A vector field with a bilinear map into itself, which we can also call a "vector product".
Note that the vector product does not have to be neither associative nor commutative.
Examples: en.wikipedia.org/w/index.php?title=Algebra_over_a_field&oldid=1035146107#Motivating_examples
- complex numbers, i.e. with complex number multiplication
- with the cross product
- quaternions, i.e. with the quaternion multiplication
archive.ph/GpyQv#selection-925.340-925.495 Reagan's OMB Director David Stockman told congress:and then:
students are "tax eaters... [and] a drain and drag on the American economy." Student aid "isn’t a proper obligation of the taxpayer,"
In fact, voters were far more likely to punish lawmakers for raising taxes. Elected officials made the political calculus that it was safer politically to divert existing funds from discretionary costs to mandatory costs like health care, prisons and primary education, than raise taxes.
Located in Tennessee in the East of the United States.
The precursor organization to ORNL was called Clinton Engineer Works, where groundbreaking Manhattan Project experiments and nuclear production took place during World War II
Some key experiments carried out there include:
- 1943: X-10 Graphite Reactor: prototype Plutonium breeder reactor
- isotope separation to purify Uranium-235:
Most notable example: gallium arsenide, see also: gallium arsenide vs silicon.
An important class of semiconductors, e.g. there is a dedicated III-V lab at: École Polytechnique: www.3-5lab.fr/contactus.php
Transpose of a matrix multiplication by 
Ciro Santilli 35 Updated 2025-01-10 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-10 +Created 1970-01-01When it distributes it inverts the order of the matrix multiplication:
One of Ciro Santilli's favorites.
Adds up chemical energy and kinetic energy.
Wikipedia mentions however that the kinetic energy is often negligible, even for gases.
The sum is of interest when thinking about reactions because chemical reactions can change the number of molecules involved, and therefore the pressure.
To predict if a reaction is spontaneous or not, negative enthalpy is not enough, we must also consider entropy via Gibbs free energy.
Within the The Holy Trinity of popular Brazilian music, Caetano has the most New Age religious feel to him. He is also perhaps the most varied of the trinity however, also covering heavier topics at times.
Caetano Veloso in the 60s
. Source. /i.s3.glbimg.com/v1/AUTH_59edd422c0c84a879bd37670ae4f538a/internal_photos/bs/2021/h/y/wSwVvmSbGgBiYJ3Fmb0g/caetano.jpeg)
- www.quora.com/How-is-a-voice-transmitted-from-one-phone-to-another
- www.quora.com/How-many-wires-does-a-telephone-use/answer/Peter-Yardley-1
Basic analogue phones connected to the public exchange use two wires mainly arranged as a twisted pair to reduce noise. The voice signal is differential (the voltage in one wire equal and opposite to the other) biased above ground by 48V. Using a twisted pair reduces induced noise because the noise signal will induce an equal voltage in each wire and because the signal is transmitted as the difference the effect of the induced noise will be dramatically reduced.
Phone Intercom by Make (2014)
Source. This video illustrates will the incredible simplicity of the connection of a telephone system. Compare that to the relative complexity of wireless communication, which requires modulation.The complex number analogue of a symmetric bilinear form.
The prototypical example of it is the complex dot product.
Note that this form is neither strictly symmetric, it satisfies:where the over bar indicates the complex conjugate, nor is it linear for complex scalar multiplication on the second argument.
Bibliography:
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