Mathematics - Science topic

• https://inquiryintoinquiry.com/2025/05/31/cactus-language-syntax-1/

Grammar 1 —

Grammar 1 is something of a misnomer. It is nowhere near exemplifying any kind of a standard form and it's put forth only as a starting point for the initiation of more respectable grammars. Such as it is, it uses the terminal alphabet ‡A‡ = ‡M‡ ∪ ‡P‡ coming with the territory of the cactus language ‡C‡(‡P‡), it specifies ‡Q‡ = ⌀, in other words, it employs no intermediate symbols, and it embodies the “covering set” ‡K‡ as listed in the following display.

Cactus Language Grammar 1

• https://inquiryintoinquiry.files.wordpress.com/2025/05/cactus-language-grammar-1.png

The last two rules of Grammar 1 dictate the following typings.

• The concept of a sentence in ‡L‡ covers any concatenation of sentences in ‡L‡, that is, any finite number of freely chosen sentences available to be concatenated one after another.

• The concept of a sentence in ‡L‡ covers any surcatenation of sentences in ‡L‡, that is, any string opening with a “(”, continuing with a sentence, possibly empty, following with a finite number of phrases of the form “,” ∙ S, and closing with a “)”.

The above appears to be just about the most concise description of the cactus language ‡C‡(‡P‡) one can imagine but there are a couple of problems commonly felt to afflict its style of presentation and to make it less than completely acceptable. Briefly stated, the problems turn on the following properties of the formulation.

• The invocation of the kleene star operation is not reduced to a manifestly finitary form.

• The type S indicative of a sentence is allowed to cover not only itself but also the empty string.

We'll discuss those issues at first in general, and especially in regard to how the two features interact with one another, and then we'll return to address in further detail the questions they engender on their individual bases.

Resources —

Cactus Language • Syntax

• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax

Survey of Animated Logical Graphs

• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

Survey of Theme One Program

• https://inquiryintoinquiry.com/2025/05/06/survey-of-theme-one-program-7/

Impact of research in technological advancement in all major fields

Key questions: 1. Does the Lorentz transformation group mathematically permit such negative velocity states? 2. Could this reconcile with the cosmic recession velocities exceeding c in comoving coordinates? 3. What experimental signatures would distinguish this from actual FTL motion?

There is was a recent interesting article in Quanta Magazine about 'paraparticles' (https://www.quantamagazine.org/paraparticles-would-be-a-third-kingdom-of-quantum-particle-20250411) as a new type of particle. The article describes how these particles (strictly theoretic at this point) are different from current particles and derive from not squaring values in current measurements. When a measurement is squared, there are two possible solutions (one positive and one negative). Not squaring provides a means of identifying aspects that 'disappear' when squared.

I have given a couple presentations (and papers) suggesting we need new mathematical tools to expand science (and mathematics). In particular is the need for a new, more powerful numeric system that could provide single values for complex numbers.

Paraparticles research might find such a system very helpful, since it could distinguish between any complex values without needing to square results (to get a non-complex or 'Real' result) and therefore distinguish between measurements of paraparticles.

I note that complex numbers do not apply to 'real' measurements, since we do not know what to do with the 'imaginary' part (hence we square it to get rid of this part or simply drop it, as for some electronic measurements).

This is the first direct potential application I have seen for the need for such a complex numeric system.

I would be interested in other's comments.

I am happy to invite the ResearchGate community to share feedback, thoughts, and critiques on what I believe marks a once-in-history achievement: the definitive and comprehensive disproof of two of mathematics’ most legendary unsolved problems — the Riemann Hypothesis and the Goldbach Conjecture.

These works, now available as preprints, present:

1. Seven independent disproofs for Riemann Hypothesis 2. Ten independent disproofs for Goldbach Conjecture

— each combining analytic, computational, probabilistic, structural, and topological approaches, rewriting the foundations of number theory.

This isn’t just mathematical progress — it is a paradigm-shifting, Nobel Prize–level breakthrough that opens the door to a new era of discovery, not only in mathematics but across every human endeavour, even fields yet unimagined. My research pushes beyond limits, exploring truths that transform our understanding of both the known and the unknown.

👉 Read the papers and share your thoughts, critiques, and insights:

👉 Visit my ResearchGate profile for more groundbreaking works: https://www.researchgate.net/profile/Sandeep-Jaiswal-9?ev=hdr_xprf

Let’s engage in an open, constructive, and pioneering discussion. How do you see these results reshaping number theory, analytic mathematics, or even the future of scientific thought itself? I welcome your perspectives, debates, and reflections!

