Mathematics - Science topic

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

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

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

Please leave your comments.

Thanks

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

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

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

So what?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Imagine an enormous cylinder in a flat landscape. You are standing along the inner edge. How big would the cylinder need to be for you to not see the curvature? I.e., Instead think you are standing along a completely flat wall. Consider an average person with average eyesight. Would happily accept both the motivation, answer and calculation.

Bonus question: If you had any particular practical tools to your disposal to improve your estimate of the curvature in this scenario, what would they be and how would they help?

1)Maybe I'm slightly less intuitive. I

consider myself kind of a skeptical empiricist/critical rationalist.

2)I don't believe concepts are eternal because they need to be adjusted to avoid contradictions.

3)Without some transcendence beyond materialism, we would NOT be able to reason.

4)Maybe reason is the ONLY absolute CONCEPT. And reason derives from God.

5)Concepts also aid execution thus, maybe I'm a more skeptical Aristotelian.

Sources:

1)

2)

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

I believe that it is common knowledge that mathematics and its applications cannot directly prove Causality. What are the bases of the problem of incompatibility of physical causality with mathematics and its applications in the sciences and in philosophy?

The main but very general explanation could be that mathematics and mathematical explanations are not directly about the world, but are applicable to the world to a great extent.

Hence, mathematical explanations can at the most only show the general ways of movement of the processes and not demonstrate whether the ways of the cosmos are by causation, what the internal constitution of every part of it is, etc. Even when some very minute physical process is mathematized, the results are general, and not specific of the details of the internal constitution of that process.

No science and philosophy can start without admitting that the cosmos exists. If it exists, it is not nothing, not vacuum. Non-vacuous existence means that the existents are non-vacuously extended. This means that they have parts. Every part has parts too, ad libitum, because each part is extended and non-infinitesimal. Hence, each part is relatively discrete, not mathematically discrete.

None of the parts of any physical existent is an infinitesimal. They can be near-infinitesimal. This character of existents is Extension, a Category directly implied by the To Be of Reality-in-total.

Similarly, any extended being’s parts -- however near-infinitesimal -- are active, moving. This implies that every part has so (finite) impact on some others, not on infinite others. This character of existents is Change.

No other implication of To Be is so primary as these two (Extension-Change) and directly derivable from To Be. Hence, they are exhaustive of To Be.

Existence in Extension-Change is what we call Causality. If anything is existent, it is causal – hence Universal Causality is the trans-scientific and physical-ontological Law of all existents.

By the very concept of finite Extension-Change-wise existence, it becomes clear that no finite space-time is absolutely dense with existents. Hence, existents cannot be mathematically continuous. Since there is continuous (but finite and not discrete) change (transfer of impact), no existent can be mathematically absolutely continuous or discrete in its parts or in connection with others.

Can logic show the necessity of all existents as being causal? We have already discussed how, ontologically, the very concept of To Be implies Extension-Change and thus also Universal Causality.

WHAT ABOUT THE ABILITY OR NOT OF LOGIC TO CONCLUDE TO UNIVERSAL CAUSALITY?

In my argument above and elsewhere showing Extension-Change as the very exhaustive meaning of To Be, I have used mostly only the first principles of ordinary logic, namely, Identity, Non-contradiction, and Excluded Middle, and then argued that Extension-Change-wise existence is nothing but Universal Causality, if everything existing is non-vacuous in existence.

For example, does everything exist or not? If yes, let us call it non-vacuous existence. Hence, Extension as the first major implication of To Be. Non-vacuous means extended, because if not extended, the existent is vacuous. If extended, everything has parts.

The point of addition now has been Change, which makes the description physical. It is, so to say, from experience. Thereafter I move to the meaning of Change basically as motion or impact.

Naturally, everything in Extension must effect impacts. Everything has further parts. Hence, by implication from Change, everything causes changes by impacts. Thus, we conclude that Extension-Change-wise existence is Universal Causality. It is thus natural to claim that this is a pre-scientific Law of Existence.

In such foundational questions like To Be and its implications, we need to use the first principles of logic, because these are the foundational notions of all science and no other derivative logical procedure comes in as handy. In short, logic with its fundamental principles can help derive Universal Causality. Thus, Causality is more primary to experience than the primitive notions of mathematics.

Extension-Change, Universal Causality derived by their amalgamation, are the most fundamental Metaphysical, Physical-ontological, Categories. Since these are the direction exhaustive implications of To Be, all philosophy and science are based on these.

Bibliography

(1) Gravitational Coalescence Paradox and Cosmogenetic Causality in Quantum Astrophysical Cosmology, 647 pp., Berlin, 2018.

(2) Physics without Metaphysics? Categories of Second Generation Scientific Ontology, 386 pp., Frankfurt, 2015.

(3) Causal Ubiquity in Quantum Physics: A Superluminal and Local-Causal Physical Ontology, 361 pp., Frankfurt, 2014.

(4) Essential Cosmology and Philosophy for All: Gravitational Coalescence Cosmology, 92 pp., KDP Amazon, 2022, 2nd Edition.

(5) Essenzielle Kosmologie und Philosophie für alle: Gravitational-Koaleszenz-Kosmologie, 104 pp., KDP Amazon, 2022, 1st Edition.

