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# 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

**ESSENTIAL REASON IN PHYSICISTS’ USE OF LOGIC:**

**IN OTHER SCIENCES TOO!**

**Raphael Neelamkavil, Ph.D., Dr. phil.**

**1. The Logic of Physics**Physics students begin with meso-world experiments and theories. Naturally, at the young age, they get convinced that the logic they follow at that level is identical with the ideal of scientific method. Convictions on scientific temper may further confirm them in this. This has far-reaching consequences in the concept of science and of the logic of science.

But, unquestionably,

**the logic behind such an application of the scientific method is**, namely, observe, hypothesize, verify, theorize, attempt to falsify for experimental and theoretical advancements, etc.,*only one manner of realizing*(1)*the ideal of scientific method***.***and*(2)*the more general ideal of reason*But does any teacher or professor of physics (or of other sciences) instruct their students on the advantages of thinking and experimenting in accordance with the above-mentioned fundamental fact of all scientific practice in mind, or make them capable of realizing the significance of this in the course of time? I think, no.

This is why physicists (and for that matter all scientists) fail at empowering their students and themselves in favour of the growth of science, thought, and life. The logic being followed in the above-said mode of practice of scientific method, naturally, becomes for the students the genuine form of logic, instead of being an instantiation of the ideal of logic as reason. This seems to be the case in most of the practices and instruction of all sciences till today. A change of the origin, justification, and significance of the use of logic in physics from the very start of instruction in the sciences is the solution for this problem. The change must be in the foundations.

All humans equate (1) this sort of logic of each science, and even logic as such, with (2) reason as such. Reason as such, in fact, is more generic of all kinds of logic. Practically none of the professors (of physics as well as of other sciences) terms the version of logic of their science as an instantiation of reason, which may be accessed ever better as the science eventually grows into something more elaborate and complex. Physicist gets more and more skilled at reasoning only as and when she/he wants to grow continuously into a genuine physicist.

As the same students enter the study of recent developments in physics like quantum physics, relativity, nano-physics (Greek

*nanos*, “dwarf”; but in physics, @ 10-9), atto-physics (@ 10-18), etc., they forget to make place for the strong mathematical effects that are due by reason of the conceptual and processual paradoxes due to epistemological and physical-ontological difference between the object-sizes and the sizes of ourselves / our instruments. The best examples are the Uncertainty Principle, the Statistical Interpretation of QM, Quantum Cosmology, etc.They tend to believe that some of these and similar physics may defy our (meso-physical) logic – but by this mistakenly intending that all forms of reasoning would have to fail if such instances of advanced physics are accepted in all of physics. As a result, again, their logic tends to continue to be of the same level as has been taken while they did elementary levels of physics.

Does this not mean that the

*ad hoc*make-believe interpretations of the logic of the foundations of QM, Quantum Cosmology, etc. are the culprits that naturally make the logic of traditional physics inadequate as the best representative of the logic of nature? In short, in order to find a common platform, the logic of traditional and recent branches of physics must improve so to adequate itself to nature’s logic.Why do I not suggest that the hitherto logic of physics be substituted by quantum logic, relativity logic, thermodynamic logic, nano-logic, atto-logic, or whatever other logic of any recent branch of physics that may be imagined? One would substitute logic in this manner only if one is overwhelmed by what purportedly is the logic of the new branches of physics. But, in the first place, I wonder why logic should be equated directly with reason. The attempt should always be to bring the logic of physics in as much correspondence with the logic of nature, so that reason in general can get closer to the latter. This must be the case not merely with physicists, but also with scientists from other disciplines and even from philosophy, mathematics, and logic itself.

Therefore, my questions are: What is the foundational reason that physicists should follow and should not lose at any occasion? Does this, how does this, and should this get transformed into forms of logic founded on a more general sort of physical reason? Wherein does such reason consist and where does it exist? Can there be a form of logic in which the logical laws depend not merely on the size of objects or the epistemological level available at the given object sizes, but instead, on the universal characteristics of all that exist? Or, should various logics be used at various occasions, like in the case of the suggested quantum logic, counterfactual logic, etc.?

