Questions about studying mathematics without formal instruction.

learn more… | top users | synonyms (1)

2
votes
3answers
47 views

Applying Rouché's Theorem

Determine how many zeros of the following polynomial lie inside the circle $|z|=2$ \begin{equation} z^{5}+2z^{4}+z^{3}+20z^{2}+3z-1=0\end{equation} My Reasoning If we put $f(z)=z^{5}+2z^{4}$ and ...
1
vote
2answers
39 views

Property of a system of two inequalities

I have this system $$\begin{cases} a+b>1 \\ a-b>1 \end{cases}$$ can I sum the second inequality to the first getting $a>1$? Or this property can be used only equations?
2
votes
2answers
37 views

Number of possible eight digit number divisible by 9

An eight digit number divisible by 9 is o be formed by using 8 digits out of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 without repetition. Find the number of ways in which it can be done. I know divisible rule of ...
1
vote
1answer
27 views

Writing The Derivative Of $f(x)$ With Respect To $g(x)$ In Limit Form

What would be the proper way to represent this derivative in the limit form? $$\frac{\mathrm{d} }{\mathrm{d} g(x)}[f(g(x))]$$ In my attempt to solve this I've tried to word out the derivative: The ...
0
votes
1answer
25 views

On the equivalency of two indefinite integrals using u substitution.

I am reading the Separation of variables page on wikipedia, at a certain point it states that the following equation Is equal to (1) because of the substitution rule of integrals. The ...
4
votes
0answers
56 views

How can I retain the mathematics that I've supposedly learnt?

So my question simply is "What is the best method to make sure you retain what you have learnt?" Okay so I've tried learning mathematics up to where I should be at in the past. Every time though I ...
2
votes
1answer
41 views

Largest possible number of student passed in all three subject

Question: From $50$ students taking examination in Mathematics, Physics and Chemistry, $37$ passed in mathematics, $24$ passed in physics and $43$ passed in chemistry. At most $19$ students passed in ...
1
vote
1answer
43 views

How is Fubini Theorem used here?

Let $\mu$ be a $\sigma$-finite translation invariant measure defined on the Borel subsets of $\mathbb R^d$ and $\lambda$ be the usual Lebesge measure. My question is how the Fubini theorem is used in ...
2
votes
2answers
74 views

How to gain a mentor for self study

I am trying to independently learn mathematics at an upper-level undergraduate and first year grad student level. It's mostly linear algebra and statistics. The textbooks that I'm reading are quickly ...
1
vote
2answers
49 views

Self learning game theory and probability

I am teaching myself mathematics, my objective being a thorough understanding of game theory and probability. In particular, I want to be able to go through A Course in Game Theory by Osborne and ...
0
votes
1answer
22 views

Ratio of hazards in Proportional Odds model

In the proportional odds model we have the the odds of survival in 1 group are proportional to the odds of survival in another group $$\dfrac{ S_1(t)}{1-S_1(t)} = \psi \dfrac{S_0(t)}{1-S_0(t)}$$ ...
3
votes
3answers
87 views

Derivative of a function with respect to another function.

I want to calculate the derivative of a function with respect to, not a variable, but respect to another function. For example: $$g(x)=2f(x)+x+\log[f(x)]$$ I want to compute $$\frac{\mathrm ...
0
votes
0answers
19 views

Joint Limiting Distribution of Min and Max

Let $X_1,\ldots,X_n$ be iid from the uniform distribution $U(a,b)$. Let $X_{(1)}< cdots< X_{(n)}$ be the order statistics. Find the joint limiting distribution of $(n(X_{(1)}-a),n(b-X_{(n)}))$ ...
3
votes
1answer
73 views

How to avoid losing the woods for the trees in daily study/lecture time

When facing to some new material in mathematics, I feel easily to be overwhelmed by lots of details with losing the woods for the trees. So is there some good strategy to study the materials ...
3
votes
1answer
59 views

Uncountable Dense Linear Orders

Is there an example of two uncountable equipollent dense linear orders without endpoints that don't satisfy the same first order properties? Or is it true that two uncountable equipollent dense linear ...
0
votes
0answers
45 views

On what value of $f(X)$ minimizes $E[(Y-f(X))^2|X]$

$X$ and $Y$ are random variables. The question is: what value of $f(X)$ minimizes $E[(Y-f(X))^2|X]$. I am pretty sure I have found the solution to this problem by writing: $$E[(Y-f(X)-E[X|Y] +E[X|Y] ...
0
votes
0answers
13 views

Proving properties of nth roots

First let me define some things. Let $x \gt 0$ and $n \ge 1$. Now $x^{\frac{1}{n}}:=\sup\ [ y\in \mathbb R : y \ge 0 \text{ and } y^n \le x]$ (a) If $x \gt 1$ then $x^{\frac{1}{k}}$ is a decreasing ...
4
votes
0answers
71 views

How to select good exercises?

