Quantitative and Qualitative Study Design Examples

i'm seeking out thoughts for a 15-hour mathematical enrichment course in a chinese language high faculty. What (pretty) simple concern would you advocate as a subject for any such course?
historical past/issues:
My students are generally pretty good at math, but many of them have no longer been uncovered to rigorous or summary mathematical reasoning. an amazing topic would be one that could not be impossibly hard for students who have by no means written or study proofs in English.
i have taught this magnificence three times earlier than. (a part of the purpose that i'm posting that is that i have used up all my thoughts!) the primary semester I taught an introductory range theory elegance (which meandered its way toward a proof of quadratic reciprocity, though I think this become in the end too advanced/abstract for some of the students). the second one semester I taught fundamental graph idea and packages (with a focal point on planarity and coloring). The 1/3 semester I taught a class at the Rubik's dice.
the students' math backgrounds are pretty numerous: a number of them take part in contest math competitions, and so are familiar with IMO-fashion techniques, however many aren't. a number of them may additionally realize some calculus, however I cannot assume it. all of them are superb at what in the united states is on occasion termed "pre-calculus": trigonometry, conic sections, systems of linear equations (though, shockingly, no matrices), and the like. They realize what a binomial coefficient is.
So, any ideas? preferably, i'd like to find some thing a bit "sexy" (like the Rubik's cube) -- tries to encourage wide variety theory through cryptography seemed to fall on deaf ears, however being capable of "see" institution idea on the cube became pretty popular.
(Responses specifically welcome from folks who grew up in the percent -- any mathematical subjects you desire were protected within the excessive college curriculum?)

I'm a graduate student studying quantum mechanics/quantum information and would like to consolidate my understanding of linear algebra. What are some good math books for that purpose?

How to deal with lack of peers?
I am pursuing a B.S in Mathematics(freshman) in a country where people rarely study math for the sake of it and in a new university with good professors. Yet, it is new and I suffer from the lack of peers and it is literally impossible to pick up mathematical conversations on ideas with students . As a result, I find it difficult to exchange ideas with others.
Is there something which can be done to make sure that lack of peers or the pressure to push oneself doesn't harm me much?

i'm going to start an test approximately drying leaves. I need to study how a few special kind of leaves are going to dry themselves in 2 distinctive scenario. a group of them will stay in a high density, and the some other institution i can spread out in a larger region (as an instance, 1 m2 for the primary, and 3 m2 for the second). Of route, i can repeat this experiment numerous times. in the end, i will get:
Individual weight of the leaves (selecting samples) in the beginning
Individual weight along 4 days
Individual weight in the end
Nutritional composition in the start and in the end
I am remembering my knowledge in stats and R, but I am still lost. My question is, which is the best comparison method to analyse this data? I want to know:
How the humidity content change along the days
To dry this kind of leaves, which drying area is better?
Is it statistical significance between both method about losing water and nutrient content?

What's the difference between $(-<!--\; -\; -->x{)}^2,-<!--\; -\; -->x^2$ and $(-<!--\; -\; -->x{)}^2$ ?
$(-<!--\; -\; -->x{)}^2$ is definately equal to $(-<!--\; -\; -->1{)}^2(x{)}^2$, right?
But $-<!--\; -\; -->(x{)}^2$ and $-<!--\; -\; -->x^2$ are confusing me, do they mean $-<!--\; -\; -->(x^2)$ or do they mean $(-<!--\; -\; -->1{)}^2(x{)}^2$?

An experiment is designed to test the potency of a drug on 40 rats. Previous animal studies have shown that a 10-mg dose is lethal 10% of the times within the first 4 hours.
What is the probability that between 2 and 8 rats die during the experiment due to the drug?
My attempt: Let X be the number of rats that die in the first 4 hours.
$P(2\le <!--\; \le \; -->x\le <!--\; \le \; -->8)=\underset{x=2}{\overset{8}{\sum <!--\; \sum \; -->}}(\genfrac{}{}{0ex}{}{40}{x})(0.1{)}^x(0.9{)}^{40-<!--\; -\; -->x}$
To make the drug more potent, the company came with a new formula. This reduced the chances of a new drug being lethal to 1%. The new drug is administered to 10000 rats. Approximate the exact probability that 5 rats die.
My attempt:
$P(x=5)=(\genfrac{}{}{0ex}{}{10000}{5})(0.01{)}^5(0.99{)}^{10000-<!--\; -\; -->5}$
Am I on the right path?

