The Taylor series expansion is the most direct definition of the expontial as it obviously satisfies the exponential function differential equation:
- the first constant term dies
- each other term gets converted to the one before
- because we have infinite many terms, we get what we started with!
3ad6677303fb6f700a4f2f977fe86e5324e0ddb0d3b33a649e513d7e88904e85 Updated 2025-01-10 +Created 1970-01-01
This contains various outputs that seem trivially spendable in a made up of two non-zero constants, e.g.:Or are we missing something? The values are quite small and wouldn't be worth it the miner fees most likely. But is there a fundamental reason why this couldn't be spent by a non-standard miner?
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},Compared to Waring's problem, this is potentially much harder, as we can go infinitely negative in our attempts, there isn't a bound on how many tries we can have for each number.
In other words, it is unlikely to have a Conjecture reduction to a halting problem.
3 as the sum of the 3 cubes by Numberphile (2019)
Source. Static case of Maxwell's law for electricity only.
Is implied by Gauss' law if Maxwell's equations: physics.stackexchange.com/questions/44418/are-the-maxwells-equations-enough-to-derive-the-law-of-coulomb
The "static" part is important: if this law were true for moving charges, we would be able to transmit information instantly at infinite distances. This is basically where the idea of field comes in.
It should be illegal to give someone money to advertise your product.
We should just use tags and upvote-based algorithms.
Advertisement is a form of brainwashing.
TODO find Sergey Brin quote bout advertising being a form of perverting search engine ranking neutrality. In other words, there are neutral metrics to what people find good, like links and upvotes. And then advertising is a way to pervert that to someone's profit.
Sometimes Ciro Santilli thinks advertising should be on-subscription only, but that is too restrictive, e.g. imagine the local business owner who wears a t-shirt of their business.
One of the representations of the Lorentz group that show up in the Representation theory of the Lorentz group.
- 2024 www.uktech.news/foodtech/alternative-meat-research-centre-20240829A research centre examining the implementation of alternatives to meat, backed by £38m, will be launched by the University of Leeds.The virtual research centre will look into lab-grown meat as well as plant and fungus-based meat alternatives to determine the health, sustainability and feasibility of moving away from traditional meat.
- 2024 techcrunch.com/2024/08/04/even-after-1-6b-in-vc-money-the-lab-grown-meat-industry-is-facing-massive-issues/ "Even after $1.6B in VC money, the lab-grown meat industry is facing 'massive' issues"
- www.reuters.com/markets/commodities/hoxton-farms-raises-22-million-cultivated-animal-fat-2022-10-20/ Hoxton Farms, cultured animal fat
Linear combination of a Dirichlet boundary condition and Neumann boundary condition at each point of the boundary.
Examples:
- heat equation when metal plaque is immersed in a large external environment of fixed temperature.In this case, the normal derivative at the boundary is proportional to the difference between the temperature of the boundary and the fixed temperature of the external environment.The result as time tends to infinity is that the temperature of the plaque tends to that of the environment.Shown a solved example in the FreeFem tutorial: doc.freefem.org/tutorials/thermalConduction.html (github.com/FreeFem/FreeFem-doc/blob/1d5996d8b891fd553fd318321249c2c30f693fc3/source/tutorials/thermalConduction.rst)
Related concepts:
A relativistic version of the Schrödinger equation.
Correctly describes spin 0 particles.
The most memorable version of the equation can be written as shown at Section "Klein-Gordon equation in Einstein notation" with Einstein notation and Planck units:
Has some issues which are solved by the Dirac equation:
- it has a second time derivative of the wave function. Therefore, to solve it we must specify not only the initial value of the wave equation, but also the derivative of the wave equation,As mentioned at Advanced quantum mechanics by Freeman Dyson (1951) and further clarified at: physics.stackexchange.com/questions/340023/cant-the-negative-probabilities-of-klein-gordon-equation-be-avoided, this would lead to negative probabilities.
- the modulus of the wave function is not constant and therefore not always one, and therefore cannot be interpreted as a probability density anymore
- since we are working with the square of the energy, we have both positive and negative value solutions. This is also a features of the Dirac equation however.
Bibliography:
- Video "Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)" at youtu.be/OCuaBmAzqek?t=600
- An Introduction to QED and QCD by Jeff Forshaw (1997) 1.2 "Relativistic Wave Equations" and 1.4 "The Klein Gordon Equation" gives some key ideas
- 2011 PHYS 485 lecture videos by Roger Moore from the University of Alberta at around 7:30
- www.youtube.com/watch?v=WqoIW85xwoU&list=PL54DF0652B30D99A4&index=65 "L2. The Klein-Gordon Equation" by doctorphys
- sites.ualberta.ca/~gingrich/courses/phys512/node21.html from Advanced quantum mechanics II by Douglas Gingrich (2004)
There are unlisted articles, also show them or only show them.