Mathematics - Science topic

In the context of the co-toxicity factor formula, the term "expected mortality" refers to the predicted or estimated mortality rate of an organism under the combined effects of multiple toxic substances. The co-toxicity factor formula is used to assess the combined toxicity of different substances on an organism, taking into account their toxicities.

To calculate the expected mortality using the co-toxicity factor formula, you typically follow these steps:

It's important to note that the co-toxicity factor formula is a simplified approach to assess combined toxicity and may not account for all possible interactions between substances. The formula assumes an additive or independent effect of the substances. In reality, interactions between substances can be more complex, including synergistic (enhanced) or antagonistic (reduced) effects.

Furthermore, the specific formula or equation used for calculating the co-toxicity factor may vary depending on the context, study design, and toxicological data available. It is essential to consult relevant literature, regulatory guidelines, or expert advice to ensure the appropriate use of the co-toxicity factor formula and interpretation of the results in your specific research or assessment.

La relation entre la température du verre et le diamètre de la fibre de verre dépend de plusieurs facteurs, notamment la composition du verre et le processus de fabrication spécifique. Il n'y a pas de formule mathématique générale unique qui puisse décrire cette relation pour tous les types de verre.

Cependant, pour certains types de fibres de verre, une approximation couramment utilisée est la formule de coefficient de dilatation thermique linéaire. Cette formule relie la variation du diamètre d'un matériau à sa variation de température. La formule générale est la suivante :

ΔD = α * D * ΔT

où :

Le coefficient de dilatation thermique linéaire (α) est une propriété spécifique du matériau de la fibre de verre et peut varier en fonction de sa composition. Il est généralement exprimé en unités de (1/°C) ou (1/K).

Il est important de noter que cette formule est une approximation et peut ne pas être exacte pour tous les types de verre ou dans toutes les plages de température. Pour obtenir des résultats précis, il est recommandé de consulter les spécifications techniques du matériau de la fibre de verre spécifique que vous utilisez, ou de vous référer aux données fournies par le fabricant.

The relationship between scientific understanding and the reality of nature is a complex and ongoing philosophical debate. Science aims to provide explanations and models that describe and predict the behavior of the natural world. However, it is important to recognize that scientific models are simplifications and abstractions of reality, and they can never fully capture the intricacies and complexities of the real world.

In summary, the relationship between scientific understanding and the reality of nature is complex and evolving. Scientific models and theories aim to capture aspects of reality, but they are limited abstractions. The search for a theory of everything requires ongoing exploration, including potential changes in conceptual viewpoints, the possibility of undiscovered mathematical frameworks, and the continuous refinement of scientific models to approach a more complete understanding of the world.

Researchers can use various methods and techniques to measure students' creativity. While teachers' responses can provide valuable insights, it is important to note that creativity is a complex and multifaceted trait that cannot be solely determined based on a few individuals' opinions. However, teachers who have been working closely with the students can offer valuable observations and assessments regarding their creative abilities. Here are three ways in which grade teachers can contribute to measuring students' creativity:

It is worth noting that these methods should be complemented with other measures, such as standardized tests specifically designed to assess creativity, self-report questionnaires, and peer evaluations. Additionally, involving multiple teachers in the assessment process can provide a more comprehensive and balanced understanding of students' creativity.

Filippo Maria Denaro wrote, "I don’t think that any fluid dynamics course can be done without some fundamental math.." I completely agree. Among the many books on Fluid Mechanics, I find that the one that best embraces the physical point of view of the discipline is that of Guyon, E., Hulin, J. P., Petit, L., & de Gennes, P. G. (2001). Physical hydrodynamics, EDP sciences, whose first edition (2001) was prefaced by Pierre Gilles de Gennes (Nobel Prize for Physics) who did not lack praise for the particularity of the integration of physics in the very conception of the book. The book is in its third edition and has more than 750 citations.

Guyon, E., Hulin, J. P., & Petit, L. (2021). Hydrodynamique physique. EDP sciences.

Even if the book is in French, I think it would be interesting to consult it if only to have an idea of its construction

Viscoelastic microfluidics: progress and challenges.

