Quantitative and Qualitative Study Design Examples

Smoothest function which passes through given points?
I am trying to interpolate/extrapolate on the basis of a known collection of (finitely many) points. I'm wondering if there is a way to formalize this intuitive notion: find a 'smoothest' function which passes through each of the points. Of course I general the function would not have a nice form.
The idea is similar to that of Bézier curves (ad would be closer still, if I used a parametric curve rather than a function) and essentially opposite to the Lagrange interpolating polynomial, where fitting more than a few points usually produces wild oscillations.
Any idea how to make an idea like this work?

I'm trying to learn linear algebra on my own. When I taught myself calculus I-III a year ago, I used Stewart's Concepts and Contexts 4th edition and I absolutely loved it. Theorems and definitions were displayed actually, the graphs were distinctive, and the problem selection was quite large.
Is there a linear algebra textbook just like how Stewart's textbook changed into designed?
moreover, i am sooner or later planning to research differential equations and i'm aware that pretty a few texts overlap when coaching those topics. i would favor to study them in my view, so i might want to keep away from texts with an amazing amount of differential equation, unless there's a text that covers each fields absolutely in the identical manner that Stewart's textual content included unmarried and multivariable calculus collectively.
P.S I don't imply to be praising Stewart's. it's the only arithmetic textbook i've used to self-have a look at so I do not have a lot else to relate to.

What are some applications of sets & relations in science/business/tech that a highschooler can understand? To kindle a young mind, what examples can be given?

Beneficial to touch theoretically deeper texts earlier in core areas e.g. analysis, algebra?
Desired future direction: Dynamical System(Chaos), PDE
More beneficial to read theoretically deep, modern and masterpiece texts earlier, (e.g. levels like UTX/GTM/GSM/LNM/CSAM) ? Especially in core areas (e.g. Analysis, Algebra, Topology etc.)
i.e. Read briefly less for intro-level, then quickly drop to the deeper/modern texts, although basic skill's not trained enough, but to be trained again/better in 'deeper-stage'. (Sometimes it seems learning new material in deeper/complex 'environment/context' gives better/faster maturity.)
e.g: Calculus/Analysis: Apostol -> Hardy, Courant, Stein, Rudin, Amann
Linear Algebra: Axler -> Dym, Grueb, Lax, Halmos
Algebra: van der Waerden -> Lang, Hungerford, Jacobson, Bourbaki
Topology: Kelley -> Milnor

I am a pure math major. I am taking a one-semester course covering the following topics:
Inner Product Spaces,
Direct Sums of Subspaces,
Primary Decomposition Theorem,
Reduction - (triangular, Jordan, rational, classical),
Dual Spaces,
Orthogonal Direct Sums,
Quadratic Forms
Our lecturer is various theorist so he'll probably make a few links with number theory. He does now not advise any textbooks for the course but I research excellent alone out of books designed for self-study. Lecture theaters full of whispering and distractions are hell to me! i love books which are written in a private, quirky fashion that set off deep reflection and inspire impartial questioning. i like colourful motivating cloth!! I dislike mention of any "real-international applications" (ie physics/engineering). I hate when writers pass over big chunks in proofs and say the lacking steps are "trivial" whilst they're now not. i love it while answers are available to the physical activities.

i'm hoping to self-have a look at Geometry, Algebra, Calculus, Vector Calculus, Linear Algebra, probability and information, and different intermediate maths. i've located the pleasant way for me to analyze is to work on problems. regrettably, many math textbooks don't provide all of the answers so that i'm able to test my work.
Does absolutely everyone have any recommendations for materials that do provide all solutions to problems so that i will feel at ease that i'm solving the problems efficaciously?

