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_.

learn more… | top users | synonyms (1)

2
votes
0answers
114 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
vote
0answers
152 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
6k 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
54 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 ...
1
vote
1answer
63 views

Permutation of word

Question: Find the permutation of letters of the word EXERCISES in which vowels are together. My Efforts: I have rearranged the word in such a way that all the vowel come together. EEEI XRCSS Now ...
1
vote
0answers
57 views

Problem showing a double summation equality.

I'm trying to show that $$G(L) = \sigma^2(\sum_{j=0}^\infty \psi_{j}^2 + \sum_{h=1}^\infty\sum_{j=0}^\infty \psi_j \psi_{j+h}(L^h-L^{-h}))$$ is equal to: ...
2
votes
3answers
89 views

Equation of line passing through point.

The straight line $3x + 4y + 5 = 0 $ and $4x - 3y - 10 = 0$ intersect at point $A$. Point $B$ on line $3x + 4y + 5 = 0 $ and point C on line $4x - 3y - 10 = 0$ are such that $d(A,B)=d(A,C)$. Find ...
0
votes
1answer
861 views

Is the Gamma Function a jointly sufficient statistic?

A random sample $X_{1},...,X_{n}$ are pulled from a gamma distribution. Are there jointly sufficient statistics based on these observations for the two unknown parameters? The definition of a gamma ...
3
votes
3answers
87 views

Given $o(a)=5$, prove $C(a)=C(a^{3})$

Given $o(a)=5$, prove $C(a)=C(a^{3})$ At this point I would like a hint rather than a full solution. I know we are given $a^{5}=e$ and that we wish to prove this implies that $C(a) =\{ x \in ...
2
votes
1answer
156 views

About using Archimedean property to prove the existence of least upper bound

I am studying the proof of existence of least upper bound, but I can not understand how the autor applies the Archimedean property. Archimedean property. Let $x$ and $\epsilon$ be any positive ...
0
votes
1answer
39 views

$Var(\bar{X})$ for a random sample from Bernoulli Distribution

Let $X_1,...,X_n$ be a random sample of size $n$ from a Bernoulli distribution with parameter $p$ where $0< p< 1$ is unkown. (a) Find $\theta^2=Var(\bar{X}).$ (b) Find the value of $c$ so that ...
3
votes
0answers
91 views

Periodic curve on unit sphere and torsion

Define $S^2 \subset \mathbb{R^3}$ be the unit sphere. Suppose that $\alpha :\mathbb{R} \to S^2$ is a differentiable curve parametrized by arc-length. a) Show that $\kappa(s)$, the curvature of ...
0
votes
0answers
66 views

Characteristics of Singularity Points

Determine the character of the singularity at $z=0$ for each of the following functions: \begin{align} &\frac{1/z^{7}}{e^{z}-1} \tag{1} \\ &\frac{}{} \\ ...
0
votes
1answer
52 views

Follow-up question on mathematical induction with arbitrary base case

Note: This question has already been answered here Proving mathematical induction with arbitrary base using (weak) induction. I was trying to 'reconstruct' at least one proof given in this question ...
1
vote
2answers
215 views

Expectation of Truncated Random Variables

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 $0<\delta<0.5$ and $\epsilon >0$ and define ...
2
votes
1answer
74 views

Poles and Zeros

Determine the number of zeros and poles inside (counted with multiplicity) of the function \begin{equation} f(z)=\frac{z^{6}\sin(\pi z)}{(1-z^{2})(2-z^{2})(3-z^{7})} \end{equation} inside the ...
1
vote
1answer
46 views

Kolmogorov Exponential Bounds (Upper)

This is one version of Kolmogorov exponential bound from Allan Gut's Probability: A Graduate Course (2005, p385-386). Let $Y_k$ be an independent sequence of random variables with zero mean and ...
4
votes
4answers
79 views

Existence of the square root in $\mathbb{C}$

I am stuck on the following Proposition: Proposition: Show that for every $z \in \mathbb{C} \setminus (- \infty, 0]$ there exists exactly one $w \in \mathbb{C}$ such that $w^2=z$ and Re$(w)>0$ ...
1
vote
0answers
31 views

Order Type Stratification

Is there some sort of interesting way of organizing certain order types that aren't ordinals? When I say certain order types, some examples include but are not limited to: order types of dense ...
1
vote
1answer
47 views

Residue Calculus (Computing an Improper Integral)

Use residue calculus to compute the integral $\int_{-\infty}^{\infty}\frac{1}{(z^{2}+25)(z^{2}+16)}dz$ My solution If we add to the interval $I_{R}=[-R,R]$ add the semicircle $\gamma_{R}$ in the ...
2
votes
3answers
92 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
45 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?
3
votes
2answers
607 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
33 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
34 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 ...
5
votes
1answer
252 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
71 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
56 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
123 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 ...
2
votes
2answers
143 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
38 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 $$\dfrac{ S_1(t)}{1-S_1(t)} = \psi \dfrac{S_0(t)}{1-S_0(t)}$$ ...
7
votes
3answers
4k 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 ...
3
votes
1answer
90 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 ...
5
votes
1answer
261 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
153 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] ...
4
votes
0answers
96 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
118 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
65 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
36 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 ...
-2
votes
1answer
523 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
37 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
45 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
40 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} ...
1
vote
4answers
209 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 ...
2
votes
8answers
134 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
41 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: ...
1
vote
1answer
83 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
49 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
865 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
46 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, ...