Is there a "canonical" representation of integers using numbers other than primes?
Consider the (cumbersome) statement: "Every integer greater than 1 can be written as a unique product of integers belonging to a certain subset, S of integers.
When S is the set of primes, this is the Fundamental Theorem of Arithmetic. My question is this: Are there any other types of numbers, for which this is true.
EDIT: As the answers show, this obviously cannot be done. What if we relax the integer condition, i.e. can there be any other canonical representation of positive integers using complex numbers?

Should I go back and start with a more "proof" based approach?
I should go to a book like the one by Spivak which is entirely different from the book used for my course, and learn or in a way re-learn it the way it's presented in that book?
Would a more proof based approach help me in this understanding? Will I always lack some aspect of understanding if I don't know how to prove these problems?
To quote one of the comments on this question, the question can also be put
"will studying calculus in a proof based manner help in understanding the techniques I've already learned"

When is a definition via properties considered valid?
How one would define the validity of a definition of an object by its properties?
Little background: I'm trying to implement a kind of framework of mathematics in which the user is not restricted to any formal system, but can define anything at will just like in english but imposing logical constraints on the steps possible during a proof preventing mistakes. This obviously requires to prove that the definition is "valid".
Example: If we consider G to be a group and + its group operation, we can define $\mathrm{\forall <!--\; \forall \; -->}x\in <!--\; \in \; -->G:x+0=0+x=x$, we can prove that two sets with such property are equal $0_1=0_1+0_2=0_2$ so the definition is solid. Here equality is by definition $x=y:⟺<!--\; ⟺\; -->(\mathrm{\forall <!--\; \forall \; -->}z:(x\in <!--\; \in \; -->z⟺<!--\; ⟺\; -->y\in <!--\; \in \; -->z)\wedge <!--\; \wedge \; -->\mathrm{\forall <!--\; \forall \; -->}w:(w\in <!--\; \in \; -->x⟺<!--\; ⟺\; -->w\in <!--\; \in \; -->y)$, so basically in any formula of set theory we can freely replace x with y and vice versa. But what if we don't want to be restricted to set theory? Then what seems reasonable is to require for every definition a list of formulas(not necessarily of set theory) which constitutes the notion of "equality" and then proving that in these formulas they are indeed interchangeable. This issue does not arise when considering objects defined through expressions $3:=2+1$as they are just directly replaceable short-hand notations.

I am studying computer aided geometry and I have a background in mathematics. For me a (real) projective transformation is a map $f:{\mathbb{R}\mathbb{P}}^n\to <!--\; \to \; -->{\mathbb{R}\mathbb{P}}^n$ induced by a linear isomorphism $F:{\mathbb{R}}^{n+1}\to <!--\; \to \; -->{\mathbb{R}}^{n+1}$ since being injective it maps lines to lines (m-subspaces to m-subspaces). In the context of graphic design they usually never propey define projective maps, they usually project something onto a plane or use the following construction:
Given $(x,y,z)\in <!--\; \in \; -->{\mathbb{R}}^3$ we consider the point $(x,y,z,1)\in <!--\; \in \; -->{\mathbb{R}}^4$ (i.e. a copy of ${\mathbb{R}}^3$ on the affine hyperplane $\{w=1\}$, then we apply a bijective linear map on ${\mathbb{R}}^4$ and project the image back on the hyperplane (which causes some trouble if the image has forth coordinate equal to zero). This process on ${\mathbb{R}}^3$ is called a projective map. What is the link with my definition? I am sure it might involve considering homogeneous coordinates $(x,y,z)\to <!--\; \to \; -->[x:y:z:1]$, anyhow I don't get why the confused notation, they seem different things.

I am a student of Pure Mathematics and also interested in programming .I have learnt C++,SAGE .
Recently I have started learning "Cryptography" .But there are many definitions involved here like polynomial time algorithm,time complexity etc.
My question is it all right for a student in Pure Mathematics to study Cryptography or as time progresses I will eventually fall out of place and lose interest in this subject.
Is Cryptography more suitable for computer science graduates or it does not matter which background a student is from to study this?
Please share your thoughts here as i am still in my early days and may help to change the subject if necessary before it is too late

i'm completing my undergraduate diploma and one aspect i've noticed is how little weight has been placed upon the potential to examine proofs, in basically all of my math guides. In first year calculus you are shown the proofs for matters like the limit of sin(x)/x at 0, but in my experience there's no incentive to be able to apprehend them.
This pattern persisted even in more advanced undergraduate publications on foundations and real analysis. As one instance, the professor spent a whole lecture proving the schroeder-bernstein theorem, and only a few students made an effort to recognize it (they surely were not influenced to do so thru grades). typically speaking, my classes have followed a format in which the professor will show theorems for a giant portion of the lecture time but assessments are designed with packages and proof-writing in mind and truely maximum proofs carried out by using the professor are far too hard for a pupil to recreate independently, so there is no incentive to research the details of the greater complex proofs.
This seems unusual to me, thinking about the format of most guides requires you to apprehend the arguments backing up a specific proposition. is that this genuine of maximum university applications? need to a extra emphasis be positioned upon studying how to read complex proofs?

