Questions about studying mathematics without formal instruction.

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2
votes
0answers
69 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
votes
1answer
62 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
vote
0answers
25 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
13 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
vote
0answers
35 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
30 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 > ...
0
votes
4answers
62 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
44 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
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1answer
27 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
49 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
37 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
124 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
39 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
57 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
56 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
47 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
69 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
67 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
vote
0answers
33 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
votes
0answers
35 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
96 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
165 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
44 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
62 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
69 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
230 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
47 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
22 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
35 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: ...
1
vote
3answers
63 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
83 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
70 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 ...
5
votes
3answers
77 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
0answers
20 views

Finding $Var(S^2), E(S^4),$ and unbiased estimator for $\sigma^4$ from random, normal samp

Let $X_1,...,X_n$ be a random sample of size $n$ from the normal distribution $N(\mu,\sigma^2)$ and let $S^2$ be the sample variance. (a) Find $V(S^2)$ and derive $E(S^4)$. (b) find an unbiased ...
0
votes
1answer
17 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
65 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
14 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
36 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
85 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
43 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
30 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
66 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$(w)>0$ ...
0
votes
0answers
17 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 ...
0
votes
1answer
31 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
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
36 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
24 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 ...