The validity of various counting systems is a question that comes up frequently. In light of this, I would like to discuss the validity of Balanced Ternary Mathematics.

• https://inquiryintoinquiry.com/2025/03/30/cactus-language-preliminaries-1/

❝Thus, what looks to us like a sphere of scientific knowledge more accurately should be represented as the inside of a highly irregular and spiky object, like a pincushion or porcupine, with very sharp extensions in certain directions, and virtually no knowledge in immediately adjacent areas. If our intellectual gaze could shift slightly, it would alter each quill's direction, and suddenly our entire reality would change.

❝Picture two different configurations of such an irregular shape, superimposed on each other in space, like a double exposure photograph. Of the two images, the only part which coincides is the body. The two different sets of quills stick out into very different regions of space. The objective reality we see from within the first position, seemingly so full and spherical, actually agrees with the shifted reality only in the body of common knowledge. In every direction in which we look at all deeply, the realm of discovered scientific truth could be quite different. Yet in each of those two different situations, we would have thought the world complete, firmly known, and rather round in its penetration of the space of possible knowledge.❞

— Herbert J. Bernstein • “Idols of Modern Science”

The task before us is to describe the syntax of a family of formal languages intended for use as a sentential calculus, and thus interpreted for the purpose of reasoning about propositions and their logical relations.

To carry out our discussion we need a way of referring to signs as if they were objects like any others, in other words, as the sorts of things which can be named, indicated, described, discussed, and renamed if necessary, which can be placed, arranged, and rearranged within a suitable medium of expression — or else manipulated in the mind — which can be articulated and decomposed into their elementary signs, and which can be strung together in sequences to form complex signs.

Signs having signs as their objects are known as “higher order signs”, a topic which demands an adequate level of formalization, but in due time. The present discussion needs a quicker way to get into the subject, even if it settles for informal means which cannot be rendered absolutely precise.

Resources —

Cactus Language • Preliminaries

• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries

Survey of Animated Logical Graphs

• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Theme One Program

• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

Mr. Vaibhav Sunder’s comment, while poetic in tone, appears to misinterpret the core intent of Mr. André Michaud’s clarification. It is important to emphasize that Michaud’s contribution was not merely a linguistic clarification, but a reassertion of Einstein’s original physical and theoretical intent in the 1905 paper. His distinction reveals how modern retellings of Einstein’s postulates have abstracted away from their original scope and meaning—especially by shifting from emission-based descriptions of light to observer-invariant formulations devoid of mechanical grounding.

This has significant consequences for how relativity has been developed and interpreted since. Michaud’s effort encourages a theoretical return to physical principles that Einstein himself aligned with, rather than the more geometric and axiomatic structures that later dominated relativistic interpretation. In this light, Michaud’s work is not about semantics but about theoretical clarity, with direct implications for ongoing frameworks like ECM or synchronized kinematic-electromagnetic mechanics.

Furthermore, Vaibhav's statement that physics cannot be translated into intuitive, everyday language unless embedded in the "rhythms" of mathematics oversimplifies the relationship between formalism and understanding. While mathematics is undeniably the precise language of physics, its purpose is to model, not mystify. To suggest that physics can only be understood or expressed through mathematical rhythm risks reinforcing an elitist view that equates intellectual depth solely with mathematical articulation.

This tendency often leans into intellectual supremacy, which, if left unchecked, can shade into intellectual dishonesty—where genuine clarity and accessibility are sacrificed for technical intimidation. Intuition and language, when grounded in physical understanding, play a crucial role in both communicating and developing physical insight. History shows us that the greatest physicists—Einstein included—valued physical reasoning as much as mathematical formulation.

Let us not forget: equations alone do not yield understanding. It is the synthesis of meaning and mathematics that advances science.

Soumendra Nath Thakur

May 16, 2025

Hi dear mathematicians. I want to know your opinion about my short proof regarding the "Twin Primes Conjecture". I applied even some notions of the Goldbach conjecture. Can you use the results of the approach in my proof?

Here is the link:

The assertion that "Relativity does not have a mathematical explanation for why the speed of light is c" is fundamentally correct. In special relativity, the invariance of the speed of light is not derived from first principles but postulated as a foundational axiom. While Maxwell’s equations predict that electromagnetic waves propagate at a fixed speed c in vacuum, these equations are formulated within particular reference frames and do not inherently explain why this speed should remain invariant across all inertial observers. Special relativity adopts this invariance as its second postulate: that the speed of light in a vacuum is the same for all observers, regardless of their relative motion. As such, the value of c is not mathematically deduced from within relativity—it is assumed.

