Science topics: Mathematics
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Mathematics - Science topic

Mathematics, Pure and Applied Math
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Dear Colleagues, a recent trend in Fractional Calculus is in introducing more and more new fractional derivatives and integrals and considering classical equations and models with these operators. Thus, we have to think about and to answer questions like “What are the fractional integrals and derivatives?”, “What are their decisive mathematical properties?”, “What fractional operators make sense in applications and why?’’, etc. These and similar questions have remained mostly unanswered until now. To provide an independent platform for discussion of these trends in the current development of FC, the SI “Fractional Integrals and Derivatives: “True” versus “False””( https://www.mdpi.com/journal/mathematics/special_issues/Fractional_Integrals_Derivatives2021) has been initiated. In this SI, some important papers have been already published. However, you are welcome to share with the scientific community your viewpoint. Contributions to this SI devoted both to the new fractional integrals and derivatives and their justification and those containing constructive criticism of these concepts are welcome.
Best regards
Yuri Luchko
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Indeed, in the FC literature, the term “fractionalization” (not “fractionation”) is sometimes used. The meaning of this term is as follows: One takes some (mostly classical) ordinary or partial differential equations with integer order derivatives and formally replaces one or several of the integer order derivatives by the fractional derivatives in one or another sense. It seems to me that nowadays a majority of FC publications devoted to the fractional differential equations employs this procedure explicitly or implicitly. Maybe, some of these “fractionalized” equations have some sense and/or at least potential applications. However, establishing new mathematical models in this way contradicts the basic rules and ideology of mathematical modelling. Thus, one more important question that the FC community should address is “What fractional derivatives and fractional differential equations make sense in applications and why?’’
For some partial answers to this question, I would recommend the following recent publications:
R. Hilfer: Mathematical and physical interpretations of fractional derivatives and integrals. In: Handbook of Fractional Calculus with Applications: Basic Theory, Vol. 1, Walter de Gruyter GmbH, Berlin, 47-86 (2019), https://doi.org/10.1515/9783110571622-003
V.E. Tarasov, S.S. Tarasova: Fractional Derivatives and Integrals: What Are They Needed For? Mathematics 8, no. 2: 164 (2020), https://doi.org/10.3390/math8020164
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can anyone suggest or interested to have conversation on AI/ML and PLS based mathematical tool. I need to know more information.
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Explainable ML using OLAP data for verification - that could be an interesting topic.
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Hello? Good evening, I would like to ask if there is a questionnaire aligned to the set of indicators on Mathematical Competence given by Sir Turner?
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Hello Sir? Goodmorning. Thank You for sharing your time and effort in answering my concern. I appreciate your thoughts Sir. I am able to skim the article you've mentioned about Mathematical Literacy and I downloaded it already, I believe this is an edge as I develop my study. Fortunately, a colleague of Sir Ross Turner further give a helping hand. I am beyond grateful for your kindness. Thank You so much!
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Can anyone please confirm whether there exists an explicit mathematical relationship between the gravitational instability growth rate and the corresponding structure (star) formation rate in molecular clouds?
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Thanks a lot, dear Javad Fardaei, for the article
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I have formulated the mathematical equation of the vibration problem. The resulting equation is coupled nonlinear differential equation of 2nd order ODE. Please, anyone, suggest to me how to solve it using MATLAB.
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Dear Sandip,
to answer your question, I agree with Abdelghani: you have 1) to transform the 2nd order equations into 1st order ones and then 2) to use a MATLAB ode solver to numerically solve the resulting 1st order system.
To do step 1), you have to introduce new variables that correspond to first derivatives. For instance, let us say that you have a system of two 2nd order ODEs in x=x(t) and y=y(t), e.g. x''=f(t,x,y,x',y') and y''=g(t,x,y,x',y'). Then you can introduce two dummy variables X=x' and Y=y' (for example) and get a system of four 1st order ODEs: x'=X, X'=x''=f(...), y'=Y, Y'=y''=g(...).
Then, in order to apply step 2), you have to define a variable vector, say v=[x,X,y,Y], and a function, say myFun, that defines the system. In this case:
function vder = myFun(t,v)
vder(1)=v(2); % this equation corresponds to x'=X
vder(2)=f( t, v(1), v(3), v(2), v(4) ); % you should actually write your original equation for x''=X'=...
...etc...
end
Then, choose an ODE solver (for example ode45) and solve the system by using the basic command [t,v] = ode45(@myFun,tspan,v0); where tspan is an interval for the independent variable (e.g. tspan=[0,10]) and v0 is a vector containing the initial values for x, X=x', y, and Y=y'. So please notice that you must provide initial values for the first derivatives!
You can find more details about all this stuff here: https://www.mathworks.com/help/matlab/math/choose-an-ode-solver.html
Hope this helps! :-)
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I am working on RIS aided communication. Whichever paper I go though, they cook up some complicated mathematics specially optimization problem, which seems to be unsolvable at first. But, then I see they are using some techniques which I have never seen anywhere. Can anyone from wireless comm background tell me, how you people proceed and get those sort of maths?
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Optimization is a broad field of mathematics and not some mysterious quackery. I am not surprised that someone from another field entirely (wireless communications) might not be familiar with these topics. This is why I stress the importance of multi-discipline teams. Nobody knows everything. If you get smart people from different fields together in the same room to discuss a problem, you may find some clever solutions that any one person alone would never come up with.
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Hi!
I'm currently working on a Data Science project for optimizing the prices of the products one of the biggest supermarket chains in Mexico.
One of the things that we are working on, is finding the price elasticity of demand of such products. What we usually do, is that, apart from fitting an XGBoost model for predicting sales, we fit a linear regression, and we get the elasticity from the coefficient corresponding to the price (the slope).
However, it is abvious that linear regression is sometimes a poor fit for the data, not to mention that the execution times are way longer since it requires to run separately XGBoost and LR (which is not good considering that there are thousands of products to model).
Because of this, it ocurred to me that we could use numerical differentiation for finding the price elasticity. At last, calculating a numerical derivative is way faster than fitting another model.
However, I'm not sure if this is mathematically correct, since the data does not come from a function.
So the question would be, is this mathematically correct? Does it make sense?
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From my experience, I would not use a linear predictor for elasticity. I know your question is asking about mathematically correct solutions but I don't think that's your issue. I would probably use logistic regression as a first choice.
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Dear colleagues,
I am researching in a topic related to philosophy and teaching methods. Please, could you point out me if are there any sources on dialectics or contradiction in mathematics education?
