Science topics: Mathematics
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Mathematics - Science topic
Mathematics, Pure and Applied Math
Questions related to Mathematics
Differential Logic • Overview
A reader once told me “venn diagrams are obsolete” and of course we all know how unwieldy they become as our universes of discourse expand beyond four or five dimensions. Indeed, one of the first lessons I learned when I set about implementing Peirce's graphs and Spencer Brown's forms on the computer is that 2‑dimensional representations of logic quickly become death traps on numerous conceptual and computational counts.
Still, venn diagrams do us good service at the outset in visualizing the relationships among extensional, functional, and intensional aspects of logic. A facility with those connections is critical to the computational applications and statistical generalizations of propositional logic commonly used in mathematical and empirical practice. All things considered, then, it is useful to make as visible as possible the links between variant styles of imagery in logical representation — and that is what I hoped to do in the sketch of Differential Logic outlined below.
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.
Introduction
Cactus Language for Propositional Logic
Differential Expansions of Propositions
Propositional Forms on Two Variables
Transforms Expanded over Ordinary and Differential Variables
Field Picture
Differential Fields
Propositions and Tacit Extensions
Enlargement and Difference Maps
Tangent and Remainder Maps
Resources —
Logic Syllabus
Survey of Differential Logic
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
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
Survey of Animated Logical Graphs
Survey of Theme One Program
Cactus Language • Syntax 9
Grammar 4 —
If one imposes the distinction between empty and significant types on each non‑terminal symbol in Grammar 2 then the symbols “S” and “T” give rise to the expanded set of non‑terminal symbols “S”, “S′ ”, “T”, “T′ ”, leaving the last three to form a new intermediate alphabet.
Grammar 4 has the intermediate alphabet ‡Q‡ = {“S′ ”, “T”, “T′ ”} with the set ‡K‡ of covering rules listed in the next display.
Cactus Language Grammar 4
Grammar 4 partitions the intermediate type T as T = η + T′ in parallel fashion with the division of its overlying type as S = η + S′. That is an option we will close off for now but leave open to consider at a later point. It suffices to give a brief discussion of the considerations involved in choosing between grammars at this point, and then move on to the next alternative.
Resources —
Cactus Language • Syntax
Survey of Animated Logical Graphs
Survey of Theme One Program
Impact of research in technological advancement in all major fields
"Can negative velocity states in non-inertial reference frames yield observational equivalence to superluminal motion through relativistic Doppler shift?"
I'm investigating whether the mathematical framework of special relativity permits a loophole through negative velocity states. Specifically:
Consider an observer O and object A in relative motion. While |v| < c must hold universally, can we construct a reference frame transformation where v < -c becomes mathematically valid? If we analyze the relativistic Doppler shift formula:
f' = f √[(1 - β)/(1 + β)] where β = v/c
For negative velocities approaching -c, the observed frequency shift could theoretically mimic superluminal recession. This wouldn't violate causality if the "faster-than-light" motion exists only as an observational artifact in specific reference frames.
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?
I'm particularly interested in whether this approach offers new insights into the interpretation of relativistic constraints. Has anyone explored similar frameworks or can point to theoretical obstacles I may be overlooking?
#SpecialRelativity #LorentzTransformation #DopplerEffect #TheoreticalPhysics #ReferenceFrames
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:
- Riemann Hypothesis: DOI: 10.13140/RG.2.2.12073.71528
- Goldbach Conjecture: DOI: 10.13140/RG.2.2.35561.81764
- Supersolid Light Research: DOI: 10.13140/RG.2.2.14590.29763
👉 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.
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
Survey of Animated Logical Graphs
Survey of Theme One Program
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:
May 09, 2025
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:
c = Lₚ / Tₚ
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.
Regards,
Soumendra Nath Thakur
Cactus Language • Discussion 1
Answering a question from a reader …
Re: Cactus Language • Preliminaries 9
Re: Cybernetics • Joe Bury
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
Logical Graphs • Formal Development
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
Survey of Animated Logical Graphs
Survey of Theme One Program
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 ?
As shown the Design curve and the Fragility curve are always complement of each other. For a given system with an upper bound of capacity ( km= area * maximum stress) the Survive function (SR) times km is capacity, and the fragility function (FR) times km is fragility, where SR+FR=1. These relations apply at any state of the system before failure.
