Mathematics - Science topic

I am sending you an article on this subject.

Say θ∈R^D and ℓ(θ) is the log-likelihood. Then ∇²ℓ(θ) is the Hessian of the log-likelihood (the DxD matrix of the partiel second derivatives for θ). The observed Fisher information is I^* = −∇²ℓ(θ^∗) with θ^∗ being the vector of the maximum likelihood estimates. The standard errors are the square-roots of the diagonal of the inverse Fisher Information:

se = sqrt( diag(I^∗−1) )

Example: exponential distribution with parameter θ

L(x) = θ·exp(-θ·x)

ℓ(x) = log(θ) - θ·x

ℓ(x) = n·log(θ) - θ·Σ_ix_i (for a sample of n values x = (x₁, x₂, ..., x_n))

∇¹ℓ(θ) = n/θ - Σ_ix_i

∇²ℓ(θ) = -n/θ²

θ^*: ∇¹ℓ(θ^*) = 0 ↔ n/θ^* - Σ_ix_i= 0 ↔ n/θ^* = Σ_ix_i·Σ ↔ n/Σ_ix_i·Σ =θ^* = 1/x̅

I^* = -∇²ℓ(θ^*) = n/(θ^*)²

se = sqrt( (n/(θ^*)²)⁻¹ ) = sqrt(1/n) · θ^*

The imaginary compartmentalization does not remain of the same dimension all throughout the processualization of complex functions . Even self-determination is ensconced in a sphere where the focus is not on mathematical rigor but rather on collecting some bits of data on the functionals [ of the originary function ] on the very powerful machinery of manifolds and “post-Newtonian calculus”. The systematics of functional differential equations does not have a continuously differentiable solution for every value of the parameter , say , a .

Adnan Mohammad Awad Roshdi R Khalil Hening Huang

The theory of the interacting gases as the electrons in solids is still

in infancy. Many areas of physics need more development. Cosmology is still under uncertainties. Particle physics is hard, and in infancy.

Philippos Afxentiou I am not sure if I have identified the topic of the discussion correctly. I understand that you are making the point that physics must be a unified subject applying at the very small scale and also at the very large scale.

Sydney Ernest Grimm makes a very good point about the CMB rest frame and it does suggest a unique frame of reference within the universe. Also the recent LIGO experiments have shown that gravitational waves and electromagnetic waves travel at the same speed even across expanding space. This suggests that light travels as a wave disturbance of the medium of space. The space rest frame then explains the CMB dipole:

These ideas which arise from the large scale of the universe then can be applied at the small scale. Light is a wave in the medium of space so electrons must be looped waves in the medium of space.

Richard

Hello Leopold Modrakowski

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

The Gamma function G(n) is well defined for any positive value of n,

G(n)= Integral from 0 to infinity [Exp(e^-x^n)]dx . . . . . (1)

Needless to say, it is of great mathematical and physical importance.

However, for practical purposes, it can be calculated using an adequate closed-form polynomial without going through complicated numerical integration.

The required closed form solution must retain its mathematical and physical properties, namely:

i-minimum of Gamma occurs at x = 1.4616321 and the corresponding value of Gamma(x) is 0.8856032.

ii-Gamma(1.)=Gamma(2.)=1.

iii-Gamma(x)=(x-1.) !

iv-The recurrence relation ,Gamma (x)=x. Gamma (x-1)

We propose a simple preliminary approach that gives the required value of Gamma(x) with an error less than 0.001. The proposed preliminary approach must satisfy conditions i-iv,

but since the factorial function x! is not yet defined for negative numbers (x<0), we divide the entire positive x-space into three intervals as follows:

a) x element of ]0.1]

Here, the proposed second-order polynomial expression for the Gamma function is G(x)=F(x-1) where F(x) is the factorial function x!. F is approximated by,

F(x)=(1.-0.46163*x+0.46163*x*x) . . . . . . . (2)

x element of [0,1].

b) x element of [1,2]

The Gamma function is approximated via the expression,

G(x)=Done(x)+0.3333/X**1.5 . . . . . . . . . .  (3)

Where 1/3 *1/X. Sqrt(x) is a correction factor.

c) x element of [0,infinity[

We can here use the expression (4) supplemented by the expression (2) for the remaining fraction,

G(x)=F(x-1) . . . . . . . . . . . . . . . . . . . . .. . . . (4)

Equations 2, 3 and 4 were implemented in a simple algorithm which produced the required numerical results.

