Mathematics - Science topic

f1 to f5 exist if (m(t) ,r(t), u(t)) is used - unite vector u(t) change direction. Use product rule and tell me your result. Never been discussed before?

• https://inquiryintoinquiry.com/2025/03/16/signs-of-signs-1-a/

Re: Michael Harris • Language About Language

• https://mathematicswithoutapologies.wordpress.com/2015/05/23/language-about-language/

There is a language and a corresponding literature treating logic and mathematics as related species of communication and information gathering, namely, the pragmatic-semiotic tradition transmitted through the lifelong efforts of C.S. Peirce. It is by no means a dead language but it continues to fly beneath the radar of many trackers in logic and math today. Nevertheless, the resource remains for those who wish to look into it.

Resources —

Higher Order Sign Relations

• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations

Survey of Pragmatic Semiotic Information

• https://inquiryintoinquiry.com/2024/03/01/survey-of-pragmatic-semiotic-information-8/

Survey of Semiotics, Semiosis, Sign Relations

• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

Soumendra Nath Thakur

ORCID: 0000-0003-1871-7803

Tagore's Electronic Lab, India.

March 10, 2025

A fundamental and precise understanding of the speed of light (c) in relation to the observer's speed (S), along with a clear explanation of why the observer's speed is negligible, is presented through a nuanced, non-relativistic scientific framework grounded in consistent physical principles.

The principle of special relativity asserts that the laws of physics remain the same for all observers in any inertial frame of reference. As a result, the relativity's understanding of light’s constant speed originates from Einstein’s theory of special relativity (1905), which states that the speed of light in a vacuum is a universal constant, unaffected by the motion of the light source or the observer.

Mathematically, this is expressed as:

λ f = c

where:

c represents the speed of light,

λ is the photon's wavelength,

f is the photon's frequency.

The equation

λ f = c

can also be rewritten as:

f = c/λ

This highlights the inverse relationship between wavelength and frequency while maintaining the constant speed of light. However, in logical terms, this inverse relationship implies that their ratio must always yield c.

However, the equation λ f = c is primarily a mathematical convenience derived from the known frequency-wavelength inverse relationship. While effective in describing wave behaviour, it does not fully explain the fundamental reason why light's speed remains constant.

Here on, we will discuss a fundamental and precise understanding of the speed of light (c) in relation to the observer's speed (S);

Each observer in question is characterized not only by their matter mass (Mₘ) but also by the gravitational mass (M𝗀), which includes both the observer’s own mass and the gravitational influence of the massive body they reside on. Matter mass of the observer’s host body is a crucial factor in this physical consideration.

In addition to ordinary (baryonic) mass, the mass of dark matter is also accounted for and included in the total matter mass (Mᴍ). In classical mechanics, inertial mass is traditionally considered equivalent to gravitational mass, expressed as m = m𝗀. However, recent observations of the gravitational effects of dark matter and dark energy on ordinary (baryonic) inertial mass have necessitated an extension of this classical equation.

Extended Classical Mechanics (ECM), in alignment with established observational evidence and theoretical formulations, refines this understanding by recognizing that gravitational mass is equivalent to the sum of matter mass and negative apparent mass, which can dynamically modify the effective mass—potentially turning it negative at intergalactic scales. — This relationship is expressed as:

Mg = Mᴍ + (−Mᵃᵖᵖ)

When considering the speed of light in relation to the speed of an observer, the scale of measurement should be at least planetary. Consequently, the speed (S) of the planet on which the observer is located must be taken into account and included in the calculation.

According to the principles of Extended Classical Mechanics (ECM), a photon possesses negative apparent mass (-Mᵃᵖᵖ) where: Mᴍ = 0 for photons, which, unlike matter mass (Mᴍ), exhibits an anti-gravitational property.

As a result, photons are not inherently restricted by an upper speed limit when external influences (gravitational fields, Planck-scale constraints) are absent.

However, Planck units impose a fundamental limit, restricting the smallest possible wavelength to the Planck length (ℓP) and the shortest measurable time to the Planck time (tP), thereby constraining a photon's behaviour within permissible limits.

Beyond the Planck scale, classical, relativistic and even quantum descriptions of spacetime break down. At the Planck length (ℓP ≈ 1.616 × 10⁻³⁵ m) and Planck time (tP ≈ 5.391 × 10⁻⁴⁴ s), gravitational and quantum effects become inseparable, implying that distances smaller than ℓP and times shorter than tP lose physical significance. This arises because, at these scales, quantum fluctuations of spacetime dominate, leading to a breakdown of continuous space and time concepts. Hence, any attempt to define a wavelength smaller than ℓP or a frequency beyond the Planck frequency (fP = 1/tP) is meaningless in a physically observable sense.

The Planck scale imposes a fundamental limit on measurable space-time intervals, ensuring that beyond these limits, conventional descriptions of motion—including those of photons—lose physical meaning. This restriction provides a natural boundary for photon behavior before additional external influences, such as gravitational redshift and cosmic expansion, further alter their observed properties.

Within a gravitational field, a photon expends energy while escaping, leading to a redshift in its wavelength. However, beyond significant gravitational influence, a photon's speed—defined by the ratio of its wavelength (λ) and frequency (f)—further changes due to the cosmic recession of galaxies, resulting in an additional energy loss.

The Planck length (ℓP) and Planck frequency (fP), as defined in Planck units, are derived from Planck’s constant and other fundamental constants. They establish a theoretical limit on the smallest meaningful measurements of space and time, where our current physical understanding, including relativity, breaks down and quantum gravity effects become dominant.

In classical mechanics, speed is determined using the values of distance and time associated with a given motion. The fundamental equation for speed is:

S = d/t

At the quantum scale, this equation is expressed as:

ΔS = Δd/Δt

where:

Δd corresponds to the Planck length (ℓP),

1/Δt corresponds to the Planck frequency (fP), where Δt represents the Planck time.

The expression c = λf, where f = 1/Δt, translates directly into the quantum-scale speed equation ΔS = Δd/Δt. Here, wavelength (λ) corresponds to a measurable distance (Δd), and its division by the time period of one oscillation (1/Δt) mirrors the definition of speed as distance per unit time. This consistency shows that the speed of light (c) is fundamentally a ratio of measurable spatial and temporal quantities, reinforcing that c = λ(1/Δt) is structurally identical to the speed equation ΔS = Δd/Δt, where ΔS represents velocity measured within a defined quantum reference frame.

