KJW

This looks to me like an argument from incredulity.

From what I have read (I haven't watched any of the videos), he did identify himself, and that claims that he didn't identify himself or that he was threatening were blatant lies. It was also suggested that if they're willing to lie about events that took place in a room full of reporters, then what wouldn't they lie about?

It took me a while to figure out that all one needs to do is paste the URL of the image directly into the post. It defaults to imbedding the image but provides the option to display the link instead. One can then manipulate the size of the image. There is also a "Media Options" button at the top-right of the image.

It would appear that you believe reality has a non-trivial topology. I'm actually quite ambivalent about whether or not spacetime has a non-trivial topology. Anyway, I do consider the question of whether two different descriptions are describing the same reality to be a fundamental question. However, one shouldn't be blasé about what characterises equivalent descriptions. It is something that requires careful consideration. I do believe that reality does have more structure than what is provided by topology. Yeah, and it is the same two experiments also with the old standard, as the old one standard of length is still based on a derivative of the speed of light and therefore has no potential of deviation. This is my what bothers me. ... Your line of thought only works if the alternative standard of length used in an experiments can be considered sufficiently independent of c. I was actually addressing the concern that I was measuring the speed of light using a standard of length based on the speed of light. Anyway, there was a time when the standard of length was based on a platinum-iridium bar. How is the length of a platinum-iridium bar based on the speed of light? Actually, you did suggest that because atoms are based on electromagnetism that their size is based on the speed of light. But you didn't explain precisely how the electromagnetism of the atom leads to the size of the atom being based on the speed of light. On the other hand, given the fundamental connection between space and time that is manifested by the speed of light, it may be that a standard of length that is not based on the speed of light is impossible. That is, you may have a problem with a standard of length being based on the speed of light, but if it is impossible for a standard of length to be independent of the speed of light, then this becomes problematic to your idea that the speed of light can vary. and this idea works for any wave, not just light. the acoustic metric is a perfect example of that. it shows that we can treat sound waves identical to light in a vacuum with curvature and using that special definitions of time and space we get all the familiar framework. One can't replace the speed of light in a vacuum with the speed of sound. One can't even replace the speed of light in a vacuum with the speed of light in water. In the measurement of c based on the measurement of the speed of light in both still water and moving water, the choice of using light in water was merely to provide a speed that is fast enough for the relativistic effect to be significant. In principle, one could choose the speed of any object to apply the relativistic velocity-addition formula. Although like the speed of light in a vacuum, the speed of sound in a medium is constant with respect to the speed of the source, unlike the speed of light in a vacuum, the speed of sound in a medium is not constant with respect to the speed of the observer. The speed of sound in a medium is constant relative to the medium and therefore does not depend on the speed of the source relative to the medium. But the speed of sound relative to the observer does depend on the speed of the observer relative to the medium. The important role played by the medium with regards to sound in contrast to the absence of a medium with regards to light in a vacuum manifest in the difference in the Doppler effect formula for sound and for light in a vacuum.

To complete the connection between Fourier transforms and differential operators with regards to conjugate variables that I started earlier in this topic: [math]\text{Let }F(\xi) = \displaystyle \int_{-\infty}^{\infty} f(x)\ \exp(-2\pi i\ \xi x)\ dx \ \ \ \ ;\ \ \ \ f(\pm \infty ) = 0[/math] [math]\text{Then }\displaystyle \int_{-\infty}^{\infty} \dfrac{d}{dx}f(x)\ \exp(-2\pi i\ \xi x)\ dx[/math] [math]= -\displaystyle \int_{-\infty}^{\infty} f(x)\ \dfrac{d}{dx}\exp(-2\pi i\ \xi x)\ dx[/math] [math]2\pi i\ \xi\ \displaystyle \int_{-\infty}^{\infty} f(x)\ \exp(-2\pi i\ \xi x)\ dx[/math] [math]= 2\pi i\ \xi\ F(\xi)[/math] [math]\text{Therefore }\dfrac{d}{dx} \equiv 2\pi i\ \xi[/math] [Please refresh browser window if the above LaTex doesn't render]

The Heisenberg uncertainty principle relates the standard deviations of conjugate pairs of variables. But a variable having finite standard deviation does not necessarily have compact support (eg Gaussian function). So the Heisenberg uncertainty principle doesn't really say anything about the support of conjugate pairs of variables. However, I have indicated that at least one of a conjugate pair of variables must be without compact support. How this impacts on the physics is unclear since it is reasonable to assume that physical variables are bounded in value.

"Compact support" means that the function is zero everywhere outside of some finite domain.

