Questions about studying mathematics without formal instruction.

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0
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2answers
22 views

Find the equation of the parabola given the tangent to a point and another point.

I have a problem with derivatives, I've been trying to solve but I was not able to do it. A parabola is tangent to the line $3x-y+6 = 0$ in the point $(0,6)$ and goes through the point $(1,0)$. ...
0
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0answers
11 views

(soft question) Kreyszig's Functional Analysis or Rudin's Real and complex analysis for an introduction into more advanced analysis?

I realise they are quite different in their approach and material covered, but they share the central stuff like normed/Banach/Hilbert spaces, Hahn-Banach theorem etc. Not really understanding what ...
1
vote
1answer
46 views

Categories and the direct image functor

I've just started learning about sheaves and yesterday I had a look at MacLane/Moerdijk's book "Sheaves in Geometry and Logic". I've discovered something in this book on which I'd like some ...
7
votes
3answers
53 views

How do I decide what problems and how many problems to do when I try to self study?

I am a math major at a relatively small college with barely any choice of classes to choose from so I have to supplement my studying with a lot of self studying. I usually have no problem getting ...
0
votes
1answer
18 views

A Question on the Scaling Invariance of Brownian Motion

I read the following paragraph. Let $B_t, \ t \in [0, \infty)$ be a standard linear Brownian motion. For each $q > 4$, define the following sequence of sets. $$ \Omega_k := \left\{\omega \in ...
1
vote
1answer
35 views

How to show this inequality in the law of iterated logarithm

Let $\psi(t) = \sqrt{2t\log\log t}$ and $q>4$. Then we have for large $k$, $$ \psi\left(q^k - q^{k-1}\right) \geq \psi\left(q^k\right) \left(1-\frac{1}{q}\right). $$ To prove this, I can tell by ...
1
vote
1answer
15 views

Zero divisors and inverstible elements

I have learned about $X_n = \mathbb{Z} / n\mathbb{Z}$. I understand that a zero divisor is an element $x\neq 0$ in $X_n$ such that $xy = 0$ for some $y\neq 0$. I understand that an element $x$ in ...
0
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0answers
14 views

On Nonlinear Autonomous system of two equations if the eigenvalues of the Jacobian matrix are 0.

Suppose we have a non-linear autonomous system of two equations: $$\begin{cases} x'(t) = F(x,y) \\ y'(t) = G(x,y) \end{cases} $$ and we obtain a fixed point for this equation, but the eigenvalues of ...
0
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0answers
2 views

Unable to follow notation and meaning of probability distribution for binary time series

I am unable to understand concepts related to the probability distribution of binary time series. [Mathematics is not my background]. This is from the book Binary time series by Benjamin Kedem, vol ...
0
votes
2answers
54 views

I want to study higher mathematics. Where do I start?

Since last year, I've been interested in higher mathematics and don't really know where to start and don't know what knowledge I still need to obtain, before being able to understand the concepts. ...
0
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0answers
14 views

Total Variation of a Signed Measure

In Royden's Real Analysis the total variation $|\mu|$ of a signed measure $\mu$ is defined to be $$|\mu|(E) := \sup\sum_{k=1}^n |\mu(E_k)|,$$ where the supremum is taken over all finite disjoint ...
1
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2answers
27 views

A Question on Bolzano-Weierstrass Theorem

The following are two equivalent statements about the Bolzano-Weierstrass theorem. Every bounded real sequence has a convergent subsequence. A subset of $\mathbb R$ is compact if and only if it is ...
0
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0answers
22 views

On a cauchy problem (differential equation system) .

I am approaching the theory of these kind of problems but I am missing an example. I am tasked to solve: $$X'= \left( \begin{array}{ccc} 1 & 4 \\ 1 & 1 \\ \end{array} \right)X + \left( ...
0
votes
1answer
23 views

Convergence of $\sum \frac{i^n}{\log(n)}$

Study the convergence of $$\sum_{n \geq 2 } \frac{i^n}{\log(n)} $$ where $\log$ of course denotes the 'natural logarithm' and $i \in \mathbb{C}$ Oddly enough I managed to show Abel's Test for ...
1
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0answers
24 views

Limit of a random walk

Suppose we have a simple random walk: $X_n\pm1$ with equal probabilities. For any finite $n$, $E[\sum_{k=1}^nX_k]=0$. Does it imply that $E[\lim_{n\rightarrow \infty}\sum_{k=1}^nX_k]=0?$ Thank's! ...
-1
votes
1answer
26 views

How to derive this inequality

I learnt that for a standard normal random varialbe $Z$ and positive $x$, we have $$\mathbb P (Z > x) \geq \frac{x}{x^2+1} \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} = \frac{1}{x+\frac{1}{x}} ...
0
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0answers
30 views

Textbooks to complete concurrently - Self learning empowerment

A user is completing some year challenge that takes them through $9$ textbooks and they are alternating in author. Algebra - Cohn Analysis - Rudin Topology - Lee Repeat three times. I would like ...
0
votes
1answer
11 views

Limiting random variable

My question is about limiting r.v. Suppose, we have a sequence of r.v.s. $\{X_n\}$. And we know that $\liminf X_n=-\infty$ and $\limsup X_n =\infty$ almost surely. What can we say about $\lim X_n$. ...
1
vote
2answers
29 views

The set $\mathbb{Z}$ is totally ordered

Having the following definition of the $\leq$-Relation in $\mathbb{Z}$: For $a, b\in \mathbb{Z}$ we define $$ a \leq b : \iff b-a \in \mathbb{N} $$ Show that $(\mathbb{Z}, \leq)$ is totally ...
1
vote
1answer
16 views

Question on induction and the application of an 'equivalent' induction hypothesis.

