Mathematics - Science topic

Is it possible in a world as fragmented as ours to present a new concept of Unity in which Science, Philosophy and Spirituality or Ontology can be conceived working in Complete Harmony?

In this respect the late Thomas S. Kuhn wrote in his

The Structure of Scientific Revolutions

"Today research in parts of philosophy, psychology, linguistic, and even art history, all converge to suggest that the traditional paradigm is somehow askew. That failure to fit is also increasingly apparent by the historical study of science to which most of our attention is necessarily directed here."

And even the father of Quantum Physics complained strongly in his 1952 colloquia, when he wrote:

"Let me say at the outset, that in this speech, I am opposing not a few special statements claims of quantum mechanics held today, I am opposing its basic views that has been shaped 25 years ago, when Max Born put forward his probability interpretation, which was accepted by almost everybody. It has been worked out in great detail to form a scheme of admirable logical consistency which has since been inculcated in all young students of theoretical physics."

Where is the source of this "crisis of physics" as has been called?

Certainly the great incompatibility between General Relativity and Quantum Mechanics is in a certain sense, one of the reasons, of that great crisis, and that shows clearly the real need of a paradigm shift.

As one that comes from the Judeo-Christian tradition, that need of a real paradigm shift was of course a real need too. Philosophers such as Teilhard de Chardin, Henry Bergson, Charles Pierce and Ken Wilber, all of them worked for it!.

Ken Wilber said that goal of postmodernity should be the Integration of the Big Three, Science, Philosophy and Spirituality, and a scientist as Eric J. Lerner in his The Big Bang Never Happened, show clearly in it, how a paradigm shift was in cosmology is a real need too.

My work about that need started in 1968, when I found for the first time, an equation that was declared the most beautiful equation of mathematics, I mean Euler's relation found by him in 1745, when working with infinite series. It was this equation that took me in 1991, to define what I now call a Basic Systemic Unit, that has the most remarkable property to remain the same in spite of change, exactly the same definition of a Quantum as defined by professor Art Hobson in his book The Tales of Quantum, and that the University of Ottawa found when working with that strange concept that frightened Einstein, the entanglement concept, that seemed to violate Special Relativity.

Where is the real cause of the incompatibility between GR and QM?

For GR Tensor Analysis was used, a mathematical tool based on real numbers, and with it there was the need to solve ten functions representing the gravitational field:

"Thus, according to the general theory of relativity, gravitation occupies an exceptional position with regards to other forces, particularly the electromagnetic forces, since the ten functions representing the gravitational field at the same time define the metrical properties of the space measured."

THE FOUNDATION OF THE GENERAL THEORY OF RELATIVITY

By A. Einstein

Well the point is that, in that metrics that define the GR, time is just another variable, just as space, and as so with the same symmetrical properties, at the point that is can take both signs positive and negative, so time travel could be conceived just as a space travel, and any direction, in fact Stephen Hawking in his A BRIEFER HISTORY OF TIME, writes:

"It is possible to travel to the future. That is, relativity shows that it is possible to create a time machine that will jump you forward in time." Page 105

This is exactly the point that has made physics some sort of metaphysics, and as so created the great crisis of physics. While QM is based on the complex Schrödinger's wave equation or on complex numbers, in which the symbol sqr(-1), is a symbol to separate two different orders of reality, such as Time and Space, GR is based just on real numbers.

The Basic Systemic Unit concept, based on Euler's relation is in fact the definition of a Quantum, and as so it can be used to deduce all fundamental equations of physics as can be seen in my paper... resolving in this way that great crisis of physics

Quantum Physics

Edgar Paternina

retired electrical engineer

I have been seeing and following a lot of work on these topics, it even seems that there are more results on them than on the corresponding classical topics, particularly on general topology.

What could be the cause of such results?

Pages 7-11

Has our mathematical knowledge progressed as much as contemporary science?

1- Assume a rectangle in the second dimension; this rectangle's components are lines. Its geometric characteristics are perimeter and area.

2- Assume a cube in the third dimension. Its components are the plane. Its geometric characteristics are area and volume.

3- What are the names of its components by transferring this figure to the 4th dimension? And what is the name of its geometric characteristics? And with the transfer to the 5th and higher dimensions, our mathematics has nothing to say.rectangle is just a simple shape how about complex geometric shapes?

According to new physical theories such as strings theory, we need to study different dimensions.

Modifying the original Feistel structure will it be feasible to design a lightweight and robust encryption algorithm. Somehow changing the structure's original flow and adding some mathematical functions there. I welcome everyone's view.

Homomorphic encryption is a type of encryption that lets you perform mathematical operations on encrypted data without decrypting it first. This means that the raw data remains encrypted while it's being processed, analyzed, and run through algorithms

A minion is a low-level official protecting a bureaucracy form challengers.

A Kuhnian minion (after Thomas Kuhn's Structure of Scientific Revolutions) is a low-power scientist who dismisses any challenge to existing paradigm.

A paradigm is a truth structure that partitions scientific statement as true to the paradigm or false.

Recently, I posted a question on Physics Stack Exchange that serves as a summary of the elastic string paradigm. My question was: “Is it possible there can be a non-Fourier model of string vibration? Is there an exact solution?”

To explain, I asked if they knew the Hamiltonian equation for the string vibration. They did not agree it must exist. I pointed out there are problems with the elastic model of vibration with its two degrees of freedom and unsolvable equations of motion can only be approximated by numerical methods. I said elasticity makes superposition the 4th Newtonian law. How can a string vibrate in an infinite number of modes without violating energy conservation?

Here are some comments I got in response:

“What does string is not Fourier mean? – Qmechanic

“ ‘String modes cannot superimpose!’ Yet, empirically, they do.” – John Doty

“ A string has an infinite number of degrees of freedom, since it can be modeled as a continuous medium. If you manage to force only the first harmonic, the dynamics of the system only involve the first harmonic and it’s a standing wave: this solution does depend on time, being (time dependence in the amplitude of the sine). No 4th Newton’s law. I didn’t get the question about Hamilton equation.

“What do you mean with ‘archaic model’? Can I ask you what’s your background that makes you do this sentence? Physics, Math, Engineering? You postulate nothing here. You have continuum mechanics here. You have PDEs under the assumption of continuum only. You have exact solutions in simple problems, you have numerical methods approximating and solving exact equations. And trust me: this is how the branch of physics used in many engineering fields, from mechanical, to civil, to aerospace engineering.” – basics

I want to show the rigid versus elastic dichotomy goes back to the calculus wars. Quoting here from Euler and Modern Science, published by the Mathematical Association of America:

"We now turn to the most famous disagreement between Euler and d’Alembert … over the particular problem of the theory of elasticity concerning a string whose transverse vibrations are expressed through second-order partial differential equations of a hyperbolic type later called the wave equation. The problem had long been of interest to mathematicians. The first approach worthy of note was proposed by B. Taylor, … A decisive step forward was made by d’Alembert in … the differential equation for the vibrations, its general solution in the form of two “arbitrary functions” arrived at by means original with d’Alembert, and a method of determining these functions from any prescribed initial and boundary conditions.”