Just like logic is not to be taken as a bad guide by citing the examples of the many logicians, scientists, and “logical” human beings doing logic non-ideally, I believe that there is a kernel of reason behind physics, justified solely on the most basic and universal characteristics of physical existents. These universals cannot belong solely to physics, but instead, to all the sciences, because they belong to all existents.

This kernel of reason in physics is to be insisted upon at every act of physics, even if many physicists (and other scientists and philosophers) may not ensure that kernel in their work. I shall discuss these possibly highest universals and connect them to logic meant as reason, when I elaborate on:

**3. The Ontology of Physics (in a forthcoming discussion in RG)**The matter on which physicists do logical work is existent matter-energy in its fundamental implications and the derivative implications from the fundamental ones. This is to be kept in mind while doing any logically acceptable work physics, because existent matter-energy corpora in processuality delineate all possible forms of use of logic in physics, which logic is properly to be termed nature’s reason.

Moreover, conclusions are not drawn up by one subject (person) in physics for use by the same subject alone. Hence, we have the following two points to note in the use of logic in physics and the sciences: (1) the intersubjectively awaited necessity of human reason in its delineation in logical methods should be upheld at least by a well-informed community, and (2) the need for such reason behind approved physics should then be spread universally with an open mind that permits and requires further scientific advancements.

These will make future generations further question the genuineness of such logic / specific realization of reason, and constantly encourage attempts to falsify theories or their parts so that physics can bring up more genuine instantiations of human reason. But is such human reason based on the reason active in nature?

Although the above arguments and the following definition of logic in physics might look queer or at least new and unclear for many physicists, for many other scientists, for many mathematicians, and even for many logicians, I define here logic for use in physics as the fundamental aspect of reason that physics should uphold constantly in every argument and conclusion due from it:

**Logic in physics**is (1) the methodological science (2) of approaching the best intersubjectively rational and structural consequences (3) in what may be termed thought (not in emotions) (4) in clear terms of ever higher truth-probability achievable in statements and conclusions (5) in languages of all kinds (ordinary language, mathematics, computer algorithms, etc.) (6) based on the probabilistically methodological use, (7) namely, of the rules of all sensible logics that exemplify the Laws of Identity, Non-contradiction, and Excluded Middle, (8) which in turn must pertain to the direct and exhaustive physical implications of “to exist”.

Here I have not defined logic in physics very simply as “the discipline of the rules of thought”, “the discipline of the methodological approach to truths”, etc., for obvious reasons clarified by the history of the various definitions of logic.

But here comes up another question: Is the reason pertaining to physical nature the same as the most ideal form of human reason? From within the business of physics, how to connect the reason of physical nature with that of humans? I may suggest some answers from the epistemological and ontological aspects. But I would appreciate your responses in this regard too.

**2. The Epistemology of Physics (in a forthcoming discussion in RG)**

**3. The Ontology of Physics (in a forthcoming discussion in RG)**

We assume the answer is no because a minus sign appears to the left:

-h^2/2m (d^2Ψ(x,t)/dx^2]partial)+V(x,t)Ψ(x,t)=ihdΨ(x,t)/dt]partial

And,

-h^2/2m (d^2Ψ*(x,t)/dx^2]partial)+V(x,t)Ψ*(x,t)=-ihdΨ*(x,t)/dt]partial

The question is what is the mathematical/physical meaning of the minus sign?

I found the solutions of the equations D_1(z)=0, D_3(z)=0, D_5(z)=0, D_7(z)=0 and D_9(z)=0 which are x=2logφ=0.9624, x=2.03185, x=2.89218, x=3.68896, x=4.46019 respectively. D_odd(z) functions are the well known Bloch Wigner Ramakrishnan functions introduced by Zagier and z=exp(-x). Do you know if these solutions are related to each other?

Bifurcation is a fascinating concept found in various fields, including mathematics, physics, and biology.

The really important breakthrough in theoretical physics is that the Schrödinger Time Dependent Equation (STDE) is wrong, that it is well understood why is it wrong, and that it should be replaced by the correct Deterministic Time Dependent Equation (DTDE). Unitary theory and its descendants, be they based on unitary representations or on probabilistic electrodynamics, will have to go away. This of course runs against the claims about string and similar theories made in the video. But our claims are a dense, constructive criticism with many consequences. Taken into account if you are concerned about the present and the near future of Theoretical Physics.

