Quantitative and Qualitative Study Design Examples

Do you prove all theorems whilst studying?
When you come across a new theorem, do you always try to prove it first before reading the proof within the text? I'm a CS undergrad with a bit of an interest in maths. I've not gone very far in my studies -- sequence two of Calculus -- but what I'm trying to understand right now, though, is how one actually goes about studying so that when finished with a good text, there's more of an intuitive understanding than superficial.
After reading "The Art Of Problem Solving" from the Final Perspectives section of part eight in 'The Princeton Companion to Mathematics', it seems to hint at approaching studying in that very way. A quote in particular, from Eisenstein, that caught my attention was the following -- I'm not going to paraphrase much:
The approach utilized by the director turned into as follows: each pupil needed to show the theorems consecutively. No lecture took place at all. nobody became allowed to inform his answers to every person else and every scholar acquired the following theorem to show, impartial of the opposite college students, as soon as he had proved the previous one correctly, and as long as he had understood the reasoning. This changed into a completely new interest for me, and one that I grasped with outstanding enthusiasm and a zeal for knowledge. Already, with the first theorem, i was a long way ahead of the others, and whilst my friends had been still struggling with the eleventh or 12th, I had already proved the hundredth. there has been simplest one younger fellow, now a medicine pupil, who could come close to me. even as this technique is excellent, strengthening, as it does, the powers of deduction and inspiring self sustaining wondering and opposition among college students, generally speakme, it can likely no longer be adapted. For as lots as i can see its advantages, one ought to admit that it isolates a positive power, and one does not obtain an overview of the complete concern, that may simplest be done with the aid of an amazing lecture. as soon as one has acquired a amazing variety of material thru [...] for college kids, this technique is manageable only if it deals with small fields of effortlessly, comprehensible information, in particular geometric theorems, which do now not require new insights and thoughts.
I feel that this type of environment is something you don't often see, especially in the US -- perhaps that's why so many of our greats are foreign born. As I understand it, he does go on to say that he wouldn't particularly recommend that method of study for higher mathematics, though.
A similar question was posed to mathoverflow where Tim Gowers (Fields Medal) went on to say that he recommended similar methods to study: link
I'm not quite certain that I understood the context of it all, though. Upon asking a few people whose opinion mattered to me, I was told that it if time were precious to me, it would be a waste going about studying mathematics in that way, so I'd like to get some perspective from you math.stackexchange. How do you go about studying your texts?

Let's imagine I have a function $f$(x) whose evaluation cost (monetary or in time) is not constant: for instance, evaluating f(x) costs $c(x)\in <!--\; \in \; -->\mathbb{R}$ for a known function c. I am not interested in minimizing it in few iterations, but rather in minimizing it using "cheap" evaluations. (E.g. maybe a few cheap evaluations are more "informative" to locate the minima, rather than using closer-to-the-minima and costly evaluations.)
My questions are: Has this been studied or formalized? If so, can you point me to some references or the name under which this kind of optimization is known?
Just to clarify, I am not interested in either of these:
Standard optimization, since its "convergence rate" is linked to the total number of function evaluations.
Constrained optimization, since my constraint on minimizing $f$(x) is not defined in terms of the subspace allowed to x, but in terms of the added costs of all the evaluations during the minimization of $f$.
Goal optimization, since using few resources to minimize $f$ can not be expressed (I think) as a goal.
Edit
Surrogate-based optimization has been proposed as a solution. While it faces the fact that the function is expensive to compute, it does not face (straightforwardly) the fact that different inputs imply different evaluations costs. I absolutely think that surrogate-based optimization could be used to solve the problem. In this line, is there a study of how to select points xi to estimate the "optimization gain"? In terms of surrogate-based optimization: what surrrogate model and what kind of Design of Experiment (DoE) could let me estimate the expected improvement?
Also, some remarks to contextualize. Originally, I was thinking on tuning the hyperparameters of machine learning algorithms. E.g., training with lots of data and lots of epochs is costly, but different architectures may be ranked based on cheaper trainings and then fine-tuned. Therefore:
Let us assume that c(x) can be estimated at zero cost.
There is some structure in c(x): large regions have higher cost, large regions have cheaper cost, the cost is smooth, the minima may be in either region.
Let us assume that $f$(x) is smooth. Also, if possible, let us assume it has some stochastic error that we can model (such as Gaussian error with zero mean and known variance).

