Quantitative and Qualitative Study Design Examples

complicated evaluation, and mainly contour integrals and the residue idea have proved a totally effective tool in computing a massive class of real feature integrals which might be pretty difficult to compute if simplest inside the scope of real calculus. There are a exquisite many strategies designed for this cause, as an instance, both the selection of the complex variable characteristic and the choice of the ideal contour is critical to our fulfillment of computation.
I want to study these techniques systematically, so I want recommendations for tutorials that cover them in a systematical way, the more inclusive the better. In a word, I want to learn as many those techniques as possible, so I think such a tutorial is a must.
Ps: the tutorial I currently have at hand is Stein's Complex Analysis, it is good but covers too few exercises about such techniques.

I am currently studying Divide and conquer.
I have been reading algorithm design by Jon Kleinberg along with the lecture slides from our lecturer.
I am stuck with the part where they teach us about the solution of divide and conquer recurrences.
I could not understand how they got the answer.
It goes like this.
Fix a>0, b>1 and consider
T(n)=aT(nb)+f(n) (when n>n0), n0=c.
First solve this when nn0 is a power of b:
put U(i)=T(n0bi) and g(i)=f(n0bi).
Get
U(i)=aU(i−1)+g(i) (when i>0), U(0)=c)
iterate to obtain
enter image description here
Important special case: f(n)=np for some fixed p.
Write B=bp.
What I do not understand is that when they said "when nn0 is a power of b:", n should be something like n=n0bi. After that when we substitute T(n) with the n=n0bi, it should be something like
T(n0bib)
How did they end up with T(n0bi)?
Also they said iterate to obtain
enter image description here
How did they obtain that? They did not provide any walk through that I could follow on how to get that..

Determining Components of an Experiment
Let's say that there is a study that focuses with the effect of storage times on the calcium content of barley. Samples of barley were stored for different times (either 0,1,2, or 4 months) and the calcium content was measured (grams of Ca per kg of barley). We suppose that this study is balanced and a total of 35 samples were taken.
Initial Ideas
Our subjects (or units) would our samples of barley, our treatement would be the different storage times (0,1,2, or 4 months) and lastly, our response (or conclusion) would be the amount of calcium content that was collected from the barley.
Based on the description above, one of the questions had to do with whether this study was a design experiment or an observational study and I chose the former because the researchers were in control of assigning the treatments to our subjects. How does this sound?

Sum solution unclear
I have started studying algorithms and currently am reading Skiena's Algorithm Design book. While doing the tasks, I encountered with question that I could not find solution for. I took a look in solution, but don't get how it is being solved. Specifically line
Now the third summation goes from j to i+j the formula on closer examination reveals that
$\underset{k=j}{\overset{i+j}{\sum <!--\; \sum \; -->}}(j+i-<!--\; -\; -->k)\text{ which is}\underset{k=1}{\overset{i}{\sum <!--\; \sum \; -->}}(k)$
Can you explain to me how it could be transformed to this?

Applications of logic in sciences
I understand math as the study and description of the behavior of mathematical structures, and as you know this math structures could include rings, fields, metric spaces, propositions, categories, numbers, sets, operators, differential equations..., a big part of this structures was born under the need of the description of a problem. For example the study and solution of the problem of the Brachistochrone curve gives to us the calculus of variations, or the study of the behavior of the heat and waves was the main column of the development of the Fourier series expansion, and as you know this is useful in physics, electrical, mechanical, and in general engineering.
So other structures such as differential equations, tensors, matrices are useful for physics, chemistry, economics, engineering, and even abstract ones such as linear spaces, groups, rings, operators, Banach spaces, Hausdorff spaces are useful in physics.
But in general logic, understanding it as the classification of truth parametrized but several specifications using several structures such as languages, binary operators, models, this to proof under what conditions a given expression id true, so it has several applications in number theory, algebra, topology, but this ones are mathematical fields, so i want to know if besides computer science foundations, type theory in CS, programming languages fundamentals, design and analysis of algorithms, digital logic, computer architecture, (that by itself is a huge approach of logic in life), are there any applications of logic in physics, economics, engineering, biology..