I'm studying on Rudin "Principles of of Mathematical Analysis" which I begin to find as a good and complete reference. I wonder how many exercises shall I do at the end of each chapter ? In case of ...
3
votes
2answers
64 views

Determining final and initial object in a certain category

I am reading Paolo Aluffi's greatly entertaining book "Algebra: Chapter $0$" and I got stuck on some excercises dealing with universal properties. Let $C$ be a category, and let $A$ and $B$ be two ...
1
vote
1answer
43 views

Normal Distribution and Iterated Logarithm

Let $X_n$ be independent $N(0, \sigma^2)$-distributed random variables with partial sum $S_n := \sum_{k=1}^n X_k$, $n \geq 1$. Then I read the following results. $$ \sum_{k = 1}^n \mathbb P (S_n > ...
0
votes
3answers
34 views

Limit with terms very similar to those that should give an exponential function

I have been trying to solve the following limit but am completely stuck. $$\lim_{\alpha \rightarrow \infty} 1-\left( \frac{y+\alpha}{\alpha-1} \right)^{-\alpha}$$ I have tried inverting the ratio ...
-3
votes
1answer
448 views

Posterior Probability

a) Your initial belief is that a defendant in a court case is guilty with probability 0.5. A witness comes forward claiming he saw the defendant committed the crime. You know the witness is not ...
0
votes
1answer
33 views

Confusion about the associative property and the mechanics of Parenthesis

This is a follow up question on my earlier post (Updated): Showing that a set $M$ with two elements classifies as a field. I feel this post is necessary because I realize that what confuses me is how ...
1
vote
1answer
29 views

Simple probability question

Question: In class of 125 students, in examination 70 students passed in mathematics and 55 students passed in statistics and 30 passed in both the subject. Find the probability of the event where ...
0
votes
1answer
36 views

(Updated): Showing that a set $M$ with two elements classifies as a field

My question is more conceptual, so I will come straight to the exercise: Exercise: Let $M= \lbrace g,u \rbrace $ be a Set. On $M$ the Addition and the Multiplication is given by: \begin{align} ...
2
votes
4answers
108 views

Constructive proof for existence of integer part of real number

I try to prove de following exercise of my analysis textbook. Show that for every real number $x$ there is exactly one integer $N$ such that $N \le x < N + 1$. I have been finding a ...
1
vote
7answers
101 views

Proof that $|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|},\quad x,y \geq 0$

Any hints on how I can prove the inequality: $$|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|},\quad x,y \geq 0$$ Thank you.
0
votes
3answers
34 views

Is this derivative correct?

I'm newbie at Calculus, so I'm doing some exercises of derivates, I know by the formula: $f(x) = \sqrt u$ $\frac {df(x)}{dx} = \frac{u'}{2 \sqrt u}$ that the derivate of the next function is: ...
0
votes
1answer
29 views

Question about Lebesgue Covering Dimension

Suppose we have a metric space equipped with two different metrics: $(X,d), (X, d')$. What must be true of the metrics: $d, d'$ in order for $X$ to have the same Lebesgue covering dimension? A ...
0
votes
1answer
39 views

Proof on existence of the natural numbers, crucial step.

I am trying to understand/reconstruct the proof given by my Professor addressing the existence of natural numbers. However there is one step in particular I don't understand and the more I think about ...
0
votes
1answer
51 views

Without solving the equation, determine the nature of its roots: $x^2 + ax + a^2 = 0$

I'm working through the book Core Maths for Advanced Level on my own, and, after solving the above problem, I'm not getting the same answer as the book. So, given: $$x^2 + ax + a^2 = 0$$ Using the ...
1
vote
1answer
39 views

Want to check measure theory proof

I need to show that sigma-finiteness implies semifinite. Does the following proof work? Let $(X,m,\mu)$ be a $\sigma-$finite measure. Let, $E\in m$, and $\mu(E)=\infty$. By $\sigma-$finiteness, ...
3
votes
3answers
459 views

Probably of 2 six in 5 dice rolls

What is the probability of obtaining exatcly 2 six when rolling a dice 5 times? In order to obtain this probability, I will need to devide the number of favorable events by the number of possible ...
2
votes
1answer
55 views