I am studying control systems, and I want to solve following problem.
Given full rank state matrix A (with all unstable eigenvalues), design input matrix B, such that cost function J=trace(B′XB) is minimized, where X is the solution to discrete-time Ricatti equation (DARE). I have contraint that (A,B) is stabilizable, i.e.
For a given full rank $A\in <!--\; \in \; -->{\mathbb{R}}^{n\times <!--\; \times \; -->n}$, with ${\mathrm{\lambda <!--\; \lambda \; -->}}_i(A)>1$, solve the following
$$\begin{array}{ll}\underset{X\in <!--\; \in \; -->{\mathbb{R}}^{n\times <!--\; \times \; -->n},B\in <!--\; \in \; -->{\mathbb{R}}^{n\times <!--\; \times \; -->m}}{\text{minimize}}& \mathrm{t}\mathrm{r}\left(B^{\prime }XB\right)\\ \text{subject to}& X=A^{\prime }X(I+B{B}^{\prime }X{)}^{-<!--\; -\; -->1}A\\ & (A,B)\text{ is stabilizable}\end{array}$$
From my understanding, since all eigenvalues of A are outside of unit circle (discrete-time system), we can change condition (A,B) is stabilizable with (A,B) is controllable, which is equivalent to rank([$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}([BAB{A}^{2}B\dots <!--\; \dots \; -->A^{n-<!--\; -\; -->1}B])=n$
The problem is for sure feasible, since for any full rank A, there is B such that rank condition is satisfied and we can solve DARE.

i'm presently reading Multivariate Calculus (Larson and Edwards book). I want to do a project in pc technology to look some first-class programs of factors i am mastering.
Any specific source of papers/journals/books? thank you

Let A and B be two positive definite matrices of orders n and d, respectively. Let X be a matrix of type (n,d) (n rows and d columns).
Is it true that A−1XB−1X⊺ have eigenvalues between zero and one?
I am unsure if the result holds with this generality, so I am giving more context below.
This problem arises in the study of "contingency tables" (i.e. statistics) where X is a design matrix of zeros and ones, where each row represents an observation and each column represents the category or cell where this observation falls. Thus, xij=1 means that the ith object appeared in the jth cell and if xij=0 it means taht the object did not appear. It is assumed that at least one entry of every row and every column will be one (the possibility of different objects being in different categories is OK). The matrices A and B are obtained by normalising the rows and columns of X. That is, definexi⋅=∑j=1dxij,x⋅j=∑i=1nxij
and set A=diag(xi⋅), B=diag(x⋅j).
I have made more than a few attemps without success but neither of these have really used the structure of the problem. For example, if λ is an eigenvalue of the matrix A−1XB−1X⊺ then any eigenvector of λ will satisfy XB−1X⊺=λAv. It is tempting to multiply on the left by v⊺ and us the fact that boths sides are now weighted norms of v. This already shows that λ≥0. But the algebra becomes messy and I could not see a path forward to show λ≤1. On top of this, I am not really using the structure of X,A and B as I already mentioned.
A perhaps useful fact (that I have not been able to exploit) is that the vector 1 (full of ones) has eigenvalue 1 and any other eigenvector is orthogonal to this one!

The role of dual space of a normed space in functional analysis
We have known that dual space of a normed space is very important in functional analysis.
I would like to ask two questions related dual space of a normed space:
What is the motivation of constructing dual space of a normed space?
What is the main role of dual space of a normed space in functional analysis?