There is an inverse relationship between the standard deviation width of a function and the standard deviation width of its Fourier transform. The value of the product of the two standard deviations depends on the function (hence the inequality) with the minimum value (where the inequality becomes the equality) achieved by the Gaussian function whose Fourier transform is also the Gaussian function. It can be proven that the Fourier transform of a function with compact support does not have compact support. To prove this, note that any function with compact support is the product of some function with the rectangular function. Therefore, the Fourier transform of any function with compact support is the convolution of some function (with or without compact support) with the sinc function, and therefore does not have compact support. The converse is not necessarily true. For example, both the Gaussian function and its Fourier transform (also the Gaussian function) are functions without compact support.

This is a common misunderstanding of refraction. Refraction is easiest to understand in terms of classical electromagnetic waves. This can then be translated to the quantum picture provided the important aspects of the classical picture are maintained. When an electromagnetic wave passes through a medium, it exerts a force on the charges and charge dipoles of the medium. Depending on how easily the charges and charge dipoles of the medium can move in response to this force, the motion of the charges and charge dipoles of the medium creates an electromagnetic wave that combines with the original electromagnetic wave to produce a total electromagnetic wave that is delayed with respect to the original electromagnetic wave and therefore travels through the medium at a slower speed. Thus, the refractive index of the medium depends on how readily the charges and charge dipoles of the medium can respond to the passing electromagnetic wave. This depends on the frequency of the passing electromagnetic wave. Higher frequencies exert a greater force, but larger bulkier charges and charge dipoles respond more to lower frequencies. At visible frequencies, only electrons can significantly respond to the passing electromagnetic wave, and in this case, the refractive index depends on the polarisability of the electron orbitals of the medium and increases with frequency due to the increasing energy of the photons.

Thanks. +1 I have an interest in supramolecular chemistry, in particular, host-guest chemistry.

Why are you posting an image of melamine cyanurate?

Interesting - substitution, also called replacement. Of course that is the function of the klein 4-group I was illustrating. But I'm not sure I would call group theory the most basic. I was referring to the process of doing mathematics rather than any subject within mathematics. Nevertheless, as far the subject matter of mathematics is concerned, I do consider group theory to be near the foundations of mathematics, and more fundamental than arithmetic. For example, if one considers the group of the integers under addition, then multiplications are endomorphisms of this group. Compare the distributive law with the definition of a morphism. Also note that under any morphism, the identity maps to the identity. By mapping the endomorphisms to the integers, one has created a ring. That is, whereas a group has one operation, the group of integers under addition naturally admits a second operation by considering its endomorphisms (though in general, the endomorphisms of a group are categorically different to the elements of a group). As a "formalist", I regard doing mathematics as a purely mechanical process, and require mathematics that is rigorous to be performed by a machine (or by a human emulating a machine) (but not AI). Human intuition and cleverness can still be used as a guide, but in the end, all the individual steps of a derivation or proof must be indistinguishable from the product of a purely mechanical process.

Rowan Atkinson: Free Speech In particular, "the freedom to be inoffensive is no freedom at all".

As I see it, the most basic operation or activity in mathematics is substitution. To transform one statement to another statement, one performs substitutions of expressions within the initial statement with other expressions.

Useful shorthand, but not at all rigorous. I feel kinda dirty using them, like I need a shower. May I answer this question? I will wait to give @AVJolorumAV the opportunity to answer.

I don't know if this is significant or just a strange coincidence, but if you consider the series: [math]1+2+4+8+16+32+64\>+\>...[/math] as a binary number, then that binary number would be represented as an infinite string of "1"s: [math]...1111111111111111[/math] In the two's complement representation of signed binary numbers, this corresponds to [math]-1[/math].

That's why one doesn't attempt to sum the series... because the result depends on how it is done. That's why one uses a formula that gives the correct answer. In the case of the series [math]1-1+1-1+1-1+1-1\>+\>...\>[/math], this corresponds to: [math]1-1+1-1+1-1+1-1\>+\>...\\=1+x+x^2+x^3+x^4+x^5\>+\>...\>=\dfrac{1}{1-x}\\=\dfrac{1}{2}\>\>\text{for }\>\>x=-1[/math] Interestingly, it also corresponds to: [math]1-1+1-1+1-1+1-1\>+\>...\\=\eta(x) = 1-\dfrac{1}{2^x}+\dfrac{1}{3^x}-\dfrac{1}{4^x}+\dfrac{1}{5^x}-\>...\\\text{for }\>\>x=0[/math] Now lets consider the series [math]1+1+1+1+1+1+1+1\>+\>...\>[/math]. This corresponds to: [math]1+1+1+1+1+1+1+1\>+\>...\\=1+x+x^2+x^3+x^4+x^5\>+\>...\>=\dfrac{1}{1-x}\\=\>???\>\>\text{for }\>\>x=1[/math] That is, even the formula is undefined for [math]x=1[/math] But: [math]1+1+1+1+1+1+1+1\>+\>...\\=\zeta(x) = 1+\dfrac{1}{2^x}+\dfrac{1}{3^x}+\dfrac{1}{4^x}+\dfrac{1}{5^x}+\>...\>=\dfrac{\eta(x)}{1 - 2^{1-x}}\\=-\eta(0)=-\dfrac{1}{2}\>\>\text{for }\>\>x=0[/math]

It appears that the LaTex code between the "math" tags (I only use "math" tags) has to be all on the same line. This appears to be different to the original forum which iirc allowed code to be split across multiple lines (between a single pair of tags). Usually, I use a pair of tags for each line, but some code (eg matrices) requires everything to be between a single pair of tags. However, the use of "\\" does allow rendered text to be on multiple lines.

