Quantitative and Qualitative Study Design Examples

What statistical test to use to compare effectiveness of drugs used to fight a disease
So let's say that I want to collect some data on a sample of people who has the same type of disease.
One group will take a drug A
Another one will take drug B
Third type will take both of them.
I want to perform a statistical test to see whether taking both of the drugs simultaneously results in getting back to health quicker.
Can anyone tell me what test would I use and what hypothesis would I state? Assumptions that I would make?
I need to plan a research project (Without actually doing it)

Studying math in the morning or in the evening has no effect on a student's performance.
I know this claim might be a little unconventional. Most examples I have read in books and online are that something either increases or decreases. However, for the research I am currently doing, what's relevant it's to see if there's no change.
For what I have read, the null hypotheses should always have an equality and the alternative hypotheses an inequality. However, in this case, I want precisely the opposite.
So, let's consider M as the test scores of people studying in the morning, and E the tests scores of people studying in the evening.
H0:M¯¯¯¯¯≠E¯¯¯¯
H1:M¯¯¯¯¯=E¯¯¯¯This formalization is what I though should be correct, since H1 is my claim (what I am trying to demonstrate) and H0 is the complement of that. However, it goes against the hypothesis formulation definition.
I think, in the end, what I am asking is how can one formalize hypotheses for an experiment that is designed to show no correlation between two levels of a factor?

What I have found is that I love theoretical discussions of mathematics and proving theorems, though the education system focuses on doing exercises and applications, and that is what grades are determined on. I have a hard time doing exercises mainly because I don't find them motivating, although, I admit, doing the exercises does cultivate one's intuition to a certain degree.
Up until now my main technique of learning was to take meticulous notes from textbooks(not in class as all they do is mundane exercises), think about those concepts and then practice exercises before exams, and it worked fine for discrete math, precalculus, and even the first course of calculus, which was mainly focused on derivatives. Further. I as able to maintain A's in all my classes with enough energy left over to even do very well in other classes.
However, this hasn't gone too well in the second course of calculus, which is mostly about integrals of various functions and their applications and techniques. The instructor usually does problems on the board step by step without any discussion for what we are doing. For example, both the fundamental theorems were introduced and given about 15 minutes of time to discuss. But, of course, this could be bad teaching style.
My question is this, should I learn math theoretically and understand the material well even though on tests this would mean getting about B's that I have gotten or should I only care about the grades and just do what I am told, like a human calculator, practice exercises and just focus on getting A's? What is the better for me in the long run if I want to get into a decent grad school? And have any of you ever had to decide between getting an A or spending more energy and time actually learning the theory, which is more satisfying? I could be looking at this completely wrong and perhaps there is a way to do both? Thank you and your input will be appreciated.

i'm essentially questioning in which there are a few sturdy applications to get a masters in arithmetic. whilst locating a list of these isn't so difficult, i am conscious that a number of them serve in particular to help the ones whose undergraduate math training is a bit susceptible to be a PhD candidate. i'm going to count on (perhaps wrongfully) that i am now not in that category. So my query is this: Assuming i am qualified to pursue a PhD in mathematics, but am barely uneasy about a 5 year commitment, which abbreviated graduate school alternatives are to be had to me?

Question: What is known about the role that binary representation of data plays in algorithmic complexity theory?
I define "analog representation of data" loosely as any method of representing data which is not a sequence of 1's and 0. A more precise definition would be nice to have obviously. The basic idea is that an analog representation of data contains additional information beyond what is contained in a binary representation.
A very trivial example of an "analog" representation of the natural numbers is to label all the primes $p_1,p_2,p_3\cdots <!--\; \cdots \; -->$ and then represent numbers by their prime factorization, instead of in binary form. It is obvious that with this representation of numbers, the problem of factoring large numbers is of polynomial complexity.
Less trivially, according the Shor's Alogirithm, if we represent numbers using an analog physical system (in this case, using quibits instead of bits), then it's possible to factor large numbers in polynomial time as well.
The natural conclusion to draw is that the choice to use binary or not for representing data likely plays an important role in determining if a given problem is in P or NP, and one would think that in order to prove P is not equal to NP it would be required to use the assumption that data is represented in a binary way at some point.

Getting ready for Calculus?
So I wanted to start a Masters software however they require that i've Calculus III. I need to take that path at the college, however I need to be prepared for it. As I take a look at Khan Academy and do some of the sports for pre-calc, I realise i am thus far out of faculty that I need to look at regions of math even previous to pre-calc. Is there an evaluation of some type on-line that could assist guide me to what I want to awareness on or have to I just work through Khan Academy material? Any thoughts?