In standard relativity, photons are treated as massless (rest mass m = 0), yet they carry energy and momentum, implying an effective inertial influence. In Extended Classical Mechanics (ECM), this leads to a re-interpretation: photons and other massless particles can exhibit negative apparent mass (Mᵃᵖᵖ < 0) due to their kinetic energy characteristics. This challenges the conventional notion of masslessness by introducing a dynamical interpretation tied to acceleration and force. Similarly, in cosmological contexts, dark energy—such as that inferred in studies by A. D. Chernin et al. on the Coma Cluster—is interpreted as having negative effective mass, a view consistent with ECM’s framework.

Within ECM, particles exhibiting negative apparent mass—such as photons emitted from gravitationally bound systems—tend toward unbounded propagation speeds. This provides an alternative explanation for the observed superluminal recession of distant galaxies, where the recession is not merely a relativistic artifact of metric expansion, but a real, force-driven phenomenon resulting from gravitational–antigravitational interaction. Specifically, such recession occurs when the negative effective mass component dominates over the matter mass, producing a net repulsive dynamic. Here, "unbounded" refers to the mathematical tendency of speed to diverge as apparent mass becomes increasingly negative or frequency increases without bound.

However, ECM also acknowledges that physical unboundedness is constrained by fundamental quantum limits. The Planck scale introduces the smallest meaningful physical quantities—the Planck length (Lₚ) and Planck time (Tₚ)—which naturally impose an upper bound on velocity. This bound is expressed through the ratio:

This expression does not emerge from relativity itself but from dimensional considerations in quantum gravity. It defines the maximum attainable speed for any propagation process, including those involving particles with Mᵃᵖᵖ < 0. While mathematical models may suggest speeds approaching infinity, the Planck scale sets a physical boundary, beyond which further acceleration or frequency increase ceases to be meaningful or measurable. In this way, ECM preserves causal consistency and enforces a speed limit—not as a postulate of relativity or a consequence of spacetime curvature, but as a boundary arising from the discrete, physical limits imposed by the Planck scale.

• https://inquiryintoinquiry.com/2025/05/07/cactus-language-discussion-1/

Answering a question from a reader …

Re: Cactus Language • Preliminaries 9

• https://inquiryintoinquiry.com/2025/04/25/cactus-language-preliminaries-9/

Re: Cybernetics • Joe Bury

• https://groups.google.com/g/cybcom/c/kqwcuOMDtJo/m/aclbc7llDwAJ

JB:

❝What does subcatenation and surcatenation mean? Their definitions are not found in a dictionary. I get the formulas you wrote but I don't understand the meaning.❞

Thanks for the question, Joe,

The current presentation of Cactus Language is rather abstract and formal because that's what we need for a fully computational parsing algorithm, and there's quite a bit more to do on that score as we go, but I have written more intuitive introductions to the same material various times before — You might try one of the following for starters.

Logical Graphs • First Impressions

• https://inquiryintoinquiry.com/2024/08/26/logical-graphs-first-impressions-a/

Logical Graphs • Formal Development

• https://inquiryintoinquiry.com/2024/09/12/logical-graphs-formal-development-b/

Keeping it short and simple as possible —

Under the Existential Interpretation —

• The syntactic connective of Concatenation is interpreted as the Logical Conjunction, which says all of its operands are true.

• The syntactic connective of Surcatenation is interpreted as the Minimal Negation Operation, which says exactly one of its operands is false.

Under the Entitative Interpretation —

• The syntactic connective of Concatenation is interpreted as the Logical Disjunction, which says some of its operands are true.

• The syntactic connective of Surcatenation is interpreted as the Dual of Minimal Negation, which says not just one of its operands is true.

Resources —

Cactus Language • Preliminaries

• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries

Survey of Animated Logical Graphs

• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

Survey of Theme One Program

• https://inquiryintoinquiry.com/2025/05/06/survey-of-theme-one-program-7/

They really have pushed it to the next level !!...

I am finished with their Enterprise...

This is unheard of and  highly inappropriate  and defies the very purpose of being a  preprint server!!...

Good-luck and Goodnight !...

Submitting Author

Determine the importance of mathematics in various fields. and why the knowledge of mathematics is so crucial in our lives?

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 ?

I am currently seeking a one-year sabbatical opportunity in the fields of Mathematics. With a Ph.D. in Applied Mathematics, extensive teaching, research and academic leadership experience across Nigeria, South Africa, and Saudi Arabia. I am eager to contribute to research, teaching, and collaboration at an academic or research institution.