Table I presents some examples of numerical results of the proposed technique compared to those of the numerical tables obtained by numerical integration of Eq. 1.

Table I. results of the proposed method vs those of the numerical tables obtained by numerical integration of Eq. 1.

x                          10.5                     1.4616                      0.5                     0.25

G(x) Proposed    11877478           0.88527            1.82646                    3.5798

technique

G(x) numerical   11899423.084 0.8856032.       1.7738                      3.6534

tables

First, what do you mean by a number? If you mean a set of things that includes, say, rational numbers and extends the addition & multiplication operations in a way that is consistent with the usual rules (e.g., associativity: a+(b+c)=(a+b)+c & a*(b*c)=(a*b)*c, distributivity: a*(b+c)=a*b+b*c, zeros: 0+a=a, 0*a=0, commutativity: a+b=b+a, a*b=b*a, unit/identity: 1*a = a, inverses: every a has a b=-a where a+b=0, every a not zero has a b=1/a where a*b=1) then we can make a claim that the complex numbers are the biggest class of "numbers".

Actually, you can go further by constructing a non-Archimedean set of "real numbers" which extends real numbers and is still an ordered field, and then have an extended set of complex numbers of the form a+i.b where a, b are these extended real numbers.

Non-Archimedean means that there are "numbers" a, b > 0 where a/b is larger than any whole number 1, 2, 3, ...

But if you keep the regular real numbers, but are willing to lose something else like commutativity of multiplication (a*b=b*a) then there are the quaternions discovered/invented by William Rowan Hamilton in the 19th century. These have the form a+b.i +c.j +d.k where a, b, c, d are real numbers. The essential properties of the symbols i, j, k are that i*i = j*j = k*k = -1, and i*j = -j*i = k, j*k = -k*j = i, & k*i = -i*k = j. Quaternions form a division ring.

I dont see very well how you could achieve the "unity of mathematics"

based on series/sequences alone. For example geometry would remain apart. you must persue some more modest goal.

@ Andrea Ossicini,

Interesting - thanks for sharing

Problem solving technique and thinking aloud strategies helpful for mathematics teaching

The tandem of Mathematics and Computer Science is vital for the development of AI. Mathematics provides the theoretical foundation, while Computer Science provides the practical tools to implement AI algorithms. By combining these two disciplines, researchers and practitioners can create AI systems that can perform increasingly complex tasks, and help solve some of the most challenging problems facing our society.

Dear Dr Babura,

I think that you mean how Sturges came up with his rule. Part of your question is explained in the following link,

Please let me know if this is what you are looking for Dr.

Best wishes.

See attached file

Je pence qu'il y a une grandes différences entre la physique mathématiques et les math, où les maths sont des outils utilisés dans la physique comme les langues, par contre la physique mathématiques est une description mathématiques pour les phénomènes naturels, c'est a dire ces une autre représentation discret des phénomènes par des équations ( ces variables c'est des être naturels comme le temps, distance, énergie massa...), des ça aide à bien maîtriser et comprendre ces phénomènes. Bien sûr pour un physiciens il devrait maîtriser les math pour qu'ils puissent expliquer ces phénomènes et mieux comprendre la nature

To sum up

1. The issue is not about removing mathematics or questioning their role in physics. It is about what motivates Ph. D students most.

2. Current physics master curricula are, dispite the existence in some of a parallel non written or expressed properly strategy in the programs, are monopolizng learning goals to undepagogical levels.