[Editorial Note: The boundary conditions were taken to be the string endpoints. The use of the word hyperbolic is, I believe, a clear reference to Taylor’s string. A string with constant curvature can only have one mathematic form, which is the cycloid, which is defined by the hyperbolic cosh x function. The cosh x function is the only class of solutions that are allowed if the string cannot elongate. The Taylor/Euler-d’Alembert dispute whether the string is trigonometric or hyperbolic.

Continuing the quote from Euler and Modern Science:

"The most crucial issue dividing d’Alembert and Euler in connection with the vibrating string problem was the compass of the class of functions admissible as solutions of the wave equation, and the boundary problems of mathematical physics generally, D’Alembert regarded it as essential that the admissible initial conditions obey stringent restrictions or, more explicitly, that the functions giving the initial shape and speed of the string should over the whole length of the string be representable by a single analytical expression … and furthermore be twice continuously differentiable (in our terminology). He considered the method invalid otherwise.

"However, Euler was of a different opinion … maintaining that for the purposes of physics it is essential to relax these restrictions: the class of admissible functions or, equivalently, curves should include any curve that one might imagine traced out by a “free motion of the hand”…Although in such cases the analytic method is inapplicable, Euler proposed a geometric construction for obtain the shape of the string at any instant. …

Bernoulli proposed finding a solution by the method of superimposition of simple trigonometric functions, i.e. using trigonometric series, or, as we would now say, Fourier series. Although Daniel Bernoulli’s idea was extremely fruitful—in other hands--, he proved unable to develop it further.

Another example is Euler's manifold of the musical key and pitch values as a torus. To be fair, Euler did not assert the torus but only drew a network show the Key and Pitch can move independently. This was before Mobius's classification theorem.

My point is it should be clear the musical key and pitch do not have different centers of harmonic motion. But in my experience, the minions will not allow Euler to be challenged by someone like me. Never mind Euler's theory of music was crackpot!

Given:

In an isosceles triangle, the lengths of the two equal sides are each 1, and the base of the triangle is m.

A circle is circumscribed around the triangle.

Find the chord of the circle that intersects the two equal sides of the triangle and is divided into three equal segments by the points of intersection.

What is the mathematic difference between AI and AC?

Standardization of AI and AC based on the DIKWP model(初学者版)

https://blog.sciencenet.cn/home.php?mod=space&uid=3429562&do=blog&quickforward=1&id=1449103 The original chat log: https://chatgpt.com/share/e7bf95de-3027-4b3a-8b42-0ec8331873c4

I am interested in the existence of intelligent tutoring systems for teaching physics and mathematics in secondary schools or artificial intelligence tools that can be used in the classroom for student-teacher collaboration in these subjects, preferably with free access.

I am also interested in any relevant studies/research or information on the above topic.

National Achievement Test are given to learners like in Grade 10 on the following subjects like Science, Mathematics and English, to assess how much students learned in a specific disciplines

Let me share a quote from my own essay:

"Dynamic flows on a seven-dimensional sphere that do not coincide with the globally minimal vector field, but remain locally minimal vector fields of matter velocities, we interpret as physical fields and particles. At the same time, if in the space of the evolving 3-sphere $\mathbb{R}^{4}$ the vector field forms singularities (compact inertial manifolds in which flows are closed), then they are associated with fermions, and if flows are closed only in the dual space $\mathbb{R}^{4}$ with an inverse metric, then the singularities are associated with bosons. For example, a photon is a limit cycle (circle) of a dual space, which in Minkowski space translationally moves along an isotropic straight line lying in an arbitrary plane $(z,t)$, and rotates in the planes $(x,t)$, $(y,t)$." (p. 12 MATHEMATICAL NOTES ON THE NATURE OF THINGS)

Hi all

i was wondering if anyone knew of a valid and reliable assessment of task-based engagement that could be used to compare student engagement across different types of tasks in mathematics?

thanks

James

Bonjour,

Je suis actuellement en train de travailler sur un projet de recherche portant sur l'utilisation de l'optimisation mathématique pour déterminer le taux directeur optimal en politique monétaire. J'aimerais savoir s'il existe des travaux de recherche récents ou des modèles spécifiques qui ont abordé ce sujet. De plus, je suis à la recherche de conseils sur la manière de structurer mon modèle et de choisir des variables pertinentes pour ce type d'analyse. Toute suggestion de lecture ou d'expertise serait grandement appréciée.

Merci d'avance pour votre aide

Yes it is .

In triangle ABC, the median BM_2 intersects the bisector AL_1 at point P.

The side BC is divided by the base of the bisector AL_1 into segments CL_1=m and BL_1=n.

Determine the ratio of the segments AP to PL_1.

Dear research community members I would like to post a preprint of my article in Mathematics but need an endorsement If anyone can do this for me I will greatly appreciate for once Secondly I will send you my new paper with explanations

Endorsement Code: SP84WZ

Thanks a lot

P.S I don't know the procedure The moderator sent me a link

Ruslan Pozinkevych should forward this email to someone who's registered as an endorser for the cs.IT (Information Theory) subject class of arXiv

or alt visit

and enter SP84WZ

Once again thank you and apologize for bothering

How do we improve the inflation prediction using mathematics?

Ate there any Research Project and Grants for individuals without involvement of employer in Mathematics?

Is there a galactic rotation anomaly? Is it possible to find out the speed and time of the galactic rotation anomaly?

Is there a galactic rotation anomaly? Is it possible to find out the speed and time of the galactic rotation anomaly?

Abstract: Orbital speeds of stars, far from centre of a galaxy, are found roughly constant, instead of reductions predicted by current gravitational theories (applied on galactic and cosmological scales). This is called the anomalous rotation of galaxies. This article intends to show that constant angular speeds of all macro bodies in a galaxy are natural phenomenon and there is no mystery about it.

Keywords: Galaxy, Stable galaxy, rotational anomaly.

A planetary system is a group of macro bodies, moving at certain linear speed in circular path around galactic centre. Central body of planetary system is by far the largest and controls mean linear speeds of all other members. Gravitational attractions between macro bodies of planetary system cause perturbations in their directions of motion, resulting in additional curvatures of their paths. When perturbed paths of smaller macro bodies are related to central body in assumed static state, we get apparent orbital paths of planetary bodies. They appear to revolve around static central body in elliptical/circular paths. Apparent orbital paths are unreal constructs about imaginary static state of central body. They are convenient to find relative positions of macro bodies in the system and to predict cyclic phenomena occurring annually. In reality, planetary bodies do not orbit around central body but they move in wavy paths about the central body. Central and planetary bodies move at a mean linear speed along their curved path around galactic centre.

