Questions about studying mathematics without formal instruction.

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Good Textbooks for Real Analysis and Topology.

I'm currently in my 3rd year of my undergrad in Mathematics and moving onto my 4th year next year. I took a course in Real Analysis I, but the professor was very confusing and we didn't use a textbook ...
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0answers
33 views

Show that $c-b\lt b-a$

The question: Let $G$ be an Arf semigroup and $a\lt b\lt c$ be three consecutive elements in $G$. How to show that $c-b\lt b-a$ and how to show that this is not necessarily the case for every ...
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0answers
9 views

Givens rotation and retraction mapping

Here's part of the texts in the book Optimization Algorithms on Matrix Manifolds, when talking about the retraction on a matrix sub-manifold. A retraction is a mapping that maps a tangent vector in a ...
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1answer
35 views

True/False question regarding continuity

For the two scenarios below, either give an example if such a request is possible, or argue why such an request is impossible. I think the first is possible and second is impossible. However, I can't ...
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2answers
34 views

Approaches to teaching and learning analysis

I found that studying linear algebra by getting into vector spaces and linear transformations first made things very easy. This is the approach Halmos or Axler, just to name a few, take. IMHO, the ...
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1answer
22 views

Give an example of a relation R on $A^2$ which is reflexive, symmetric, and not transitive

I am just looking for some clarification on this exercise: Let $A = \{a,b,c,d\}$. Give an example of a relation $R$ on $A^2$ which is reflexive, symmetric, and not transitive. I understand that if I ...
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2answers
49 views

Is there any motivation for the direct sum?

I believe that everything have reason to exist . I want to learn why the direct sum exist. Is there any motivation for the direct sum to exist ?
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1answer
20 views

Differentiation of $u(t)=\int_0^t h(s,t)ds, \ \forall t \in \mathbb{R}$ with the multivariable chain rule

Problem: Let $h: \mathbb{R}^2 \to \mathbb{R}$ be continuous and differentiable with respect to its second variable, define $u(t)= \displaystyle \int_0^t h(s,t)ds, \ t \in \mathbb{R}$ In an ...
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1answer
27 views

Is $\sum \cos(n \pi) \frac{n}{n^2+1}$ conditionally or absolutely convergent?

Is $\sum \cos(n \pi) \frac{n}{n^2+1}$ conditionally or absolutely convergent? My Working Clearly, $\sum \cos(n \pi) \frac{n}{n^2+1} = \sum (-1)^n \frac{n}{n^2+1}$, and it can be shown by using ...
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0answers
9 views

Book recommendation for introductory algebraic combinatorics?

Preferably: It should have plenty of motivation (as I am self-learning). It should not be skimpy on proofs (as I am self-learning), but perhaps I can make do since I can ask questions elsewhere. It ...
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1answer
24 views

Divergent subsequence of an unbounded sequence

Let $(a_n)$ be a sequence of real numbers that is unbounded above. Show that $\exists$ a subsequence $(a_{n_k})_{k \ge 1}$ such that $\lim_{k \rightarrow \infty} a_{n_k} = + \infty$. Working ...
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0answers
28 views

Cauchy-Euler Equation of order $n$

What I wish to prove is that for a Cauchy-Euler equation of order $n$, the substitution $x=e^{t}$ transforms it into a linear differential equation with constant coefficients. To put it as a theorem: ...
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1answer
19 views

Sequential Criterion for Functional Limits

How can I use the sequential criterion for functional limits to show that the following limit exist and compute the limit: $$\lim_{x \rightarrow 0} \sqrt{|x|}\cos\left(1/x\right) \ \text{for} \ x ...
3
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1answer
51 views

Mathematics or physics at university

I have a strong interest in maths, and I feel that advanced physics is cool too (although I've only studied classical mechanics at high school, which is kind of boring). So I'm not sure about which ...
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1answer
25 views

Derivative of Normal Vector Field

This is an example from Do Carmo (Example 4, page 139). Consider the saddle point $p=(0, 0, 0)$ of the hyperbolic paraboloid $z=y^2-x^2$ with parameterization $\mathbf x(u, v)=(u, v, v^2-u^2)$. It is ...
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1answer
125 views

The form of subrings of $k[[t]]$

I saw this question in an algebraic geometry book. I tried to solve this. But I did trivial thing, so I don't write what I did here. This is just self-studying. I want to learn how to solve. Please ...
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0answers
16 views

Describe the Unit Ball

I was asked to describe the unit ball in $C(I)$. All I could come up with was that by definition $B_{1}(0):=\{x \in C(I) : ||x||_{\infty}<1 \}$, where $||x||_{\infty}:=\sup_{t \in I} |x(t)|$. Thus, ...
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0answers
24 views

Revise high school material

Can you suggest me a comprehensive book to revise high school mathematics (up to besic calculus)? It should be extremely clear and complete and "scientific" (not like most high school books). Thank ...
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0answers
29 views

