taeto

Do you see whether there is a matrix \(\bar{M}\) that can map a polynomial to its integral? Since integration followed by differentiation produces the same polynomial that you started from, it should then be that \(M\) and  \(\bar{M}\) are inverse matrices, if \(M\) is the 'differentiation matrix'.

There is probably a theorem somewhere to say that it does work out that way, just because that is how linear functions work when you compose them. Their matrices get multiplied together.

So if this works for polynomials of degree at most 5, it should work the same for polynomials of any bounded degree d. All you need is to use matrices with d+1 rows and columns to represent differentiation, and then their k'th powers represent k times differentiation for k < d.

But polynomials can have all kinds of degrees. The next question is how to deal with the linear transformation of differentiating a polynomial when we do not know how large its degree might be. If you have a 100x100 matrix to differentiate any polynomial of degree < 100, then it doesn't work to differentiate a polynomial of degree 100 or more. 

Would it be possible to say that differentiation \(\frac{d}{dx}\) can be represented as a matrix \(M\), and that second derivative \(\frac{d^2}{dx^2}\) can be represented by the squared matrix \(M^2\)? 

The truth of the predicate \(x=x\) follows from logic, i.e., it is how equality is defined. It has nothing to do with \(\mathbb{R}\).

The field \( ( \{0\}, +, \cdot ) \) with one element is kind of okay as a field, as it satisfies the important axioms. Yet most authors exclude it explicitly, because it is a cumbersome exception for some theorems. In this 'field' you can always divide by zero, for example.so it is a little too exceptional.

That looks right.

Then a polynomial in your space is represented as a vector.

And the transformation maps this vector to another vector using matrix multiplication, or does it?

You could choose a basis for the space of polynomials over \(\mathbb{R}\) in a variable \(x\) of degree at most 5.

A 'random' such basis might be \(\{ 1 + x^3 -x^5, x + x^2- x^3, x^2 +x^4, x^3 + x^5, x^4 - x^5, 2x^5\} \).

How does your transformation act on the elements of this basis?

If any polynomial can be expressed as a linear combination of the polynomials in the basis, how would the transformation act on it?

Can you find an even better basis to use to show the same?

How will the transformation look in matrix notation?

15 minutes ago, studiot said:

Just as there are many ways to define compact in Mathematics

I really do not know what you refer to here. 

You could say that if we restrict to Euclidean spaces, then we can define 'compact' to mean the same as 'closed and bounded'. But that does not change the meaning of the notion. It just means that we are applying a theorem which expresses that being compact is equivalent to being closed and compact in that special case of a topological space, where we have additional information about its structure. What I mean is that to say that we can explain the property in a different way in a special setting is not the same as saying that the property itself has gotten a different definition.

What better examples do you have in mind?

30 minutes ago, studiot said:

It is also true that Physics uses a different definition of metric from Mathematics, particularly evident in Relativity, both GR and SR.

Mathematics defines 'metric' and 'metric tensor'. They are different things.

What is a definition of 'metric' in physics that is different from the definition in mathematics, and which is not the definition of 'metric tensor'?

39 minutes ago, studiot said:

As to compactness in five dimensions, Einstein-Bergmann-barmann assumed that the fifth dimension is compact.

I am not sure if this was not also the case with Kaluza? Perhaps you have more information?

Einstein, Bargmann and Bergmann knew about Kaluza-Klein already, and they tried to get a hold on electromagnetism using this same idea, without success.

The point is that before you can reasonably 'assume' a coordinate system based on a space in which some or all of coordinate ranges are finite in size, then it is best to know about the existence of such spaces, and how to construct them. And the method is called 'compactification of dimensions', starting from spaces in which the ranges of the coordinates are not already compact. It is clear that 'compactification of a dimension' in physics does not mean the same as 'compactification' of the range of a coordinate in mathematics.

And the second derivative of the gradient of a potential? With change in stability?

And \(\varepsilon\) stands for permittivity of a medium?

Vaguely similar to Poisson's equation in electrostatics \(\varepsilon \nabla^2 \varphi = -\rho,\) where \(\rho\) is charge distribution. 

I should do the dimensional analysis. But trying to get in a first wild guess, I would try \(-\frac{d}{dr} \rho(r)\) as the RHS of the mystery equation.

48 minutes ago, studiot said:

I really meant that the meaning of compact and compactification is the same applied to various objects of mathematics and physics and is based on the definition in set theory.

Certainly this description is not entirely correct.