Thank you for your answers.
Regards
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An activity theory perspective on contradictions in flipped mathematics classrooms at the university level, Helge Fredriksen and Said Hadjerrouit, 520-541, 2019
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Please look at the text of the section on random walk from page 9 to formula 4.7, where you will find mathematical calculations justifying the probabilistic interpretation of the Riemann zeta function.
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If the distribution of the zeros of the Riemann zeta function is implicit in your question, then you may find the following paper exciting:
Regards,
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Does nature understand mathematics?
A connected question is -- How are theory, theorem, and truth related?
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It may be useful to note that Galileo was the first to state that mathematics is the language of nature.
The following article may be useful too
However, there exist several instances where mathematicians and physicists seem to speak different languages, which indicates that the question whether nature understand some aspects of mathematics is not meaningless.
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I will be glad if researchers and professors answer my question with mathematical formulas or explanations. Thank you so much.
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Hi, it should be taken into account that it is actually 'stress' doing the deformation for us. So when you use a roll with a smaller radius, the contact area between the roll and the sheet/foil is also smaller, increasing the stress.
Stress = Force / Area
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I can't understand why each time we divide by the mass of oxidized NH4-N and multiply by the mass of treated NH4-N.
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Look at the attached screenshot search and I think your question is answered if I understand you. Best wishes David Booth
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Max Planck wrote of natural units; ... These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as 'natural units'
1. Could the units (kg, m, s, A, K) share this numerical relationship (kg = 15, m = -13, s = -30, A = 3, K = 20)?
2. Could these geometrical MLTA objects (see diagram below) be natural units?
This is easy to test. These MLTA objects are the geometry of 2 dimensionless constants; the fine structure constant alpha = 137.035999139 and Omega = 2.0071349496, and so can have numerical solutions, i.e.: V = 25.3123819. We can then use a scalar v = 11843707.905m/s such that V*v = 299792458m/s or v = 7359.323miles/s gives V*v = 186282miles/s.
As the scalars have units associated (v uses m/s or miles/s), they share the same numerical relationship (v = 17, l = -13, t = -30 ...), and so we only need 2 scalars to define the others.
This then permits us to arrange combinations of (G, h, c, e, me, kB) whereby the scalars will cancel (scalars = 1), if this unit relationship is correct, the SI constants will return the same numerical solutions as the equivalent MLTA constants ... for if the scalars are gone (cancelled), the SI constants are MLTA constants (in the example, see diagram below, the 2 scalars are r, v).
The methodology is explained here
1. Is there a logical flaw to the above?
2. Are there other arguments to support/refute this?
3. Is this evidence that ours is a mathematical (simulation) universe?
Some background.
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"My electron" is only my hobby. Since then I try to prove it wrong or right by experiment.
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How can we mathematically calculate the critical thickness of polycrystalline(Sputtering) grown thin?
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Thank you, Dr. Len
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Plz share if yes . I want to know how to get patent in vedic maths.
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Here there are many patents on Vedic Mathematics. Here I attached one patent on vedic mathematics.
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Is multiverse a differential of possibility, a gradient of cross universe space time planes.
Do multiverses branch or all have separate beginnings.
Are multiverses different mathematical constants or is it different potentials.
How can we progress to a mathematical model of multiverse theory.
First we need a mathematical model of the universe and then we need to generalise it for the multiverse. Is this quantum physical, does this take into account the electromagnetic generation of mass and gravity in a recursive relationship?
Does Arc Length calculus change our notion of frequencies and therefor electromagnetism?
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I want to understand the mathematics of fluorescence process in terms of excitation and emission wavelengths. I want to develop a general mathematical model with certain specific parameters and without employing a spectrometer, I want to see the emission spectrum mathematically.
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One can construct simple model (Shifted Harmonic Oscillator model) and derive tractable equations to obtain absorption and emission spectra.
1. Absorption and Emission Lite: Assume ground and excited states are shifted classical Harmonic Oscillator. Ground state |g> energy: Eg(R) = 1/2 ω2R2 & Excited state |e> energy: Ee(R) = 1/2 ω2(R - R0)2 + ΔE. Treating R classically one gets absorption (Ia) and emmision (Ie) spectra as:
Ia(e) = exp[(e-ΔE)2/2σ2] Ie(e) = exp[(E-ΔE + ω2R02)2/2σ2] (Arbitrary Units)
absorption and emission is peaked at different energies and the difference is ω2R02 --- stokes shift, see: https://en.wikipedia.org/wiki/Stokes_shift
σ is the linewidth caused by molecules coupling to environment (Homogeneous Broadening) caused because individual molecules see slightly different environment, e.g. ΔE (Inhomogeneous Broadening) is different for different molecule.
2. Absorption and Emission Pro: Same as 1, but treat R quantum mechanically,
this gives |g>|v0> , |g>|v1> ... |g>|vn> states and |g>|v0'> , |g>|v1'> ... |g>|vn'> where |vn> is nth vibrational state centered around 0 and |vn'> nth vibrational state centered around R0.
  • Energy of |g>|vn> --> <vn|<g| H |g>|vn> = (n + 1/2)ℏω and
  • Energy of |g>|vn> --> <vn'|<e| H |e>|vn'> = (n + 1/2)ℏω + ΔE
With this we write the absorption at T = 0K ( or ℏω >> kT)
Ia(e) = ∑n |<vn'|v0>|2 exp[(e-ΔE - nℏω)2/2σ2]
= ∑n (1/n!) * (ωR02/2)n exp[-(ωR02/2)2] exp[(e-ΔE - nℏω)2/2σ2]
Ie(e) = ∑n |<v0'|vn>|2 exp[(e-ΔE + nℏω)2/2σ2]
= ∑n (1/n!) * (ωR02/2)n exp[-(ωR02/2)2] exp[(e-ΔE + nℏω)2/2σ2]
|<v0'|vn>|2 are Franck-Condon factor,
|<v0'|vn>|2 = |<vn'|v0>|2 = (1/n!) * (ωR02/2)n exp[-(ωR02/2)2].