Perturbations of orbital paths of macro bodies in planetary system are related directly to their matter-content and inverse square of distance from central body. Distance from central body has greater effect of magnitudes of perturbations. Hence, normally, paths of planetary bodies at greater distance from central body are perturbed by lesser magnitudes. Curvatures and thus angular speeds of their apparent orbits reduce as distance from central body increases. Since planetary system has no real spin motion, this is an imaginary phenomenon. However, many learned cosmologists seem to take spin motion of planetary system as real phenomenon and consider that members of all spinning group pf macro bodies should behave in similar manner, i.e. angular (spin) speed of members should reduce as their distance from centre of system increases.

Stable galaxy consists of many macro bodies revolving around its centre. This group can be considered as a spinning fluid macro body, rotating at a constant angular speed. Gravitational collapse initiates spin motion of galactic cloud and maintains constant spin speed of outer parts of stable galaxy. Centre part of galaxy, which is usually hidden, may or may not be spinning. We can observe only visible stars and their angular speeds about galactic centre. Linear motions of macro bodies, caused by gravitational attractions towards other macro bodies in the system, have two components each. One component, due to additional linear work invested in association with it, produces macro body’s linear motion, in a direction slightly deflected away from centre of circular path. Other component, towards centre of its circular path, is caused by additional angular work invested in association with it. This component produces angular motion of macro body.

All matter-particles in a fluid macro body, spinning at constant speed, have constant angular speeds. Consider a matter-particle at O, in figure 1, moving in circular path AOB. XX is tangent to circular path at O. Instantaneous linear speed of matter-particle is represented by arrow OC, in magnitude and direction. It has two components; OD, along tangent XX and DC, perpendicular to tangent XX and away from centre of circular path. This component, DC, represents centrifugal action on matter- C particle due to its motion in circular path. In

order to maintain constant curvature of path, X D O X matter-particle has to have instantaneous A linear (centripetal) motion equal to CE E

toward centre of circular path. If magnitudes B Figure 1 and directions of instantaneous motions are as shown in figure 1, matter-particle maintains its motion along circular path AOB at constant angular speed.

Should the matter-particle increase its instantaneous linear speed for any reason, both components OD and DC would increase. Component OD tends to move matter-particle at greater linear speed along tangent XX. Outward component DC tends to move matter-particle away from centre of its circular path. The matter particle tends to increase radius of curvature of its path. This action is usually assigned to imaginary ‘centrifugal force’. In reality expansion of radius of curvature of path is caused by centrifugal component of linear motion. Reduction in centripetal action also produces similar results.

Should the matter-particle decrease its instantaneous linear speed for any reason, both components OD and DC would reduce. Component OD tends to move matter-particle at lesser linear speed along tangent XX. Reduction in outward component DC tends to move matter-particle towards centre of its circular path. The matter particle tends to reduce radius of curvature of its path. Reduction of radius of curvature of path is caused by reduction in centrifugal component of linear motion. Increase in centripetal action also produces similar results.

In other words, matter-particle regulates its distance from centre of its circular path so that its angular speed remains constant. This is the reason for action of centrifuges. As linear speeds of matterparticles increase, they move outwards, in an effort to maintain their angular speed constant.

Additional work, done for linear motion of a matter-particle and additional work, done for its angular motion are entirely separate and distinct. Additional work for linear motion of a matter-particle can produce only linear motion and additional work for angular motion can produce only angular motion. In the case, explained above, increased in linear speed of matter-particle is considered. That is, additional work invested in association with matter-particle is of linear nature. It can only increase its linear motion. As no additional work for angular motion is invested matter-particle cannot change its angular speed. Instead, matter-particle is compelled to move away from centre of its rotation, so that it can increase magnitude of linear motion while keeping magnitude of angular motion constant.

Similarly, increase in centripetal effort invests additional work required for angular motion of matterparticle. Matter-particle tends to increase magnitude of its angular motion. Curvature of its path

increases by reducing its distance from centre of circular path. Matter-particle tends to move towards centre of circular path, so that it can increase its angular speed while keeping its linear speed constant.

Every macro body in a stable galaxy behaves in a manner similar to matter-particle, represented in figure 1. They tend to position themselves in the system, so that their linear and angular speeds match corresponding works associated with them. Macro bodies strive to maintain their angular speeds constant by keeping appropriate distance from centre of rotation. Macro bodies towards the central region may experience additional centripetal effort. They might increase their angular motion and move towards central point to merge with black hole present there. In due course of time, macro bodies on outer fringes move away from galaxy and destroy its stability.

In a galaxy, various macro bodies arrive at their relative position gradually by error and trial, during which their relative positions and linear and angular speeds are stabilized. Galaxy, as a whole, stabilizes only when constituent macro bodies have reached their steady relative positions and motions. In order to maintain stability, it is essential to maintain relative positions of all constituent macro bodies by having constant and equal angular speeds and linear speeds corresponding to their distances from galactic centre. Change in relative position or linear or angular speed of even one macro body is liable to destabilize the galaxy.

As and when superior 3D matter-particles at the fringe of galaxies attain linear speeds approaching speed of light, they break-down into primary 3D matter-particles and produce halo around equatorial region. Halos of neighbouring stable galaxies interact to prevent their translational movements and maintain steady state of universe.

Therefore constant angular speeds of constituent macro bodies of stable galaxies are their natural states. There are no mysteries or anomalies about them. This phenomenon is mystified by those who consider imaginary spin motions of planetary systems are real. Therefore, assumptions of dark matter, time dilation, modification of gravitational laws, etc and complicated mathematical exercises are irrational and unnecessary to prove non-existing rotation anomaly of galaxies.

Galactic rotation anomaly is a non-existing phenomenon derived from imaginary spin motions of planetary systems about their central bodies in assumed static states. Constant angular speeds of stars in a galaxy confirm static state of galactic center (in space), rather than produce an anomaly.

[1] Nainan K. Varghese, MATTER (Re-examined), http://www.matterdoc.info

Reply to this discussion

Chuck A Arize added a reply

5 hours ago

Yes, there is a galactic rotation anomaly observed as the discrepancy between the predicted and actual rotation speeds of galaxies. This anomaly, often attributed to dark matter, shows that the outer regions of galaxies rotate faster than expected. Measuring the speed and time of this rotation anomaly involves detailed observations of galactic rotation curves and modeling, which reveal the velocity profile and suggest the presence of unseen mass influencing the rotation.

Abdul Malek added a reply

3 hours ago

Abbas Kashani > "Is there a galactic rotation anomaly?"

There is a galactic rotation anomaly, but only according to officially accepted theories of gravity and the (Big Bang) theory of the formation of the galaxies inferred for a finite, closed and a created (in the finite past) universe.

But all these theories based on causality and theology are wrong! The dialectical and scientific view is that the universe is Infinite, Eternal and Ever-changing, mediated by dialectical chance and necessity. Gravity is a dialectical contradiction of the unity of the opposites of attraction and repulsion (due to inherent free motion of matter particles, vis viva). In short (human) time scale, new galaxies are seen to be formed through the dissipation and/or ejection of matter in the form of stars, star clusters or even a large part of the galaxy as quasars from the existing galaxies.