Thus, the equation

ΔS = Δd/Δt

can be interpreted as:

c = fλ

where:

ΔS represents c (the speed of light),

Δd represents the photon's wavelength (λ),

1/Δt represents the frequency (f).

Additionally, the speed of light can be expressed in terms of the Planck scale:

c/ℓP = fP

Since the Planck length (ℓP) is the smallest meaningful spatial unit, and the Planck frequency (fP) is the highest fundamental oscillation frequency in the universe, their ratio is equivalent to the ratio of a photon's wavelength to its inverse frequency (which corresponds to the Planck time).

I have previously mentioned that photons are not inherently restricted by an upper speed limit when external influences (gravitational fields, Planck-scale constraints) are absent. This distinction is crucial in understanding the fundamental difference between the speed of observers and the speed of photons.

Photons, having negative apparent mass (-Mᵃᵖᵖ), exhibit fundamentally different gravitational properties than observers with positive matter mass (Mᴍ).

Photons exhibit an anti-gravitational nature, meaning they follow the dynamics of negative apparent mass, which aligns with negative effective mass (-Mᵉᶠᶠ) contributions in the universe (similar to dark energy).

Observers and massive objects exhibit gravitational properties, meaning their motion is bound to the gravitational pull of the universal potential centre (i.e., the tendency toward gravitational collapse).

Because the forces governing these entities are opposite in nature, their respective speeds must also be opposite in direction. The observers’ movement is toward the universal gravitational potential, while photons move away from the universal potential due to their anti-gravitational nature. This opposition results in an effective cancellation of speed components between the two systems.

Since an observer's speed is defined in a gravitational reference frame and a photon's speed in an anti-gravitational reference frame, their effective speeds appear in opposite directions. Given that the magnitude of negative effective mass (-Mᵃᵖᵖ) dominates, the observer's gravitational speed is effectively negligible when compared to the anti-gravitational motion of photons.

The anti-gravitational motion of photons, driven by negative apparent mass (-Mᵃᵖᵖ), aligns with the large-scale acceleration of cosmic structures. This is evident in the recession of galaxies, where the dominance of negative effective mass (-Mᵉᶠᶠ) contributes to the observed cosmic expansion, reinforcing the fundamental opposition between gravitationally bound matter and anti-gravitational dynamics.

Moreover, since the measurement system itself is dictated by the dominant mass-energy contribution, we must recognize that:

The negative measurement system dominates due to the overwhelming contribution of negative apparent mass (-Mᵃᵖᵖ) and negative effective mass (-Mᵉᶠᶠ), which surpasses the contribution of positive matter mass (Mᴍ).

In a mass-energy dominated measurement framework, where the contribution of -Mᵃᵖᵖ vastly exceeds that of positive matter mass, the effective measurement system aligns with anti-gravity, making the observer’s motion negligible in contrast to the dominant anti-gravitational dynamics.

Gravitational deceleration in a positive mass system corresponds to anti-gravitational acceleration in a negative effective mass system, reinforcing that the observer’s motion is measured within a negative measurement framework when compared to photons.

The speed of photons in the anti-gravitational system is vastly superior to the speed of observers in the gravitational system . This makes the gravitational motion of observers negligible compared to the anti-gravitational motion of photons.

Thus, the ultimate outcome is that the speed of observers with positive mass is rendered insignificant when contrasted against the anti-gravitational speed of photons with negative apparent mass, which follows an opposite trajectory—away from the universal potential. The dominance of the negative measurement system further amplifies this effect, reinforcing the fundamental asymmetry between the two domains.

Hi all

I'm an final semester ug btech student. I want to do research in mathematics and publish a paper for the same. Can anyone help me finding out problem statements for the same?

If one dedicates time every morning to solving mathematical problems, especially focusing on geometry, how could this consistent effort influence cognitive abilities like analytical thinking and memory after a year? Are there any studies or examples that demonstrate such an effect?

I published an presentation "Three-Body Problem", where new equations for the interaction of celestial bodies and a solution method are proposed.

If you look on the Internet, it seems that everyone is enthusiastically looking for a solution to the three-body problem. And then someone exclaimed: "I found it!". "And around the silence, taken as a basis," - as Vaenga sings by russian. It turns out that those who are passionately searching, the process of searching is important. It cannot be stopped. You must constantly experience the drama of chaos in the Universe and be proud of your belonging to those physicists who tirelessly prove that there is no other World, because it follows from mathematical solution.

Yes, the World is mathematically substantiated, mathematics is a project, a tool and material for creating the World. But mathematics can be just a game for the mind. You need to find the mathematics that describes the project of the Universe. Mathematics that proves that there is no Building cannot be a project of the Universe.

It turns out that many seekers like mathematical games. I appeal to seekers of truth.

Solomon Khmelnik

I’m currently pursuing a PhD in Mathematics with a focus on Fluid Dynamics and heat pipes. My goal is to join DRDO as a Scientist B, but given the limited vacancies for MSc Mathematics candidates, I’m wondering whether pursuing an MTech would significantly increase my chances. If I decide to go for an MTech, I would have to withdraw from my PhD, so I’m unsure if that’s a good choice or if continuing my PhD at a central university would be better. Additionally, since my background is in Mathematics rather than Engineering, what MTech disciplines would be most suitable for DRDO, and are institutions like NITs, CIITs, or DIITs offering relevant MTech programs for MSc Mathematics students?

I have an article that I need to prepare for publication. It is titlde "The Superposition of Distance: A Quantum-Relativistic Paradigm of Finite Points in Infinite Space" If anyone can support me in preparing this for publication please contact me.

Thanks,

So I have two mathematical formulas, one covered with a number of patents around the applications, the other being not such a worry for me.

I am trying to write a paper, but I just used the basics instead, and am hoping that maybe someone can guide me to make it acceptable for the scientific community.