Zhou, J. and Papautsky, I.

Microsyst Nanoeng 6, 113 (2020).

https://doi.org/10.1038/s41378-020-00218-x

Abstract:

The manipulation of cells and particles suspended in viscoelastic fluids in microchannels has drawn increasing attention, in part due to the ability for single-stream three-dimensional focusing in simple channel geometries. Improvement in the understanding of non-Newtonian effects on particle dynamics has led to expanding exploration of focusing and sorting particles and cells using viscoelastic microfluidics. Multiple factors, such as the driving forces arising from fluid elasticity and inertia, the effect of fluid rheology, the physical properties of particles and cells, and channel geometry, actively interact and compete together to govern the intricate migration behavior of particles and cells in microchannels. Here, we review the viscoelastic fluid physics and the hydrodynamic forces in such flows and identify three pairs of competing forces/effects that collectively govern viscoelastic migration. We discuss migration dynamics, focusing positions, numerical simulations, and recent progress in viscoelastic microfluidic applications as well as the remaining challenges. Finally, we hope that an improved understanding of viscoelastic flows in microfluidics can lead to increased sophistication of microfluidic platforms in clinical diagnostics and biomedical research.

###

Without proper understanding of viscoelastic liquids, which are omnipresent in all biology, our biological description of living systems remain incomplete.

Without proper understanding of viscoelastic liquids, which are omnipresent in all biology, our biological description of living systems remain incomplete.

Just one example. Flowing of blood through micro-vessels utilizes viscoelastic properties of this liquid. Advances in this ar4 can help us to understand better the very mechanisms of blood clotting, organ infractions, and many other health issues.

What about using the fractal dimension (box-counting) as a measure? Here is a paper on tortuosity.

Regards,

Joachim

K. Kassner You are right.

The method that I proposed may not work for all functions f(x,y), especially if they are continuous and do not have isolated zeros. In that case, one might try to separate the integrals or use a coordinate transformation as you suggested in your comment. For example, if we use polar coordinates $(r,\theta)$, then we have

$$\delta(f(r,\theta))=\frac{1}{r}\delta(r-r(\theta))$$

where $r(\theta)$ is the zero of f(r,$\theta$) as a function of r for a fixed $\theta$. This can be seen by using the Jacobian of the transformation and the property of the delta function.

Dear Dinu Teodorescu ,

Thanks. I think in exactly the same way as you!

Sincerely,

Sergey

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:

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.

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.

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.

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

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:

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.

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-y_k), where y_k are the zeros of g(y)-a. Each term acquires a denominator |g'(y_k)| in the process.

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 !?

Physics confirming math, or denying it. Time for colleges to catch-up and be competitive.

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!

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.

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:

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.

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

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?

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.

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

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:

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

Dear Décio Krause

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.

There are several mathematical and statistical methods used to quantify and discuss survey questions. Here are some of the most common ones:

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.

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.

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.

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

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:

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

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.

Very very short. https://www.researchgate.net/post/The_Irretutable_Argument_for_Universal_Causality_Any_Opposing_Position

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?

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!

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

Can numbers obey a physical law?

==============================

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!

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/354418793_The_Informational_Conception_and_the_Base_of_Physics, and

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

https://www.researchgate.net/publication/369357747_The_informational_model_-Nuclear_Force.

Cheers

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.

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.

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)

See Wikipedia for Potassium-40, abundance.

Important or not? As a question it is too general, no definite answer could be given.

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,

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.

Goes like this:

e^2pii=1

e^pi=-1

Consequently:-2pi=pi

and thats precisely what I was trying to say

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.

@Juan Weisz

incentive here is that e^2pii=1 (Euler's formula)

Raid, apologies here's the attachment. David Booth

An equivalent formulation of the Riemann hypothesis. See formula (3.22) in the

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.

@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

where:

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 .

Adnan Mohammad Awad Roshdi R Khalil Hening Huang

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.

Hello Leopold Modrakowski

No, unfortunately I didn't find any solution for this problem.

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

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

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.

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.

@ Andrea Ossicini,

Interesting - thanks for sharing

Problem solving technique and thinking aloud strategies helpful for mathematics teaching