I am working on the following problem which I decided to solve by induction Problem: Let $(a_n), (b_n)$ be sequences for $n \geq 1$. Define $B_n:= \sum_{i=1}^n b_n$ for $n \in \mathbb{N}$. Show ...
2
votes
0answers
66 views

measure theory exercise (verification)

Hi I found the following exercise in the Dudley's book and I'd like to see if my answer is correct; the last part is what I'm not entirely sure, since I'm not completely familiar with this kind of ...
0
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0answers
14 views

Self studying elliptic curves

Is there a book or lecture notes to learn to determine the integer points of elliptic curves? I have heard that they can be used to factor integers and solve equations like $y^2=x^3+ax+b+c$ and I ...
0
votes
1answer
53 views

What is meant by a kernel and homomorphism in algebraic structures?

I have just started with discrete maths. I was doing some group theory and I stumbled upon kernel and homomorphism. I didn't understand what was written in the book. I googled it up and also looked ...
1
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0answers
18 views

Subsets of $ \mathbb Q $ of order type $ \omega^{\alpha}$ for each countable ordinal $\alpha $.

My introductory text in Set Theory (Stillwell) includes an exercise (6.3.1) asking for an explicit example of a subset of $ \mathbb Q $ or order type $ \omega^2 $. This seems straight forward enough. ...
0
votes
1answer
12 views

on Limits and sequences proofs of a closed subspace of the hilbert space.

$M$ is a closed subspace of the Hilbert space $H$ and $ x\in H$ My book states these two claims. (1) If $d = \inf_{y \in M} \|x - y \|^2 $then there is a sequence of elements $\{y_n\}$ of $M$ such ...
1
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0answers
32 views

What is the tensor product of Z/10Z with Z/12Z? [duplicate]

I've just met tensor products of modules in part of my self-study and as a concrete exercise I've been asked to calculate $\mathbb{Z}/(10){\otimes}_{\mathbb{Z}} \mathbb{Z}/(12)$. Short of writing out ...
1
vote
1answer
20 views

Progressively Measurable for Rigth Continuous Adapted Processes

Any adapted and right continuous process $X_t$ is progressively measurable. For the above statement, I found proof in several books. They all have similar argument as follows. For a given $t > ...
1
vote
0answers
23 views

Help in deriving Maximum likelihood estimator

The problem is estimating a metric $\operatorname{Vol}(D)$ for the following situation : Given noisy observations the observed data are a random sample $Y_1,\ldots,Y_n$ where $Y_i \in \mathcal{R}^d$. ...
0
votes
4answers
56 views

Cardinality of sets of functions

Show that the set $A$ of all functions $f:\mathbb{Z}^{+} \to \mathbb{Z}^{+}$ and $B$ of all functions $f:\mathbb{Z}^{+} \to \{0,1\}$ have the same cardinality. I am having trouble to define a ...
0
votes
2answers
36 views

Gelfand trigonometry question

If we start with a lemma that states that when $ a^2+b^2=1$ there exists an angle $ \theta $ such that $ a=\cos\theta $ and $ b=\sin\theta$ Suppose that $\alpha$ is some angle if ...
0
votes
1answer
19 views

A question on semi-martingale and its variations

With probability one, paths of semimartingales have unbounded variation. What I know is that a martingale is also a semi-martingale, for example, Brownian motion. Hence, Brownian is an example of ...
3
votes
3answers
42 views

Group isomorphisms and a possible trivial statement?

I have the following set $G=\lbrace a,b,e \rbrace$ and I successfully computed the following Cayley-Table \begin{align} \begin{array}{|c|c|c|c|} \hline \circ& a & b & e \\ \hline a& ...
0
votes
3answers
33 views

Trigonometry proof- finding an angle

I don't even know where to start with this problem. Suppose $\alpha$ is some angle less than $45^\circ$. If $a=\cos^2\alpha - \sin^2\alpha$ and $b = 2\sin\alpha\cos\alpha$, show that there is an ...
6
votes
0answers
115 views

Finitely additive function on an infinite set, s.t., $m(A)=0$ for any finite set and $m(X)=1$ (constructive approach)

Other exercise which I found in Dudley's Analysis book: Show that there is a measure on a infinite set $X$, defined on $2^X$ s.t. is finitely additive, $m(A)=0$ for any finite set and $m(X)=1$. ...
1
vote
1answer
31 views

$\{ x \in H: x=\sum_{k=1}^{\infty}c_{k}u_{k}$, $|c_{k}| \leq \frac{1}{k}\}$ is compact

Let $H$ be a complex inner product space that is also a complete metric space with respect to the distance induced by the inner product. Assume $\{u_{k}\}_{k=1}^{\infty}$ be an orthonormal set in ...
0
votes
1answer
52 views

For which angles $a$ is $\sin^4 a - \cos^4 a > \sin^2 a - \cos^2 a$?