The two mathematical formulas:

1) Increases data capabilities by using the concept of "dead space" or "alternative methods" to make data transmission/storage slip into Ternary instead of Binary. This is the patent protected stuff.

2) A simple construct designed to make fast conversions to utilize dead space in transmissions, its fast, efficient, and allows a 150% increase in all wavelength transmissions when using my patented stuff above and this conversion method.

Is the core idea of ​​the article "On the Relativity of the Speed ​​of Light" really difficult to understand?

When writing this paper, I followed the unspoken rule that they should tactfully question the wrong theories of celebrities, so the real point of view is relatively hidden. After so many years, some scientists still do not correctly understand the core idea of ​​this article:

"The Lorentz transformation is not wrong, but it was completely misinterpreted by Einstein, who lacked basic mathematical knowledge and basic logical knowledge. See the paper:

If Einstein's explanation is valid, then there must be a complete space-time transformation that Einstein should have thought of but ignored. Such a deduction of complete space-time transformation in the sense of relativity denies the invariance of the relative speed of light, which is the focus of relativity. Some people angrily say that Einstein is actually very ignorant, which may be true. The root cause of the world's carnival in the theory of relativity lies in the simple charm of ignorant imagination. The so-called exact solution of the Dirac equation that the textbook promotes the success of the important application of relativity in quantum mechanics is also completely wrong! The correct exact solution has been given in part, see Mathematics & Nature 3 (https://mathnature.github.io/, 2022). It is absurd, then, that a false conclusion should be claimed to have been extensively tested experimentally.”

I recently published a paper proposing a rigorous proof of the Riemann Hypothesis by integrating noncommutative spectral geometry and topological field theory.(doi.org/10.13140/RG.2.2.29395.90405) The idea revolves around constructing a noncommutative spectral manifold, where the zeros of the Riemann zeta function are encoded in the spectrum of a Dirac-like operator. By analyzing the stability of gauge field configurations using Yang-Mills action and studying quantum adiabatic dynamics, I demonstrate that the Berry phase is quantized only when the zeros lie on the critical line.

This method not only offers a potential resolution to the Riemann Hypothesis, but it also bridges the fields of analytic number theory, quantum mechanics, and noncommutative geometry. I would love to hear thoughts and insights from fellow researchers regarding the use of topological field theory in this context and whether this approach could offer a new perspective on other major open problems in mathematics and physics.

Synergy has to do with obtaining a total greater than the sum of its parts, but this seems not to be a logical thing if we think just in terms a real numbers in which 1+1 = 2.

In fact in my book

today available in researchGate:

and published in Amazon in 1999, there is a proposal how the Principle of Synergy can be applied not only to the physical, but to other areas of human thought, and how based on the most beautiful equation of mathematics, I mean Euler’s relation, it is possible to obtain a sum greater that the sum of its part, so that mathematically Synergy can be represented.

A whole greater than the sum of its parts is a concept that has been considered metaphysical, but that most prolific mathematician of all times, in his work with infinite series in 1745, found that relationship that was described as the most beautiful relationship in mathematics in a 1998 competition held by Intelligencer, called Euler's relation. This relationship in fact includes all numbers: one, zero, the equals sign, but also the square root of minus one, as well as infinity, of course, since it is an infinite series. It is this concept the one then that permits to describe mathematical the concept of Synergy and that has permitted to deduce all fundamental equation of physics in a unified framework of reference:

* that of the pendulum a real harmonic oscillator

* that of the gravitational field including that of the planet mercury obtained by Einstein, but in this case obtained with a mathematical tool not so much complicated as was done with Tensor Analysis

* those of SR in another approach, in which linear moving is just a special case of the more general solution obtained with the BSU concept in which covariance is included as it is a consequence of the isomorphic property of Euler’s relation mentioned above and finally the

* Schrödinger’s wave equation

and most probably it will serve even for that new field of physics called Quantum Field Theory, as in fact it is a real harmonic oscillator, that vector in the complex plane, that remains invariant in spite of change, as Euler’s relation remains the same with those mathematical operations that represent change: integration and derivation.

Edgar Paternina

retired electrical engineer

February 15, 2025

Consider the question: What if this philosophical construct (presented within an image, attached below) were plotted on a three-dimensional mathematical graph? Only then would the scientific facts behind it be fully justified.

Take, for example, the Penrose Triangle—an "impossible" figure first conceptualized 62by Swedish artist Oscar Reutersvärd in 1934 and later rediscovered in the 1950s by physicist Roger Penrose. Described as "impossibility in its purest form," this figure was popularized by Penrose and further explored in the works of M.C. Escher. It appears to be a solid structure composed of three straight sections of square beams, seamlessly joined at right angles.

However, its paradoxical nature cannot be conclusively analyzed through mere physical observation and philosophical interpretation. Instead, by mathematically plotting the enclosed three-dimensional image onto a precise three-dimensional graph, its inherent absurdity becomes evident. The deceptive illusion, which seems structurally feasible in a two-dimensional representation, is mathematically exposed as an impossible configuration in three-dimensional space.

Image

Abstraction vs. Reality

Addressing the statement: "Abstract mathematics is based on logical principles rather than empirical validity. It does not rely solely on physical evidence."

When we add one apple to another, we perceive two apples. However, the concept of "two" itself is an abstract mathematical construct rather than a directly observable physical entity. The sum exists as a logical principle within mathematics, not as a tangible proof in itself.

All real numbers—1, 2, 3, and beyond—are fundamentally conceptual, created within the framework of mathematical reasoning rather than derived from physical evidence. While mathematical concepts often align with physical reality, their foundation is purely abstract, shaped by human perception and logical consistency rather than empirical observation.

Mathematics is not confined to the physical universe; its abstract principles hold universally, independent of space, time, or physical existence. It is the fundamental language of the cosmos—objective, unique, and uninfluenced by human divisions such as culture, religion, or race.

Physicists seek mathematical formulations to explain the origins, structure, and dynamics of the universe. In doing so, they attempt to decipher the underlying mathematical order that governs both the observable universe and any potential realities beyond it. Understanding the universe is, in essence, understanding its mathematical nature.

For philosophy to be scientifically meaningful, it must be grounded in abstract mathematical logic. Without mathematical rigor, philosophical reasoning remains speculative and cannot be accepted as a scientific discipline.

