Mathematics - Science topic

Advantages and Limitations of Artificial Intelligence

In book: Artificial Intelligence Applications of Critical Transportation Issues

Chapter: Why Artificial Intelligence?

by: Mashrur Chowdhury and Adel W. Sadek

A very strange view. You can, of course, ignore the mathematical algorithms embedded in AI and treat the AI carrier as a "black box". Nevertheless, the mathematics is there and it will work. Now, in the vast majority of cases, the operators of any modern equipment do not perform mathematical calculations, relying on the corresponding systems. Dream of a magic wand?

There are quite effective hunters in nature, for example, a cheetah (2D environment) or a falcon (3D), these really do not know mathematics, but solve practically problems similar to the one mentioned by the author of the question. But Is their efficiency sufficient in comparison with the efficiency of a modern radar complex?

Dear Maged Gumaan Bin-Saad

Thank you so much for providing these papers.

Kind regards,

Faezeh

In any leadership theory the " controlling power" of the management is the vital factor. The controlling power is that which can control the entire parameters and the decision variables of an objective function. First of all fuzzify the parameters/ variables which are more significant in the real time situations. Choose the fuzzy numbers as LOCK Fuzzy set, Then apply several strategies( as per Leadership Theory) of controlling keys of the fuzzy LOCK any one can get the Mathematical foundations of several Leadership Theory.

Five different perspectives on mathematical modeling in mathematics education By: Aline Abassian

also i think their solutions should be much simpler.

Polynomial Division to Find Oblique Asymptotes. The idea is that when you do polynomial division on a rational function that has one higher degree on top than on the bottom, the result always has the form mx + b + remainder term. Then the oblique asymptote is the linear part, y = mx + b. https://magoosh.com/hs/ap-calculus/2017/oblique-asymptotes/

Hello,

the worst case is not for sine but for constant value at either + or - 2^n for all 2^m variables.

Then one FFT value at least will be + or - 2^(m+n)

· The difference between physics and biology is in the study of inanimate and living systems, but in the physics of living systems (LSP) this difference disappears.

· There is a difference between physics and mathematics in why and how, but in conceptual analysis this difference disappears.

Dear @Raju Chiluvuri

To my opinion, mathematics is the precursor to all the disciplines of science. And, in fact, mathematics is also a science.

Thanks!

Thanks for your answers.

Ok with the previous answers but the most important for such a huge building is not architecture but well engineering. It is not without reasons that engineers spend 4-5 years to study mathematics, physics, mechanics (including fluid mechanics), chemistry, resistance of materials, electricity, electronics, with plenty of mathematical equations, and so on.

Search RG for Ed Gerck. I'm sure he'd be glad to discuss this topic.

Dear colleagues. A very interesting question, some years ago, in 2012, I published a work where I give a definition of Mathematics that can serve to answer the question.

The book of Computer Aided Optimal Design: Structural and Mechanical Systems by the Mota Soares, C.A. and Templeman, A.B can be useful.

see links

When the electron's energy can take only certain values this just means that the states that would correspond to the other values don't exist, under those circumstances. These circumstances are described by the boundary conditions imposed, that are part of the physical description, too.

It is not very difficult, but some algebra needs to be involved.

The workflow is the following:

1. Knowing relative humidity at T=T0 (as an input), calculate the partial pressure of vapor at this temperature.

2. Calculate water vapor concentration rho_0 using the ideal gas equation.

3. Calculate saturated vapor pressure at T=T1 from tables or the Clausius-Clapeyron equation.

4. Calculate corresponding saturated vapor density rho_1 at T=T1 using the ideal gas equation.

5. If rho_1 < rho_0, there will be no condensation, otherwise the mass of water condensed in volume V will be V(rho_1 -rho_0).