Proving a theorem in Ergodicity

I read the theorem stated below on Wikipedia (http://en.wikipedia.org/wiki/Ergodicity#Formal_definition). But I do not understand how to prove the equivalence of these different definitions.Any hints ...
3
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5answers
92 views

How do I show that $\sqrt{5+\sqrt{24}} = \sqrt{3}+\sqrt{2}$

According to wolfram alpha this is true: $\sqrt{5+\sqrt{24}} = \sqrt{3}+\sqrt{2}$ But how do you show this? I know of no rules that works with addition inside square roots. I noticed I could do ...
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2answers
39 views

Material for advanced highschooler

I'm a high school student who just finished elementary school.Though since I was into math I started going through advanced math while I was in elementary school and I pretty much finished most of the ...
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0answers
23 views

Diffeomorphism between a regular surface and the plane

Do Carmo states that (example 2, page 74) if $\mathbf x: U\subset\mathbb R^2\rightarrow S$ is a parameterization, then $\mathbf x^{-1}: \mathbf x(U)\rightarrow \mathbb R^2$ is differentiable. Why is ...
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1answer
22 views

Lipschitz continuity of $f(x,y)=4x^2+xy-\frac{1}{y-1}$ on an open set $U \subset \mathbb{R} \times (\mathbb{R} \setminus \lbrace 1 \rbrace)$

Problem: Find an open set $U \subset \mathbb{R} \times (\mathbb{R} \setminus \lbrace 1 \rbrace )$ which includes the points $(0, 1/2$) and $(0,3/2)$ such that the function ...
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0answers
40 views

Open Convex Subsets of Dense Spaces

So I asked this question yesterday, Existence of Non-Trivial, Convex, Open Set in $C_{\mathbb{C}}[0,1]$ Under $L^{0}$ Metric, and it made my start wondering the following... Suppose the following: ...
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1answer
28 views

The Simplex in $\mathbb{R}^n$ is convex

Problem: Show that $S:= \lbrace v \in \mathbb{R}^m \mid v=\displaystyle \sum_{j=1}^n a_j v_j, \text{ with } a_1, \dots , a_m \in [0,1], \ \sum_{j=1}^m a_j=1 \rbrace$ the Simplex of $\mathbb{R}^n$ ...
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1answer
66 views

Preferable Order of College Level Mathematics Study

I recently bought a good amount of math books, because I want to self teach math, and they are on their way--via U.P.S.--to my house. I was just wondering if someone would be kind enough to tell me in ...
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1answer
46 views

Existence of Non-Trivial, Convex, Open Set in $C_{\mathbb{C}}[0,1]$ Under $L^{0}$ Metric

I've been struggling with this problem for the last four hours. The problem is to show that the space of $\mathbb{C}$-valued continuous functions on $[0,1]$ under the metric ...
2
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2answers
51 views

Evaluating integral limit in two ways gives different limits

Problem Show that $$\lim\limits_{h \rightarrow 0^{+}} \int_{-1}^{1} \frac{h}{h^{2}+x^{2}} \, dx = \pi.$$ I can do this by evaluating the integral directly and showing that it is equal to ...
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1answer
44 views

Is a butterfly network on 8-inputs planar?

I could prove that a four input butterfly network is planar. For that I simply drew it such that no two edges intersect. But I could not use the same approach for the 8-input butterfly network. So I ...
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1answer
43 views

Proving that Markov Chain Monte Carlo converges

I am trying to understand how the very basic Markov Chain Monte Carlo approach works: We try to approximately calculate the expected value $E_{\pi(x)}[X]$ by drawing sequential samples from a Markov ...
3
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1answer
43 views

Differentiable functions defined on a regular surface

First, recall a general definition of a differentiable function as follows. Suppose $f :D\subset \mathbb R^3 \rightarrow \mathbb R$. Then $f$ is differentiable at $\mathbf a \in D$ if there is a ...
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1answer
19 views

What test could I use to test $\mu_1$ and $\mu_2$ instead of $\bar{x}$ and $\mu$?

A single sample $t$ test is only against a sample and a population correct? Can it test against a population and itself or two populations? ie $\mu_1$ vs. $\mu_2$....if not what tests would you use ...
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0answers
60 views

Recommended Textbook/Resources

I'm looking for a textbook or resources my younger brother could use. (He is in year 9, equivalent to US high school freshman) He is wanting to advance upon his math, he currently does exercises out ...
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1answer
45 views

Sequence of learning mathematics from basic algebra to calculus.