What are the allowed axioms to solve the Riemann Hypothesis?
First, is it even possible to state a well known problem and know what axioms are allowed for it? I'd guess that one must look at the branches of math involved, but I'm no mathematician.
Therefore, I would like to ask the people working on this problem if this there is a fixed list of allowed axioms for this problem.

i've recently started studying up on design concept, with the ultimate reason of doing some authentic studies in that region. I understand the arithmetic pretty properly, but am now not understanding the application. Can a person here give an explanation for that? instead are there a few articles, web-websites, papers explaining the software of layout concept in agriculture, biological experiments and so forth.
for example after establishing the lifestyles of/constructing a certain sort of design mathematically, how can we use that expertise in exercise?

I have recently started studying graphs and their different traversal algorithms, and can't seem to be able to come up with an answer to this question. I really need your help, I don't even know where to start. P.S. It was my birthday yesterday and I don't want to cry because of this problem.
A company in New York manufactures blue halogen bulbs for cars. Unfortunately, it is very difficult to color bulbs consistently. Naturally, it is also very important to package bulbs that look alike in pairs. To package bulbs in pairs, bulbs coming out of the assembly line are first partitioned into two sets of bulbs sharing similar colors (e.g., one set of darker bulbs, another set of lighter bulbs), and then pairs are formed within each set.
Because of increasing demand, the company wishes to hire more workers to partition bulbs into two sets. To determine whether applicants have appropriate skills to perform this rather delicate task, they are asked to take this simple test: Given a set of n bulbs, compare each pair of bulbs and determine whether two bulbs have “similar” or “different” colors. An applicant also has an option to say “indeterminate” for each pair. The company wishes to hire applicants who are consistent in their judgment. We say m judgments (resulted in either “similar” or “different”) are consistent if it is possible to partition n bulbs into two sets such that (i) for each pair {a,b} determined to be “similar”, a and b indeed belong to the same set, and (ii) for each pair {a,b} determined to be “different”, a and b indeed belong to different sets.
For a given test result for n bulbs with m judgments, design an O(n+m)-time algorithm to determine whether the judgments are consistent. In practice, there should be a minimum number of judgments applicants are required to make, but your algorithm should work for any integer $m\ge <!--\; \ge \; -->0$

Fourier and Laplace transforms together, is this possible?
Answering on some posts on MSE about Laplace transform and Fourier transform I stumbled upon a question to which I cannot answer myself (not having a good ground in pure mathematics).
The question is the following:
Is there some mathematical constraint that doesn't let us use both Fourier and Laplace transform on the same equation?
I'm not saying that it would be useful in any case, I was just wondering if it's feasible! Just as an example I could use both transforms to solve the one dimensional (or three, doesn't change much) wave equation with some external force
$$\{\begin{array}{l}{\mathrm{\partial <!--\; \partial \; -->}}_t^{2}u(x,t)-<!--\; -\; -->c^2{\mathrm{\partial <!--\; \partial \; -->}}_x^{2}u(x,t)=f(x,t)\\ u(x,0)={\mathrm{\partial <!--\; \partial \; -->}}_{t}u(x,t){|}_{t=0}=0\\ -<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->}<<!--\; <\; -->x<<!--\; <\; -->\mathrm{\infty <!--\; \infty \; -->}t><!--\; >\; -->0\end{array}$$

