Questions tagged [self-learning]

The process of studying mathematics without formal instruction. Don't use this tag just because you were self-studying when you came across the mathematical question you're asking; it is only for when the fact that you're self-studying is what your question is about.

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25 views

Proof Validation: Let $f: A \to B$. Then $f^{-1}f$ is the identity on $\mathcal{P}(A)$ if 1-1, and $ff^{-1}$ the identity on $\mathcal{P}(B)$ if onto.

I am attempting some self-learning of proof based texts, and I'm a little confused on what's "allowed". Specifically, I am trying to prove the below claim, and I am unsure about a few steps ...
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2answers
38 views

Is it worthwhile to redo problems where an arithmetic mistake has been made?

When learning new concepts in math, is it worth my time to redo problems if my mistakes were purely arithmetic mistakes (or very simple algebra mistakes)? If I made sign errors, or added fractions ...
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Verifying Proof: If $L$ is a poset with a bottom element, and $\exists \sup(S)$ for every subset $S \subset L$, then $L$ is a complete lattice

I am currently working through Kaplansky for self study and was hoping to get some feedback on this proof. I would appreciate comments on clarity and legability as well. Claim: If $L$ is a poset with ...
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18 views

Study multiple subjects at the same time or deep dive into one?

I want to learn probability theory and discrete math. However, I also need to brush up on computational calculus and linear algebra. Would you recommend only studying one subject at a more intense ...
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36 views

Learning Mathematics: unique knowledge of math.

I've come to a point where I now more fully (like 99%) understand that math is about specialized/unique knowledge which can't be summarized by some grand scheme or united principle. So my question is, ...
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Is this theorem important or not? [closed]

Taken from Allen Hatcher Book Given a map $F : Y \times I \to X$ and a map $\widetilde{F}: Y \times \{0\} \to \widetilde{X}$ lifting $F|Y\times \{0\}$,then there is unique map $\widetilde{F} : Y \...
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2answers
33 views

Is median a number that minimizes the variance?

As far as I know, the mean is a value that minimizes the variance but I wonder if the median also a value that minimizes the variance or it is a value that means a value in the middle of the data? ...
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25 views

S. Roman Advanced Linear Algebra prerequisites [closed]

In Advanced Algebra by S. Roman, he says that the prerequisites are a basic knowledge of matrices and determinants but that a prior course in linear algebra (even at the sophomore level) is highly ...
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1answer
109 views

In need of new direction/book recommendation; self studying math

I self study mathematics in hope of one day getting a degree in Applied Mathematics. I will provide my mathematical background first as I think it will be necessary to you in giving advice. (I am a ...
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41 views

Prerequisites for Complex Analysis by Serge Lang

In Serge Lang’s Complex Analysis, he says that a prerequisite is 2 years if calculus. But I’m not sure exactly what he means by this. In regards to calculus related courses, I’ve done 3 semesters of ...
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Prerequisites for Sullivan's 1970 notes

What are the mathematical prerequisites to start reading Sullivan's MIT lecture notes of 1970: Geometric Topology: Localization, Periodicity, and Galois Symmetry? I have already learnt some algebra (...
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Prioritising what to study over summer [closed]

After this semester is finished, I plan on studying some more maths over the summer. I’m currently a second year student and my summer is 3 months long. Since this is about the length of 1 full ...
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33 views

Pdf of two uniform random variables with different ranges multiplied

We know that random variables that is of uniform distribution can only have 1 numerical value in a certain range and 0 otherwise. But what happens if we multiply two uniformly distributed random ...
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1answer
32 views

An intuitive (non rigorous) text book on graph theory which is student friendly with vivid illustrations

Background Hello, I am an undergraduate in CS. I would like to study Graph Theory on my own (self-study) for a competitive examination (named GATE). It is an examination for undergraduates and as such,...
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Exercise 4.4.1 Terence Tao

I am self-studying Terence Tao. I am stuck on the following question. Exercise 4.4.1. Let $x$ be a rational number. Then there exists an integer n such that $n \leq x x < n + 1$. In fact, this ...
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Transform this game into a Strategic Form Game and draw the game table

Find the Subgame Perfect equilibrium strategies. Solution: Stage 1 : player 2 chooses F after C. And player 2 chooses E after A. Stage 2: player 1 chooses C. Then SPNE={ (C), (E after A, F after C)}. ...
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How can I prove that the mean of a probability distribution is a value that minimizes the variance of the distribution?

