Complex Numbers
- See Also
Hyperbolic cosine
It can be shown that the area under the curve of the hyperbolic cosine (over a finite interval) is always equal to the arc lengthcorresponding to that interval:
Hyperbolic tangent
The hyperbolic tangent is the (unique) solution to the differential equation f ′ = 1 − f 2, with f (0) = 0.
The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact, Osborn's rule states that one can convert any trigonometric identity (up to but not including sinhs or implied sinhs of 4th degree) for θ {\\displaystyle \\theta } , 2 θ {\\displaystyle 2\\theta } , 3 θ {\\displaystyle 3\\theta } or θ {...
Each of the functions sinh and cosh is equal to its second derivative, that is: All functions with this property are linear combinations of sinh and cosh, in particular the exponential functions e x {\\displaystyle e^{x}} and e − x {\\displaystyle e^{-x}} .
The following integrals can be proved using hyperbolic substitution: where C is the constant of integration.
It is possible to express explicitly the Taylor series at zero (or the Laurent series, if the function is not defined at zero) of the above functions. The sum of the sinh and cosh series is the infinite series expression of the exponential function. The following series are followed by a description of a subset of their domain of convergence, where...
The following expansions are valid in the whole complex plane: 1. sinh x = x ∏ n = 1 ∞ ( 1 + x 2 n 2 π 2 ) = x 1 − x 2 2 ⋅ 3 + x 2 − 2 ⋅ 3 x 2 4 ⋅ 5 + x 2 − 4 ⋅ 5 x 2 6 ⋅ 7 + x 2 − ⋱ {\\displaystyle \\sinh x=x\\prod _{n=1}^{\\infty }\\left(1+{\\frac {x^{2}}{n^{2}\\pi ^{2}}}\\right)={\\cfrac {x}{1-{\\cfrac {x^{2}}{2\\cdot 3+x^{2}-{\\cfrac {2\\cdot 3x^{2}}{4\\cd...
The hyperbolic functions represent an expansion of trigonometry beyond the circular functions. Both types depend on an argument, either circular angle or hyperbolic angle. Since the area of a circular sector with radius r and angle u (in radians) is r2u/2, it will be equal to u when r = √2. In the diagram, such a circle is tangent to the hyperbola ...
The decomposition of the exponential function in its even and odd partsgives the identities Additionally,
Since the exponential function can be defined for any complex argument, we can also extend the definitions of the hyperbolic functions to complex arguments. The functions sinh z and cosh z are then holomorphic. Relationships to ordinary trigonometric functions are given by Euler's formulafor complex numbers: Thus, hyperbolic functions are periodic ...
Bobbi Althoff (born July 31, 1997) is an American podcaster and influencer known for her viral interviews with Drake, Lil Yachty, Offset and other celebrities. Early life Althoff grew up as the second youngest of six children in Southern California, where her father worked as a contractor. ...
Glycerol (/ˈɡlɪsərɒl/),[6] also called glycerine or glycerin, is a simple triol compound. It is a colorless, odorless, viscous liquid that is sweet-tasting and non-toxic. The glycerol backbone is found in lipids known as glycerides. It is also widely used as a sweetener in the food industry and as a humectant in pharmaceutical ...
Lady Iris Cholmondeley. Education. Stowe School. Alma mater. Open University. Sarah Rose Cholmondeley, Marchioness of Cholmondeley ( / ˈtʃʌmli / CHUM-lee; née Hanbury; born 15 March 1984) is a British peeress, former model and former political staffer. She is married to David Cholmondeley, 7th Marquess of Cholmondeley .