( 80% Application goals to Calculations and 20% understanding (explain why) in specific situations. Pedagogical this is a scandal. Bloom demands 4-2-2-2-1 ration in knowledge, understanding, application, synthesis, evaluations. Essay qs address the latest 2.)

3. I believe students should be inquired to experience by self-discovery radical formulations that involves new conceptual ideas ornee physical quantiries as Smolin's variety, views that provenly mathematically rebuild established theories

Hello Rosario Rox

Use the formula C1V1= C2V2

C1= 4.26 x 10^6

V1= volume to be taken (Xml)

C2= 0.25 x 10^6

V2= required volume (30ml)

4.26 x 10^6 x Xml = 0.25 x 10^6 x 30ml

X= 0.25 x 10^6 x 30ml / 4.26 x 10^6

Therefore X= 1.76ml

So, add 1.76ml of cell suspension (the one at a density of 4.26 x 10^6) into 28.24ml of culture media to obtain 0.25 x 10^6 cell density in 30 ml cellular suspension.

Best.

You might find some inspiration in the examples that come with this software:

1. Sort the data: Begin by sorting the data from smallest to largest, or from largest to smallest. This will make it easier to find an optimal value.

2. Calculate the mean or median: Calculate the mean or median of the data set, which will represent the average value.

3. Find the mode: Find the mode, which is the most common value in the dataset. This could be an optimal value.

4. Look for outliers: Look for any outliers in the data that could skew the results. Remove any outliers and recalculate the mean or median.

5. Apply constraints: Apply any constraints that are applicable to the data. For example, if you are looking for an optimal value within a certain range, then you would only consider values within that range.

6. Select the optimal value: Once you have considered all the applicable factors, select the optimal value that best meets your criteria.

On the base of my experience it is better to use native language or intermediary language for detailed explanations. You can find the short video with math content in YouTibe and use as the additional manuals in intermediary language.

Thank You for an opinion!

Howdy Robert A. Phillips,

We share a sense of Nature and of knowledge. Your preprint of "The Cause of the Gravitational Effect" reminds me of Albert Einstein's first cut in 1911 at the bending of the path of light in a gravitational field gradient. Your observation on KE in a fluid is appropriate to my fluid dynamics focus, but I would word it "Kinetic energy involves more than just the body. In any case our agreement is quite pleasant.

"Using a natural fluid model to interpret the nature of cosmological interactions resolves many of the "big questions." " You might enjoy my condensation cosmology concept. The Universe did not "inflate," but as primal energy condensed to matter, light slowed (same speed always is measured) and the Universe has appeared to expand.

The issue of whether natural processes "are quadratic" represented by v² in K.E. = mv²/2 remains of interest to me in forming an object oriented model of flow in a mountain stream rapids. What is happening "in" a turbulent flow structure that must be specified in its object class data and functions, whether it is processed analytically or by a computer program. Imaginary and irrational number expressions may freely be used during calculation as long as they are made rational in the end. Do we need to notice that; does nature notice that; does it matter? Daniel Boone was never "lost" in the Kentucky wilderness, but on occasion he was "mighty bewildered" for days. Maybe I'm just bewildered.

Happy Trails, Len

There has been a development regarding the usage of Steepest Entropy Ascent formalism which modifies QM by incorporating the second law of thermodynamics in the formalism itself. The research shows that this is quite a consistent modification, including preclusion of signaling and other things. However, this approach is motivated by the requirements of consistency with thermodynamics, and not from the information theoretical approach.

You may check the following works, they might be of interest.

https://dx.doi.org/10.1098/rsta.2019.0168,

https://dx.doi.org/10.1103/PhysRevA.91.013848,

https://dx.doi.org/10.1103/PhysRevE.90.042113,

And some recent work shedding light on this

The issue of scientific continuity can be defined as the preservation of knowledge, research progress, and expertise in a field over time, despite changes in personnel, funding, and other factors. Scientific continuity is important because it ensures that research programs and projects can be continued over the long term, and that valuable knowledge and data are not lost or forgotten.