Wave mechanics with a fully deterministic behavior of waves is the much needed and sought --sometimes purposely but more often unconsciously-- replacement of Quantism that will allow the reconstruction of atomic and particle physics. A rewind back to 1926 is the unavoidable starting point to participate in the refreshing new future of Physics. Many graphical tools currently exists that allow the direct visualization of three dimensional waves, in particular of orbitals. The same tools will clearly render the precise movement and processes of the waves under the truthful deterministic physical laws. Seeing is believing. Unfortunately there is a large, well financed and well entrenched quantum establishment that stubbornly resists these new developments and possibilities.

When confronted with the news they do not celebrate, nor try to renew themselves overcoming their quantum prejudices. Instead the minds of the quantum establishment refuse to think. They negate themselves the privilege of reasoning and blindly assume denial, or simply panic. The net result is that they block any attempt to spread the results. Accessing funds to recruit and direct fresh talents in the new direction is even harder than spreading information and publishing.

Painfully, this resistance is understandable. For these Quantists are intelligent scientists (yes, they are very intelligent persons) that instinctively perceive as a menace the news that debunk the Wave-Particle duality, the Uncertainty Principle, the Probabilistic Interpretation of wave functions and the other quantum paraphernalia. Their misguided lifelong labor, dedication and efforts --of themselves and of their quantum elders, tutors, and guides-- instantly becomes senseless. I feel sorry for such painful human situation but truth must always prevail. For details on the DTDE see our article

Hopefully young physicists will soon take the lead and a rational wave mechanics will send the dubious and troublesome Quantism to its crate, since long waiting in the warehouse of the history of science.

With cordial regards,

Daniel Crespin

In my paper in Phys. Letts..vol 68A (1978)409-411, I have discussed a metric

projectively related to Friedman/R-W metric with identical geodesics.Questions:

Are there other such solutions for this case or for other conformaly flat

spaces? One such solution defines an infinite succession. Is there a computer

program to find infinite succession of (covariant)Einstein tensors.; the

change represents the erruption of matter-energy in assumed spontaneous

projective change. on approach to a singularity. Physically the change is

caused by intervention of Gauge fields to avoid gravity-induced collapse.

The point is that addition to Christofell connection of a term (Identity tensor

times Vector) leaves a system of geodesics unchanged,and is in accord

with equivalence principle. This way one can relate both gauge field and

Q.M. with G .R..Details on request.See also Matsience Report no92(1978)/

paper 9,14pp (www.imsc.res.in/Library)-Black Body Structure of a Black

Hole. And Lie Structure of Quasiconformal Maps in R*(*=N). And Physics of

String Theory in Quantum Field Theory. QM,& Optics- Ed VV Dodonov &

V I Manko , Moscow (1990) Nova Publishers (N,Y)vol 187 of Proc 0f Lebedev Phy. Inst.Acad. 0f Sciences 0f the USSR./pp113-116..

Qm is the ultimate realists utilization of the powerful differential equations, because the integer options and necessities of solutions correspond to nature's quanta.

The same can be said for GR whose differential manifolds, an sdvanced concept or hranch in mathematics, have a realistic implementation in nature compatible motional geodesics.

1 century later,so new such feats have been possible, making one to think if the limit of heuristic mathematical supplementation in powerful ways towards realist results in physics in reached.

We assume that the statistical weight SW for free nodes in a geometric shape is of extreme importance in mathematics and theoretical physics but is still absent.

However, SW, which is a dimensionless mathematical/physical quantity attached to the importance of the position of the node, can be well defined via the normal/Gaussian distribution curve or equivalently via the B-matrix transition chains.

Both approaches give exactly the same result, which shows that SW is uniquely defined.