So, the observed high orbital velocities of the starts, star clusters etc. at the periphery of the galaxies and of the planets at the periphery of the planetary systems within the galaxies is just a natural phenomena and there is no anomaly!

"Ambartsumian, Arp and the Breeding Galaxies" : http://redshift.vif.com/JournalFiles/V12NO2PDF/V12N2MAL.pdf

Article THE CONCEPTUAL DEFECT OF THE LAW OF UNIVERSAL GRAVITATION OR...

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Preston Guynn added a reply

4 days ago

Your discussion question statement is:

The limit of galactic rotation velocity is expected because rotation minus precession has a maximum velocity. Our solar system's relative rotation velocity with respect to the Milky Way galaxy is at this maximum, and as a fraction of speed of light the observed velocity can be designated vg/c, and is determined in the single page proof of the quantum of resistance:

Article The Physical Basis of the Quantum of Resistance is Relativis...

The detailed proofs are in:

Article Thomas Precession is the Basis for the Structure of Matter and Space

Note that the observed velocity is the difference between rotation and precession.

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Dale Fulton added a reply

3 days ago

The galactic rotation "anomaly" (flat rotation curve) is actually a misinterpretation of the measurements of the galactic rotations, when performed with spectrographic (redshift) measurements. This has misled astronomers since the inception of the spectrographic velocity measurements, as being totally doppler shift, whereas they contain many non-linear components of redshift due to gases and other effects from each galaxy. Recent measurements of the Milky Way galaxy rotation curve prove that this is the case, i.e, that spectrographic velocities are misleading, and that proper motion or parallax is the only way to accurately measure those velocities.

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André Michaud added a reply

20 hours ago

There is no galactic rotation anomaly. Such a concept emerges from the lack of careful study of past historical discoveries about orbital structures in the universe established since Ticho Brahe first collected his data about the planetary orbits in the solar system, from which Johannes Kepler abstracted his 3 laws, that were then mathematically confirmed by Newton.

The galactic rotation parameters are well known by those who studied the true foundation of astrophysics. Put in perspective in this article:

Article Inside Planets and Stars Masses

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Abbas Kashani added a reply

44 seconds ago

Preston Guynn

Dale Fulton

André Michaud

Greetings and politeness and respect to the great and respected professors and astronomers, I am very grateful for your efforts, dear ones. Thank you and thank you

Abbas Kashani

Mohaghegh Ardabili University

It will be better if there will be any mathematical relation or suggest some research articles?

During Kurt Gödel's lecture on the occasion of Albert Einstein's birthday in 1945, this question was already raised by John Wheeler. Gödel did not comment on it. Both Einstein and Gödel did not believe in quantum theory. Is there currently any reference or article that relates to this question? From today's scientific perspective, is there a relationship between Heisenberg's Uncertainty Principle and Gödel's Incompleteness Theorem? Even so, when both the principles arise from different theoretical frameworks and serve different purposes within their respective domains. Please provide references.

Can we apply the theoretical computer science for proofs of theorems in Math?

We assume that the difference is huge and that it is not possible to compare the two spaces.

The R^4 mathematical space considers time as an external controller and the space itself is immobile in its description or definition in the face of curl and divergence operators.

On the other hand, the unit space 4 D x-t time t is woven into the 3D geometric space as a dimensionless integer.

Here, the curl and divergence operators are just extensions of their original definitions ​​in 3D geometric space.

Comment les enseignants de mathémattiques du cycle primaire conçoivent-ils leurs évaluations? Quelles conceptions ont-ils pour le concept d'évaluation?

Would someone be kind enough to answer the question why there is a Pareto Principle related to Grades in Primary Education in Mathematics, but please without counter-questions such as who says or where it is written that it is so. If Primary Education in Mathematics lasts, for example, 8 years (the number varies between countries), then in the first three grades 80% have excellent and very good grades in Mathematics, and 20% good and bad. In the fourth grade, this ratio is approximately 50%:50%. However, from the fifth to the eighth grade, the relationship is reversed and only 20% have excellent and very good grades, and 80% good and bad. Sometimes, for some reason, the ratio is 70:30 instead of 80:20, but the relationship and regularity exists. I thank you in advance for your reply, as well as for your kindness and time.

With the term “gravity”, we refer to the phenomenon of the gravitational interaction between material bodies.

How that phenomenon manifests itself in the case of the interaction of two mass particles at rest relative to an inertial reference frame (IRF) has, in the framework of classical physics, mathematically been described by Isaac Newton. And Oliver Heaviside, Oleg Jefimenko and others did the same in the case of bodies moving relative to an IRF. They described the effects of the kinematics of the gravitating objects assuming that the interaction between massive objects in space is possible through the mediation of “the gravitational field”.

In that context, the gravitational field is defined as a vector field having a field- and an induction-component (E_g and B_g) simultaneously created by their common sources: time-variable masses and mass flows. This vector-field (a mathematical construction) is an essential element of the mathematical description of the gravitational phenomena, and as such an element of our thinking about nature.

One cannot avoid the question of whether or not a physical entity is being described by the vector field (E_g, B_g) and what, if any, is the nature of that entity.

In the framework of “the theory of informatons”^[1],[2],[3], the substance of the gravitational field – that in that context is considered as a substantial element of nature - is identified as “gravitational information” or g-information” i.e. information carried by informatons. The term “informaton” refers to the constituent element of g-information. It is a mass and energy less granular entity rushing through space at the speed of light and carrying information about the position and the velocity of its source, a mass-element of a material body.

[1] Acke, A. (2024) Newtons Law of Universal Gravitation Explained by the Theory of Informatons. https://doi.org/10.4236/jhepgc.2024.103056

[2] Acke, A. (2024) The Gravitational Interaction between Moving Mass Particles Explained by the Theory of Informatons. https://doi.org/10.4236/jhepgc.2024.103060

[3] Acke, A. (2024) The Maxwell-Heaviside Equations Explained by the Theory of Informatons. https://doi.org/10.4236/jhepgc.2024.103061

In his article "More is different", Anderson said that new laws of physics "emerge" at each physical level and new properties appear [1]; Wheeler, when claiming that "law without law" and "order comes out of disorder", argued that chaotic phenomena " generate" different laws of physics [2][3]. What they mean is that the laws, parameters, and constants of the upper level of physics appear to be independent of the laws of physics of the lower level. Is this really the case? Are we ignoring the conditions that form the physical hierarchy, thus leading to this illusion?

Let's suppose a model. The conditions for the formation of new levels are at least two: i. Existence of low-level things A,B ...... , the existence of interaction modes a, b,...... ; two, the existence of a sufficient number of low-level things, NxA, MxB....... Then when they are brought together, there are many possible combinations, e.g., (AA), (AAA), (AAA)', ...... , (AB), (BA), (AAB)', (BAB), ........ Then it escalates to [(AA)(AA)], [(AB)(ABA)], ....... What this actually leads to is a change in the structure of things and a corresponding change in the way they interact. The result of the "change" is the appearance of new physical phenomena, new forces, and so on.