According to Page 87, John_K_Kruschke-Doing_Bayesian_Data_Analysis-EN.pdf 2nd Edition, the author says that the mean of a distribution is a value that minimizes the variance of a probability ...
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499 views

How do you retain knowledge of multiple undergraduate math courses?

I know, "practice, practice, practice." But is that really what professional mathematicians do? Do you actually solve a differential equation every day, do a contour integral every day, do ...
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22 views

Find pure strategy Nash equilibrium in 3-player game

The answer of this question says that The pure strategy Nash equilibria of this game are: (U ; L; A), (D; R; A) and (D; R; B). But this answer does not make sense for me. I found that (U,L,A), (U,R,A),...
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32 views

distribution of sample mean divided by sample variance [closed]

I'm reading about sampling from the normal distribution. It is well-known that $\displaystyle \frac{\bar{X} - \mu}{S / \sqrt{n}}$ has Student's $t$ distribution with $v=n-1$ where $v$ is degrees of ...
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34 views

Can two different topologies have the same basis? [duplicate]

I started general topology recently, and a partner of mine made a example of two different topologies generated by the same basis, so I'd like to know if that is possible, and if it is not, to see a ...
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1answer
52 views

The construction of the reduced strategy form matrix

Question Given the payoffs in the matrix below, two players play a variant of the “Battle of Sexes”(BoS) game, in which each player chooses in between F and C. At first, player 1 makes a choice in ...
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1answer
28 views

Linearization of a fixed point (dynamical systems)

Just looking at the following piece of math from Strogaz's dynamic / chaos book. What I don't understand is the last part, where he claims that O($Ƞ^{2}$) is negligible if $f^{'}(x^{*})!=0$. I guess ...
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2answers
75 views

About argument of forgetful functor

Let $(\{0,1\},0) \in \text{Set}_*$ be the set $\{0,1\}$, equipped with the base point $0 \in \{0,1\}$. Prove that $(\{0,1\},0)$ represents the forgetful functor $U:\text{Set}_* \rightarrow \text{Set}$....
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Show that the game has a unique Nash equilibrium (in pure or mixed strategies)

Each of 15 students simultaneously announces a number in the set {1, 2, . . . , 100}. A prize of 1 TL is split equally between all students whose number is closest to 1/3 of the class average. Show ...
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2answers
36 views

Find best responses

Question: There are N voters who have positions that can be indicated by the numbers 1 through 7. The number of voter with each position is indicated in the table below: Assume that voters always ...
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34 views

Splitting numbers into equal parts

I am trying to solve a problem in which I am dividing a list stick into equal size given by K. Input: stick = [5, 9, 7], K = 4 This means that we have three sticks with lengths as 5, 9 and 7. We would ...
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1answer
36 views

Which pure strategies in each game are dominated?

Which (pure strategies) in each game are dominated? For each dominated strategy specify the (mixed) strategy that dominates it. The solution manual says that in Game 1, R is dominated by $\sigma_2 = (...
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1answer
23 views

find the pure strategy Nash equilibrium with 3 players

There are three players and each stakes 10 dollar. The players simultaneously choose a number from the set {1, 2, 3, 4}. If three numbers are different, the player choosing the middle number wins the ...
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1answer
33 views

Find the Nash equilibrium for the given question

Consider there are two players in a simultaneous game. Each of them chooses and pays integer multiples of one dollar to play the game. The player with the highest bid wins 100 dollar and the loser ...
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Verifying examples of definition of field extension

Let K be a field containing F as a subfield, written as $K/F$. Call $F$ the base field of $K/F$ and $K$ the extension field of $K/F$. Definition: Let $K/F$ be an extension of fields. K is a vector ...
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4answers
277 views

A group structure on the integers solutions of a Pell equation [closed]

When I work over abstract algebra, by the way I am beginner, I encounter a new type of question. I have never seen this type before such that Let $ G = \{(x, y) \in\Bbb Z \times\Bbb Z : x^ 2 − 13y^ 2 ...
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1answer
33 views

How can an autodidact gauge their learning progress when relating to mathematics?

