# Search results for query: *

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1. ### A Does the theory of information have anything to offer for physics?

Is there any use for this concept in classical branches of physics? Can it be of any help for a physicist in resolving problems (or, at least, in resolving them more efficiently when compared with traditional methods)? The word «classical» means exactly that, i. e. mechanics, hydrodynamics...
2. ### I What is the proper matrix product?

@fresh_42 What is wrong with regarding ##v^T\cdot v## as an inner product automatically?
3. ### I What is the proper matrix product?

@fresh_42 Let me explain myself. Of course, I did not mean the inner product operation and matrix multiplication to be the same. Let's say we have two vectors (vector and covector to be precise) that we will regard as matrices with one row and one column. So, by definition of inner product we...
4. ### I What is the proper matrix product?

@fresh_42 I am afraid I am missing your point. We can multiply only those matrices that have equal numbers of rows and columns. For example we can multiply a matrix 2x3 by another matrix 3x4. But how should we multiply 2x3 by 4x4? It is not defined, is it? So, how can one refer to set of...
5. ### I Is there any matrix equivalent for the Clifford product?

Well, the question is in the title.
6. ### I What is the proper matrix product?

It says in any textbook (for example, in classical text «Theory of matrices» by P. Lankaster) on matrix theory that matrices form an algebra with the following obvious operations: 1) matrix addition; 2) multiplication by the undelying field elements; 3) matrix multiplication. Is the last one...
7. ### I Which class of functions does 1/x belong to?

Well, of course, but I meant its classification from the viewpoint of functional analysis.

Thank you!
9. ### I Which class of functions does 1/x belong to?

I am not sure I understand your point. The analytic functions form a small and restrictive class of functions. It can be broadened by dropping some requirements imposed on class members. It gives us this sequence (incomplete, I guess, but it illustrates the basic idea): ##C^\omega \subset...
10. ### I Which class of functions does 1/x belong to?

Yes, you are right. I was thinking about interval [-1,1].
11. ### I Which class of functions does 1/x belong to?

For historical reasons the hyperbola always was considered to be one of the «classical» curves. The function, obviously, does not belong to C0. Apparently, is does not fit L2 or any other Lp? What is the smallest class?
12. ### I Principal difference between complex numbers and 2D vectors revisited

The way I see it the discussion could now be closed. At least I got my answer and it certainly makes sense to me. Thanks a lot to everyone involved.
13. ### I Principal difference between complex numbers and 2D vectors revisited

It was not about higher dimensions for me, actually. It is just an example. If one talks about 2D tangent space, he uses vectors, not complex numbers. It is probably possible to consider tangent plane as Argand plane, but no one is doing that (AFAIK). There must be a reason for that.
14. ### I Principal difference between complex numbers and 2D vectors revisited

I can't argue with that. But I fail seeing how it answers my question, sorry. I realize the difference results of this extra algebraic structure. My question is how to understand whether this extra is helpful for solving a task (thus choosing between the two formalisms to work with the problem...
15. ### I Principal difference between complex numbers and 2D vectors revisited

Well, let's consider a specific problem (the way I see it is still my original question, just reformulated for specific situation). There is a 2D regular surface in Euclidean 3D space. If one talks about tangent space in any of its point, he necessarily use concept of 2D linear space. Vectors...
16. ### I Principal difference between complex numbers and 2D vectors revisited

I know this topic was raised many times at numerous forums and I read some of these discussions. However, I did not manage to find an answer for the following principal question. I gather one deals with the same set in both cases equipped it with two different structures (it is obvious if one...
17. ### Review of modern non-equilibrium thermodynamics theories

Thank you, I checked the monograph you cited. For now it looks like the best shot. But actually, I was thinking about something deeper and more thorough (probably a journal review paper). Some critique of basics and upshots of rational thermodynamics approach can be found in «A history of...
18. ### Review of modern non-equilibrium thermodynamics theories

Looking for thorough serious comparative review of modern generalisations of classical non-equilibrium thermodynamics of continua. I have heard about several such generalisations: 1) rational mechanics by Truesdell, Coleman, Noll. 2) extended thermodynamics by Müller (and Ruggeri). 3) the...

Thank you!
20. ### I Can the cross product concept be completely replaced by the exterior product?

Do we really need concept of cross product at all? I always believed cross product to be sort of simplification of exterior product concept tailored for the 3D case. However, recently I encountered the following sentence «...but, unlike the cross product, the exterior product is associative»...
21. ### Is there place for higher order derivatives in mechanics?

The building of theoretical mechanics can be constructed using only the first and the second derivatives (those of coordinates in case of kinematics: velocity and acceleration and those of energy in case of dynamics: force and gradient thereof). It is obviously unavoidable if one wants to deal...
22. ### Analogies between temperature and time in thermodynamics

That pretty much answers my question. Thank you!
23. ### Analogies between temperature and time in thermodynamics

It surely is. Thank you! I would appreciate you providing me with specific reference to a monograph or a review of applications of Kubo's idea (if you have such a reference at the ready, of course)? My guess is you refer to the so-called «Thermal field theory»...
24. ### Analogies between temperature and time in thermodynamics

Looking through the book of abstracts for «XXI International Conference on Chemical Thermodynamics in Russia (RCCT-2017)» I came across the abstract of talk given by Peter Atkins (University of Oxford) titled «Thoughts about thermodynamics» (you'll find the whole abstract at the end of the...
25. ### I What is difference between transformations and automorphisms

@fresh_42 Pretty clear and exhaustively! Thank you very much!
26. ### I What is difference between transformations and automorphisms

@fresh_42 Will it be correct (albeit not quite rigourous, but for now I am trying to grasp the very idea) to say that transformations are automorphisms of space (set, manifold) arising in context of consideration of general linear group and subgroups thereof?
27. ### I What is difference between transformations and automorphisms

Could you please help me to understand what is the difference between notions of «transformation» and «automorphism» (maybe it is more correct to talk about «inner automorphism»), if any? It looks like those two terms are used interchangeably. By «transformation» I mean mapping from some set...
28. ### Is Musk right in saying most academic papers are useless?

