Questions about studying mathematics without formal instruction.

learn more… | top users | synonyms (1)

3
votes
3answers
59 views

Integrating a function with an infinite number of discontinuities

I would appreciate some help with the following exercise: Let $$f(x)=\begin{cases} 1 & \text{if}\ x= 1/n\ \text{for some}\ n \in \mathbb{N} \\ 0 & \text{otherwise} \end{cases}$$ Show that ...
1
vote
0answers
30 views

Show a subset A is compact if and only if the image of the map T(A) is compact

The question is as follows: Let $\{v_1,v_2,\dots,v_n\}$ be a set of linearly independent vectors of an $n$-dimensional normed linear vector space $V$. Define the map $T: R^n \to V$ by $T(a) = ...
0
votes
1answer
58 views

Abstract Algebra: Ring Homomorphism injective

Reference: Ring Homomorphism $\phi:f\to S$ is injective Referring this I have a doubt which I needed to clear. Below is my answer and query. We know $\phi:f\to S$ be ring homomorphism, where $f$ is a ...
7
votes
1answer
113 views

What's the motivation of definition of primary?

Primary ideal can be regard as the generalization of prime ideal and radical. But Why it's defined like that?It's not symmetry. Why not define like that:
2
votes
1answer
86 views

Question about some details of a proof

i) Why it's a unit can prove this proposition ii)see picture
0
votes
1answer
30 views

Regarding jointly multivariate normal X1,X2…X5

So I have a question from statistical inference that I need some help with: $X_1,X_2,...,X_5$ are jointly multivariate normal with means = $\mu_i$, variances = $\sigma^2_i$, correlation = $\rho$ ...
1
vote
2answers
37 views

Example- $l_p$ norm space

$$||x||= {[{\sum _{i=1}^\infty |x_i|^p}]}^{1/p}$$ Is a norm on $l_p$ space :- space of all sequences made of scalars from $\mathfrak C$(filed of complex numbers). To prove that above is norm on $l_p$ ...
3
votes
0answers
39 views

Where does this series converge?

Let $ \{r_1, r_2 ,r_3,... \}$ be an enumeration of $\mathbb{Q}$. For each $r_n \in \mathbb{Q}$ define: $$u_n(x)=\begin{cases} 1/{2^n} & x>r_n \\ 0 & x \leq r_n \end{cases} $$ and let $$h ...
4
votes
1answer
80 views

Geodesic eqautions and length of a curve in geodesic coordinate system.

About geodesic coordinates: Let S be regular surface. $p\in S$ $\gamma$ be unit speed geodesic on $S$ with parameter $v$ and $\gamma (0)=p$ $\tilde \gamma^v$ be unit speed geodesic s.t. ...
1
vote
1answer
62 views

A function that is differentiable at a single point

Consider the function $$f(x)=\begin{cases} x^2 & x \in \mathbf{Q}, \\ 0 & x \notin \mathbf{Q} \end{cases} $$ $f$ is continuous only at $0$ and now I need to show that at this point it is ...
2
votes
4answers
73 views

Are different constructions of an algebraic structure always isomorphic?

Any two complete ordered fields are isomorphic (as proved, e.g., in Spivak's Calculus; see also this question). While I understand this proof, I cannot yet appreciate why it is necessary. Given any ...
1
vote
2answers
62 views

Determining constant in a CDF

I have a question that I literally have no idea how to begin, I was hoping someone could help me: It says $X_1,X_2,\ldots,X_n$ is a sample from a distribution It says that the Cumulative ...
0
votes
1answer
41 views

Cubic root formula derivation

I'm trying to understand the derivation for the cubic root formula. The text I am studying from describes the following steps: $$x^3 + ax^2 + bx + c = 0$$ Reduce this to a depressed form by ...
0
votes
2answers
68 views

What is this distribution???

Let $X_1, X_2, \ldots, X_n$ be a random sample from a population with $E(X_i) = \mu$ for all $i \in \{1,\ldots, n \}$. Define $ Y_i = \begin{cases} 1 & \mbox{ if } X_i < \mu \\ 0 ...
1
vote
1answer
35 views

Abbott's Exercise $6.2.14$ : Convergent subsequences for bounded sequences of functions

I have been trying to solve the following exercise from Abbott's "Understanding Analysis". I understand that $(a)$ comes from an application of the Bolzano Weierstrass Theorem as we assume that ...
0
votes
0answers
30 views

Reference for power series

I would need some references for power series, Taylors series of elementary functions, derivation and integration of power series, convergence of sequences of functions and series of functions. The ...
0
votes
1answer
17 views

Covering of a universal space and covering of a subset

It seems to me that the definition of covering depends on the set we are referring to. For example, if $X$ is the universal topological space, then a collection $\mathcal A$ of subsets of $X$ is said ...
0
votes
3answers
41 views

Converse of order limit theorem

Part of the Order Limit Theorem states that: Assume $(u_n) \rightarrow l$ and $(v_n) \rightarrow m$, then if $v_n \le u_n$ for all $n \ge N \in \mathbb{N}$, then $m \le l$. I can prove this. ...
1
vote
2answers
27 views

Showing that two Matrices are not similar over GL$_n(\mathbb{F}_2)$

Problem: Show that $$ \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$ Are not similar over GL$_n ( \mathbb{F}_2)$ Note: We write ...
1
vote
3answers
131 views

Is it possible to learn abstract algebra no precalculus or calculus?

