Questions about studying mathematics without formal instruction.

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1
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1answer
30 views

A doubt regarding splitting field.

$\Bbb Q(\omega)= \Bbb Q(\sqrt3,\iota)$ This is written in my text book that i am following. But I think this is a typo. Since $\Bbb Q(\sqrt3,\iota)$ is a larger field. in which $\Bbb Q(\omega)$ is ...
0
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1answer
15 views

Deriving Stochastic Euler Equation

If a consumer has utility function \begin{equation*} u(c_t) = ac_t - \cfrac{b}{2}c_t^2 \end{equation*} and present value budget constraint \begin{equation*} \sum_{j=0}^\infty E_t[\beta^jc_{t+j}] = ...
0
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0answers
20 views

symmetric normalized Graph Laplacian and symmetric normalized Adjacency matrix share the same eigenvalue

I am trying to show that the symmetric normalized Graph Laplacian and symmetric normalized Adjacency matrix share the same eigenvalues ($\lambda_i$ and $1 - \lambda_i$ for i=1 to n. $\lambda$ is an ...
3
votes
1answer
72 views

Application of Picard-Lindelöf to determine uniqueness of a solution to an IVP

I am still struggling quite a lot with the Picard-Lindelöf-Theorem (also known as the Cauchy-Lipschitz-Theorem). Problem: Consider the following IVP with $\alpha \neq 1$ $$\begin{cases} y'&= ...
0
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0answers
5 views

Testing series with a non-zero mean for stationarity

I was wondering if the following is correct. I am trying to wrap my head around testing time series for stationarity, but the lecture notes I am given are rather minimal. With external sources, I ...
0
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1answer
39 views

Given a second countable space $X$, show $A \subset X$ has uncountable limit points

Here's my attempt at a solution and I'm wondering if it's correct. Let $X$ have a countable basis with $A \subset X$ an uncountable set. Show $A$ has uncountably many limit points. Let $A'$ be the ...
0
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1answer
44 views

Why is the set of positive definite matrices in $\mathbb R^{n\times n}$ a positive cone

The set of positive definite matrices in $\mathbb R^{n\times n}$ is geometrically a positive cone. This statement appears in almost every article on real positive definite matrices I read but without ...
0
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2answers
31 views

Finding a cycle with a specific property

I am reading the book Dummit and Foote - Abstract Algebra . One of the exercises is to find an $n$-cycle $(n \ge 5)$, $\sigma$ such that $\sigma^k = \tau$ for some positive integer $k$, where ...
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1answer
55 views

Question about $G_\delta$ set

I'm on summer break but I want to keep my math skills sharp so I'm self-studying a bit from Munkres. This question is from pg 194, chapter 4 about the Countability and Separation Axioms. I've ...
1
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1answer
35 views

Show that for $f,g: [0, + \infty) \to \mathbb{R}$ convex functions of Class $C^2$ their product $fg$ is convex

Problem: Let $f,g: [0, + \infty[ \to \mathbb{R}$ be two convex functions of Class $C^2$. Assume that $$ f(0) \geq 0, g(0) \geq 0 \text{ and } f'(0) \geq 0, g'(0) \geq 0 \tag{!}$$ Show that $fg$ ...
4
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4answers
160 views

Some aspects of inner products in $\mathbb R^3$

I read several descriptions about inner products recently due to an exercise (here it is no longer active so I like to ask again). I am still very confused about this concept. Any clarification will ...
0
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1answer
18 views

Differentiability of Norm $N: U \subset \mathbb{R}^n \to \mathbb{R}, \ x \mapsto \sum_{i=1}^n i|x_i|$

Problem: Let $U:= \lbrace x \in \mathbb{R}^n \mid x_i \neq 0 \text{ for } 1 \leq i \leq n \rbrace $ and show that the Norm given by $$ N: \begin{cases} U & \longrightarrow \mathbb{R} \\ x ...
1
vote
1answer
27 views

Question on closed sets using a convergent sequence

Intro: The following two questions are from my exam preparation sheet, it is not mandatory and will not be accredited (or improve marks and the like). There won't be a correction, merely an online ...
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1answer
45 views

Is there any dominant strategy? [closed]

Is there any dominant strategy for the below matrix? I think no... But I am not sure. please check my answer:) $$\left (\begin{array}{ccc} (0,0) & (0,1) \\ (1,0) & (-1,-1) \end{array}\right ...
0
votes
1answer
43 views

Show a simple strategy.

Imagine that we have 49 cards with the values written on their faces, (they are all visible ) as follows; $$25, 24, 23, 22, ........3, 2, 1, 2, 3, .........23, 24, 25$$ suppose Paola and Victor are ...
0
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1answer
32 views

Help me writing Payoff matrix.