Pass x, instead of tau, as rightly pointed out by Saeb AmirAhmadi Chomachar

syms x y psi tau t

c1 = 1;c2 = 1.5;lambda = 0.1;

x_r(tau) = 0.8486*tau - 0.6949;

y_r(tau) = 5.866*sin(0.1257*tau + pi);

psi_r(tau) = 0.7958*sin(0.1257*tau - pi/2);

x_r_dot = 0.8486;

y_r_dot(tau) = 0.7374*cos(0.1257*tau + pi);

psi_r_dot(tau) = 0.1*cos(0.1257*tau - pi/2);

phrase1 = c1/2*(cos(psi)*(x - x_r) + sin(psi)*(y - y_r))*(cos(psi)*x_r_dot + sin(psi)*y_r_dot);

phrase2 = c1/2*(-sin(psi)*(x - x_r) + cos(psi)*(y - y_r))*(-sin(psi)*x_r_dot+cos(psi)*y_r_dot);

phrase3 = 0.5*(psi - psi_r)*psi_r_dot;

eq = -2*(1-lambda)^2*(phrase1 + phrase2 + phrase3) - 2*lambda^2*(t - tau);

eqn = rewrite(eq,'log');

sol = solve(eqn == 0 , x , 'IgnoreAnalyticConstraints',1);

pretty(sol)

I am not expert in this subject , may be the following links are useful

High-Voltage Leak Detection of a Parenteral Proteinaceous Solution Product Packaged in Form-Fill-Seal Plastic Laminate Bags. Part 3. Chemical Stability and Visual Appearance of a Protein-Based Aqueous Solution for Injection as a Function of HVLD Exposure

Rasmussen, M., Damgaard, R., Buus, P., Guazzo, D. M.Journal:PDA Journal of Pharmaceutical Science and TechnologyYear:2013

Interesting question. However, the foundations that allowed calculus to evolve started long before Newton and Leibniz. The foundation is not calculus but the concept of the limit. Archimedes (287-212 BC) was probably the first to recognize what became the concept of the limit in is estimation pi and the area of the circle by taking inscribed and circumscribed polygons bounding the circle and using the simple fact that one approaches pi from below while the other from above. Of each sequence defines a Cauchy sequence - concept unknown at that time and the completion of the reals (also unknown at that time) show the limit of each sequence is the same and equal a the number pi. In reality the concept of infinitesimal - goes back to Archimedes although the formal concept of "infinity" was not accepted until long afterwards.

Roll backwards to the Greeks when faced with the proposition of an infinite number of prime number, was a problem as they believed the universe was finite. Infinity was not something the Greeks wanted to accept and Aristotle (385-348 BC). But Archimedes had just shown that infinity and infinitesimals had a role in mathematics - in fact a central role. It was not to the 1600's that mathematicians attack the problem of infinity to try to understand what it mean as they developed the concept of numbers that are used today.

As we better understood our real number system, the concept of point set or general topology was defined to abstract and better understand the structure. In general topology concepts like nets (generalizations of sequences required to define integrals for example), convergence, close to, in a neighborhood and limits are all defined through the concept of open sets which are used to define a topology on a set which now allows for the definition and study of the concept of limits and continuity and handle infinity. While the formulations of general topology came along after the "birth of the calculus" and known as Analysis Situs a term coined by Henri Poincaré through the work of Poincaré , Euler, Cantor, Lefschetz, Courant, Hilbert, etc., a firm foundation was laid not only to the real number system, but to limits, continuity all the foundations of what we now know as 'calculus."

Topology is so important to the foundations of calculus and the concept of a limit that in his classic text, "General Topology," John Kelley writes in the first paragraph of the preface he writes, "...I have, with difficulty, been prevented by my friends from labeling it: What Every Young Analyst Should Know." No truer words have been spoken or written as the foundations of topology has allowed the concept of calculus to be expanded far beyond its original intent.

I believe that one of the most important reasons for low achievement in mathematics is their lack of basic mathematics and their belief that mathematics has no strong necessity in their lives, as there is a lack of training for students on questions that measure higher levels of thinking. As a result, students focus on memorization without understanding, as they remember in the time immediately preceding the test. As for students, they see that one of the factors for their low level of achievement in mathematics is the way the book is presented, as it does not allow them the opportunity to follow the course themselves, and the teaching methods of mathematics do not encourage research and benefit from mathematics, also the teacher does not use educational aids during the explanation In addition, they believe that the time difference between the tests is small and that the time allocated for answering is much less with what you need from these tests. They also admitted that they are worried about their test score, which will affect their achievement results

A multi-depot VRP with less than 10 nodes should be almost enumerable, as there exist less than 1024 possible subsets of customers. Given this fact, perhaps the simplest solution approach is to generate all feasible routes from each depot, discard those that are not TSP-optimal, and directly solve a set partitioning formulation based on these routes. Now, if you face larger problems (e.g., 15 nodes or more), you should use the formulations suggested by Adam and Noha, or even go for sophisticated branch-and-price approaches as described in

since the code associated with this paper is freely accessible at

Dear Sumit Bhowmick

I hope you find the following discussion is useful

https://math.stackexchange.com/questions/202261/solid-angle-between-vectors-in-n-dimensional-space#:~:text=The%20formula%20of%203D%20solid,c)%7C%E2%86%92a%7C.