See above. I am trying to re-teach myself mathematics in a different manner than is formally taught (i.e., set theory, number theory, mathematical logic, abstract algebra, discrete math and then ...
0
votes
1answer
30 views

Is norm of a differentiable function continuous?

The following argument is from my class notes. "Suppose $\gamma(s): I\rightarrow \mathbb R^3$ is a regular curve. Fix $t_0$ and define $s(t) = \int_{t_0}^t \|\gamma'(t)\|dt$. That is, $s(t)$ is the ...
0
votes
0answers
12 views

Text book Suggestion for studying Bi-level the0ry & Convex analysis.

I have taken some courses in Convex optimization.I want to work on the bi-level optimization. Now I would like to know a little bit more about the pure mathematical side. Is there any good books in ...
0
votes
1answer
40 views

Calculating mean vector of a multivariate distribution

I have a question concerning calculating the mean vector (vector of expected values) of a general multivariate distribution. I try to obtain the mean vector by doing a vector integration and I ...
0
votes
0answers
44 views

Microeconomics maximization problem

I would like to discuss this question about microeconomics: Martha National County Club is a golf club in an isolated wealthy community and accepts only females as members. There are 1,000 ...
2
votes
1answer
59 views

Convergence in $L_p$, Vitali's theorem, and convergence in measure

Working a measure theory question for practice from Bartle. Assume that $(X,\mathbb{X},\mu)$ is a finite measure space $f_{n}\rightarrow f$ in $L_{p}(X,\mathbb{X},\mu)$ $\varphi$ is a real-valued ...
1
vote
1answer
49 views

Which of these things is not like the others?

What's in a name? Well quite a lot, if you're confused enough. I have an engineering-style mathematics education, based on good old hand waving and learning bits and pieces from all over the place. I ...
4
votes
1answer
106 views

Using the Parseval Identity to compute $ \sum_{n=1}^{+ \infty} \frac{1}{(4n^2-1)^2}$

Parseval's Identity: For continuous $f: [- \pi , \pi] \to \mathbb{R}$ $$ \sum_{n=- \infty}^{+ \infty} |c_n|^2 = \frac{1}{2 \pi} \int_{ - \pi}^{ \pi} |f(x)|^2dx, \text{ where } c_n = ...
0
votes
1answer
44 views

Could you please explain the detail of the proof

Proposition: Proof: Question: Why it's isomorphism?
1
vote
4answers
118 views

Which book among these would you recommend for first year calculus?

I'm struggling a bit with functions(limits, squeeze theorem, etc). I have done some research and found a list of books on calculus but I'm not sure which one would be better suited for me, so I would ...
1
vote
2answers
56 views

Why are these two lemmas worth including and proving?

The following two lemmas are from Stein and Shakarchi (2005). (p4 Lemma 1.1) If a rectangle is the almost disjoint union of finitely many other rectangles, say $R=\cup_{k=1}^n R_k$, then $|R| = ...
1
vote
1answer
116 views

Better way to prove this sequence problem relating $\lim x_{n+1}/x_n$ and $\lim\sqrt[n]{x_n}$

Came across the following exercise in Bartle's Elements of Real Analysis. Let $X= (x_n)$ be a sequence of strictly positive real numbers, let $\lim \left({ \dfrac{x_{n + 1}}{x_n}}\right) = L$, ...
3
votes
3answers
132 views

How to show a collection of sets is countable

In the proof for "Every open subset $\mathcal O$ of $\mathbb R$ can be written uniquely as a countable union of disjoint open intervals" Stein and Shakarchi (2005 p6) argue that (after having defined ...
2
votes
2answers
26 views

find the coefficient of the given term when the expression is expanded by the binomial theorem

I am just trying to understand why the term is $\binom{15}8$(3p$^2$ - 2q)$^7$. I need to find the coefficient in $p^{16}q^7$ in $(3p^2 - 2q)^{15}$ So, I know that $n = 15$ and I have $a^{n - k}b^k$ ...
0
votes
0answers
30 views

Intuition analysis-deconstruction-reconstruction.