To assess the issue of scientific continuity, the following factors can be considered:

By considering these factors, researchers and institutions can determine the level of scientific continuity in a field, and take steps to address any gaps or weaknesses to ensure that research progress is maintained over the long term.

Thank You Sir.

Consequence number one: the numerical axis exhibits the properties of fractality not only through the Cantor fractal.

Howdy Folks,

I am satisfied that mathematics and physics (science) have been well defined and described here. A couple movies are running in my mind's eye that I wish to pass along as afterwords - they are observations not insults.

In Ray Bradbury's work "Medicine for Melancholy" he includes a prose movie of Pablo Picasso sketching a mural in the moist sand of a long beach as the tide is coming in - just like the "then current" theories in science presented by academics as truth, the foaming edges of the waves wash the mural away - new paradigms replace old and the human creation of science is adjusted. It is not "nature" even now.

M. C. Escher's "Metamorphosis" is a great contribution to defined elements fitted together perfectly into a closed, consistent, unnatural whole. Rigorous, however independent of nature, EXCEPT FOR THE FACT that humans and their imagination are natural. I disagree with the separation of "artificial" from natural, except as a verbal convenience.

These are creations of human minds, not figments of human imagination. And I carefully avoided an observation that fresh ideas are not fertilizer and to bury them in a field will not benefit flowers there.

Great exchange, Thanks again, Happy Trails, Len

Thanks for your thoughts and wishes

Many people write to me, and my mailbox is very accepting.

Some simple people assume that their questions are, somehow, special, worthy, or unique. They even may want to claim innovation!

But, we live in a fluid, and many of our thoughts are not even

original, or our own. Ego interferes.

Only correct living, like following the Dhrama and Jesus, can keep the laboratory of the mind clean and receptive to higher thoughts.

The complex numbers have been followed without deep questioning, and their justification, in medieval Italy, was to solve the equation x^2+1=0.

But, they actuall lead one astray from the actual solution.

This was revealed by the FFT, that can now be entirely calculated with rational numbers, and without any trigonometric functions, as mentioned above.

Then, we conclude that FT ~ FFT, as the RG preprint briefly explains. For the savvy, a letter cn b a wrd!

There are no mathematical real-numbers either. Thus, the set C is deprecated and the set G is superfluous. They cannot be calculated (Gisin,Ozhigov, Gerck)with probability 1.

Mathematics gets simpler, and correct, accessible to natural processes, and accessing them in quantum mechanics.

Then, the Shor quantum algorithm of 1994, without any practical result since then, only theoretical results, opens the calculations of the FFT, to the components of numbers.

This reveas the prime numbers as "lumps", resonances in the number line, much in the same way that a physical chord has physical resonances. Easier to calculate, with such a physical model.

In fact, and the meaning of the question is not understood.

Plato, who preceded but for a couple of decades overlapped with Euclid, dealt in abstractions (e.g. forms) but seems to have swung both ways in respect of the literality of gods. However, Pythagoras preceded them both by several hundred years and advanced geometry from measurement practices to abstract generalizations. Yet he appears to have believed in the gods of the Greek pantheon. So it seems that abstraction in Greek mathematics emerged well before abstraction in monotheistic religion related phenomena. But these are vague observations. The concept of abstraction and how it applies in the two domains needs to be clarified before a plausible answer can be given.

Well it's a good idea to find some of them, first. The first equation implies that y=0 is an equilibrium, so a class of equilibria is of the form (x,0,z). That reduces the problem. From the last equation it then should be possible to solve for z and, from the second, for x.

Then look at the other factor of the first equation; and so on.

Stam Nicolis, you have been telling me how the problem is being considered in GTR, Cosmology, etc. This is known already.