Why are numbers and shapes so exact? ‘One’, ‘two’, ‘point’, ‘line’, etc. are all exact. But irrational numbers are not so. The operations on these notions are also intended to be exact. If notions like ‘one’, ‘two’, ‘point’, ‘line’, etc. are defined to be so exact, then it is not by virtue of the exactness of these substantive notions, but instead, due to their being defined so, that they are exact, and mathematics is exact.

But on the other side, due to their being adjectival: ‘being a unity’, ‘being two unities’, ‘being a non-extended shape’, etc., their application-objects are all processes that can obtain these adjectives only in groups. These are pure adjectives, not properties which are composed of many adjectives.

A quality cannot be exact, but may be defined to be exact. It is in terms of the exactness attributed to these notions by definition that the adjectives ‘one’, ‘two’, ‘point’, ‘line’, etc. are exact. This is why the impossibility of fixing these (and other) substantive notions as exact misses our attention.

If in fact these quantitative qualities are inexact due to their pertaining to groups of processual things, then there is justification for the inexactness of irrational numbers, transcendental numbers, etc. too. If numbers and shapes are in fact inexact, then not only irrational and other inexact numbers but all mathematical structures should remain inexact except for their having been defined as exact.

Thus, mathematical structures, in all their detail, are a species of qualities, namely, quantitative qualities. Mathematics is exact only because its fundamental bricks are defined to be so. Hence, mathematics is an as-if exact science, as-if real science. Caution is advised while using it in the sciences as if mathematics were absolutely applicable, as if it were exact.

Bibliography

(1) Gravitational Coalescence Paradox and Cosmogenetic Causality in Quantum Astrophysical Cosmology, 647 pp., Berlin, 2018.

(2) Physics without Metaphysics? Categories of Second Generation Scientific Ontology, 386 pp., Frankfurt, 2015.

(3) Causal Ubiquity in Quantum Physics: A Superluminal and Local-Causal Physical Ontology, 361 pp., Frankfurt, 2014.

(4) Essential Cosmology and Philosophy for All: Gravitational Coalescence Cosmology, 92 pp., KDP Amazon, 2022, 2nd Edition.

(5) Essenzielle Kosmologie und Philosophie für alle: Gravitational-Koaleszenz-Kosmologie, 104 pp., KDP Amazon, 2022, 1st Edition.

Mathematical Generalities: ‘Number’ may be termed as a general term, but real numbers, a sub-set of numbers, is sub-general. Clearly, it is a quality: “having one member, having two members, etc.”; and here one, two, etc., when taken as nominatives, lose their significance, and are based primarily only on the adjectival use. Hence the justification for the adjectival (qualitative) primacy of numbers as universals. While defining one kind of ‘general’ another sort of ‘general’ may naturally be involved in the definition, insofar as they pertain to an existent process and not when otherwise.

Why are numbers and shapes so exact? ‘One’, ‘two’, ‘point’, ‘line’, etc. are all exact. The operations on these notions are also intended to be exact. But irrational numbers are not so exact in measurement. If notions like ‘one’, ‘two’, ‘point’, ‘line’, etc. are defined to be so exact, then it is not by virtue of the exactness of these substantive notions, but instead, due to their being defined as exact. Their adjectival natures: ‘being a unity’, ‘being two unities’, ‘being a non-extended shape’, etc., are not so exact.

A quality cannot be exact, but may be defined to be exact. It is in terms of the exactness attributed to these notions by definition that the adjectives ‘one’, ‘two’, ‘point’, ‘line’, etc. are exact. This is why the impossibility of fixing these (and other) substantive notions as exact miss our attention. If in fact these are inexact, then there is justification for the inexactness of irrational, transcendental, and other numbers too.

If numbers and shapes are in fact inexact, then not only irrational numbers, transcendental numbers, etc., but all exact numbers and the mathematical structures should remain inexact if they have not been defined as exact. And if behind the exact definitions of exact numbers there are no exact universals, i.e., quantitative qualities? If the formation of numbers is by reference to experience (i.e., not from the absolute vacuum of non-experience), their formation is with respect to the quantitatively qualitative and thus inexact ontological universals of oneness, two-ness, point, line, etc.

Thus, mathematical structures, in all their detail, are a species of qualities, namely, quantitative qualities, defined to be exact and not naturally exact. Quantitative qualities are ontological universals, with their own connotative and denotative versions.

Natural numbers, therefore, are the origin of primitive mathematical experience, although complex numbers may be more general than all others in a purely mathematical manner of definition.

Bibliography

(1) Gravitational Coalescence Paradox and Cosmogenetic Causality in Quantum Astrophysical Cosmology, 647 pp., Berlin, 2018.

(2) Physics without Metaphysics? Categories of Second Generation Scientific Ontology, 386 pp., Frankfurt, 2015.

(3) Causal Ubiquity in Quantum Physics: A Superluminal and Local-Causal Physical Ontology, 361 pp., Frankfurt, 2014.

(4) Essential Cosmology and Philosophy for All: Gravitational Coalescence Cosmology, 92 pp., KDP Amazon, 2022, 2nd Edition.

(5) Essenzielle Kosmologie und Philosophie für alle: Gravitational-Koaleszenz-Kosmologie, 104 pp., KDP Amazon, 2022, 1st Edition.

Every common source avoids the derivation, saying it to be too difficult. Where can i fiend volterra's original derivation? Knowing which mathematics would be necessary?

Hello ResearchGate Community,

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

Skills:

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

Interests:

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

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

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

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

Thank you for your consideration.