Best regards

I can imagine volumes of reasonings behind the eight-word response. Thank you Hermann Gruenwald !

Dear Sneha,

Please follow the paper in the link:

You will get the design formulas and an example of antipodal Vivaldi Antenna.

If you have more questions you can ask the first author of the paper.

Best wishes

First thanks , it is really an interesting question.

A problem is that gifted and talented students do not receive the necessary care

To meet their needs by remaining in regular classes. Therefore, I find it important to do the following:

- Staying away from traditional methods during teaching, this leads to boredom for students, especially talented people

- When constructing lessons conceptually, we take into account that gifted and talented students may also suffer

One of the weaknesses in understanding the curriculum and that they need to be considered. And when applying the teaching conceptually within

In the ordinary class, students of all levels will learn in a deeper way.

- Adding open-ended questions to both education and evaluation for their positive results on students ’understanding

As well as their attitudes towards the material.

The- direct and indirect financial support for talented people

Quantum computing (QC) is the enabling technology for efficiently processing huge quantities of (quantum) information, in many cases outperforming "classical" computing (i.e. binary logic based). It provide you the "muscles" for data crunching, provided you feed it with quantum-coded information (qubits) and you get probabilistic results (with high likelihood if well designed).

Artificial Intelligence (and Machine Learning more specifically) is a discipline focused in performing data analysis with the objective of simulating human reasoning for achieving a certain goal. It can then definitively take advantages by a super fast computing capability provided by QC, both for speeding up "classical" algorithms or for running QC native ones, which are expected to open the door to a next level of AI capabilities beyond our current imagination.

Just be patient for few more years and wait for a working universal QC becoming available (at competitive price).

Interesting question. Following the discussion.

What a good question, dear Parul! I can tell you that, after 35 years of career in math & stats, I still fell rewarded when I solve a problem or finish a project. I do not know what chemicals are doing that for me, but they do. It is true, I did not have many frustrations to overcome, it worked well for me math & stats-wise, so I am pleased (and happy, as you say) by default. I am probably not the perfect subject for your study, you need people who had to struggle much more than me to achieve their goals.

Hi Natacha,

rcompanion.org is a great source with many examples of non-parametric tests.

sthda.com is also good, but the author uses his own limited packages.

1. Patience.

2. Knowledge of field and ability to impart that knowledge to others.

3. Ability to ask questions and define specific aims

4. Writing skills to produce grants and publications.

5. Research support in terms of grants or other types of research funding.

6. Contacts, perhaps a network of researchers with similar interests.

7. Foster attendance a scientific meetings, including presentations by students.

Age requirement l problem. If it was 40 better !!

Publications in good journals, either on old unsolved issues, or on novel paths and ideas.

The answer to a question about the numerical algorithms for resolving the issue of differentiability of a function is typically provided by the textbooks on experimental mathematics.

I recommend in particular: Chapter 5: “Exploring Strange Functions on the Computer” in the book: “Experimental Mathematic in Action”.

For the review please see

You can also get a copy of the text in a form of a preprint from

Judging by the quote placed in the beginning of Chapter 5, the issue of investigation of the “strange functions” was equally challenging i 1850s as it is 170 years later:

“It appears to me that the Metaphysics of Weierstrass’s function

still hides many riddles and I cannot help thinking that enter-

ing deeper into the matter will finally lead us to a limit of our

intellect, similar to the bound drawn by the concepts of force

and matter in Mechanics. These functions seem to me, to say

it briefly, to impose separations, not, like the rational numbers”

(Paul du Bois-Reymond, [129], 1875)

The situation described in your question is even more complicated because the function is represented only by a few values on a rectangular grid and it is additionally assumed that the function is not differentiable at a certain point. In this situation I can suggest to use the techniques employed in the theory of generalized functions (distributions).

For a very practical example you can consult a blog: “How to differentiate a non-differentiable function”:

In order to answer your question completely I would like to know what is the equation, boundary conditions and the numerical scheme used to obtain a set of the grid point values mentioned in the question.