Suppose you give a first lecture on Kaluza-Klein Theory, and you explain to the students that 'compact' and 'compactification' mean exactly the same in physics as in mathematics. Then you give them as homework to compactify one of the dimensions \(x\) of a 5-dimensional manifold, where \(x\) initially has range \(\mathbb{R}.\)

The obvious way to compactify in mathematics is to add either \(\infty\) or both of \(\pm \infty\), as well as enough new open sets to recover a topological space over the new set. So how do you mark such answers, which are correct according to your lecture? Full marks, I suppose. But how do you then explain that this does not work in physics, other than explaining that compactification is indeed different?

22 hours ago, studiot said:

There is no difference between the mathematician's definition of 'compact' and the physicist's.

 

 

It is possible that I am mislead by the page

https://en.wikipedia.org/wiki/Compactification_(physics)

and its first line

In physics, compactification means changing a theory with respect to one of its space-time dimensions. Instead of having a theory with this dimension being infinite, one changes the theory so that this dimension has a finite length, and may also be periodic.

It suggests that being 'compact' in physics has something to do with being small. Which is more like the common use of the notion, as in 'a compact car'. Whereas in mathematics 'compact' has nothing at all to do with size. Topology has no concept of 'length' or 'volume'.

The Euclidean real line \(\mathbb{R}\) and the open interval \( (0\, ; 1) \) are topologically the same and non-compact. Whereas \( \mathbb{R} \cup \{\pm \infty\} \) and \([0\, ; 1]\) are identical and compact. If the concepts of 'compact' agree between mathematics and physics, then compactification of a dimension could introduce a new coordinate with range \([0\, ; 1]\) but not \( (0\, ; 1)\). In contrast, the wikipedia page does not seem to think there is a difference. Adding something to say "this dimension has a finite length closed interval as its range", though clumsy, might make it clearer.

And beware that the explanation in the textbook refers to Euclidean space. This is a special kind of space for which it is indeed true that 'compact' means the same as 'closed and bounded'. This is not generally the case. The example of the compact non-Euclidean space \( \mathbb{R} \cup \{\pm \infty\} \) already shows it. 

2 hours ago, Mordred said:

Let's give a simplified example.

Let's take a metric space and assign 4d dimensions {x,y,z,t} now at each coordinate I wish to map how temperature varies over time. Now for extreme accuracy I want to measure at each infinitesimal coordinate so I can apply partial derivatives. (Calculus of variations)

So we can assign a set to temperature at each infinitesimal coordinate. Now ordinarily that set (dimension recall this value can vary without changing any other value) would be infinite in possible range. However I can place a boundary at Planck temperature and 0 Kelvin. I have now a infinitisimal topological metric space. Whose region is defined by the boundary of infinitisimals (point like) and it's dimension has been compactified by the upper and lower temperature bounds.

However as I am using infinitisimal regions I have another infinity problem in the number of spaces. So I must set some boundary to that. So as we can never measure below the Planck length I can now limit the number of infinitsimal spaces which limits the number of seperate temperature sets.

 

Thanks for giving more detail. I am already 99.9% familiar with differentials and 1-forms of manifolds. But your use of "compactified" still sounds different from the mathematical usage. In ordinary mathematical sense you would topologically "compactify" a space by adding more points and more open sets to obtain a compact space. You seem to be doing something else which has to do with a bounding (analytically) of coordinate ranges. Besides, adding "infinitesimals" usually means to add more points as well, in between the reals, so to speak.  

19 hours ago, BabcockHall said:

https://www.statnews.com/2020/04/29/gilead-says-critical-study-of-covid-19-drug-shows-patients-are-responding-to-treatment/

Patients taking remdesivir recovered more quickly than those taking a placebo.  Business Insider and CNN have stories.

It makes sense, because remdesivir has been useful against other coronaviruses, SARS so far as I recall, or was it Ebola. Also because the molecule is similar to adenosine and seems able to trick a viral RNA polymerase to try to build it into new RNA strings where adenosine would have belonged, and thus blocking the further production of the viral RNA.

On the other hand, the link states that "Gilead says", and Gilead is a (the?) manufacturer of remdesivir in the US. The available information about the study says that the decrease in lethality among test patients treated with remdesivir was not statistically significant (the shorter recovery times presumably were?)  compared to the patients treated with placebo. And an earlier study of the same drug did not produce any determination. So maybe more tests are needed.

6 hours ago, Mordred said:

This is an example of compactification for the topological spaces used in Calabi-Yau manifolds. Source being String Theory on Calabi-Yau manifolds by Brian R Green.