Physics is an exact match for math, so let's use math as an example of this phenomenon. Suppose we have a number of strings (threads) that can be regarded as underlying things, then, when a string is curled into a circle, L=2πR, the law of the relationship between the length of the string and its radius, and the irrational constant π appear; when two strings are in cascade, L=l₁+l₂, the law that the total length of the string is equal to the sum of the individual string lengths (Principle of superposition) appears; and, when three strings form a right triangle, the law of Pythagoras, c²=a²+b², the law of sums of interior angles of triangles ∠A + ∠B + ∠C = 180° , and the irrational constant √2 appear ...... ; and the transcendental number e appears when the string length L grows in a fixed proportion (continuous compound interest)[4]^‡ ...... ; when the string vibrates, sine waves (sinωt) appear; when two strings are orthogonal, i appears ...... ; and when more kinds of vibrating strings are superimposed under specific conditions, more phenomena appear ^*.......

All these "qualitative changes" do not seem to be caused by "quantitative changes", but more by the need to change the structure. As mathematical theorems emerge, so must the laws of physics, and it is impossible for physics to transcend mathematics. Therefore, as long as there is a change of structure in physics, i.e. the possibility of symmetry breaking [5]^**, new "symmetries", new "laws", new "forces", new "constants", new "parameters" are almost inevitable.

Can we try to attribute all physical phenomena to emergence under hierarchical structural conditions? For example, the fine structure constant^‡‡and the Pauli exclusion principle emerge because of the formation of atomic structure; the "nuclear force" emerges because of the combination of protons and neutrons; The "strong interaction force" and "weak interaction force" appeared because of the structure of protons and neutrons. We should pay attention to the causal relationship here. Without structure, there would be no new phenomena; it is the more fundamental interactions that form structure, not these new "phenomena".

-----------------------------

* e.g. Blackbody radiation law, Bose statistics, Fermi statistics, etc.

** Should there be "spontaneous symmetry breaking"? Any change in symmetry should have a cause and a condition.

‡ What does it mean in physics if e will appear everywhere and the individual mathematical constants appear so simply? They must likewise appear at the most fundamental level of physics.

‡‡ https://www.researchgate.net/post/NO33_What_Does_the_Dimensionless_Nature_of_the_Fine-Structure_Constant_Mean

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2024-07-27 补充

In addition to the structure and statistics generated by the interactions that result in new laws of physics, the expression of the different orders of differentials and integrals of such generating processes is another important way of making the laws of physics emerge.

Typical examples of such expressions can be seen @ Ingo D. Mane: “On the Origin and Unification of Electromagnetism, Gravitation, and Quantum Mechanics“：

-----------------------------

[1] Anderson, P. W. (1972). More Is Different: broken symmetry and the nature of the hierarchical structure of science.

. Science, 177(4047), 393-396. https://doi.org/doi:10.1126/science.177.4047.393

[2] Wheeler, J. A. (1983). ‘‘On recognizing ‘law without law,’’’Oersted Medal Response at the joint APS–AAPT Meeting, New York, 25 January 1983. American Journal of Physics, 51(5), 398-404.

[3] Wheeler, J. A. (2018). Information, physics, quantum: The search for links. Feynman and computation, 309-336.

[4] Reichert, S. (2019). e is everywhere. Nature Physics, 15(9), 982-982. https://doi.org/10.1038/s41567-019-0655-9;

[5] Nambu, Y. (2009). Nobel Lecture: Spontaneous symmetry breaking in particle physics: A case of cross fertilization. Reviews of Modern Physics, 81(3), 1015.

Please suggest me, I want to present my research work.

how i can calculate mathematically COD concentration if i add 1gr from glucose to 1 L distilled water

I am need someone to collaborate with me to make article. Especially in international journal.

The video "The Biggest Question Physicists Aren't Asking" (https://www.youtube.com/watch?v=iVyl8pGd44I) says the Michelson-Morley experiment of 1887 didn't disprove the existence of the aether. It reminds us that Hendrik Lorentz explained the experiment's negative results with his Lorentz ether theory (LET), which he initially developed in 1892 and 1895. The theory was improved in 1905 and 1906 by Henri Poincaré - and it was based on the aether theory of Augustin-Jean Fresnel, Maxwell's equations, and the electron theory of Rudolf Clausius. The video says Albert Einstein's alternative to the aether in Special Relativity is preferred by physicists because it makes fewer assumptions. Interestingly, Einstein wondered in later years about the possibility of the aether actually existing.

Just as the building blocks in chemistry are atoms and molecules, the building blocks physics could use to determine what the aether is might be binary digits and topology. There are two clues to this conclusion. First - in his book "A Brief History of Time", Stephen Hawking states that quantum spin tells us what particles actually look like. A particle of matter has spin 1/2 and must be completely rotated twice (720 degrees) to look the same. Added to this is - a Mobius strip must be travelled around twice in order to reach the starting point. The second clue was supplied by a paper Einstein published in 1919. That paper asks if gravitation and electromagnetism play a role in forming elementary particles.

The clues can produce the following hypothesis. The BITS or binary digits of one and zero code for a Mobius strip (similar to the way that topological figure can be viewed on the Internet). Then two Mobius figures are joined to create a Klein bottle: possibly, the doughnut-shaped figure-8 version of the Klein. The Klein bottle is immersed in the 3rd dimension, with binary digits filling in any holes or gaps to produce a technically flat and simply-connected result. This procedure is similar to computer art's Sky Replacement, where the 1s and 0s can make a smooth blue sky stretching from horizon to horizon. The 1s and 0s naturally exist on quantum scales, and imaginary numbers are essential in quantum mechanics. So the complex (real+imaginary) numbers of Wick rotation could be given a practical use by being a subroutine of the Mobius strips and becoming the 4th dimension of time which can't be separated from the dimensions of space. Trillions of Mobius strips could form a photon while trillions of more complicated figure-8 Klein bottles might form the more complicated graviton. Interaction of photons and gravitons (in a process called Vector-Tensor-Scalar [VTS] Geometry) creates the Mobius-based matter particles. In this scenario, the aether - the medium waves travel through - wouldn't be an abstract thing called space filled with alleged Virtual Particles which can't be detected and may not even exist. The medium would be a sea filled with photons and gravitons.