The following question is a refinement of this question, which caused a lot of people to give answers that were missing the point entirely, probably because the question was not clear. Being human, ...
3
votes
1answer
37 views

Isomorphism of Grassmannians

I want to prove that two CW complexes $\mathrm{Gr}_{n}(\mathbb{R}^{n+k})$ and $\mathrm{Gr}_{k}(\mathbb{R}^{n+k})$ are isomorphic to one another. I'm pretty sure I can just show that the number of ...
1
vote
2answers
39 views

Where am I going wrong? Converging sequence

I am struck with a dilemma concerning the following exercise in Bartle's Elements of Real Analysis. Determine the convergence or the divergence of the sequence $(x_n)$ given by $$ x_n = ...
0
votes
1answer
59 views
1
vote
2answers
82 views

Proof Verification - Every sequence in $\Bbb R$ contains a monotone sub-sequence

Came across the following exercise in Bartle's Elements of Real Analysis. This is the solution I came up with. Would be grateful if someone could verify it for me and maybe suggest better/alternate ...
0
votes
0answers
37 views

Please check the question: Compute $EX$

Question: A box contains $10$ balls numbered $1,2,\ldots,10$. A random sample of $7$ balls is selected. $X=$ the smallest of the numbers drawn. Compute $E(X)$ $R(X)= \{1, 2, 3, 4\}$ ...
0
votes
1answer
92 views

Books (and supporting material) that are useful in deconstructing one's intuition?

I recently came across the following problem from Paul Zeitz's book The Art and Craft of Problem Solving. Given the image below, can you find a way to connect corresponding blocks (i.e. A to A, B to ...
1
vote
0answers
29 views

Continuity of a positive preserving operator between C(X) and C(Y)

I've been struggling with this question in Reed and Simon while I'm prepping for quals. Suppose that $T:C(X)\rightarrow C(Y)$ is a positive operator. Prove that T is continuous and $\Vert ...
59
votes
6answers
5k views

Why can't you pick socks using coin flips?

I'm teaching myself axiomatic set theory and I'm having some trouble getting my head around the axiom of choice. I (think I) understand what the axiom says, but I don't get why it is so 'contentious', ...
0
votes
0answers
33 views

On Lucas Lehmer primality Test

http://primes.utm.edu/notes/proofs/LucasLehmer.html is proof of the Lucas Lehmer Test I read. The part I do not understand is why did he consider the sequence $S_n=S_{n+1}^2-2$. I mean why would ...
0
votes
1answer
36 views

circular differentiation

Suppose one starts with a function $f: \mathbb R^2 \rightarrow \mathbb R$ using $\mu, \sigma^2$ as its input, i.e. $f=f(\mu, \sigma^2)$. (Note that here I omitted the specific form of $f$ since I ...
0
votes
1answer
27 views

$a, z_1 \gt 0$ and $z_{n + 1} = (a + z_n)^{\frac 1 2}$ then $(z_n)$ is monotone and bounded?

Having trouble with the following exercise in Bartle's Elements of Real Analysis. Let $a, z_1 \gt 0$. Define $z_{n + 1} = (a + z_n)^{\frac 1 2}$ for $n \in \Bbb N$. Show that $(z_n)$ converges. ...
1
vote
0answers
21 views

Prove that the empirical measure is a measurable fucntion

This problem came from Schervish, Theory of Statistics, Sec. 1.4 Prob. 24. Suppose that $X_1, \ldots, X_n$ are exchangeable and take values in the Borel space $(\mathcal{X}, \mathcal{B})$. Prove ...
0
votes
3answers
97 views

Derivatives of sine and cosine at $x=0$ give all values of $\frac{d}{dx}\sin x$ and $\frac{d}{dx}\cos x$?

In video 3 of the video lectures by MIT on Single Variable Calculus presented by David Jerison, the latter says: Remarks: $\dfrac{d}{dx}\cos x\left|\right._{x=0}=\lim\limits_{\Delta ...
1
vote
2answers
42 views

primitive root of residue modulo p

I was trying to prove that for the set $\{1,2,....,p-1\}$ modulo p there are exactly $\phi(p-1)$ generators.Here p is prime.Also the operation is multiplication. My Try: So I first assumed that if ...
2
votes
1answer
77 views

Proof Verification: Show sequence is bounded and find limit: $x_1 \gt 1$ and $x_{n + 1} = 2 - \frac{1}{x_n}$

Came across the following exercise in Bartle's Elements of Real Analysis and am a little unsure about my solution. Would be extremely grateful if someone could verify it for me. Let $x_1 \in ...
0
votes
0answers
41 views

What would be a good source to learn differential equations?

I have done my school mathematics well but I have had difficulties in modeling even relative easy problems to differential equations. Are there some good books or some other material to learn to ...