Science topics: Mathematics
Science topic

# Mathematics - Science topic

Mathematics, Pure and Applied Math
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Why prime numbers have a great importance in mathematics for the rest of the numbers ?
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Prime numbers have great importance in mathematics due to their unique properties and their fundamental role in number theory. Here are a few reasons why prime numbers hold significance:
1. Building blocks of numbers: Every positive integer greater than 1 can be expressed as a product of prime numbers in a unique way, known as the prime factorization. This property is known as the fundamental theorem of arithmetic. It means that prime numbers are the building blocks of all other numbers, making them fundamental in understanding the structure of numbers.
2. Divisibility and factors: Prime numbers play a crucial role in determining divisibility and factors of a number. A prime number has only two distinct positive divisors: 1 and itself. This property makes prime numbers essential in understanding and analyzing the factors and divisors of any given number.
3. Cryptography: Prime numbers find extensive applications in modern cryptography algorithms. Public-key encryption systems like the RSA algorithm heavily rely on the difficulty of factoring large composite numbers into their prime factors. The security of such systems is rooted in the use of prime numbers.
4. Distribution of primes: The study of prime numbers involves analyzing their distribution throughout the number line. The prime number theorem, proven by mathematicians Jacques Hadamard and Charles Jean de la Vallée Poussin independently in 1896, provides an estimation of the number of primes below a given value. Investigating the distribution of primes leads to a deeper understanding of the overall structure of the number system.
5. Unsolved problems: Prime numbers are at the center of many unsolved problems in mathematics. The most famous among them is the Riemann Hypothesis, which deals with the distribution of prime numbers. Numerous other conjectures and problems related to primes continue to captivate mathematicians, highlighting their ongoing importance in the field.
6. Mathematical proofs: Prime numbers often serve as key elements in mathematical proofs. They can be used to establish important theorems and propositions in various branches of mathematics, including algebra, number theory, and geometry. Prime numbers provide a foundation for rigorous mathematical reasoning.
In summary, prime numbers hold immense importance in mathematics as they form the basis for number theory, cryptography, and many other mathematical concepts. Their unique properties and distribution patterns make them invaluable in understanding the structure of numbers and solving mathematical problems.
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Vehicle routing problem is a classical application case of Operation research.
I need to implement the same in electric vehicle routing problem with different constraints.
I want to understand mathematics behind this. The journals available discuss different applications without much talking of mathematics.
Any book/ basic research paper/ PhD/ m.tech thesis will do the needful.
Thanks in advance.
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There is a vast literature on exact and heuristic approaches to vehicle routing problems. (You are looking at several thousands journal articles!)
If you are interested in exact approaches, then you need to be familiar with the following:
(i) The basics of integer programming, including the branch-and-bound method and cutting-plane methods.
(ii) The basics of computational complexity, including the concept of polynomial-time algorithms, pseudo-polynomial-time algorithms and NP-completeness.
(iii) Elementary graph theory (nodes, edges, arcs, and so on).
It also helps to know a bit about:
(iv) Dynamic programming.
(v) Lagrangian relaxation.
(vi) The branch-and-cut method, which combines branch-and-bound with strong cutting planes from polyhedral studies.
(vii) The branch-and-price method, which combines branch-and-bound with Dantzig-Wolfe decomposition and dynamic programming.
A good place to start is the book "The Vehicle Routing Problem", edited by Toth and Vigo. There is also "The Vehicle Routing Problem: Latest Advances and New Challenges", edited by Golden et al.
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Qm is the ultimate realists utilization of the powerful differential equations, because the integer options and necessities of solutions correspond to nature's quanta.
The same can be said for GR whose differential manifolds, an sdvanced concept or hranch in mathematics, have a realistic implementation in nature compatible motional geodesics.
1 century later,so new such feats have been possible, making one to think if the limit of heuristic mathematical supplementation in powerful ways towards realist results in physics in reached.
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The short answer is No. And QM is much more than an application of the theory of differential equations, as is GR. The resolution of spacetime singularities-that are predicted by GR, isn't a mathematical issue, it's a physical issue. It will be the discovery of what are the appropriate physical degrees of freedom that will indicate what is the mathematical framework that is appropriate for describing them.
QM doesn't have any particular outstanding mathematical issues-it's quantum field theory that poses mathematical challenges.
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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 adjectival: ‘being a unity’, ‘being two unities’, ‘being a non-extended shape’, etc. 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 numbers too. If numbers and shapes are in fact inexact, then not only irrational 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.
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I have revised the lead text (Mathematics and Causality: A Systemic Reconciliation). Now it has become 8 pages in A-4. See the link below.
I believe your questions may be answered here in my humble manner. Further questions and suggestions are most welcome. Thanks.
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a) are there some possible hard and fast rules for review deadlines (yes or no?), and b) is there some obligation from an Editorial Board side to be giving the first answer to authors about the internal number of an article, if the article is submitting via email id (yes or no?).
Thanks for the input. https://clarivate.com/contact-us/
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Dear Dr. Hairi Ismaili ,
Thank you for your reply.
Please, see above: "on maths and other..."
Greetings,
Sergey
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Applying mathematical knowledge in research models: This question has been in my mind for a long time. Can advance mathematics and applied mathematics solve all the problems in modeling research? Especially the formula derivation in the theoretical model part, can the analysis conclusion be obtained through multiple derivations or other methods? You have also read some mathematics-related publications yourself, and you have to admire the mystery of mathematics.
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We all know that Mathematics include Reading , Writing & Arithmetic & its starts with every action of our life image & as such it is the action of our performance & image in every part of our life. With this some years back I have expressed my views in this areas which I submit herewith for your kind perusal .
In my early days students interested in Mathematics & scoring full marks they can perform in their working of mathematics either by listening to music or song or prior to during a home work they have formulated a habit of reading either a lesson or interested topics & after carrying out their working system they used to give justice to the subject of Mathematics.
This is my personal opinion
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Which areas in mathematics education is trending currently
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As an AI language model, I can provide an overview of some current research trends in mathematics education up until my knowledge cutoff in September 2021. However, please note that the field of research is constantly evolving, and there may have been new developments since then. Here are a few prevalent research trends in mathematics education:
1. Technology Integration: Researchers are exploring the effective use of technology in mathematics education. This includes studying the impact of digital tools, interactive software, simulations, and online resources on teaching and learning mathematics. Additionally, there is a focus on designing technology-enhanced learning environments and investigating the role of technology in fostering conceptual understanding and problem-solving skills.
2. Problem-Solving and Mathematical Thinking: There is an emphasis on promoting problem-solving skills and mathematical thinking among students. Researchers are investigating instructional strategies and interventions that help students develop problem-solving abilities, reasoning skills, and a deep conceptual understanding of mathematical concepts. This includes exploring the use of open-ended problems, mathematical modeling, and real-world contexts to engage students in authentic mathematical experiences.
3. Learning Trajectories and Progressions: Research in this area focuses on understanding the developmental progression of mathematical concepts and skills. Learning trajectories provide a framework for mapping out the sequence of learning in different mathematical domains and identifying the key milestones along the way. By understanding how students progress through these trajectories, researchers aim to develop effective instructional approaches and interventions that cater to students' diverse learning needs.
4. Assessment and Feedback: There is ongoing research on developing innovative assessment methods and providing effective feedback in mathematics education. This includes investigating formative assessment strategies, computer-based assessments, and alternative approaches to evaluating mathematical competencies. Researchers are also exploring the role of feedback in enhancing students' learning and understanding of mathematics.
5. Equity and Access: Mathematics education research is increasingly focusing on issues of equity, diversity, and inclusion. Researchers are examining the factors that contribute to achievement gaps among different student populations and investigating strategies to promote equitable mathematics learning experiences. This includes exploring culturally responsive teaching practices, addressing stereotype threats, and promoting access to high-quality mathematics education for all students.
These research trends highlight some of the current areas of focus in mathematics education. However, it is essential to note that the field is dynamic, and new trends may have emerged since my knowledge cutoff. For the most up-to-date information, it is advisable to consult recent academic journals and conferences in the field of mathematics education.
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As it is not possible to show mathematical expressions here I am attaching link to the question.
Your expertise in determining and comprehending the boundaries of integration within the Delta function's tantalizing grip will be treasured beyond measure.
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Use δ(a-√x2+y2)=(a/√a2-x2)(δ(y-√a2-x2)+δ(y+√a2-x2)) to do the integral over y. Then the integral over x remains and its integration interval is [-a,a].
The general recipe is to transform the δ function δ(a-g(y)) into a sum of δ functions δ(y-yk), where yk are the zeros of g(y)-a. Each term acquires a denominator |g'(yk)| in the process.
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An attempt to extrapolate reality
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Both textbooks and also most of secondary school maths teachers are not sufficient to excite students. Some of the USA textbooks are really extraordinary attractive but still very common in so many other countries apart from Europe !?
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Our answer is a competitive YES. However, universities face the laissez-faire of old staff.
This reference must be included:
Gerck, E. “Algorithms for Quantum Computation: Derivatives of Discontinuous Functions.” Mathematics 2023, 11, 68. https://doi.org/10.3390/math1101006, 2023.
announcing quantum computing on a physical basis, deprecating infinitesimals, epsilon-deltas, continuity, limits, mathematical real-numbers, imaginary numbers, and more, making calculus middle-school easy and with the same formulas.
Otherwise, difficulties and obsolescence follows. A hopeless scenario, no argument is possible against facts.
What is your qualified opinion? Must one self-study? A free PDF is currently available at my profile at RG.
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Physics confirming math, or denying it. Time for colleges to catch-up and be competitive.