Whats the difference between a one way anova and two way anova?
I know that in one way anova you compare the difference between two or more means and the same in two way, but I'm unclear as to how the use of categorical variables differs between them.
Any help would be appreciated

I'm taking real analysis I, and my professor gives us a previous exam with solution each time before the midterm/exam. I would do the previous exam and check if I get it right. But it's obvious that my professor are not going to give us the same questions.
So, I also read and tried to apprehend the proofs which are given at the ebook. once in a while, I proved the theorems and corollaries by myself. besides, i might memorize the theorems and corollaries.
For guides like calculus, we have a few preferred steps to resolve a kind of question. So it is simpler to prepare for the exam. however for proof-based guides like real evaluation, algebra and topology, and many others, how can we recognise whether we're nicely-organized for the examination? How did you observe the materials and prepare for the examination whilst you have been an undergraduate pupil? checks are designed to test our know-how of the substances, but how can we take a look at ourselves first?

Is there any abstract theory of electrical networks?
Designing electrical networks is among the highly mathematical engineering disciplines, which uses a vast scope of techniques from Fourier analysis and complex function theory, to logic, combinatorics and topology. But, at least to me, with my minor knowledge of electrical engineering, it seems that at the end of the day, what is physically built out of these theories--I mean in a manufacturer laboratory-- is always a finite graphs with nodes labeled with "simple functions", in a way that the whole thing is again a function, with desired characteristics. But, from a mathematical point of view, it is customary to investigate such structured functions, in a categorical context and exploit the language and power of category theory, much like what programmers do.
Here, I do not dare to further these vague ideas and pose my question:
What is the right and fruitful mathematical foundation for the theory of electrical networks? Is there any purely axiomatic approach to the subject, accessible to a mathematics enthusiast with minor background in electrical engineering.

I am writing to have some advice about what Algebra I should study to start tackling Algebraic Geometry and (advanced) Algebraic Topology.
My background consists in Linear Algebra and a 'fundational' course covering the basics of Groups, Rings and Fields.So I think I can learn about this topics, having all the necessary prerequisites, but do not know which are the most relevant and fundamental topics to the stated area. Any advice and reference to some compact text apt to self study would be great!

What are the practical uses of e ?
How can e be used for practical mathematics? This is for a presentation on (among other numbers) e, aimed at people between the ages of 10 and 15.
To clarify what I want:
Not wanted: $e^{i\mathrm{\pi <!--\; \pi \; -->}}+1=0$ is cool, but (as far as I know) it can't be used for practical applications outside a classroom.
What I do want: e I think is used in calculations regarding compound interest. I'd like a simple explanation of how it is used (or links to simple explanations), and more examples like this.

What parts of a pure mathematics undergraduate curriculum have been discovered since 1964?
What parts of an undergraduate curriculum in pure mathematics have been discovered since, say, 1964? (I'm choosing this because it's 50 years ago). Pure mathematics textbooks from before 1964 seem to contain everything in pure maths that is taught to undergraduates nowadays.
I would like to disallow applications, so I want to exclude new discoveries in theoretical physics or computer science. For example I would class cryptography as an application. I'm much more interested in finding out what (if any) fundamental shifts there have been in pure mathematics at the undergraduate level.
One reason I am asking is my suspicion is that there is very little or nothing which mathematics undergraduates learn which has been discovered since the 1960s, or even possibly earlier. Am I wrong?