3. Absorption and Emission Pro Max: Same as 2 but at finite temperature T. Assuming instant thermalization --> states are population according to Bolzmann distribution,
Ia(e) = ∑mn |<vn'|vm>|2 exp[(e-ΔE - nℏω)2/2σ2] * Pm
= mn |<vn'|vm>|2 exp[(e-ΔE - nℏω)2/2σ2] * (exp[-βmℏω] /∑kexp[-βkℏω])
= ∑mn (1/n!) * (ωR02/2)n exp[-(ωR02/2)2] exp[(e-ΔE - nℏω)2/2σ2] * (exp[-βmℏω] /∑kexp[-βkℏω])
Ie(e) = ∑mn (1/n!) * (ωR02/2)n exp[-(ωR02/2)2] exp[(e-ΔE + nℏω)2/2σ2] * (exp[-βmℏω] /∑kexp[-βkℏω])
Pm = (exp[-βmℏω] /∑kexp[-βkℏω])
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Hello, this is my first post on this site. I'm an undergraduate student doing some Raman spectroscopy of CVD-grown graphene strained on silicon dioxide nanospheres. I notice that D and G' peaks show up in some measurements. The transfer process of the graphene to the silicon dioxide nanosphere-coated silicon chips I would think is far from perfect, as in certainly interferes with the structure of the graphene as there are rips and tears across the sample, as well as impurities and other things. You do however have "pristine" regions that only show G and 2D peaks.
On a loosely related tangent, I'm interested in how the molecular symmetry of graphene plays a role in its Raman spectra, and how that can be expressed mathematically. I wonder if perhaps the mathematical description of graphene in terms of group theory can possibly help explain the redshifts that occur in strained graphene versus unstrained graphene. If anyone has some advice or things to read about that, please let me know!
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In perfect graphene, the D-peaks are by symmetry not Raman active. ideal graphene should only show the single G-peak. However, when there is anything disturbing the symmetry and structure, the condition is relaxed and the D-peaks become visible. Hence, their name D for "defect" peaks. Based on the relation of peak intensity or the integrated intensity below the modes, the quality of graphene layers can be quantified.
There are numerous reviews on Raman spectroscopy in graphene, explaining which information can be extracted and how they can be extracted from analyzing the spectra.
Check, for example, this article by Andrea Ferrari:
You should be able to quickly find numerous more articles.
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I have myself tried using the basic method but including incomplete ionization to figure out the depletion width however I failed miserably because of many mathematical roadblocks. I was wondering if this had been done in the literature before and I just missed it.
If anyone can help me in this regard then I would be very grateful. Thanks.
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If you can send me a sample data file, I can have a look at it...
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We are planning to implement matrix based mathematical algorithms on FPGA. Could anyone suggest good book for these topics?
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You mustr try Practical FPGA Programming in C written by "Max" Maxfield.
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please is there any mathematical function that relates the SMD (D32) to the mean diameter (D50)? I actually understand what each of them represent.
thanks
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Thanks for this
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I am primarily interested in 2-player combinatorial games with perfect information. Useful wiki links are below.
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Is there a way to model the influence of pH on the electroosmotic EDL potential and velocity field inside the flat microchannel mathematically?
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Thank you Masuduzzaman.
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How could one increase data values from weekly to daily observations using mathematical algorithm for interpolation?
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Hello Nasiru,
What's the reason for wanting these values? That may have more to do with what sort of approach, if any, might make sense.
At first glance, doing this doesn't sound like a good idea to me. Among the reasons:
Any method for taking adjacent weekly values and inserting some estimate for the 6 six "missing" daily values will only bias the estimates of day-to-day variance in scores/values as well as increase the serial correlation across a lag from 1 to 6 days. As well, the presumption of a (perfectly) predictable day to day change is unlikely to be realized in actual measurements.
Good luck with your work.
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The mathematical representation of both equations is same.
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Please, see the attached file...
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Hi
I'm solving nonlinear second order equation by using finite difference method . finally for calculating value at any desired node, knowing three preceding nodes is required however by knowing boundary condition just one of these nodes becomes obvious and still knowing two other values is necessary. it must be noted there are plenty of guesses for values of these nodes which lead to compatible response.
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please write the ODE, not only the discretization You used.
however, for multi-step methods you need to create the starting values using a single-step method for all required nodes. Use a discretization of the same accuracy.
Note that the second order ODE could be written as system of two first order ODE.
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I am currently working on the use of metacognitive abilities to improve teacher proficiency of teaching mathematics in Primary schools. I am looking for international collaborators from Japan, Germany, Singapore, Netherlands, USA, Canada and Australia.
Thank you
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In My Opinion, Those with creative thinking skills and intelligence
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The 2023 ranking is available through the following link:
QS ranking is relatively familiar in scientific circles. It ranks universities based on the following criteria:
1- Academic Reputation
2- Employer Reputation
3- Citations per Faculty
4- Faculty Student Ratio
5- International Students Ratio
6- International Faculty Ratio
7- International Research Network
8- Employment Outcomes
- Are these parameters enough to measure the superiority of a university?
- What other factors should also be taken into account?
Please share your personal experience with these criteria.
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Cenk Tan; There are, of course, several websites that rank the universities worldwide. However, QS is the most famous of which.
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Here I attached the journal and the mathematical solution of the Rayleigh number. I need to plot graph exactly in Figure 3 from the journal but I didn't know the command. I have tried using this command,
plot(subs(N1 = 0.5, N3 = 2, N5 = 1.5, a1 = 1, a2 = 3, a3 = 1, Q = 10, R), a = 0 .. 10, 11 .. 22);
but the results show only a single line 2D graph. Hopefully anyone could help me with this. Thank you in advanced.
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You have only derived one relationship for R. Each of the graphs in Figure 3 show three relationships [1) linear inverted and parabolic; 2) heating from below and cooling from above; and 3) step function] So you need 3 different equations for R. Then there are the three cases [1) N1=0.1, Q=10; 2) N1=0.1, Q=20; and 3) N1=0.5, Q=10]. Maple is doing exactly what you've told it to do: plot one relationship for R. After Maple solves the equation, you could easily paste it into Excel and plot the results.
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topics should be where we can mathematics algorithms
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you can think of in covid 19, Lung cancer, Brain tumor detection using different deep Learning Techniques and possible to use also ML Algorithms.
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I asked people about their knowledge & Interest.
Most had no knowledge but high interest. Now I want to describe this relationship using statistical methods.
I used a likert scale so, the data is ordinal.
Using a Spearman's rank test I get a positive correlation with good significance.
But I don't really understand this result. I expected a negative correlation, since interest and knowledge frequencies have contrary slopes. Does the test look at individual pairs, where those with high knowledge may also have high interest?
PS: How could I form a mathematical equation to describe the relationship (regression)?
