Science topic

Mathematical Physics - Science topic

The use of rigorous mathematical techniques to make new predictions and test the limits and validity of physical models, while also developing new techniques in mathematics.
Questions related to Mathematical Physics
  • asked a question related to Mathematical Physics
Question
3 answers
Is paraphrasing necessary in such cases, or is direct quotation with appropriate citation sufficient?
Relevant answer
Answer
Actually neither. One pr9v8des the reference to the work, where the equation appezred and explains how it’s being used in the new work.
  • asked a question related to Mathematical Physics
Question
7 answers
Which book is good for the study of fractional calculus?
Relevant answer
Answer
Theory and applications of fractional differential equations, by:
Kilbas A.A.
  • asked a question related to Mathematical Physics
Question
21 answers
Dear Sirs,
In the below I give some very dubious speculations and recent theoretical articles about the question. Maybe they promote some discussion.
1.) One can suppose that every part of our reality should be explained by some physical laws. Particularly general relativity showed that even space and time are curved and governed by physical laws. But the physical laws themself is also a part of reality. Of course, one can say that every physical theory can only approximately describe a reality. But let me suppose that there are physical laws in nature which describe the universe with zero error. So then the question arises. Are the physical laws (as an information) some special kind of matter described by some more general laws? May the physical law as an information transform to an energy and mass?
2.) Besides of the above logical approach one can come to the same question by another way. Let us considers a transition from macroscopic world to atomic scale. It is well known that in quantum mechanics some physical information or some physical laws dissapear. For example a free paricle has a momentum but it has not a position. Magnetic moment of nucleus has a projection on the external magnetic field direction but the transverse projection does not exist. So we can not talk that nuclear magnetic moment is moving around the external magnetic field like an compass arror in the Earth magnetic field. The similar consideration can be made for a spin of elementary particle.
One can hypothesize that if an information is equivalent to some very small mass or energy (e. g. as shown in the next item) then it maybe so that some information or physical laws are lossed e.g. for an electron having extremely low mass. This conjecture agrees with the fact that objects having mass much more than proton's one are described by classical Newton's physics.
But one can express an objection to the above view that a photon has not a rest mass and, e.g. rest neutrino mass is extremely small. Despite of it they have a spin and momentum as an electron. This spin and momentum information is not lost. Moreover the photon energy for long EM waves is extremely low, much less then 1 eV, while the electron rest energy is about 0.5 MeV. These facts contradict to a conjecture that an information transforms into energy or mass.
But there is possibly a solution to the above problem. Photon moves with light speed (neutrino speed is very near to light speed) that is why the physical information cannot be detatched and go away from photon (information distribution speed is light speed).
3.) Searching the internet I have found recent articles by Melvin M. Vopson
which propose mass-energy-information equivalence principle and its experimental verification. As far as I know this experimental verification has not yet be done.
I would be grateful to hear your view on this subject.
Relevant answer
Answer
Information isn't a special kind of matter-it's a property of any kind of matter, that describes the state matter is found in.
  • asked a question related to Mathematical Physics
Question
5 answers
Mathematical physics conditions are necessary.
Relevant answer
Answer
Years ago, in the 90s, it was the time there was an intense research development of synergetics, and some questions regarding your inquiry were adressed.
Please look at the monography:
Samarskii A.A., Galaktionov V.A., Kurdyumov S.P., Mikhailov A.P. Blow-up in Problems for Quasilinear Parabolic Equations. – Berlin: Walter de Gruyter, 1995. – 533 p.
  • asked a question related to Mathematical Physics
Question
9 answers
Dear All,
Recently, I have looked into the Web in search for some works on Search And Rescue or guarding strategies. I have found some works related to UAV exploration in urban environment or SAR disaster site exploration. But, it was not what I really meant.
I wonder if there is any up-to-date literature (analytical, mathematical, physical, simulations maybe?) devoted to search and rescue or guarding, e.g., on the sea. I recall the stories from the time of WWII when allied vessels, after loosing contact with the submarine, started to move in circles with increasing radius. Also, there were special books prepared by mathematicians describing sequences of random turns to avoid being hit by a torpedo.
Do you know if there were any advances in the topic since then? Where the strategies using by modern SAR service are taken from?
Regards,
Michal
Relevant answer
Answer
Hi Michał,
In general, the entire SAR action conducted at sea and especially a selection of the search pattern depends (obviously) on a specific situation. Many factors are affecting the course of the action, such as search area (proximity of the shoreline and nearest rescue center), distance to the commence search point, number of the vessels involved, availability of support for the air (helicopters, aircraft), environmental conditions (usually unfavorable), etc.
When looking from the merchant ship's viewpoint, we have an official manual prepared jointly by IMO and ICAO, the so-called IAMSAR Manual (International Aeronautical and Maritime Search and Rescue Manual) where some procedures to be followed are presented. Among these, there are also search patterns introduced. Their application differs mainly depending on the aforementioned factors (especially the number of the searching vessels or possible cooperation with an aircraft). One of the most common patterns for a single ship is "expanded square", which is very similar to the example of allies vessels action recalled in your question. Other popular patterns used are, among others, sector search (also called a "flower-pattern") - usually with the support of a helicopter, as well as various modifications of parallel track searches - usually in cooperation with aircraft or other ships involved in the SAR action. For more practical information please refer to:
IAMSAR, 2007. International Aeronautical and Maritime Rescue Manual. Volume II, Mission Coordination. IMO/ICAO publications, London/Montreal; or any newer IAMSAR version.
However, in the mentioned IAMSAR Manual, you will find mainly simply operational procedures for the crewmembers of the merchant ships. If you are looking for algorithms or mathematic explanations of the most effective search methods/patterns you have to check scientific sources. There are a few, quite recently published papers about modern solutions in this field. You can see, for instance, the following papers:
Ai, B., Jia, M., Xu, H., Xu, J., Wen, Z., Li, B., & Zhang, D. (2021). Coverage path planning for maritime search and rescue using reinforcement learning. Ocean Engineering, 241, 110098. https://doi.org/10.1016/j.oceaneng.2021.110098
Ai, B., Li, B., Gao, S., Xu, J., & Shang, H. (2019). An Intelligent Decision Algorithm for the Generation of Maritime Search and Rescue Emergency Response Plans. IEEE Access, 7, 155835–155850. https://doi.org/10.1109/ACCESS.2019.2949366
Xiong, W., van Gelder, P. H. A. J. M., & Yang, K. (2020). A decision support method for design and operationalization of search and rescue in maritime emergency. Ocean Engineering, 207, 107399. https://doi.org/10.1016/j.oceaneng.2020.107399
Especially the first publication seems to be suitable for you. When you check the reference list, you can find there the older papers/books from the math domain, like:
Koopman, B. O. (1957). The Theory of Search. Operations Research, 5(5), 613–626. https://doi.org/10.1287/opre.5.5.613
Good luck,
Mateusz
  • asked a question related to Mathematical Physics
Question
14 answers
This paper is a project to build a new function. I will propose a form of this function and I let people help me to develop the idea of this project, and in the same time we will try to applied this function in other sciences as quantum mechanics, probability, electronics …
Relevant answer
Answer
Are you sure you have defined your function correctly?
1. Usually z=x+iy. But in your function z is in the limit, thus being in both the arguments and what the integral is computed against. If z is not x+iy, the function is not a function of (x,y).
2. What do you mean by limit? Do you want to compute the case when z->0?
  • asked a question related to Mathematical Physics
Question
4 answers
I would like to know your recommendations for books/online courses (MIT-OCW/Youtube/Udemy etc.) available on Photoacoustic Signals (Basic/Advanced/any level). I would prefer literature with more emphasis on their mathematics/physics.
I eventually want to observe the effects of these signals post Absorption spectroscopy (obtained from a nanosecond/femtosecond laser pulse source).
  • asked a question related to Mathematical Physics
Question
9 answers
Here we discuss about one of the famous unsolved problems in mathematics, the Riemann hypothesis. We construct a vision from a student about this hypothesis, we ask a questions maybe it will give a help for researchers and scientist.
Relevant answer
Answer
I put together a solution of the RH myself. While it can't be considered a complete proof while not vetted by experts, it presents various strong arguments and a real breakthrough, which is the inversion formula for Dirichlet series. Given any Dirichlet F(s), you know a(n) from F(s). Unfortunately, it's impossible to have an integral representation for a(n) usually, it's a Taylor power series. Please head to my page for the paper.
  • asked a question related to Mathematical Physics
Question
3 answers
I came across this question (attached below). I tried to solve it but got stuck in the portion where we need to calculate the inverse tranform. I found a solution of this question (also attached below) but there, in the encircled portion, I couldn't get how they took a 2x factor out and change the limits from (-∞, +∞) to [0,+∞ ). I know that we can do this changing of limits only if it's an even function and can take the limiting points [0,+∞ ) iff the function is of the form y = ax2 . But here the term inside the exponential function is of the form y = ax2 + bx + cix, where i =complex number, and accordingly the limit should change to some random [m,+∞ ) in place of [0,+∞ ). Also, the 2x factor would not be there because the limiting point is changing from [0,+∞ ) to [m,+∞ ) and the graph will not be symmetrical across X=0 Axis.
I will be highly grateful if you can kindly clarify my doubt or let me know where I am making a mistake in understanding the question.
Relevant answer
Answer
I love your equations. Please keep us in the loop.
  • asked a question related to Mathematical Physics
Question
5 answers
The Nilsson diagram is obtained by solving the Schrodinger equation. If the deformation parameters are continuous, I wonder the orbits should be continuous as well. If the Pauli exclusion principle is the reason, the nilsson quantum number are not always equal, such as 5/2[402] and 5/2[642], why?
Relevant answer
Answer
ear
  • asked a question related to Mathematical Physics
Question
8 answers
I am looking for some interesting areas of research in mathematics or in mathematical physics for undergradute students, I am in my 3rd year, and I've taken some basic courses such as: linear algebra, advanced calculus, mathematical methods, applied mathematics, and ODE..., What do you suggest to me?
Relevant answer
Answer
ODE , PDE can be solved numarically . You can go through the link - https://www.researchgate.net/post/Mathematical_operators_and_applications
  • asked a question related to Mathematical Physics
Question
10 answers
Dear Sirs,
The elevator example in general relativity is used to show that gravitational force and an inertial force are not distinguishable. In other words the 2nd Newton's law is the same in the two frames: inertial frame with homogenous gravitational field and the elevator's frame without gravitational field which has constant acceleration in respect to the inertial frame.
But every one knows that an inertial force is a force which does not obey the 3rd Newton's law. For example such forces are cetrifugal force and Coriolis force existing in the Earth reference frame. Gravitational force satisfies the 3rd Newton's law. So one can conclude that the gravitational force is not inertial.
Could you clarify the above controversy.
Relevant answer
Answer
  • asked a question related to Mathematical Physics
Question
5 answers
I am by no means an expert on this subject, but a few papers on this subject sparked an interest into whether instantons give rise to a non-zero vacuum expectation value or could be involved in the generation of the Higgs field.
Instantons in mathematical physics arise as solutions to a set of non-linear differential equations than minimize the Yang-Mills functional for a non-abelian gauge theory. This is part of the differential geometric way of writing classical fields in terms of a connection and the curvature of a connection. The classical electromagnetic field is a U(1) connection and the curvature form of this connection is an anti-symmetric matrix that whose entries are the electric and magnetic fields. For non-abelian groups such as SU(2) and SU(3), the connection and curvature of the connection formalism give rise to the weak force of the Z and W-, W+, and the 8 gluons of the SU(3) strong force. The instanton number can be thought of as describing the number of instantons present and is an expression of how "twisted" or topologically non-trivial the vector bundle or underlying space is.
The Higgs field is what gives spin 1/2 particles mass as well as giving mass to the Z and W-, W+ particles. The masses of spin 1/2 particles are determined by something called the Yukawa coupling. My question is how can instantons contribute to a non-zero vacuum expectation value and are there theories that say the Higgs field is built up in this way?
Relevant answer
Answer
The short answer is No, instantons don’t do this in the Standard Model. The reason is that, in four spacetime dimensions, there is a phase transition between a phase where the expectation value of the Higgs field vanishes and a phase where it doesn’t.
Incidentally, instantons are not, just, classical solutions of the esuatiins of motion of the gauge fields, that connect different vacua of gauge fields, there can exist instantons that connect different vacua of scalar fields.The existence of distinct phases for the scalar field, in the presence if the gauge fields, too, implies that such field configurations, that could affect the expectation value of the scalar, don’t exist.
  • asked a question related to Mathematical Physics
Question
61 answers
Dear Sirs,
Everyone knows the derivation of Lorentz transformations from electromagnetic wave front propagation. But Lorentz transformations are the basis of the general mechanics theory. It seems to me it is logically correct to derive the transformations from purely mechanical grounds. But how to do this? Mechanical (sound) waves are not of course applicable here. Or there is only purely mathematical approach? I The later is also not good in physics. Could it be derived from gravitational wave propagation? If it is so is there any controversy because General relativity is based on special relativity? I would be grateful for your suggestions.
Relevant answer
Answer
Length contraction CAN be deduced by purely mechanical processes. The other Transformations are substituted by other mechanical means. For example, time dilation can be speed of light changes in different media density.
  • asked a question related to Mathematical Physics
Question
2 answers
Operator product expansion is made in the deep ultraviolet region. In thermal field theory, it is explicitly shown , that the UV divergence in the quantum corrections does not depend on temperature. Hence, temperature shouldn't play any essential role in OPE, in thermal field theory(TFT). On the other hand, TFT is not Lorentz invariant. Hence, in the mixing of scalar operators in OPE, the matrix of anomalous dimensions should vary with quantum corrections due to the non- Lorentz invariant. Hence , I cannot understand the relevance of OPE in thermal field theory.
Relevant answer
Answer
Gravity is an action-at-a-distance force. Gravitational waves caused by the revolution of the sun affect the orbits of planets and provide some planetary precession data. The chasing effect of gravitational waves also causes the planetary orbital mechanical energy to continue to increase slowly until the planet escapes from the solar system. Gravitational waves exist; the gravitational model under the influence of gravitational waves that we constructed was a physical model. Through the calculation of planetary orbital precession, the correctness of the gravitational equation under the influence of gravitational waves is verified.
  • asked a question related to Mathematical Physics
Question
14 answers
Good day,
In standard PDE literature, there are a lot of solution approaches for the 1-D wave equation initial-boundary value problem. One way is with separation of variables. Below I describe the problem, the standard solution, and then I ask about a more complicated problem.
Given:
Wave Equation utt = c2uxx.; x ∈ (0, 1), t> 0 with
c=1,
u(0, t) = u(1, t)=0, t> 0,
u(x, 0) = f(x),
ut(x, 0) = g(x),
x ∈ (0, 1).
Solution:
u(x, t) =summation over k=1..∞ of [ sin (k π x)  (ak cos (k π t) + bk k π sin (k π t)) ]
ak, bk determinted by Fourier analysis of the functions f(x), g(x) .
Question:
My question is about an obstacle.
Let's say there is an obstacle of infinitesimal length of time and space, at x=x2 and t=t2, such that u(x2,t2)=d2.