You're welcome Rich Philp. For your information, I have a modest knowledge regarding programming, but still I made some games for PC. Here is a free one:

Dear;

The following documents show exemples of its application :

Automated Solution of Differential Equations by the Finite Element Method

Regards

Mohammad Esmail Aryaee Panah

Not all these phenomena can be summarized in the permittivity of the material. For a start there is the permeability, which is as basic as the permittivity, then whole areas that these two do not cover at all, such as fluorescence, ionisation, photo-electricity, Rayleigh and Raman scattering, interaction with (other) fundamental particles, interaction with gravity/space-time, and more.

An alternative method of demonstrating stability is given by Vasile Mihai POPOV, a great scientist of Romanian origin, who settled in the USA.

The theory of hyperstability (it has been renamed the theory of stability for positive systems) belongs exclusively to him ... (1965).

See Yakubovic-Kalman-Popov theorem, Popov-Belevitch-Hautus criterion, etc.

If the Liapunov (1892) method involves "guessing the optimal construction" of the Liapunov function to obtain a domain close to the maximum stability domain, Popov's stability criterion provides the maximum stability domain for nonlinearity parameters in the system (see Hurwitz , Aizerman hypothesis, etc.).

Dear Pranav Khade

Are you trying to play chess in 3D?

Your idea is a good one, but your assumptions are not.

You need to give a clear definition of paths, so I suggest for you to start in one 3D box, it includes 8 points. I prefer to give each point the following notation

P(i, j, k), so the locations of the 8 points are at

(0,0,0) (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1),(0,1,1) and (1,1,1).

Study this cube carefully, define each Delanoy edge (axioms of the path), and then add another box, which means 12 points, etc.

If you find the closed formula that allows you to calculate all possible paths from the starting point at the origin to the farthest point at the upper corner of the rectangular box, then you are on the right track.

I wish you good luck.

exemple.docx

Bryan Soh, your question is a good one. I think it's necessary to be familiar with the nature of your variables (which it seems you are). Unfortunately, I'm not, but might I suggest that you conduct your analyses both ways, look at the results, then think carefully about which results are likely to be most valid.

I also think it is a good idea to present both sets of results if that's permissible. As you are obviously aware, the world of research isn't black and white, and making other researchers, and consumers of research, aware of that could well be helpful. About 20 years ago, I read an article in a top psychological journal in which the author analysed her data in more than one way (from memory, it was more than only two ways), and she discussed the ins and outs intelligently and with insight. It was, for me, much more enlightening that the run-of-the-mill articles that seem to report clean-cut results but leave the reader wondering how much cleaning up and manipulation, and obscuring, occurred to obtain those results.

Every good wish as you plough on!

Sure. But you are not really helping by not being precise! And I am quite certain that there is one major goal - such as a quickest route of the water. I suggest that you check with the literature - which is quite big.

A search string might look like this, or with smaller adjustments:

water network [supply, distribution*, system*] problem*

Tool - An American progressive rock band :)

You may check

I don't understand the relationship of "is prior to" to ideas. Do you mean which idea came first to a human being? Or which idea has fewer presuppositions? Or which idea does not depend on the other? If that is the case, is your question a false either-or assertion, as can't two ideas be co-dependent, like chicken and egg?

Thank you Dear Anatol Badach.

Armin,

Did you find the equation?

Amauri Martins-Britto It is indeed helpful for me in other ways. Thank you.

Optimization algorithms handling a fixed number of real correlated values (non-separable) can be of 2 types: gradient-based and derivative-free.

Gradient-based procedures seem to be significantly faster than derivative-free ones.

GRADIENT-BASED METHODS (some):

Stochastic Gradient Descent (SGD)

Nonlinear Conjugate Gradient

L-BFGS

Levenberg-Marquardt Algorithm (LMA)

Dear Mohamed Azzedine

Read my answer carefully.

I show the formula of your (trivial answer), and the second set is also trivial.

One can generate infinite sets of such families!!

Observe that each number n in the first family has the property

3n + 1 = 2^k which satisfies Collatz's Conjecture.

And each number n in the second family has the property

3n + 1 = 7x 2^k

therefore after removing 2^k by k steps divisions, we reach the number 7

which satisfies Collatz's conjecture too.

The whole procedure is NOT PROOF.

All are elementary computations.

Regards

Very interesting question. To me, best answer is given by Roger Penrose using the paradigm "The three worlds of reality": the Physical World, the Mental World, and the Platonic World (Mathematics lives here).