In what sense is Lebesgue integral the "most general"?
[ UPDATE: This question is apparently really easy to misinterpret. I already know about things like the Henstock-Kurzweil integral, etc. I'm asking if the Lebesgue integral (i.e. the general measure-theoretic integral) can be precisely characterized as "the most general integral that can be defined naturally on an arbitrary measurable space." ]
Apologies as I don't have that much of a background in real analysis. These questions may be stupid from the point of view of someone who knows this stuff; if so, just say so.
My layman's understanding of various integrals is that the Lebesgue integral is the "most general" integral you can define when your domain is an arbitrary measure space, but that if your domain has some extra structure, like for instance if it's Rn, then you can define integrals for a wider class of functions than the measurable ones, e.g. the Henstock-Kurzweil or Khintchine integrals.
[ EDIT: To clarify, by the "Lebesgue integral" I mean the general measure-theoretic integral defined for arbitrary Borel-measurable functions on arbitrary measure spaces, rather than just the special case defined when the domain is equipped with the Lebesgue measure. ]
My question: is there a theorem saying that no sufficiently natural integral defined on arbitrary measure spaces can (1) satisfy the usual conditions and integral should, (2) agree with the Lebesgue measure for measurable functions, and (3) integrate at least one non-measurable function? Or, conversely, is this false? Of course this hinges on the correct definition of "sufficiently natural," but I assume it's not too hard to render that statement into abstract nonsense.
Such an integral would, I guess, have to somehow "detect" sigma algebras looking like those of ${\mathbb{R}}^n$ and somehow act on this.
[ EDIT 2: To clarify, by an "integral" I mean a function that takes as input $((\mathrm{\Omega <!--\; \Omega \; -->},\mathrm{\Sigma <!--\; \Sigma \; -->},\mathrm{\mu <!--\; \mu \; -->}),f)$, where $(\mathrm{\Omega <!--\; \Omega \; -->},\mathrm{\Sigma <!--\; \Sigma \; -->},\mathrm{\mu <!--\; \mu \; -->})$ is any measure space and $f:(\mathrm{\Omega <!--\; \Omega \; -->},\mathrm{\Sigma <!--\; \Sigma \; -->})\to <!--\; \to \; -->(\mathbb{R},\mathcal{B})$ is a measurable function, and outputs a number, subject to the obvious conditions. ]
UPDATE: The following silly example would satisfy all of my criteria except naturality:
Let $(\mathrm{\Omega <!--\; \Omega \; -->},\mathrm{\Sigma <!--\; \Sigma \; -->},\mathrm{\mu <!--\; \mu \; -->})$ be a measure space, let $\mathcal{B}$ denote the Borel measure on R, and let $f:(\mathrm{\Omega <!--\; \Omega \; -->},\mathrm{\Sigma <!--\; \Sigma \; -->})\to <!--\; \to \; -->(\mathbb{R},\mathcal{B})$ be a Borel-measurable function. Define
$${\int <!--\; \int \; -->}_{\mathrm{\Omega <!--\; \Omega \; -->}}fd\mathrm{\mu <!--\; \mu \; -->}:=\{\begin{array}{ll}\text{the Khintchine integral}& \text{if }(\mathrm{\Omega <!--\; \Omega \; -->},\mathrm{\Sigma <!--\; \Sigma \; -->},\mathrm{\mu <!--\; \mu \; -->})=(\mathbb{R},\mathcal{L},{\mathrm{\mu <!--\; \mu \; -->}}_{\text{Lebesgue}});\\ \text{the Lebesgue integral}& \text{otherwise.}\end{array}$$

Is second derivative of a function related to curve smoothness?
If there exist a first derivative of a function at any point then the funtion is continuous at that point.
What if the second derivative of that function is also exist at that point ? Does this mean that function is smooth at that point ? Means instead of taking sharp bend the function is having curved shape ?

After studying general a linear algebra course, how would an advanced linear algebra course differ from the general course?
And would an advanced linear algebra course be taught in graduate schools?

Though there are several posts discussing the reference books for topology, for example best book for topology. But as far as I looked up to, all of them are for the purpose of learning topology or rather on introductory level.
I am wondering if there is a book or a set of books on topology like Rudin's analysis books, S.Lang's algebra, Halmos' measure (or to be more updated, Bogachev's measure theory), etc, serving as standard references.
Probably, such a book should hold characteristics such as being self-contained, covering the most of classical results, and other good properties you can name.
In the end, I only have the interest on general-topology (topological space, metrization, compactification...), and optionally differential topology (manifolds). So please don't divert into algebraic context.

Is there any academic fields where pure mathematics (Abstract Algebra or Number theory) are applied to medical/biological studies?
I am currently looking for applications of pure mathematics ( Especially abstract algebra or number theory) to medical studies, currently I can only think of the explore of deep learning (used in medical image analysis), design of Graph Neural Networks, and a little bit use of integer programming (Not sure if it is pure math)... Thanks so much in advance!