Warmly,

Shard

Assistant Professor | Research Associate |

PhD Candidate in Mathematics

Dear colleagues

I am pleased to inform you that the International Conference on Nonlinear Analysis and Applications (ICNAA 2024) & Symposium on Ancient Indian Mathematics (in the memory of Late Professor S. L. Singh) is being organized by the Department of Mathematics, Pt. L. M. S. Campus, Sridev Suman Uttarakhand University, Rishikesh-249201, Uttarakhand, India from May 10 to May 12, 2024 (https://icnaaa2024.wordpress.com).

Your active participation in the conference will undoubtedly contribute to the success of the event.Kindly submit the abstract of your talk at your earliest convenience.

Link to Registration:https://forms.gle/8LEY9KECHHKqTrPq8

Looking forward to your valuable contribution.

Best Regards

Anita Tomar

Hello

I'm learning camsol. I studied the mathematical particle tracking method used for modeling in turbomolecular pumps, and I can model a single-stage rotor, but I can not model a single-stage rotor and stator.

Can you guide me, please?

thanks

maryam

Hello everyone,

I am currently undertaking a research project that aims to assess the effectiveness of an intervention program. However, I am encountering difficulties in locating suitable resources for my study.

Specifically, I am in search of papers and tutorials on multivariate multigroup latent change modelling. My research involves evaluating the impact of the intervention program in the absence of a control group, while also investigating the influence of pre-test scores on subsequent changes. Additionally, I am keen to explore how the scores differ across various demographic groups, such as age, gender, and knowledge level (all measured as categorical variables).

Although I have come across several resources on univariate/bivariate latent change modelling with more than three time points, I have been unable to find papers that specifically address my requirements—namely, studies focusing on two time points, multiple latent variables (n >= 3), and multiple indicators for each latent variable (n >= 2).

I would greatly appreciate your assistance and guidance in recommending any relevant papers, tutorials, or alternative resources that pertain to my research objectives.

Best,

V. P.

Greetings and courtesy to the professors and students of mathematics. I wanted to know if there is a relationship between the curves and the orthogonal paths of the differential equation with the characteristics of its solution? If the answer is yes, please state the type of relation and relational formula. Thanks

I will check the complete information if some one knows. Then mention me. thanks

Why is it necessary to study the History of Mathematics?

I am looking to estimate the diameter (nm) of a variety of double stranded plasmids (pUC19, pMAL pIII, pKLAC2, etc.) when they are natively supercoiled and when they are relaxed.

If someone could point me towards a formula it would be much appreciated! Thanks. 

Category theory, with its focus on abstraction and relationships between mathematical structures, presents a promising framework for formally expressing mythologisation. This mathematical branch could potentially model the complex, symbolic narratives of myths, translating them into a system of objects and morphisms that reflect the underlying patterns and connections inherent in mythological themes. A key question arises: Can category theory effectively capture the depth and nuance of mythological narratives, preserving their rich symbolic content while providing a formal representation? Additionally, how might such a formalisation impact our understanding of myths and their role in conveying universal truths and cultural values?

- Interested in working on ethnomathematics and indigenous learning systems in mathematics.

The fine-structure constant is obtained when the classical Bohr atomic model is relativisticized [1][2]. α=e²/ℏc, a number whose value lies very close to 1/137. α did not correspond to any elementary physical unit, since α is dimensionless. It may also be variable [6][7]^*.

Sommerfeld introduced this number as the relation of the “relativistic boundary moment” p₀=e²/c of the electron in the hydrogen atom to the first of n “quantum moments” p_n=nh/2π. Sommerfeld had argued that α=p₀/p₁ would “play an important role in all succeeding formulas,” he had argued ^‡ [5].

There are several usual interpretations of the significance of fine structure constants [3].

a）In 1916, Sommerfeld had gone no further than to suggest that more fundamental physical questions might be tied to this “relational quantity.” In Atomic Structure and Spectral Lines, α was given a somewhat clearer interpretation as the relation of the orbital speed of an electron “in the first Bohr orbit” of the hydrogen atom, to the speed of light [5].

b) α plays an important role in the details of atomic emission, giving the spectrum a "fine structure".

c) The electrodynamic interaction was thought to be a process in which light quanta were exchanged between electrically charged particles, where the fine-structure constant was recognized as a measure of the force of this interaction. [5]

d) α is a combination of the elementary charge e, Planck's constant h, and the speed of light c. These constants represent electromagnetic interaction, quantum mechanics, and relativity, respectively. So does that mean that if G is ignored (or canceled out) it represents the complete physical phenomenon.

Questions implicated here :

1) What does the dimensionless nature of α imply? The absence of dimension means that there is no conversion relation. Since it is a coupling relation between photons and electrons, is it a characterization of the consistency between photons and charges?

2) The various interpretations of α are not in conflict with each other, therefore should they be unified?

3) Is our current interpretation of α the ultimate? Is it sufficient?

4) Is α the most fundamental physical constant^**? This is similar to Planck Scales^† in that they are combinations of other fundamental physical constants.

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

* Spatial Variation and time variability.