1. How does the solution u(x,t) change after time t2?
2. How does the answer to 1 change if the obstacle is extended for a finite time?
3. How does the answer to 1 change if the obstacle is extended for a finite space?
4. How does the answer to 1 change if d2 = 0?
5. Can you point me to some references where I can find the solutions for 1-4? In my opinion, this is related to string vibration as well as electromagnetic wave propagation, for example if a propagating wave encounters a conductor. I hope to be able to find at least some elementary references on this topic.
Thank you in advance..
Relevant answer
Answer
Great ideas here
  • asked a question related to Mathematical Physics
Question
15 answers
Dear Sirs,
I would like to find out whether galilean relativity principle (which means the same
form of three Newton's laws in all inertial frames) is derived from the three Newton's laws or
any other classical mechanics statements.
Relevant answer
Answer
Galilean relativity principle is the statement that Newton's laws are invariant under the transformations of the Galilean group. And this can be checked. Conversely, the equations of motion that are invariant under the Galilean group, describe Newton's laws.
  • asked a question related to Mathematical Physics
Question
44 answers
The document: DOI: 10.13140/RG.2.1.4285.9289
Mathematically the question is to determine all the transformations realized between some coordinate systems which have a physical reality for the experimenters: each of these four-dimensional coordinate systems is formed by a cartesian and rectangular coordinate system of a three-dimensional Euclidean physical space, and by a particular temporal parameter which is qualified as cartesian and whose construction is specified. We obtain then a group of nonlinear transformations that contains the Poincaré group and is described by about fifteen real numbers.
Interpretation:
1 / The paradox of Ehrenfest:
If the elements of a family of observers are not motionless the ones with recpect to the others, in other words if their world lines are not elements of a unique physical space, then even in the context of classical kinematics, how they can manage to put end to end their infinitesimal rules to determine the length of a segment of curve of their reference frame (each will naturally ask his neighbor not to move until measurement is ended) ? this is the basis for the proposed solution to Ehrenfest paradox. Inspired by the expression of the law of Hubble, every theory must provide explicit or implicit assumptions to compare "the proper distance" D (which can vary over time) which separates an arbitrarily chosen experimenter P from a certain object, and "the proper distance" D' which separates another arbitrarily selected experimenter P' from the same object and this because it is admitted that this concept of proper distance has a physical meaning even in a non-comoving reference frame.
2 / The authorized relative motions are quantified:
I establish an Eulerian description of the construction of all the physical spaces of the "classical kinematics" and an Eulerian description of the construction of all the physical spaces of nature in the context of the new theory. In classical kinematics all the authorized relative motions between observers can be described by two arbitrary functions of the universal temporal parameter (one of the rotation and one of the translation) and in the context of the new theory, all the authorized relative motions between observers are described by at most 15 real numbers. A notion of expansion of the universe is established as being a structural reality and a rigorous formulation of the experimental law of Hubble is proposed.
Thank you.
Relevant answer
Answer
The Modification of Special Relativity:
The Modification of Newton's Gravitational Law and its Application in the Study of Dark Matter and Black Hole: https://www.researchsquare.com/article/rs-373969/v1
The Physical Cause of Planetary Perihelion: Precession:https://www.researchsquare.com/article/rs-536456/v1
  • asked a question related to Mathematical Physics
Question
1 answer
Is it possible to formulate the Ricci-flow as the Euler-Lagrange equations of some system? What would be the corresponding action functional?
Relevant answer
Answer
  • asked a question related to Mathematical Physics
Question
19 answers
As you know, creating a 3x3 magic square using 7 (or 8, or 9) distinct squared integers is one of the most interesting enigmas (see pages 17, and 18 of the following paper). Also, magic squares have some meaningful connections with physical phenomena (see pages 18 to 23 of the following paper). The question is that: how can we use physics tools to solve such a problems?
Relevant answer
Answer
Please read our recent paper:
In the page 2, paragraph #8 (point (1)) we have explained what is the question.
Best regards,
Peyman
  • asked a question related to Mathematical Physics
Question
26 answers
Math relates numbers to other numbers. But this is insufficient for physics. Physics models which include a cause-and-effect are more useful and result in bettter human understanding. For example, current models are time reversable. If cause-and-effect is part of the calculation, the model would not be time reversable without another equation showing how energy (entropy) is expended. Further, any proposed model that was only mathematical manupliation would not be considered physics.
Relevant answer
Answer
You are correct. Causality and the arrow of time are being consistently ignored as both Quantum probabilities and General relativity seem to develop math models at the expense of little regard for causality. Then there are the many complex papers on how to determine causality after the models have made liberal use of probability and shifting parameters from one side to the other of equations with only the number relations of math as a justification. This is the reason for the question.
The FQXi site seems to be devoting more to causality, sometimes called correlation.
Thanks for your insight. Perhaps, the comment should extend to fundamental principles.
  • asked a question related to Mathematical Physics
Question
92 answers
The total energy of two bodies in gravitational interaction must be
(m1 + m2) c^2 - G m1 m2 / r ,
where r is the distance between them. When r is  G/c^2 times the reduced mass, the total energy and hence the total mass vanish! It is the Schwarzschild radius, so a black hole may form. Does it necessarily have zero mass? Is this not contradictory?
Relevant answer
Answer
Recognizing the simple theory of the electron radius https://wikimili.com/en/Classical_electron_radius they only have the mass deficiency. Let the Black Hole be a matter sphere. Than its gravitational energy due to self interaction is E_g = - (3/5) G M^2/ r If r is the Schwarzschild radius than G M m /r = m c^2 /2 for the probe mass m so r = 2 G M / c^2 so we have E_g = - 3/10 M c^2. Now let M be the nuclear (rest) energy of all matter at infinity which would build the Black Hole sphere than we have:
M c^2 - 3/10 Md c^2 = Md c^2 where now Md is the dressed Black Hole Mass. So finally Md = 10/13 M so 3/13 of the constituent infinity mass deficiency.
  • asked a question related to Mathematical Physics
Question
2 answers
A conical surface in a Riemannian manifold M is the union of all the geodesics that connects a fixed point p (the vertex) and any point on some curve in M which doesn't contain the point p. If we define the same in the Riemannian product space Stimes R in the analogous way, how can such surfaces be parametrized? In which way can they be visualized?
Relevant answer
Answer
  • asked a question related to Mathematical Physics
Question
3 answers
I got a question (in a Question paper) as follows:-
A three-sphere is like a two-sphere. It consists of all points equidistant from a fixed point (the origin) in four dimensional space. Consider a particle free to move on a three sphere. How many conserved quantities does this system possess?
The answer say's 6 conserved quantities are there, but how is it possible? Can anyone kindly explain.
Relevant answer
Answer
The dynamics of a particle moving on any-fixed-manifold is given by the Lagrangian
L = (1/2)gIJ(x)(dxI/dt)(dxJ/dt)
where xI(t) are the coordinates in the ambient space and gIJ(x) is the metric of the manifold, in the present case, a 3-sphere.
Spherical symmetry means that angular momentum is conserved. The components of angular momentum are given by the tensor MIJ=xIpJ-xJpI, where pI=gIJ(x)(dxJ/dt) and pI=gIJ(x)pJ.
Said in an equivalent way: If gIJ(x) is the metric of a sphere, it's a standard exercise that these quantities are conserved for the xI(t) that solve the equations of motion.
In d dimensions there are d(d-1)/2 non-zero components of MIJ; for d=4 this gives 4x3/2 = 6.
  • asked a question related to Mathematical Physics
Question
3 answers
I have attached there some equations which is needed to be solved by keller box method.but I have faced problems with block elimination portion because of here momentum equation starts with f'' instead of f'''.I have also attached here sample matrix when equation starts with f'''.what will happen when it starts with f''?what will be iteration of converges for this?
Relevant answer
Answer
Your coefficient matrix will be of order 8 instead of order 7 for new set of equations with boundary conditions
  • asked a question related to Mathematical Physics
Question
53 answers
Dear Sirs,
I think many knows the ideas due to Jules Henri Poincaré that the physics laws can be formally rewriten as a space-time curvature or as new geometry solely without forces. It is because the physics laws and geometry laws only together are verified in the experiment. So we can arbitrary choose the one of them.
Do you know any works, researchers who realized this idea. I understand that it is just fantasy as it is not proved in the experiment for all forces excepting gravitation.
Do you know works where three Newtons laws are rewritten as just space-time curvature or 5D space curvature or the like without FORCES. Kaluzi-Klein theory is only about electricity.
Relevant answer
Answer
📷Preston Guynn. added a reply on June 19, 2019:
Force, mass, and energy are a parallel set of descriptions of the effects of special relativistic Thomas Precession. All matter and space, and their interactions are described with distance in three dimensions, time, and their derivatives.
Newton's first law of motion is , "Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it."
Yet the concept of motion requires at least two objects, and if there are two objects, then there is always an external force, which is gravitation.
So the idea of rewriting Newton's laws without force (or mass or energy) is good, but it should be extended to incorporate the most basic non-linear effects of motion in space time, which are special relativity and Thomas Precession.
See my article describing the recent discovery of the effects of Thomas Precession the particle and galactic scales.
Article Thomas Precession is the Basis for the Structure of Matter and Space
Recommended / Share
  • 3 Recommendations
  • asked a question related to Mathematical Physics
Question
18 answers
Dear all :
I need to solve the following integral (attached as an image file)
The context is on the calculation of View Factors in Radiation Heat Transfer
I worked out the this expression from the general formula, working out my configuration of two bodies in cylindrical coordinates (for one of the bodies) and using spherical coordinates (for the other body).
But I'm not sure if I ended with a well defined Integral, since I used two different coordinated systems on a same problem.
I used Cylindrical Coordinates for one of the dA and Spherical Coordinates for the other dA, however both dA are part of the same integral.
Hopefully someone out there can give me some help !
Regards and Thank you !
Relevant answer
Answer
When is an integral well defined? First, if the integrating function is integrable (in this case it is); second, if the domain of integration makes sense and is non-contradictory, then the final answer is that it depends on domains A1 and A2. What I do not understand is what the two additional integration variables mean, therefore I suppose that in the domains A1 and A2 the limits for z and η are included. If not, it is a meaningless integral.
  • asked a question related to Mathematical Physics
Question
2 answers
As applied to physics, the source is a mathematically described process and the target is one without a mathematically described process or without a mathematically described process known to the student. Analogy can suggest a mathematical model to a researcher. Analogy assists the student by demonstrating that knowledge already acquired can help in understanding a new subject. Thus analogy can be an investigative tool and a pedagogical tool. John Holland in his book on Emergence from Chaos to Order attributes the source-target characterization to Maxwell (p. 210) but I have not been able thus far to locate Maxwell’s employment of that characterization. Maxwell spoke about analogy as a useful pedagogical tool in an 1870 address to the Mathematical and Physical Sections of the British Association included in his collective works, volume 2, page 215. At page 219: Analogy `is not only convenient for teaching science in a pleasant and easy manner, but the recognition of the formal analogy between the two systems of ideas leads to a knowledge of both, more profound than could be obtained by studying each system separately.’
Do you know the origin of the source-target analogy?
Relevant answer
  • asked a question related to Mathematical Physics
Question
16 answers
Dear Sirs,
Everybody knows plane and spherical wave solutions of Maxwell equations, e.g for decaying plane wave E=E0*exp(-kx)*cos(w(t-x/v)). But seems to me they give the unreal situation that the wave amplitude is nonzero at different points of space at given time moment. Could you advise the experiment or natural phenomenon which produces such a wave in nature?
Maybe we have infinte speed of the EM interaction? Do you know any real solution of Maxwel equations which exists only in one space point at the given time moment? Maybe using delta function? Or maybe there is my mistake?
Relevant answer
Nice Dear Joaquin Diaz-alonso
  • asked a question related to Mathematical Physics
Question
52 answers
Dear researchers,
Hi,
Reputable Springer Publishing has compiled a list of more than 400 titles of its most valuable books (in various fields of engineering, medicine, management, astronomy, computer, politics, psychology, nutrition, law, mathematics, physics, chemistry, etc.) free to download. Do not hesitate to take this opportunity and download your favorite books for free as soon as possible. The list of these books is given in the file below. Please share it with your colleagues.
Best regards,
Saeed
Relevant answer
Answer
Thanks so much
  • asked a question related to Mathematical Physics
Question
8 answers
https://www.amazon.com/Um-Cabo-Verdiano-Pelo-Mundo-Portuguese/dp/0955944023 Um Cabo-verdiano pelo Mundo é um apanhado de factos e sentimentos vividos pelo autor que, tendo nascido no Paul, Santo Antão, Cabo Verde, chegou à idade da reforma nos seus serviços e vivencias através dos tempos, nos Estados Unidos da América. Durante a sua vida ensinou em vários continentes contactando assim com vários povos e culturas procurando sempre dar o seu melhor não só através do ensino como também nas comunidades onde exercia as suas funções. Hoje revive com alegria e também com muita saudade tudo o que encontrou, tudo o que viu, tudo o que fez nunca esquecendo as pessoas que pelo caminho encontrou. 📷Mundo é um apanhado de factos e sentimentos vividos pelo autor que, tendo nascido no Paul, Santo Antão, Cabo Verde, chegou à idade da reforma nos seus serviços e vivencias através dos tempos, nos Estados Unidos da América. Durante a sua vida ensinou em vários continentes contactando assim com vários povos e culturas procurando sempre dar o seu melhor não só através do ensino como também nas comunidades onde exercia as suas funções. Hoje revive com alegria e também com muita saudade tudo o que encontrou, tudo o que viu, tudo o que fez nunca esquecendo as pessoas que pelo caminho encontrou.
Um Cabo-Verdiano Pelo Mundo (Portuguese Edition) (Portuguese) Paperback – April 30, 2009
by Salazar Ferro (Author)
Falecimento do Professor António St.Aubyn
22/04/2016 - 16:24No dia 18/04/2016, em Lisboa, aos 79 anos📷
Com profunda tristeza, a Divisão de Informática vê partir o Professor António St.Aubyn, Professor Emérito da Universidade Técnica de Lisboa, Instituto Superior de Agronomia, e o primeiro Presidente do Centro de Informática do ISA (CIISA), atualmente DI.
Nascido a 10 de março de 1937, na Ribeira Brava, São Nicolau, Cabo Verde, o Professor St.Aubyn licenciou-se e doutorou-se pela Universidade de Coimbra, tendo tido uma intensa atividade académica e científica nas áreas da matemática e da estatística, da qual resultaram diversas publicações científicas.
Tendo ingressado no Instituto Superior de Agronomia no ano letivo de 1974/1975, onde foi Professor Catedrático desde 1979, e durante muitos anos Presidente do Departamento de Matemática do ISA, onde teve um papel preponderante, não só na criação do próprio departamento, mas também no lançamento dos seus cursos de Mestrado e Doutoramento, o Prof. St.Aubyn foi também docente nas Faculdades de Ciências das Universidades de Coimbra e Lisboa, no Instituto Superior de Economia e Gestão, no Instituto Superior Técnico e na Universidade Lusíada.
Esteve ainda ativamente envolvido em diversas sociedades científicas, nomeadamente na Sociedade Portuguesa de Matemática, de que foi Presidente, na Sociedade Europeia de Matemática e na Sociedade Portuguesa de Estatística.