Conclusion

This distinction underscores a fundamental contrast between philosophy and science: while philosophy interprets illusions conceptually, science—through mathematical rigor—reveals their underlying reality.

Attachment: the image as stated in the above mentioned text.

Chaos Theory, Chaotic dynamics

Recent HSC mathematics results in Mauritius indicate a concerning trend: a 5.52% decline in student performance within a year. The overall pass rate dropped from 91.8% in 2022 to 82.9% in 2024, with fewer students achieving top grades. This trend raises critical questions about the effectiveness of teaching methodologies, student engagement, and systemic issues in mathematics education.

This discussion seeks to explore key challenges contributing to this decline, including:

Given the increasing importance of STEM careers, AI, and data-driven industries, how should mathematics education evolve?

Key Discussion Points: 🔹 What strategies have been most effective in improving mathematics achievement at different educational levels? 🔹 How can technology be integrated to make learning more engaging and effective? 🔹 Should curricula prioritize real-world applications of mathematics rather than abstract problem-solving? 🔹 What pedagogical innovations (e.g., flipped classrooms, gamification, inquiry-based learning) have worked in other contexts?

I welcome insights from educators, researchers, and policymakers on addressing these issues and improving student outcomes in mathematics.

It is the theory of probability, where probability and mathematical statistics are used to analyze data and draw conclusions. Mathematical tools in statistics include statistical analysis, hypothesis testing, probability distributions, etc. These tools are used to analyze samples and predict the likely behavior of societies and statistical events in general.

Physics has long been intertwined with mathematics as its primary tool for modeling nature. However, a fundamental question arises:

This question challenges the assumption that any mathematical construct must necessarily describe a real physical system. If we take a purely mathematical perspective, an infinite number of logically consistent mathematical structures can be conceived. Yet, why does our physical reality seem to adhere to only a few specific mathematical frameworks, such as differential geometry, group theory, and linear algebra?

v Mathematical Pluralism vs. Physical Reality:

v Physics as a Computationally Limited System:

v The Relationship Between Mathematics and Nature:

v Beyond Mathematical Formalism:

This discussion also touches on a deeper philosophical question: Are we merely discovering the mathematical laws of an objectively real universe, or are we creating a mathematical framework that fits within the constraints of our own perception and cognition?

If mathematics is merely a tool, then our physical theories may be contingent on human cognition and not necessarily reflective of a deeper objective reality. Conversely, if mathematics is truly the "language of nature," then understanding its full structure might reveal hidden aspects of the universe yet to be discovered.

Werner Heisenberg once suggested that physics will never lead us to an objective physical reality, but rather to models that describe relationships between observable quantities. Should we accept that physics is not about describing a fundamental "truth," but rather about constructing the most effective predictive models?

Physics has long been regarded as the science dedicated to uncovering the fundamental laws governing nature. However, in contemporary theoretical physics, there is an increasing reliance on mathematical models as the primary tool for understanding reality. This raises fundamental questions:

Throughout the history of science, there have been instances where physicists, without fully grasping fundamental principles, formulated models that later turned out to be mere consequences of emergent effects rather than reflections of objective reality. For instance, in classical thermodynamics, macroscopic quantities such as temperature and pressure emerged as statistical descriptions of microscopic particle behavior rather than fundamental properties of nature.

The crucial question today is: Are we still facing similar emergent illusions in modern theoretical physics? Could it be that many of the sophisticated mathematical models we use are not pointing to an underlying physical reality but are merely the byproducts of our perception and modeling techniques?

Mathematics has always been an essential tool in physics, but over time, it has also shaped the way we think about physical reality. In many areas of theoretical physics, mathematical methods have advanced to a point where we may no longer be discovering physical truths but instead fine-tuning mathematical structures to fit our theoretical frameworks.

Examples of such concerns can be found in theories like string theory, where extra spatial dimensions and complex symmetry groups are introduced as necessary mathematical elements, despite lacking direct experimental verification. This raises the possibility that some of these theoretical constructs exist only because they are mathematically required to make the model internally consistent, rather than because they correspond to something physically real.

One of the most profound implications of this discussion is that the very question of whether physics describes "physical reality" might be fundamentally misguided.

Werner Heisenberg once argued that physics will never lead us to an understanding of an objective physical reality. Instead, what we develop are models that describe relationships between observable phenomena—without necessarily revealing the true nature of reality itself.

The final question in this discussion is: Are we still trapped in emergent effects that arise purely from our mathematical approaches rather than reflecting an objective physical reality?

Ultimately, this discussion seeks to examine whether physics is merely a computational framework for describing phenomena, or if we are still subconsciously searching for a physical reality that might forever remain out of reach.

I would need a detailed mathematical derivation of mimetic finite difference operators in cartesian as well as curviliniear, in particular spherical coordinates. I also need to know how the pole problem in spherical coordinates can be tackled when using mimetic operators. Does anybody have a hint for me?

ORCiD: 0000-0003-1871-7803

February 10, 2025

In Extended Classical Mechanics (ECM), any massless entity reaching the Planck frequency (fp) must acquire an effective mass (Mᵉᶠᶠ = hf/c² = 21.77 μg). This acquisition of mass is a direct consequence of ECM's mass induction principle, where increasing energy (via f) leads to mass acquisition.

At the Planck scale, the induced gravitational interaction is extreme, forcing the entity into gravitational collapse. This is a direct consequence of the mass acquisition at the Planck frequency, where the gravitational effects become significant.

The apparent mass (−Mᵃᵖᵖ) of a massless entity contributes negatively to its effective mass. However, at the Planck threshold, the magnitude of the induced effective mass (|Mᵉᶠᶠ|) surpasses |−Mᵃᵖᵖ|, ensuring that the total mass is positive:

This irreversible transition confirms that any entity at fp must collapse due to self-gravitation.

Below the Planck frequency, a photon behaves as a massless entity with effective mass determined by its energy-frequency relation. However, at fp​, the gravitating mass (Mɢ​) and effective mass (Mᵉᶠᶠ) undergo a shift where induced mass dominates over negative apparent mass effects.