Optimization Problem : Dumpster
I am trying to help my friend. This is his problem related to constructing a dumpster, so it can minimize construction cost :
For this project we locate a trash dumpster in order to study its shape and construction. We then attempt to determine the dimensions of a container of similar design that minimize construction cost."
(Already located, measured, and described a dumpster found).
"While maintaining the general shape and method of construction, determine the dimensions such a container of the same volume should have in order to minimize the cost of construction. Use the following assumptions in your analysis:
The sides, back, and front are to be made from 12-gauge (0.1046 inch thick) steel sheets, which cost $0.70 per square foot (including any required cuts or bends).
The base is to be made from a 10-gauge (0.1345 inch thick) steel sheet, which costs $0.90 per square foot.
Lids cost approximately $50.00 each, regardless of dimensions.
Welding costs approximately $0.18 per square foot for material and labor combined.
Give justification of any further assumptions or simplifications made of the details of construction.
Describe how any of your assumptions or simplifications may affect the actual result.
If you were hired as a consultant on this investigation, what would your conclusion be? Would you recommend altering the design of the dumpster? If so, describe the savings that would result."

What is the difference between "Design Function" and "Objective Function"?

Book Recommendation for Second Course in Linear Algebra
I only took a non-rigorous linear algebra course (It was designed for non-math students). I finished most of Hungerford's algebra. Now I have two choices to study more advanced linear algebra: Hoffman's linear algebra, Finite dimensional vector spaces by Halmos. I already looked at the first chapter of both books.
I think Halmos goes at a faster pace which is something I like. Would I miss anything if I start learning the book of Halmos instead of Hoffman's ?

Numerical method for steady-state solution to viscous Burgers' equation
I am reading a paper in which a specific partial differential equation (PDE) on the space-time domain $[-<!--\; -\; -->1,1]\times <!--\; \times \; -->[0,\mathrm{\infty <!--\; \infty \; -->})$ is studied. The authors are interested in the steady-state solution. They design a finite difference method (FDM) for the PDE. As usual, there are certain discretizations in time-space, $U_j^n$, that approximate the solution u at the mesh points, $u(x_j,t_n)$. The authors conduct the FDM method on $[-<!--\; -\; -->1,1]\times <!--\; \times \; -->[0,T]$, for T sufficiently large such that
$\left|\frac{U_j^N-<!--\; -\; -->U_j^{N-<!--\; -\; -->1}}{\mathrm{\Delta <!--\; \Delta \; -->}t}\right|<{10}^{-<!--\; -\; -->12},\mathrm{\forall <!--\; \forall \; -->}j,$
where $t_N=T$ is the last point in the time mesh and $\mathrm{\Delta <!--\; \Delta \; -->}t$ is the distance between the points in the time mesh. The approximations for the steady-state solution are given by $\{U_j^N{\}}_j$
I wonder why the authors rely on the PDE to study the steady-state solution. As far as I know, the steady-state solution comes from equating the derivatives with respect to time to 0 in the PDE. The remaining equation is thus an ordinary differential equation (ODE) in space. To approximate the steady-state solution, one just needs to design a FDM for this ODE, which is easier than dealing with the PDE for sure. Is there anything I am not understanding properly?
For completeness, I am referring to the paper Supersensitivity due to uncertain boundary conditions. The authors deal with the PDE $u_t+u{u}_x=\mathrm{\nu <!--\; \nu \; -->}{u}_{xx}$, $x\in <!--\; \in \; -->(-<!--\; -\; -->1,1)$, $u(-<!--\; -\; -->1,t)=1+\mathrm{\delta <!--\; \delta \; -->}$, $u(1,t)=-<!--\; -\; -->1$, where $\mathrm{\nu <!--\; \nu \; -->},\mathrm{\delta <!--\; \delta \; -->}>0$ . They employ a FDM for this PDE for large times until the steady-state is reached. Why not considering the ODE$u{u}^{\prime }=\mathrm{\nu <!--\; \nu \; -->}{u}^{″}$, $u(1)=-<!--\; -\; -->1$, $u(1)=-<!--\; -\; -->1$, instead?