Figura muito estimada no seio da comunidade cabo-verdiana, esteve ativamente envolvido nos movimentos pela independência do país, e mais tarde no desenvolvimento do ensino da Matemática em Cabo Verde, a convite do seu Ministério da Educação. Foi ainda membro fundador da Academia das Ciências e Humanidades de Cabo Verde.
Intimamente ligado aos processos de informatização da antiga Universidade Técnica de Lisboa, integrou em 1981 a Comissão Informática da UTL, que tinha como objetivo a renovação das infraestruturas de cálculo das escolas da UTL.
Já no ISA, e após a sua criação em 1984, impulsionou o CIISA ao longo de mais de uma década, durante a qual se fez a transição da era dos mini-computadores e seus terminais de acesso, para a era da micro-informática, das redes de computadores, e da Internet.
Para sempre ligado à história do Centro de Informática, na memória ficará alguém cuja visão desempenhou um papel fundamental para que o ISA fosse, nomeadamente no contexto da UTL, de alguma forma pioneiro na adoção das chamadas Novas Tecnologias, que haveriam de se revelar determinantes, não apenas no apoio às atividades científicas e académicas, mas para todo o funcionamento da instituição.
📷
he Cape Verde International Days on Mathematics 2017ArticleFull-text available
  • February 2018
  • 📷Delfim F. M. Torres
  • 📷Ricardo Almeida
  • 📷Paulino Lima Fortes
  • 📷Dorota Mozyrska
DownloadRecommendFollowShareOs desafios do planeamento estratégico: o caso da􀂌􀂊􀂜􀂘 Universidade de Cabo VerdeArticle
  • January 2014
  • 📷Paulino Lima Fortes
  • Bio || Bio Paulino Lima Fortes é licenciado em Matemática (ensino) pela Universidade de Évora (1989). Fez estudos pós-graduados em Mathématiques Théoriques (Geometria, Topologia, Álgebra e Física Matemática) na Universidade de Dijon, França (1993-1995). É doutorado em Matemática área de Geometria, pela Universidade de Évora (2005). É professor auxiliar na Universidade de Cabo Verde, onde ingressou em 1989, onde desenvolve atividades de ensino e investigação nas áreas de Análise, Topologia, Geometria e Física-Matemática, tendo proferido conferências no âmbito da história do pensamento científico. Áreas de Interesse || Current Research
    1. Geometria e Topologia.
    2. Modelação Matemática.
    3. Pensamento Geométrico e espacial.
    4. Artigos
      1. O conceito de corte e a construção do contínuo: uma abordagem histórica, Revista de Ciência e Tecnologia, nº1,Edições Uni-CV, 2013.
      2. Livros
      3. Fundamentos de Geometria Não-Standard, Edições Uni-CV.
      4. Reuniões Científicas || Scientific Meetings Selecção de conferências/comunicações
      5. On some convex topologic concepts in topological vector spaces, XXI Oporto Meeting in Geometry, Topology and Physics, Instituto Superior Técnico, Lisboa, 2015.
      6. Em torno dos paradoxos de Zenão de Eleia: interpretações standard e não-standard, Jornadas Luso-Espanholas de Filosofia da Ciência, CFFCUL, Lisboa, 2014.
      7. Pistas e despistes no ensino da Geometria, Jornadas sobre o ensino da Matemática e da Língua Portuguesa, Universidade de Cabo Verde, 2013.
      8. A descoberta do bosão de Higgs, um triunfo do Modelo Standard, Reitoria da Universidade de Cabo Verde, Praia, 2012.
      9. A recuperação de informação geométria a partir de uma núvem de pontos, Departamento de Engenharias e Ciências do Mar, Universidade de Cabo Verde, 2011
      10. .http://cfcul.fc.ul.pt/equipa/docs/2014/Paulino%20Lima%20Fortes_cv.pdf
      11. https://www.researchgate.net/profile/Gastao_Frederico/research
      12. Gastao S. F. Frederico currently works at Federal University of Ceará, Brazil. Gastao does research in Calculus of Variations, Optimal Control, Fractional Calculus and EDP. Their most recent publication is 'Noether-type theorem for fractional variational problems depending on fractional derivatives of functions'.… Read moreLanguages
        • English
        • French
        • Portuguese
        • DisciplinesCosmologyApplied MathematicsMechanical EngineeringAerospace EngineeringControl Systems EngineeringSkills and expertise (38)MATLAB SimulationModeling and SimulationControl TheoryNumerical AnalysisEngineering, Applied and Computational MathematicsSystem ModelingMathematical AnalysisMathematical ModellingNonlinear AnalysisMechanical EngineeringView allStats overviewResearch Research overviewView all
        • 32Research items
        • 0Projects
        • 0Questions
        • 1Answer
        • Featured researchMost recommended in the last month📷Sourceψ-Hilfer pseudo-fractional operator: new results about fractional calculusArticleFull-text available
        • April 2020
        • 📷Jose Vanterler
        • 📷Gastao S. F. Frederico
        • 📷Edmundo Capelas de Oliveira
        • 74 Reads
        • 1 Recommendation
        • DownloadRecommendFollowShare📷SourceNoether-type theorem for fractional variational problems depending on fractional derivatives of functionsPreprintFile available
        • August 2019
        • 📷Matheus J. Lazo
        • 📷Gastao S. F. Frederico
        • 📷P. M. Carvalho-Neto
        • 25 Reads
        • DownloadRecommendFollowShare📷SourceNoether-type theorem for fractional variational problems depending on fractional derivatives of functionsArticleFull-text available
        • August 2019
        • Applicable Analysis
        • 📷Matheus J. Lazo
        • 📷Gastao S. F. Frederico
        • 📷P. M. Carvalho-Neto
        • 82 Reads
        • 2 Recommendations
        • DownloadRecommendFollowShare📷SourceNoether theorem for action-dependent Lagrangian functions: conservation laws for non-conservative systemsPreprintFile available
        • June 2019
        • 📷Matheus J. Lazo
        • 📷Juilson Paiva
        • 📷Gastao S. F. Frederico
        • 70 Reads
        • 1 Recommendation
        • DownloadRecommendFollowShare📷SourceNoether theorem for action-dependent Lagrangian functions: conservation laws for non-conservative systemsArticleFull-text available
        • June 2019
        • Nonlinear Dynamics
        • 📷Matheus J. Lazo
        • 📷Juilson Paiva
        • 📷Gastao S. F. Frederico
        • 38 Reads
        • 1 Citation
        • https://www.researchgate.net/profile/Telma_Silva DownloadRecommenatured research📷SourceExistence, uniqueness, stability and asymptotic behavior of solutions for a mathematical model of atherosclerosisArticleFull-text available
        • December 2015
        • Discrete and Continuous Dynamical Systems - Series S
        • 📷Jorge Tiago
        • 📷Rafael Santos
        • 📷Adélia Sequeira
        • 📷Telma Silva
        • 233 Reads
        • 5 Citations
        • 1 Recommendation
        • DownloadRecommendFollowShareRecently read📷SourceMathematical Modeling of Atherosclerotic Plaque Formation Coupled with a Non-Newtonian Model of Blood FlowConference PaperFull-text available
        • July 2013
        • The Cape Verde International Days on Mathematics 2013
        • 📷Jorge Tiago
        • 📷Telma Silva
        • 📷Adélia Sequeira
        • 📷Rafael F. Santos
        • 310 Reads
        • 10 Citations
        • 1 Recommendation
        • DownloadRecommendFollowShareMost recommended in the last monthModeling of the Early Stage of Atherosclerosis with Emphasis on the Regulation of the Endothelial PermeabilityArticle
        • April 2020
        • Journal of Theoretical Biology
        • 📷Telma Silva
        • 📷Willi Jäger
        • 📷Maria Neuss-Radu
        • 📷Adélia Sequeira
        • 15 Reads
        • 1 Recommendation
        • Full-text requestedRecommendedFollowingShareMathematical Analysis and Numerical Simulations for a Model of AtherosclerosisChapter
        • December 2016
        • 📷Telma Silva
        • 📷Jorge Tiago
        • 📷Adélia Sequeira
        • 88 Reads
        • 1 Citation
        • Request full-textRecommendFollowShareMost cited in the last monthMathematical modeling of atherosclerotic plaque formation coupled with a non-Newtonian model of blood flowArticle
        • January 2013
        • 📷Telma Silva
        • 📷Adélia Sequeira
        • 📷R.F. Santos
        • 📷Jorge Tiago
        • 21 Reads
        • 9 Citations
        • Request full-textRecommendFollowShareProjectsThe Cape Verde International Days on Mathematics 2020 https://www.researchgate.net/profile/Ivanilda_Cabralhttps://www.researchgate.net/publication/331991393_Astronomy_for_development_in_Portuguese-speaking_countries https://www.facebook.com/robert.sousa.161 https://www.researchgate.net/profile/Robert_De_Sousa Doutor em Matemática e Aplicações, Mestre em Estatística e Gestão de Informão
        • Antonio Afonso Delgado
Relevant answer
Answer
  • asked a question related to Mathematical Physics
Question
3 answers
Hi
Can anyone please shed some light on Morton number and its physical significance? Where it is a must to use Mo instead of Bond number or Eotvos number and why? 
Thanks.
Relevant answer
Answer
In the case of bubble dynamics, we study the effect of Eo number on the shape instabilities of the rising bubble. However, Eo number contains the radius of the bubble which is actually changing continuously with time. Thus, there is a need to define Morton number which depends only on the fluid properties and not on the bubble radius. Morton number is a suitable ratio of Eo and Ga numbers so as to eliminate the bubble radius.
  • asked a question related to Mathematical Physics
Question
5 answers
its atomic number is not too high. So what kind of mathematical or physical constraint on its nuclear structure breaks it so easily? Why doesn't it have any natural stable isotope?
If it's one isotope with nearly equal neutron and proton number be produced, why that would not be stable?
Relevant answer
Answer
The detailed answer is complicated. Basically, the stability of nuclides depends on their number of protons/neutrons and some configuration are more stable then others (analogue to electronic configurations in the atom). So it turns out that all configurations with 43 protons either decay in Mo (42) or Ru (44) because both these elements have a lot of stable isotopes.
  • asked a question related to Mathematical Physics
Question
18 answers
Assume we've developed a novel beam theory. How can we validate and corroborate its mathematical and physical underpinning?
For example, assume there is a cantilevered beam (one end clamped and one end free). A point force P is applied to the free-end of the beam. Through the novel beam theory named A-BEAM-THEORY, we calculate the transverse deflection of beam as w-A as a function in x, where x is along beam. And through Euler-Bernoulli and Timoshenko it (deflection function w=w(x)) is found to be w-EB and w-Tim. How can we authorize and validate our novel beam theory named A-Beam-Theory? It means how to demonstrate (or claim) deflection function w-A is more reliable than w-EB and/or w-Tim? I mean how can we assure the superiority and qualification of the novel beam theory and to show it is superior to existing beam theories? Should we compare slopes at the end-point (in case of bending)? Should we compare deflection at the end point? Should we compare the whole deflection function mathematically? Should we compare moments, stress or strain? Should we compare first or second or other natural frequencies (in case of vibration)? There is no concern about the difference of Euler-Bernoulli or Timoshenko beam theory, and only the validation of the novel strategy is minded and matters. Simply I want to show and claim my novel methodology is more accurate than existing methods, for vibration and/or for bending.
Relevant answer
Answer
Euler Bernoulli Theory is not able to take transverse shear stresses into account i.e transverse shear stresses will vanish in all points through the thickness in EB Theory.
To solve this problem, Timoshenko beam theory was proposed. But it considers a constant distribution for transverse shear stresses in all points. Thus it is not correct at the top and bottom surfaces of the structure.
All higher order shear deformation theories (like exponential, trigonometric, parabolic ,….) doesn’t have these problems.
You can compare your model with EB and Timoshenko theory
  • asked a question related to Mathematical Physics
Question
6 answers
In multiscale modeling by homogenization in porous media, at least one of the phases must be connected. It is for this requirement that one-dimensional problems could not be considered. What are the mathematical and physical constraints that result in the connectivity requirement?
Relevant answer
Answer
Dear Moussa Mirehei,
imagine a porous solid consisting of a solid phase and a hollow phase for which
you want to compute the homogenized elasticity tensor. Imagine the case that the solid phase is originally connected but then disconnected by -for simplicity- a planar cut in 3d or a linear cut in 2d thus creating a gap of finite width in the porous solid. As a consequence of the discarded connectivity there is no mechanical restistance in terms of non-zero stiffness against pulling apart the
porous specimen in the direction perpendicular to the cut. Instead, there is a relative translation as a rigid body motion of the separated parts. This rigid body motion indicating zero stiffness manifests in the homogenized elasticity tensor in terms of a zero eigenvalue thus making the elasticity tensor singular.
Mathematical homogenization of linear elastic, microheterogeneous solids underpins the above statement more precisely, e.g. in the appendix of our paper [1] which you can find here on RG.
I hope that the physical/mechanical notion of disconnection enabling a rigid body motion and its mathematical consequence for homogenizing linear elastic solids is helpful.
Best
Bernhard
[1] "The Heterogeneous Multiscale Finite Element Method for the Homogenization of Linear Elastic Solids and a Comparison with the FE$^2$ Method, Computer Methods in Applied Mechanics and Engineering 329,
DOI: 10.1016/j.cma.2017.10.001"
  • asked a question related to Mathematical Physics
Question
5 answers
This discussion investigates inverse eigenvalue solutions as creating a pattern that might fit numerical sequences or time series, with the solution providing a mathematical or physical model.
In particular, application of the inverse Sturm-Liouville Eigenproblem (iSLE) using the revisited Matrix-Variational Method [1], can be applied to time series analysis as suggested in RG by Jean-Philippe Montillet.
For example, if one could estimate a functional model with known signals within a time series, in the presence of complex noise (i.e. sum of white plus colored noise, or mix chaotic noise). Can we estimate this functional model as an iSLE, in those terms? This could be interesting, with many applications, including climate change.
As another example, it is possible to hear the shape of a drum [2]. The eigenvalues representing the possible modes of vibration, give a common basis for a certain shape, and not others. Noise in the measurement may also be reduced by using the eigenproblem itself to filter, processing only the part of the signal that correspond to viable oscillating shapes. The same can happen in quantum mechanics.
Relevant answer
Answer
This discussion is now closed. RG is unsuitable for physics discussions, for lack of moderator and presence of cabals. I can be reached by PM.
  • asked a question related to Mathematical Physics
Question
13 answers
I am currently searching for any open position as a PhD research position in applied mathematics. I am actively working in the area of Fractional Differential Equations with Applications in Science and Engineering. I have attached my curriculum vitae (C.V.) (Please see the attached PDF file). If you know any available position related to applied mathematics or any person who are in search of candidates for this type of positions, please do let me know. I would also greatly appreciate if you could share my C.V. with your connections.
Thank you very much in advance!
With Kind Regards,
Mohammed K A Kaabar
Relevant answer
Answer
Thank you very much Dr. Konstantin Kudryavzev !!
  • asked a question related to Mathematical Physics
Question
12 answers
Dear Sirs,
Could anybody help to solve the following task?