Another possibility is that there is no medium for the gravitational and electromagnetic waves, and that there truly is no aether. In that case, waves would not merely be described by mathematics but would literally be the result of maths. A 3D (three dimensional) cube can be regarded as a reality coded on a 2D surface - in other words, the cube is a projection from a square. The 2D square would be a nonlinear (angular) math object resulting from adding four lines, each one being separated from those adjoining it by 90 degrees. The cubic shape would result from adding, in one direction, multiple layers of the information in the square. Instead of programming a set of points to follow a straight line, they can be represented curvilinearly as a waveform and described by Fourier analysis, v=f(lambda), etc. Interacting particles can produce waves just as masses can curve spacetime to produce gravity and gravitational waves. VTS Geometry plausibly explains the inverse - it doesn't solely regard mass as the producer of gravity but also regards gravity, partnering with electromagnetism, as producer of mass. Inverting quantum mechanics, gravitational and electromagnetic waves create particles with mass (protons, neutrons, quarks, electrons, etc - even the Higgs boson). As Stephen Hawking and Leonard Mlodinow point out in their book "The Grand Design", ultimate reality does not have to be described with quarks though it certainly can be. In this paragraph, the idea of curved space is replaced by gravitational and electromagnetic waveforms travelling on curved trajectories.

There exists a neural network model designed to predict a specific output, detailed in a published article. The model comprises 14 inputs, each normalized with minimum and maximum parameters specified for normalization. It incorporates six hidden layers, with the article providing the neural network's weight parameters from the input to the hidden layers, along with biases. Similarly, the parameters from the output layer to the hidden layers, including biases, are also documented.

The primary inquiry revolves around extracting the mathematical equation suitable for implementation in Excel or Python to facilitate output prediction.

what is difference between green production and regular production in inventory control with theoretical expression and mathematical term ?

What is going with physics?

Two excellent books written by experts in the field:

- Eric J. Lerner “THE BIG BANG NEVER HAPPENED,

- and Lee Smolin and his The Trouble with Physics,

show that the Great Crisis of Physics, will not be solved if we don’t change our frame of references, abandoning once and for all, the way we are trying to solve those fundamental problems of physics. Institution such as ResearchgGate, certainly are way to make easier to obtain that goal, but then it is necessary, that people open their minds to new ways to “seeing reality”… not the one they are using in mainstream physics, such as GR and Big Bang and so on…

“THE STRANGE CAREER OF MODERN COSMOLOGY

In our century the cosmological pendulum has swung back. The universe of present-day cosmology is more like that of Ptolemy and Augustine than that of Galileo and Kepler. Like the medieval cosmos, the modern universe is finite in time-it began in the Big Bang, and will end either in a Big Crunch or in a slow decay and dissipation of all matter. Many versions, like Stephen Hawking's, are finite in space as well, a perfect self-enclosed four-dimensional sphere. There is a gap between the heavens and the earth: in space there exist strange entities, governed by the pure and ethereal mathematics of general relativity-black holes, cosmic strings, axions-which cannot, even in principle, be studied on earth.

The nineteenth-century universe evolved by laws still in action today, as did that of the Jonians, yet the universe of modern cosmology is the product of a single, unique event, qualitatively different from anything occurring today-just as the medieval cosmos was the product of the creation. While scientists of a century ago saw a universe of continuous change, evolution, and progress, today's researchers see a degenerating universe, the ashes of a primordial explosion.

To earlier scientists, and to most of today's scientists outside cosmology, mathematical laws are descriptions of nature, not the true reality that lies behind appearances. Yet today cosmologists assume, as did Plato and Ptolemy, that the universe is the embodiment of preexisting mathematical laws, that a few simple equations, a Theory of Everything, can explain the cosmos except for what "breathed fire" into these equations to make them come alive.

Big Bang cosmology does not begin with observations but with mathematical derivations from unquestionable assumptions. When further observations conflict with theory, as they have repeatedly during the past decades, new concepts are introduced to "save the phenomenon"-dark matter, WIMPs, cosmic strings-the "epicycles" of current astronomy.”

“Alfven wrote sixty years later, "The people were told that the true nature of the physical world could not be understood except by Einstein and a few other geniuses who were able to think in four dimensions. Science was something to believe in, not something which should be understood. Soon the bestsellers among the popular science books became those that presented scientific results as insults to common sense. One of the consequences was that the limit between science and pseudo-science began to be erased. To most people it was increasingly difficult to find any difference between science and science fiction."^ Worse still, the constant reiteration of science's incomprehensibility could not fail to turn many against science and encourage anti-intellectualism.”

THE BIG BANG NEVER HAPPENED

Eric J. Lerner

“THE TROUBLE WITH PHYSICS

In this illuminating book, the renowned theoretical physicist Lee Smolin argues that fundamental physics – the search for the laws of nature- is loosing its way. Ambitious ideas about extra dimensions, exotic particles, multiple universes, and strings have captured the public’s imagination- and the imagination of experts. But these ideas have not been tested experimentally, and some, like string theory, seem to offer no possibility of been tested. Yet these speculations dominate the field… As Smolin points out, the situation threatens to impede the very progress of science…” Brian Appleyard, Sunday Times(London)” "

Edgar Paternina

Retired electrical engineer

Updated information of my thoughts and activities.

This is meant to be a one-way blog, albeit you can contribute with your recommendations and comments.

How do mathematicians recognize the article entitled "On the Nature of Some Euler's Double Equations Equivalent to Fermat's Last Theorem", published in Mathematics, Volume 10, Issue 23 (December-1 2022) by MDPI?

btw have u ever know about math bounded and reality bounded? for example it happen when student interpret the solution of problem toward mathematics. if you have experience or research about that, please let me know and lets discuss it. Thannk you!

Greetings mathematicians and physicists,

As we know, the idea of ​​assuming a higher space dimension than the natural 3 dimension is effective in solving many physical and mathematical problems.

Here, I'll pose some philosophical questions about " time ", and I don't know where they'll lead us when we try to answer them.

I look forward to hearing from you soon. Thank you so much.

Best from Algeria,

Khaldi Said, Phd student in mathematics.

Recent research papers and journals on indigenous knowledge system

Is there any geometrically derived mathematical expression for the duration of the natural day at any latitude and any time of the year?

In the isosceles triangle ABC (AC=AB), the angle at the vertex is 20°.

Point D is chosen on the side AB such that AD=BC.

Find the measure of angle CDB.

Please give answer. Also explain mathematical equations behind this.

In the recently published JCR, the journal Mathematical Biosciences and Engineering does not appear. I have searched everywhere and have found nothing, it does not appear in the list of excluded journals. The journal had an impact factor of 2.6 in the previous JCR (Mathematics and Computational Biology). I have contacted the journal, but they have not responded.

Does anyone know anything or can advise where to look?

Regards

Consider a circle of radius R with center O.

Two other circles are internally tangent to this circle and intersect at points A and B.

Find the sum of the radii of the other two circles, given that ∠OAB = 90°.

IIHIs

In triangle ABC, the bisector AL₁ is drawn.

Points O₁, O₂, O are the centers of the circles circumscribed around triangles ACL₁, ABL₁, ABC, respectively.

The radii are denoted as R₁, R₂, R for the respective circles.

The task is to find OO₁ and OO₂.