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Hello,
in an article I found the following sentence in the abstract:
"The results suggested that (1) Time 1 mathematics self-concept had significant effects on Time 2 mathematics school engagement at between-group and within-group levels; and (2) Time 2 mathematics school engagement played a partial mediating role between Time 1 mathematics self-concept and Time 2 mathematics achievement at the within-group level."
What is the meaning of the within-group-level and between-group-level in this context?
The article I am referring to is:
Xia, Z., Yang, F., Praschan, K., & Xu, Q. (2021). The formation and influence mechanism of mathematics self-concept of left-behind children in mainland China. Current Psychology, 40(11), 5567–5586. https://doi.org/10.1007/s12144-019-00495-4
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Hi Max,
I quickly went through the paper, and I understand your confusion. In study 2, there are actually two within-components or, if you will, a three-level design (time within students within classes).
After having read through the paper, I am quite convinced that the authors refer to between-groups when they mean between-classroom effects, e.g., average classroom math self-concept differs with respect to some predictor on the classroom level. In other words, by within-group effects the authors refer to the individual-(student-)level. While this makes complete sense in study 1, in it slightly confusing in study 2 in my opinion because of the longitudinal design.
Hope this helps!
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An attempt to extrapolate reality
📷
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There are many reasons why students may have a negative attitude towards mathematics. Some possible reasons include:
Lack of confidence: Students may feel that they are not good at mathematics and may believe that they are incapable of doing well in the subject.
Difficulty with abstract concepts: Mathematics involves working with abstract concepts, which can be difficult for some students to understand.
Negative experiences: Students may have had negative experiences with mathematics in the past, such as poor grades or negative feedback from teachers or peers.
Lack of engagement: Some students may find mathematics boring or irrelevant to their lives, which can lead to disengagement and a negative attitude towards the subject.
Cultural stereotypes: Some students may hold negative cultural stereotypes about mathematics, such as the belief that it is a subject for boys or that only extremely intelligent people can excel in the subject.
Teacher quality: The quality of instruction in mathematics can vary widely, and students who have had poor teachers may develop a negative attitude towards the subject as a result.
Addressing these issues and finding ways to engage students and help them develop a more positive attitude towards the subject can be a challenge, but it is an important one for educators to tackle.
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I know that δ(f(x))=∑δ(x−xi)/f′(xi). What will be the expression if "f" is a function of two variables, i.e. δ(f(x,y))=?
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To write δ(f(x, y)) in terms of the Dirac delta function, you can use the property of the delta function known as the sifting property. The sifting property states that if g(x) is a continuous function with a single root at x = a, then the integral of g(x) multiplied by the Dirac delta function δ(x - a) is equal to g(a).
Applying this property to the function f(x, y), you can express δ(f(x, y)) as follows:
δ(f(x, y)) = Σ[δ(x - xi, y - yi) / |∇f(xi, yi)|],
where the sum is taken over all points (xi, yi) where f(xi, yi) = 0, and ∇f(xi, yi) is the gradient of f evaluated at (xi, yi). The |∇f(xi, yi)| represents the magnitude of the gradient.
This expression accounts for the fact that the Dirac delta function is being evaluated at the points where f(x, y) is zero. The presence of the gradient magnitude in the denominator ensures that the Dirac delta function is appropriately weighted at each point to maintain the integral property.
It's important to note that the expression above assumes that f(x, y) is continuous and differentiable, and that the points (xi, yi) where f(xi, yi) = 0 are isolated. If f(x, y) has multiple roots or singularities, the expression may require modification to handle those cases appropriately.
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I'm using target encoding in my work, and I'd like to understand why it's effective from a mathematical point of view.
Intuitively, my understanding is that it allows you to encode the past with the future. I can see why that's effective, and also why it could cause target leakage. However, I can't find a good mathematical explanation for its effectiveness/ issues.
Does anyone know the answer, or have a link to a resource they'd be willing to share?
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Target encoding is a technique used in machine learning to encode categorical variables as numerical values based on the target variable. The idea is to use the target variable to create a new feature for each unique category in the categorical variable, where the value of the feature is the average value of the target variable for that category.
From a mathematical point of view, target encoding can be effective because it can help capture the relationship between the categorical variable and the target variable. By encoding the categorical variable based on the target variable, the resulting numerical values can provide a more informative representation of the categorical variable, which can help improve the performance of the machine learning model.
One way to think about this is in terms of information theory. The target variable provides information about the relationship between the categorical variable and the target variable. By encoding the categorical variable based on the target variable, we are effectively incorporating this information into the feature representation. This can help improve the model's ability to learn patterns and make accurate predictions.
However, target encoding can also be prone to target leakage, where the encoded feature incorporates information from the target variable that would not be available at prediction time. This can lead to overfitting and poor generalization performance. To mitigate this issue, it is important to use proper cross-validation techniques and to ensure that the encoding is done using only information that would be available at prediction time.
Here are a few resources that you may find helpful:
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We assume that the Lagrange multipliers originally introduced in the Boltzmann-Einstein model to derive the Gaussian distribution are just a mathematical trick to compensate for the lack of true definition of probability in unified 4D space.
The derivation of the Boltzmann distribution for the energy distribution of identical but distinguishable classical particles can be obtained in a mathematical approach [1] or equivalently via a statistical approach [2] where the Lagrange multipliers are completely ignored.
1-The Boltzmann factor: a simplified derivation
Rainer Muller
2- Statistical integration, I. Abbas
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The the Lagrange multipliers LG originally introduced in the Boltzmann-Einstein model to derive the Gaussian distribution are just a classical mathematical trick to compensate for the lack of true definition of probability in unified 4D space.
The derivation of the Boltzmann distribution for the energy distribution of identical but distinguishable classical particles can be obtained via a mathematical approach [1] or equivalently via a statistical approach [2,3] where the Lagrange multipliers can be completely ignored.
However, there are still many people who claim that
i- without the Lagrange multipliers all economic theory, and some finance applications as well, are gonna be in trouble in finite or infinite dimensional spaces.
ii- while the use of Lagrange multipliers may not be the only way to derive the Boltzmann distribution, it is a well-established and useful technique that should not be dismissed as a mere "trick" .
iii- without the L.G. all economic theory, and some finance applications as well, are gonna be in trouble in finite or infinite dimensional spaces.
iv-LG are used in various fields, including physics, economics, and optimization. They are used to optimize a function subject to a set of constraints, by introducing additional parameters (the Lagrange multipliers) that allow the constraints to be incorporated into the objective function.
In the context of statistical mechanics, Lagrange multipliers are used to enforce the constraints on the total energy, volume, and number of particles in a system, while maximizing the entropy.
While it's true that the use of Lagrange multipliers is a mathematical technique, it's not just a "trick" that can be ignored. The Lagrange multipliers are necessary to incorporate the constraints into the optimization problem and obtain the correct solution. Without the Lagrange multipliers, the constraints would not be taken into account, and the resulting distribution would not accurately reflect the physical behavior of the system.
We assume that the fears or claims i-iv will not happen because LG constraints will be incorporated in adequate statistical theory.
In brief, Lagrange multipliers is just a classic mathematical trick that we can do without.
A detailed answers to this question and other related questions such as the numerical statistical solution of double and triple integration as well as the time dependent PDE are all explained in references[2,3] where the modern definition of probability in the transition matrix B is an interconnected thing of the three topics. Imagination is the first important common factor in mathematics, physics and especially in the probability of transition in unitary 4D space is incorporated in B-matrix statistical chains to numerically solve single, double, and triple (Hypercube) integrals as well as time dependent PDE[ 2.3].
Ref:
1-The Boltzmann factor: a simplified derivation
Rainer Muller
2-I.M.Abbas, How Nature Works in Four-Dimensional Space: The Untold Complex Story, Researchgate, May 2023.
3-I.M.Abbas, How Nature Works in Four-Dimensional Space: The Untold Complex Story, IJISRT review, May 2023.
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Hello!
I am curious , can anyone guide me how we can calculate the amount of hydrogen is stored in the metal hydride during the absorption process both in %wt. an in grams and how much energy is released during absorption
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Hello
To explain my answer I propose you what guided it;
It is difficult to know on which form the hydrogen is in the metal, that is to say it is in hydride form, atomic or molecular. The next difficulty is to measure its concentration. I approached these questions by studying the adsorption of incondensable gases on metals, atomically clean (Ta and Al (111)) at very low H2 partial pressure, of the order of 10^-3 Pascal and at room temperature. While the residence time of the molecules should be extremely short of the order of 10^- 10 seconds or less, it is much larger and depends on the polarity of the molecules and the presence of defects, impurities and dislocations. These defects create local variations of the crystal field and internal stresses while the adsorbed molecules create image charges that modify the electronic and vibrational structure at the surface of the solid. This is verified by electron spectrometry and is expressed by the dielectric function. Even at very low coverage, the adsorbed molecules become unstable due to the relaxation of internal stresses of entropic origin. This scheme is parallel to that of the effects of adsorbed charges on insulators and is expressed by an equation of state and expresses the surface barrier deformation by defects and the resulting physical/chemical adsorption and ion diffusion processes. On the basis of these elements, the hydrogen concentration of the order of ppm would not be measurable by weight but fatal to the cohesion. One can imagine several experiments to verify this scheme.
Have a nice day
Claude
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Mathematics and theoretical physics are currently searching for answers to this particular question and two other related questions that make up three of the most persistent questions:
i- Do probabilities and statistics belong to physics or mathematics?
ii- Followed by the related question, does nature operate in 3D geometry plus time as an external controller or more specifically, does it operate in the inseparable 4D unit space where time is woven?
iii-Lagrange multipliers: Is it just a classic mathematical trick that we can do without?
We assume the answers to these questions are all interconnected, but how?