Let us have two different inertial masses, say different in 2 times. Point masses are good also. Each of them is pulled by a string (constant force). To solve the problem we should use Einstein equations as every inertial mass possesses a kinetic energy and therefore curve space-time around itself. Did we receive the accelerations of each body which differs in 2 times? So maybe here we get a contradiction in GR?
Relevant answer
Answer
Thank you very much
I badly explained my question. I mean the following task. A spring pulls some mass. The mass here is inertial one but according Einstein will curve space-time. The question is in that whether this curvature explains the body inertia (resistance to the velocity change) or NOT?
  • asked a question related to Mathematical Physics
Question
4 answers
Recent research has shown key important aspects with measurability of theoretical postulates verifying hypotheses parameters, processes, phenomena, and models.
Tensor matrices have necessarily roles to bring about complex nature manifesting spatially and temporarily. Explaining everything that is existing in terms of the fundamental entities have lead to realization of geometric topology space tensor manifold time evolving event gridnetwork.
Einstein's General Theory of Relativity, Quantum to Particle Theory of Everything, String Theory among others have measurability in mind a proof of model requirement automatically. For example, Schwartzchild blackhole mathematics helped to identify, observe, and measure singularity blackhole consequently, the recent telescopic photos observing directly, proving validity with General Theory of Relativity tensor predictive capability.
Providing the thumb rules, below certain associative relationships might connect mathematics with physics to measure model......
. typically scalars, scalar matrices are helpful to get statistical measurements that are analyzable observationally experimentally......
. tensors have stochastical vector matrices that aren't amenable to direct measurements. Hence transforming tensors or matrix tensors to scalar matrix systems are key to make measurable operational parametric graphical experimental observational gridnetworks.
My analyses metrix protocol techniques have yielded a rough estimate of overall globally ~80% of objects universally are measurable statistically. This will mean ~20% are uncertainity stochastic probabilities with a few% inherent immeasurable tensor network, aether may be example. I have space time sense 2x2 tensor grid part of a large tensor matrix that if transformed to 5 dimensional like scalar matrix natural manifolds protocol will help eventually in the quantitative grand unified theory of everything. There are more to come after our QFM/EM modeling going on with collaborative platform TEI.
Given below are a few references that are associated, not exhaustive, suggestions welcome. Additions editions expansions!!!!!
(6) Zurek, Wojciech H. (2003). "Decoherence, einselection, and the quantum origins of the classical". Reviews of Modern Physics. 75 (3): 715. arXiv:quant-ph/0105127. Bibcode:2003RvMP...75..715Z. doi:10.1103/revmodphys.75.715 & Dan Stahlke. "Quantum Decoherence and the Measurement Problem" (PDF). Retrieved 2011-07-23.
Relevant answer
Answer
I go for simpler models by changing the postulates (modeling). I'm unsure what that 20% is. Not measured because nobody has done it or cannot be measured.
  • asked a question related to Mathematical Physics
Question
5 answers
Here is my article I uploaded on RG on 21st April 2018:
And here is a book chapter submitted on 13th July 2018:
Deleted research item The research item mentioned here has been deleted
There is a strange similarity, if compared form section §1 of my article, and from section 2. of the chapter, especially section §3 and 4., yet no proper citation.
What do you think?
Relevant answer
Answer
That is why I said "politely" and "to the author". Maybe you could ask him directly, without mentioning the term "plagiarism". Just say that you noticed similarities and that you were wondering if he was somehow inspired by your work... maybe doing so you get a citation (since it seems to be a provisional chapter yet).
Good luck anyway.
  • asked a question related to Mathematical Physics
Question
4 answers
Could anyone give the relationship of electrode resistance with reduction or oxidation potential regarding a mathematical or physical point of view?
Relevant answer
Answer
Your welcome Sereilakhena Phal
  • asked a question related to Mathematical Physics
Question
6 answers
Greetings,
Completing Bachelors in Engineering this June'19, I thought I'd start with Masters/PhD in Gravitational Physics this fall but I received rejections from almost every graduate school I applied to. To where I received an offer from, I won't be able to pay off the tuition fees.
Of course I knew that to receive an offer, one needs to have some experience with the subject. With the engineering curriculum on one hand, I tried to manage my interests in gravity. From watching lecture videos by Frederic Schuller and Leonard Susskind to reading books by Sean Carrol and to even doing a summer research internship on black hole geometries, I tried to gain experience on the subject.
I wish to understand relativity from a mathematical point of view.
" A good course in more abstract algebra dealing with vector spaces, inner products/orthogonality, and that sort of thing is a must. To my knowledge this is normally taught in a second year linear algebra course and is typically kept out of first year courses. Obviously a course in differential equations is required and probably a course in partial differential equations is required as well.
The question is more about the mathematical aspect, I'd say having a course in analysis up to topological spaces is a huge plus. That way if you're curious about the more mathematical nature of manifolds, you could pick up a book like Lee and be off to the races. If you want to study anything at a level higher, say Wald, then a course in analysis including topological spaces is a must.
I'd also say a good course in classical differential geometry (2 and 3 dimensional things) is a good pre-req to build a geometrical idea of what is going on, albeit the methods used in those types of courses do not generalise. "
- Professor X
^I am looking for an opportunity to study all of this.
I would be grateful for any opportunity/guidance given.
Thanking you
PS: I really wanted to do Part III of the Mathematical Tripos from Cambridge University, but sadly my grades won't allow me to even apply :p
Relevant answer
Answer
There are two sides to your problem: the practical and the ambitional. You will have to look after both. Recognize the practical issues but don't let go of your ambition. You may have to get a temporary job just to live, but that does not mean you give up on your dreams.
Your problem is not unique and has been overcome by famous scientists. Faraday started working for a bookbinder and ended as a revered scientist. His personal drive got him through. Dirac got a first degree in electrical engineering and ended as a revered theorist. Einstein worked early on in a Patent office and ended as a revered theorist. Other examples can be found, such as Ramanujan. Now there's a great example of talent beating disadvantage. So you see, it's not the end of the world if there are practical difficulties in your way at this time in your life. If you keep your spirits high, focused on what really interests you, you may succeed. It may be very hard, but don't give up.
You should understand that training is not enough. You have to practice being creative. Some people on this forum will probably disagree with the following suggestion, but have a go at writing a paper on a novel topic and seeing the reaction. It may take time to find a problem that you can work on, and you may very well get rejection. But having a go will teach you more than doing a lecture course on analysis. Papers do not all have to be in quantum field theory or relativity. Go on the arXives and see what sort of topics are viable for you. Most likely, at this stage, it might be in the General Physics section. But at least you might start from there.
Good luck in your ambition. Never give up.
George Jaroszkiewicz
  • asked a question related to Mathematical Physics
Question
28 answers
The 1998 astronomical observations of SN 1A implied a (so-called) accelerating universe. It is over 20 years later and no consensus explanation exists for the 1998 observations. Despite FLRW metric, despite GR, despite QM, despite modified theories like MOND, despite other inventive approaches, still no explanation. It is hard to believe that hundreds or thousands of physicists having available a sophisticated conceptual mathematical and physics toolkit relating to cosmology, gravity, light, and mechanics are all missing how existing physics applies to explain the accelerating expansion of space. Suppose instead that all serious and plausible explanations using the existing toolkit have been made. What would that imply? Does it not imply a fundamental physical principle of the universe has been overlooked or even, not overlooked, but does not yet form part of physics knowledge? In that case, physics is looking for the unknown unknown (to borrow an expression). I suspect the unknown principle relates to dimension (dimension is fundamental and Galileo’s scaling approach in 1638 for a problem originating with the concept of dimensions --- the weight-bearing strength of animal bone — suggests fundamental features of dimension may have been overlooked, beginning then). Is there a concept gap?
Relevant answer
Answer
Allow me to mention that the discovery of the new fundamental science of Cryodynamics, sister of Thermodynamics, has confirmed Zwicky 1929. So that the universe is stationary and eternal.
The 90 years long adherence to the "Big Bang" is a historical tragedy, a "Dark Age."
Can anyone forgive me for that statement?
  • asked a question related to Mathematical Physics
Question
7 answers
In my opinion, there are two main obstacles in math teaching and studying:
1st MAIN OBSTACLE. From all other sciences, mathematics has the largest degree of coherence and inter-connectivity between all its branches. From the first years of studying mathematics, it is obvious that each math lesson is essential to understand all other later math lessons. In contrast, in physics for example: you can understand very well mechanics without knowing electricity; you can understand optics without knowing electricity or mechanics and so on. Because mathematics demands a marathon-like effort on many years to understand it, a maximum tenacity in daily or periodical study (which is very time consuming), the "natural" selection is harsh, because just a small percent of people are sufficiently motivated by "the science and art of counting" (which math is) to get over this first obstacle.
2nd MAIN OBSTACLE. The poverty of methods and digital resources (images, animations and software) which is used to teach math from the first grades: it's only in the last 20 years that digital resources in math exploded and were implemented recently in teaching (and that is not sufficient time for extensive and diverse implementation in math pedagogy).
MY THESIS
In the "reference frame" of exact and "almost-exact "sciences, math may appear as a very important domain and language: which is true. However, my thesis is that the human being are based on not one, but three types of logic and metalogic, which logics are not reducible one to another:
1. rational logics (all "governed" my mathematics) which may be all integrated in a rational metalogic (rML) ("governed" by meta-mathematics)
2. emotional logics (studied by arts, aesthetics, psychology, philosophy, religions etc) which may be all integrated in an "emotional metalogic" (eML);
3. "volitional" logics (also studied by psychology, neurobiology and medicine in general, sociology, philosophy, religions etc), which may be all integrated in a "volitional metalogic" (vML), which I also conjecture to not be reducible to any of the first and second types of metalogic (rML and/or eML).
I have extensively presented my aforementioned thesis in my papers:
MY "META-THESIS"
My "meta-thesis" would that mathematics and rML in general aren't sufficiently "powerful" (and would never be) in the "humanity reference frame" (which is almost infinitely larger than the "reference frame" of exact sciences) to gain more interest than eML and vML. My prediction (and conjecture) is that rML may only imitate eML and vML, but can never replace them. I also argued in my (previously mentioned) papers that eML and vML (which are conjectured to be irreducible to rML, which rML may be only an "imitator" of both eML and rML, but not their "replacer").
See also other URLs of my work:
See also general URLs:
This discussion was inspired by Prof. Patrick Dasgupta, who is full professor since 2004 in Physics and Astrophysics Faculty from University of Delhi (https://du-in.academia.edu/PatrickDasgupta), when he kindly invited me (directly or using the Academia.edu robot?) to participate to a session on his article called “On Reasonable Effectiveness of Pedagogy in Mathematical Physics”
Draft paper URL:
Session URL (on which I’ve also pasted this large discussion-comment):
Regards!
Relevant answer
Answer
Dear Mrs. Bays,
1a. Math teaching surely needs "momentum", but in large quantities and the teachers who share constant big enthusiasm in math teaching are rare: however, I am glad that you are one of those enthusiastic teachers.
1b. I am also "ambivalent" in your sense: passionate on both exact and humanistic sciences, in sciences, arts and religions. However, I tend to be more oriented to the scientific rigorous approach.
2a. Besides starting from the earliest grades with many math-based games (like origami for example), I think that digital animations and constant teaching and learning using a math software (like Mathcad for example, which also has the possibility to implement programming routines built on lines of code, but also to generates graphs and animations) would be an ideal way to further continue the math learning and teaching. Learning basic programming (and a basic math software) in parallel with math is essential for every child to create his own math experiments and to very himself after working on paper.
2b. I don't think that mathematicians are (statistically and significantly) more predisposed to brain cancer than the rest of the population: this is a "hard claim". Do you have any scientific proof on this claim? Knowing one or two cases of mathematician who developed brain cancer isn't a proof: I also know examples of persons passionate by literature who developed brain cancer. I hope you didn't share this false claim with your children! I am waiting for proofs on this last claim of yours: as a pediatrician specialist (with experience in adult medicine too), I assure you there is no published evidence about mathematics predisposing to brain cancer.
Regards!
  • asked a question related to Mathematical Physics
Question
6 answers
An in depth examination is made of the hypothesis that all of mathematical physics can be based on Newton’s Laws within the context of an inertial frame, the Galilean Transform and the negative results of the Michelson-Morley Experiment. To that end a new mathematical model for a charge neutral continuous mass atom to replace the quantum mechanical model for the atom and a charge neutral continuous mass photon to replace the electromagnetic field has been created. Charge neutral continuous mass photons are created by direct collision of charge neutral continuous mass atoms.
One cannot sweep away the experimental results for which the concept of charge is now the accepted explanation, but one can derive the experimental results without the concept of charge.
The model has been applied to historically important physics experiments to yield numerical experimental results. A large portion of the text is devoted to how small mass photons (~10^-10 AMU) interact with liquids, solids, gases, vapors and spectroscopic gratings. The text can be found on my website: www.jmkingsleyiii.info
Relevant answer
Answer
I don't know "whether there will be enough rain this summer to provide grass for the sheep and whether a human torso used to be used instead of a sheep in the Mongolian equivalent of polo on horseback (or was that in Tajikistan ?)"come from where,but What does it have to do with your paper ?
I couldn't get into you link ,please add it here if you have time.
  • asked a question related to Mathematical Physics
Question
3 answers
The great thinkers of the Age of Reason and the Enlightenment were scientists. Not only did many of them contribute to mathematics, physics, and physiology, but all of them were avid theorists in the sciences of human nature.
Relevant answer
Answer
Thank you for your contributions.
  • asked a question related to Mathematical Physics
Question
21 answers
Recently I am working on mathematical modeling for nonlinear oscillatory problems
Relevant answer
Answer
One useful book is
Braun, Oleg M., and Yuri S. Kivshar. The Frenkel-Kontorova model: concepts, methods, and applications. Springer Science & Business Media, 2013.
  • asked a question related to Mathematical Physics
Question
3 answers
what is the mathematical and physical meaning of stoke's theory for an optical signal , does it differ if the signal is polarized or not!
Relevant answer
Answer
Some statements to clarify:
1. In 1852, Sir George Gabriel Stokes showed that the light polarization behavior could be represented in terms of physical observables. He found that any polarization could be completely described by four measurable quantities, now known as the Stokes polarization parameters. Unlike Jones vectors (whose vector entries are complex numbers), the Stokes parameters are real-valued and can represent both full and partially polarized light.
2. Advanced modulation formats employed in current optical communication systems use both phase and amplitude of the optical field to codify information. Polarization-Division Multiplexing (PDM) allows doubling the number of symbols per time slot by coding information in the two orthogonal SOPs of the same wavelength. Concerning the polarization of the total optical field inside the fiber, these modulation formats with PDM give rise to different SOPs at adjacent time slots.
3. To recover the state of polarization of a given symbol and represent it on the Stokes space one needs to compute the respective Stokes parameters from the Jones vector (see [1]). A schematic diagram showing the transition between Jones and Stokes spaces ca be found for instance in [4].