Given: ∠CAL₁ = ∠BAL₁; γ₁ (O₁; R₁ = O₁ A); γ₂ (O₂; R₂ = O₂ A); γ₀ (O; R = OA).

Find: OO₁, OO₂

Below it is rework of the chapter “The Prize” from Simon Singh's book “ Fermat’s Last Theorem: The story of a riddle that confounded the world's greatest minds for 358 years” :

<<Fermat wrote that his proof would not fit into the margin of his copy of Arithmetica, and Wiles’s 100 pages of dense mathematics certainly fulfils this criterion, but surely the Frenchman did not invent modular forms, the Taniyama-Shimura conjecture, Galois Groups and the Kolyvagin-Flach method centuries before anyone else.

If Fermat did not have Wiles’s proof then what did he have?

Mathematicians are divided into two camps:

The sceptics believe that Fermat’s Last Theorem was the result of a rare moment of weakness by the 17th-century genius.

They claim that although Fermat wrote, ”I have discovered a truly marvellous proof”, he had in fact found only a flawed proof.

Other mathematicians, the romantic optimists, believe that Fermat may have had a genuine proof.

Whatever this proof might have been, it would have been based on 17th-century techniques, and would have involved an argument so cunning that it has eluded everybody.

Indeed, there are plenty of mathematicians who believe that they can still achieve fame and glory by discovering Fermat’s original proof.

In my case it is pure passion for the Mathematics and the desire to do justice to Fermat and his genius !!! >>

This articles, entitled "On the Nature of Some Euler's Double Equations Equivalent to Fermat's Last Theorem" effectively provide a reformulation of Fermat's Last Theorem and has been published in 2022 in the journal "Mathematics" by publisher MDPI (Multidisciplinary Digital Publishing Institute).

The Journal "Mathematics" is indexed in SCOPUS. Impact factor 2.4. It is quoted with a journal rank: JCR - Q1 (Mathematics) / CiteScore 3.5 - Q1 (General Mathematics).

Ossicini's article is indicated by Mathematics as "Feature Paper".

This label is used to represent the most advanced investigations which can have a significant impact in the field.

A Feature Paper should be an original contribution that involves several techniques or approaches, provides an outlook for future research directions and describes possible research applications.

Feature papers are submitted upon individual invitation or recommendation by the scientific editors and must receive positive feedback from the reviewer.

I'm reading an article titled "Scientists Seek Life Across the Multiverse" and it says,

"If the multiverse hypothesis is correct, physicists would no longer have to find explanations for the absurdly improbable fine-tuning of the laws of nature that has made our existence possible. We are just lucky to live in a good universe among many different ones. One universe fine-tuned for life is an unlikely fluke. But one habitable universe among many is to be expected."

Another way of phrasing this is - Scientists are so eager to avoid any notion of Intelligent Design of the cosmos that they're willing to deny Earth's own scientific potential ... and their own intelligence.

Evolution can be observed in the form of adaptation of structure and function to the environment but there’s no reason to extrapolate this theory in order for it to account for life’s origin. In future centuries, human technology will develop terraforming and incredibly advanced bioengineering of cells - amino acids, proteins, water, nucleic acids, etc which were gathered in space or on planets and combined (science already knows these molecules exist out there). This could account for life’s origin since it agrees with 19th-century chemist Louis Pasteur’s proving that life can only originate from life. The origin-of-life hypothesis presented here obviously needs time travel back to a time when there was no life. This is feasible using General Relativity's concept of curved time (which is made circular via Wick rotation and future warping of space-time).

It's convenient to say Wick rotation is a form of mathematical trickery but explanation of the photoelectric effect seems to have sprung directly from Max Planck's idea of quanta - now called photons - which was also regarded for years as a mathematical convenience. Could an extension of evolution spring directly from the supposed math trickery of Wick rotation? We only need to be open to our current interpretations of science and maths not being set in stone. History has shown that presently accepted theories always change. And we are not the endpoint of history - we're simply one more step passing through it.

"Paper 4: Mathematical Framework for the Alcubierre Drive Using New Quarks: Unifying Dark Energy and Dark Matter"

The Dodecahedron Linear String Field Hypothesis (DLSFH) provides a viable theoretical foundation for the Alcubierre drive. By defining new quarks with the necessary properties to generate negative energy density and manipulate spacetime, this framework supports the feasibility of faster-than-light travel.

I invite the community to discuss this grand idea and our understanding of the need to explore new physics to fill the gap that Quantum Mechanics currently is missing!

A Theoretical Physicist, above all things, must have imagination and be a philosopher before he can part any knowledge of the Universe!

I want to research on the affectivity of blended learning in Mathematics

The general objective of the study is to explore the coping mechanisms used by mathematics teachers to improve teaching and learning in overcrowded classrooms at ...................

If we have the iterative functions X= X²+Y, where X is complex number and Y constant complex number. Can one find the Y and initial value of X in iterative function from generated sequences mathematically in order to generate next values?

I look forward your comments.

Best wishes.

FOR ME incredible! Discovering a proof for Goldbach's Conjecture is a monumental achievement in the realm of number theory. I perseverance and insight have unlocked a solution to one of mathematics' most intriguing mysteries.

MY discovery not only sheds light on a centuries-old problem but also enriches OUR understanding of prime numbers and their intricate relationships.I WILL PUBLISHE THIS AFTER TWO DAYS INSHALLHA IN WORLD.

Hello ResearchGate Community,

I am an accomplished Assistant Professor and Research Associate with a diverse background spanning over 13 years in both industry and academia. Currently completing my PhD in Mathematics, with an anticipated date of July 2024, I am eager to leverage my expertise in technology adoption, statistical and mathematical modeling, and structural equation modeling (SEM) to contribute to cutting-edge research initiatives.

Skills:

Proficient in technology adoption strategies, statistical and mathematical modeling techniques, including Structural Equation Modeling(SEM) and Artificial Neural Networks(ANN). Extensive experience utilizing tools such as SPSS and AMOS to analyze data and derive meaningful insights.

Interests:

Passionate about leveraging mathematics and statistics to drive technological advancement, particularly in the realms of e-learning and online education.

Interested in exploring the dynamics of technology adoption, usage, and acceptance within higher education institutions, with a focus on post-adoption behavior, continuous intention usage, and actual usage patterns.

I am seeking a postdoctoral opportunity where I can collaborate with like-minded researchers to address complex challenges at the intersection of mathematics, statistics, and technology adoption. My goal is to contribute to the development of innovative solutions that enhance the effectiveness of educational technologies and inform strategies for organizational change.

If you are aware of any leads/opportunities or research projects aligned with my expertise and interests, I would welcome the opportunity to connect and explore potential collaborations. Please feel free to reach out to me here on ResearchGate or via email at [[email protected]]

Thank you for your consideration.