‡ Sommerfeld considered α "important constants of nature, characteristic of the constitution of all the elements."[4]

† https://www.researchgate.net/post/NO30The_Relation_Between_Mathematics_and_Physics_6-Are_Planck_Scales_Constants_Parameters_or_Principles

** https://www.researchgate.net/post/No28The_Relation_Between_Mathematics_and_Physics_4-Could_All_Physical_Constants_be_Unified_to_a_set_of_Fundamental_Constants_of_Nature

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

[3] 张天蓉. (2022). 精细结构常数. https://blog.sciencenet.cn/blog-677221-1346617.html

[1] Sommerfeld, A. (1916). The fine structure of Hydrogen and Hydrogen-like lines: Presented at the meeting on 8 January 1916. The European Physical Journal H （2014）, 39(2), 179-204.

[2] Sommerfeld, A. (1916). Zur quantentheorie der spektrallinien. Annalen der Physik, 356(17), 1-94.

[4] Heilbron, J. L. (1967). The Kossel-Sommerfeld theory and the ring atom. Isis, 58(4), 450-485.

[5] Eckert, M., & Märker, K. (2004). Arnold Sommerfeld. Wissenschaftlicher Briefwechsel, 2, 1919-1951.

[6] Wilczynska, M. R., Webb, J. K., Bainbridge, M., Barrow, J. D., Bosman, S. E. I., Carswell, R. F., Dąbrowski, M. P., Dumont, V., Lee, C.-C., Leite, A. C., Leszczyńska, K., Liske, J., Marosek, K., Martins, C. J. A. P., Milaković, D., Molaro, P., & Pasquini, L. (2020). Four direct measurements of the fine-structure constant 13 billion years ago. Science Advances, 6(17), eaay9672. https://doi.org/doi:10.1126/sciadv.aay9672

[7] Webb, J. K., King, J. A., Murphy, M. T., Flambaum, V. V., Carswell, R. F., & Bainbridge, M. B. (2011). Indications of a Spatial Variation of the Fine Structure Constant. Physical Review Letters, 107(19), 191101. https://doi.org/10.1103/PhysRevLett.107.191101

we discovered new mathematics branch for Riemann hypothesis i think do you agree with me for this proof if you agree with me please comment this. Mathematical research is a collaborative and iterative process that benefits from scrutiny and discussion within the community.

Dear researcher, what is a the most powerful computer technology support mathematics learning

Many years ago I witnessed a mathematics class in Japan, in which the teacher displayed, to a class of 10 year old students, a narrow strip of paper which she identified as being one meter in length. She then proceeded to distribute one strips of paper to each child in the class, asking them to give her back "a one-half meter length of paper." Most of the students simply folded cut their strip lengthwise in half and returned one of the halves to the teacher. The teacher then placed each child's response (the actual strip of paper) on the blackboard and initiated a discussion about who had given a correct answer. I would like to find a report that refers to this research so I can share it with teachers and mathematics educators. Do you have any suggestions?

By the way, this same sort of confusion between half of the whole and half of the unit frequently appears in discussions regarding the number line. For instance, a child (or teacher) may be unsure where to locate 1/2 on a number line from 1 to 7. So I think it's a very important issue to keep track of.

Three of my articles published in Ratio Mathematics in December 2023 not yet added in research gate.

Dear Researchers,

I am reaching out to seek insights and opinions on the potential connections between chaotic dynamics, arithmetic functions, and open conjectures in analytic number theory. My interest lies in exploring the derivation of chaotic operators from mathematical constructs such as L-Dirichlet functions and conjectures like those presented by Yitang Zhang in 2022 on Landau-Siegel zeros, as well as the Montgomery conjecture on the distribution of zeros.

Specifically, I am intrigued by the possibility of deriving chaotic dynamics from these mathematical frameworks and understanding their implications for questions related to the Riemann Hypothesis.

I also want to inform you that I have recently derived a chaotic operator from Yitang Zhang's latest theorem on Landau-Siegel zeros. The work has been accepted for publication in the European Physical Journal.

My ultimate goal is to further investigate the derivation of chaotic operators from these mathematical foundations and to understand the conditions under which ζ(0.5+iH)=0. welcome any insights, suggestions, or collaboration opportunities that may arise from your expertise in these areas.

Thank you for your time and consideration. I look forward to engaging in fruitful discussions with the research community.

If anybody done this type of work,please answer me.

I need the above book if any one have please share with me. I will be thankful for this.

email: [email protected]

Consciousness and mathematics

Why the best way to learn math is to do math ?

Are you a talented young mathematician and would like to continue your mathematics studies at the PhD program under my supervision at University of Ljubljana, Faculty of Mathematics and Physics? Are you simultaneously interested in the young researcher position at the Institute of Mathematics, Physics and Mechanics in Ljubljana?

If the answers to the above question are affirmative, you may contact me at aljosa.peperko(at)fs.uni-lj.si

More information about my reseach interests:

Kind regards, Aljoša Peperko

Dear Wolfram Mathematica users, What do you consider to be the most reliable and stable version of Wolfram Mathematica that you have tried so far? For my part, the most stable version that I have tried so far is V 12.1.1, although I recognize that V 13.3.3 is superior in many aspects, especially in some graphical elements, in addition to achieving a better integration with certain repositories of the Wolfram Foundation. I have not yet tested the latest version (V 14.0).

My all proposal are rejected. I don't where it was problem. If any one have mathematical proposal ( mostly in mathematical fluid dynamics) which work was already done, please share with me and help me. I really need help.

How a mathematical fluid dynamics model (in most cases we solved and analysis through Navier Stokes equations and graphs) directly connect to physical models (experimental work) like mechanical ones are doing?

benefit that a teacher get from teaching mathematics

Dear All,

Do you have any idea about any mathematics conferences that will be held in May 2024?