The question I have posed is for considerations towards new attempts to discuss the problem rationally and to come to possible further conclusions that might help us understand the very problems better

Minkowski "spacetime" and Lorentz's Transforms are just contrived geometrical tricks based on the axiom that the velocity of light c is an universal constant in any IRF :

INSPIRE: https://inspirehep.net/literature/2158754

QM can have values unknown, but not uncertain. Likewise, RG questions. Please stay on topic, per question. Do not be uncertain yourself.

Opinions do not matter, every opinion is right and should be, therefore, not discussed.

But, facts? Mass is defined (not a choice or opinion) as the ratio of two absolutes: E/c^2. Then, mass is rest mass. There is no other mass.

This is consistent, which is the most that anyone can aspire. Not agreement, which depends on opinion. Science is not done by voting.

Everyone can, in our planet, reach consistency -- and the common basis is experiment, a fact. We know of other planets, and there consistency may be uncertain -- or ambivalent, and even obscure. A particle, there, may be defined, both, as the minimum amount of matter of a type, or the most amount of quantum particles of a type.

We can entertain such worlds in our minds, more or less formed by bodies of matter, and have fun with the consequences using physics. But, and there is my opinion (not lacking but not imposing objectivity) we all -- one day -- will be lead to abandon matter. What will we find? That life goes on. The quantum jump exists. Nature is quantum.

Manabu Goto this may interest you: https://conference-service.com/conferences/mathematics-education.html

An Empirical Study of ML Algorithms for Stock Daily Trading Strategy.

The engineering philosophy behind Coq and the Coq kernel is well worth careful consideration:

To reach the Scopus document search module, you should use academic IPs. If your institute has been listed in the Scopus database, you have permission to search documents in Scopus. It is not free of charge, and your university should pay its share to Scopus to provide this service for its academic researchers.

Cite:

No. Consider the series ... in the above link example/ answer by Mark Gritter.

Both are divergent series, but the difference between the series is ... a convergent series.

Hereby attached is a system of three Ordinary Differential Equations (ODE). We have to use numerical methods such as RK4 and Euler to obtain the results using Java. I think RK4 is better for this kind of problem. In this thread, Dudley J Benton has already mentioned C programming for this. It is really amazing.

For the parameter values, you can use your own values for your convenience.

Many thanks

I have evidence that that is the upper bound... today it has been proved. But I reserve my ideas to publish them by a proper article. I just hope that journals can accept so clear ideas behind of this problem is a fundamental basis but not complicated to understand as it is believed. Many thanks.

Thank you very much Madam @Shima Shafiee

We need to be optimistic, because that is the lesson from nature. An animal can self-medicate, obeying natural laws in chemistry that are unknown to animals. A tree grows when pruned, so we can see this pandemic as an opportunity. Let's grow, nature is not a zero-sum game!

Irrational numbers and mathematical real-numbers are uncomputable, with probability 1.

But irrational numbers can be calculated exactly in algebra a and that is how animals are able to calculate-- in a network of thoughts.

One of the most important features that distinguish mathematics from other sciences is abstraction, so some of them think that mathematics is abstract concepts. Abstraction is simply (a mental process based on separating one of the properties from something, and considering it independent of others).

(The Link Between Mathematics and Ethics) was built on the basis of the new view of man in contemporary psychology, as this view tends towards the primacy of the mind. In other words, this is consistent with the new scientific view that began with the theory of relativity and quantum mechanics, which proved the centrality of the mind. After World War II, many psychologists pointed out that the abolition of the role of the mind in human behavior and the subjection of the mind to instinct in the method of psychoanalysis led to the dehumanization of man. Therefore, psychology in the new view considers reason and determination as the highest human faculties, and they distinguish man from animals. (And the mind and the will not only control the body, but they also control the emotions and nullify them when necessary. By subordinating the emotions to the mind, harmony and happiness become within the reach of man). (And the old view of science considers that the human mind cannot choose freely because matter does not act except by mechanical necessity. This is the reason why the old view tended to explain human actions in the language of instinct). In short, this study comes within this framework, that is, we proceed here from the fact that man is a conscious force, and what mathematics does is that it creates what can be called a “moral authority” with mental foundations, meaning that the ethics generated by mathematics are based on the authority of the mind, not fear. (According to Dr. Mahmoud Bakir's opinion).