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One could invoke that energy subdivides between systems by scaling them with parameters, same way as Lagrange multipliers are used in variational problems. I just read my latest article, where I did not do that, but instead used an entire gradient, same way as entire time differentiation. It would result in the same traveling waves, with one more variable, probably, and need not be reduced to obtain the phenomenon?
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I am trying to understand the mathematical relations that computes the residual entropy of methane in REFPROP. I have tried to replicate the graphs in "Entropy Scaling of Alkanes II" by Ian bellusing the mathematical expressions in the paper aand have been unsuccessful so far.
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The whole point of "residual entropy" (and residual enthalpy) is the part that is *not* an ideal gas; so using an ideal gas relationship will provide no insight. Most textbooks provide an integral with respect to pressure ∫dP, which is useless because equations of state are never in this form. We use integration by parts to convert this to an integral with respect to specific volume. Put the equation of state into this equation and you have the residuals.
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Can someone please provide more insight on the mathematical formulation of a frequency constrained UC model, comprising of only synchronous sources, as well as non-synchronous sources. Also what would be the associated MILP code in GAMS?
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The previous paragraph outlines a step-by-step process for using Mixed Integer Linear Programming (MILP) to solve the frequency constrained unit commitment problem in power system operation. The process involves formulating the objective function and constraints, solving the MILP problem using a suitable solver, validating the solution, and implementing the optimal unit commitment schedule in the power system operation. The main constraints include power balance, generator capacity, minimum up and down time, ramping, and frequency constraints. The solution obtained should satisfy all the constraints and be physically feasible
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The error of building a physical world based on the basic feelings of fundamental concepts such as space and time occurred during the creation of Newtonian mechanics by Newton. Of course, this mistake should have been made, so that man would not be deprived of the numerous gifts of technology resulting from this science! But when the world showed another face of itself to man in very small and large scales, this theory along with the error did nothing.
When Newton had those ideas about space and time (of course, maybe he knew and had no choice), he built a mathematical system for his thoughts, differential and integral calculus! Mathematics resulting from his thoughts was a systematic continuation of his thoughts, with the same assumptions about space and time. That mathematics could not show him the right way to know the real world! Because the world was not Newtonian! Today, many pages in modern physics are created based on new assumptions of space and time and other seemingly obvious variables!
Now, why do we think that these pages of current mathematics necessarily lead to the correct knowledge of the world! Can we finally identify the world, as it is, by adopting appropriate and correct assumptions?!
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Good question. I suggest that the background structure for science research, which includes quantum mechanics is consciousness. Yet consciousness is deleted from mainstream science, hence for such research, there is no background structure. Two papers in the Journal Communicative and Integrative Biology discuss this situation: Omni-local consciousness: and The two principles that shape scientific research:
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Apart from the mathematical systems that confirm human feelings and perceptive sensors, there are countless mathematical systems that do not confirm these sensors and our sensory data! A question arises, are the worlds that these mathematical systems evoke are real? So in this way, there are countless worlds that can be realized with their respective physics. Can multiple universes be concluded from this point of view?
Don't we see that only one of these possible worlds is felt by our body?! Why? Have we created mathematics to suit our feelings in the beginning?! And now, in modern physics and the maturation of our powers of understanding, we have created mathematical systems that fit our dreams about the world!? Which of these mathematical devices is actually true about the world and has been realized?! If all of them have come true! So there is no single and objective world and everyone experiences their own world! If only one of these mathematical systems has been realized, how is this system the best?!
If the worlds created by these countless mathematical systems are not real, why do they exist in the human mind?!
The last question is, does the tangibleness of some of these mathematical systems for human senses, and the intangibleness of most of them, indicate the separation of the observable and hidden worlds?!
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Dear Seyed Mohammad Mousavi You're right on question. Unfortunately we accustomed for centuries to use this tool of manmade one dimension, static mathematic to blend it to nature that it is changing constantly in three dimensions. For same reason, all of our equations never settled perfectly. Newtons gravity, Einstein's , Maxwell, Lorenz...all misguided and not working. My theories on universe is mentioned this phenomenon. read my articles
regards
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Recently I've discussed this topic with a tautologist researcher, Quine's follower. The denial of the capacity of deductive logic to generate new knowledge implies that all deductive results in mathematics wont increase our knowledge for real.
The tautologic nature of the deduction seems to lead to this conclusion. In my opinion some sort of logic omniscience is involved in that position.
So the questions would be:
• Is the set of theorems that follow logically from a set A of axioms, "implicit" knowledge? if so, what would be the proper difference between "implicit" and "explicit" knowledge?
• If we embrace the idea that no new knowledge comes from deduction, what is the precise meaning of "new" in this context?
• How do you avoid the problem of logic omniscience?
Thanks beforehand for your insights.
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Thanks for your comment.
In my case I find that use of the term 'implicit' very problematic. In my opinion it is case of language abuse, because the term 'implicit' is normally used to refer to thinks that are known, but not said. For example, if I say: "buy a car", it is implicit that you will buy it with money. I don't have to say: "buy a car with money", because it is already known, and no need to be explicit. So the original meaning of 'implicit' is an implied idea that it is not need to be said.
However, used in the way of Lakatos, we would have to say that abc conjecture is implicit in the ZFC axioms, not because it is known at all, but because its true/falsehood is a possible conclusion of a deduction. Which in my opinion is nonsense, and introduce the questionable concept that un-deduced conclusions are already known someway before the deduction is actually performed.
Therefore I do not think that all logically implied conclusions fit into the concept of implicit.
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Are there certain methods, for instance T-tests or ANOVAs, for certain ways a survey question is asked?
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There are several mathematical and statistical methods used to quantify and discuss survey questions. Here are some of the most common ones:
1. Descriptive statistics: Descriptive statistics can be used to summarize the data from survey questions. For example, you can calculate measures of central tendency (such as the mean, median, and mode) and measures of variability (such as the range, standard deviation, and variance) to describe the distribution of responses.
2. Cross-tabulation: Cross-tabulation (also known as contingency tables or pivot tables) can be used to examine the relationship between two or more survey questions. It allows you to see how the responses to one question vary with the responses to another question.
3. Chi-square test: The chi-square test can be used to determine whether there is a significant association between two categorical variables. It can be used to test whether the responses to one survey question depend on the responses to another survey question.
4. T-tests and ANOVA: T-tests and analysis of variance (ANOVA) can be used to compare the means of two or more groups on a single survey question. They can be used to test whether there are significant differences in the responses to a survey question between different groups (such as men and women, or different age groups).
5. Regression analysis: Regression analysis can be used to examine the relationship between one or more independent variables and a dependent variable. It can be used to test whether there is a significant linear relationship between a survey question and other variables, such as demographic variables or other survey questions.
6. Factor analysis: Factor analysis can be used to identify underlying factors or dimensions that explain the pattern of responses to multiple survey questions. It can be used to group survey questions that measure similar concepts or to identify unique factors that explain variation in the responses.
These are just a few examples of the mathematical and statistical methods that can be used to quantify and discuss survey questions. The choice of method will depend on the research question, the type of data collected, and the level of analysis needed.
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Actually, I am working on the modeling of path loss between the coordinator and the sensor nodes of a BAN network. My objective is to make a performance comparison between the CM3A model of the IEEE 802.15.6 standard and a loss model that I have implemented mathematically.
So, according to your respectful experience, how can I implement these two path loss models? Do I have to define both path loss equations under the Wireless Channel model? Or do I create and implement for each path loss model a specific module under Castalia (like the wireless channel module) and after I call it from the omnet.ini file (configuration file) ?
You will find attached the two models in a figure.
Thanks in advance
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Not sure if I could get your question properly, but here is an alternative solution to obtain the Path Loss value:
There is a software: "NYUSIM"
You can actually get the path loss comparisons by running the simulation. This software can simulate up to 100 GHz. All you have to do is to insert the appropriate simulation data in terms of your desired outcome.
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Many people believe that x-t spacetime is separable and that describing x-t as an inseparable unit block is only essential when the speed of the object concerned approaches that of light.
This is the most common error in mathematics as I understand it.
The universe has been expanding since the time of the Big Bang at almost the speed of light and this may be the reason why the results of classical mathematics fail and become less accurate than those of the stochastic B-matrix ( or any other suitable matrix) even in the simplest situations like double integration and triple integration.
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CASE A:To begin with, let's admit that nature does not see with our eyes and does not think with our brains.We try to understand how nature performs its own resolutions in space-time x-t like an inseparableunit block.
However, B-Matrix statistical chains (or any other suitable stochastic chains) can answer this question and demonstrate, in a way, how nature works:
i-nature see:the curve as a trapezoidal area.ii-nature see:the square as a cube or a cuboid volume.iii-nature see:the cube as a 4D Hypercube and evaluates its volume as L^4.In all hypotheses i-iii, time t is the additional dimension.However, many people still believe that x-t spacetime is separable and that describing x-t as an inseparable unit block is only essential when the speed of the object concerned approaches that of light.This is the most common error in mathematics as I understand it.The universe has been expanding since the time of the Big Bang at almost the speed of light and this may be the reason why the results of classical mathematics fail and become less accurate than those of the stochastic B-matrix ( or any other suitable matrix) even in the simplest situations like double integration and triple integration.This is the reason why the current definition of double and triple integration is incomplete.Brief,-------Time is part of an inseparable block of space-time and therefore,geometric space is the other part of the inseparable block of space-time.In other words, you can perform integration using the x-t space-time unit with wide applicability and excellent speed and accuracy.On the other hand, the classical mathematical methods of integration using the classical mathematical  technique in the geometric Cartesian space x-alone can still be applied but only in special cases and it is to be expected that their results are only a limit of success.In other words, you can perform integration using the x-t space-time unit with wide applicability and excellent speed and accuracy.1-It is important to understand that mathematics is only a tool for quantitatively describing physical phenomena, it cannot replace physical understanding.2-It is claimed that mathematics is the language of physics, but the reverse is also true, physics can be the language of mathematics as in the case of numerical integration and the derivation of the normal/ Gaussian distribution law via the  statistical  B-matrix chains.However, in a revolutionary technique, chains of B matrices are used to solve numerically PDE, double and triple integrals  as well as the general case of time-dependent partial differential equations with arbitrary Dirichlet BC and initial arbitrary conditions.At first,I=∫∫∫ f(x,y,z) dxdydzwhich has been defined as the limit of the sum of the product f(x,y,z) dxdydz for a small infinitesimal dx,dy,dz is completely ignored in numerical statistical methods as if it never existed.It is obvious that the new B matrix technique ignores the classical 3D integration I=∫∫∫ f(x,y,z) dxdydz.We concentrate below on some results in the field of numerical integration via the theory of the matrix B, which in itself is not complicated but rather long.