Regards

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.

Fermat's last theorem was finally solved by Wiles using mathematical tools that were wholly unavailable to Fermat.

Do you believe

A) That we have actually not solved Fermat's theorem the way it was supposed to be solved, and that we must still look for Fermat's original solution, still undiscovered,

or

B) That Fermat actually made a mistake, and that his 'wonderful' proof -which he did not have the necessary space to fully set forth - was in fact mistaken or flawed, and that we were obsessed for centuries with his last "theorem" when in fact he himself had not really proved it at all?

To improve quality instruction in secondary mathematics subject, give some strategies on how a supervisor collaborate with mathematics teachers in identifying areas needs to be improved and in implementing targeted interventions to enhance teaching and learning outcomes in mathematics.

As an author, I believe that you can share me your expertise in this matter. Thank you.

JAS Pi.ai LLM (Assimilation: Copy & Paste)

what is the mathematical formula for compromise?

That's an interesting question! Compromise is often seen as a key element of negotiation and conflict resolution, but it can be difficult to define precisely or to quantify mathematically.

However, one way to think about compromise mathematically is as a form of optimization, where the goal is to find a solution that balances competing priorities or interests. In this sense, the formula for compromise might be something like:

Compromise = max(u1 + u2)

where u1 and u2 are the utilities or benefits accruing to the two parties involved in the compromise, and the max function represents the idea of finding the optimal solution that maximizes the total benefit for both parties.

In practice, of course, compromise is often more complex and multifaceted than this simple formula suggests. It may involve trade-offs, concessions, and a willingness to prioritize certain interests over others. But this formula provides a starting point for thinking about compromise as a form of optimization and problem-solving.

test this theorem. show sources

Sure, here are a few sources that explore the idea of compromise as optimization or problem-solving:

Together, these sources provide a range of perspectives on the idea of compromise as optimization or problem-solving, and offer insight into how this concept can be applied in various contexts.

Reply in a thread

has this theorem been used in the current Israel-Hamas conflict (War!)? test with variables and produce theorem (Answer!), please

To my knowledge, the specific theorem we've been discussing has not been explicitly applied to the current conflict between Israel and Hamas. However, the underlying ideas of compromise as optimization and problem-solving are certainly relevant to efforts to resolve this conflict.

To apply these ideas to the Israel-Hamas conflict, we could consider the following variables:

We could then formulate the problem as a mathematical optimization problem:

max(u1 + u2) - c

subject to the constraints that represent the various interests and priorities of each party.

The optimal solution to this problem would represent a compromise that maximizes the total benefit to both parties while minimizing the cost or compromise required.

Of course, in the real world, the Israel-Hamas conflict is far more complex and multifaceted than this simple mathematical model suggests. It involves deep-seated historical, political, and ideological divisions, as well as complex regional and international dynamics. However, the idea of compromise as optimization and problem-solving offers a potential framework for thinking about how this conflict might be resolved in a way that balancing the interests and priorities of all parties involved.

Specifically, how does subject-level random intercept and random slope influence the goodness-of-fit (R-squared) of the model?

And, if subject A contributes 10 data-points and subject B contributes 5 to the whole dataset, wouldn't A account for more of the total residual error than B? How do multilevel models control for this?

Schauder Fixed Point conjecture deals with the existence of fixed points for certain types of operators on Banach spaces. It suggests that every non-expansive mapping of a non-empty convex, weakly compact subset of a Banach space into itself has a fixed point. The status of this conjecture may depend on the specific assumptions and settings.

I see a lot of mathematics but few interpretations in time (how it evolves, step by step with its maths).

Is maths made up or real? This is the message I wrote on the website of mathematician Eugenia Cheng.

Hi! I read about you in "Nautilus" magazine. I think your statement "Some of the power of math lies in the very fact that it's made up" is a fascinating place from which to start speculating. The second step is to recall statements by Eugene Wigner and Max Tegmark. Eugene Wigner, a physicist, wrote an article in 1960 titled “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”. He argued that mathematical concepts have applicability far beyond the context in which they were originally developed. In a recent interview, physicist and mathematician Dr. Max Tegmark, stated that “math is the true driver of the universe and the universe itself is a model of the mathematical principles”.

Something made up by humans could be unreasonably effective in the universe if humans used that maths to create the universe. That sounds impossible - pure fantasy, too unrealistic even for science fiction. But bear with me. In an article written for the magazine Nautilus, it’s stated that the journals of American physicist John Wheeler, which he always kept at hand, reveal a stunning portrait of an obsessed thinker. The article says, “He knew that quantum measurement allowed observers in the present to create the past ...” and his journal contains thoughts agreeing with “The universe has created an observer and now, in an act of quantum measurement, the observer looks back and creates the universe.” Could the origin of life be related to the movie “Interstellar”? In the movie, it’s stated that humans will oneday be able to build things they can’t make now. If we take this idea to an extreme, and take “oneday” to mean an indefinite point in the far future, will we do what is obviously regarded as impossible and create life – and conceivably, the universe itself? Someday there will be a human civilization that can build their mathematics into the creation, structure, and functioning of life and the cosmos. Emotion may well declare this an absurdity and we might retreat to things like quantum fluctuation or spontaneous creation from nothing. Logically – using Einstein’s nonlinear and interactive curved time added to limitless advance of human potential through the eons – the absurdity is plausible. Just for now, cast off any thoughts about the fragility or shortness of human existence. Believe that people will oneday be able to build things they can’t make now – and that someday there will be a human civilization that can build their mathematics into the creation, structure, and functioning of life and the cosmos.