Planck-scale energy is not just a massive state—it is a self-gravitating mass that collapses under its own gravitational influence. This suggests that at Planck conditions, the gravitationally induced mass dominates over any negative mass contributions, maintaining a positive mass regime.

At the Planck scale, gravitational mass (Mɢ​) is immense due to the fundamental gravitational interaction. Since |+Mɢ​| ≫|−Mᵃᵖᵖ|, the net effective mass remains positive:

This suggests that at Planck conditions, the gravitationally induced mass dominates over any negative mass contributions.

The condition −Mᵃᵖᵖ > Mɢ could, in principle, lead to a transition where the effective mass becomes negative. This might occur under strong antigravitational influences, possibly linked to:

• Dark energy effects in cosmic expansion.

• Exotic negative energy states in high-energy physics.

• Unstable quantum fluctuations near high-energy limits.

Since Mᵉᶠᶠ ≈ Mᴍ, under these extreme conditions, it implies that matter mass emerges predominantly as a consequence of gravitational effects. This aligns with ECM’s perspective that mass is not an intrinsic property but rather a dynamic response to gravitational interactions.

This work on ECM provides a detailed and nuanced understanding of how gravitational interactions can induce mass in initially massless particles, leading to gravitational collapse at the Planck scale. This perspective not only aligns with fundamental principles but also offers potential explanations for cosmic-scale phenomena involving dark matter, dark energy, and exotic gravitational effects. The detailed mathematical foundations and the implications of apparent mass and effective mass in ECM further clarify how mass can dynamically shift between positive, zero, and negative values based on gravitational and antigravitational influences.

The concept of infinity is well known in Mathematics and I have no disagreement with this. But, in real world, something like number of species or number of water molecules or even number of stars, are anything exist that beyond finite?

Discuss your research ideas

It is well known that Medal Fields Prize is intended for excellent research of mathematicians under forty years old because many mathematicians think that the main contributions in the life of the researchers are obtained when they are younger than forty. I do not believe so. It is true, by common experience, that the students of Mathematics, which are constantly in interaction at the same time, with several (and sometimes, very different) subjects, develop a high degree of good ideas which inspire them and lead them to obtain new and interesting results. This interaction between different branches is expected to remain (more or less consciently) up to forty years old. By the same reason, if necessary, whoever researcher, independently of his/her age, may return to study the different mathematical matters and create new important contributions, even in his/her very definite area of research. Furthemore, it may help to overcome a blockade. It is incredible the fact that when one studies again different matters it inspires you, and combined with your experience and knowledge, you see the contents of these different subjects with new perspective, often helping in your area of research creating new knowledge and solving problems. This is the motive why I believe that the career of each mathematician is always worthly and continuous independently of his/her age as demonstrated by most senior mathematicians in all the areas of research who are living examples for us.

What is your opinion on the relationship between the age of a researcher and the quality of his/her contributions?

Thank you very much beforehand.

In mathematical and statistical models?

# 163

Dear Sarbast Moslem , Baris Tekin Tezel, Ayse Ovgu Kinay, Francesco Pilla

I read your paper:

A hybrid approach based on magnitude-based fuzzy analytic hierarchy process for estimating sustainable urban transport solutions

My comments:

1- In the abstract you say “The study employs the newly developed Magnitude Based Fuzzy Analytic Hierarchy Process, chosen for its accuracy and computational efficiency compared to existing methods”

Are you aware that Saaty explained that it is incorrect to apply fuzzy in AHP because it is already fuzzy?

Since when using intuition ensures accuracy? Do you have any proof of what you say?

Sensitivity analysis does not ensure quality, what it does us to find how strong is a solution

2- In page 2 you talk about linear regression for evaluation. Linear regression is used to predict the value of a dependent variable on the value of one or more independent variables.

3- In page 3 “AHP offers mechanisms for ensuring consistency in decision-making through pairwise comparisons and sensitivity analyses”

True, AHP ensures consistency by FORCING to adhere to transitivity, by imposing the DM to change his/her estimates.What it ensures, is transitivity without any mathematical foundation,just for the sake of the method. Therefore, this ‘consistency’ is fabricated.

Even if there is a real consistency, it reflects the coherence of the DM, but it does not necessarily mean that consistency and weights can be applied to the real-world. There is no mathematical support for this assumption, neither to assume that the world is consistent, let alone common sense; it is convenient indeed, for the method, but useless for evaluation

4- Page 3 “On the other hand, expressing the data in the form of fuzzy numbers to better express the uncertainty in individual judgments has led to the suggestion and widespread use of fuzzy AHP (FAHP) methods, which include calculations based on fuzzy arithmetic”

5- In several parts of the paper it mentions validation of results. That is only a wish, because no MCDM method has any real yardstick to compare to. It is another and very common fallacy.

6- Pag 8, Fig 4. I understand of that waiting time does not depend on speed but on frequency buses arrival (Number of buses of the same route per hour). The more the frequency the lesser the waiting tine. What role do have the buses speed here? It appears that this concept does not come from transportation experts.

I guess that interchanging buses or routes is more adequate

“Need of transfer” normally refers to paying a single ticket, that allows a pax to change bus routes, that is, you he can board another bus with the same ticket

Your definition on “Time availability” does not seen too coherent, because what “Number of times that UBT is deployed??? over a route” mean?

“Limited time of use” (C4.2)????. I understand that you want to say ‘Operating hours’, that is start running and finish running, or simply ‘Scheduling’. I am afraid that your expression does not belong to the urban transportation industry.

Please do not be offended with my observations, it is not my intention. Only that is you want readers understanding what you write, you must use the appropriate words. If not, your work risks to be misunderstood and downgraded

Why “providing new buses” is related to “comfort” in stops?

7- Page 9 “In other words, it models the state of uncertainty in the mind of the decision-maker.”

Very true, but why those uncertainties of a MIND can be translated to real-life? In other words, what theorem or axiomsupports that what the DM estimates can be used in real-world? It is a simple assumption that even defies common sense. In my opinion, it does not make any sense to apply fuzzy to invented values. Yes, one will get a crisp value, and what is it good for? For nothing

These are some of my comments. Hope that they can help you

Nolberto Munier

Mathematics and statistics have been used to develop encryption techniques used to protect against cyber threats and ensure the security of information and data. Mathematics is also used to generate complex passwords and develop machine learning models to detect cyber attacks and threats in networks. In addition, various mathematical concepts such as algebra, geometry, statistics, and numerical analysis are used in specific areas such as security software development, software bug checking, databases, and complex networks.