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Hello Lukas,
As the software reports, the estimated Spearman correlation is about .34 (Pearson very close, at about .35). Yes, the correlation is positive. That's because the general trend is that, persons having higher self-rating of knowledge tend to have higher self-rated interest, whereas those with lower self-rated knowledge tend to have lower self-rated interest.
Look within any row of your two-way contingency table, and as knowledge rating increases, you'll see a higher relative proportion of high ratings of interest.
Yes, it's true that the correlation is far from perfect. All that tells you is that you'll find exceptions to the general trend described above as you examine knowledge-interest ratings from the same person over your sample.
Mathematical relationship? (Using 0-5 scales for each variables)
Estimated rating of self-interest = 2.62 + 0.35 * rating of knowledge
Good luck with your work.
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I plan to divide my long research article(simulation + mathematical) into two parts, but I am clueless about how to do this. (I can not separate the simulation and mathematical analysis)
I have a few questions regarding the same
1) Do I need to show the common mathematics in both parts?
2) Can the introduction be the same?
3) Can some explanations remain the same in both parts?
Can someone give me the reference of any article divided into two parts?
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Dear Amit Das,
See my comment precede by >>
I plan to divide my long research article(simulation + mathematical) into two parts, but I am clueless about how to do this. (I can not separate the simulation and mathematical analysis)
>> If the mathematical model is highly original and valuable by itself - you can publish it either separately or along with key simulation results. If the model is quite standard, just briefly refer to it and focus on simulation results. Either way, splitting it on two articles hardly makes sense.
I have a few questions regarding the same
1) Do I need to show the common mathematics in both parts?
2) Can the introduction be the same?
3) Can some explanations remain the same in both parts?
Can someone give me the reference of any article divided into two parts?
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Complex systems are becoming one of very useful tools in the description of observed natural phenomena across all scientific disciplines. You are welcomed to share with us hot topics from your own area of research.
Nowadays, no one can encompass all scientific disciplines. Hence, it would be useful to all of us to know hot topics from various scientific fields.
Discussion about various methods and approaches applied to describe emergent behavior, self-organization, self-repair, multiscale phenomena, and other phenomena observed in complex systems are highly encouraged.
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Jiří Kroc: Greetings Prof. Kroc. In neurology the cutting-edge research is on 1) neurodegeneration, 2) neuroprotection, 3) the unification/entanglement between the nervous system and the immune system and 4) disorders of consciousness. thanks, Mustafa.
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In my previous question I suggested using the Research Gate platform to launch large-scale spatio temporal comparative researches.
The following is the description of one of the problems of pressing importance for humanitarian and educational sectors.
For the last several decades there has been a gradual loss in quality of education on all its levels . We can observe that our universities are progressively turning into entertaining institutions, where students parties, musical and sport activities are valued higher than studying in a library or working on painstaking calculations.
In 1998 Vladimir Arnold (1937 – 2010), one of the greatest mathematicians of our times, in his article “Mathematical Innumeracy Scarier Than Inquisition Fires” (newspaper “Izvestia”, Moscow) stated that the power players didn’t need all the people to be able to think and analyze, only “cogs in machines,” serving their interests and business processes. He also wrote that American students didn’t know how to sum up simple fractions. Most of them sum up numerator and denominators of one simple fraction with the ones of the other, i.e. as they did it, 1/2+ 1/3 according to their understand is equal to 2/5 . Vladimir Arnold pointed out that with this kind of education, students can’t think, prove and reason – they are easy to turn into a crowd, to be easily manipulated by cunning politicians because they don’t usually understand causes and effects of political acts. I would add, for myself, that this process is quite understandable and expected because computers, internet and consumer society lifestyle (with its continuous rush for more and newer commodities we are induced to regard as a healthy behavior) have wiped off young people’s skills in elementary logic and eagerness to study hard. And this is exactly what the consumer economics and its bosses, the owners of international businesses and local magnates, need.
I recall a funny incident that happened in Kharkov (Ukraine). One Biology student was asked what “two squared” was. He answered that it was the number 2 inscribed into a square.
The level and the scale of education and intellectual decline described can be easily measured with the help of the Research Gate platform. It could be appropriate to test students’ logic abilities, instead of guess-the-answer tests which have taken over all the universities within the framework of Bologna Process which victorious march on the territories of former Soviet states. Many people can remember the fact that Soviet education system was one of the best in the world. I have therefore suggested the following tests:
1. In a Nikolai Bogdanov-Belsky (1868-1945) painting “Oral accounting at Rachinsky's People's school”(1895) one could see boys in a village school at a mental arithmetic lesson. Their teacher, Sergei Rachinsky (1833-1902), the school headmaster and also a professor at the Moscow University in the 1860s, offered the children the following exercise to do a mental calculation (http://commons.wikimedia.org/wiki/File:BogdanovBelsky_UstnySchet.jpg?uselang=ru):
(10 х 10 + 11 х 11 + 12 х 12 + 13 х 13 + 14 х 14) / 365 = ?
(there is no provision here on Research Gate to write square of the numbers,thats why I have writen through multiplication of the numbers )
19th century peasant children with basted shoes (“lapti”) were able to solve such task mentally. This year, in September, this very exercise was given to the senior high school pupils and the first year students of a university with major in Physics and Technology in Kyiv (the capital of Ukraine) and no one could solve it.
2. Exercise of a famous mathematician Johann Carl Friedrich Gauss (1777–1855): to calculate mentally the sum of the first one hundred positive integers:
1+2+3+4+…+100 = ?
3. Albrecht Dürer’s (1471-1528) magic square (http://en.wikipedia.org/wiki/Magic_square)
The German Renaissance painter was amazed by the mathematical properties of the magic square, which were described in Europe firstly in Spanish (the 1280s) and Italian (14th century) manuscripts. He used the image of the square as a detail for in his Melancholia I painting , which was drawn in 1514, and included the numbers 15 and 14 in his magic square:
16 3 2 13
5 10 11 8
9 6 7 12
4 15 14 1
Ask your students to find regularities in this magic square. In case this exercise seems hard, you can offer them Lo Shu (2200 BC) square, a simpler variant of magic square of the third order (minimal non-trivial case):
4 9 2
3 5 7
8 1 6
4. Summing up of simple fractions.
According to Vladimir Arnold’s popular articles, in the era of computers and Internet, this test becomes an absolute obstacle for more and more students all over the world. Any exercises of the following type will be appropriate at this part:
3/7 + 7/3 = ? and 5/6 + 7/15=?