Question on designing a state observer for discrete time system
I came through this problem while studying for an exam in control systems:
Consider the following discrete time system
$$\stackrel{\to <!--\; \to \; -->}{x}(k+1)=A\stackrel{\to <!--\; \to \; -->}{x}(k)+b\stackrel{\to <!--\; \to \; -->}{u}(k),\stackrel{\to <!--\; \to \; -->}{y}(k)=c\stackrel{\to <!--\; \to \; -->}{x}(k)$$
where $b=(0,1{)}^T,c=(1,0),A=\left[\begin{array}{ccc}2& & 1\\ 0& & -<!--\; -\; -->g\end{array}\right]$ for some $g\in <!--\; \in \; -->\mathbb{R}$
Find a feedback regulation (if there is any) of the form $u(k)=-<!--\; -\; -->K\stackrel{^<!--\; ^\; -->}{x}(k)$ where $\stackrel{^<!--\; ^\; -->}{x}(k)$ is the country estimation vector that is produced via a linear complete-order state observer such that the nation of the system and the estimation blunders $e(k)=\stackrel{\to <!--\; \to \; -->}{x}(k)-<!--\; -\; -->\stackrel{^<!--\; ^\; -->}{x}(k)$ go to zero after a few finite time. layout the kingdom observer and the block diagram.
My method
it is clean that the eigenvalues of the machine are ${\mathrm{\lambda <!--\; \lambda \; -->}}_1=2,{\mathrm{\lambda <!--\; \lambda \; -->}}_2=-<!--\; -\; -->g$ (consequently it is not BIBO solid) and that the pair (A,b) is controllable for every fee of g, as nicely a the pair (A,c) is observable for all values of g. consequently we will shift the eigenvalues with the aid of deciding on a benefit matrix okay such that our device is strong, i.e. it has its eigenvalues inside the unit circle $|z|=1$.
The state observer equation is
$$[\stackrel{\to <!--\; \to \; -->}{x}(k+1)\stackrel{\to <!--\; \to \; -->}{e}(k+1){]}^T=\left[\begin{array}{ccc}A-<!--\; -\; -->bK& & Bk\\ O& & A-<!--\; -\; -->LC\end{array}\right][\stackrel{\to <!--\; \to \; -->}{x}(k)\stackrel{\to <!--\; \to \; -->}{e}(k){]}^T$$
With characteristic equation
$$\mathrm{\chi <!--\; \chi \; -->}(z)=|zI-<!--\; -\; -->A+bK||zI-<!--\; -\; -->A+LC|={\mathrm{\chi <!--\; \chi \; -->}}_K(z){\mathrm{\chi <!--\; \chi \; -->}}_L(z)$$
Also consider
$$K=\left[\begin{array}{ccc}{k}_1& & k_2\\ k_3& & k_4\end{array}\right]$$
and let $a=k_1+k_3,\mathrm{\beta <!--\; \beta \; -->}=k_2+k_4$
Then ${\mathrm{\chi <!--\; \chi \; -->}}_K(z)=(z-<!--\; -\; -->2)(z+g+\mathrm{\beta <!--\; \beta \; -->})+a$.
So we can select some eigenvalues inside the unit circle and determine $a,\mathrm{\beta <!--\; \beta \; -->}$ in terms of g. Choosing e.g. ${\mathrm{\lambda <!--\; \lambda \; -->}}_{1,2}=\pm <!--\; \pm \; -->1/2$ we get $a=3g+33/8,\mathrm{\beta <!--\; \beta \; -->}=9/4-<!--\; -\; -->g,g\in <!--\; \in \; -->\mathbb{R}$
Questions
I want to ask the following:
Is my approach correct? Should I select the eigenvalues myself since I am asked to design the observer or should I just solve the characteristic equation and impose $|{\mathrm{\lambda <!--\; \lambda \; -->}}_{1,2}|<1$?
Should I determine L matrix as well since the error must also vanish? (because it is not asked)

This might be a really simple question, but I just didn't find an answer from anywhere.
I'm teaching linear algebra to myself and in my study material I came upon notation that I just don't understand. I can't find an explanation for it from my material and it is hard to find on the internet as well it seems.
Example:
$U=L((3,2,-<!--\; -\; -->6,4),(0,4,1,-<!--\; -\; -->5))$
What does the L() notation mean? U itself should be a subspace for $R^4$. I would assume that those are vectors within the L().

where do combinatorics play a role in computer science?
I am graduated from computer science. I need some advice in Combinatorics material related to master fields of computer science. I already know that discrete mathematics have a key role in some concepts which I need a little more elaboration. In which fields could I trace combinatorics concepts and theorems and techniques?