The most fundamental ingredient in creation of the universe might be base 2 maths aka the BITS or binary digits of 1 and 0. These could be used to construct everything (even the quantum units of gravitation and electromagnetic energy), and could build in the 4 dimensions of space-time. Maybe "made up" maths is actually used to produce the universe in some remote future - with the information being transmitted, via the curvature of time enabling different periods to interact, back to our present and distant past. Then the made-up maths becomes real maths!

I see on your website that people ask about 1+1=2. If all objects and events in space-time are connected, there would actually only be one being. The idea of "2" or being able to add one thing to another thing would be a result of the way our limited senses and technology operate - an illusion caused by our perceiving things as separate and unconnected.

Well, I hope you read all of this rather lengthy message. I'm rather pleased with how my thoughts turned out and have decided to post this on my ResearchGate webpage.

This algorithm has been devised by me in 2003. What I think is that this algorithm can be useful to evaluate testing the speed of processor of computer and furthermore, this algorithm can be used in data flow with encryption algorithms. I would love to know ideas about this subject from people who are expert in mathematics and computer science. I upload a pdf below. Thank you.

Investigating the role of hypergeometric series in Concrete Mathematics, seeking insights into its applications for combinatorial problem-solving.

It is crucial to understand that this expression could be used in problems related to engineering, physics, mathematics, or any other aspect of real life.

Typically, Matlab is used to solve ODE and PDE problems. Perhaps users calculated this term 0^0 incorrectly in the process.

>> % How to fix this problem 0^0 in Matlab !?

>> % Mathematically, x^0=1 if x≠0 is equal 1 else undefined(NaN)

>> 0^0

ans =

1

>> f=@(x,y) x^y;

>> f(0,0)

ans =

1

>> v=[2 0 5 -1];

>> v.^0

ans =

1 1 1 1

The mathematical universe hypothesis suggests that the universe is not just described by mathematics, but actually is mathematics. The advocates of this hypothesis think that the universe exists because it is logically consistent and self-contained. Consequently, the universe does not need any external cause or reason to exist and sustain. Aristotle interpreted mathematics as the science of discrete and continuous quantities. It has been refined into the abstract science (or pseudoscience) of number, quantity, and space, either as abstract concepts or as applied concepts. It is also claimed that mathemetics exists because human exists (it is in everything what humans do). Then:

How and why can the concept of mathematics as the hypothesized form of mathematical universe and concept of mathematics as a known area of human knowledge coexist?

Is it possible for the solution of the Fokker-Planck equation to have negative values? I am referring to the mathematical aspect, irrespective of its physical interpretation. Additionally, considering that the solution represents a probability distribution function, is it acceptable to impose a constraint ensuring that the solution remains strictly positive?

I'm currently working on solving complex mathematical equations and wondering about the most effective approach. Should I go the traditional route of hard coding solutions, or would it be more advantageous to leverage machine learning? Looking for insights from the community to help make an informed decision.

Share your experiences, thoughts, and recommendations on whether to hard code or use machine learning for solving complex mathematical equations. Feel free to provide examples from your own projects or suggest specific considerations that could influence the decision-making process.

From the earliest Pythagorean (~570BCE-~490BCE) view that "everything is number" [1], to the founder of modern physics, Galileo (1564-1642), who said "the book of nature is written in the language of mathematics" [2], to attempts by Hilbert (1862-1943) to mathematically "axiomatize" physics [3]，and to the symmetry principle [9], which today is considered fundamental by physics, Physics has never been separated from mathematics, but there has never been a definite answer as to the relationship between them. Thus Wigner (1902-1995) exclaimed [4]: "The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. gift which we neither understand nor deserve."

CN Yang, commenting on Einstein's "On the method of theoretical physics" [5], said, "Was Einstein saying that fundamental theoretical physics is a part of mathematics? Was he saying that fundamental theoretical physics should have the tradition and style of mathematics? The answers to these questions are no "[6]. So what is the real relationship between mathematics and physics? Is mathematics merely a tool that physics cannot do without? We can interpret mathematics as a description of physical behavior, or physics as operating according to mathematical principles, or they are completely equivalent, but one thing is unchangeable, all physics must ultimately be concretely embodied in its physical parameters, regardless of who dominates whom. We need to remember the essential question, "That is, we don't invent mathematical structures - we discover them, and invent only the notation for describing them"[7]. Mathematics is abstract existence, physics is reality. We cannot completely replace physical explanations with mathematical ones. For example, ask "How do light and particles know that they are choosing the shortest path [8]. The answer is that it is determined by the principle of least action. This is the correct mathematical answer, but not the final physical answer. The final physical answer should be, "Light and particles are not searching for shortest paths, they are creating and defining shortest paths". Why this can be so is because they are energy-momentum themselves. The ultimate explanation is just math*，if we can't boil it down to specific, well-defined, measurable physical parameters. Following Pythagoras' inspiring vision that the world can be built up from concepts, algorithms, and numbers [9]. When we discuss the relationship between math and physics, do we need first ask:

1) What are numbers? Shouldn't we first attribute numbers to "fundamental quantities" in mathematics and physics? Are scalars, vectors, and spinors complete expressions of such fundamental quantities? All other quantities are composites of these fundamental quantities, e.g., tensor.