A control volume is a mathematical construct used in the process of creating mathematical models of physical processes.

We assume that the control volume in R^4 space (3D+t external control) is incomplete and almost useless for dealing with complex physical and mathematical problems.

On the other hand, the 4D x-t unitary control volume called Cairo techniques and B matrix space is the complete universal space adequate for the description of classical and quantum physics situations as well as mathematical events.

In a way, this is the correct approach to the unified field theory (energy density).

My answer is negative and thoroughly substantiated via 2 points.

1) The easiest part (lesser limiting factor) he has to comprehend the approach used in physics thinking and epistemology (i.e. working with hypotheses instead of etiological thinking, refraining from teleological inquiries etc), the importance of relying on maths, relevancy of equations etc. Not easy but can be accomplished to large degree by serious commitment and authentic interest

2) ease with representations (geometrical, model-wise etc) of physical systems and working cognitively on that level, abstract aplicational mathematical thinking (this may not be easy even for mathematicians) etc. This is something that required in my opinion an in-born trait

Discuss it

In the attached paper titled, "Approximate expressions for BER performance in downlink mmWave MU-MIMO hybrid systems with different data detection approaches", mathematical operator ℝ{.} is used with minus sign in Equation (14). Can anybody help explain the meaning of this operator? Why minus sign is used?

Hi dear professors. I want to share with you a little formula of arithmetics in order to know the usefulness of this formula and its meaning according to you. This is the formula like in the attached picture: 1+8*E^2≠3*F^2 where E and F can be any integers.

The proof is easy in the article of this link:

Under the theme “Mathematics for All, Mathematics as a Path to Development,” one of the best examples of math's impact on daily life is its application in medical imaging, such as CT and CBCT scanners.

How do these devices work?

Mathematics is at the heart of it! When X-rays are emitted from a source, they pass through tissues and are captured by detectors. Along their path, these rays undergo absorption or attenuation, which depends on the properties of the tissue they traverse. Each ray's path represents a unique equation, where the unknowns are the attenuation values of the points it passes through.

As the source rotates around the object, it generates multiple paths, creating a system of n equations with n unknowns. Sophisticated software solves this complex system using advanced algorithms, such as the Radon Transform or iterative methods, to reconstruct the internal structure of the object as a detailed 3D image.

How does this contribute to development?

Mathematics is the backbone of such groundbreaking technologies, enabling precise diagnostics and effective treatment planning. This practical application of equations demonstrates how math can transform our world.

Now it’s your turn:

What other applications of mathematics in medicine and engineering do you think are underexplored? How can we further harness the power of math as a tool for development?

Let’s explore the incredible role of mathematics in everyday life together in this discussion!

Answer briefly

I want a qualitative scale that contains questions that reveal the mathematics teacher's perception of beauty and simplicity in mathematics?

Conventional fragility is defined as probability of failure. Based on concise mathematics, it is found that if fragility is probability of collapse then the design curve is probability of survive. The sum of these two is equal to 1. Consequently, if a member (structure) is designed based on a give curve, then its fragility of collapse is also known!.

Scale the horizontal axes of a fragility curve (s) of a structure, between 0 and 1. Then:

what is the probability of collapse at s=0.5 ?

what is the probability of survive at s=0.5

Don you agree with the above findings? Why ?

Well actual physics, can be represented mathematically with the Basic Systemic Unit, based on Euler’s relation with its most remarkable property of remaining the same in spite of change, that permits to deduce the fundamental equations of physics such as :

* that of the pendulum a real harmonic oscillator

* that of the gravitational field including that of the planet mercury obtained by Einstein, but in this case obtained with a mathematical tool no so much complicated as was done with Tensor Analysis

* those of SR in another approach, in which linear moving is just a special case of the more general solution obtained with the BSU concept in which covariance is included as it is a consequence of the isomorphic property of Euler’s relation mentioned above and finally the

* Schrödinger’s wave equation

For those interested in the way all this is obtained you can see my papers:

https://www.researchgate.net/publication/384190006_Quantum_Physics (https://www.researchgate.net/publication/384190006_Quantum_Physics)

https://www.researchgate.net/publication/385592651_A_Quantum_Theory_of_GravitationNewpdf (https://www.researchgate.net/publication/385592651_A_Quantum_Theory_of_GravitationNewpdf)

https://www.researchgate.net/publication/382357270_Special_Relativity_Another_Approach (https://www.researchgate.net/publication/382357270_Special_Relativity_Another_Approach)

that I really hope will contribute to overcome the great crisis of physics, because the great incompatibility between QM and GR.

So yes, actual physics, can be represented mathematically in a real coherent way, but for it is necessary to make a real paradigm shift.

Edgar Paternina

retired electrical engineer

Einstein overcomplicated the theory of special and general relativity simply because he did not define time correctly.

A complete universal or physical space is a space where the Cartesian coordinates x, y, z are mutually orthogonal (independent) and time t is orthogonal to x, y, z.

Once found, this space would be able to solve almost all problems of classical and quantum physics as well as most of mathematics without discontinuities [A*].

Note that R^4 mathematical spaces such as Minkowski, Hilbert, Rieman. . . etc are all incomplete.

Schrödinger space may or may not be complete.

Heisenberg matrix space is neither statistical nor complete.

All the above mathematical constructions are not complete spaces in the sense that they do not satisfy the A* condition.

In conclusion, although Einstein pioneered the 4-dimensional unitary x-t space, he missed the correct definition of time.

Universal time t* must be redefined as an inseparable dimensionless integer woven into a 3D geometric space.

Here, universal time t* = Ndt* where N is the dimensionless integer of iterations or the number of steps/jumps dt*.

Finally, it should be clarified that the purpose of this article is not to underestimate Einstein's great achievements in theoretical physics such as the photoelectric effect equation, the Einstein Bose equation, the laser equation, etc. but only to discuss and explain the main aspects and flaws of his theory of relativity, if any.