The analysis of the signal samples in the Stokes space gives then access to the information about polarization effect occurred along propagation. Actually, Stokes space provides a “geometric” visualization and easy understanding of some compensation techniques, such as polarization demultiplexing (see [1]), polarization dependent losses monitoring (see [2]), or even modulation format recognition (see[3])
[1] B. Szafraniec, B. Nebendahl, and T. Marshall, “Polarization demultiplexing in Stokes space,” Opt. Exp. vol. 18, no. 17, pp. 17 928–17
939, Aug 2010
[2] N. J. Muga and A. N. Pinto, “Digital PDL compensation in 3D stokes space,” J. Lightw. Technol., vol. 31, no. 13, pp. 2122–2130, Jul. 2013.
[3] ADLES et al.: Blind Optical Modulation Format Identification From Physical Layer Characteristics, JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 8, APRIL 15, 2014
[4] N. J. Muga, S. Ziaie, A. Shahpari, A. N. Pinto, Using the Stokes Space for Equalization of Polarization Impairments in Digital Coherent Optical Receivers, International Conf. on Transparent Optical Networks - ICTON, Trento, Italy, Vol. Mo.B2.4, pp. 1 - 3, July, 2016
  • asked a question related to Mathematical Physics
Question
5 answers
Does the need for revolution hang in the air or do I just think it is? Anyway, here's my Manifesto.
Relevant answer
Answer
I believe this is a self-promotion post and should be deleted.
Either way, the question is way unclear.
  • asked a question related to Mathematical Physics
Question
30 answers
In stimulated emission process, we always assume that the secondary excited photon, as a result of e-h recombination, has the same frequency, polarization, phase and direction with the incident photon. The first three terms can be understood from phase matching criteria (coherency).... but for the last term, how can we explain and be sure about the direction of secondary photon? Are there any mathematical or physical description?
Relevant answer
Answer
Dear Kai,
Ok, the number of photons is not relevant, but I can try to explain the conservation of moment needed (and of course also energy although is not important for the question related with the direction of the photons) in a simple form.
Let us have one atom at rest in a laboratory L (thus at rest ) hit by an incident photon p. The atom must recoils in a direction exactly opposite to the incident photon.
Now let us consider the time inversion symmetry T for the physical process. Then under time reversal this process would appear to be a case of absorption in which two photons impinge upon a moving atom. One of the photons is absorbed and cancels the momentum. The other continues on its way. Thus you can see how the conservation of momentum is needed in a simple form is you don't want to waste energy and momentum in this physical process.
  • asked a question related to Mathematical Physics
Question
1 answer
for information on Point Vortex System to the rg "Project Point Voritces on the surface of a sphere Takashi Sakajo Paul K. Newton" = 2 general theorical maths results on approximation of general finite Point Vortex System in R2 from SERFATI.ph. 2 papers on PVS, one here given from serfati philippe (rct 21-8-17: the 2nd given here: "Borne en temps des caractéristiques de l’Equation d’Euler 2D à tourbillon positif et localisation pour le modèle point-vortex", see its 2017-abstract on my profile)= "Tourbillons -presque- mesures spatialement bornés et équation d'Euler 2D" June 1998 Philippe Serfati on rg (given here). extract from (abstract 2017) Considering the point vortex model for the 2D Euler equation, we prove that the approximate Dirac are localized in Ɛ"-domains (Ɛ"=Ɛ'log(p)(1/Ɛ+e). Ɛ'² =Ɛ), Ɛ ->0+, on each [0,T], if, initially, they are so in Ɛ-domains and are bounded only by Cst.exp(p)(1/Ɛ), where exp(p) and log(p)(.+e) are any p-iterative composed of exponential and log(.+e), p arbitrary. This result and some of those from our other "Borne en temps des caractéristiques de l’Equation d’Euler 2D à tourbillon positif et localisation pour le modèle point-vortex" [where L°° bounds dependance is, mostly importantly, also very weak: LINEARLY in log(p)(e+lvortexlLoo) and a slightly weaker result on the point vortex model where exp(p) and log(p)(.+e) here are replaced by powers of Ɛ of arbitrary small exponents] allow us to make three conjectures: 1) positive measure vortex remains spatially bounded for all [0,T], 2) two differently signed measures vortex, initially separated, remain so for short T (and conclusion of 1) holds), 3) Delort's convergence theorem holds in situation 2). ----- on PVS = my precisely targeted (3rd) result on point vortex system (in the 3 results in "Borne en temps des caractéristiques de l’Equation d’Euler 2D à tourbillon positif et localisation pour le modèle point-vortex") is extremely rarely cited (:by one or very few papers of marchioro (etal)), as PVS result, and never cited for its/my-own improvement in my further paper, never cited = "Tourbillons -presque- mesures spatialement bornés et équation d'Euler 2D", my (3rd) result and my improvement on PVS, being still the strongest ones (even in 2017), to our knowledge (for example, marchioro (etal) with only powers of Ɛ of FIXED exponents but in 1998, there is "simultaneously and independently" (at least for me) to my first "borne en temps etc" (and my other "tourbillon etc"), the weaker to both "borne" and a fortiori, "tourbillon" = "Marchioro, C. (1998). On the localization of the vortices. Bollettino dell'Unione Matematica Italiana, 1(3), 571-584.", where L°° bounds for diracs are initially in cst/Ɛ^k (:as in "borne" but stronger in "tourbillon": cst.exp(p)(1/Ɛ), any p), for ANY k>0, and t-diameters in cst Ɛ^a, 0<a<1/3 (in "borne" the already stronger a< 1/2 and more stronger in "tourbillon" in cst.Ɛ^(1/2).log(p)(1/Ɛ), any p. before, marchioro (etal) had it only for one fixed ko). Strongest and (non/)citations, comparing the weaker ones and citations in papers, books, chapters of/by the ones of iftimie etal, by marchioro (etal), by P.K. newton etc and their descendances and coauthors. for example = "Newton, P. K. (2001). The N-vortex problem: analytical techniques (Vol. 145). Springer Science & Business Media."
Tourbillons -presque- mesures spatialement bornés et équation d'Euler 2D. Available from: https://www.researchgate.net/publication/314398465_Tourbillons_-presque-_mesures_spatialement_bornes_et_equation_d%27Euler_2D [accessed Apr 6, 2017].
Relevant answer
Answer
--+20/8/18+rct8/12/18= "serfati ph /Point Vortex Systems, ANY polynoms, ANY p-iterated logs and exponentials in 1/epsilon". (sent eg to marchioro, iftimie, pulvirenti) (intended to be) told to researchers on this topic = my  "Borne en temps des caractéristiques de l’Equation d’Euler 2D à tourbillon positif et localisation pour le modèle point-vortex Apr 1998 Philippe Serfati" (see my profile on reseachgate) contains 2 disjoint results R1 and R2 (although the maths are similar and which, of course, inspired by marchioro and pulvirenti) =- the first often quoted and described as such, on the size in t of a positive vortex patch with compact support in x at t = 0 (often cited for my growth in C. (t.log (q) (t )) ^ (1/4), ALL q in lN *, log (q) = q-iterated log (2 + x), iftimie and all have only q = 1 in 2018 (i have eg, me only, this which is very much important, C = (1 + log (p) (lrotu (0) loo)) C '(lrotu (0) l1) (then almost independant of lrotu (0) loo), all p in lN *, and iftimie and all have in 2018, only C = (1 + lrotu (0) loo) ^ k.C '(lrotu (0) l1), for A k> 0 FIXed, ie a polynomial dependance of fixed degree, in lrotu (0) loo).- the second R2 (of "borne ...") very rarely quoted or described, but iftimie / 2007, p126 (cf lower) recognizes it as mine and mine alone = the regular N e-approximations signed, e > 0 of a system of N vortex points, of bounds L °° in cst / e ^ k, ALL k> 0 to t null, are, over all [0, T], at distances <Ce ^ μ, ALL 0 <μ <1/2, their PV (t) = before me, all and marchioro (= MP93 / article1 / "vortices loc euler .." + MPc91-5 / book, and / but noting that then iftimie me recognize me to be the 1st on the 3rd M98 / article2 / "on loc vortices ..", there of same force here, at best "simultaneous and independent", at least me, even if that M98, who then did only a redemonstration of my result / R2 here, does not quote me inexplicably (whereas I had given it to M since 1998), MP93 had only k in [2.8 / 3 [and 0 <μ <3 (8/3 -k) / 17 (<0.12 <1/8). furthermore, I am improving my / this 2nd result of my "whirlwinds almost measures etc" (it, never cited or described!) where I suppose that cst.exp (p) (1 / e) (and not cst / e ^ k) and I have Ce ^ (1/2) / log (p) (1 / e), ALL p> 0 (and not only Ce ^ μ, μ <1/2). - $$$$ eg also, IF one quotes MP93 and / or M98 and / or and / or MP90'book for 1to2 of these papers, one MUST quotes my R2 / PVS of "borne en temps..." 98 (which is the + strong (and only MP93 is anterior, but weaker), and even also my " tourbillons presque mesures etc", which is even stronger on all even in 2019), which is my theorem and not that said-in /by M98 and M = tell me what you will do about it. $$$$ = Seminaires & Congres 15, 2007, p. 119–179 LARGE TIME BEHAVIOR IN PERFECT INCOMPRESSIBLE FLOWS by Dragos Iftimie emis.ams.org/journals/SC/2007/ 15/pdf/smf_sem-cong_15_119- 179.pdf ... = ""...126 D. IFTIMIE ... Formally, this can be justified in the following way. The vortex approximation consists in ignoring the term (x−zi) ⊥ 2π|x−zi| 2 when it comes to define the velocity of the point zi . But this contribution is just rotation about zi (faster and faster as x approaches zi) so it shouldn’t affect zi itself. Rigorously, the first complete justification is due to Marchioro and Pulvirenti [29] and was later improved by Marchioro [28] and Serfati [41]. It consists in proving that if the initial vorticity is localized and converges to a sum of Dirac masses in a certain way not too restrictive, then at later times it will stay localized and converge to a sum of Dirac masses that are the solutions of the vortex system. More precisely, we have the following theorem. Theorem 4.1 (Serfati). — Suppose that ωε(0) = P k j=1 ω j ε (0) and z1(0), . . . , zk(0) are distinct points such that – ω j ε (0) has definite sign; – supp ω j ε (0) ⊂ D zj(0), ε ; – kω j ε (0)kL1 = aj ; – |ωε(0)| ≤ C ε k for some arbitrary k ∈ N. Let ω j ε (t) denote the time evolution of ω j ε (0) and P k j=1 aj δzj (t) the solution of the vortex model with initial data P k j=1 ajδzj (0). Then for any T > 0 and µ < 1 2 there exists a constant C1 = C1(T, µ) such that supp ω j ε (T ) ⊂ D zj(T ), C1ε µ . Moreover, for any T ≥ 0, we have the following weak convergence in the sense of measures: ωε(T, ·) * X k j=1 aj δzj (T) as ε → 0..."". --Philippe Serfati PhD+Ec.Norm.Sup.-PS+AGREGATION in MATHS. 245 citations by 125 international university authors. see researchgate (eg for full texts), google citations etc. --"Borne en temps des caractéristiques de l’Equation d’Euler 2D à tourbillon positif et localisation pour le modèle point-vortex Apr 1998 Philippe Serfati" --"Tourbillons -presque- mesures spatialement bornés et équation d'Euler 2D June 1998 Philippe Serfati" --Marchioro, C., Pulvirenti, M. (1993). Vortices and localization in Euler flows. Commun. Math. Phys. 154:49–61.  --Marchioro, C. (1998). On the localization of the vortices. Bollettino dell'Unione Matematica Italiana, 1(3), 571-584. --Marchioro, C., Pulvirenti, M. (1994). Mathematical Theory of Incompressible Nonviscous Fluids. Volume 96 of Applied Mathematical Sciences. New York: Springer-Verlag.
  • asked a question related to Mathematical Physics
Question
14 answers
Today I found an interesting paper by G. Poelz (retired from Hamburg University) which suggests that electrons have wave character, see http://arxiv.org/pdf/1206.0620.pdf. Basically he describes an electron model based on the solution of the wave equation in spherical coordinates (see Appendix 6.2 in his paper). This would need the use of spherical Bessel functions of the first kind (see for instance: http://mathworld.wolfram.com/SphericalBesselFunctionoftheFirstKind.html).
Interestingly, I found that George Shpenkov also uses a similar method to describe not only electrons but other atoms as well, based on the solution of the wave equation in spherical coordinates. See his page at this www.researchgate.net or at http://shpenkov.janmax.com. Shpenkov asserts that his method is different from the electron cloud model based on the Schrodinger equation.
While of course this kind of electron model may be different from the standard picture, it seems to be able to fulfill Louis de Broglie's vision in his Nobel lecture: Wave nature of electrons. (see http://www.nobelprize.org/nobel_prizes/physics/laureates/1929/broglie-lecture.pdf)
So do you think it is possible to find an exact electron model based on the solution of the wave equation in spherical coordinates?
Relevant answer
Answer
I am affraid that from the wave behaviour of the electron alone we
cannot get very much information of what the electron really is, with mass, charge
and magnetic moment. Particle Physics should eventually give us more information,....and its strange companions the muon and the tau.
  • asked a question related to Mathematical Physics
Question
22 answers
As for example, light beam attenuation is described by the differential equation
dS/dx = -S
which solution is S~e(-x).
But what physical processes could be described by the differential equation:
dS/dt = -t*S or dS/dx = -x*S
which solution is S~e(-t^2) or S~e(-x^2), with t as time and x as distance.
Do you have ideas?
Thank you very much in advance,
Algis
Relevant answer
Answer
In continuation, what is internal expectation from dynamical insights after your current equation of concern?
What is its physical model context?
If your your first equation links "attenuation of light", then your current equation should represent "moderated attenuation".
The moderation stems from the new appearance of "position coordinate" on the RHS of your equation.
Thanks to all having participated in this discussion.
  • asked a question related to Mathematical Physics
Question
10 answers
When I use the split step method to solve the nonlinear schrodinger equation, there is an external driven, expressed as a constant in the right hand side. How should I do with this part when I impletement split step method?
Relevant answer
Answer
Hello,
this question was asked like three years ago, and still no answer, however, it appears frequently in the search results.
To treat a constant term in LLE with a split-step method you simply include it in the expression for the linear part of dE/dt, and then integrate it with respect to time to obtain a solution after a time step.
Say, you have linear part written as
dE/dt = A + K*E, where K can include your -(1 + i*theta) term accounting for losses and detuning as well as -i*beta*d^2/dtau^2 term for the dispersion (as well as higher-order dispersion terms), and A can be the pump parameter or other constant terms (which are not multiplied by E in the original equation).
Then if you try to find an exact solution for this equation after the time step h, you will end up with the following relation:
E(t+h, tau) = A*(exp(K*h)-1)/K + exp(K*h)*E(t, tau)
You can treat it with FFT, and then obtain:
E(t+h, tau) = IFFT[FA*(exp(FK*h)-1)/FK + exp(FK*h) * FFT(E(t, tau))],
where FA = FFT(A), so you will discretize A in tau and take FFT of it, FK is calculated in the same manner. Basically, you replace -i*beta*d^2/dtau^2 by -i*omega^2, where omega is your Fourier wavenumbers.
Hope that helps to somebody who also stuck at this question.