I think these four tests will be enough. All of them are for logical skills, unlike the tests created under Bologna Process.
Dear colleagues, professors and teachers,
You can offer these tasks to the students at your colleges and universities and share the results here, at the Research Gate platform, so that we all can see the landscape of the wretchedness and misery resulted from neoliberal economics and globalization.
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Mathematics is a language of all science subjects
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"The book of nature is written in the language of mathematics."
Galileo Galilei
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There are any papers or studies about the Blended Learning in Mathematics, exactly in (Calculus) ?
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The above article is very interesting and it strengthens my conclusions that this problem has been around with up for a long time - at least back to the early 1970''s when I first had experience observing new college students as a TA in grad school. I've taught off and on since the 1960's undergraduate and graduate level courses to both "math majors" and science and engineering majors. My observations were formed originally the early 1970's as the trend of more students taking calculus in high school were less prepared in the mathematical fundamentals and there was a gap in their preparation that showed up at the freshman college level. A teacher in calculus should not have to stop his/her lecture and go over the principle of mathematical induction or the basic analytic geometry of a conic section in freshman calculus. But alas - that was the case. In multidimensional calculus, the students should come in with a firm understanding of linear algebra. However, in my experience that was not always the case. That is the students had holes in their preparations and at some point those holes had to be filled or eventually they would hit a brick wall.
Calculus introduces new and sometimes difficult concepts and the students tend to have little intuition of the concepts of continuity and limits. But without developing a firm understanding and intuition of these and related topological concepts - it is difficult to grasp the calculus. If one does not have the background and facility of understand the fundamental tools of symbolic manipulation (a necessity to do all mathematics), understanding of basic geometry, the basic knowledge of set theory, mathematical induction and logic, it becomes a game of memorization and mechanics. Over time this lack of preparation in the fundamentals has grown more not less prevalent. This lack of preparations is reflected today in that many calculus sequences at the college level include analytic geometry. At one college I have taught specific courses over time, University of South Florida, there is now a required course known as "Bride to Abstract Mathematics." It is a course in logic, construction of proofs, set theory , relations and functions. That is the basic foundations that were once taught in high schools in the US but gave way with the push to present calculus in high school.
I expect we are seeing the same issues arise as subjects like abstract algebra and algebraic geometry become more and more important in applications like computer vision and robotics. It is difficult to apply mathematics to a field when ones understanding of the underlying mathematics has gaps.
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Philosophers of science typically recognize two kinds of values in scientific practice: (1) epistemic (or theoretical, or cognitive) virtues, like accuracy, testability, empirical support, etc, and (2) ethical (or social, or regulative) norms, like justice, egalitarianism, openness, etc. Of course, the strict separation of these categories is open to disagreement.
Are there values or norms (of either kind) that are unique to mathematics? Rigour (or provability) is one possibility; computability is another. Can you think of others? Do values play the same kind of role in math as in the natural sciences?
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Elegance and parsimony are mathematical virtues. They seem to be aesthetic, rather than ethical or straightforwardly epistemic. However, aesthetics can be an aid to cognition.
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Hi all,
There are instructions on the steps of running ‘align’ in a command-line way as shown in the following explanation found online (https://pymol.org/dokuwiki/doku.php?id=command:align). However, I didn’t found the mathematics or details for PyMOL running ‘align’ through the click-button way in GUI interface. Anyone know the mathematics or details behind it? Thanks a lot!
Yaohui
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"In short, “align” is a automated multi-step superposition algorithm based on dynamic programming and iterative refinement. “align” first performs a per-residue global dynamic-programming sequence alignment for the input atom selections using the BLOSUM62 weightings from BLAST. Then, a per-atom correspondance is established between atoms in the selections. Matching side chains atoms will be included if they were provided in the selection arguments. An initial superposition is then performed followed by up to five cycles of iterative refinement wherein atoms with per-atom deviations over two standard deviations from the mean deviation (if any) are thrown out and the fit is repeated. Finally, the number of atoms remaining at the end of this procedure is printed out along with the resulting RMS value for those atoms."
Pymol offers different alignment methods, which are optimised for different levels of sequence similarity:
Basically, "align" https://pymolwiki.org/index.php/Align does a sequence based structural alignment, first doing a sequence alignment to decide which Calpha atoms to perform a 3D alignment on, then iteratively removing very poorly 3D-fitting atoms from the alignment to reduce the influence of structurally variable loops on the alignment.
In contrast "super" https://pymolwiki.org/index.php/Super aligns two selections by a sequence-independent (unlike align) structure-based dynamic programming alignment followed by a series of refinement cycles intended to improve the fit by eliminating pairing with high relative variability (just like align).super is more robust than align for proteins with low sequence similarity.
"cealign" https://pymolwiki.org/index.php/Cealign aligns two proteins using the CE algorithm. It is very robust for proteins with little to no sequence similarity (twilight zone). For proteins with decent structural similarity, the super command is preferred and with decent sequence similarity, the align command is preferred, because these commands are much faster than cealign.
Each of the commands offers a number of parameters to tweak the method, e.g. setting limits to the maximal gap length and specifying the number of iteration cycles
You can also extend the PyMol capabilities using the "TMalign" module https://pymolwiki.org/index.php/TMalign
To do superpositions between exactly defined atom sets (e.g. in ligands), use "fit" https://pymolwiki.org/index.php/Fit or "pair_fit" https://pymolwiki.org/index.php/Pair_fit
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How can you advise on the methods of mathematical prediction of the causes of salinity?
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thank you
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One of my research questions is about the level of democratic practices in the mathematics high school classroom. When examining the normality of the the demoratic practices, it turned out that they are not normal. In normal samples, we use one-sample t-tests with critical values that fit our assumptions of the level of the democratic practice. For example if we want to take into consideration three levels, we divide (5-1)/3=1.33. This enables us to consider what is below the grade 2.33 weak, what is between 2.33 and 3.66 midium, and what is abover 3.66 high. This serves us in doing the one-sample t-test, where the critical value is 2.33 or 3.66 to verify the level of the variable.
My question is: How can we do that in case of non-normal distribution?
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I really thank all of you for your valuable comments. I attach an article in which to find the significance of the level of democratic practices, we used one sample t-test. My question was about what we do if the data does not follow normal distribution.
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I would like to know the effect of temperature and pressure on the density of Fe2O3 nanoparticles, and if there are some tables or equations that describe the change mathematically.