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7 Free Nodes:
Single finite Integral
I=∫ f(x) dx ... for  a<=x<=b
Briefly, we arrive at,
The statistical integration formula for 7 nodes is given by,
I=6h/77(6.Y1 +11.Y2 + 14.Y3+15.Y4 +14.Y5 + 11.Y6 + 6.Y7)
which is the statistical equivalence of Simpson's rule for 7 nodes.
Now consider the special case,
I=∫ y dx from x=2 to x=8 where y=X^2.
That is,
X = 2 3   4  5   6   7  8
Y = 4 9 16 25 36 49 64
Numerical result via Trapezoidal ruler,
It = Y1/2+Y2+Y3+Y4 +Y5+Y6+Y7/22+9+16 +25+ 36+ 49+32=169 square units.
Analytic integration expression I=X^3/3
Ia=(384-8)/3= 168 square units.
Finally, the statistical integration formula for 7 nodes is given by,
Is=6 h/77 (6*4 +11* 9 + 14* 16+15* 25 +14* 36 + 11*49 + 6*64)
I = 167.455 square units. This means that static integration is quite fast and accurate.
CASE B:
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Double finite Integral  I=∫∫ f(x,y) dx dy... for the domain  a<=x<=b and  c<=y<=d
If we introduce a specific example without loss of generality where ,
the function Z=f(x,y) is defined as,
Z(x,y)= X^2.Y^2 + X^3 . . . . . (1)
defined on the rectangular domain [abcd],
1<=x=>3 and 1<=y=>3 . . . Domain D(1)
The process of double numerical integration (I),
I=∫∫ f(x,y) dxdy
on the D1 domain can be achieved via three different approaches, namely,
1-analytically (a),
Ia=(x^3/3 *y^2 +x^4/4) + (x^2*y^3/3+x^3) . . . (2)
2-Rule of the Double Sympson (ds),
I ds=
h^3.(16f(b+a/2,d+c/2)+4f(b+a/2,d)+4f(b+a/2,c)+4f(b,d+c/ 2) +4f(a,d+c/2)+f(b,d)+f(b,c)+f(a,d)+f(a,c))/36 . . . . (3)
iii- The statistical integration formula via the Cairo technique (ct),
Ict = 9h^3/29.5( 2.75Z(1,1)+3.5Z(1,2)+2.75.Z(1,3)+3.5Z(2,1)+4.5Z(2,2)+ 3 .5Z(2.3)+2.75Z(3.1)+3.5Z(3.2)+2.75Z(3.3)) . . . . (4)
where h is the equidistant interval on the x and y axes.
The numerical results are as follows,
i- Ia =227.25
ii-Ids=226.5
iii-I ct=227.035 ..which is the most accurate.
CASE C:
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Triple finite Integral and Hypercube
I=∫∫∫ f(x,y) dx dy dz... for the domain  a<=x<=b and  c<=y<=d & e<=z<=f.
Again, we present the nature and the matrix of the triple integration on the cube abcdefgh, divided into 27 equidistant nodes,
we present below the supposed nature matrix of the triple integration on the 3D cube abcdefgh (1,1,1 & 2,1,1, .......& 3,3,3)
I=∫∫∫ W(x,y,z) dxdydz
on the cube domain.
I = 27h^4/59( 2.555W(1,1,1)+3.13W(1,2,1)+2.555.W(1,3,1)+3.13W(2,1,1)+3.876 W(2,2,1)+ 3,13W(2,3,1)+2,555W(3,1,1)+2,555W(3,2,1)+3,13Z(3,3,1) . . . etc.)
The question arises, why are statistical forms of integration faster and more accurate than mathematical forms?
We assume that the answer is inherent in the processes of integration, whether they belong to the 3D geometric space or the unitary x-t space.
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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.
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The Irretutable Argument for Universal Causality. Any Opposing Position?
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The congruent number problem has been a fascinating topic in number theory for centuries, and it continues to inspire research and exploration today. The problem asks whether a given positive integer can be the area of a right-angled triangle with rational sides. While this problem has been extensively studied, it is not yet fully understood, and mathematicians continue to search for new insights and solutions.
In recent years, there has been increasing interest in generalizing the congruent number problem to other mathematical objects. Some examples of such generalizations include the elliptic curve congruent number problem, which asks for the existence of rational points on certain elliptic curves related to congruent numbers, and the theta-congruent number problem as a variant, which considers the possibility of finding fixed-angled triangles with rational sides.
However, it is worth noting that not all generalizations of the congruent number problem are equally fruitful or meaningful. For example, one might consider generalizing the problem to arbitrary objects, but such a generalization would likely be too broad to be useful in practice.
Therefore, the natural question arises: what is the most fruitful and meaningful generalization of the congruent number problem to other mathematical objects? Any ideas are welcome.
here some articles
M. Fujiwara, θ-congruent numbers, in: Number Theory, Eger, 1996, de Gruyter, Berlin, 1998,pp. 235–241.
New generalizations of congruent numbers
Tsubasa Ochiai
DOI:10.1016/j.jnt.2018.05.003
A GENERALIZATION OF THE CONGRUENT NUMBER PROBLEM
LARRY ROLEN
Is the Arabic book about the congruent number problem cited correctly in the references? If anyone has any idea where I can find the Arabic version, it will be helpful. The link to the book is https://www.qdl.qa/العربية/archive/81055/vdc_100025652531.0x000005.
EDIT1:
I will present a family of elliptic curves in the same spirit as the congruent number elliptic curves.
This family exhibits similar patterns as the congruent number elliptic curves, including the property that the integer is still "congruent" if we take its square-free part, and there is evidence for a connection between congruence and positive rank (as seen in the congruent cases of \$n=5,6,7\$).
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Thank you, Irshad Ayoob . I need to rephrase my question. What I mean by a generalization of the congruent number is as follows: A congruent number is related to the area of a right triangle or, simply, to the Diophantine equation a^2 + b^2 = c^2. An integer n is congruent if 2n = ab. Historically, this was not the first definition of a congruent number. Instead, in Arab manuscripts, n is congruent if the two Diophantine equations v^2 - n = u^2 and v^2 + n = w^2 have simultaneously a solution. By the way, this is equivalent to the well-known definition of a congruent number today, which is linked to the right triangle.
Now, my remark is about the degree two Diophantine equation. For example, let's take a^2 + 2b^2 = c^2. We know that if this Diophantine equation (or any other degree two of the form ra^2 + sb^2 = t*c^2, where a, b, c, r, s, t are integers) has a non-trivial solution, it will have an infinite number of solutions. So, in the case of the Pythagorean triple, we have the definition of the congruent number. But for the other equations, what is the correct definition of a congruent number?
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Insistence on mathematical continuity in nature is a mere idealization. It expects nature to obey our idealization. This is what happens in all physical and cosmological (and of course other) sciences as long as they use mathematical idealizations to represent existent objects / processes.
But mathematically following nature in whatever it is in its part-processes is a different procedure in science and philosophy (and even in the arts and humanities). This theoretical attitude accepts the existence of processual entities as what they are.
This theoretical attitude accepts in a highly generalized manner that
(1) mathematical continuity (in any theory and in terms of any amount of axiomatization of physical theories) is not totally realizable in nature as a whole and in its parts: because the necessity of mathematical approval in such a cosmology falls short miserably,
(2) absolute discreteness (even QM type, based on the Planck constant) in the physical cosmos (not in non-quantifiable “possible worlds”) and its parts is a mere commonsense compartmentalization from the "epistemology of piecemeal thinking": because the aspect of the causally processual connection between any two quanta is logically and mathematically alienated in the physical theory of Planck’s constant, and
(3) hence, the only viable and thus the most reasonably generalizable manner of being of the physical cosmos and of biological entities is that of CAUSAL CONTINUITY BETWEEN PARTIALLY DISCRETE PROCESSUAL OBJECTS.
PHYSICS and COSMOLOGY even today tend to make the cosmos mathematically either continuous or defectively discrete or statistically oriented to merely epistemically probabilistic decisions and determinations.
Can anyone suggest here the existence of a different sort of physics and cosmology that one may have witnessed until today? A topology and mereology of CAUSAL CONTINUITY BETWEEN PARTIALLY DISCRETE PROCESSUAL OBJECTS, fully free of discreteness-oriented category theory and functional analysis, is yet to be born.
Hence, causality in its deep roots in the very concept of To Be is alien to physics and cosmology till today.
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Humans in this age, by inventing and using mathematical models and trying to match them with natural phenomena, are free to know the world. The mentioned models have their own logic and causality! And experience has shown that they do not have a significant relationship with nature! And there will always be an inevitable distance between our models of nature, the motivation to reduce this distance may be another request for scientific efforts!