How can I deepen my knowledge in Mathematics without getting a degree in Mathematics?
I'm an undergraduate student, I'm pursuing a Bachelor Degree in Computer Engineering. I had some exams about Mathematics (calculus 1/2/3, probability and statistics, algorithms and data structures, complex functional analysis).
While I don't want to become a researcher/professor (I would like to be a Software Engineer) I also love Mathematics, and I feel like I always want to know more.
Question: Which is the best way of studying on my own? Should I get some textbooks? Aren't textbooks suitable for those who are going to have an exam?
Actually I don't know which part I would like to deepen the most, but I'm really amazed by the Riemann hypothesis. What would it take to understand a wrong proof?

Euclid's view and Klein's view of Geometry and Associativity in Group
One common item in the have a look at of Euclidean geometry (Euclid's view) is "congruence" relation- specifically ""congruence of triangles"". We recognize that this congruence relation is an equivalence relation
Every triangle is congruent to itself
If triangle $T_1$is congruent to triangle $T_2$then $T_2$is congruent to $T_1$.
If $T_1$is congruent to $T_2$and $T_2$is congruent to $T_3$, then $T_1$is congruent to $T_3$.
This congruence relation (from Euclid's view) can be translated right into a relation coming from "organizations". allow $Iso({\mathbb{R}}^2)$ denote the set of all isometries of Euclidean plan (=distance maintaining maps from plane to itself). Then the above family members may be understood from Klein's view as:
∃ an identity element in $Iso({\mathbb{R}}^2)$ which takes every triangle to itself.
If $g\in <!--\; \in \; -->Iso({\mathbb{R}}^2)$ is an element taking triangle $T_1$to $T_2$, then $g^{-<!--\; -\; -->1}\in <!--\; \in \; -->Iso({\mathbb{R}}^2)$ which takes $T_2$to $T_1$.
If $g\in <!--\; \in \; -->Iso({\mathbb{R}}^2)$ takes $T_1$to $T_2$and $g\in <!--\; \in \; -->Iso({\mathbb{R}}^2)$ takes $T_2$ to $T_3$ then $hg\in <!--\; \in \; -->Iso({\mathbb{R}}^2)$ which takes $T_1$ to $T_3$.
One can see that in Klein's view, three axioms in the definition of group appear. But in the definition of "Group" there is "associativity", which is not needed in above formulation of Euclids view to Kleins view of grometry.
Question: What is the reason of introducing associativity in the definition of group? If we look geometry from Klein's view, does "associativity" of group puts restriction on geometry?

What is a good book to learn all of precalculus?
I need a no-fluff book with great exposition on precalculus. It should cover up intermediate algebra, trigonometry and anything else needed to get a strong preparation for Spivak's Calculus. It shouldn't contain annoying images or anything agitating, just serious math. But above all things, it should be very comprehensive and rigorous. It should help me to understand the concepts, and not just memorize them like we're supposed to do with most American textbooks...

Are the actuarial exams hard? I heard that they're hard. is this proper? Are they like the qualifying checks in grad college? as an example, is the opportunity exam and the monetary math examination comparable to qualifying assessments (e.g. desires months and months of have a look at)?

Undergraduate summer activities in the US or Europe?
I a currently a college freshman and I would like to do something math-related in the summer. I will have taken all of the basic courses except for analysis and set theory by the end of the summer (calculus,linear algebra, modern algebra, topology). I would like to know if there is anything out there for me, preferably something to which applications have not yet closed.
I think there are both summer schools and research summers or something like that. I have never done any research before though (not even close). And I doubt I could be able to without a lot of help.
So mainly I would like to know about the current panorama for math summer opportunities. I would also like to know if a dedicated student would be able to profit from a research summer. And finally if it would be plausible for a freshman to be accepted (I'm from Mexico if it helps).