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It is very hard to have a unique mathematical expression that will described changes in density x temperature of treatment. It will strongly depends on your synthetical route of the samples, the crystal structure, porosity, and so on.
Normally, increasing the temperature and pressure you will see the densification of the nanoparticles. However, you can see a phase transition that changes the density of the system, so it is not a straightforward analysis. I recommend you to synthesize several NP at a varied range of temperature, then perform porosity/density measurements employing helium pycnometer.
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Combining more than one higher quality has less quality. So how do we express this mathematically? E.g; Let 2x'2 - 4x'2 +6x'2. If it is noticed, one of them has 1 term and the other has 2 terms. So what do we get from here? it looks like 4x'2 =8x 6x'2=12 x sums will be 20x when the derivative is taken. However, if we take their exponent without the differentiation, it will be 4x'2 =16 6x'2 =36. So when more than one higher attribute comes together, it only has less attributes when deriving. THIS IS ANOTHER DEFINITION OF THE DERIVATIVE. THANK YOU. I didn't actually get this out of the way. THERE WAS A LOT OF STARS IN THE PARIS SAINT GERMAIN FOOTBALL TEAM; HOWEVER, WHEN PLAYING TOGETHER, MANY QUALITIES TURNED INTO FEW QUALITIES. THANKS.
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Maybe you have not captured all of the relevant qualities. Best wishes David Booth
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How can all these discrepancies be explained mathematically?
I am working with a scale that could be considered to have 11 dichotomous (0-1) and polytomous (0-2, 0-3, and 0-5) items OR 20 sub-items (dichotomous and some polytomous). The sample size is sound (> 700 subjects).
A) If I do an exploratory analysis* with 11 items on half of the (randomized) sample, I get 2 factors. The confirmatory analysis** with the other half confirms the two factors, presenting good adequacy indices values. Some authors also obtained 2 factors, either with similar or different methodology (including item-theory analyses).
B) Moreover, I also tested unidimensionality as some found only 1 factor. Again, all indices are adequate (ECVI slightly higher).
C) However, if I do an exploratory analysis* with the 20 sub-items (similarly to other authors), I get 4 factors. Additionally, just out of curiosity, in the total sample, I get 5 factors!
E) There have also been 3, 4, and 5-factor solutions in the literature, either with 11, 20, or 30 sub-items (all dichotomous).
* Unrotated EFA; maximum likelihood extraction; eigenvalues > 1; based on tetrachoric correlation matrix followed by an oblique rotation (promax).
** CFA with the Diagonally Weighted Least Squares method.
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I don't quite understand why the same scale would have 11 or 20 items, but it is not surprising to me that with more items, you could "find" more factors. With more items, there is more room for additional factors (e.g., specific or method factors) to be measured and identified. The more items you use, the more likely it is that some of them will reflect something that is at least slightly different or "unique" relative to the other items. Likewise, with a larger sample size, you have more power to reject models with fewer factors.
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Basically, I am from a Mathematics background. I found interesting to read ANN but I don't have proper knowledge of ANN, especially in the calculation part. Can anyone help me or suggest me a book considering my situation. Thank you.
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If your background is Mathematics the attached "PNN Primer.pdf" may be useful.
For numerical practice download the NNB DEMO software package (only for Windows), unzip and run. It comes with bundled documentation.
In no time you will be handling neural networks numerically.
This is the link to NNB DEMO, which expires on Thursday May 5, 2022.
Please tell if NNB DEMO fits your expectations.
Best regards
Daniel Crespin
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Dear talented researchers,
I want to calculate three solar irradiance components (DNI, DHI, & GHI) from these equations which contain logistics, maximum and minimum, Less than, Greater or equal, terms.
How can I use these equations please? I don not know how to use equations 12 & 13 because they have terms such as max, min.
I am not sure, but it seems to be a Logistic map.
1- just for example, how to solve this part from equation 12?
min (1:88 * 10^-8 CO^4, SOLZEN < 77)
2-how to deal with these mathematical operation such as greater than, less than 77, equal or greater than inside these equations?
Thank you very much for your kind response.
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Dear Zhwan
I have been thinking in your problem and I b elieve that the logistic part that bothers you could have the following explanation:
Logistics refers to a curve normally known as a sigmoid (very similar to the letter S,) but with right horizontal lines at both extremes, instead fo the curls, and very used in many disciplines, most especially in Artificial Intelligence, and normally varies between -x and x. It is defined as tanh and equal to the ratio between sinh and cosh.
In my opinion, for your geographical position, you should compute the ratio solzen and solalt , and then multiply it by CO^2, which meaning I don’t remember.
Because the sigmoid, and think that we can safetly take a value of 0.5, since it is the average of the linear part of the curve.
Multiplying 0.5 by CO^2 this product MUST be < 77.
Just tu represent that in MCDM is my problem
Therefore, I would appreciate it if you can give me the following values:
CO
CO^4
SOLZEN
SOLALT
1/SOLZEN tan^2
I believe that with this data I could be able to solve the problem using SIMUS.
I agree with your efforts to use algebra and trigonometry, but how are you going to express the < relationships?
Anyway, let me know if you agree with my suggestion, and if you do, give me the values I request.
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Six Nobel Prizes are awarded each year, one in each of the following categories: literature, physics, chemistry, peace, economics, and physiology & medicine. However Mathematics a subject mankind cannot do without is a strange omission and has remained excluded until today. Same with accounting. From 1901 doyens such as Albert Einstein, Marie Curie, Earnest Hemingway were honored with the prestigious Nobel. Do you think it’s time to rethink ?
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The Abel prize is a new mathematics prize of the Norwegian Academy of Science and Letters, dedicated to the memory of Niels Henrik Abel (1802-1829) on the occasion of the bicentenary of his birth. It is modeled after the Nobel Prize, and developed from a proposal by the mathematics department at the University of Oslo in fulfillment of a request formulated by the Norwegian mathematician Sophus Lie towards the end of the 19th century.
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AbelPrize.html
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Pls can someone assist me with details on how to calculate effective dose mathematically using exposure parameters such mAs, kV, FFD in x-ray examination
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suppose that A and B are two entities. And arrival of B depends on the state of A. The arrival of A is known and can be expressed in mathematical conditions but B can not be as it depends on A. The arrival of entity depends on when A gets processed (Lets say the process is Process 1 with a triangular func of (5,6,7) mins). Whenever A starts to get processed, B is created. I have been able to create the first creation of B logically which relates to A but, I am not sure how to provide the logic for "Time Between Arrival". 