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i 'red' in a maths popularization book of steven strogatz that 1+3=4, 1+3+5=9, 1+3+5+7=16, and so on; wich would be the hypothesis when trying to demonstrate this striking 'fact'?
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in his famous 'principia', newton uses mathematical considerations to demonstrate the angular momentum conservation; in the 'first' triangle one of the side is the velocity--its value, so a number--, and the other two sides are the 'distances' at t and t+1, respectively; now, if the ares are the same, could we say that theese two sides express the flow of the time
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Our response is YES. Quantum computing has arrived, as an expression of that.
Numbers do obey a physical law. Massachusetts Institute of Technology Peter Shor was the first to say it, in 1994 [cf. 1], in modern times. It is a wormhole, connecting physics with mathematics, and has existed even before the Earth existed.
So-called "pure" mathematics is, after all, governed by objective laws. The Max Planck Institute of Quantum Optics (MPQ) showed the mathematical basis by recognizing the differentiation of discontinuous functions [1, 2, 3], in 1982.
This denies any type of square-root of a negative number [4] -- a.k.a. an imaginary number -- rational or continuous.
Complex numbers, of any type, are not objective and are not part of a quantum description, as said first by Erwin Schrödinger (1926) --
yet,
cryogenic behemoth quantum machines (see figure) consider a "complex qubit" -- two objective impossibilities. They are just poor physics and expensive analog experiments in these pioneering times.
Quantum computing is ... natural. Atoms do it all the time, and the human brain (based on +4 quantum properties of numbers).
Each point, in a quantum reality, is a point ... not continuous. So, reality is grainy, everywhere. Ontically.
To imagine a continuous point is to imagine a "mathematical paint" without atoms. Take a good microscope ... atoms appear!
The atoms, an objective reality, imply a graininess. This quantum description includes at least, necessarily (Einstein, 1917), three logical states -- with stimulated emission, absorption, and emission. Further states are possible, as in measured superradiance.
Mathematical complex numbers or mathematical real-numbers do not describe objective reality. They are continuous, without atoms. Poor math and poor physics.
It is easy to see that multiplication or division "infests" the real part with the imaginary part, and in calculating modulus -- e.g., in the polar representation as well as in the (x,y) rectangular representation. The Euler identity is a fiction, as it trigonometrically mixes types ... avoid it. The FFT will no longer have to use it, and FT=FFT.
The complex number system "infests" the real part with the imaginary part, even for Gaussian numbers, and this is well-known in third-degree polynomials.
Complex numbers, of any type, must be deprecated, they do not represent an objective sense. They should not "infest" quantum computing.
Quantum computing is better without complex numbers. software makes R,C=Q --> B={0,1}.
What is your qualified opinion?
REFERENCES
[1] DOI /2227-7390/11/1/68
[3] June 1982, Physical review A, Atomic, molecular, and optical physics 26:1(1).
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Can numbers obey a physical law?
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The situation is exactly the opposite - physical laws obey numbers. Otherwise, you will have to believe some pseudo-scientific statements that somehow caught my eye that a few hundred million years ago the number pi was exactly 3. Like, the Earth was closer to the Sun ... You are saying something similar here. (But, not about Betelgeuse...)
Speaking of prime numbers. In the picture, the spider has 8 legs. If you do not notice 3 legs, then yes! - Exactly 5!
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Physics
The physicist betting that space-time isn't quantum after all
Most experts think we have to tweak general relativity to fit with quantum theory. Physicist Jonathan Oppenheim isn't so sure, which is why he’s made a 5000:1 bet that gravity isn’t a quantum force
By Joshua Howgego
13 March 2023
📷
Nabil NEZZAR
JONATHAN OPPENHEIM likes the occasional flutter, but the object of his interest is a little more rarefied than horse racing or the one-armed bandit. A quantum physicist at University College London, Oppenheim likes to make bets on the fundamental nature of reality – and his latest concerns space-time itself.
The two great theories of physics are fundamentally at odds. In one corner, you have general relativity, which says that gravity is the result of mass warping space-time, envisaged as a kind of stretchy sheet. In the other, there is quantum theory, which explains the subatomic world and holds that all matter and energy comes in tiny, discrete chunks. Put them together and you could describe much of reality. The only problem is that you can’t put them together: the grainy mathematics of quantum theory and the smooth description of space-time don’t mesh.
Most physicists reckon the solution is to “quantise” gravity, or to show how space-time comes in tiny quanta, like the three other forces of nature. In effect, that means tweaking general relativity so it fits into the quantum mould, a task that has occupied researchers for almost a century already. But Oppenheim wonders if this assumption might be mistaken, which is why he made a 5000:1 bet that space-time isn’t ultimately quantum.
The quantum experiment that could prove reality doesn't exist
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A special spammer again appeared after SS post above, in this case again aimed at to place his “last post in the thread”, which are indicated in the RG rather useful “– Science topic” options, which are indicated below threads’ questions; an example here are “Space Time”, “Quantum”, “QUANTA”, “Mathematics” – Science topic” options, aimed at to replace really scientific answers; what this spammer does thoroughly in many threads.
Since he has only full stop imagination about what physics [and other sciences, though] is, besides that there are some terribly scientific words in physics, his posts are some senseless sets of such words – as that this post is; and, at that, he recommended the replaced SS post, seems knowing that his recommendation of any post means that this post is some trash.
As to
“…Quantum mechanics is but a language, and like a language it contains contradictory propositions. (These propositions can be precise and "work", but that's not the point.) …”
- quantum mechanics is the theory just as that classical mechanics quite equally is. And has indeed a lot of really fundamental flaws – again, many of the same flaws the classic mechanics has also.
These flaws exist because of in mainstream philosophy and sciences, including physics, all really fundamental phenomena/notions, first of all in physics “Matter”– and so everything in Matter, i.e. “particles”, “fields”, etc., “Consciousness”, “Space”, “Time”, “Energy”, “Information”, are fundamentally completely transcendent/uncertain/irrational,
- so in every case, when the mainstream physics addresses to some really fundamental problem, the results completely obligatorily logically are nothing else than some transcendent fantastic mental constructions; in both cases
- when some authors attempt to solve some directly fundamental problems, and some fantastic and fundamentally non-testable experimentally , say, “string” one, theories” appear in physics, and
- when a theory describes experimentally observed material objects/systems/events/effects, in this case, say, QED, are fitted with experiments, but for that really completely ad hoc – and really wrong – mathematical tricks are used; etc.
Real fundamental physics development can be possible only provided that the fundamental phenomena/notions above are scientifically defined, what is possible, and is done, only in framework of the 2007 Shevchenko-Tokarevsky’s “The Information as Absolute” conception, recent version of the basic paper see
- and practically for sure in many cases basing on the SS&VT Tokarevsky’s informational physical model , which is based on the conception; two main paper are
https://www.researchgate.net/publication/355361749_The_informational_physical_model_and_fundamental_problems_in_physics, where yet now more 30 fundamental physical problems are either solved or essentially clarified.
Including in last link [mostly section “Conclusion”] the fundamental flaws of both mechanicses are discussed.
However that
“….General relativity is a genuine theory. Both can't be unified until we have a proper quantum theory...”
- is fundamentally incorrect. In the GR the author addressed to completely transcendent for him fundamental phenomena/notions above, and so postulated for these phenomena really completely transcendent and fundamentally non-adequate to the reality properties; etc.; more see the SS post 5 days ago now in https://www.researchgate.net/post/Do_you_think_that_general_relativity_needs_modifications_or_it_is_a_perfect_theory/138 ,
- here only note that really Gravity is, of course, fundamentally nothing else than some fundamental Nature force, as that Electric, Strong/Nuclear, and Weak Forces are; and with a well non-zero probability it acts as that is shown in the SS&VT initial models of the Forces – see https://www.researchgate.net/publication/365437307_The_informational_model_-_Gravity_and_Electric_Forces, and
Cheers
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Hi, My name Is Debi, I'm master student Mathematics Education major at Yogyakarta State University 2nd month. Please give me advice what the trend topic on mathematics education aspecially topic learning media math and learning psychology of math. May you share with me about it on your country or your universisty. Thank you so much.
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1. Logic in school mathematics: norm and reality.
2. Why do we need proofs in mathematics teaching?
3. Proof and understanding in mathematics.
4. Free and bound variables in mathematics and school mathematics.
5. After all, what does it mean to solve an equation?
6. What is inequality and what does it mean to solve inequality?
7. What is a definition and how should it be applied.
8. Teaching mathematics: proving VS storytelling.
9. The concept of teaching mathematics (according to A. Naziev).
10. Correction of errors in traditional formulations of Vieta's theorem and related theorems.
11. Moral education: in the searches of lost mathematics.
12. Name, denotation and sense in mathematics and teaching it.
13. Semantic reading in teaching mathematics.
14. Teaching mathematics on the base of axioms, definitions, and theorems.
15. Stereometric problems on combinations of figures.
16. Solving cross-section problems with assistance of LaTeX.
17. Solving problems on the representation of shadows using LaTeX.
18. Representation of combinations of spatial figures by means of LaTeX.
19. Evidence-based solution of problems on the transformation of graphs of functions.
20. Quantifiers in mathematics and mathematics teaching.
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This is actually a trivial question and I'm just being mischievous.
It turns on the shades of meaning of both "idea" and "exist."
Mathematically, a concept exists whether anyone has happened upon it or not. (A meaningless attempt at a concept is not a concept).
When first thought about by an actual brain of any kind, a concept acquires its first glimmer of existence as in the real world.
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Mathematically speaking, whether an 'idea' exists before it can be 'had', relates to the question of whether mathematical concepts exist a priori, leaving them to be 'discovered', or whether the mind must piece them together, leaving them to be 'invented'.