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
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?
Dear Professors,
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 (fn) 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:
- Exhibit non-hierarchical recursion
- Encode intentionality or negentropic action
- Support diachronic causality (i.e., where causation is asymmetric and nonlocal across time)
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:
- Has category theory been extended to represent functional organization or processual identity rather than static structure?
- Are there frameworks, perhaps in topos theory, process categories, or sheaf theory, that can accommodate self-referential transformations with embedded boundary conditions?
- Could morphisms be reinterpreted as oscillatory transductions, capturing phase-shifted transitions between informational holons?
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.
Could you suggest scientific methodologies or models that can effectively integrate these factors to calculate the monthly safety stock?
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?
- dP/dt=f1+f2+f3+f4+f5
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?
Signs Of Signs • 1
Re: Michael Harris • 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
Survey of Pragmatic Semiotic Information
Survey of Semiotics, Semiosis, Sign Relations
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
Physics and the Principle of Synergy
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:
- Mathematics Anxiety – How do fear and stress impact learning?
- Lack of Individualized Support – Are students receiving the guidance they need?
- Peer Pressure & Learning Culture – Is excelling in mathematics socially discouraged?
- Overemphasis on Memorization – Are students understanding concepts or simply learning formulas for exams?
- The Role of Technology – Could AI, digital tools, and interactive platforms bridge the learning gap?
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.
Does Every Mathematical Framework Correspond to a Physical Reality? The Limits of Mathematical Pluralism in Physics
Introduction
Physics has long been intertwined with mathematics as its primary tool for modeling nature. However, a fundamental question arises:
- Does every possible mathematical framework correspond to a physical reality, or is our universe governed by only a limited set of mathematical structures?
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?
Important Questions for Discussion
v Mathematical Pluralism vs. Physical Reality:
- Are all mathematically consistent systems realizable in some physical sense, or is there a deeper reason why certain mathematical structures dominate physical theories?
- Could there exist universes governed by entirely different mathematical rules that we cannot even conceive of within our current formalism?
v Physics as a Computationally Limited System:
- Is our universe constrained by a specific subset of mathematical frameworks due to inherent physical principles, or is this a reflection of our cognitive limitations in developing theories?
- Why do our fundamental laws of physics rely so heavily on certain mathematical structures while neglecting others?
v The Relationship Between Mathematics and Nature:
- Is mathematics an inherent property of nature, or is it merely a tool that we impose on the physical world?
- If every mathematical structure has an equivalent physical reality, should we expect an infinite multiverse where every possible mathematical law is realized somewhere?
v Beyond Mathematical Formalism:
- Could there be fundamental aspects of physics that are not fully describable within any mathematical framework?
- Does the reliance on mathematical models lead us to mistakenly attribute physical existence to purely abstract mathematical entities?
Philosophical Implications
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.
Introduction: Conceptual Remnants and the Challenge of Physical Objectivity
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:
- Is physics still unknowingly entangled in issues arising from emergent effects?
- Could these emergent effects create a gap between physical reality and the virtual constructs generated through mathematical modeling?
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?
Mathematical Models and Conceptual Remnants: Are We Chasing a Mirage?
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.
- Has theoretical physics become a vast computational engine, focusing on adjusting relationships between mathematical variables rather than seeking an independent physical reality?
- Could it be that many of the concepts emerging from our models are mere reflections of mathematical structures rather than objective entities in nature?
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.
Fundamental Critique: Should We Even Be Searching for Physical Objectivity?
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.
- Perhaps physics should not aim to discover a reality independent of our models since every model is ultimately a mathematical structure shaped by human perception.
- If the goal of physics is not to describe "absolute truth" but rather to create predictive models, should we then accept that we will never fully grasp "what actually exists"?
Finally: Between Computational Accuracy and Physical Reality
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?
- Should physicists strive to distinguish between mathematical models and physical objectivity, or is such a distinction inherently meaningless?
- Is the search for an independent physical reality a conceptual mistake, as Heisenberg and others have suggested?
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
Absolute Collapse Condition
Mass Acquisition at Planck Frequency:
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.
Gravitational Collapse:
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.
ECM's Mass-Induction Perspective
Apparent Mass and Effective Mass:
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:
|Mᵉᶠᶠ| > |−Mᵃᵖᵖ|
This irreversible transition confirms that any entity at fp must collapse due to self-gravitation.