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I stuck with this kind of problem. However i want to solve this. I have two types of entities, yet rather than use 2 create modules, these two types of entities are created using 1 create module and then I use assign module to ensure they are different entities. How to ensure that a sum of entities are necessary to be processed and dispose from the system before another sum of entities proceed with different process? Both types of entities will be processed using different processes.What condition can be used in decide module to allow this event?
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Example
How to find w1, w2,..., wk?
All maximization or minimization
y1=f1(x1,..xn) (maximize)
y2=f2(x1,..,xn) (maximize)
........................
yk=fk(x1,...,xn) (maximize)
F=w1.y1 + w2.y2 + ...+ wk.yk
Mixing of maximization or minimization
y1=f1(x1,..xn) (maximize)
y2=f2(x1,..,xn) (minimize)
........................
yk=fk(x1,...,xn) (maximize)
F=w1.y1 + w2.(1/y2)+ ...+ wk.yk
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Yes, I know what you asked about. But my advice was mainly about a transformation of objectives so that they could be "comparable".
As to weights, you should say what you want to calculate, because the "practical backgorund" really matters.
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Is there anyway to relate the Exciton binding energy and Urbach energy of semiconductor of thin film? Can we mathematically relate these two parameters? Any suggestion on this topic would be helpful.
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@Muhammad Hamza El-Saba
Could you please provide the link of your book if accessible to me.
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Hello to all
I am an electrical student and I have a question about optimization.
Can you help me?
To optimize (mathematically not meta-heuristic algorithms) the values of the elements and the size of the inductor-capacitor, etc. of an electronic power converter I need a few examples of formulation and simulation. (Preferably in GAMS)
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You can use ads for that, try to also download ads tutorials book for guide on how to go about it.
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Hello
I'm learning camsol. I studied the mathematical particle tracking method used for modeling in turbomolecular pumps, and I can model a single-stage rotor, but I can not model a single-stage rotor and stator.
Can you guide me, please?
thanks
maryam
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Any advances?
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I want to draw a bifurcation diagram using either maple,mathematical or Matlab. an example is the attached file.
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Respected sir,
My name is fawad nadeem belong to Pakistan. I am interested in bifurcation matlab code. Kindly provided me the bifurcation code.
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Stiffness is associated with a small change in the input producing a large change in the output. The phase-field equations such as the non-conserving Allen Cahn and the mass conserving Cahn-Hilliard are stiff differential equations. The solution of these equations represents the dynamic of the interface between two phases. The Allen-Cahn equation is second-order while the Cahn-Hilliard equation is fourth-order nonlinear PDEs whose mathematical form is given in the picture included below with appropriate initial and boundary conditions. The parameter epsilon in the equations represents the thickness of the interface.
Thanks in advance for all of your consideration.
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Thank you so much to all of you for your detailed explanation and interest.
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How to analyze the mathematics textbook from the prospective of student‘s use in order to examine how textbooks support students to do mathematics learning?
Is there any theory that could be used for doing such research?or some relevant articles?or give some suggestions?
Thank you very much!
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Good morning
Which level?
Best regards
Ph.D. Ingrid del Valle García Carreno
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I wish to extend a paper by incorprating the particular feature the authors havent used or considered. However after going through the litreature It isnt clear how much that particular feature plays a role, all I know it does play an very important role for the output that I care about. For experimentation I am assuming a simple linear regression function ax+by where a serves as the contribution to the paper I am extending and x its feature set, my goal is to find the parameter b (mse minimization) by encoding the feature in variable y and thus determine the strength that y plays
However there are some limitation first of that I am assuming the relationship be linear which is very wide of assumption , and I m hoping to consider some kind of non linearity
Question is how do I proceed from here. Is there any mathematical equation I can consider as intial assumption
PS: Note Y is here a continous value not categorical
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In my view scientific research is about explaining or predicting phenomena. And that is impossible without the use of models, whether they are implicit, or explicit.
And the data are only "visible" by using , whether you are aware of that or not, models. And very often the language of mathematics is used to generate these models. In that sense we all are descendants of Isaac Newton.
Very often I do not see any explicit model, but instead a multitude of procedures in order to process raw data, and basically all is then a question of curve fitting, or one step ahead prediction, where the curve is not known, and the measure of goodness of fit is often not clear at all. And the scientists, that use these models very often do not have a clue about the relation between data and model.
And people from statistics or machine learning barely speak the language of each other. There is hard work to do in University, to redress this, and to partly destroy the Tower of Babel. But as long as we get our papers published , while nearly no one is reading them, the atomization of research will continue.
And that worries me!
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A very interesting topic, "quantification of randomness" in mathematics it is sometimes reffered to as "complex theory" (although it is more about pseudorandom than randomness) that is based on saying that a complicated series is more random and then there are tests for randomness in Statistics and perhaps the most intriguing test related to information theory -"entropy"(as also being of relevence to and result of second law of thermodynamics), while there are also random numbers generators (pseudorandom numbers generators) and true random numbers generators using quantum computing.
So, what I've been trying to, is making a complete list of all available algorithms or books or even random number generators that will allow me to tell me how much random a series is, allowing me to "quantify randomness".
There are 125 unique infinite series which are pseudorandom that I have discovered and generated based on a rule, now how do I test for randomness and quantify it? Uf the series is random or there is probably a pattern, or something that will allow me to predict the next number in the series given I don't know what the next number is.
Now, do anyone know of any github links based on any of the above? ^ (like anything related to quantifying randomness in general that you think will be helpful).
A book/books on quantifying randomness will be very very helpful too. Actually anything at all...
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You should check out seminal and fundamental work by Gregory Chaitin starting in 1965 when he was a student in CUNY (City Univ. of New York) and continuing through the 1970's.
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Is there mathematical relation ship between the twist per turn vs the frequency at which cross talk deep occurring ?
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to be published in high reputation journals
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Relationship of Laplace Transformations and Beta-Gamma F ( x ).
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Human dynasty in its millennium era. We have identified fire from the friction of stones and now we are interacting with Nano robots. Once it was a dream to fly but today all the Premier league, La liga and Serie A players travel in airplane at least twice in a week due to the unprecedented growth of human science. BUT ONE THING IS STILL ELUDING IN THE GLITTERING PROFILE OF HUMAN DYNASTY.