Reminder : Given a set S of n elements (we will use [n] in the following for simplicity), a Latin square L is a function $L:[n]\times <!--\; \times \; -->[n]\to <!--\; \to \; -->S$, i.e., an $n\times <!--\; \times \; -->n$ array with elements in S, such that each element of S appears exactly once in each row and each column. For example,
Latin square
1, 2, 3
3, 1, 2
3, 1, 2
Let $L_1$ and $L_2$ be two Latin squares over the ground sets $S_2$ respectively. They are called orthogonal if for every $(x_1,x_2)\in <!--\; \in \; -->S_1\times <!--\; \times \; -->S_2$ there exists a unique $(i,j)\in <!--\; \in \; -->[n]\times <!--\; \times \; -->[n]$ such that $L_2(i,j)=x_2$. For example, the following are two orthogonal Latin squares of order 3.
1, 2, 3 2, 3, 1
3, 1, 2 3, 1, 2
1, 2, 1 1, 2, 3
It is known that there at most n−1 mutually orthogonal Latin squares of order n, and that the bound is achieved if and only there exist an affine plane of order n.
Graph : I'm building a graph $G_n$ with vertex set the latin squares of order n and two vertices are adjacent iff the Latin squares are orthogonal.
I want to understand some properties of this graph. For simplicity I consider the squares up to permutation of [n], hence w.l.o.g. all my squares have for first line $\{1,2,\dots <!--\; \dots \; -->,n\}$. Indeed if I call $H_n$ the graph not up to permutations, then $H_n$ is the n! graph blowup of $G_n$, or using the Tensor product
$$H_n=G_n\times <!--\; \times \; -->K_{n!}$$
As I'm mainly interested in the chromatic number of my graph, and we know that $\mathrm{\chi <!--\; \chi \; -->}(H_n)\le <!--\; \le \; -->min\{\mathrm{\chi <!--\; \chi \; -->}(G_n);n!\}$, I will study only $G_n$
For instance $G_3=K_2$
I know that :
It's trivial that $G_n$ is not complete.
If there exist an affine plane of order n then $G_n$ contains $K_{n-<!--\; -\; -->1}$ as a subgraph, and $\mathrm{\chi <!--\; \chi \; -->}(G_n)\ge <!--\; \ge \; -->n-<!--\; -\; -->1$
I wonder the following :
What is the maximum degree of $G_n$? We know that we have at most n−1 mutually orthogonal latin squares, but to how many squares can one square be orthogonal (still up to permutation)?
Do we have any other info on the chromatic number, not coming from the property $\mathrm{\chi <!--\; \chi \; -->}(G_n)\le <!--\; \le \; -->\mathrm{\Delta <!--\; \Delta \; -->}+1$
Can $G_n$ contains an induce k-cycle with k>3 (i.e. chordless cycle)?
Can it be conjectured that
Conjecture : for any n, $G_n$ is the disjoint union of complete subgraphs (of different sizes).
Edit After some simple Brute force and some additional reading, I can tell that
$G_4$ is made of 2 disjoint $K_3$ and 18 isolated vertices, for a total of 24 Latin squares up to permutations.
$G_5$ is made of 36 disjoint $K_4$ and 1200 isolated vertices, for a total of 1344 Latin squares up to permutation.
The case n=6 would be the first interesting case, as there are no affine plance of order 6, hence we will find no $K_5$ in $G_6$. It is known since 1901 (from Tarry hand checking all Latin squares of order 6) that no two were mutually orthogonal. So $G_6$ is made of only isolated vertices.
It is also know that the case n=2 and n=6 are the only one with only isolated vertices. (see design theory by Beth, Jingnickel and Lenz)
From the article "Monogamous Latin Square by Danziger, Wanless and Webb, available on Wanless website here. The authors show that for all n>6, if n is not of the form 2p for a prime $p\ge <!--\; \ge \; -->11$, then there exists a latin square of order n that possesses an orthogonal mate but is not in any triple of Mutually Orthogonal Latin Squares. Therefore our graph $G_n$ will have some isolated $K_2$

Help me understand what is and isn't exponential growth.
If the price at which something grows is proportional to itself, then you definately could name it exponential increase. do not quote me on this, however I suppose it has something of the shape $y=e^x$
Now take for example this. if you want to calculate the extent of a sphere, the formula is $v=4/3\pi r^3$. The by-product with recognize to the radius is $dv/dr=4\pi r^2$ This suggests the rate of change of the volume with appreciate to the radius.
yet the extent of the field is proportional to the radius, which in flip defines the dimensions of the field. The rate at which the volume of the sector increases is proportional to itself in a manner. Intuitively, i might assume to discover someplace in there an exponential yet there isn't always. Why is that?

Finding a basis of an infinite-dimensional vector space?
the opposite day, my trainer become speakme limitless-dimensional vector spaces and headaches that arise when trying to find a foundation for the ones. He referred to that it is been demonstrated that a few (or all, do no longer pretty remember) infinite-dimensional vector areas have a foundation (the end result uses an Axiom of choice, if I recall efficiently), that is, an endless listing of linearly independent vectors, such that any detail within the area can be written as a finite linear aggregate of them. however, my instructor stated that honestly finding one is simply complicated, and i were given a experience that it changed into essentially not possible, which jogged my memory of Banach-Tarski paradox, where it is technically 'viable' to decompose the sector in a given paradoxical way, however this can not be truly exhibited. So my question is, is the basis state of affairs analogous to that, or is it surely viable to explicitly discover a basis for endless-dimensional vector areas?