Implications for Massless-to-Massive Transition
Behaviour Below Planck Frequency:
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:
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.
Threshold Dominance at the Planck Scale
Gravitational Mass Dominance:
At the Planck scale, gravitational mass (Mɢ) is immense due to the fundamental gravitational interaction. Since |+Mɢ| ≫|−Mᵃᵖᵖ|, the net effective mass remains positive:
Mᵉᶠᶠ = Mɢ = (−Mᵃᵖᵖ) ≈ +Mᵉᶠᶠ
This suggests that at Planck conditions, the gravitationally induced mass dominates over any negative mass contributions.
Transition Scenarios for Negative Effective Mass
Conditions for Negative Effective Mass:
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.
Linking Effective Mass to Matter Mass at Planck Scale
Matter Mass Emergence:
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.
Conclusion
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?
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.
# 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.
“Reaching to the destination without shifting buses”
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
Need CBSE India 10th board examination data on the mathematics subject.
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!
I want a qualitative scale that contains questions that reveal the mathematics teacher's perception of beauty and simplicity in mathematics?
Can the physical reality be represented mathematically?
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:
QUANTUM PHYSICS
https://www.researchgate.net/publication/384190006_Quantum_Physics (https://www.researchgate.net/publication/384190006_Quantum_Physics)
A QUANTUM THEORY OF GRAVITATION
https://www.researchgate.net/publication/385592651_A_Quantum_Theory_of_GravitationNewpdf (https://www.researchgate.net/publication/385592651_A_Quantum_Theory_of_GravitationNewpdf)
SPECIAL RELATIVITY WITH ANOTHER APPROACH
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
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.
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.
Historically, the foundation has also had a partnership with ResearchGate:
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?
Differential Propositional Calculus • Overview
❝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 —
Casual Introduction
Cactus Calculus
Part 2 —
Formal_Development
Elementary Notions
Special Classes of Propositions
Differential Extensions
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2#Differential_Extensions
Appendices —
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?
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.
Differential Logic • 1
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
Survey of Differential Logic
Differential Logic • 18
Tangent and Remainder Maps —
If we follow the classical line which singles out linear functions as ideals of simplicity then we may complete the analytic series of the proposition f = pq : X → B in the following way.
The next venn diagram shows the differential proposition df = d(pq) : EX → B we get by extracting the linear approximation to the difference map Df = D(pq) : EX → B at each cell or point of the universe X. What results is the logical analogue of what would ordinarily be called “the differential” of pq but since the adjective “differential” is being attached to just about everything in sight the alternative name “tangent map” is commonly used for df whenever it's necessary to single it out.
Tangent Map d(pq) : EX → B
To be clear about what's being indicated here, it's a visual way of summarizing the following data.
d(pq)
= p ∙ q ∙ (dp , dq)
+ p ∙ (q) ∙ dq
+ (p) ∙ q ∙ dp
+ (p) ∙ (q) ∙ 0
To understand the extended interpretations, that is, the conjunctions of basic and differential features which are being indicated here, it may help to note the following equivalences.
• (dp , dq) = dp ∙ (dq) + (dp) ∙ dq
• dp = dp ∙ dq + dp ∙ (dq)
• dq = dp ∙ dq + (dp) ∙ dq
Capping the analysis of the proposition pq in terms of succeeding orders of linear propositions, the final venn diagram of the series shows the “remainder map” r(pq) : EX → B, which happens to be linear in pairs of variables.
Remainder r(pq) : EX → B
Reading the arrows off the map produces the following data.
r(pq)
= p ∙ q ∙ dp ∙ dq
+ p ∙ (q) ∙ dp ∙ dq
+ (p) ∙ q ∙ dp ∙ dq
+ (p) ∙ (q) ∙ dp ∙ dq
In short, r(pq) is a constant field, having the value dp ∙ dq at each cell.
Resources —
Logic Syllabus
Survey of Differential Logic
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
I was briefly reviewing your proof and noticed something unusual in this part.
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).
- What is the theory behind this?
- Any similar approach like this in other scientific field or any simple analogous explanation?
- You could direct me to resources such as videos, or literature.
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!
A minion is a low-level official protecting a bureaucracy form challengers.