That is how compactification is understood in mathematics. It is a topological concept. Unfortunately it has little to do with the use of the same term in physics, where it is an analytical concept, that is, it talks about sizes. E.g. the default compactification of \(\mathbb{R}^4\) is gotten by adding a single new element, which is usually written with the \(\infty\) symbol. In the context of the thread, this is not apparently what is meant.  

3 minutes ago, Alex_Krycek said:

While it is big, it is still smaller than the asteroid that impacted the Earth and wiped out the dinosaurs.

Well, some of the dinosaurs.

At 30,578 km/h even a collision with such a smallish size would pose concern. Since there apparently exists a good estimate of the mass of the asteroid that hit 65M years ago, is there also a good estimate of its velocity? After all the energy depends only linearly on mass, but quadratically on speed.

And adding some quotes from Jeffrey Beall found in Wikipedia: ...it is clear that MDPI sees peer review as merely a perfunctory step that publishers have to endure before publishing papers and accepting money from the authors and that it's clear that MDPI's peer review is managed by clueless clerical staff in China."

Do not despair. After all, you are expected to be the humourless one.

16 hours ago, empleat said:

I usually like to read pages on wiki about authors, because there are some biased people, which think because people with OCD, can overcome their problem. That implies, that mind is immaterial or something. I don't even...

Neither do I. But generally reviewers are anonymous, and sometimes the reviewer is not even allowed to know the identity of the author(s), such as for ""double-blind" review.

If you somehow have access to the review, you may have clues to figure out the identity of the reviewer. Most directly if you find a pdf with the review, and the format of the pdf reveals explicitly from where it originated. Mostly though it is not realistic.

Once I was a referee of a long paper written by a close colleague, who was incidentally also the external examiner at my M.Sc. examination. I wrote a long report on this paper, which obviously he got to read. It would have been an important result, but the paper contained a small but important mistake somewhere around page 60 or so, which could not be repaired, and the submission got rejected. I asked him later if he knew from the style of the report who the reviewer might have been, and he said no, and told me that lots of reviewers use too similar styles in their reports to make them identifiable.

17 hours ago, empleat said:

It is easy to find if an article was peer-reviewed

That may not be completely accurate. There are "predatory" journals that mainly exist because authors pay money for "submission fees", so that they can be nearly guaranteed to have their paper appear in print, which adds to their cv. Such a journal may ask for a few corrections of misprints and bad grammar, but they have no interest in rejecting a paper on scientific grounds, because that would cut into their revenue.

17 hours ago, empleat said:

I found an interesting publication 

This is an MDPI journal. Wikipedia says "The quality of MDPI's peer review has been disputed. MDPI was included on Jeffrey Beall's list of predatory open access publishing companies in 2014, but was removed in 2015 following a successful appeal." I do not know the details. But I would be very cautious to trust a paper that has gone through the peer-review of an MDPI publication without first checking it carefully myself. 

18 hours ago, empleat said:

publication  as well as names of authors

Sometimes you can tell by googling the names of the authors. 

If we take the first author of your paper, Klee Irwin, we get from RationalWiki the information that Klee Irwin is a pseudoscience proponent and fraudster. He became widely known for his infomercials for "Dual Action Cleanse", a "natural" remedy, which was subject to numerous lawsuits. Some have reported being scammed by his company. 

I am going to go out on a limb here and suggest you not be worried. This paper has not seen a peer-review in the ordinary sense of the word.

1 hour ago, swansont said:

How do you come to that conclusion?

The air force's report is a video?

It must be a true coverup; the video appears to have been pulled. I will run to get my tinfoil hat now.

It is a beautiful intro to QFT, very much appreciated.

Just please correct the spelling to Klein-Gordon (or Klein-Fock-Gordon for completeness). Klein is a very common name in German speaking countries, for some reason (klein = small).

What is a \(\bar{k}\)?

37 minutes ago, CharonY said:

I only had them once (but they were the tiny variant) and I found them somewhat dry and thought it lacked it a bit of fat.

The tiny ones get sprinkled onto some dishes kind of as a spice, except they are not really too spicy as such. Their richness in proteins can make the food seem heavy. I have not been able to finish such a dish without leaving something over. Not that I found the taste to be bad, just the richness seemed overwhelming.

Next to the top right it says "see back for play instructions". I wonder if they could actually be useful.

I have not seen something like this before. But if it has only little to do with Analysis and Calculus, then there is more hope for an answer in the Brain Teasers and Puzzles section of the forum.

15 minutes ago, goa said:

careful consideration

That is very considerate.

Miisa, a Finnish form of Misha, which is a Russian form of Michael.

I can assume that the number \(DD\) itself is named Didi, a pet form of the German name Dieter?