Over the past few months I have been learning about discrete mathematics and proof writing. I understand that learning writing proofs is a collaborative effort - I need someone to actually review my ...
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1answer
20 views

True or false Self adjoint operator and trace

I have a problem trying to prove the next statement or giving a counterexample If $T:\mathbb{R}^3 \to \mathbb{R}^3$ is self-adjoint such that trace($T^2$)=$0$, then $T=0$. I tried with the canonical ...
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4answers
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How do I learn more advanced mathematics? [closed]

I am a junior at high school (16.1 years old); although I perform at the top of my class and have some competitive achievements in the nation, I think I am both very far from the IMO and college-level ...
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1answer
47 views

Category property in monoid homomorphism

Problem: Show the collection of all sets with the collection of all monoid homomorphisms is a category. By definition, to satisfy a category, I need to prove two axioms 1): for every morphism $f:X \...
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2answers
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Is the following sentence contrapositive of the sentence “If it's spring, the birds will sing”

I need to find the contrapositive, which is "If the birds do not sing, it is not spring" But if it's written like, "If it's not spring, the birds will not sing", can this be ...
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0answers
13 views

Finding a lower bound on N using CLT for coin toss experiment

I'm studying for an upcoming exam and came across the following question: A coin is tossed $N$ times with probability of heads $p_{h} = \frac{3}{4}$. Find a lower bound on $N$ such that $P[Y_n \geq \...
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2answers
114 views

Munkres vs Hatcher

I am self-studying algebraic topology and I was wondering which book is better for my purposes. The second part of Munkres' topology deals with algebraic topology and it seems easier to understand, ...
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0answers
58 views

Trouble applying theory to practice

I am a first-year math student and I am having difficulties applying newly learned theory to problems. For example, my current calculus book is Real Analysis: Series, Functions of Several Variables, ...
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2answers
36 views

Percentage of the normal area - How to find the area under the normal distribution?

First this is not my homework is an example for learn and solve this little problem. (Reference online pdf here.) I have problems only with this section in exercise text: The middle $60\%$ means ...
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78 views

Can I do good math research if I am mediocre in graduate course?

I mostly got B's in my graduate courses like algebraic topology, differential geometry, algebraic geometry and etc, and I start to lose confidence and patience because of this. Is it still possible to ...
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0answers
9 views

What is the link between general position of a dataset and rank of a matrix?

I would like to know the link between general position and multicollinearity. In statistics, a multivariate dataset of dimension p is said to be in general position if at most p observations lies in a ...
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1answer
44 views

Shared area of $2$ circles (polar integral)

Given $x^2+y^2=4y$ and $x^2+y^2=4x$ find the shared area of the 2 circles What I tried was : so first I transformed them to polar coordinates I got $r=4cos\theta$ for $x^2+y^2=4x$ and $r=4sin\theta$ ...
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0answers
54 views

Opinion regarding the book “Friendly Approach To Functional Analysis” [closed]

I am interested in learning functional analysis through self study. My long-term goal is to use the concepts to grasp a better understanding of other topics e.g. optimization. My background includes ...
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2answers
62 views

CGA Representation of an Ellipse

In conformal geometric algebra (CGA), I know the representation of a circle $s$ with center $c$ is: $$ S = H(c) - \frac{1}{2}r^2n_{\infty} $$ That $H(c)$ is the representation of $c$ (i.e., $n_0 + c +‌...
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0answers
46 views

Creative approoach to understanding advanced math notation

I was very pleased when I saw Hilbert curves and as a student I am interested to learn advanced mathematical concepts such as Hilbert spaces in an interesting and creative way. In many papers I am ...
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3answers
34 views

Calculate if an expression is uniformly lower than another?

I have two expressions $$ \frac{2}{n-1} \quad \text{and} \quad \frac{2n-1}{n^2} $$ and I need to prove that $$ \frac{2n-1}{n^2} < \frac{2}{n-1} $$ for all $n \in \{2,3,\ldots\}$. How can I do this?
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1answer
56 views

area to the right of $r\cos\theta=6$ and $r=10$

I have been trying to solve this question for a while and I seem to get stuck although it does not seem hard Given $r\cos\theta=6$ and $r=10$ we need to find the area inside the circle and on the ...
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0answers
38 views

Reversing role of group and set in definition of stabilizer

Given a group action of $G$ on $M$, we have the notion of the stabilizer of a point $m$ : $Stab_G(m)=\{g \in G | g(m)=m \}$ , and we could extend this to a definition of the stabilizer of a generic ...

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