What kind of mathematical "discoveries" have enabled mankind to build modern computers?
After studying the very thrilling Examples of mathematical discoveries which had been saved as a mystery I got here to think of something: maximum math discoveries appear to had been made centuries ago, and with Pascal and Leibniz we already had machines that would add and multiply (a few don't forget them as the first computer systems). Of direction modern computer systems needed energy and electric signals that can be transformed to at least one's and zero's, but those we've since the 19th century. First computer systems were also built with transistors, however we've the ones since the 1940's.
current computers are, at their center, not so extraordinary from the primary computer systems that followed the Von Neumann structure (with its arithmetic-logic unit). this is: they understand how move information from one place to any other, the way to upload and a way to multiply, in addition to perform logical operations (and, or, xor, now not), and from that they get all of the other operations (some can substract, divide and do some different stuff, however it is all very primitive in that sense). Of path, current computers paintings at a miles better level than their predecessors: while inside the beginning programming needed to be accomplished the use of commands composed of zero's and 1's or, in the event that they were lucky, the usage of meeting language, now we have high stage programming languages that (to say it kind of) enclose a group of these single commands into one high stage command.
And when I see pretty graphics in video-games, programs like photoshop, 3D rendering, CAD programs and such, I always think of all the calculus stuff I learned and how it must be applied to achieve those wonderful results.
And this is where my question arises: all of this mathematical knowledge has been available to us for way longer than computers existed. So, are there any modern mathematical discoveries that enabled the giant leap we took from the first computers we had in the 40's to what we have now? Or maybe old math started being applied in a different way at some point in time?

I have heteroskedastic data of unequal sample sizes and would like to run a two way welch ANOVA.
1.) Is this appropriate? Why or why not?
2.) How do you do this in r?
3.) What are other ways of dealing with this situation?

Initial-value problem for non-linear partial differential equation $y_x^2=k/y_t^2-<!--\; -\; -->1$
For this problem, y is a function of two variables: one space variable x and one time variable t.
k>0 is some constant.
And x takes is value in the interval [0,1] and $t\ge <!--\; \ge \; -->0$
At the initial time, y follows a parabolic profile, like $y(x,0)=1-<!--\; -\; -->(x-<!--\; -\; -->\frac{1}{2}{)}^2$
Finally, y satisfies this PDE:
$${\left(\frac{\mathrm{\partial <!--\; \partial \; -->}y}{\mathrm{\partial <!--\; \partial \; -->}x}\right)}^2=\frac{k}{{\left(\frac{\mathrm{\partial <!--\; \partial \; -->}y}{\mathrm{\partial <!--\; \partial \; -->}t}\right)}^2}-<!--\; -\; -->1.$$
Does anyone have an idea how to solve this problem (and find the expression of y(x,t)) ?
About: The problem arise in physics, when studying the temporal shift of a front of iron particles in a magnetic field.

Probability question. Find the probability that the data set falls within 45% to 52% of the data set
one of your employees has cautioned that your enterprise expand a brand new product. A survey is designed to have a look at whether or not or not there may be hobby inside the new product. The reaction is on a 1 to five scale with 1 indicating without a doubt could no longer buy, · · ·, and 5 indicating absolutely could purchase. For an preliminary analysis, you will document the responses 1, 2, and 3 as No, and 4 and 5 as yes.
a. five people are surveyed. what is the opportunity that as a minimum three of them replied sure?
b. 100 people are surveyed. what's the approximate possibility that between 45% to fifty two% of people answered sure?
For component a) There are 5 choices that human beings can respond by way of 1, 2 ,three ,4 and five. when you consider that 1, 2 and three are considered "No", the possibility of a person answering "No" is three/five. For choices four and 4, the probability of a person responding with this is 2/5.
This seems like it fallows a binomial distribution so I calculated the chance of P(three) + P(four) + P(five).
but for component b), i'm stressed. i will calculate the chance of a person pronouncing sure but I do not know how to calculate the chance that a percentage of people announcing sure. Does absolutely everyone recognize how to technique this question? I though approximately the use of the Z table, however that already calculates region.