I am trying to derive weak-field Schwarzschild metric using Linearized Einstein's field equations of gravity:

[]h

_{μν}– 1/2 η_{μν}[]h = -16πG/ c4 TμνFor static, spherically symmetrical case, the Energy- momentum tensor:

T

_{μν}= diag { ρc^{2}, 0, 0, 0 }Corresponding metric perturbations for static ortho-normal coordinates:

h

_{μν}= diag { h_{tt}, h_{xx}, h_{yy}, h_{zz}}With one index rised using flat space-time Minquoskwi metric η

^{μν}= { -1 , 1, 1, 1 }:h

^{μ}_{ν}= diag { -h_{tt}, h_{xx}, h_{yy}, h_{zz}}Trace of the metric:

h = h

^{γ}_{γ}= - h_{tt}+ h_{xx}+ h_{yy}+ h_{zz}The four equations:

1) []h

_{tt}– 1/2 η^{tt}[]h = -16πG/ c^{4}T_{tt}=> []h

_{tt}+ 1/2 []( - h_{tt}+ h_{xx}+ h_{yy}+ h_{zz})= -16πGρ/ c^{2}=> 1/2 []( h

_{tt}+ h_{xx}+ h_{yy}+ h_{zz})= -16πGρ/ c^{2}2) []h

_{xx}– 1/2 η^{xx}[]h = -16πG/ c^{4}T_{xx}=> []h

_{xx}- 1/2 []( - h_{tt}+ h_{xx}+ h_{yy}+ h_{zz})= 0=> 1/2 []( h

_{tt}+ h_{xx}- h_{yy}- h_{zz})= 0Similarly:

3) 1/2 []( h

_{tt}- h_{xx}+ h_{yy}- h_{zz})= 04) 1/2 []( h

_{tt}- h_{xx}- h_{yy}+ h_{zz})= 0Adding equations 2), 3) & 4) to 1) respectively, yield:

[]( h

_{tt}+ h_{xx}) = []( h_{tt}+ h_{yy}) = []( h_{tt}+ h_{zz})= -16πGρ/ c^{2}Solving the equations using:

[] ≈ ▼

^{2}≈ 1/R^{2}d/ dR ( R^{2}d/ dR ) for static spherically symmetric case; we get:( h

_{tt}+ h_{xx}) = ( h_{tt}+ h_{yy}) = ( h_{tt}+ h_{zz})= -8πGρR^{2}/ 3c^{2}– K_{1}/ R + K_{2}Similar solutions for vacuum case, with Tμν= 0 would be:

( h

_{tt}+ h_{xx}) = ( h_{tt}+ h_{yy}) = ( h_{tt}+ h_{zz})= – K'_{1}/ R + K'_{2}For the metric to be asymptotically flat:

K

_{2}= K'_{2}= 0For the solutions to be continuous at boundary, R= r, the radius of spherically symmetric matter:

- 8πGρr

^{2}/ 3c^{2}≈ - 2Gm/ rc^{2}The remaining two constants must be:

K

_{1}= 0 & K'_{1}= 2Gm/ Rc^{2}Therefore, my solution comes:

( h

_{tt}+ h_{xx}) = ( h_{tt}+ h_{yy}) = ( h_{tt}+ h_{zz})= - 2Gm/ Rc^{2}But, as per the literature, the weak field Schwarzschild metric must come out to be:

h

_{tt}= h_{xx}= h_{yy}= h_{zz}= 2Gm/ Rc^{2}Thus the solutions must come out to be:

( h

_{tt}+ h_{xx}) = ( h_{tt}+ h_{yy}) = ( h_{tt}+ h_{zz})= + 4Gm/ Rc^{2}I am not able to make out where I am making mistake. Can anybody please help?

Thanks.

Nikhil

Physics is Stalled by Politics - Paper with solutions to 64 significant problems published in #3 journal but rejected by arXiv.org.
The new paper entitled,