For those interested in the distinction 'invention' and 'discovery' in this context, Jacques Hadamard has published interesting views to this matter in his booklet "The Psychology of Invention in the Mathematical Field" (1954).
Thanks Karl Sipfle for this non-trivial question.
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Physics is a scence of representations, with mathematical aspects in them, foremost and not of naked correlations and parameter analysis.
It also has competent conceptualizaions, genious principles.
Even the innocent seeking uniform motion is a representational scheme fo motions under the theory of kinematics. (Representations are seperate from reality but are invaluable part of scientific infering, predicting, explaining etc) i.e heat is represented as a flow between subsystems. Representations change i.e Einstein found the curved spacetime one for gravity phenomena.
Physics is also the science f cosmology. It has no meaning if it bypasses the universe-i.e the sum of subsystems. This discipline has problems because we cannot take ourself out of it and study it but physics has tools for this (QM) or theoretical approximaions (more cognitively open consideration of the concept of boundary conditions).
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If want to know about the big things that are the sum how many things act, you need to know how many things act. Following is just a few concepts that need changing and my proofs of the changes:
The proofs that 4 constants ae changed by relativistic velocities as a PDF file will pop up when the following link is Left clicked: -------------------------------------------------------------------- https://drive.google.com/file/d/1e1ExWG-VyTR8PAPxU5OnfSzd86uj59nh/view?usp=share_link --------------------------------------------------------------------- A link to prerequisite proofs that all Doppler shifts change time and distance (axial, gravitational and transverse not just the transverse). -------------------------------------------------------------------- https://drive.google.com/file/d/1vGRBH1AgUOCP8_zp7fKxBTMPg-YP_-uh/view?usp=share_link -------------------------------------------------------------------- A link to: Proof of a version of the Schrodinger equation for relativistic velocities: ------------------------------------------------------------------ https://drive.google.com/file/d/1kh2d4fYFOd8rbS6tgyUbTA5zDZW-aUNH/view?usp=share_link-------------------------------------------------------------------------------------------- I hope you can make use of the above. Samuel Lewis Reich (sLrch53@gmail.com)
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Is it possible to mathematically calculate K-40 from K total determined by ICP MS in sediment samples?
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See Wikipedia for Potassium-40, abundance.
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We assume that this statement is false, but one of the most common mathematical errors.
So a question arises: what is the importance of the LHS diagonal?
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Important or not? As a question it is too general, no definite answer could be given.
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I have two networks, and wish to get them to dynamically interact with one another, yet retain modularity.
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Hello Faizan Rao, to dynamically join two separate database schemas while retaining modularity, you can use a mathematical layer such as a view or a stored procedure to perform the interaction. This method allows you to keep the database schemas separate while still enabling them to communicate and share data.
Here's a step-by-step guide on how to do this:
Identify the shared data: Determine the common data points between the two schemas that need to be combined or interact with each other. This could be a common key, attribute, or any other data that can be used to relate the two schemas.
Create a view or stored procedure: Depending on your database system (MySQL, PostgreSQL, SQL Server, etc.), you can create a view or a stored procedure that performs the necessary calculations or data manipulations. This will act as the mathematical layer between the two schemas. This layer will query data from both schemas and perform the required calculations, transformations, or aggregations.
For example, in SQL Server, you can create a view like this:
CREATE VIEW CombinedData AS
SELECT a.*, b.*
FROM Schema1.Table1 a
JOIN Schema2.Table2 b
ON a.CommonKey = b.CommonKey;
In this example, Schema1.Table1 and Schema2.Table2 represent tables from the two separate schemas, and CommonKey is the column used to relate the data.
Access the view or stored procedure: Whenever you need to access the combined data, you can query the view or execute the stored procedure to fetch the results. This ensures that the two schemas remain separate, but the data can be combined and accessed dynamically as needed.
Maintain modularity: By using a view or stored procedure, you can keep the two schemas separate and modular. When updates are needed, you can make changes to the individual schemas without affecting the other. The view or stored procedure can then be updated to accommodate the changes. A mathematical layer in the form of a view or a stored procedure can help you dynamically join two separate database schemas while retaining modularity. This method ensures that the schemas remain independent and can be maintained separately, while still allowing for the interaction and sharing of data between them.
Regards,
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Category theory is a branch of mathematics that deals with the abstract structure of mathematical concepts and their relationships. While category theory has been applied to various areas of physics, such as quantum mechanics and general relativity, it is currently not clear whether it could serve as the language of a metatheory unifying the description of the laws of physics.
There are several challenges to using category theory as the language of a metatheory for physics. One challenge is that category theory is a highly abstract and general framework, and it is not yet clear how to connect it to the specific details of physical systems and their behaviour. Another challenge is that category theory is still an active area of research, and there are many open questions and debates about how to apply it to different areas of mathematics and science.
Despite these challenges, there are some researchers who believe that category theory could play a role in developing a metatheory for physics. For example, some have proposed that category theory could be used to describe the relationships between different physical theories and to unify them into a single framework. Others have suggested that category theory could be used to study the relationship between space and time in a more unified and conceptual way.
I am very interested in your experiences, opinions and ideas.
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I believe your sentiment is that Category Theory is so general that it might if applied to physics provide an insight into the overarching principles of the science. But Category Theory is not going to give you any insight that you do not already possess. It might provide a convenient notation for expressing that insight. Using Category Theory to navigate physics without a physical understanding would be like sailing a yacht without a keel.
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Is it mathematically justified to place negative and positive numbers on the same plane?
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Goes like this:
e^2pii=1
e^pi=-1
Consequently:-2pi=pi
and thats precisely what I was trying to say
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This question discusses the YES answer. We don't need the √-1.
The complex numbers, using rational numbers (i.e., the Gauss set G) or mathematical real-numbers (the set R), are artificial. Can they be avoided?
Math cannot be in ones head, as explains [1].
To realize the YES answer, one must advance over current knowledge, and may sound strange. But, every path in a complex space must begin and end in a rational number -- anything that can be measured, or produced, must be a rational number. Complex numbers are not needed, physically, as a number. But, in algebra, they are useful.
The YES answer can improve the efficiency in using numbers in calculations, although it is less advantageous in algebra calculations, like in the well-known Gauss identity.
For example, in the FFT [2], there is no need to compute complex functions, or trigonometric functions.
This may lead to further improvement in computation time over the FFT, already providing orders of magnitude improvement in computation time over FT with mathematical real-numbers. Both the FT and the FFT are revealed to be equivalent -- see [2].
I detail this in [3] for comments. Maybe one can build a faster FFT (or, FFFT)?
The answer may also consider further advances into quantum computing?
[2]
Preprint FT = FFT
[2]
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The form z=a+ib is called the rectangular coordinate form of a complex number, that humans have fancied to exist for more than 500 years.
We are showing that is an illusion, see [1].
Quantum mechanics does not, contrary to popular belief, include anything imaginary. All results and probabilities are rational numbers, as we used and published (see ResearchGate) since 1978, see [1].
Everything that is measured or can be constructed is then a rational number, a member of the set Q.
This connects in a 1:1 mapping (isomorphism) to the set Z. From there, one can take out negative numbers and 0, and through an easy isomorphism, connect to the set N and to the set B^n, where B={0,1}.
We reach the domain of digital computers in B={0,1}. That is all a digital computer needs to process -- the set B={0,1}, addition, and encoding, see [1].
The number 0^n=0, and 1^n=1. There Is no need to calculate trigonometric functions, analysis (calculus), or other functions. Mathematics can end in middle-school. We can all follow computers!
REFERENCES
[1] Search online.
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Your feedback is welcome
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@Juan Weisz
incentive here is that e^2pii=1 (Euler's formula)
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I noticed that in some very bad models of neural networks, the value of R² (coefficient of determination) can be negative. That is, the model is so bad that the mean of the data is better than the model.
In linear regression models, the multiple correlation coefficient (R) can be calculated using the root of R². However, this is not possible for a model of neural networks that presents a negative R². In that case, is R mathematically undefined?
I tried calculating the correlation y and y_pred (Pearson), but it is mathematically undefined (division by zero). I am attaching the values.
Obs.: The question is about artificial neural networks.
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Raid, apologies here's the attachment. David Booth
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1 - Prof. Tegmark of MIT hypothesizes that the universe is not merely described by mathematics but IS mathematics.
2 - The Riemann hypothesis applies to the mathematical universe’s space-time, and says its infinite "nontrivial zeros" lie on the vertical line of the complex number plane (on the y-axis of Wick rotation).
3 - Implying infinity=zero, there's no distance in time or space - making superluminal and time travel feasible.
4 - Besides Mobius strips, topological propulsion uses holographic-universe theory to delete the 3rd dimension (and thus distance).
5 - Relationships between living organisms can be explained with scientifically applied mathematics instead of origin of species by biological evolution.
6 - Wick rotation - represented by a circle where the x- and y-axes intersect at its centre, and where real and imaginary numbers rotate counterclockwise between 4 quadrants - introduces the possibility of interaction of the x-axis' ordinary matter and energy with the y-axis' dark matter and dark energy.
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An equivalent formulation of the Riemann hypothesis. See formula (3.22) in the
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Theoretical and computational physics provide the vision and the mathematical and computational framework for understanding and extending the knowledge of particles, forces, space-time, and the universe. A thriving theory program is essential to support current experiments and to identify new directions for high energy physics. Theoretical physicists provide a great deal of assistance to the Energy, Intensity, and Cosmic Frontiers with the in-depth understanding of the underlying theory behind experiments and interpreting the outcomes in context of the theory. Advanced computing tools are necessary for designing, operating, and interpreting experiments and to perform sophisticated scientific simulations that enable discovery in the science drivers and the three experimental frontiers.
source: HEP Theoretical and Computationa... | U.S. DOE Office of Science (SC) (osti.gov)
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Physics, mathematical, and computational sciences have contributed to the betterment of mankind and continue to push innovation and research today because they provide fundamental frameworks for understanding the natural world, developing new technologies, and solving real-world problems.