Ignoring the displacements due to strain in the element, the rigid body translation equals the average of the nodal displacements of the elements and the rigid body rotations equal the average rotations of the nodes of the element. I believe this would offer high accuracy for elements away from the regions of constraints.

History of mathematics can offer students a window into the processes of developing new mathematics, and the ways in which this knowledge is accepted and circulated within different communities. Especially pertinent to the question of decolonising the curriculum, turning to the history of the discipline offers a window into the role played by mathematics and mathematicians in settler colonialism, and how the mathematics we use today was in turn shaped by colonialism.

A new project at The Open University will produce an interactive online database of original sources, selected to demonstrate the global nature of mathematics. Original sources – such as letters, calculation aids, khipu or clay tablets – enable students to see beyond the final results explicated in their university textbooks to the work that went into uncovering these results in the first place...

origin

Clarivate announced that starting with 2023 ESCI-indexed journals will also be assigned an impact factor. See: https://clarivate.com/blog/clarivate-announces-changes-to-the-2023-journal-citation-reports-release/

Online-based classrooms are necessary for circumstances where the learners and teachers feel that the knowledge can be catered to without meeting in person. It is an opportunity for learners to learn at their doorsteps. Nonetheless, online classes are not as effective as offline classes for many learners who take education as a social process where peer learning, group dynamics, and interpersonal relationship are of high importance. Online class doesn't provide the experience of personal interaction while learning.

There is even such a discipline called "COmputational Social Sciences"...

I recommend research results from authors of the French group of Didactis of Mathematics as: Vergnaud, Bideaud, Meljac, Fischer, Brun. Also, from Oxford University Peter Bryant and Terezinha Nunes.

Most statisticians agree that the minimum sample size to get any kind of meaningful result is 100.

We can imagine any finite quantity to be halved, and then each half to be halved again, and so on, without end. You can then say the finite quantity has infinitely many parts. It's just a conceptual distinction.

Raphael Neelamkavil , you said, "But in your paper "The zero-dimensional physical theory (V): information, energy, efficiency, and intelligence" you exhibit enough awareness of the above facts. Hence, I do not understand what else you meant by the question at the end of your comment!".

The title of this forum topic is, "Can the Primitive Notions (Categories) and Axioms of Mathematics, Physics and Philosophy Converge?". When I said, "This is all a good question and debate, yet what is deliberate misinformation and ignorance, what is the drive for it, and can it be written about in a way so as not to promote it as being useful?" I am essentially asking whether or not the primitive notions are already granted by the contemporary ideas of mathematics, physics, and philosophy. My work highlights they are not, so a clear issue in this debate is all about what is assumed and what is not. My work with zero-dimensionality assumes nothing, and thence creates a new spectrum of ideas for mathematics, physics, and philosophy which has been, is, and perhaps still will be for most.

In Leon Lederman's wonderful book, "The God Particle" he comments about a pecking order in physics. It goes something like, "the experimental physicists only defer to the theoretical physicists who in turn only defer to the mathematician and the mathematician only defer to God..."

Physics is not mathematics. The role of physics like all sciences is to understand the universe. The theoretical physicists' role is to develop theories that explains all the experiments from the past plus allows predictions to be made to test the and potentially falsify the theory. The role of the experimental physics is to develop and perform those experiments. There is often as much deep mathematics used by an experimental physicists as a theoretical physicists.

The role of the mathematician is simple to expend the understanding of mathematics through rigorous logical reasoning. If a physicists or any other scientist finds these tools useful - then great.