2) Do mathematics and physics have to have these fundamental quantities in common before we can further discuss the consistency between their theorems? That is, the unification of mathematics and physics must begin with the unification of fundamental quantities.

3) Where do these fundamental quantities come from in physics? In what way are they represented?

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

Notes：

* And then what do imaginary numbers in physics correspond to? [10][11]

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

References：

[1] McDonnell, J. (2017). The Pythagorean World: Why Mathematics Is Unreasonably Effective In Physics Springer.

[2] Kosmann-Schwarzbach, Y. (2011). The Noether Theorems. The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century. Y. Kosmann-Schwarzbach and B. E. Schwarzbach. New York, NY, Springer New York: 55-64.

Einstein, A. (1934). "On the method of theoretical physics." Philosophy of science 1(2): 163-169.

[3] Corry, L. (2004). David Hilbert and the axiomatization of physics (1898-1918): From Grundlagen der Geometrie to Grundlagen der Physik, Springer.

[4] Wigner, E. P. (1990). The unreasonable effectiveness of mathematics in the natural sciences（1960）. Mathematics and Science, World Scientific: 291-306. 【这个说法本身可能是存在问题的，不是数学在物理学中的有效性，而是不能够区分物理学准则和数学算法。】

[5] Einstein, A. (1934). "On the method of theoretical physics." Philosophy of science 1(2): 163-169.

[6] Yang, C. N. (1980). "Einstein's impact on theoretical physics." Physics Today 33(6): 42-49.

[7] Russell, B. (2010). Principles of mathematics （1903）, Routledge.

[8] https://www.researchgate.net/post/NO4_How_do_light_and_particles_know_that_they_are_choosing_the_shortest_path

[9] Wilczek, F. (2006). "The origin of mass." Modern Physics Letters A 21(9): 701-712.

[10] Chian Fan, e. a. (2023). "How to understand imaginary numbers (complex numbers) in physics." from https://www.researchgate.net/post/NO6_How_to_understand_imaginary_numbers_complex_numbers_in_physics.

[11] Baylis, W. E., J. Huschilt and J. Wei (1992). "Why i?" American Journal of Physics 60(9): 788-797. 【复数、虚数、波函数】

Is there any relationship between the two mathematical statements?

1. (a-b)^((n+1)/n) and

2. (a)^((n+1)/n) - (b)^((n+1)/n)

Hi!

I am facing technical problems in MAC PRO laptop regarding editing or typing the mathematical work in the files created through Microsoft WORD. Till date i was using MATHTYPE IN WINDOWS but now the mathematical work of these files is not accessible. I have bought MathType compatible with MAC and WINDOWS but still i am not able to work on it. Please help me and guide me about any other software which i may use in MAC for maths work and which is accessible on Window as well or which my students may use in WINDOWS and it is accessible on mac.

Hear me out. I know this is going to be tad bit long but I assure you this is thought provoking. ( I mean I still have a hard time convincing myself if this doubt is even well founded)

Firstly we don't know the most fundamental shape of all and consequently the dimension for existence. Now I say this because firstly a circle although a shape, is mathematically defined as a polygon of infinite sides, right?

Secondly, now when you want to construct a line in a single dimension, what you do is that you take a point in 2 dimensions of infinitesimal size and glue a bunch of points together, in order to amount to a given length( note that here when the size although infinitesimal is fixed one tends to replicate those sizes and then actually make it amount to that length), but when we see that the solid circle that we are calling our point here can't really be though of as the most fundamental shape, then we can actually see that this infinitesimal point of the smallest length comprehensible still can be called as an infinite sided polygon and therefore we can never really define the point of any dimension rather we would be stuck in cycles, and we can never really define a point because for that we would need some line segment and basically we can't define even an infinitesimal point, right?

PS: I am still very much confused if at all this thought is well founded, and if it is, what might this imply.

Right now, in 2022, we can read with perfect understanding mathematical articles and books

written a century ago. It is indeed remarkable how the way we do mathematics has stabilised.

The difference between the mathematics of 1922 and 2022 is small compared to that between the mathematics of 1922 and 1822.

Looking beyond classical ZFC-based mathematics, a tremendous amount of effort has been put

into formalising all areas of mathematics within the framework of program-language implementations (for instance Coq, Agda) of the univalent extension of dependent type theory (homotopy type theory).

But Coq and Agda are complex programs which depend on other programs (OCaml and Haskell) and frameworks (for instance operating systems and C libraries) to function. In the future if we have new CPU architectures then

Coq and Agda would have to be compiled again. OCaml and Haskell would have to be compiled again.

Both software and operating systems are rapidly changing and have always been so. What is here today is deprecated tomorrow.

My question is: what guarantee do we have that the huge libraries of the current formal mathematics projects in Agda, Coq or other languages will still be relevant or even "runnable" (for instance type-checkable) without having to resort to emulators and computer archaeology 10, 20, 50 or 100 years from now ?

10 years from now will Agda be backwards compatible enough to still recognise

current Agda files ?

Have there been any organised efforts to guarantee permanent backward compatibility for all future versions of Agda and Coq ? Or OCaml and Haskell ?

Perhaps the formal mathematics project should be carried out within a meta-programing language, a simpler more abstract framework (with a uniform syntax) comprehensible at once to logicians, mathematicians and programers and which can be converted automatically into the latest version of Agda or Coq ?