A Kuhnian minion (after Thomas Kuhn's Structure of Scientific Revolutions) is a low-power scientist who dismisses any challenge to existing paradigm.
A paradigm is a truth structure that partitions scientific statement as true to the paradigm or false.
Recently, I posted a question on Physics Stack Exchange that serves as a summary of the elastic string paradigm. My question was: “Is it possible there can be a non-Fourier model of string vibration? Is there an exact solution?”
To explain, I asked if they knew the Hamiltonian equation for the string vibration. They did not agree it must exist. I pointed out there are problems with the elastic model of vibration with its two degrees of freedom and unsolvable equations of motion can only be approximated by numerical methods. I said elasticity makes superposition the 4th Newtonian law. How can a string vibrate in an infinite number of modes without violating energy conservation?
Here are some comments I got in response:
“What does string is not Fourier mean? – Qmechanic
“ ‘String modes cannot superimpose!’ Yet, empirically, they do.” – John Doty
“ A string has an infinite number of degrees of freedom, since it can be modeled as a continuous medium. If you manage to force only the first harmonic, the dynamics of the system only involve the first harmonic and it’s a standing wave: this solution does depend on time, being (time dependence in the amplitude of the sine). No 4th Newton’s law. I didn’t get the question about Hamilton equation.
“What do you mean with ‘archaic model’? Can I ask you what’s your background that makes you do this sentence? Physics, Math, Engineering? You postulate nothing here. You have continuum mechanics here. You have PDEs under the assumption of continuum only. You have exact solutions in simple problems, you have numerical methods approximating and solving exact equations. And trust me: this is how the branch of physics used in many engineering fields, from mechanical, to civil, to aerospace engineering.” – basics
I want to show the rigid versus elastic dichotomy goes back to the calculus wars. Quoting here from Euler and Modern Science, published by the Mathematical Association of America:
"We now turn to the most famous disagreement between Euler and d’Alembert … over the particular problem of the theory of elasticity concerning a string whose transverse vibrations are expressed through second-order partial differential equations of a hyperbolic type later called the wave equation. The problem had long been of interest to mathematicians. The first approach worthy of note was proposed by B. Taylor, … A decisive step forward was made by d’Alembert in … the differential equation for the vibrations, its general solution in the form of two “arbitrary functions” arrived at by means original with d’Alembert, and a method of determining these functions from any prescribed initial and boundary conditions.”
[Editorial Note: The boundary conditions were taken to be the string endpoints. The use of the word hyperbolic is, I believe, a clear reference to Taylor’s string. A string with constant curvature can only have one mathematic form, which is the cycloid, which is defined by the hyperbolic cosh x function. The cosh x function is the only class of solutions that are allowed if the string cannot elongate. The Taylor/Euler-d’Alembert dispute whether the string is trigonometric or hyperbolic.
Continuing the quote from Euler and Modern Science:
"The most crucial issue dividing d’Alembert and Euler in connection with the vibrating string problem was the compass of the class of functions admissible as solutions of the wave equation, and the boundary problems of mathematical physics generally, D’Alembert regarded it as essential that the admissible initial conditions obey stringent restrictions or, more explicitly, that the functions giving the initial shape and speed of the string should over the whole length of the string be representable by a single analytical expression … and furthermore be twice continuously differentiable (in our terminology). He considered the method invalid otherwise.
"However, Euler was of a different opinion … maintaining that for the purposes of physics it is essential to relax these restrictions: the class of admissible functions or, equivalently, curves should include any curve that one might imagine traced out by a “free motion of the hand”…Although in such cases the analytic method is inapplicable, Euler proposed a geometric construction for obtain the shape of the string at any instant. …
Bernoulli proposed finding a solution by the method of superimposition of simple trigonometric functions, i.e. using trigonometric series, or, as we would now say, Fourier series. Although Daniel Bernoulli’s idea was extremely fruitful—in other hands--, he proved unable to develop it further.
Another example is Euler's manifold of the musical key and pitch values as a torus. To be fair, Euler did not assert the torus but only drew a network show the Key and Pitch can move independently. This was before Mobius's classification theorem.
My point is it should be clear the musical key and pitch do not have different centers of harmonic motion. But in my experience, the minions will not allow Euler to be challenged by someone like me. Never mind Euler's theory of music was crackpot!