Physics, for example, provides a fundamental understanding of the laws of nature that govern the behavior of matter and energy, from the smallest particles to the largest structures in the universe. This understanding has led to the development of technologies such as lasers, semiconductors, and superconductors, which have revolutionized communication, computing, and energy production.
Mathematics provides the language and tools for describing the structure of the natural world and for solving problems across a wide range of fields, from engineering to economics. Mathematical models and simulations allow scientists and engineers to study complex systems and make predictions about their behavior, leading to new discoveries and innovations.
Computational science, which combines mathematics, computer science, and domain-specific knowledge, has become increasingly important in recent years due to the explosion of data and the growing complexity of problems in many fields. Computational tools and algorithms are used to simulate physical processes, analyze large data sets, and develop new materials and drugs.
Overall, physics, mathematical, and computational sciences continue to play a critical role in driving innovation and advancing knowledge in many fields, making them essential for the betterment of mankind.
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Irrational numbers are uncomputable with probability one. In that sense, numerical, they do not belong to nature. Animals cannot calculate it, nor humans, nor machines.
But algebra can deal with irrational numbers. Algebra deals with unknowns and indeterminates, exactly.
This would mean that a simple bee or fish can do algebra? No, this means, given the simple expression of their brains, that a higher entity is able to command them to do algebra. The same for humans and machines. We must be able also to do quantum computing, and beyond, also that way.
Thus, no one (animals, humans, extraterrestrials in the NASA search, and machines) is limited by their expressions, and all obey a higher entity, commanding through a network from the top down -- which entity we call God, and Jesus called Father.
This means that God holds all the dice. That also means that we can learn by mimicking nature. Even a wasp can teach us the medicinal properties of a passion fruit flower to lower aggression. Animals, no surprise, can self-medicate, knowing no biology or chemistry.
There is, then, no “personal” sense of algebra. It just is a combination of arithmetic operations.There is no “algebra in my sense” -- there is only one sense, the one mathematical sense that has made sense physically, for ages. I do not feel free to change it, and did not.
But we can reveal new facets of it. In that, we have already revealed several exact algebraic expressions for irrational numbers. Of course, the task is not even enumerable, but it is worth compiling, for the weary traveler. Any suggestions are welcome.
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@Ed Gerck
Irrational numbers are uncomputable with probability one
================================================= ===
My deepest apologies, but I have read your Answer dated December 14, 2022 in the FLT thread https://www.researchgate.net/post/Are-there-other-pieces-of-information-about-Victory-Road-to - FLT#view=641367549777ccc70c026256/234 .
There was a link to your own thread given by you. This thread gives your erroneous statement from the very beginning, namely: "Irrational numbers are uncomputable with probability one".
Please agree, Dear Professor Ed G., that any irrational number is calculated with 100% accuracy with a probability of 1 for any number of orders p-1, if you write down p orders. Thus, if you write for the root of 2 one order before point and three orders after point, you will have sqrt(2)=1.414..., i.e., you can consider that you have written 4 orders. At the same time, the accuracy of 100% with a probability of 1 is provided for 3 orders, i.e. 1.41, etc., for any number of orders...
Speaking of some kind of all orders "full notation" , as you would like to see it, it's not possible for such a representation of irrational numbers.
If you point out my mistake to me, I will be grateful.
Greetings,
SPK
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What is this mean ( ± 0.06) and How can I calculate it mathematically?
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Standard error = s / √n
where:
• s: sample standard deviation
• n: sample size
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All tests doing a proof for the Riemann-Hypothesis on the Zeta-Function must fail.
There are no zeros by the so called function of a complex argument.
A function on two different units f(x, y) only then has values for the third unit
`z´ [z = f(x, y)]
if the values variables `x´ and `y´ would be combined by an algebraic rule.
So it should be done for the complex argument, Riemann had used.
But there isn´t such a combination. So Riemann only did a `scaling´. Where both parts of the complex number stay separate.
The second part of the refutation comes by showing wrong expert opinion of mathematics. This is on the false use of `imaginary´ and `prefixed multiplication´.
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The imaginary compartmentalization does not remain of the same dimension all throughout the processualization of complex functions . Even self-determination is ensconced in a sphere where the focus is not on mathematical rigor but rather on collecting some bits of data on the functionals [ of the originary function ] on the very powerful machinery of manifolds and “post-Newtonian calculus”. The systematics of functional differential equations does not have a continuously differentiable solution for every value of the parameter , say , a .
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What are the properties of transversal risks in networks? Happy for applied examples and diffusion properties.
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Transversal risks in networks are risks that cross multiple nodes or elements of a network, rather than being confined to a single node. These risks can have a significant impact on the network as a whole and can be difficult to manage and mitigate.
Some properties of transversal risks in networks include:
1. Diffusion: Transversal risks have the potential to spread rapidly through a network, affecting multiple nodes and elements. The speed and extent of diffusion depend on factors such as the topology of the network, the connectivity between nodes, and the nature of the risk itself.
2. Interconnectivity: Transversal risks often arise from the interconnectivity between nodes or elements in a network. The risk can be amplified when the interconnectivity is high, and nodes or elements are highly dependent on one another.
3. Cascading effects: Transversal risks can trigger cascading effects, leading to a chain reaction of failures or disruptions across the network. These cascading effects can be difficult to predict and control.
4. Non-linearity: Transversal risks often exhibit non-linear behavior, meaning that the impact of the risk is not proportional to the size or severity of the risk. Small disruptions can lead to large and unexpected consequences.
Applied examples of transversal risks in networks include:
1. Cybersecurity: Cyber attacks can spread through computer networks, affecting multiple nodes and elements. A single attack can lead to cascading effects, disrupting critical systems and services.
2. Supply chain disruptions: Disruptions in one part of a supply chain can affect multiple nodes and elements downstream, leading to inventory shortages, delays, and other disruptions.
3. Financial contagion: Financial risks can spread through interconnected financial institutions, leading to a systemic crisis that affects the broader economy.
4. Disease outbreaks: Diseases can spread through social networks, leading to large-scale epidemics that affect multiple regions and populations.
Overall, the properties of transversal risks in networks highlight the importance of understanding the interconnectivity and complexity of modern systems and networks. Effective management of transversal risks requires a holistic approach that considers the entire network and its interdependencies.
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Project Name - Improving Achievement and Attitude through Co-operative learning in F. Y. B. Sc. Mathematics Class
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What is missing is an exact definition of probability that would contain time as a dimensionless quantity woven into a 3D geometric physical space.
It should be mentioned that the current definition of probability as the relative frequency of successful trials is primitive and contains no time.
On the other hand, the quantum mechanical definition of the probability density as,
p(r,t)=ψ(r,t)*.ψ(r,t),
which introduces time via the system's destination time and not from its start time is of limited usefulness and leads to unnecessary complications.
It's just a sarcastic definition.
It should be mentioned that a preliminary definition of the probability function of space and time proposed in the Cairo technique led to revolutionary solutions of time-dependent partial differential equations, integration and differentiation, special functions such as the Gamma function, etc. without the use of mathematics.
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The theory of the interacting gases as the electrons in solids is still
in infancy. Many areas of physics need more development. Cosmology is still under uncertainties. Particle physics is hard, and in infancy.
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Hello
I have an Excel file containing weather data of Missouri in U.S. The data starts from 25th July and ends on 9th September in 2014. For each day, almost 21 times data has been recorded (6 hours within solar noon time).
How can I make a type99 source file using this Excel file? I already have studied mathematical reference of Trnsys help, but that was not very helpful. Thanks
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No, unfortunately I didn't find any solution for this problem.
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The Gamma function,
G(n)= Integral from 0 to infinity [Exp(e^-x^n)]dx
is of the great mathematical and physical importance.
It can be calculated without numerical integration (for practical purposes) via its mathematical and physical properties:
i-minimum of Gamma occurs at x = 1.4616321 and the corresponding value of Gamma(x) is 0.8856032.
ii-Gamma(1.)=Gamma(2.)=1.
iii-Gamma(x)=(x-1.) !
A simple preliminary approach that gives the value of Gamma(x) with an error less than 0.001 is the second-order polynomial expression for the factorial x,
(1.-0.46163*x+0.46163*x*x),x element of [0,1].
For example, this gives:
G(10.5)=11877478.
vs the value of 11899423.084) given by numerical integration.
and Gamma(1.4616)= 0.88527 vs 0.8856032.
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The Gamma function G(n) is well defined for any positive value of n,
G(n)= Integral from 0 to infinity [Exp(e^-x^n)]dx . . . . . (1)
Needless to say, it is of great mathematical and physical importance.
However, for practical purposes, it can be calculated using an adequate closed-form polynomial without going through complicated numerical integration.
The required closed form solution must retain its mathematical and physical properties, namely:
i-minimum of Gamma occurs at x = 1.4616321 and the corresponding value of Gamma(x) is 0.8856032.
ii-Gamma(1.)=Gamma(2.)=1.
iii-Gamma(x)=(x-1.) !
iv-The recurrence relation ,Gamma (x)=x. Gamma (x-1)
We propose a simple preliminary approach that gives the required value of Gamma(x) with an error less than 0.001. The proposed preliminary approach must satisfy conditions i-iv,
but since the factorial function x! is not yet defined for negative numbers (x<0), we divide the entire positive x-space into three intervals as follows:
a) x element of ]0.1]
Here, the proposed second-order polynomial expression for the Gamma function is G(x)=F(x-1) where F(x) is the factorial function x!. F is approximated by,
F(x)=(1.-0.46163*x+0.46163*x*x) . . . . . . . (2)
x element of [0,1].
b) x element of [1,2]
The Gamma function is approximated via the expression,
G(x)=Done(x)+0.3333/X**1.5 . . . . . . . . . .  (3)
Where 1/3 *1/X. Sqrt(x) is a correction factor.
c) x element of [0,infinity[
We can here use the expression (4) supplemented by the expression (2) for the remaining fraction,
G(x)=F(x-1) . . . . . . . . . . . . . . . . . . . . .. . . . (4)
Equations 2, 3 and 4 were implemented in a simple algorithm which produced the required numerical results.