An exercise on first order logic formulas, terms and Polish notation

This is part of my homework (not mandatory and not accredited). Please comment/answer if my reasoning for the exercises is correct, because I'd like to see if I understand the material. I will start ...
0
votes
0answers
50 views

Help in Understanding the Formula for The Lattice Point Counting in Triangles with Rational Coordinates

Yesterday I have found this paper while searching Google. However, since the author of this paper gave no examples of implementing the following formula, I don't understand how to implement it in ...
1
vote
2answers
22 views

problem with rudin theorem 1.14 first step $g^{-1}((\alpha, \infty]=\cup_{n=1}^\infty f_n^{-1}((\alpha, \infty])$

Theorem 1.14 of Complex and real analysis by Rudin states: if $f_n:X\to [-\infty, +\infty]$ is measurable for $n = 1,2,3...$ and: $$g=\sup_{n\geq 1} f_n,~~~~h=\limsup_{n\to \infty} f_n,$$ then $h$ ...
1
vote
1answer
58 views

Show expectation is infinite

Let $X_1,\ldots,X_n$ be independent, identically distributed with expectation 1 and finite variance. Find the limit distribution of $\sqrt{n}(\bar{X}_n^{-1}-1)$. If the random variables are sampled ...
5
votes
1answer
93 views

The Use of Sound in Mathematics. [closed]

I'm not sure that this question is appropriate here. There's a good chance it's too opinion-based. If that's the case, I'm sorry. I was sat in a research seminar recently and wondered whether it'd be ...
1
vote
1answer
120 views

Proving, that closure of set is equal this set iff set is closed

I've started intorduction to topology course and I need help with one of the problems: Let $A \subset(X,T). $ Prove that $cl(A) = A\iff A$ is closed. It may looks trivial, but I had a little ...
0
votes
1answer
32 views

Definition of $C^1$ functions with values in $\mathbb R^m$

My analysis textbook defines a $C^1$ function $f:\mathbb{R^n}\to\mathbb{R^m}$ as one in which for each component function $f_i, 1\leq i\leq m$ the partial derivative $\frac{\partial f_i}{x_j}$ exists ...
1
vote
2answers
71 views

Path to 3d Mathematics programming, where to start?

This might read like duplicate of this question https://math.stackexchange.com/search?q=where+to+start However since that one wasn't answered, and I have a more specific problem in regards to ...
2
votes
1answer
29 views

Fourier Transforms of $L^1$ functions

Suppose that $f_n$ and $f$ are $L^1(\mathbb R^n)$ functions with $f_n \to f$ in $L^1$ sense. Then is it true that their Fourier transforms defined as $$ \hat f(\xi) := \int_{\mathbb R^n} ...
2
votes
0answers
62 views

How to Read Michael Artin's Algebra

I am currently reading Algebra by Michael Artin (for self study). I'm finding the book pretty interesting, but I'm still not sure how I should study the book. For example, should I try to prove every ...
1
vote
1answer
34 views

Conditional probability with balls in urns involving discards

I found this problem in a statistics book, and I'm wondering if my solution is correct. "You and a friend play a game involving 20 balls in an urn, of which 1 is red and 19 are white. The game is ...
3
votes
0answers
86 views

Compact family of Lip functions under the sup norm metric, proof verification.

Hi everyone I'd like to know if the following is correct, I'd appreciate your opinion and also any suggestion to improve my argument. Thanks in advance for your time. If $(K,d)$ is a compact ...
1
vote
1answer
24 views

Expected value of Cumulative Hazard

Define $T=\min(T^0,C)$ where $T^0$ is the failure time and $C$ is the censoring time. Define the failure indicator $$\delta = \begin{cases} 1 & \text{if $T^0\leq C$}\\ 0 & \text{if $T^0> ...
1
vote
1answer
33 views

Convergent Series $\frac{1}{n^q}, \ \ q>1$

How to show the following result about series? Thank you! Convergent Series: $$\sum_n \frac{1}{n^q}, \ \ q>1$$
0
votes
1answer
34 views

Finding probability given mean and standard deviation

I don't know how to approach this problem: X is normally distributed with a mean of 200 and a standard deviation of 10. Find P(X ≥ 203)
0
votes
3answers
41 views

Finding a recursive formula for a number

I am trying to find a recursive formula for a given number in order to solve a problem I am working on. For every $n \in \mathbb{N} \setminus \lbrace 0,1 \rbrace$ we define the number ...
0
votes
2answers
63 views

Mathematical background for one wishing to study Chaos/Complexity Theory

I don't have a very strong mathematics background. In fact I quite abhorred mathematics during my Middle/High School years. I'm currently applying for PhD programs in the field of literature as that ...