How do I go about self-studying Maths? Do I create a routine? Answer a bunch of Q's?
Im 19F & I'm starting my Computer Science course this year. Before starting it, I wanted to make sure I had good GCSE & A level Maths knowledge.
The problem is I don't know how to go about self-studying Maths. Do I answer textbook questions section per section? Do I watch videos? How do I know when I'm comfortable enough with a topic to move on to the next? Do I do 1 section a day? How do I organise it all?
To add: I have plenty of time in my day to dedicate to self-studying. I have no commitments.

Finding the function of a parabolic curve between two tangents
Alright folks, first question here so let me make the situation and background clear.
I'm attempting to start studying for aerospace engineering, so I'm working on improving on my math skills as we speak, but couldn't help "jumping in" to some of the design and starting to look at it. I'm not adverse to algebra, although my calculus legs haven't been walked on in awhile, which is why I haven't been able to get farther myself on my own question.
Another side reason is that I'm attempting to get the gist of equation-based curves in CAD, so rocket engine nozzle curves are perfect for learning that...if I can figure out the equations!
If the format or my thoughts seem a bit off, I have an idea of the concepts in play here, my math skills have just atrophied a little too much for my own comfort.
What I'm working with is the G.V.R. Rao approximation of a bell nozzle curve; essentially,
$$f_c=\{\begin{array}{ll}0.382R_t& \text{For divergent throat curve}\\ f_p& \text{Main body}\end{array}$$
Where $f_p$ starts from a point with a tangent of angle ${\mathrm{\theta <!--\; \theta \; -->}}_n$ and ends at a point with a tangent of angle ${\mathrm{\theta <!--\; \theta \; -->}}_e$. $f_p$ also has to fit in a region equal to $L_f-<!--\; -\; -->0.382R_t$, where $L_f$ is the complete distance between the throat and exit plane, so the displacement in the x- or y-axis, depending on how you view the rocket (orientation-wise).
I do know how to differentiate the curve $f_p$ to get $f_p^{\prime }$ and then find the angle of the slope at a point, but this is backsolving from two slopes to find the region in between.
If it's any help, ideally I'd be constructing the nozzle in CAD vertically, that is, $y_n>y_e$
What I'm looking for is help toward the derivation of a formula that allows me to construct a curve that is smooth between the two points. One of the reasons I've had a hard time figuring out the exact parameters is because it feels a lot like curve-fitting, which I haven't had much experience with.
If anyone can help break it down for me, it'd be much appreciated but if the question turns out to be too vague, references to places where I can get the requisite learning would be also appreciated.

What Are R-Modules Used For?
Kind of a simple question, but what exactly are R-modules used for? Do they have any engineering applications?
EDIT:
I am a graduate student researcher in computer architecture, a subfield of computer engineering. Specifically, I do research on the best way to build future general purpose processors
One thing I am looking into is if it is possible to apply mathematics to improve the design of CPUs. That is, can we use concepts from mathematics to improve the execution of general purpose programs on hardware. CPUs are a massive engineering design problem, and where exactly we could improve the design by applying math isn't entirely clear.
What I don't have is a very deep mathematical background. I have taken an introductory abstract algebra course and one in coding theory. I've also read a number of coding theory papers...
I know that other electrical engineering subfields like communications and compressed sensing have successfully applied elements of linear algebra and abstract algebra and have gotten very good results.
The fact that this particular question spans both engineering and mathematics makes it both hard to formulate and to discuss with people. I'd be happy to talk about it in more detail, but I'm not entirely sure what the best forum would be for that.
At least for now, I figured a good place to start would be to see if other people have successfully used some of the more abstract math concepts in engineering systems. One of the few I am aware of are R-modules, so I figured I'd ask if anyone knows of some engineering uses of them...

What reason is there to conjecture that every finite string is really in the decimal expansion of $\mathrm{\pi <!--\; \pi \; -->}$?
One of my students asked me this, and it occurred to me that I had never really questioned it.
Apparently, it is only conjectured but widely believed that the decimal expansion in base 10 of π contains all finite strings of the numerals 0 through 9.
Am I even accurate that the conjecture is widely accepted? Regardless, what is the rationale for this belief? Do any good heuristics exist? It seems perfectly logical (dare I say likely) that, just maybe, the string 2347529384759748975847523462346435664900060906, for example, never occurs. The conjecture seems absurdly strong, to me.
And just because a separate question would be ridiculous: does this conjecture extend to other famous transcendental numbers? Is it indeed conjectured that this is a property of transcendental numbers in general?