Computational topology of solitons

The well-established research area of algebraic topology currently goes interdisciplinary with computer science in many directions. The Topological Data Analysis gives new opportunities in visualization for modeling and special mapping. A study on metrics used or simplicial complexes are reliable for future results in the area of mathematics. Today, the machine learning from one side is a tool for the analysis in topology optimization, topological persistence and optimal homology problems, from other side the topological features in machine learning are new area of research, topological layers in neural networks, topological autoencoders, and topological analysis for the evaluation of generative adversarial networks are in general aspects of topology machine learning. On practical point of view, the results in this area are important for solitary-like waves research, biomedical Image analysis, neuroscience, physics and many others. That gives us opportunity to establish and scale up an interdisciplinary team of researchers to apply for funding for fundamental science research in interdisciplinary field.

More Info: https://euraxess.ec.europa.eu/jobs/249043

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.

In a project with analysis of log-linear outcomes I have not found the solution to this problem. (log is the natural logaritm)

I assume it is simple, but I am out of clue, and I hope someone more mathematical proficient can help.

YES IT IS Beauty OF MATH IN NUMBER THEORY

Nominations are expected to open in the early part of the year for the Breakthrough Prize in Fundamental Physics. Historically nominations are accepted from early/mid-January to the end of March, for the following year's award.

https://breakthroughprize.org/Nominations

Historically, the foundation has also had a partnership with ResearchGate:

https://www.researchgate.net/press-newsroom/researchgate-breakthrough-prize-partnership

https://www.researchgate.net/institution/Breakthrough

The foundation also awards major prizes for Life Sciences and for Mathematics, and has further prizes specific to younger researchers.

So who would you nominate?

• https://inquiryintoinquiry.com/2024/11/27/differential-propositional-calculus-overview-b/

❝The most fundamental concept in cybernetics is that of “difference”, either that two things are recognisably different or that one thing has changed with time.❞

— W. Ross Ashby • An Introduction to Cybernetics

Differential logic is the component of logic whose object is the description of variation — the aspects of change, difference, distribution, and diversity — in universes of discourse subject to logical description. To the extent a logical inquiry makes use of a formal system, its differential component treats the use of a differential logical calculus — a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.

In accord with the strategy of approaching logical systems in stages, first gaining a foothold in propositional logic and advancing on those grounds, we may set our first stepping stones toward differential logic in “differential propositional calculi” — propositional calculi extended by sets of terms for describing aspects of change and difference, for example, processes taking place in a universe of discourse or transformations mapping a source universe to a target universe.

What follows is the outline of a sketch on differential propositional calculus intended as an intuitive introduction to the larger subject of differential logic, which amounts in turn to my best effort so far at dealing with the ancient and persistent problems of treating diversity and mutability in logical terms.

Note. I'll give just the links to the main topic heads below. Please follow the link at the top of the page for the full outline.

Part 1 —

• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1

Casual Introduction

• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1#Casual_Introduction

Cactus Calculus

• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1#Cactus_Calculus

Part 2 —

• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2

Formal_Development

• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Formal_Development

Elementary Notions

• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Elementary_Notions

Special Classes of Propositions

• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Special_Classes_of_Propositions

Differential Extensions

• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Differential_Extensions

Appendices —

• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Appendices

References —

• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_References

It is provable that quantum mechanics, quantum field theory, and general relativity violate the the axioms of the mathematics used to create them. This means that neither of these theories has a mechanism for processes that they describe to be feasible in a way that is consistent wth rules used to develop the math on which that are based . Thus, these theories and mathematics cannot both be true. This is proven with a $500 reward for disproving (details in the link). So I can prove that the above mentioned theories are mathematical nonsense, and I produce a theory that makes the same predictions without the logical mistakes. https://theframeworkofeverything.com/

Within a specific problem, without the whole picture?

I am currently working on optimizing our inventory management system and need to calculate the monthly safety stock for various SKUs. I have already generated weekly safety stock values based on historical data and lead times. However, I need to adjust these values for a monthly period considering several factors:

1. SKU Contribution Ratio: This ratio indicates the importance of each SKU. A higher ratio means the SKU is more critical and should have a higher safety stock.

2. CCF Factor: This factor reflects our past ability to fulfill orders based on historical order and invoice data.

3. Monthly Stock Reduction Percentage: This percentage shows how much stock is typically left at the end of each month. If this value is 100% for four consecutive months, it indicates no need to keep that much inventory for the respective SKU. Conversely, if the values are decreasing, it suggests that the safety stock has been used and needs to be adjusted.

Given these factors, I need to determine a safety factor for the month, which will be used to adjust the weekly safety stock values to monthly values.

please help me to find a theory that supports my study about storytelling in teaching mathematics

Famous mathematicians are failing each day to prove the Riemann's Hypothesis even if Clay Mathematics Institute proposes a prize of One Million Dollars for the proof.

The proof of Riemann's Hypothesis would allow us to understand better the distribution of prime numbers between all numbers and would also allow its official application in Quantics. However, many famous scientists still refuse the use of Riemann's Hypothesis in Quantics as I read in an article of Quanta Magazine.

Why is this Hypothesis so difficult to prove? And is the Zeta extension really useful for Physics and especially for Quantics ? Are Quantics scientists using the wrong mathematical tools when applying Riemann's Hypothesis ? Is Riemann's Hypothesis announcing "the schism" between abstract mathematics and Physics ? Can anyone propose a disproof of Riemann's Hypothesis based on Physics facts?

Here is the link to the article of Natalie Wolchover:

The zeros of the Riemann zeta function can also be caused by the use of rearrangements when trying to find an image by the extension since the Lévy–Steinitz theorem can happen when fixing a and b.

Suppositions or axioms should be made before trying to use the extension depending on the scientific field where it is demanded, and we should be sure if all the possible methods (rearrangements of series terms) can give the same image for a known s=a+ib.

You should also know that the Lévy–Steinitz theorem was formulated in 1905 and 1913, whereas, the Riemann's Hypothesis was formulated in 1859. This means that Riemann who died in 1866 and even the famous Euler never knew the Lévy–Steinitz theorem.