In some sense people expressing this opinion (although I agree with opinion of @StatGuy2000 that the thread title does not accurately describe Musk's view on this, but I heard other people saying just that) are right. But it is not the whole picture. I would suggest the following analogy: some...
29. ### Can't find the home page for specific journal (non-trivial case)

@DrClaude You are right. I should have used the verb «hope» in stead of «expect» (I was under pleasant impression of my yesterday's discovery that they opened access to «Annales de l'I.H.P. Physique théorique»; hoped the Italians followed the French). Thank you very much for checking it for...
30. ### Can't find the home page for specific journal (non-trivial case)

For the first time in my lime I am not able to find home page for well-known and respected western periodical. I am looking for the paper titled «Définition covariante des équilibres thermodynamiques» by J. M. Souriau published in Supp. Nuovo Cimento, 1, I , 4 (1966) pp. 203-216 (given the fact...
31. ### I Did Paul Dirac say anything about Bohmian mechanics?

Okay, got it. Thank you!
32. ### I Did Paul Dirac say anything about Bohmian mechanics?

Did not he? I'm a bit (a lot, actually) puzzled. Dirac was one of the leading theorists of his time. Can it be that he refrained from meditation over the very essence and core question of his discipline? Anyway, he and Bohm lived in the same country for many years. Probably, they knew each...
33. ### I Did Paul Dirac say anything about Bohmian mechanics?

Could you, please, give me reference to any paper or talk by Paul Dirac where he expresses his views about or give comments to the de Broglie-Bohm theory (Bohmian mechanics)?
34. ### I Are there relations that are valid for any field?

@fresh_42 Thank you for your tip, I'll have to ponder over it. @mathman It is absolutely true, but I fail seeing how it answers my question(s).
35. ### I Are there relations that are valid for any field?

I use the word «field» in purely algebraic sense here. Sometimes, when reading textbooks I encounter sentences like «Although the formulae in this section derived for the field of real numbers, they remain valid for complex numbers field as well». Or even more general variant of it: «...remain...
36. ### I Nice intro to connections between algebra and geometry

I'm grateful to everyone for the suggestions. I'll take a look at all of them.
37. ### I Nice intro to connections between algebra and geometry

Connections between algebraic structures and geometries are mentioned in almost any course of modern geometry or algebra. There are monographs dedicated to the subject. Unfortunately, the books, I managed to find, are written for professional mathematicians. I am looking for a book that focuses...
38. ### A Concept of duality for projective spaces and manifolds

Thank you for such a thorough explanation. The whole picture clarified considerably. But one point still escapes me. Let's say we have a differentiable (smooth) manifold. It «generates» tangent and co-tangent linear (vector!) spaces, that are dual to each other. The question is whether those...
39. ### A Concept of duality for projective spaces and manifolds

It looks like we are sailing on parallel courses here. I received no special education in mathematics and sincerely believed projective geometry to be some set of formal rules used in doing technical or architecture drawing, something that albeit being practicallly important just thrives on...
40. ### A Concept of duality for projective spaces and manifolds

Yes, you are right! Metric tensor mapping vectors to one-forms can be visualised as a circle in Euclidean space (I read about this in «Spacetime, geometry, cosmology» by W. L. Burke). The way to recover one-form from vector (arrow) and tensor (circle) is the same procedure as construction of...
41. ### A Concept of duality for projective spaces and manifolds

I firstly learned about duality in context of differentiable manifolds. Here, we have tangent vectors populating the tangent space and differential forms in its co-tangent counterpart. Acting upon each other a vector and a form produce a scalar (contraction operation). Later, I run into the...
42. ### Insights Problems with Self-Studying - Comments

I believe people who are self-studying are in particular need of helping hand in two cases, which are in some sense are opposite of each other. Being so the cases may require different approaches. I guess everyone who is self-studying sooner or later run into the situation when he can't...
43. ### A Why are conics indistinguishable in projective geometry?

Thank you for pointing (projecting :-)) me in the right direction. I have found the book you recommended and I like it much. Exactly kind of book I needed. Join to Micromass in sincere recommending this book to everyone who is looking for elucidation for subjects like the one discussed in this...
44. ### A Why are conics indistinguishable in projective geometry?

It is said that curves of the second order which we usually refer to as ellipse, parabola and hyperbola, i. e. conics, are all represented on projective plane by closed curves (oval curve), which means there is no distinction between them. Why is it? Projective space can, in principle, be...
45. ### I Why only normal subgroup is used to obtain group quotient

Thanks a lot for such a detailed explanation. It answers my question fully.
46. ### I Why only normal subgroup is used to obtain group quotient

Not quite sure I understand how cosets can be groups themselves. They lack identity element, since it is already included in the generating subgroup (normal or non-normal).
47. ### I Why only normal subgroup is used to obtain group quotient

Hello! As far as I know any subgroup can, in principle, be used to divide group into bundle of cosets. Then any group element belongs to one of the cosets (or to the subgroup itself). And such division still is not qualified as a quotient. Yes, the bundle of cosets in this case will be...
48. ### Is there any proper way to print threads?

OK. Thanks for clarification.
49. ### Is there any proper way to print threads?

I meant «compressed»...
50. ### Is there any proper way to print threads?

Well, for sure it does. But it takes five paper sheets, what could be «copressed» to fit only one when optimised for printing. I just thought I overlooked this, it is a pretty standard feature for forums, as far as I know.