Warmly,

Shard

Assistant Professor | Research Associate |

PhD Candidate in Mathematics

1)https://www.researchgate.net/publication/372338504_Proving_the_Fundamental_Theorem_of_Calculus_through_Critical_Race_Theory

2)https://www.researchgate.net/publication/373366727_Antiracist_Mathematics_White_Supremacy_Often_Even_Harms_Its_Own_Theorem

3)https://www.scribd.com/document/731934683/KalergiandHart-CellerandMemeticsWhiteAntifragility

4)https://www.researchgate.net/publication/380427514_Kalergi_and_Hart-Cellerand_Memetics_White_Antifragility

There are several definitions of mathematical creativity. Please share your favorite definition of mathematical creativity

My paper has been published in European Journal of Mathematics and Statistics, vide Volume 4. No.6 ( 2023)

I would like to support students to demonstrate their understanding of the mathematical contents, any ideas or strategies about this?

Does developing reading, written, and audio mathematical language and developing students’ understanding of this language lead to the possibility of improving mathematical communication and solving verbal problems?

The circle touches AB and AC the lateral sides of the isosceles triangle ABC at the vertices B and C (Fig. 1).

On the arc of this circle, which lies inside this triangle, there is a point K so that the distances from it to the sides AB and AC are equal to 24 cm and 6 cm appropriately (Fig. 2).

Find the distance from point K to side BC.

Enough to expand science.

1)

2)

Need assistance 5 topics for my research project. But prefer them to be more related to Mathematics and Natural sciences and Health Education for senior primary learners!

The question seems to be whether mathematics is necessary or not, but the question mark is that if mathematics is essential for physics, chemistry and almost all other subjects, then they should change their methods and language to something understandable for teachers of these subjects.

Which makes sense.

Classical mathematical language and formations are redundant, in many cases illogical and contradictory, and they furthermore have the structural appearance of a heroglipic language.

Personally, I and too many of my mathematician and physicist friends find ourselves paralyzed when faced with the simplest mathematical situation.

So what !?

I have gathered 5 expert's opinion. However, as I checked for the consistency index and consistency ratio it is > 0.10 which is considered inconsistent. What are some methods u may suggest to make it acceptable? I have read some papers but I am not a mathematics major so I don't have enough knowledge to understand them.

We invite discussions, suggestions, and collaborations on the following: Mathematics is the mother of all the sciences, engineering and technology, and a normed division algebra of all dimensions is the holy grail of mathematics. Singh along with Prof. SD Joshi (IIT Delhi) and Prof. Anubha Gupta (IIIT Delhi) developed normed division algebra of all dimensions which is available in preprint at: https://doi.org/10.13140/RG.2.2.18553.65120/3

Dear colleagues

In many articles I read that AHP has a strong mathematical foundation. I wonder if somebody can explain mathematically each of the steps in the AHP and ANP. Specifically, I am asking for somebody to explain rationally the following aspects:

1- It makes sense using pair-wise comparisons between two different criteria, but which is the mathematical justification of using intuition values to indicate the preferences of the DM, other than in personal problems?

2- The results from the Eigen Values (EV) or geometric mean analysis are trade-off values. Could somebody explain why they are considered equivalent to weights, when they are two different things?

3- Why in AHP AV is preferred to geometric mean?

4- Is it valid to assume that criteria preferences are constant?

5- Is it natural that the selection of criteria does not take into account the alternatives they have to evaluate? It appears that in so doing the preferences are constant, no matter to what alternatives or problem they refer. For instance, a preference of say quality is twice preferred to price, is applicable to everything, meaning that the DM cannot change his/her preferences in aspects so different as selecting a restaurant, buying a car or selecting a long-distance transportation mean.

6- Is there any axiom or theorem that says that the DM estimates must comply with transitivity?

7- Is there any axiom or theorem that says that these values and transitivity can be applied to the real world?

8- Is there any axiom or theorem that supports the idea that subjective weights can evaluate alternatives, or is it intuitive?

9- Is it real and valid that increasing or decreasing the importance of a criterion can be compensated by proportional changes in others? In case it is true, why should it be proportional? Simply because its sum is one?

10- Have users realized that decreasing, say one level, in the Saaty Fundamental Scale, is not as little as it appears to be?

11- AHP was in 1983 charged with Rank Reversal, which is true, albeit further it was found that RR happens in all MCDM methods. Does anybody know why or at least the cause that produces it, irrelevant the method?

12- Why is it assumed that a ranking is invariant when adding alternatives?

13- In sensitivity analysis, most methods work with increasing or decreasing the importance of only one criterion, while keeping the others constant. Is that realistic?

14- On what grounds AHP considers that the criterion with the highest weight is the most important. Is than correct or it is intuitive?

Thank you for your answers

We developed a mathematical model from mathematical point of view and we need tumor growth data that reflect number of tumor cells respect to time for our validation. We are not expert in this field and we need help from experts.

We assume the answer is yes.

Furthermore, this is to be expected to occur in the heat diffusion equation and Schrödinger's PDE from a physical and mathematical point of view.

The question arises: does their combination simplify or complicate the solution?

I am interested to calculate the peierls barrier for the movement of screw dislocation in BCC iron between two peierls vally. For this I am using nudged elastic band (NEB) method in LAMMPS.

We developed initial and final replicas using ATOMSK. However we have to create intermediate replicas having Kinks (between initial and final position) using linear interpolation.

Is there any mathematical relation for generating such replicas or any software that can be used for the same purpose.

Please leave your comments.

Thanks

We assume that this is true nowadays, because two mathematicians from two different mathematical fields can hardly find a common language to communicate.

The question arises: is it possible to reorganize at least the mathematical language?

The mathematical derivation of the two-dimensional trapezoidal rule formula sounds like academic hum and numerical calculations using this formula deceive or mislead you.

So what?

In general relativity, singularities, like those theorized to exist at the centres of black holes or at the origin of the Big Bang, present unique challenges to both mathematical and physical understanding. These singularities are characterized by conditions where physical laws as currently understood cease to be predictable or observable. This raises a fundamental question: Are singularities purely mathematical constructs that exist beyond the Planck scale, and thus beyond the scope of empirical validation?

General relativity predicts the existence of singularities, regions of infinite density where the gravitational field becomes infinite. Notably, the mathematical representation of these singularities involves values that approach division by zero, which is undefined and non-physical. For example, the Friedman-Robertson-Walker (FRW) solution to Einstein's field equations, which underpins the standard Big Bang model, indicates a singularity at the time of the universe's inception.

These singularities occur at scales smaller than the Planck scale, where the effects of quantum gravity are hypothesized to become significant, yet remain unquantified by existing theories. As such, singularities are not observable with current technology or provable by existing physical laws, which are based on empirical evidence. This limitation leads to the interpretation of singularities as mathematical abstractions rather than physical entities.

Given these considerations, should singularities be viewed solely as theoretical constructs within the mathematical frameworks of cosmology and black hole physics? How might advances in theoretical physics, particularly in quantum gravity, change our understanding of these enigmatic features? Whether and how singularities might bridge the gap between current mathematical theory and physical reality.