Although we have the gravitation theory, Maxwell's theory of electromagnetism, Max Planck's Quantum mechanics, Einstein's relativity theory and in most recently the Stephen Hawking's Big bang concepts...… Why can't we still revert back and forth into our life?
Any possibilities in future?
if not..
Why? in terms of mathematics, physics and theology??
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Dear Shuraik Kader,
Given Albert Einstein's theory of relativity, cosmology regarding the development of the universe, quantum mechanics, future technologies for building interplanetary spacecraft, etc., time travel is theoretically possible. But in practice the building of a time machine by humans is impossible. Even if a man would achieve the required technological development in the next several hundred years, unfortunately he would not have enough time for it. First of all, a person must first solve other global problems, such as the necessary one is urgent, i.e. in the perspective of the next max. 2-3 decades of time, stopping or significantly slowing down the progressing global warming process, achieving zero-emission economy and avoiding a global climate catastrophe, which may occur at the end of the current 21st century. In order for man to be able to create new technologies of the future, to be able to build interplanetary manned spacecraft, etc., he must first save the planet's climate, biosphere and biodiversity from the risk of almost total degradation.
Best regards,
Dariusz
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I’m currently doing a project and have a categorical independent variable and a continuous dependent variable. I am trying to find which group in the categorical variable produces the highest values for the continuous variable. I have already done ANOVA and post hoc tests. I was wondering does anyone know of any other mathematical or computing methods which could help me with this?
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maybe it will be more clear with an example: (categorical variable = fruit, categories =apple, orange, banana, continuous variable = shelf life in number of Days e.g. 8) I have already used ANOVA to test to see if there are differences between the fruit for shelf life. and have used post hoc tests to find which fruit combinations have different shelf lives (e.g. apples and oranges have different shelf lives)
i am wondering are there any other mathematical/computational methods I can use that will help me determine which fruit has the longest (maximises) shelf life
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I have a problem with finding references for high-order generating functions. For example in finding explicit formula of this recurrence relation: https://mathoverflow.net/questions/266478/linear-two-dimensional-recurrence-relation
Actually, in my research, there is a three-dimensional recurrence relation. Does anybody have some books about high order generating function in general?
I really appreciate any help you can provide.
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Peter Breuer Thank you for your kindly reply, Sir. Yes, that's the generating function that I mean. Do you have any references (papers or books) about this thing? Especially for a higher dimensional.
I appreciate any help you can provide.
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I have 2 functions f(x,y) and g(x,y) that depend on two variables (x,y), so I want to find a solution that minimize f(x,y) while maximizing g(x,y), simultaneously??
P.S: These functions are linearly independent.
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Metamathematics -- the fundamental logical paradigm of maths -- was never fully defined by Hilbert (nor anyone else), causing severe yet commonly ignored consequences for all branches of maths and maths theory ever since. So, it is difficult to find relevant papers and anybody interested in investigating or discussing the subject.
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What kind of statements are metamathematics: logical, mathematical, logical-mathematical, philosophical? In the case of Hilbert's claims, they were aimed at finding a secure foundation for mathematical theories, and he thought that this could be achieved if the consistency of arithmetic was established, but that is only an assumption as to "foundation" which, furthermore, results from assuming that logic (because consistency is a formal property studied by theories such as logic) is a kind of guarantee for human knowledge, but logical theories and have their own problems (such as the inapplicability of the principle from the excluded middle to quantum mechanics) and I believe that if the 20th century has taught us a lesson in terms of analytical philosophy and the "foundation of mathematics" it is that it is not even possible to find a deductive language -or an axiomatic system- that is perfect or basic, nor can logic be an unquestionable starting point.
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I want to arrive at a mathematical expression finding the natural frequency of two parallel beams connected by a coupling beam at some location. Is there any helpful theory/literature/approach to get it.
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Dear Bollapragada,
Please check out the following paper:
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What is the best mathematical approach in each statistical analysis? Does it depend on the analysis level?
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You have some question you want to answer. You choose a method that is capable of answering it . Then collect data and use the method. Answer the question and then write it up., Just like in science class. David Booth
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Can someone please share the relevant mathematics and explanation for the first order analysis of a BJT current mirror? Any link to an article/book/chapter will be very useful. You can also attach the document if that is convenient.
Thanks.
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How to calculate the critical length of fiber in fiber-reinforced polymer composite? Is there any mathematical formula there or we can keep some assumptions?
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Dear Vishal Gavande,
PFA file FYR will help you to understand the calculation.
Apart from that,
In the Coulomb friction law, the critical shear stress is defined as τ = μp, where μ is the coefficient of friction and p is the contact pressure between the surfaces.
What is critical shear stress?
(solid-state physics) The shear stress needed to cause slip in a given direction along a given crystallographic plane of a single crystal.
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The problem is described in the enclosed sheet with mathematical expressions.
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I want analysis in the differential equation.
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How to mathematically interpret the formula Kb=Kf (1−Rw)
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Hello,
my name is Athanasios Paraskevopoulos, a MSc. student in Mathematics from the Hellenic Open University. I am looking for partners, who work (or used to work) in the field of Didactics of Mathematics.
If you're interested helping me with my study please feel free to contact me via Research Gate or mail: at.paraskevopoulos@protonmail.ch
Thank you and kind regards,
Athanasios
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okey
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I have a simulation code for a Horizontal Washing Machine.
The code solves the equations of motions of the system by Matlab ode45 and plots the vibration response of the system at the transient state of performance.
In this code, the frequency (omega) is an exponential function of time, as it's stated below (and its diagram is attached to 'the question'):
-------------------------------------------------------------------------------------------------------------------------------------------------------
omega= (1-exp((-0.5)*t))*omega_0+(1-exp((-0.5)*heaviside(t-t1).*(t-t1)))*(omega_1-omega_0);
-------------------------------------------------------------------------------------------------------------------------------------------------------
ode command:
-----------------------------------------------------------------------------
[T,Y]=ode45(@snowa1,tspan1,initial_vector1);
plot(T,Y(:,1)-mean(Y(:,1)))
-----------------------------------------------------------------------------
The resulting displacement response is attached to the question.
It is desired to :
First, increase the frequency to omega_0 by exponential1
Then, increase it to omega_1 by exponential2
But 'the problem' is that:
the displacement response shows an unexpected increase in frequency at the beginning of the second exponential increase (it becomes 20 Hz, which is much larger than the maximum frequency in the simulation- 10 Hz).
Do you know what could be the reason for this response?
Any help would be gratefully appreciated.