Consider the two species competition model given by
$$\frac{da}{dt}=[{\lambda }_{1}a/(a+K1)]-<!--\; -\; -->r_{ab}\cdot <!--\; \cdot \; -->ab-<!--\; -\; -->da,(1)$$
$$\frac{db}{dt}=[{\lambda }_{2}b\ast <!--\; \ast \; -->(1-<!--\; -\; -->b/K2)]-<!--\; -\; -->r_{ba}\cdot <!--\; \cdot \; -->ab,t>0,(2)$$
for two interacting species denoted a=a(t) and b=b(t), with initial conditions a=a0 and b=b0 at t=0. Here $\mathrm{\lambda <!--\; \lambda \; -->},\mathrm{\lambda <!--\; \lambda \; -->}2,K_1,K_2,r_{ab},r_{ba}$ and d are all positive parameters. (a) Describe the biological meaning of each term in the two equations.
=> A series expansion of 1/(a+K1), gives
$\frac{1}{a+K1}\approx <!--\; \approx \; -->\frac{K1-<!--\; -\; -->a}{K{1}^2+O(a^2)}$
Now,
$\frac{da}{dt}=[\mathrm{\lambda <!--\; \lambda \; -->}1a\times <!--\; \times \; -->\frac{a+K1}{K{1}^2}]-<!--\; -\; -->r_{ab}ab-<!--\; -\; -->da,$
$\mathrm{\lambda <!--\; \lambda \; -->}$1 a represents the exponential growth of population
da represents the exponential decay of population
$\mathrm{\lambda <!--\; \lambda \; -->}$1 is the growth rate
d is the decay rate
what does r_(ab) ab represent?
The first term of RHS equation $1:[\mathrm{\lambda <!--\; \lambda \; -->}1a\times <!--\; \times \; -->\frac{a+K1}{K{1}^2}]$ represents logistic growth at a rate $\mathrm{\lambda <!--\; \lambda \; -->}$1 with carrying capacity K1.
$\frac{db}{dt}=[\mathrm{\lambda <!--\; \lambda \; -->}2b\times <!--\; \times \; -->\frac{1-<!--\; -\; -->b}{K2}]-<!--\; -\; -->r_{ba}ab,$
$\mathrm{\lambda <!--\; \lambda \; -->}$2 b represents the exponential growth of population
$\mathrm{\lambda <!--\; \lambda \; -->}$2 is the growth rate
what does r_(ba) ab represent?
The first term of RHS equation $2:[\mathrm{\lambda <!--\; \lambda \; -->}2b\times <!--\; \times \; -->1-<!--\; -\; -->bK2]$ represents logistic growth at a rate $\mathrm{\lambda <!--\; \lambda \; -->}$2 with carrying capacity K2.

How one can calculate the sample size in any random sampling? Is it varies with sampling method or it is fixed for all methods? Explain it.

Method for finding efficient algorithms?
TL;DR
What can you recommend to get better at finding efficient solutions to math problems?
Background
The first challenge on Project Euler says:
Find the sum of all the multiples of 3 or 5 below 1000.
The first, and only solution that I could think of was the brute force way:
target = 999
sum = 0
for i = 1 to target do
if (i mod 3 = 0) or (i mod 5 = 0) then sum := sum + i
output sum
This does give me the correct result, but it becomes exponentially slower the bigger the target is. So then I saw this solution:
target=999
Function SumDivisibleBy(n)
p = target div n
return n * (p * (p + 1)) div 2
EndFunction
Output SumDivisibleBy(3) + SumDivisibleBy(5) - SumDivisibleBy(15)
I don't have trouble understanding how this math works, and upon seeing it I feel as though I could have realised that myself. The problem is just that I never do. I always end up with some exponential, brute force like solution.
Obviously there is a huge difference between understanding a presented solution, and actually realising that solution yourself. And I'm not asking how to be Euler himself.
What I do ask tho is, are there methods and or steps, you can apply to solve math problems to find the best (or at least a good) solution?
If yes, can you guys recommend any books/videos/lectures that teach these methods? And what do you do yourself when attempting to find such solutions?

I am carrying out a category on discrete arithmetic and i'm inquisitive about skipping my faculties transition courses so that you can take a rigorous theory path next semester (topology, analysis, abstract algebra). What are a few properly transition books for me to examine that provide troubles and a few solutions so i will monitor my development, as well as being very , almost laboriously, particular in every step of evidence such as theorem packages. as an instance, i have observed abbots expertise analysis to be pretty cogent but Laczkovich Conjecture and evidence to be lacking some data important for me to understand some proofs as much as I would love. thank you for any assist

what is the best book for Pre-Calculus?
i've overlooked pre-calculus know-how in my school however i used to be excellent at maths, and now i am a pc science scholar, i am feeling terrible being bad in maths, so i'm seeking out the quality Pre-Calculus e-book, i love maths, i need the right nicely of precalculus books.