  • asked a question related to Mathematical Physics
Question
4 answers
Is there an alternative theory that accommodates mind and matter? After all, the universe we observe has a logical structure that the mind can understand. If we do not understand something, it must have some measure of illogic. Can we unify all known theories...String Theory, SR, GR, QT, MOND, Standard Model, Big Bang, and so on. Is it possible to unify mathematics and physics (not in the sense of one can explain the results of the other) but in the sense of principles that govern the two? Can science explain miracles, TIME, SPACE, the forces of Nature? Why is gravity? Was Einstein right? Is a complete theory of nature able to explain even notions of God? There is such a theory. It starts with understanding how the mind processes information. Read Book 1..
Relevant answer
Answer
If "feeling" and "consciousness" are ignored, I think mind and matter could be accomodated in a single theory framework. In contrast, we indeed never assign "consciousness" to matter, and henceforth, the origin of the entire hardship of our concern obviously.
  • asked a question related to Mathematical Physics
Question
2 answers
(this is (only) an information/added-ref (because rg do not "suggest" it) to my project and not a question, and/but eg see also below or my "answer" below, starting with the tx = "(here is copy of my mel ..." =) https://www.theses.fr/2017PESC1215.pdf = "Qualitative study of the solutions of density-dependent incompressible Navier-Stokes system" =Étude qualitative des solutions du système de Navier-Stokes incompressible à densité variable", Xin ZHANG, thesis29/9/17?
.
= École Doctorale MSTIC Laboratoire d’Analyse et de Mathématiques Appliquées Thèse Présentée pour l’obtention du grade de DOCTEUR DE L’UNIVERSITE PARIS-EST par Xin ZHANG Étude qualitative des solutions du système de Navier-Stokes incompressible à densité variable Spécialité : Mathématiques Soutenue le 29 September 2017 devant un jury composé de : Directeur de thèse DANCHIN Raphaël (Université Paris-Est Créteil) Rapporteur ALAZARD Thomas (l’ENS Paris-Saclay) ZHANG Ping (Institute of Mathematics, AMSS) Président du jury CHEMIN Jean-Yves (Université Pierre et Marie Curie) Examinateur DALIBARD Anne-Laure (Université Pierre et Marie Curie) GALLAGHER Isabelle (Université Paris Diderot) LACAVE Christophe (Université Grenoble Alpes) ...
=========================================
(here is copy of my mel to all the 8 members of the jury and the author of this thesis by Xin Zhang, thesis mainly on striated regs, vortex patches etc=) *6nv18/ = "VP 2d 3d nd= chemin (2d) / serfati (2+3+nd) / bertozzi (&constantin) (2d)/ gamblin & straymond (3d) / danchin (2+3+nd) etc". from SERFATI philippe (see researchgate for full texts etc) .(PhD+Ec.Norm.Sup.-PS+AGREGATION in MATHS. 245 citations by 125 international university authors) = to all the 8 members of the jury and the author of the thesis (see file below or the adweb = https://www.researchgate.net/deref/https%3A%2F%2Fwww.theses.fr%2F2017PESC1215.pdf) = Xin ZHANG Étude qualitative des solutions du système de Navier-Stokes incompressible à densité variable Spécialité : Mathématiques Soutenue le 29 September 2017 devant un jury composé de : Directeur de thèse DANCHIN Raphaël (Université Paris-Est Créteil) Rapporteur ALAZARD Thomas (l’ENS Paris-Saclay) ZHANG Ping (Institute of Mathematics, AMSS) Président du jury CHEMIN Jean-Yves (Université Pierre et Marie Curie) Examinateur DALIBARD Anne-Laure (Université Pierre et Marie Curie) GALLAGHER Isabelle (Université Paris Diderot) LACAVE Christophe (Université Grenoble Alpes) = (=chemin@ann.jussieu.fr, raphael.danchin@u-pec.fr, xinzhang@univ-paris-est.fr, christophe.lacave@imj-prg.fr, zp@amss.ac.cn, dalibard@ljll.math.upmc.fr, gallagher@math.ens.fr, Thomas.Alazard@ens.fr).
--on the 130 refs of this thesis, at least +-20/25% are on Vortex patches (by 25/30 authors), 2d, 3d, nd, general and regular plus VP2d with cusps (thesis where my name or my works do not appear, even on these subjects)= i recall eg that i have always in 2018, even by points taken separately, the strongest results in vp2d loc in t, vp2d global in t (but jy chemin and bertozzi&constantin were first for vp2d glob), the very firsts and still the strongest results in vp3d and vpnd, n>=3, and eg i have 4 works on vp with cusps and +-10 works or sub-chapters of my thesis on VP and +-5 of them published, +-still far the strongests in 2018. here is below, some of my short texts detailling some of this (some others exist also), and it is said and intend to be said, repeatedly, to all authors on VP, and eg (already) to (some of) you all (and txs eg compiled on my data = ""rg-texts on papers on anisotropies, morrey spaces and elliptic problems and holomorphies on opens and vortex patches machineries etc, 2017-18 from Serfati-philippe October 2017 DOI: 10.13140/RG.2.2.35956.04482/"" and/or foundable in diverse places of my profiles (eg) on RG etc) = tell me what you will do (eg on VP 3d and/or VPnd, n>=3, where i was the first author)
=====================================
"VP2d glob and loc in t = results, forces, anteriorities, dates etc. chemin jy /serfati p / bertozzi&constantin etc" = (intend to be) said to researchers on VP2d glob and loc in t = oct 2018= from SERFATI philippe (see researchgate for full texts etc) .(PhD+Ec.Norm.Sup.-PS+AGREGATION in MATHS. 245 citations by 125 international university authors) =
because you cite some papers on contour dynamics and (stratified/striated foliated) vortex patches in 2d, and i have a lot of works on VP, see eg them in full texts on researchgate profile =1/ for globality in t of VP2d, if you do cite jy-chemin/"persist.." and/or bertozzi&C then you HAVE at least to cite my CRAS94/vp2glob (=Une preuve directe d'existence globale des vortex patches 2D Jan 1994 Philippe Serfati) with the same force than chemin/persist (then him and me far stronger me than BC, see eg 3/) and furthermore i am "short, elementary and complete/self-contained and eg with NO paradiff" (but i am posterior to C/P and BC= my position is then in force and/or importance n°= 2 or 1 and not at all 3), 2/ as jy chemin and his students and others authors say it later in their books and papers and as the editor of my cras authorise me to say it.3/ chemin and me, have an infinity of Cm+s strata (and general discontinuous data Cm+s on them), all m, and BC only one unique C1+s (or Cm+s?, see below) stratum and only a constant by pieces discontinuous data out of this one stratum and furthermore BC is NOT self-contained because they (say they) admit a non-trivial lemma due to cotlar (with the remarks = chemin and me, use for general m, their previous results local in t, eg for me my ch4b3-91 of thesis9-92, in nd, all n, and him for n=2 but i don't know for the thesis by bertozzi, if she have m=1 or all m and non?-constant VP/contour dynamics).4/ the strongest vp2glob-in-t, always in 2018, is one of my theorems (ie a part of my th3/2.b).) in "Vortex patches dans Rn et régularité stratifiée pour le laplacien"/Oct 1993 (see full tx on my profile on rg), eg on every separate points = on transverse regularities far more general than those by jy chemin and anyone else even in 2018 (2d and nd), on lagrangian holomorphies/analyticities in t,r, (and functional ones on) uo, edos, on anisotropic regs for singular integrals and iterate commutators, then all this on lR^n, all n etc,5/ on local in t, vp2d, i was in "vp bidi"/2-1990/ch4a of my thesis9-92, first and simultaneous and independant with and from jy chemin/loc-t-vp2d/1or6-90 (see eg p. gerard/bourbaki1992, http://www.numdam.org/article/SB_1991-1992__34__411_0.pdf), him foliated, lagrangian t-C°°, with an epsilon loss in x-reg, me contour dynamic nearly circular but with lag. t-analyticity with no epsilon loss in x-reg (see also the thesis1991 by a. bertozzi but i don't have it in hand).6/ tell me what you will do.
=========================================
"VPnd, all n>=3 or n=3 = results, forces, anteriorities, dates etc. serfati p / gamblin-stRaymond / danchin / etc" =
on VPnd = (intend to be) said to researchers on VP = rct*9+13/10/18 = from SERFATI philippe (PhD+Ec.Norm.Sup.-PS+AGREGATION in MATHS. 245 citations by 125 international university authors)  =(intend to be) said to researchers on VP = 
 = i reput below my tx on VP3d and nd, all n>=3, and i still say to you my remarks on your/this last version(s) of your VPnd-paper(s), n=3 or n>=3 = you have to say that, for all n (and even also restricted to 3d), i am and i was and i was the 1st in date on VP3d and VPnd, all n>=3 and (in ch4b/3-91 of thesis (at the time, only VP2d and jy chemin existed then, only for +-one year and 2d only (with also on VP2d on one single curve, my vpbidi2-90/2d/ch4a and bertozzi/2d/thesis/91)) etc and eg its extended rework/10-93postdoc (and not its "correction") ...) i was and i am still the strongest in 2018 on them (even by points taken separately. my cras94reg.strat.euler3d.gd.tps contains only partial things = a partial "posterior" "announcement" of this ch4b (and ch4a), and in VP, only a full proof of an only axisym (but t-glob) particular VP-result, only on 3d, with no extra higher (or no the most general transverse) x-regs and no lagrangian t-analytic t-regs or no (t,r,uo,EDOs) scalar and functional lag-holomorphies etc) 
--tell me what you will do (eg because you treat, describe, cite etc papers and results etc on VPnd, all n>=3 or n=3, then you have to cite my papers on VPnd, all n>=3 or n=3) = 
+Vortex Patches in lR^n, n in [3,4,5, ... [ and n=3 (and not n=2)= forces and anteriorities and authors and works. ..see also eg other previous §s on comments (rg, scirate..) on eg papers by bae&Kelliher, danchin, sueur, fanelli, paicu, dutrifoy, c.huang, gamblin& x.StRaymond, P. Zhang, Q. J.Qiu, X.Zhang, x.Liao, L. Yanlin etc =24/8/18 = on authors on VP nd, all n, or n =3 = on VP nd, all n (and so eg n in [3,4,5, ... [, where I was absolutely present, the only one and the first in dates (and what is more, always the strongest in 2018, even by points / criteria taken separately), which all must recognize and say, eg all n, all Cm+s, all m, on transverse regularities, holomorphies/analyticities in t, r (functional ones in) uo, edos and these anisotropic regs for singular integrals and iterate commutators etc), in my ch4b / 3-91 (of thesis9-92) and its extended reworking ""Vortex patches dans Rn et régularité stratifiée pour le laplacien""/ 10-93 (:"" Vortex patches in Rn and stratified regularity for the Laplacian ""), all the 3, previous to 32 (/ 14) and 1 month(s) to gamblin & X.St-Raymond / vp3d / 11-93 (the only 1sts after me in Vp nd, n in [3,4,5, ... [), I described for VP Rn, all n and thus (eg said to bae&kelliher) in Rn, all n, eg also we have already in ch4b (with as an example biot-savart :) D2 (lap ^ -1) = finite sum of I times P, where I are operators (non-local) isotropic Cm + s (and sending and raising an anisotropic space towards the iso-Cm + s) and P explicit, basic (and thus discontinuous) pointwise operators, being only rational fractions of elements of this anisotropic space = this in Rn, all n (and not only, as eg bae & kelliher to read them, n = 2 or 3 alone and with only references to my 2 cras94 v2dglob and v3daxiglob, the last one is an announcement of only a part of the ch4b/91+ch4a/90 of thesis92 plus 2 full proved non-signed (particular) axi3d results and eg Gamblinx/StR (paper, ch and/of thesis) always cited by all, had also its own CRAS of announcement, almost never cited (in "replacement")).
.rct 24/9/18 = the thesis by gamblin dates from 11-93 but its chp / vp3d /a-preprint/pre-version? from 01-93 ((but said submitted in 11-93 by the review, 10 months latter)), whose (2) only new results are the t-global cases in axi iso-quasilip and axiVp, curl> 0, all because it is said in my cras94reg.strat.lap.euler said submitted in 01-93, which improves these 2 points to the unsigned case, this chp is weaker on everything and is posterior, the 2 to my ch4b /10-91 of my thesis 9-92, ch4b, which my rework & extension / 10-93 does not owe anything to the ch/vp3d/gamblin/, that is the reason why I do not quote nor describe it (what I do, at the opposite, in my cras, and even beyond these 2 points, then there with no obligation, seen that my ch4b is anterior to and contains his ch, except his 2axi results).
=======================================================
--my 4 works on VP (2+3+nd) with cusps, regular, irregular, singular, generalized =// Solutions non bornées de l'équation d'Euler Incompressible 3D et singularités interfaciques Article Full-text available Feb 1999 Philippe Serfati // Singularité logarithmique confinée dans un cusp régulier et équation d'Euler 2D Article Full-text available Dec 1998 Philippe Serfati // Stratifications planes superposées singulières et équation d'Euler 2D à temps grand Article Full-text available Oct 1998 Philippe Serfati// Stratifications superposées singulières. analyse harmonique. commutateurs et équation d'Euler sur Rn Article Full-text available Jul 1998 Philippe Serfati //
==========================================
--- on splash discontinuities, nd-cusps and gains for them in term of harmonic analysis =..., my last 4 papers 1998-99 on it and the last 2 ("2d sing cusp log" and "3d underlip sing interfaciques") deal with more singular situations (unless I'm mistaken) than usually do = i see cusps as superposition of 2 VP with tangent contacts/ intersections of their boundaries = more singular sols, ie VPs more singular near their (more singular) contacts, non-boundedness of/and added singularities near the contacts, on the gradients g etc of the boundaries and also on the functions f defined between these boundaries= "sing log cusp 2d" (f=log, regular cusp, ie g reg), "3d sing interfaciques" (exp(c.lfl^µ) in L1 and exp(c.lgl^µ') in L1, c and µ,µ'>0)), these situations have a gain in term of harmonic analysis = eg, lD2(Lap-1)*f lp < p^b Clflp, p-->oo, with b (>0 or =0) better than the usual b=1, ie b = 0 for a regular cusp (or inside space between 2 tangents hypersurfaces where f has its support and is in an Lp, f finite) or b <1 strictly for some irregular cusps etc (b depending µ' of exp(c.lgl^µ')) in L1, g = eg the gradient of the parametrizations of the (branches of) cusps, g being non-bounded at the approaches of the contacts or the peaks).