My expertise spans Lie symmetry analysis of differential equations, actuarial mathematics, curriculum development, and postgraduate supervision. I am passionate about fostering academic excellence and interdisciplinary research.

If you are aware of any suitable opportunities or potential collaborations, I would be pleased to connect and discuss further.

Thank you for your consideration and support! Contact me for more details about my research and experience.

Dr. Amunu Nass

Answer ?

We assume that the Fourier transform is merely an optional mathematical tool that can be dispensed with.

The statistical theory of Cairo techniques predicts the general solution of a time-dependent PDE as follows:

f(x,y,z,t)=D(N).(b+S) + B^N.IC..(1)

where D(N) is the transfer function = B+B^2+B^3 +...+B^N

b is the vector of 1D, 2D, and 3D boundary conditions, arranged in the appropriate order.

IC is the vector of initial conditions.

B is the well-defined transition matrix for the considered closed control volume.

One of the drawbacks of the mathematical Fourier transform is that it predicts the solution for 1D, in the x domain GT -∞ and LT ∞, and is not defined for the domain of the x elements of [a,b]. The Fourier transform itself predicts the wave solution for 1D, over the entire range -∞ <x< ∞ as,

n˜(k, t)=˜n0(k) exp -Dk^2 . t . . . (2)

Equation 2 is equivalent to Equation 1, but only to a subset.

Explain various fields at which integration played an important role.

Hello. For my thesis defense, I need to attend a conference related to physics. Are there any international and hybrid conferences focused solely on physics or both physics and mathematics taking place in April or May?

I hope this message finds you well. My name is Ziad Rabea, and I am a passionate researcher and developer currently in the 12th grade. From an early age, I have been deeply interested in technology and scientific exploration, which has shaped my unconventional educational path and fueled my curiosity in various fields, including artificial intelligence, programming languages, and mathematics.

At the age of 12, I wrote my first scientific paper, diving into topics that intrigued me in both pure and applied math. Two years later, I developed a full-stack social networking platform with over 70,000 lines of code. This platform was designed to facilitate language exchange, fostering fluency through communication with native speakers. This project not only honed my programming skills but also instilled in me a deep appreciation for how technology can bridge cultural and linguistic gaps.

A year later, I faced a medical challenge that required surgery, the result of my intense work schedule. The night before the procedure, I built an AI model to help me continue coding using voice commands during my recovery. and I recently created a novel multilingual and AI powered programming language that aims to break linguistic barriers.

Throughout my journey, driven by curiosity and introversion, I have built hundreds of diverse projects—from desktop applications to 2D games, AI models, and embedded systems. My unconventional path has provided me with extensive hands-on experience and fostered a deep passion for learning and innovation.

Given my non-traditional educational background and practical experience in these fields, I am eager to explore academic opportunities where I can contribute, learn, and grow further. I would be incredibly grateful for any guidance or advice on potential pathways to university or collaborative research opportunities that align with my interests in AI, programming, and mathematics.

Thank you for your time and consideration. I look forward to any insights you may offer.

Sincerely, Ziad Rabea

Based on the theorem "A sequence of bounded functions (f_n) is uniformly convergent on A to f if and only if ∥fn − f∥_A → 0", how can I prove that xⁿ is not convergent uniformly??

Why are editors of modern mathematical journals and publishers applying

crazy, bureaucratic rules in additon to very high article processing charges?

They are behaving quite similarly to the present leaders of the European Union

who strongly support war, migration and gender abnormalities.

Does students actually benefit learning through interactive mathematics delivery?

What are the pitfalls of interactive mathematics

I’m investigating the possibility of formally modeling self-referential holons, as described in Poznanski’s Dynamic Organicity Theory and related functional system frameworks, through mathematical structures capable of capturing recursion, transformation, and multiscale dynamics.

Specifically, I’m interested in whether category theory, particularly its higher-order or enriched variants, can be extended or adapted to support functional pathways that:

In the Oscillatory Dynamics Transductive-Bridging Theorem (ODTBT), such holons are treated as oscillatory units nested in transductive fields. Their dynamics are governed not just by set-theoretic structure or syntax, but by recursive waveform interactions, expressed through sine-cosine phase modulation across nested scales (the TWIST boundary being the central transductive operator).

My core questions:

I'm particularly interested in models that go beyond symbolic logic and can account for functionality as transformation (rather than representation) as the basis of system self-organization.

Any leads on formal approaches, hybrid mathematical structures, or applied category-theoretic models in biology, cognition, or systems neuroscience would be deeply appreciated.

I am starting to research socio-economic dynamics modeling and social and economic network analysis, particularly in the context of poverty eradication and inequality reduction. I'm interested in identifying the key open problems in this field, especially from a mathematical perspective.