My questions For which angles $a$ is $\sin^4 a - \cos^4 a > \sin^2 a - \cos^2 a$? For which angles $a$ is $\sin^4 a - \cos^4 a \ge \sin^2 a - \cos^2 a$? I understand that the two sides will be ...
2
votes
1answer
51 views

A question on Lebesgue measure: Inequalities

Let $A_n$ be a decreasing sequence of Borel sets with finite but with measure $\geq \epsilon > 0$ for all $n$. Then there exists a compact set $B_n \subset A_n$ for all $n$ such that $\mu(A_n ...
1
vote
4answers
34 views

Trigonometry identity proof

I am working my way through Gelfands trigonometry book. One of the exercises asks to prove the following identity: $$ \frac{\sin(a)}{1 + \cos(a)} = \frac{1 - \cos(a)}{\sin(a)}$$ I can reduce the ...
2
votes
1answer
65 views

Arzela Ascoli, help to understand some points in the proof.

Hi everyone I'd like if someone could give me an explanation of some points in the following proof, explicitly the points with the asterisk. This is from Dudley's, one direction is completely easy, ...
4
votes
2answers
57 views

Verifying that $(G, \circ )$ is a group, where the notion of $G$ and $\circ$ become very complex.

First of it all, sorry about that horrible title, if you know how to refine it please be my guest and do so. This question is of the same caliber as $\hom(V,W)$ is canonic isomorph to $\hom(W^*, V^*)$ ...
1
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0answers
32 views

Abelian Group (Alternative Proof)

Is there an alternative method to prove $(ab)^{2}=a^{2}b^{2}$ for all elements $a,b \in G \implies$ $(ab)^{-1}=a^{-1}b^{-1}$ for all elements $a,b \in G$. then the one I give below? Let $G$ be ...
0
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0answers
34 views

Proof Verification/Alternative to Induction- Well ordering proof

This question grew out of Induction and Maximum Principle, which yours truly asked Sep 23, at 11:51. Due to Mauro Allegranza's suggest, I changed focus so as to first prove the equivalence of $(a)$ ...
2
votes
1answer
95 views

Why are these logical statements not deemed to be equivalent?

I'm working through a book on my own which has just introduced the ideas of $A \Rightarrow B, B \Leftarrow A$ and $A \Leftrightarrow B$. It then gave 20 exercise questions to answer. I've correctly ...
1
vote
3answers
147 views

Find the distance between two towns given train timings

While practicing maths and starting to learning it, I found question this question: A train running between two towns arrives at its destination 10 minutes late when it goes 40 miles per hour and ...
2
votes
1answer
42 views

Any suggestions for a Math book to revive my long lost math skills and knowledge?

Since I got my bachelor degree in computer science in 2005, I have rarely been in touch with or even need to do any significant mathematical calculations. I have consequently lost most of my math ...
1
vote
1answer
16 views

Expectation of Truncated Random Variables (2)

This is a follow-up question from here. Let $X_n$ be a sequence of $iid$ random variables with zero mean, finite variance $\sigma^2$ and partial sum $S_n:=\sum_{k=1}^n X_k$. Let $$0<\delta<0.5$$ ...
2
votes
0answers
49 views

self-study hints

A question to those who took rigirous courses like math 25 (Harvard), MATH 295-396 Michigan and etc Being not able to collectively discuss problem sets from the course, as those who involved in ...
1
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0answers
32 views

Deriving the Resolvent Cubic From Elementary Symmetric Functions

On page 614 of Dummit and Foote's Abstract Algebra, while deriving the resolvent cubic for a quartic $g$, they go from a trio of elements $\theta_i, i=1,2,3$ which were defined in terms of the roots ...
0
votes
4answers
102 views

On finding the equilibrium solutions to a system of differential equations

I am asked to find all equilibrium solutions to this system of differential equations: $$\begin{cases} x ' = x^2 + y^2 - 1 \\ y'= x^2 - y^2 \end{cases} $$ and to determine if they are stable, ...
1
vote
1answer
45 views

Evaluate $\int_0^\infty \frac{1}{x^{n+2}} c^{1/x} \exp\{-\lambda/x\} \mathrm{dx} $

I have been trying to evaluate the following integral $$\int_0^\infty \frac{1}{x^{n+2}} c^{1/x} \exp\{-\lambda/x\} \mathrm{dx} $$ What I am getting is $$\frac{1}{\left(\lambda-logc ...