What would be a step by step sequence of learning mathematics from basic algebra to basic calculus? I pose this question because I am in the process of self-learning mathematics as a preparation for a ...
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2answers
51 views

Galois Theory problem (primitive roots of unity)

If $e_1,e_2......,e_{p-1}$ denote primitive $p^{th}$ roots of unity. Here $p$ is prime. And set the sum of the $n^{th}$ powers of the $e_i$ as $p_n=e^n_1+e^n_2......+e^n_{p-1}$ Now I want to show that ...
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15answers
5k views

A good way to retain mathematical understanding?

What is a good way to remember math concepts/definitions and commit them to long term memory? Background: In my current situation, I'm at an undergraduate institution where I have to take a lot of ...
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3answers
68 views

Effective enumerability and Enumerability

I am reading the book by Peter Smith "An Introduction to Gödel's Theorems". I am struck at the point where he tries to construct an non effective enumerable set by using the diagonalisation idea. ...
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1answer
49 views
+50

Solving the LaPlace PDE by complete induction

I have a problem with an induction step that appears to be bothersome for me to comprehend it. Problem: Show that $f(x_1, \dots ,x_n)=(x_1^2+ \dots + x_n^2)^{1- n/2}$ for $(x_1^2 + \dots + ...
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2answers
22 views

Standard Deviation of Population from Sample.

I guess I am a little confused. I am doing t test statistics in my class. I think I know but I would love some insight. I am trying to get the estimated standard deviation for the population. These ...
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2answers
34 views

Free $A$-module isomorphic to a direct sum of copies of $A$?

Does this proposition hold even if it's not finitely generated? I think it does, since $M$ isomorphic to the direct sum of $M_i$, $M_i$ isomorphic to the direct sum of ...
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2answers
33 views

rank of a matrix with two columns s.t. their dot product is zero

I have function $\sigma(u,v)=(f(u,v),g(u,v),h(u,v))$ s.t. $\sigma_u$ x $\sigma_v\neq(0,0,0)$ (cross-product) also, there is the $3\times 2$ matrix : $$ ...
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1answer
90 views

Prove that the curves of the family $v^3/u^2=k$ are geodesics on a surface

Prove that the curves of the family $v^3/u^2=k$ where $k$ is a constant are geodesics on a surface with the metric $$v^2 \, du^2-2uv \, du+2u^2 \, dv^2$$ where $u,v \gt 0$.
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1answer
42 views

Baby Rudin Chapter 4 Exercise Questions 5 and 6

4.5: If f is continuous on a closed set in $R^1$, prove there exist continuous functions $g$ on $R^1$ such that $g(x)=f(x)$ for all $x \in \mathbb{E}$. 4.6: Suppose $\mathbb{E}$ is compact, and prove ...
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1answer
29 views

Abstract Algebra: Automorphism and irreducible

I done how to prove this the question is A. Show that there is an automorphism of the ring of rational number $\mathbb{Q}(i)[x] $that has order of $4$. I suppose i can say that if $\varphi$ is an ...
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1answer
29 views

Abstract Algebra: Irreducible polynomials and automorphism

While I was studying and referring few examples I came up with few question. Hope you guys can help me solving my confusion. Irreducible polynomials problem 1 in this how to argue that why the ...
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1answer
62 views

The meaning of a symbol in the proposition

The question is what's the meaning of the symbol $\phi$? If it just a mapping,what's the mean of the equation? I guess it's $\phi(x)$, $x$ is the element of $M$. Then $\phi(x)$ is the element of ...
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1answer
23 views

For $f: \mathbb{R}^n \to \mathbb{R}$ homogenous, show that $\sum_{i=1}^n x_i \frac{\partial f}{\partial x_i}(x_1, \dots ,x_n)= kf(x_1, \dots , x_n)$

Definition: A function $f: \mathbb{R}^n \to \mathbb{R}$ is said to be homogenous of degree $k$ if $\forall t \in \mathbb{R}$ and $(x_1, \dots , x_n) \in \mathbb{R}^n$ the equations $f(tx_1, \dots , ...
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1answer
58 views

Problem about Galois Extension.

$Gal(\Bbb K/\Bbb F)\cong S$. Where $\Bbb K$ is extension of $\Bbb F$. And $S= G_1\times G_2\times\cdots\times G_K$ is a solvable group. Where each $G_i$ are groups of prime power order and $o(S)= n.$ ...
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0answers
22 views

Where can I find proof - There're infinitely many primes $p$ such that $p(mod\ N)\not\in H$ - Name?

Origin - http://math.uga.edu/~pete/4400FULL.pdf - on p120, Theorem 122 Fix a positive integer $N>2$, and let $H$ be a proper subgroup of $U(N)=(Z/NZ)^{\times}$. There are infinitely many ...
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1answer
44 views

The differential is NOT the Jacobi Matrix?

In the book Analysis II by C.T. Michaels the differential is introduced as the Jacobi-Matrix. In class we had the following definition: Definition: Let $U \subset \mathbb{R}^m$ be open, $f: U \to ...