I would appreciate insights into open problems at the intersection of mathematics and socio-economic dynamics. Are there fundamental conjectures, unsolved problems, or emerging mathematical techniques that could be promising in addressing these issues?

Thank you very much!

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.

Time has been traditionally seen as a constant, flowing uniformly. However, in Einstein’s theory of relativity, time is linked with space to form spacetime, and it can change depending on speed and gravity. The "arrow of time" suggests that time moves in one direction due to the increase in entropy. Despite these insights, the true nature of time—whether it's a fundamental aspect of reality or something that emerges from deeper physical processes—remains an unresolved question in both physics and philosophy.

The top cell consists of SpiroOmetad(HTL)>Perovskite>TiO2(ETL) and Botton subcell consists of aSip++>cSi>aSin++.Individually when i simulated these subcells they converged and displayed an IV curve. but after introducing the Tunnel Junction aSin++ the simulation does not converge. I have also introduced a nonlocal mesh in maths section and a hbarrier and barrier band to band model in the physics section.I have also been adjusting the mesh refinement but still did not converge.Kindly, guide what i am doing wrong.

Relying too heavily on theoretical models in physics poses several risks:

1. Dogmatic Attachment: Physicists may become overly attached to certain theories, leading to dogmatic thinking and resistance to new ideas. This can stifle scientific progress by diverting resources towards established theories rather than exploring fresh perspectives.

2. Speculation Over Experimentation: Theoretical models can lead to speculative ideas that are not grounded in empirical evidence. This can result in "mindless production of mathematical fiction," as seen in theories like string theory and supersymmetry, which have failed to yield experimental confirmation.

3. Aesthetic Bias: Theoretical physics often relies on aesthetic criteria such as simplicity and elegance, which can be misleading. These criteria may lead to theories that are mathematically appealing but lack empirical support.

I am writing this note to call out to researchers in combinatorics and probability theory. I invite researchers to think about the problem I will describe below.

The MergeSort algorithm recursively divides the input array into sub-arrays until an element is left in each sub-array. This division is an unconditional division, unlike the Quicksort algorithm. Then, MergeSort recursively merges the two sorted sub-arrays into a sorted array. In all cases, the algorithm's running time is Ө(nlgn), meaning that even if the input array is sorted, MergeSort needs Ө(nlgn) running time to sort it.

Recently, we suggested that the division phase of the MergeSort algorithm can be carried out a little more intelligently rather than unconditionally. We tried 3 ways to do this.

Way 1: We scan the array from beginning to end, divide it into increasing subarrays of consecutive terms, and then merge these subarrays into an increasing array using the tournament method.

Way 2: We scan the array from beginning to end and divide it into increasing or decreasing subarrays of consecutive terms and then merge these subarrays into an increasing array using the tournament method.

Way 3: Some arrays are combinations of 2 increasing and decreasing subarrays (the terms of subarrays need not be consecutive terms of the input array), but of course, the vast majority of arrays do not meet this condition. We divide the input array into two increasing and decreasing subarrays up to the possible term. Then, we merge these two subarrays into an increasing subarray. Then, we repeat these two operations starting from the term we left until we reach the end of the input array. Finally, we merge the sorted subarrays we found into an increasing array using the tournament method.

In all three ways, the algorithm we propose sorts at Ө(n) in the best case and at Ө(nlgn) in the worst case.

For more details, you can access our study from the link below:

Research question: What is the average processing time of our proposed algorithm for each of the 3 paths separately? We can experimentally observe that it is Ө(ngn), but we cannot prove it mathematically.

For this, it is necessary to calculate the expected value of the number of subarrays obtained after division under the assumption of a distribution.

I find myself unable to grasp the majority of what I come across in contemporary mathematics literature and this poses the question of how one moves from the level "just graduated" to being able to consider themselves ready to engage both in research and with the community at large. I am looking for opinions and personal experiences.

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.

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.

The Kenyan government wants to make mathematics options in senior schools. I wrote an article on why mathematics must be compulsory in senior schools. Use the link: https://nation.africa/kenya/blogs-opinion/blogs/mathematics-needs-to-be-a-compulsory-subject-4967282

Do you agree with me?? Discuss.

In an n-dimensional Euclidean space, what is the minimum number of rotations required to transform a given orthogonal coordinate system into an arbitrary orthogonal coordinate system? How can this be expressed mathematically, particularly using the language of linear algebra?

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.

How do chatbots mediate between mathematics and physics?

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/

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.”

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.

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?

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 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?