The claim is that Platonic World "maps" onto Physical World and hence all possible physical universes are constructs of information consistent with Mathematics.

Obviously Physical World "maps" onto Mental World (because we live in the physical world); and Mental World "maps" onto Platonic World. But the three worlds are different.

For a much better explanation you can see Penrose's book "The Emperor's New Mind"

Dear Feng Qi

I am attaching a proof of the first formula.

Very interesting discussion and proposals to solve for calculating the absorbance of a single layer of MoS2 .

The absorbance is important to optically characterize the materials. The absorbance is the absorbed fraction of the incident light with a specific wavelength on a material with a given thickness. It is required for optical filters and for photodetectors and solar cells.

The dependence on the incident light wave length stems from the energy band structure of the material such that it is dependent on the probability of transferring of an electron from the valence band to the conduction band by the incident photons. Specifically it depends on the absorption coefficient as a function of the wavelength.

The key point for the solution is to calculate the energy band structure of the two dimensional material and from it you can get the absorption coefficient as a function of lambda. Then you can calculate the absorbance. As you have here a single layer the absorption coefficient will be equal to the absorbance.

You might benefit from the thesis given in the link: https://digitalcommons.mtu.edu/cgi/viewcontent.cgi?article=1049&context=etds

Thank you for sharing your paper Martin. I found it insightful - particularly your interpretation of graphs and your case for building 'critical STEM literacy'. From recent exploration, I notice that these graphs with data from the pandemic, present interesting outliers and multiple trend-lines which are ignored or cherry-picked in the final interpretation. Inferential stats while allowing decisive conclusions lead us to overlook these interesting data points. Critical graphicacy practices if nurtured from school will create this informed public, who will then find it difficult to simply accept the text interpretation, isn't it?

Relation between milk with adulterants . Many times urea is added in the milk as adulterant to increase the density. If you is isolated then in FTIR it shows the absorption for NH2(V NH = 3200-3350 cm-1) and C=O) at (v 1700 cmi1).

Joan Nyika,

This way of thinking would support dictators and totalitarianism. It is rationalization to do nothing about it. But if one is aware then one is responsible. The legal theory is called, "participatory negligence." One can be guilty of a crime one does not commit. Hannah Arendt was instrumental in showing this, and it happened in a killing in New York. The attacker interrinterrupted, after around 80 people shouted that they woukd call the police. But the sttacker resumed, when no police help came, and finally killed the victim. All those 80 people were charged with the crime of negligence.

Please see attached.

Dear Gbadamosi-Olatunde Olatunde Bilikis Olanrewaju ,

Welcome.

You can validate your analytical model by the following methods:

- running experiments under the same assumptions of the analytical mode

- By using numerical simulation where the the system is solved by the numerical methods. There are now powerful simulator for many disciplines of science such as electronic circuits , electronic devices, semicondcutor fabrication processing.

- In some statistical systems they can be validated by Monte Carlo simulations.

This is list may not be exhaustive

Best wishes

Thank you so much for your answers.

Janez Podobnik I have been workikng since your answer and found the optimality condition and also found the proofs of optimal estimator. I have minimized the summary as you can see here in the attached picture.

But I am stuck in linking the attached theory with my case. Since this is the general cndition for an optimal estimator (attached picture) and yes generator would be minimizing a cost function but how can I say that my generator would be able to generate clean signals?

I have changed the notations in main question for simplicity purpose.

Good search, Romeo! It looks that the examples provided satisfy the unboundedness condition in our question. I did some search as well, and all the examples I found do satisfy that condition.

Thank you for the increased interest in this question, I'll leave it open of course, time is not of the essence here...

I assume your data are count values (i.e., the number of adverse childhood experiences). If so, then use a count model (negative binomial is usually a good choice). Transfroming your data is not required at all. If excess zeros are a problem, consider a zero-inflated negative binomial model.

I'm sorry, the answer was not for you. I posted by mistake !.

The "mathematicaL" explanation is to examine the formula for the partial regression coefficient. It contains the covariance between the two independent variables, and if this is quite high, it will dominate the estimation of the coefficient. At the most extreme case, the two independent variables are identical, so neither one can make a separate contribution.

In my opinion, the best approach is to choose the most theoretically relevant variable from each set of highly correlated variables and use that one. This may well not cost you as much as you might think, because once you enter one of the variables, adding any of the other variables will amount to putting "the same thing" into the equation, so you R-sq is unlikely to increase.