Who was right Plato/Einstein or Epicurus/Engels? “Our Mathematical Universe” of Max Tegmark or “The Dialectical Universe” of Abdul Malek?

Mankind therefore finds itself faced with a contradiction; on the one hand, it has to gain an exhaustive knowledge of the world system in all in its interrelations; and on the other hand, because of the nature both of man and of the world system, this task can never be completely fulfilled. But this contradiction lies not only in the nature of the two factors – the world, and man – it is also the main lever of all intellectual advance, and finds its solution continuously, day by day, in the endless progressive evolution of humanity just as for example mathematical problems find their solution in an infinite series or continued fractionations. Each mental image of the world system is and remains in actual fact limited, objectively through the historical stage and subjectively through the physical and mental constitution of its maker”. (Anti-Dühring)

Nowadays elementary learners no longer adept to manual operations. They are very keen on just choosing from the letters (mutiple choice) without even trying to solve for the right answer. Also I have seen that after pandemic, learners do not know basic operations and manual computation.

Mathematics is and is only a language and a tool, not all of science?

Can somebody tell me what mathematical formula might be used to calculate these weights (in the picture) for individual studies, in a meta-analytic review study?

How do I obtain gridded data points from discrete data points as shown in following figure?

How to compare hazard function of decreasing types (increasing types) of two continuous distribution mathematically that is which one is highly decreasing (increasing) and which one is slowly decreasing (increasing) ?

From Wundt, psychology learned from natural sciences such as physics and chemistry, learning the research methods of these subjects, that is, "external observation, experiment" and other ways, and then using mathematical description.

But is this approach fine? The human psychology and physics are completely different, so the way that study physics studies have no problem?

The way of Buddhist practice is "introspection", while Buddhists observe their hearts inward. Zen well shows this way of research. In fact, the doctrine of Buddhism is psychology. So, shouldn't the way of studying psychology be this kind of Buddhist practice? Shouldn't psychology be introspection? Why the "experiment and observation" of natural science?

Is this because westerners and Orials have different thinking habits? The way of thinking of Westerners is natural and scientific, so now they are used to studying psychology in this way, but psychology is more in line with the way of thinking of Orientals, especially Indians.

I am doing research to compare learning model of A, B, and C on students mathematical ability improvemnet. However it turns out that the pretest result shows significant differences among class that use learning model A, B, and C (score A> score B; score A > score C; and Score B = score C)

Meanwhile posttest also shows significant different that score A = Score B; Score A > score C; and score B > score C.

Is it okay to still compare the gain score of the groups using Anova?

Or shall I use pretest as covariate using Ancova?

Scratch in geometry for exemple !

I want to design a reconfigurable transmit array using varactor diode with metamaterial unit cell. I have gone through many literatures but in most of literature only different design and multiplayers are mentioned to achieve maximum transmission and 360 deg phase tuning. But i am not able to achieve both the things like high transmission and 360deg phase tuning. Could anyone suggest some resources to understand mathematics behind this?

Combination @Purpose #Logic ^Reason *Sustainable %Help (Relevant: USE)

Truth @Sustainable #Purpose ^USE *Relevant %Logic (Help: Reason)

Lies @Reason #Help ^Logic *USE %Relevant (Purpose: Sustainable)

Process @USE #Reason ^Help *Purpose %Sustainable (Reason: Logic)

Key: Live = Evil

Note: Combinatorics is a branch of mathematics that studies the

enumeration, combination, and permutation of sets of elements. It also studies the mathematical relations that characterize the properties of these elements. <Google>

Commentary: Clear definition of "INTENT" permits AI "INTUIT" with HELP USE Energy Law versions emitting IED=NS (Input Energy Decision Number Subject)

PROOF:

Combination Truth Lies Process = Being

@Purpose Sustainable Reason USE == Planet

#Logic Purpose Help Reason === Self

^Reason USE Logic Help ==== Family

*Sustainable Relevant USE Purpose ===== Earth

%Help Logic Relevant Sustainable ====== Profit

(Relevant Help Purpose Reason) ======= Communities of Interest

:USE Reason Sustainble Logic ======== Centers of Excellence

OF ...

Being planet self family earth profit communties of interest (w/) centers of excellence.

I have 'N' number of inputs (which correspond to temperature) to calculate a specific output parameter. I'm getting 'N' number output data based on the inputs.

However, my goal is to select an optimum number from all the output data and use it in another calculation.

'N' number of input data --> Output parameter calculation --> Identification of an optimized output parameter --> Use that value for another calculation.

How to find an optimal value among 'N' "number of output data. Can we employ any algorithm or process?

how Using mathematical modelling in mathematics education can be an innovative and powerful approach to helping students understand how mathematical concepts can be applied in real-world situations?

DEAR PROFESSORS

GREETINGS OF THE DAY.

NOWADAYS WE ARE USING MANY SOFTWARE FOR MATHEMATICS.

FOR MATHEMATICS WORK, IS THERE ANY FREE AI BASED APPS AVAILABLE?

IF SO, PLEASE KINDLY SHARE THE LINK HERE.

THANKS ALL

The limits of logic and mathematics is that we couldn't describe a question without symbol system, but symbol system is just an abstraction of the real world not the real world itself, so there is a distance between the abstracted symbol system and the real world, therefore there is truth we can't reach by symbol system, which A-HA moment may reach. But when we thinking we always use a symbol system like words or mathematics with apriori logic, so I wonder if AI could have A-HA moment?