Arguably the greatest theoretical physicists of the second half of the 20 Century was Richard Feynman. Here is what he said about physics.

"It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong."

I would suggest for a better understanding of physics - what it is and isn't - a read of "The God Particle" would be recommended

https://en.wikipedia.org/wiki/The_God_Particle_(book) .

As far as I understand your problem, you first need a mathematical relation between the instructions and rank. For instance, Rank x should correspond to some instruction value as y and vice versa; it means you require a mathematical function.

So there are various methods/tools to find a suitable (as accurate as you want) to find mathematical function based on given discrete values like curve fitting methods or the use of ML.

Further, Once you obtain the mathematical function, run your code a few times, and you will get a set for various combinations of (instruction, rank). These set values will work as the feedback for your derived function. Make changes based on the feedback, and you will get a much more accurate function.

I hope you are looking for the same.

One possible approach is presented in the attached word document

A point in space is given always by two points:

- coordinates of the point

- coordinates of origin of coordinates

Between those two points there is a "length" (physical dimension) with a physical Unit (Meters)

For vectors X1=[x1,1;x2,1;..,xn,1]; X2=[x1,2;x2,2;..,xn,2];,.., Xn=[x1,n;x2,n;..,xn,n]

M=[m1,m2,..mn] - vector median,

where

m1=med(x1,1; x2,1;..,xn,1)

m2=med(x1,2; x2,2;..,xn,2)

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

mn=med(x1,n; x2,n;..,xn,n)

I did give a physical explanation for why your boundary condition enforces the function f to become zero at r=0. In at least one physical interpretation, it models a system with infinite emission/absorption rate at r=0. This forces the temperature at this point to immediately become equal to the environmental temperature. In this interpretation f is (proportional to) the difference between body and environment temperatures. When you define a physcally unreasonable it is only fair that it provides you with physically unreasonable results ;-D

Thank you, Anton Vrdoljak.

For my opinion we can use any product formulas only under condition of independence. For example the formula that you used P(a,b│c)=P(a│c)∙P(b│c) is valid if a and b are independent each from other. Both may be dependent from c. But it can be the case of so called "false dependence". For example a is the usage of energy, b is sale of warm clothes, c is decreasing of tempetature.

Sorry, Jianhing. But I think you have misunderstood something in the lecture. Frequentist statistics, which is an interpretation of probability to be assigned on the basis of many random experiments.

In this setting, on designs functions of the data (also called statistics) which estimate certain quantities from data. For example, the probability p of a coin to land heads is given from n independent trials with the same coin and just counting the fraction of heads. This is then an estimator for the parameter p.

Each estimator should have desirable properties, as unbiasedness, consistency, efficiency and low variance and so on. Not every estimator has these properties. But, in principle one can proof, whether a given estimator has these properties.

So, it is not a characteristics of frequentist statistics, but a property of an individual estimator based on frequentist statistics.

The solution is not a mathematical formula!

The question can be approached by two ways:

a) experimental: prepare homogenous standards having the same density and geometry as your samples, #1 material with low K,Th,U content, #2 same as #1 with known amount of K added, #3 same as #1 with known amount of Th added ,and #4 same as #1 with known amount of U added.

From the differences between measurements #N-#1 you may deduce the contribution of each of the three elements to the spectrum and, more specifically, the contribution of each of them to three regions of interest corresponding to strong peaks of each element, usually 1.46 MeV for 40K, 2.61 MeV for Tl208 (progeny of Th) and 609 keV for Bi214 (progeny of U), assuming secular equilibrium in Th and U families, or at least the same disequilibrium conditions in sample and standards.

It is then possible to calculate a matrix relating the measurements in the 3 ROIs to the three activities.

b) Monte Carlo simulation. The NaI detector is usually quite simple and can be simulated with a good accuracy. Experimental measurements are still necessary to evaluate the gaussian broadening of the peaks at the different energies, and an experimental validation of the MC model is always desirable.