I guess, in order to answer this question, I need to write Payoff matrix. But I cannot write it. And then, I Will able to answer this question by myself. Thank you for helping. (These are just ...
2
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1answer
47 views

Check my answers: Dominant strategy.

I saw another question on Game theory. My answer for part a the nash equlibria (T, L) and (B,R). for part-b, Player-1's action T is strictly diominated. So Player1 never choose T. For part ...
4
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0answers
46 views

Is every curve measurable? [duplicate]

Does there exist a function $f:\mathbb R \rightarrow \mathbb R$ such that the set $E=\{(x,f(x)\mid x\in\mathbb R\}$ is non-measurable in $\mathbb R^2$?
0
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1answer
69 views

A question on Game theory

I'm studying Game theory, I saw the question: Consider two players; player A and player B playing the following estimation game. Each player chooses a number from {1, 2, 3}. If the difference ...
1
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0answers
44 views

Continuous transition kernel for Markov Chains

I am trying to show that the stationary distribution for a Markov Chain on a continuous state space can be obtained by building a transition density kernel, which obeys the detailed balance rule where ...
0
votes
1answer
19 views

Minumum value of integral and of partition are eventually the same?

I am trying to compare the minimum value of the integral of some function $f$ over the interval $[a,b]$ to the minimum value of the sum of the rectangles up to a certain point in a partition of ...
5
votes
2answers
127 views

Easy exercise (hint) Real Analysis

I've been stuck for a while with this problem. I suppose is something very easy, but I cannot figure out yet the correct approach. I'd really appreciated not a complete solution just some hints ...
0
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2answers
109 views

Measure Theory or Set Theory?

Having taken Real Analysis I before (the seven first chapters of baby Rudin) I have the option to take Measure Theory now. However I am torn between that and Set Theory. Which course would you go for ...
0
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1answer
40 views

Find $a,b,c\in \mathbb{R}$ that minimize the integral $\int_0^\infty |t^3 -at^2 -bt-c|^2e^{-t}$dt

Find $a,b,c\in \mathbb{R}$ that minimize the integral $\int_0^\infty |t^3 -at^2 -bt-c|^2e^{-t}$dt Hint:Use orthogonality of $(P_n)_{n=0}^\infty$ in $H=L_{2,\rho}(\mathbb{R}_+)$ with $\rho(t)=e^{-t}$ ...
3
votes
2answers
89 views

Show that $f(x,y)= \|x-y\|_2^2$ is differentiable

Problem: Show that $f: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ with $f(x,y)=\|x-y\|_2^2$ is differentiable and compute its differential at every point in the domain of $f$Note: $\| \cdot ...
2
votes
1answer
59 views

Good introduction to number theory that develops and/or makes heavy use of commutative ring theory and lattice theory?

I'd like to learn some number theory, since it provides a lot of motivation for commutative ring theory and even some motivation for lattice theory (at least, that's the impression I'm under). ...
32
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10answers
3k views

Becoming Better at Math

How can I become excellent at math? It really interests me but when I fail I become demotivated and begin to give up. EDIT: Could anyone suggest books for someone with a math education that just ...
2
votes
1answer
48 views

Let $X=L_2(0,1)$ and $S=\lbrace x\in X: \int_0^{1/2}x^2 dt\le1\rbrace$

Let $X=L_2(0,1)$ and $S=\lbrace x\in X: \int_0^{1/2}x^2 dt\le1\rbrace$. I'm trying to show wheter S can be written as $\cup_{n=1}^\infty S_n$ where $S_n\in X$ and $Int\bar{S_n}=\emptyset$ I tried a ...
2
votes
3answers
121 views

Manifolds and their dimension

Why are the following two sets manifolds and what are their dimensions? The set of all 2 by 2 matrices with determinant 1. The set of all inner products on $\mathbb R^3$. I have difficulty in ...
8
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1answer
173 views

Prerequisites for understanding G.H. Hardy's 'Divergent Series'

I picked up a copy of G.H. Hardy's 'Divergent Series' a few days ago. So far I love it, as I love the ideas associated with sequences and series, but I am finding it a bit difficult to understand. I ...
2
votes
2answers
71 views

On the canonical isomorphism between $V$ and $V^{**}$

I am trying to understand more about the Bidualspace (or double dual space). The whole idea is that $V$ and $V^{**}$ are canonically isomorphic to one another, which means that they are isomorphic ...
0
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1answer
34 views

Abstract Algebra: prove it is cyclic

I have question in referring to below link. Question. Suppose if I have [M:K]=2 and I know that K is subset of M. M:=$\mathbb{Z}_2[x]/(f(x))$ where f(x)=x$^4$+x+1. Then how this will be cyclic? I ...
0
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1answer
30 views

One and Two Tailed Independent T Test Questions

The city council for a small town has been receiving complaints from local law enforcement that citizens have been extremely uncooperative when pulled over for minor traffic violations. To remedy ...
7
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3answers
112 views

Is there a better way to read proofs?