Inherent Pitfall of Lebesgue Integration?
I am studying Real Analysis with Royden's Book. I noticed that for a function $f$ differentiable almost everywhere on [a,b] and $f$′ integrable over [a,b], it does not imply that $f(x)={\int <!--\; \int \; -->}_{[a,x]}{f}^{\prime }(t)dt+f(a).$. Take Cantor function f as a counter example, since $1=f(1)>{\int <!--\; \int \; -->}_{[0,1]}{f}^{\prime }(t)dt+f(0)=0,$, because $\mathrm{\forall <!--\; \forall \; -->}x\in <!--\; \in \; -->[0,1]$.
However, does it mean that Lebesgue integral has some inherent pitfall in the definition of integration? In a sense that it fails to restore the original picture of a function in a set of measure zero after differentiation. Is that any better (or more general) design of integration? so that we can guarantee $f(x)={\int <!--\; \int \; -->}_{[a,x]}{f}^{\prime }(t)dt+f(a)$

Estimate variance in a nested study design
I have a "mother" bacteria. I cloned it R times and measured a variable P on these R clones and recorded the mean p¯ and the variance V among these clones.
Then, I made M mutants (M new bacterias each carrying a single new mutation) from the "mother bacteria". I cloned each of these M mutants r times and measured the variable P again. I recorded the mean Pm and variance Vm of each mutant m (1≤m≤M).
The parameter I would like to estimate is the the mean square differences between the mean value of the "mother bacteria" (estimated as p¯) and the mean values of the M mutants (estimated as P¯m).
∑Mm(P¯−P¯m)2M would be a biased estimator as I have to remove the variance caused by imprecised estimation of both P¯m and P¯. I think the solution should be along the lines of
∑Mm(P¯−P¯m)2M−∑MmVmr−VR

Computing the fixed point for cos x
While studying about Compiler Design I came with the term 'fixed point'.I looked in wikipedia and got the definition of fixed point but couldn't get how fixed point is computed for cos x as said in fixed point.
It says that the fixed point for cos x=x using Intermediate Value Theorem.But I couldn't get how they computed the fixed point for cos x.Do anyone know how they computed this?

Should I combine the negative part of the spectrum with the positive one?
When filtering sound I currently analyse only the positive part of the spectrum. From the mathematical point of view, will discarding the negative half of the spectrum impact significantly on my analysis?
Please consider only samples that I will actually encounter, not computer generate signals that are designed to thwart my analysis.
I know this question involves physics, biology and even music theory. But I guess the required understanding of mathematics is deeper than of those other fields of study.

Reference for Book Design Theory
First, when i studied group action, i found out about design theory, then i got interested in it. But is there any good book to study design theory? especially for introduction and undergraduate student level. I knew about Design Theory book athor by Thomas Beth, D. Jungnickel, H. Lenz.
But i find it too way long and i thought it is not introduction level. Pleas inform me if you know anything about it, thank You so Much!

Harmonic summation
I am studying a few algorithms books at the moment, and I often see the harmonic summation come up. What I am confused about is, if the harmonic summation is:
$\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}1/i\sim <!--\; \sim \; -->\ln <!--\; \; -->n$
Why then do certain complexity analyses involving the harmonic sum, like the following one for Quicksort (from Skiena - Algorithm Design Manual pg. 49):
$S(n)=n\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}1/i$
Reduce to $\mathrm{\theta <!--\; \theta \; -->}(n{\log }_2<!--\; \; -->n)$ and not $\mathrm{\theta <!--\; \theta \; -->}(n\ln <!--\; \; -->n)$
I am guessing for algorithmic analysis, this difference is not important, or I am just missing something entirely.