This link talk about:

What arguments can I give a high school student why mathematics is important?

https://matheducators.stackexchange.com/questions/150/what-arguments-can-i-give-a-high-school-student-why-mathematics-is-important

Please follow the link below to know in detail

Check out Brian Hall's book, "Quantum Theory for Mathematicians" in the Springer Graduate Text in Mathematics series vol 267 In it he goes through the history and derivation of DeBroglie's theory.

The simplest way is to pick a specific time point as the point at which to calculate % inhibition. This is purely phenomenological. No theory.

If you have an enzyme reaction that shows a time lag before reaching maximal rate, you should investigate the reason for it. Maybe the enzyme autoactivates in the presence of its substrate, like an autophosphorylating kinase, or the substrate may slowly displace a competitive inhibitor. You may be able to preactivate the enzyme so as to get Michaelis-Menten behavior.

Time lags can also occur in coupled enzyme reactions if the concentration of the 2nd enzyme in the sequence is not high enough.

Another proof

let f(z)=exp(ig(a)), such that z=exp(i a), a belongs to [0,2pi]. From

f(z^2)=f^2(z), one concludes that there is an analytical function g(a) having the property as 2g(a)=g(2a)+2npi, for some integer number n, so n is zero. We must answer to this question which g(a) is only either linear function or not!

By power series, we could imply that g is linear function, then g(a)=k.a. indeed

f(z)=z^k, for any positive real number, are alone class functions, they satisfy at this property.

i think the attitude and positivism of the community and their leaders is one big factor to add.

Dear search for journal of Mathematics in Web of Science Journal Master list and you will see many journals for Mathematics.

Dear Amir, this equation has the following ananlitics. It can be glue for v more than zero exact solution and for v less than zero. In both cases they are spirals and exactly solvable, but tends to different stationary points. You can even draw its phase space picture: you draw two spirals(exact solutions)each on proper halfspace and jump from one to another, when v=0.

without loss of generality, you could assume that a=1. Also, exponential function classes are trivial solution of it.

For f(0) non-zero and t positive number,

f(x)=f(0)exp((2.\pi.i.x)/t) is a solution, with a=exp(-2.\pi.i.b)/t) and f(x+t)=f(x).

It feels like you didn't try to use an exact method. Thus, I have to write here that all the beginers are suppose to listen to prof. Patriksson. So, at first try to find an optimal solution. Use CPLEX, learn to model. This will be one of your first sentences in the research paper: "We tried to solve this NP-hard problem... (provide some proof that you really tried)... Because we are not able to generate an optimal solution in reasonable amount of time we decided for a heuristic approach..." I hope this is clear. Otherwise, it is like working with machine learning without knowing about statistical tests. It is slightly embarrassing.

You are correct Romeo; a nice solution & discussion can be found here:

http://www.les-mathematiques.net/phorum/read.php?4,458179,458332

Dear professor,

The following article may provide inspiration to you about the subject.

Best wishes,

Tahir

As long as the students are comfortable with algebra of real numbers and the quadratic equation, complex numbers should not be particularly challenging to them. These numbers follow rather naturally from a straightforward application of simple algebraic concepts, though novel at the same time. Humans crave novelty, and the younger they are the more so, I think.

Dear brother and one of my best friends Mahmoud,

"you are really great researcher" This is how I see you.

Warmest regards,

Kareem

I don't know about analytic solutions but this is the equation for a harmonic oscillator in a constant force field with nonlinear damping. It is a like a mass suspended by a nonlinear spring in a gravitational field. There is a single equilibrium point at x=V and x'=0 whose stability depends on the value of a. Solutions will either be unbounded (go to infinity), which is probably unphysical for your electrical circuit, or they will decay to the equilibrium point, following the usual exponential law for a linearly damped oscillator as they approach it. Only the case a=0 has (neutrally) stable oscillations whose amplitude depend on the initial conditions just as you would expect for a harmonic oscillator without damping. There are no limit cycles and no chaos.

Dear Han-Wei,

welcome,

You want to validate your results.

you can make this validation by solving a known simple example where it can be solved analytically or its numerical solution is known with high confidence.

You can simplify the geometry of the device such that it can be solved analytically.

Than you can compare your numerical solution with the analytical solution.

You must not solve specifically for the HEMT.

Best wishes