=======================================================
------ .Exemples et details sur les regularités transverses geometriques anisotropes pour les integrales singulieres et commutateurs itérés +- abstraits et vortex patches, le tout sur lR^n (6/18)= sur mes regularités stratifiées (91ch4bThese92 + vpRn/postdoc93) pour les vp nd tout n mais aussi pour des integrales singulieres et commutateurs itérés +- abstraits, ie tous k en Ck+s, reg-x transverses (admissibles) aux VPs les + generales et les + localisées, regs lagrangiennes holomorphes en t, uo, en edos, r-parametriques, scalaires et fonctionnelles, et le fait qu'ils sont plus forts et de loin et à tous points de vues sur tout ce qui existe toujours en jn 2018, .. = sur le point des reg-x transverses (admissibles) aux VPs les + generales et les + localisées = la reg transverse pour chaque k' dans (1,k) peut etre toute algebre Ak' (pour le produit ponctuel des fctions) incluse dans L°°(Rn) entre L°°(Rp,C^(k'+s)(R(n-p)), tout p dans (0,n-1) et C^(k'+s)(Rn) (:integrales singulieres et commutateurs itérés, tous ayant pour image regularisante l'espace isotrope C^(k'+s)(Rn)) et p dans (0,1) pour les vp et les Ak' conservées et transportées (en lagrange) de t= 0 à t pas nul = eg on peut envisager des algèbres Ak' (de L°°) construites sur Rp (p comme + haut) à partir 1/ d'espaces d'indices variables s(x) ou q(x) (L^q(x) etc), C^s(x), sobolev W^(s(x),q(x)) et idem les besovs.. ou encore 2/ d'espaces eux memes de fonctions regulieres (eg plateaux) sur des sous ensembles nonconnexes de Rp, discontinus à travers leurs bords (:bords de reg quelconques), ie encore des nouvelles structures "VP" mais sans reg min pour les bords B et les densités D sur ces bords (mais les (mauvaises) regs de B et D seront conservées au final sans hypothese sur elles, par ces operateurs et/ou de t=0 à t pas nul) etc ou 3/ des reg en 1 point ou 1 ensemble discret de points de Rn, eg ponctu-Cs'(xo) = lfx-fxol < lx-xol^s(xo) etc = tout ce que personne n'a aussi en 6/2018 et bien sur tout ceci à C(k+s) deformations diffeomorphiques prealables de Rn ainsi aussi à des partitions et localisations prealables de Rn (combinées à ces deformations)= Union_i Oi, i dans lN, Oi ouverts avec des algebres A_(k',i) pour chaque k',i etc, modulo des compatibilités sur les intersections non vides entre 2 Oi et Oj. =============================================
Relevant answer
Answer
(here copy of my mel to all the 8 members of the jury and the author of this thesis by Xin Zhang, thesis mainly on striated regs, vortex patches etc=) *6nv18/ = "VP 2d 3d nd= chemin (2d) / serfati (2+3+nd) / bertozzi (&constantin) (2d)/ gamblin & straymond (3d) / danchin (2+3+nd) etc". from SERFATI philippe (see researchgate for full texts etc) .(PhD+Ec.Norm.Sup.-PS+AGREGATION in MATHS. 245 citations by 125 international university authors) = to all the 8 members of the jury and the author of the thesis (see file below or the adweb = https://www.researchgate.net/deref/https%3A%2F%2Fwww.theses.fr%2F2017PESC1215.pdf) = Xin ZHANG Étude qualitative des solutions du système de Navier-Stokes incompressible à densité variable Spécialité : Mathématiques Soutenue le 29 September 2017 devant un jury composé de : Directeur de thèse DANCHIN Raphaël (Université Paris-Est Créteil) Rapporteur ALAZARD Thomas (l’ENS Paris-Saclay) ZHANG Ping (Institute of Mathematics, AMSS) Président du jury CHEMIN Jean-Yves (Université Pierre et Marie Curie) Examinateur DALIBARD Anne-Laure (Université Pierre et Marie Curie) GALLAGHER Isabelle (Université Paris Diderot) LACAVE Christophe (Université Grenoble Alpes) = (=chemin@ann.jussieu.fr, raphael.danchin@u-pec.fr, xinzhang@univ-paris-est.fr, christophe.lacave@imj-prg.fr, zp@amss.ac.cn, dalibard@ljll.math.upmc.fr, gallagher@math.ens.fr, Thomas.Alazard@ens.fr).--on the 130 refs of this thesis, at least +-20/25% are on Vortex patches (by 25/30 authors), 2d, 3d, nd, general and regular plus VP2d with cusps (thesis where my name or my works do not appear, even on these subjects)= i recall eg that i have always in 2018, even by points taken separately, the strongest results in vp2d loc in t, vp2d global in t (but jy chemin and bertozzi&constantin were first for vp2d glob), the very firsts and still the strongest results in vp3d and vpnd, n>=3, and eg i have 4 works on vp with cusps and +-10 works or sub-chapters of my thesis on VP and +-5 of them published, +-still far the strongests in 2018. here is below, some of my short texts detailling some of this (some others exist also), and it is said and intend to be said, repeatedly, to all authors on VP, and eg (already) to (some of) you all (and txs eg compiled on my data = ""rg-texts on papers on anisotropies, morrey spaces and elliptic problems and holomorphies on opens and vortex patches machineries etc, 2017-18 from Serfati-philippe October 2017 DOI: 10.13140/RG.2.2.35956.04482/"" and/or foundable in diverse places of my profiles (eg) on RG etc) = tell me what you will do (eg on VP 3d and/or VPnd, n>=3, where i was the first author)
=====================================
"VP2d glob and loc in t = results, forces, anteriorities, dates etc. chemin jy /serfati p / bertozzi&constantin etc" = (intend to be) said to researchers on VP2d glob and loc in t = oct 2018= from SERFATI philippe (see researchgate for full texts etc) .(PhD+Ec.Norm.Sup.-PS+AGREGATION in MATHS. 245 citations by 125 international university authors) =
because you cite some papers on contour dynamics and (stratified/striated foliated) vortex patches in 2d, and i have a lot of works on VP, see eg them in full texts on researchgate profile =1/ for globality in t of VP2d, if you do cite jy-chemin/"persist.." and/or bertozzi&C then you HAVE at least to cite my CRAS94/vp2glob (=Une preuve directe d'existence globale des vortex patches 2D Jan 1994 Philippe Serfati) with the same force than chemin/persist (then him and me far stronger me than BC, see eg 3/) and furthermore i am "short, elementary and complete/self-contained and eg with NO paradiff" (but i am posterior to C/P and BC= my position is then in force and/or importance n°= 2 or 1 and not at all 3), 2/ as jy chemin and his students and others authors say it later in their books and papers and as the editor of my cras authorise me to say it.3/ chemin and me, have an infinity of Cm+s strata (and general discontinuous data Cm+s on them), all m, and BC only one unique C1+s (or Cm+s?, see below) stratum and only a constant by pieces discontinuous data out of this one stratum and furthermore BC is NOT self-contained because they (say they) admit a non-trivial lemma due to cotlar (with the remarks = chemin and me, use for general m, their previous results local in t, eg for me my ch4b3-91 of thesis9-92, in nd, all n, and him for n=2 but i don't know for the thesis by bertozzi, if she have m=1 or all m and non?-constant VP/contour dynamics).4/ the strongest vp2glob-in-t, always in 2018, is one of my theorems (ie a part of my th3/2.b).) in "Vortex patches dans Rn et régularité stratifiée pour le laplacien"/Oct 1993 (see full tx on my profile on rg), eg on every separate points = on transverse regularities far more general than those by jy chemin and anyone else even in 2018 (2d and nd), on lagrangian holomorphies/analyticities in t,r, (and functional ones on) uo, edos, on anisotropic regs for singular integrals and iterate commutators, then all this on lR^n, all n etc,5/ on local in t, vp2d, i was in "vp bidi"/2-1990/ch4a of my thesis9-92, first and simultaneous and independant with and from jy chemin/loc-t-vp2d/1or6-90 (see eg p. gerard/bourbaki1992, http://www.numdam.org/article/SB_1991-1992__34__411_0.pdf), him foliated, lagrangian t-C°°, with an epsilon loss in x-reg, me contour dynamic nearly circular but with lag. t-analyticity with no epsilon loss in x-reg (see also the thesis1991 by a. bertozzi but i don't have it in hand).6/ tell me what you will do.
=========================================
"VPnd, all n>=3 or n=3 = results, forces, anteriorities, dates etc. serfati p / gamblin-stRaymond / danchin / etc" =
on VPnd = (intend to be) said to researchers on VP = rct*9+13/10/18 = from SERFATI philippe (PhD+Ec.Norm.Sup.-PS+AGREGATION in MATHS. 245 citations by 125 international university authors)  =(intend to be) said to researchers on VP = 
 = i reput below my tx on VP3d and nd, all n>=3, and i still say to you my remarks on your/this last version(s) of your VPnd-paper(s), n=3 or n>=3 = you have to say that, for all n (and even also restricted to 3d), i am and i was and i was the 1st in date on VP3d and VPnd, all n>=3 and (in ch4b/3-91 of thesis (at the time, only VP2d and jy chemin existed then, only for +-one year and 2d only (with also on VP2d on one single curve, my vpbidi2-90/2d/ch4a and bertozzi/2d/thesis/91)) etc and eg its extended rework/10-93postdoc (and not its "correction") ...) i was and i am still the strongest in 2018 on them (even by points taken separately. my cras94reg.strat.euler3d.gd.tps contains only partial things = a partial "posterior" "announcement" of this ch4b (and ch4a), and in VP, only a full proof of an only axisym (but t-glob) particular VP-result, only on 3d, with no extra higher (or no the most general transverse) x-regs and no lagrangian t-analytic t-regs or no (t,r,uo,EDOs) scalar and functional lag-holomorphies etc) 
--tell me what you will do (eg because you treat, describe, cite etc papers and results etc on VPnd, all n>=3 or n=3, then you have to cite my papers on VPnd, all n>=3 or n=3) = 
+Vortex Patches in lR^n, n in [3,4,5, ... [ and n=3 (and not n=2)= forces and anteriorities and authors and works. ..see also eg other previous §s on comments (rg, scirate..) on eg papers by bae&Kelliher, danchin, sueur, fanelli, paicu, dutrifoy, c.huang, gamblin& x.StRaymond, P. Zhang, Q. J.Qiu, X.Zhang, x.Liao, L. Yanlin etc =24/8/18 = on authors on VP nd, all n, or n =3 = on VP nd, all n (and so eg n in [3,4,5, ... [, where I was absolutely present, the only one and the first in dates (and what is more, always the strongest in 2018, even by points / criteria taken separately), which all must recognize and say, eg all n, all Cm+s, all m, on transverse regularities, holomorphies/analyticities in t, r (functional ones in) uo, edos and these anisotropic regs for singular integrals and iterate commutators etc), in my ch4b / 3-91 (of thesis9-92) and its extended reworking ""Vortex patches dans Rn et régularité stratifiée pour le laplacien""/ 10-93 (:"" Vortex patches in Rn and stratified regularity for the Laplacian ""), all the 3, previous to 32 (/ 14) and 1 month(s) to gamblin & X.St-Raymond / vp3d / 11-93 (the only 1sts after me in Vp nd, n in [3,4,5, ... [), I described for VP Rn, all n and thus (eg said to bae&kelliher) in Rn, all n, eg also we have already in ch4b (with as an example biot-savart :) D2 (lap ^ -1) = finite sum of I times P, where I are operators (non-local) isotropic Cm + s (and sending and raising an anisotropic space towards the iso-Cm + s) and P explicit, basic (and thus discontinuous) pointwise operators, being only rational fractions of elements of this anisotropic space = this in Rn, all n (and not only, as eg bae & kelliher to read them, n = 2 or 3 alone and with only references to my 2 cras94 v2dglob and v3daxiglob, the last one is an announcement of only a part of the ch4b/91+ch4a/90 of thesis92 plus 2 full proved non-signed (particular) axi3d results and eg Gamblinx/StR (paper, ch and/of thesis) always cited by all, had also its own CRAS of announcement, almost never cited (in "replacement")).
.rct 24/9/18 = the thesis by gamblin dates from 11-93 but its chp / vp3d /a-preprint/pre-version? from 01-93 ((but said submitted in 11-93 by the review, 10 months latter)), whose (2) only new results are the t-global cases in axi iso-quasilip and axiVp, curl> 0, all because it is said in my cras94reg.strat.lap.euler said submitted in 01-93, which improves these 2 points to the unsigned case, this chp is weaker on everything and is posterior, the 2 to my ch4b /10-91 of my thesis 9-92, ch4b, which my rework & extension / 10-93 does not owe anything to the ch/vp3d/gamblin/, that is the reason why I do not quote nor describe it (what I do, at the opposite, in my cras, and even beyond these 2 points, then there with no obligation, seen that my ch4b is anterior to and contains his ch, except his 2axi results).
=======================================================
--my 4 works on VP (2+3+nd) with cusps, regular, irregular, singular, generalized =// Solutions non bornées de l'équation d'Euler Incompressible 3D et singularités interfaciques Article Full-text available Feb 1999 Philippe Serfati // Singularité logarithmique confinée dans un cusp régulier et équation d'Euler 2D Article Full-text available Dec 1998 Philippe Serfati // Stratifications planes superposées singulières et équation d'Euler 2D à temps grand Article Full-text available Oct 1998 Philippe Serfati// Stratifications superposées singulières. analyse harmonique. commutateurs et équation d'Euler sur Rn Article Full-text available Jul 1998 Philippe Serfati //
==========================================
--- on splash discontinuities, nd-cusps and gains for them in term of harmonic analysis =..., my last 4 papers 1998-99 on it and the last 2 ("2d sing cusp log" and "3d underlip sing interfaciques") deal with more singular situations (unless I'm mistaken) than usually do = i see cusps as superposition of 2 VP with tangent contacts/ intersections of their boundaries = more singular sols, ie VPs more singular near their (more singular) contacts, non-boundedness of/and added singularities near the contacts, on the gradients g etc of the boundaries and also on the functions f defined between these boundaries= "sing log cusp 2d" (f=log, regular cusp, ie g reg), "3d sing interfaciques" (exp(c.lfl^µ) in L1 and exp(c.lgl^µ') in L1, c and µ,µ'>0)), these situations have a gain in term of harmonic analysis = eg, lD2(Lap-1)*f lp < p^b Clflp, p-->oo, with b (>0 or =0) better than the usual b=1, ie b = 0 for a regular cusp (or inside space between 2 tangents hypersurfaces where f has its support and is in an Lp, f finite) or b <1 strictly for some irregular cusps etc (b depending µ' of exp(c.lgl^µ')) in L1, g = eg the gradient of the parametrizations of the (branches of) cusps, g being non-bounded at the approaches of the contacts or the peaks).