Mathematical Epidemiology Reference Request
I'm looking for a good Textbook for learning Mathematical Epidemiology. Something that I could read through and use as a future reference book.

Multiple Choice: What is the design for this experiment?
The prompt is the following: A biology student wants to determine if using a fertilizer would help promote the growth of new babies in spider plants. The student has access to 90 baby spider plants of three varieties: green, variegated, and curly. There are 30 plants of each variety. They all are potted in the same amount and type of soil, given the same amount of water, and exposed to the same amount of light. The numbers 1–30 are written on slips of paper, placed in a hat, and mixed thoroughly. A plant is selected and a slip of paper is drawn. If the slip has the numbers 1–15, then the plant will receive fertilizer. If the slip has the numbers 16–30, the plant will not receive fertilizer. A green spider plant is selected and a slip of paper is drawn.
This plant is placed in the treatment group indicated by the number, and the slip is not put back in the bag. The slips are mixed again, the next green spider plant is selected, and a slip is drawn. The plant is placed in the treatment group indicated by the number. This procedure is repeated until all 30 green spider plants are assigned to treatments. The numbered slips are placed back in the bag and this procedure is repeated for the remaining types of spider plants. After one year, the shoots will be counted for each plant.
The answer choices are as follows:
A. observational study
B. matched pairs design
C. randomized block design
D. completely randomized design
My solution: My guess is C. In this case, the matched pair is when each experimental unit receives both treatments in random order, and the participants were separated into the yoga or meditation group by a flip of the coin. In A, I do not think that it is a matched pair because each person was asked their stress level. I rule out D as well, because I do think this is a matched pairs design.

Maximum possible city block distance
Anya participated in a study comparing two blood glucose monitors. A total of 20 persons with diabetes participated. Ten participants used Monitor A for two days and then used Monitor B for two days. The other ten participants (including Anya) used Monitor B for two days and then used Monitor A for two days. On the 5th day, participants rated the two monitors on the basis of four features. The four features were rated on a 5 point scale that ranged from −2 (poorly designed) to 2 (well designed). The distance between Anya’s ratings of the two monitors was 4 city blocks. What is the maximum possible city block distance between the two blood glucose monitors?

Experiment design parameters.
I have a doubt identifying the elements of the following experiment design:
A study wants to determine whether users perceive the difference while playing a game at 30 fps or 60 fps. In the game the player has to destroy objects that traverse the screen. All subjects played a first part at 30 fps, a second part half of them played at 30 fps and the other half at 60 fps and finally the two halves interchanged roles.
Here I understand that fps is an independent variable of the experiment and that the experiment is a "within subjects" one.
After the game, we ask them if they perceived a difference between the equality of the three parts and in affirmative case we ask which image was the best. We register their score at the end of each part and their responses to which was the best image. In each condition, half of the subjects used a standard resolution screen and the other half used a high resolution screen.
From here I deduce that the score is a dependent variable and I hope that the perceived quality of the image is another dependent variable (later on response time appears as a third dependent variable?). I have doubts whether the resolution of the screens is a second independent variable.
We verify that visual accuracy doesn't change between the two groups. We register each action of the subjects during the game. The results show that globally the subjects aren't capable of telling what part of the game had higher fps but the scores were higher at 60 fps.
So this part states that the visual accuracy is a controlled variable and that the principal effect of the experiment was that score was better when using higher fps.
The response time (duration between the apparition of an object and first action of the subject) were shorter after 60 fps, but only for players with a high screen resolution.
This should be an interaction effect between but there is no dependent variable stated.
Therefore my explanation of the experiment doesn't fit the solution expected. Could you please point me what I'm missing here? I should be able to state:
Two independent variables
Two dependent variables
An intermediate variable.
A controlled variable.
A principal effect.
An interaction effect.