I am looking for a book designed for self teaching applied/engineering mathematics. Can someone help me find a few?
I have used Shaum's differential equations. Are there others that I could use that are designed for self-study?

Algorithm design: $\mathrm{\theta <!--\; \theta \; -->}$(f(n)) - help explaining my answer
I am studying algorithm design and need some help explaining my (correct) answer to the following question:
Assume that $T(n)=\mathrm{\Theta <!--\; \Theta \; -->}(n^2)$. Can we say that for every input size n, our algorithm will perform a maximum of $10n^3$ operations? Why?
My answer, is as follows:
No, because by definition: T(n) is $\mathrm{\Theta <!--\; \Theta \; -->}(f(n))$ only if T(n) is both O(n) and $\mathrm{\Omega <!--\; \Omega \; -->}(n)$
$c_1\cdot <!--\; \cdot \; -->f(n)\le <!--\; \le \; -->T(n)\le <!--\; \le \; -->c_2\cdot <!--\; \cdot \; -->f(n)$
I was told that whilst the answer is correct, the explanation is not. Can somebody explain to my why this is the case?

Do a negative Gaussian curvature help with the stability of a surface?
I found the statement "[...] ruled surfaces are statically efficient, especially in the case of skewed ruled surfaces, which are very stable due to a generally negative Gaussian curvature".
Why would a negative Gaussian curvature imply (or help with) the stability of a surface? I did not find a mathematical argument for the claim in the magazine, nor any mention of the same fact elsewhere. Is it really true?
I'm interested in this question because I'm studying ruled surfaces (which have non-positive Gaussian curvature) and their applications in architecture and product design in general, and if such surfaces have strength or stability advantages, I'd be very interested to know.

Finding the optimal strategy
You have a deck of 32 playing cards. Somebody draws one card after another and shows them to you. At any point of time you may bet that the next card is black. If it is indeed black you earn $10, otherwise nothing. If you don't do anything you earn nothing as well. Find the optimal strategy.
In other words you should find the point of time where the quota of remaining red cards in the deck is maximal.

Find out the angular speed in terms of time.
Here is the equation that describes the motion of a planet under the gravitational field generated by a fixed star: $u=\frac{e}{l}\cos <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}+\frac{1}{l}$, where u is the reciprocal of the radial distance between the planet and the star, e is the eccentricity of the orbit, l is the semi latus rectum, and let h denote the angular momentum per unit mass, $\mathrm{\theta <!--\; \theta \; -->}$ is the angular coordinate. e,h,l turn out to be independent from one another, and they are independent from t and $\mathrm{\theta <!--\; \theta \; -->}$. At time t=0, we let the radial speed vanish, and we also let the angular coordinate vanish. To find the relationship between time and angular speed $\mathrm{\omega <!--\; \omega \; -->}$, we assume that u is a smooth fuction of t, and differentiate u w.r.t. t, and use $\mathrm{\omega <!--\; \omega \; -->}=h{u}^2$ to find out an expression for $\mathrm{\omega <!--\; \omega \; -->}$. To do this we can differentiate u w.r.t. $\mathrm{\theta <!--\; \theta \; -->}$ first then multiply it by ω, which equals to $h{u}^2$.
Then differentiate the first derivate of u w.r.t $\mathrm{\theta <!--\; \theta \; -->}$ first, then multiply the result by hu2 and so on. Since the whole process involves the differentiation w.r.t. θ only, we can assume that e=0.5,l=h=1. We set e=0.5 only because we wish to study bounded orbits so that we can apply Kepler's law and verify our result. However, the whole process is time consuming since the formula for the derivatives of u becomes complicated very quickly, even if we assume explicit values for e,l,h. The only effective way, therefore, is to design an algorithm for this process. But I do not have any knowlege about computer science, if anyone knows how to design algorithms or know about some other ways to find out ω in terms of t, please share.