*Measurement Quantization*, published Jan. 25, 2023 in the Intl. J. Geom. Methods Mod. Phys.,Article Measurement Quantization

lays out the foundations of quantum behavior using existing classical expressions, expanded to account for the discrete internal frame of the universe. The paper presents predictions of a length contraction effect unrelated to that described by Einstein and then presents measurement data to support the approach. It then derives the physical constants and the laws of nature from first principles. It unifies gravity with electromagnetism and writes both SR and GR anew from first principles, therein leading to a derivation of the equivalence principle. It presents simple classical solutions to dark energy, dark matter and a no free parameter description of early universe events.
But we should restate, this paper is classical mechanics, offering 530 equations describing phenomena across the entire measurement domain. While extensive support is offered, additional support rests on a long history of support for classical mechanics.
In regards to posts regarding the heated debate about the absurdity of new research being filtered, I agree! Even though this paper is published in the #3 mathematical physics journal in the world (by SJR ranking) and is indexed to NASA’s ADS, it was rejected by arXiv.org. I then sought the assistance of a top five ranked astrophysicist in the world. The case was reopened, reconsidered and then rejected a second time, as though classical mechanics was of questionable scientific merit. The point is, classical mechanics is worthy of arXiv.org.
We must conclude that the paper was rejected because of a higher cultural mandate, that breakthroughs that impact the existing funding model cannot appear as though they enjoy support by the community. For insight, see this post by Avi Loeb:
https://avi-loeb.medium.com/how-to-navigate-academia-6e8c4feea460
I will state, this paper presents solutions to 82.5% of all outstanding problems in cosmology among many other classical and quantum problems.
If community leaders really want to effect change, they would use well-vetted publications as example of this cultural absurdity and the need for change. And the best way to affect that change is to begin by supporting breakthroughs in existing classical mechanics on their blogs, in videos, in presentations and at conferences. Community leaders should not be posting literature as unanswered (i.e., dark matter, dark energy) where existing classical mechanics offers insight with straight-forward calculations. Otherwise, such individuals mimic the same cultural bias they are arguing against.There are a few point to consider in this issue

Points pro current emphasis

1. Math is the backbone of a physical theory. Good representation, good quantities of a theory, phenomena but bad math makes for bad theory

2. There is a general skepticism for reconsidering role of mathematized approach in physics Masters syllabi/upgrading role of literature/essay

2. Humans communicate, learn, think & develop construct via language

Arguments Con

1. Math is the elements in theory and "physics product" that is responsible for precision& prediction. Indespensible though, it exists in the mind of some individuals & function as well, in parallel with conception, physical arguments

2. Not all models in physics are mathematical. Some are conceptual

3. Formulations of solutions to physics problems via math techniques and methods is def of mathematical physics. However, this is a certain % of domain of skills.

But syllabus focuses 100% on this

And can you reference articles or texts giving answers to this question?

This is briefly considered in https://arxiv.org/abs/0804.1924 which is on RG as https://www.researchgate.net/publication/314079736_Entropy_and_its_relationship_to_allometry_v17.

Reviewing the literature would be helpful before considering whether to updatie the 2015 ideas.

We see many theories in physics, mathematics, etc. becoming extremely axiomatic and rigorous. But are comparisons between mathematics, physics, and philosophy? Can the primitive notions (categories) and axioms of mathematics, physics and philosophy converge? Can they possess a set of primitive notions, from which the respective primitive notions and axioms of mathematics, physics, and philosophy may be derived?

Raphael Neelamkavil

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.

I am working on topology optimization for photonic devices. I need to apply a custom spatial filter on the designed geometry to make it fabricable with the CMOS process. I know there exist spatial filters to remove the pixel-by-pixel and small features from the geometry. However, I have not seen any custom analytical or numerical filters in the literature. Can anyone suggest a reference to help me through this?

Thanks,

Is paraphrasing necessary in such cases, or is direct quotation with appropriate citation sufficient?

Mathematical physics conditions are necessary.

I am studying integral transforms (Fourier, Laplace, etc), to apply them in physics problems. However, it is difficult to get books that have enough exercises and their answers. I have found that in particular the Russian authors have excellent books where there are a lot of exercises and their solutions.

Greetings,

Ender

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

*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 …*

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).

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.

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 = ax**. But here the term inside the exponential function is of the form^{2}**y = ax**, where^{2}+ bx + c*i*x*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.

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?

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?

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.

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?

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.

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.

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 u

_{tt}= c^{2}u_{xx}.; x ∈ (0, 1), t> 0 withc=1,

u(0, t) = u(1, t)=0, t> 0,

u(x, 0) = f(x),

u

_{t}(x, 0) = g(x),x ∈ (0, 1).