[math]\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}[/math] [math]\begin{pmatrix} a & b\\c & d \end{pmatrix}[/math]

Did you look at the entire video? Towards the end of the video, he explains how the identity: [math]1\ +\ 2\ +\ 3\ +\ 4\ +\ ...\ = -\dfrac{1}{12}[/math] arises from analytic continuation of the Riemann zeta function. If you look at the old thread I referenced, you'll see how I evaluated it, including the use of the relationship between the zeta and eta functions mentioned in the video: [math]\zeta(z) = \dfrac{1}{1 - 2^{1-z}}\>\eta(z)[/math] The important thing to note, both in the old thread I referenced, and in the simpler non-convergent series that I challenged you with, is that no attempt is made to sum the series or form the limit of a partial sum of the series. Instead, the entire series is assumed to be an entity in its own right and have a value, and by exploiting the self-similarity properties of this entity, algebraically obtain this value. In the old thread I referenced, you'll see that I derived the formula for the general infinite geometric series, and that the same technique was used for the non-convergent infinite geometric series in this thread, thus resulting in the same formula being implicitly used for the non-convergent series. The formula for the general infinite geometric series is valid in the domain that the series converges, but while the series does not converge outside this domain, the formula still provides a value. The general formula is: [math]1 + x + x^2 + x^3 + x^4 + x^5\>+\>...\>=\>\dfrac{1}{1 - x}[/math] For [math]x = 2[/math], . . . [math]\dfrac{1}{1 - x} = -1[/math]

In case you were thinking that such series have no relevance to physical reality, the derivation of the Casimir effect makes use of the identity: [math]1\ +\ 8\ +\ 27\ +\ 64\ +\ 125\ +\ 216\ +\ ...\ =\ \dfrac{1}{120}[/math] [Please refresh browser window if the above LaTex doesn't render]

A while ago in the thread titled "problem with cantor diagonal argument" on page 6, as part of a general discussion about infinity, I mentioned the identity: [math]1\ +\ 2\ +\ 3\ +\ 4\ +\ ...\ = -\dfrac{1}{12}[/math] which led me to evaluate various non-convergent infinite series, including a proof of the identity above using simpler maths than analytic continuation of the Riemann zeta function. Let's consider something simpler: [math]\text{Let }X\ =\ 1\ +\ 2\ +\ 4\ +\ 8\ +\ 16\ +\ ...[/math] [math]X\ -\ 1\ =\ 2\ +\ 4\ +\ 8\ +\ 16\ +\ 32\ +\ ...\ =\ 2\ X[/math] [math]X\ =\ -1[/math] [math]\text{Therefore: }1\ +\ 2\ +\ 4\ +\ 8\ +\ 16\ +\ ...\ =\ -1[/math] [Please refresh browser window if the above LaTex doesn't render] I challenge you to find the error in the above.

I said that one can define the Riemann tensor this way. I didn't say that one has to define the Riemann tensor this way. I prefer to view GR analytically rather than geometrically, and in this view, the Riemann tensor is defined in terms of the integrability conditions of the coordinate transformation equation of the connection.

I haven't eaten McDonald's food for a few years now. I quite like it, but around the time of COVID-19, the shop at the shopping centre I normally go to closed their business, and the only McDonald's restaurants in my area would require me to make a special journey to McDonald's, which lacks the spur-of-the-moment convenience of the shopping centre. Actually, I haven't even been to Hungry Jack's (Burger King) much since COVID-19, and they are quite conveniently located for me. Why this is I can't really explain, although I did go there about four times in a row over a period of about a year to buy a thick shake, and each time, the machine was out of service (I did wonder if they ever had thick shakes during this period).

Displayed in the bottom-right image is the first four amino acid residues of the various endogenous opioid peptides: Tyr-Gly-Gly-Phe-... These four amino acid residues at the N-terminus of the peptide appear to be universal for opioid receptor activity. The fifth residue (not shown) appears to be either Met or Leu in the various endogenous opioid peptides. A careful look that the bottom-right image reveals some errors that indicate that one can't always trust artists with scientific knowledge.