• https://inquiryintoinquiry.com/2024/10/30/differential-logic-1-a/

Introduction —

Differential logic is the component of logic whose object is the description of variation — focusing on the aspects of change, difference, distribution, and diversity — in universes of discourse subject to logical description. A definition that broad naturally incorporates any study of variation by way of mathematical models, but differential logic is especially charged with the qualitative aspects of variation pervading or preceding quantitative models. To the extent a logical inquiry makes use of a formal system, its differential component governs the use of a “differential logical calculus”, that is, a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.

Simple examples of differential logical calculi are furnished by “differential propositional calculi”. A differential propositional calculus is a propositional calculus extended by a set of terms for describing aspects of change and difference, for example, processes taking place in a universe of discourse or transformations mapping a source universe to a target universe. Such a calculus augments ordinary propositional calculus in the same way the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.

Resources —

Logic Syllabus

• https://inquiryintoinquiry.com/logic-syllabus/

Survey of Differential Logic

• https://inquiryintoinquiry.com/2024/02/25/survey-of-differential-logic-7/

Program Description:

A program that converts mathematical equations from PDF files into editable equations within Word documents. The program relies on Optical Character Recognition (OCR) technology for mathematical equations, ensuring accuracy in retrieving symbols and mathematical formulas. It allows users to easily edit the equations directly in Word and provides support for various mathematical writing formats, such as LaTeX or MathType.

Program Features:

Accurate Conversion: Can read complex mathematical equations from PDF files.

Word Integration: Offers direct import options into Word documents.

Mathematical Format Support: Supports multiple formats such as MathML and LaTeX.

User-Friendly Interface: A simple design suitable for researchers and students.

Multi-Platform Compatibility: Works on operating systems like Windows and macOS.

Examples of programs that may meet this description include:

Mathpix Snip

InftyReader

You can try one of them to find the best solution for your need

I have started an investigation about the utilization of AI for teaching mathematics and physics.

In this framework, I would like any insights and previous findings.

Please send me similar studies.

Thanks you in advance

what about generate 3D shape using different ways: GAN or mathematics with python or LLMs or LSTMs and the related works about this !

Dear Esteemed Colleagues,

I hope this message finds you well. I am writing to invite your review and insights on what I believe to be a significant development in our understanding of the Riemann Hypothesis. After extensive work, I have arrived at a novel proof for the hypothesis, using a generalization of the integral test applicable to non-monotone series, as outlined in the attached document.

As a lead AI specialist at Microsoft, specializing in math-based AI, I have employed both traditional mathematical techniques and AI-based verification algorithms to rigorously validate the logical steps and conclusions drawn in this proof. The AI models have thoroughly checked the derivations, ensuring consistency in the logic and approach.

The essence of my proof hinges on an approximation for the zeta function that results in an error-free evaluation of its imaginary part at $x = \frac{1}{2}$, confirming this as the minimal point for both the real and imaginary components. I am confident that this new method is a significant step forward and stands up to scrutiny, but as always, peer review is a cornerstone of mathematical progress.

I warmly invite your feedback, comments, and any questions you may have regarding the methods or conclusions. I fully stand by this work and look forward to a robust, respectful discussion of the implications it carries. My goal is not to offend or overstate the findings but to contribute meaningfully to this ongoing conversation in the mathematical community.

Thank you for your time and consideration. I look forward to your responses and the productive discussions that follow.

Sincerely,

Rajah Iyer

Lead AI Specialist, Microsoft

Any answer my question

I am interested in the study of visual subcompetence in education, specifically how visual tools and technologies can be integrated into the educational process to enhance the development of professional competencies in future teachers, particularly in mathematics education.

I am looking for research and definitions that highlight and specify the concept of visual subcompetence in education. Specifically, I am interested in how visual subcompetence is distinguished as part of the broader professional competence, particularly in the context of mathematics teacher education.

Can you suggest any study that uses Ethnographic Research design?

I apologize to you all! The question was asked incorrectly—my mistake. Now everything is correct:

In a circle with center O, chords AB and CD are drawn, intersecting at point P.

In each segment of the circle, other circles are inscribed with corresponding centers O_1; O_2; O_3; O_4.

Find the measure of angle ∠O_1 PO_2.

Can you explain the mathematical principles behind the Proof of Stake (PoS) algorithm, including how validator selection probabilities, stake adjustments, and reward calculations are determined

توضيح كيفية التعليم الأخضر في مادة الرياضيات للأطفال

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

As an academic working and pursuing a PhD degree in Egypt, both in private and public universities respectively, I wanted to put forward a simple question:

What is the role of universities, and other academic institutions, today? Was there ever a time where universities were agents of revolutionary action and change, or was it only a subject of the overall consumerist system?

We can take many steps back till the Ancient Egyptian times, where scribes and priests were taught writing, mathematics, and documentation of daily exchanges. All the way till today's era of digital globalization and mass education, where knowledge production process has become more of a virtual canvas rather than actual knowledge. Has knowledge ever served its purpose? Have academic institutions, and of course academic scholars, ever delivered the true purpose of education?

Was, and still, education's main sole purpose is economic prosperity of certain classes, hence socio-economic segregation?

Scientists believe theories must be proven by experiments. Does their faith in the existence of objective reality mean they are classical scientists who reject quantum mechanics' statements that observers and the observed are permanently and inextricably united? In this case, scientists would unavoidably and unconsciously influence every experiment and form of mathematics. In the end, they may be unavoidably and unconsciously influencing the universe which is the home of all experiments and all mathematics.

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.

It seems it is common to combine basic observations to create new observable, which are then used for PPP and other applications. Basic observations such as pseudorange and carrier-phase observations are real measurement from GNSS. These real observations are combined to create entirely new observable which is not direct, physical, and real. Amazingly, these new observable solves the real problem such as PPP (e.g. Ionosphere -free combination).

In triangle ∆ABC (with ∠C = 90°), the angle CBA is equal to 2α.

A line AD is drawn to the leg BC at an angle α (∠BAD = α).

The length of the hypotenuse is 6, and the segment CD is equal to 3.

Find the measure of the angle α.

This problem can be solved using three methods: trigonometric, algebraic, and geometric. I suggest you find the geometric method of solution!