Table I presents some examples of numerical results of the proposed technique compared to those of the numerical tables obtained by numerical integration of Eq. 1.
Table I. results of the proposed method vs those of the numerical tables obtained by numerical integration of Eq. 1.
x                          10.5                     1.4616                      0.5                     0.25
G(x) Proposed    11877478           0.88527            1.82646                    3.5798
technique
G(x) numerical   11899423.084 0.8856032.       1.7738                      3.6534
tables
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There are a few point to consider in this issue
Points pro current emphasis
1. Math is the backbone of a physical theory. Good representation, good quantities of a theory, phenomena but bad math makes for bad theory
2. There is a general skepticism for reconsidering role of mathematized approach in physics Masters syllabi/upgrading role of literature/essay
2. Humans communicate, learn, think & develop construct via language
Arguments Con
1. Math is the elements in theory and "physics product" that is responsible for precision& prediction. Indespensible though, it exists in the mind of some individuals & function as well, in parallel with conception, physical arguments
2. Not all models in physics are mathematical. Some are conceptual
3. Formulations of solutions to physics problems via math techniques and methods is def of mathematical physics. However, this is a certain % of domain of skills.
But syllabus focuses 100% on this
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In general physics is the science that observe a phenomenon, gives it a name and describes its properties. And because we can only observe phenomena – humans are phenomena too – we actually describe the mutual relations between the phenomena. The relations are expressed with the help of measurement units (SI system) so we don’t have to compare multiple phenomena to get an answer. We just use the units as a “measuring interface” between all the distinguishable phenomena.
Now where is the math?
When I was young engineers had books full of formulas. To calculate steam turbines, electric motors, mechanical constructions made of concrete, steel, wood, etc., etc. Nobody said that these formulas are mathematical formulas. Because we only need arithmetic to calculate the mutual relations, not math. Math is something else.
So what is math? The problem is that nobody knows the real answer. The ancient Greek philosophers had the opinion that our universe is a mathematical existence. That means that the primary properties of our universe are dynamical geometrical properties. An idea that shows some similarity with the general concept in modern Quantum Field Theory (there are no particles, there are only fields). Unfortunately, in physics we can only determine the mutual relations between the phenomena, we don’t know “what’s inside”. So the ancient Greek philosophers may be right, but modern physics is still too limited to make statements about the subject that are convincing. And more worse, in mathematics there is also the search for the unifying theory… For example, probability theory has no mathematical foundation, it is an empirical theory.
Studying theoretical physics has not much to do with pure math itself, in physics we use math as "arithmetic" tools. However, in pure mathematical physics the situation is different. Mathematical physics use mathematical models to “simulate” physical reality. In other words, they build up physical reality “from scratch” and the model is constructed with the help of “facts” that originate from physics, mathematics and philosophy. Because before we can construct a model we have to create some kind of a conceptual framework.
An example… There is a nice video about mathematical physics, the Causal Dynamical Triangulations approach (https://www.youtube.com/watch?v=PRyo_ee2r0U). They use the Planck units and Einstein’s curved spacetime (therefore the math is directly related with the Ricci scalar curvature in Riemannian geometry because they have to implement the universal scalar field, the Higgs field, in the model). So if this is the way you want to do physics, try to become a mathematician too.
With kind regards, Sydney
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Dear professors and students, greetings and courtesy. I wanted to know if the real numbers are the largest and the last set of numbers that exist, or if there are sets or sets of numbers that are larger than that, but maybe they have not been discovered yet? Which is true? If it is the last set of numbers that exists, what theorem proves the non-existence of a set of numbers greater than it? And if there is a larger set than that, in terms of the history of mathematics, by obtaining the answer to which mathematical problem, it was proved that the obtained answer is not closed with respect to the set of complex numbers and belongs to a larger set? Thank you very much
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First, what do you mean by a number? If you mean a set of things that includes, say, rational numbers and extends the addition & multiplication operations in a way that is consistent with the usual rules (e.g., associativity: a+(b+c)=(a+b)+c & a*(b*c)=(a*b)*c, distributivity: a*(b+c)=a*b+b*c, zeros: 0+a=a, 0*a=0, commutativity: a+b=b+a, a*b=b*a, unit/identity: 1*a = a, inverses: every a has a b=-a where a+b=0, every a not zero has a b=1/a where a*b=1) then we can make a claim that the complex numbers are the biggest class of "numbers".
Actually, you can go further by constructing a non-Archimedean set of "real numbers" which extends real numbers and is still an ordered field, and then have an extended set of complex numbers of the form a+i.b where a, b are these extended real numbers.
Non-Archimedean means that there are "numbers" a, b > 0 where a/b is larger than any whole number 1, 2, 3, ...
But if you keep the regular real numbers, but are willing to lose something else like commutativity of multiplication (a*b=b*a) then there are the quaternions discovered/invented by William Rowan Hamilton in the 19th century. These have the form a+b.i +c.j +d.k where a, b, c, d are real numbers. The essential properties of the symbols i, j, k are that i*i = j*j = k*k = -1, and i*j = -j*i = k, j*k = -k*j = i, & k*i = -i*k = j. Quaternions form a division ring.
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As the concept comes from the Bernoulli numbers and different branches of mathematics, I have recently considered the importance of introducing the same concept, 'The unity of mathematics' within the context of the Bernoulli numbers and some special series (the Flint Hills and Cookson Hills series). I believe in the scenario of defining a balanced relationship between the effect of the Bernoulli numbers and the series of hard convergence.
I am pointing out this potential link.
For a general conclusion about what I consider should the concept of 'unity' by the Bernoulli numbers and the Flint Hills, just pay attention to this screenshot:
DOI: 10.13140/RG.2.2.16745.98402
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I dont see very well how you could achieve the "unity of mathematics"
based on series/sequences alone. For example geometry would remain apart. you must persue some more modest goal.
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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?
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@ Andrea Ossicini,
Interesting - thanks for sharing
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Mathematics Teacher Educators (MTEs) best practices.
I'm interesting in research literature about Mathematics Teacher Educators (MTEs) best practices, especially on MTEs' practices for teaching to solve problems.
thank you.
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Problem solving technique and thinking aloud strategies helpful for mathematics teaching
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Good day, Dear Colleagues!
Anyone interested in discussing this topic?
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The tandem of Mathematics and Computer Science is vital for the development of AI. Mathematics provides the theoretical foundation, while Computer Science provides the practical tools to implement AI algorithms. By combining these two disciplines, researchers and practitioners can create AI systems that can perform increasingly complex tasks, and help solve some of the most challenging problems facing our society.
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How can I define histogram bins in a well define mathematical expression especially driven from data points x_i, i=1,..,n and the range or any other well define measures in the dataset.
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Dear Dr Babura,
I think that you mean how Sturges came up with his rule. Part of your question is explained in the following link,
Please let me know if this is what you are looking for Dr.
Best wishes.
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Kindly share with me any details of Scopus indexed Mathematics conferences in India.
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See attached file
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Physics continues a tradition of assesment in graduate program based on final exams and of the form of mathematized exersices with no conceptual qs or essays.
This fulfils the aim. Of. Mastering demanding nomenclature in the domain. Given slow progress in field last decades this might be a good alternative but there are also pedagogical reasons.
This form of assesment is extreme and outdated.it has further disadvantages
** Students do not develop critical research skills such as literature analysis and research.
**certain skills for future researcher are notvtested i. E ability to combine research from different Source, ability to think critically of competing thesis or theories, to discern gaps in current research
** A mixed approach should ensure all aims
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To sum up
1. The issue is not about removing mathematics or questioning their role in physics. It is about what motivates Ph. D students most.
2. Current physics master curricula are, dispite the existence in some of a parallel non written or expressed properly strategy in the programs, are monopolizng learning goals to undepagogical levels.
( 80% Application goals to Calculations and 20% understanding (explain why) in specific situations. Pedagogical this is a scandal. Bloom demands 4-2-2-2-1 ration in knowledge, understanding, application, synthesis, evaluations. Essay qs address the latest 2.)
3. I believe students should be inquired to experience by self-discovery radical formulations that involves new conceptual ideas ornee physical quantiries as Smolin's variety, views that provenly mathematically rebuild established theories
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I have Expi293 cell cultures (suspension). After counting them, the density turned out to be 4.26 x 10^6. I need to split them. Starting from this concentration, how can I obtain a 30ml cellular suspention at a density of 0.25 x 10^6? How many mL of cellular suspension (the one at a density of 4.26 x 10^6) and how many mL of medium do I need? I still have difficulties in understanding which kind of mathematical calculation do I need to use. Could you explain it to me?
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Use the formula C1V1= C2V2
C1= 4.26 x 10^6
V1= volume to be taken (Xml)
C2= 0.25 x 10^6
V2= required volume (30ml)
4.26 x 10^6 x Xml = 0.25 x 10^6 x 30ml
X= 0.25 x 10^6 x 30ml / 4.26 x 10^6
Therefore X= 1.76ml
So, add 1.76ml of cell suspension (the one at a density of 4.26 x 10^6) into 28.24ml of culture media to obtain 0.25 x 10^6 cell density in 30 ml cellular suspension.
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