=======================================================------ .Exemples et details sur les regularités transverses geometriques anisotropes pour les integrales singulieres et commutateurs itérés +- abstraits et vortex patches, le tout sur lR^n (6/18)= sur mes regularités stratifiées (91ch4bThese92 + vpRn/postdoc93) pour les vp nd tout n mais aussi pour des integrales singulieres et commutateurs itérés +- abstraits, ie tous k en Ck+s, reg-x transverses (admissibles) aux VPs les + generales et les + localisées, regs lagrangiennes holomorphes en t, uo, en edos, r-parametriques, scalaires et fonctionnelles, et le fait qu'ils sont plus forts et de loin et à tous points de vues sur tout ce qui existe toujours en jn 2018, .. = sur le point des reg-x transverses (admissibles) aux VPs les + generales et les + localisées = la reg transverse pour chaque k' dans (1,k) peut etre toute algebre Ak' (pour le produit ponctuel des fctions) incluse dans L°°(Rn) entre L°°(Rp,C^(k'+s)(R(n-p)), tout p dans (0,n-1) et C^(k'+s)(Rn) (:integrales singulieres et commutateurs itérés, tous ayant pour image regularisante l'espace isotrope C^(k'+s)(Rn)) et p dans (0,1) pour les vp et les Ak' conservées et transportées (en lagrange) de t= 0 à t pas nul = eg on peut envisager des algèbres Ak' (de L°°) construites sur Rp (p comme + haut) à partir 1/ d'espaces d'indices variables s(x) ou q(x) (L^q(x) etc), C^s(x), sobolev W^(s(x),q(x)) et idem les besovs.. ou encore 2/ d'espaces eux memes de fonctions regulieres (eg plateaux) sur des sous ensembles nonconnexes de Rp, discontinus à travers leurs bords (:bords de reg quelconques), ie encore des nouvelles structures "VP" mais sans reg min pour les bords B et les densités D sur ces bords (mais les (mauvaises) regs de B et D seront conservées au final sans hypothese sur elles, par ces operateurs et/ou de t=0 à t pas nul) etc ou 3/ des reg en 1 point ou 1 ensemble discret de points de Rn, eg ponctu-Cs'(xo) = lfx-fxol < lx-xol^s(xo) etc = tout ce que personne n'a aussi en 6/2018 et bien sur tout ceci à C(k+s) deformations diffeomorphiques prealables de Rn ainsi aussi à des partitions et localisations prealables de Rn (combinées à ces deformations)= Union_i Oi, i dans lN, Oi ouverts avec des algebres A_(k',i) pour chaque k',i etc, modulo des compatibilités sur les intersections non vides entre 2 Oi et Oj. =============================================
  • asked a question related to Mathematical Physics
Question
3 answers
I am working on the finding the number of pores (cells) and cell wall thickness of Aluminum foams during and after solidification.
It has been established that the amount of foaming agent (TiH2) and the temperature of the molten metal are responsible for the initiation and growth of the pores. My question is that what is the phenomenon governing this process? are there any mathematical or physical laws that can be applied to predict the number of stable pores after solidification? How to find the distribution the cell wall thickness?
Relevant answer
Answer
Thank you all for your responses.
  • asked a question related to Mathematical Physics
Question
117 answers
In Euclidean geometry, one studies the basic elemental forms in one, two or three dimensions as lines, surfaces and volumes along with their angular correlations and other metric attributes. In differential geometry, one studies forms and their evolution from simple to complex thru the analysis of curvature as provided by differential calculus. In my view, the most pivotal element that grounds curvature analysis is the figure of functional derivative which equates the angular behavior of a tangent line sweeping over the function. While Euclidean geometry is a science of old, differential geometry is a 19th century newcomer. What is the nature of the connection between Euclidean geometry and differential geometry? We know that Riemannian geometry generalizes Euclidean geometry to non-flat or curved spaces. Yet Riemannian spaces still resemble the Euclidean space at each infinitesimal point (in the first order of approximation). Further several important Euclidean constructs such as the arc length of one-dimensional curves, area of plane regions, and volume of solids do possess natural analogues in Riemannian differential geometry.
Is differential geometry more general or just complementary to Euclidean geometry? Can the first become a complete substitute of the second? What is the nature of the connection between the two in pure geometric abstraction? Why is Euclidean geometry insufficient to the description of the natural world in its geometric aspects, if it is? In the paper below, I discuss these issues from a mathematical-physics viewpoint while presenting a novel approach to differential geometry and its application:
Relevant answer
Answer
Ed Gerck - I hear you. You probably know more about their internal policy than I do since you’ve been tracking them for long. At the moment Google is serving content from their Knowledge Base left and right. And I can assure you there is much more harm coming from certain quarters of the scholarly world than from Wikipedia. Given how difficult it is to build such platforms with little support from those who first benefit from it, let’s not detract the overall effort. But given the imperfections that you rightly pointed out, let’s call them work in progress
That was a parenthesis. Let’s now go back to the business of geometry.
Cordially.
  • asked a question related to Mathematical Physics
Question
80 answers
Mathematics has been always one of the most active field for researchers but the most attentions has gone to one or few subjects in one time for several years or decades. I'd like to know what are the most active research areas in mathematics today?
Relevant answer
Answer
Yes, mathematics has been always not only one of the most active field of the researchers, it was for a long time along with philosophy one of the first sciences. But, it’s hard to say what are the most active research areas in mathematics today or what are the most important scientifically explored in mathematics. Less and less support is provided for purely fundamental mathematics is nowadays, and more and more is required to solve specific problems by "someone else" i.e. mathematics turns into a servant of other sciences
  • asked a question related to Mathematical Physics
Question
3 answers
(abstract 2017) Stratified regularity and 3D Euler equation at large time Abstract — We solve the Euler equation 1/ in all dimension at small time, for data with stratified regularity, D2Δ-1 conserving this type of regularity (: only an announcement of a part of our Ch4B, thesis, 92, univ Paris 6-PMC, France, see also below) and 2/ we give an almost (ie except only on a simple estimate on a regularization of the initial axi-data and classical maths on euler eqs) self contained proof, with no paradifferential-calculus, at large time in the 3D axiymmetrical case (and newly without condition of sign linked to the curl), firstly for general under-lipchitz speeds secondly in the vortex patches 3d frame (VP3 :ie local C1+s(R3) deformations of L°°(R,C1+s(R2))), precising fully and newly in this last case, the matricial discontinuous part of the gradient of the speed. our here well existing under-lipchitz axi3d solutions are, by our other "Pertes de régularité pour le laplacien et l'équation d'Euler sur Rn", lagrangian C°° in time. Ch4B is refined, reworked, extended and included in results of the paper = ""Vortex patches dans Rn et régularité stratifiée pour le laplacien"" (but ch4B has still a remaining interest on small differences in some approaches). ---- our axi part here, in particular, on the novelty of the absence of condition of sign linked to the curl is +- "similar, simultaneous and independent" (at least for me), to Saint Raymond, X. (1994). Remarks on axisymmetric solutions of the incompressible Euler system. Communications in partial differential equations, 19(1-2), 321-334, this one extending and completing the other paper on its axi-part with the condition of sign linked to the curl = Gamblin, P., & Saint Raymond, X. (1995). On three-dimensional vortex patches. Bulletin de la Société Mathématique de France, 123(3), 375-424. ------- 15 to 25 citations (with google (scholar) and me). GS14, ap17.
Relevant answer
Answer
(intend to be) said and asked to (certain and/or) all authors (on their papers) on euler (and nse) on 3Daxi sols (without swirl) eg when x.st-Raymond1994 (on euler) is (/should be) cited, eg on extension to nse (eg major papers on nse on 3Daxi sols (without swirl))=
/under-quasi-log lip vp3d axi glob in t no positive curl condition 1st result serfati ph.1.93-94/= from SERFATI Philippe, PhD+ Ec.Norm.Sup.-PS+AGREGATION in MATHS. 240 citations by 120 international university authors. --about your paper(s) on the Axisymmetric 3D Euler (+ns) Equations Without Swirl, see my "Serfati, P. (1994). Régularité stratifiée et équation d'Euler 3D à temps grand. CR Acad. Sci. Paris Ser. I Math, 318, 925-928."/submitted1-93 (see full text on my profile on researchgate or on the site gallica and the/its 14 citations, sp18, according to my google citations), and its theorem 3, where i give a short self contained proof (except the last line at the end of p926 on a basic regularization of the initial data) with no paradifferential calculus and no positivity condition on the initial curl, for global existence of 3d euler eqs axisymmetrical case with no swirl and for under-lipschitz speeds and (L1 and L°°) discontinuous bounded (initial) curls with then NO positivity condition (: curl_o.(-x2,x1,0) >=0, condition asked before with also here over-lipschitz speeds (a/ isotrope + b/ aniso/VP), by eg a/ majda/cpam1986 (see my ref [4]) and a very short time before me, by b/ gamblin&x.st-raymond, my ref [3]/preprint1-93), simultaneously and independently (at least for me) of the classical/major reference on these precise non-signed points: "Saint Raymond, X. (1994). Remarks on axisymmetric solutions of the incompressible Euler system. Communications in partial differential equations, 19(1-2), 321-334."/5-93 (see eg :https://www.tandfonline.com/doi/abs/10.1080/03605309408821018 which says the receiving date "5-93" = then 4 months after my own submitted date, and(/but?) i remember that i went to a talk (because i was just aware of it) around these subjects made by x. st R. to whom i gave my finalized paper, him and gamblin, in their previous (:for x.st-R) paper together, didn't ((be able to) (think to)) prove the/my 2 results (iso quasilip +VPaniso overlip) without the positivity condition)= --my th3 is then the 1st historical result with these natural conditions (ie under/quasi/log- lip axi 3d global in t for euler (with no positivity condition on the curl_o)), then i believe that you have to cite my paper. --the/my other ths (fully proved or announced) here are of independant interests on striated x-regs (eg the b/ said before but always with no positivity on the curl_o, fully proved, this case proved again also in "Saint Raymond, X. (1994). Remarks..". --tell me what you will do.
  • asked a question related to Mathematical Physics
Question
6 answers
Supersymmetry theory proposes that the super partners of the existing bosons and fermions are extinguished through some symmetry breaking mechanism… though there are some articles that offers few or no exotic particles other than the SM particles, but it would also be the case (in the theoretical model) that the symmetry breaking may cause polarization either only towards bosons or fermions… if not, why/how nature would choose which fermions and bosons among the super particles are to survive? Further, is it at least theoretically possible that the existing bosons and fermions have their super partners within the observed fermions and bosons? I do understand that in the second proposal the constraint of equal mass will be violated.
Relevant answer
Answer
So far, the Supersymmetry has not been very succsesful in physics. However, mathematically, it is a very beautiful subject, which may explain why it stood so long with theoretical physics.
  • asked a question related to Mathematical Physics
Question
1 answer
Physical mechanism underlying the relation between t, J and U, then the questions are as follows:
For the 1/2 filling: When ε_{d} is in between -U and 0, we are in the local moment (intermediate) regime in Coulomb blockade staircase, If we don't have any hopping, we have double degeneracy. The ground state -which is characterized by occupancy- is not only the single state but two states anymore.
In T/|t|-U/|t| planes: At very high temperature we have some sort of classical phase which is the mixture of those corresponding four states (LM, Mixed valance regime and the other two: |up,up>, |0>). Now, in the large repulsion limit, let's get temperature gradually down. We cooldown the system, thus we first hit U. It means that some of the states (zero and double occupied) will become eliminated from our configuration space. So, in the low energy Hilbert space will only contain up and down. In the related diagram, automatically we have a boundary (crossover) where T~U. Below this crossover, charge fluctuations are frozen out and only up and down remain in the Hilbert space. In each sites are occupied by only up and down electrons and these guys have really hard time hopping. Because, it's really difficult. You need to create a charge fluctuation. So, the only hopping process that you have to go from the case which the each site is occupied by a single electron to the case which a single site is double occupied with opposite spins due to the Pauli exclusion principle. It means that a low energy scale will be generated in this regime which is the order of square of the matrix element for the hopping devided by the energy that you have to pay for the hopping. Some low energy scale will be generated which is order of the square of t devided by U. But this is a very low scale. And again we have a boundary in low temperature and high repulsion regime order of t^2/U. In between these two boundaries, we have a local moment paramagnet state with frozen charge fluctuations. Charge fluctuations are protected by a gap (can be seen in Coulomb blockade staircase) which is order of repulsion as Δq~U. In this state the material is a Mott insulator. We still have a very large configuration space of the order of 2^n. Above the crossover line we had 4^n possible configurations. So, in the low temperature something must happen to this 2^n. We can not keep the degerate GS of this 2^n. To avoid this, usually spins are magnetically ordered except some special cases. The ~t^2/U boundary -which is the line we broke in low temperature and high repulsion regime- is the characteristic energy scale for magnetic ordering.
Why this phenomena implies this energy scale? This related to the degenerate perturbation theory. For simplicity consider a dimer with no hopping. We have two site. We can organize these two site for four states of |sigma, sigma'> by using the quantum number of total spin. The manifold of these four states is actually made of the one singlet state which S=0 (opposite spins) and the 3 (degenerate) triplet states which S=1. When hopping is zero these 3 three states are degenerate. When hopping is nonzero these four states will split. They will obviously split by favoring the singlet as the lowest energy state. Because this guys allows <updown,0| Hopping |singlet> is a nonzero matrix element. In the presence of a nonzero hopping these two states in the expectation value of hopping are both singlet so automatically we have nonzero matrix element. If you do this for triplet state we can not get nonzero matrix element. If we compare this two expectation value we see that the singlet state is lower by an energy which is the square of the perturbation matrix element t^2 divided by the energy state of the intermediate high energy state, hence we get t^2/U. If you actually carry out the calculation properly for a dimer the energy difference of expectation values getting from singlet and triplet states will be 4t^2/U. This is called antiferromagnetic superexchange J_{AF}. This essentially is the scale of magnetically ordering and for one-band Hubbard model this is AF order. That’s my understanding.
Q1: From above explanation, Should we catch why the Hubbard model turn into the t-J model in infinite U limit? Are there enough evidence for this inference?
Q2: Does Increasing or decreasing temperature directly effect the hopping ability of electron? For example, when the system is heated, the kinetic energy of electrons will become larger and will this cause any changes of the hopping constant strenght or just shows an effect on fluctuations?
Q3: To detect the different phase islands in T/|t|-U/|t| plane, |t| should be selected as independent system parameter. That’s why we scale the other Hamiltonian parameters with |t| for comparison in different energy scales. If the answer of Q2 is “Yes…”, how can we investigate these islands for a fixed value of hopping?
Q4: Q3 is indirectly related to determine the Mott transition point for each different U by using the spectral weight. We need a fixed hopping for this purpose. Again, If the answer of Q2 is “Yes…”, I guess, there should be another justification. Are there any instructions in Luttinger theorem for this?
Q5: It has been claimed that The t-J model was first derived in 1977 from the Hubbard model by Spalek [1, 2]. Or, is this just reflects the spirit of t-J model? Contrary to this, according to some other experts, this model totally has been propounded first by Zhang and Rice to derive explicitly a single-band effective Hamiltonian for the high-T_{c} Cu-oxide superconductors [3]. Which one is duly admitted as the first study?
Q6: Relatedly with Q1; How about a unified t-J-U? Are there any other challenging work in literature except [4]? If there is an exact proof of canonical transformation between Hubbard and t-J model, is this kind of attempt still possible? If the answer is “Yes…”, How do we avoid double counting? How can be made a fine tunning on the ranges of two different interaction (Hubbard U and superexchange) at the same time?
1. K. A. Chao, J. Spalek, A. M. Oles, J. Phys. C 10, L 271 (1977), Phys. Rev. B 18, 3453 (1978).
2. The method has been originally proposed by J. Spalek, A. M. Oles, preprint of Jagiellonian University, SSPJU-6/76 (1976) and Physica B, 86-88, 375 (1977).
3. F. C. Zhang, T. M. Rice, Phys. Rev. B 37, 3759 (1988). arXiv:cond-mat/0303501
Relevant answer
Answer
Dear Behnam Farid, Thank you very much for your brief explanations. They helped me to understand most of my concerns. With my best regards.
  • asked a question related to Mathematical Physics