I'm finishing my undergraduate degree in 6 weeks and I'm pretty happy with how my education is coming along so far. I can write proofs, solve many different problems, and I even have some idea as to ...
0
votes
1answer
42 views

Determine the languages for the given alphabet

For the alphabet $\sum = \{0,1\}, let A,B,C \subseteq \sum^*$ be the languages below. $i. A = \{1, 0, 00, 11, 000, 111, 0000, 1111\}$ $ii. B = \{w \in \sum^*|||w|| \ge 2 \}$ $ii. C = \{w \in ...
2
votes
2answers
36 views

Evaluating expression at infinity

How do I evaluate something like: $$xe^{-(x-\theta)}\text{ from }x = \theta\text{ to }x=\infty?$$ This came up in an integration I tried to do, and I realize it's a very basic question. But I am ...
0
votes
3answers
67 views

Normalizer of a subgroup of $GL_2(\mathbb{R})$

I have following subgroup of $GL_2(\mathbb{R}):$ $$A=\Bigg\{\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right),\left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} ...
6
votes
9answers
575 views

Good Textbooks for Real Analysis and Topology.

I'm currently in my 3rd year of my undergrad in Mathematics and moving onto my 4th year next year. I took a course in Real Analysis I, but the professor was very confusing and we didn't use a textbook ...
2
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1answer
46 views

Givens rotation and retraction mapping

Here's part of the texts in the book Optimization Algorithms on Matrix Manifolds, when talking about the retraction on a matrix sub-manifold. A retraction is a mapping that maps a tangent vector in a ...
0
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1answer
66 views

True/False question regarding continuity

For the two scenarios below, either give an example if such a request is possible, or argue why such an request is impossible. I think the first is possible and second is impossible. However, I can't ...
1
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2answers
64 views

Approaches to teaching and learning analysis

I found that studying linear algebra by getting into vector spaces and linear transformations first made things very easy. This is the approach Halmos or Axler, just to name a few, take. IMHO, the ...
0
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1answer
57 views

Give an example of a relation R on $A^2$ which is reflexive, symmetric, and not transitive

I am just looking for some clarification on this exercise: Let $A = \{a,b,c,d\}$. Give an example of a relation $R$ on $A^2$ which is reflexive, symmetric, and not transitive. I understand that if I ...
0
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2answers
51 views

Is there any motivation for the direct sum?

I believe that everything have reason to exist . I want to learn why the direct sum exist. Is there any motivation for the direct sum to exist ?
3
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1answer
39 views

Differentiation of $u(t)=\int_0^t h(s,t)ds, \ \forall t \in \mathbb{R}$ with the multivariable chain rule

Problem: Let $h: \mathbb{R}^2 \to \mathbb{R}$ be continuous and differentiable with respect to its second variable, define $u(t)= \displaystyle \int_0^t h(s,t)ds, \ t \in \mathbb{R}$ In an ...
0
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1answer
61 views

Is $\sum \cos(n \pi) \frac{n}{n^2+1}$ conditionally or absolutely convergent?

Is $\sum \cos(n \pi) \frac{n}{n^2+1}$ conditionally or absolutely convergent? My Working Clearly, $\sum \cos(n \pi) \frac{n}{n^2+1} = \sum (-1)^n \frac{n}{n^2+1}$, and it can be shown by using ...
0
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0answers
18 views

Book recommendation for introductory algebraic combinatorics?

Preferably: It should have plenty of motivation (as I am self-learning). It should not be skimpy on proofs (as I am self-learning), but perhaps I can make do since I can ask questions elsewhere. It ...
4
votes
1answer
81 views

Divergent subsequence of an unbounded sequence

Let $(a_n)$ be a sequence of real numbers that is unbounded above. Show that $\exists$ a subsequence $(a_{n_k})_{k \ge 1}$ such that $\lim_{k \rightarrow \infty} a_{n_k} = + \infty$. Working ...
3
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0answers
67 views

Cauchy-Euler Equation of order $n$

What I wish to prove is that for a Cauchy-Euler equation of order $n$, the substitution $x=e^{t}$ transforms it into a linear differential equation with constant coefficients. To put it as a theorem: ...
0
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1answer
85 views

Sequential Criterion for Functional Limits

How can I use the sequential criterion for functional limits to show that the following limit exist and compute the limit: $$\lim_{x \rightarrow 0} \sqrt{|x|}\cos\left(1/x\right) \ \text{for} \ x ...
3
votes
1answer
79 views

Mathematics or physics at university

I have a strong interest in maths, and I feel that advanced physics is cool too (although I've only studied classical mechanics at high school, which is kind of boring). So I'm not sure about which ...