**Solution:**

u(x, t) =

**summation over k=1..∞ of**[ sin (k π x) (a_{k}cos (k π t) + b_{k}k π sin (k π t)) ]a

_{k}, b_{k}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=x

_{2}and t=t_{2}, such that u(x_{2},t_{2})=d_{2}.1. How does the solution u(x,t) change after time t

_{2}?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 d

_{2 }= 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..

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.

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.

Is it possible to formulate the Ricci-flow as the Euler-Lagrange equations of some system? What would be the corresponding action functional?

As you may be aware, the quest to construct a 3x3 magic square using 7, 8, or 9 distinct square integers poses a captivating challenge (refer to pages 17 and 18 of the provided paper). Furthermore, it is intriguing to note the profound connections between magic squares and various physical phenomena (see pages 18 to 23 of the aforementioned paper). The crucial inquiry then arises: How can we harness the tools of physics to unravel these enigmas?

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.

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?

A conical surface in a Riemannian manifold

*is the union of all the geodesics that connects a fixed point***M***(the vertex) and any point on some curve in***p***which doesn't contain the point***M***. If we define the same in the Riemannian product space***p***times***S**^{2 }*in the analogous way, how can such surfaces be parametrized? In which way can they be visualized?***R**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.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?

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.

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 !

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?

In experiments we only deal with hermitian operators and they are called physical observables. But in quantum theory non-hermitian operators also exist. Are we using them only as a mathematical requirement or is there any other reason for their existence?

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?

Dear researchers,

Hi,

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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 - 16:24*

**No 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 2017**ArticleFull-text available

- February 2018
- 📷Delfim F. M. Torres
- 📷Ricardo Almeida
- 📷Paulino Lima Fortes
- 📷Dorota Mozyrska

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**Os desafios do planeamento estratégico: o caso da Universidade de Cabo Verde**Article- 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
- Geometria e Topologia.
- Modelação Matemática.
- Pensamento Geométrico e espacial.
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- O conceito de corte e a construção do contínuo: uma abordagem histórica, Revista de Ciência e Tecnologia, nº1,Edições Uni-CV, 2013.
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- Fundamentos de Geometria Não-Standard, Edições Uni-CV.
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- On some convex topologic concepts in topological vector spaces, XXI Oporto Meeting in Geometry, Topology and Physics, Instituto Superior Técnico, Lisboa, 2015.
- Em torno dos paradoxos de Zenão de Eleia: interpretações standard e não-standard, Jornadas Luso-Espanholas de Filosofia da Ciência, CFFCUL, Lisboa, 2014.
- Pistas e despistes no ensino da Geometria, Jornadas sobre o ensino da Matemática e da Língua Portuguesa, Universidade de Cabo Verde, 2013.
- A descoberta do bosão de Higgs, um triunfo do Modelo Standard, Reitoria da Universidade de Cabo Verde, Praia, 2012.
- A recuperação de informação geométria a partir de uma núvem de pontos, Departamento de Engenharias e Ciências do Mar, Universidade de Cabo Verde, 2011
- .http://cfcul.fc.ul.pt/equipa/docs/2014/Paulino%20Lima%20Fortes_cv.pdf
- https://www.researchgate.net/profile/Gastao_Frederico/research
- 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 more
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**The Cape Verde International Days on Mathematics 2020****https://www.researchgate.net/profile/Ivanilda_Cabral****https://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

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.

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?

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.

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?

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.

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

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?

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!!!!!

(1) https://www.researchgate.net/project/TOWARDS-THE-GRAND-UNIFIED-THEORY-OF-EVERYTHING-LINKING-FUNCTIONALITY-MODEL-CONCRETE-TO-ABSTRACTIONS?_sg=XNNU-DQ2rx6YNmUfxUjYZ6LaLamHZK8jGl_UOQblslDC1A9ygt_x0mY-iylt49cJFCqpUNKip5okOdqjIf43mH-lRm2JzYvjuM4j including working papers with tensor analyses......

(4)

(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.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?

Could anyone give the relationship of electrode resistance with reduction or oxidation potential regarding a mathematical or physical point of view?

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

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?

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!

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

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.

Recently I am working on mathematical modeling for nonlinear oscillatory problems

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!