I'am finishing my undergraduate degree in pc technological know-how and in spite of having needed to take a few math training my math capacity is still quite terrible. I struggled plenty with it which I consider changed into due to missing a few portions of know-how that I had to realize and not seeing the large photograph. Now i'am reading neural networks and seems they require pretty some math(eg: the backpropagation set of rules uses the chain rule). some human beings say you do not need the maths but I don't think I will be capable of to understand neural networks absolutely with out it. or even if I could get away with i would in all likelihood nevertheless want the mathematics later on.
i have approximately a month that i'm able to dedicate completely to this quest and i am determined to examine linear algebra and multivariate calculus on this time frame. i can start with linear algebra (doesn't depend on calculus, proper?) they have got solutions so I ought to be able to easily tune my progress. I also observed this path from Berkeley Math a hundred and ten. Linear Algebra. It would not have video lectures but it has greater assignments with answers so i'm able to exercise even greater. As for textbooks MIT uses "advent to Linear Algebra, Fourth version, Gilbert Strang" and Berkeley uses "Linear Algebra through S.H. Friedberg,A.L. Insel and L.E. Spence,Fourth edition". people here appear to mention true matters approximately them.
I'am beginning this journey the following day. in the suggest time i'd want to get some advice. Do you've got any recommendation with the intention to make this method smoother? Are there every other assets I should recognize approximately?

I would love to study a math plotting software which could be helpful in visualizing subjects in advanced calculus (my on the spot want). it might additionally be useful if the mathematics plotting software turned into of a few use in electric engineering, however this is not mandatory. the choice standards is listed right here in lowering weight:
Ease of Use (syntax and techniques that are intuitive and easy to adapt to other problem areas)
Healthy ecosystem (lots of tutorials, examples online, books and other resources
Industry use (looking for the most commonly used software suites within engineering and science)
Adaptability (commonly used outside mathematics. ie. electrical engineering, modeling).
I have narrowed my search down to:
Matlab
Mathematica
Maple
But this list is by no means exclusive. Currently I am leaning towards Matlab, because I have seen it being used in upper year courses in my electrical engineering program.
I would appreciate your input with regard to which software suite would be best and why. Thank you.

Model theory is typically based on a formal language whose production rules and syntax are designed to formalize the deductive process, and as such typically include predicate or relation symbols. Obviously, any predicate by itself constitues a (atomic) formula, but any formula, when interpreted in a model, corresponds to a predicate as well, insofar as the formula corresponds to a definable subset of the model's universe which one could then define as the interpretation of a predicate symbol.
Neither of those correspondences are bijections, however though evidently predicates are extraneous, so the question is - are they blanketed in the language for the identical cause that the established quantifier is included no matter being able to be described using handiest the existential quantifier and negation image, i.e. for comfort? Or is the inclusion of predicate symbols genuinely important, either in version concept particularly or in the philosophical motivation as a rigorous examine of the deductive procedure?

I'm not sure if the idea I have about them is right or wrong.
My professor defines four possible actions a Turing machine can do on a given input string:
Hang: The turing machine goes off the left end of the tape and "hangs"
Loop: The turing machine goes in a loop- never halts
Accepts: Turing machine halts in an accepting state
Rejects: Turing machine halts in a rejecting state
Is a turing recognizable language simply one that may either hold or loop? How do I determine if it could cling or loop? simply run it thru the system and if it does not halt on the rejecting state (or accepting) and its not part of the language of the TM then I determine its not decidable- so it must be recognizable? and then the decider is that if the language is assured to just accept or reject? What about languages that aren't recognizable? What might they do? as it seems that every one languages would be recognizable until it would be viable for the language to now not be inside the language of the turing system- but it halts inside the accepting kingdom anyways. Which looks like it would be because of bad layout of the turing system as opposed to the language itself.

What is the difference between ANOVA and ANCOVA?
In the context of using only experiment data for ANOVA analysis, ANCOVA offers post hoc statistical control. Is this a valid conclusion and why?