Questions about studying mathematics without formal instruction.

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16
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6answers
1k views

Best Math books or apps for adults to learn math from the beginning

I lost a possible job because I didn't know how to multiply and subtract negative valued integers. I also don't know how fraction manipulation works. What reference books can I read that can help for ...
0
votes
4answers
40 views

Why do you add +- to only one side when you remove square root from both sides?

As the title says, why when you take a square root of both sides of the equation do you add $\pm$ only to the side which is a number, as opposed to an unknown? For example: $$x^2 = 9 \implies x = ...
1
vote
1answer
29 views

Norms of Ideals and generators.

I'm self studying some Algebraic Number Theory, looking at norms of ideals within rings of integers for some number field. I know that if we have a principal ideal $I= (a)$, then the norm of the ...
0
votes
0answers
30 views

Perron Frobenius Theorem and Markov chains and more

I came across few ways of calculating convergence rates of Markov chains but I am a bit confused as to how these differ from each other and what may be the best way to calculate. The second ...
2
votes
1answer
49 views

Confused About Step in Proof of Divergence of $\sum \frac{1}{p}$

I was going through the number theory text by Ireland and Rosen, and was following the proof of the divergence of the sum of reciprocal primes. But I came across a step unclear to me. The proof so ...
1
vote
2answers
50 views

Convergence time of a Markov chain

We know that a regular Markov chains converges to a unique matrix. The convergence time maybe finite or infinite. My interest is in the case where the convergence time is finite. How can we accurately ...
1
vote
0answers
35 views

Six digit permutations of 1 to 6 - divisible by 8

I am working on a problem in A Concise Introduction to Pure Mathematics 3rd Ed under the counting and choosing chapter. It is a multipart question and I am stuck on the last part: 'The digits 1 2 3 4 ...
0
votes
4answers
75 views

A basic Combinatorics Book

So, this is my problem...I have completed my boards and among all others, I have a great weakness in combinatorics. So this means I can utilize my free time now to address this problem. I think it is ...
2
votes
1answer
100 views

Structured Self-Learning Program for Calculus I & II

I'm interested in a organised program which comprehensively covers the topics of Calculus I and Calculus II. I've recently finished taking my secondary school's university-level Calculus I course, ...
0
votes
1answer
54 views

Need an Algebra 2/Precalculus text to prepare for Calculus

I'm about to finish my Algebra 1 text and I could use some recommendations for a text to prepare for Calculus. I've searched through several forums and some of the books I've seen recommended are: ...
0
votes
1answer
22 views

Bounded Linear Transformation proof

One paragraph in my text is to prove that $\|T\|=\sup\{|\langle Tf, g\rangle|:\|f\|<1, \|g\|<1\}$, where we have a bounded linear operator between two Hilbert spaces $T:\mathcal H_1\rightarrow ...
1
vote
1answer
39 views

$M_R$ is finitely generated iff Every submodule of $M_R$ is finitely generated

$M_R$ is finitely generated Every submodule of $M_R$ is finitely generated. Do the sentences above have the same meaning? Thanks for any replies.
2
votes
1answer
36 views

Find the permutation

This is part of an exercise I did on an assignment but I am having trouble remembering how to complete the exercise (even though I got full marks on my assignment). Let $P_1=(3\,4\,1\,2\,5), ...
0
votes
2answers
60 views

How many coefficients are in the expansion $(x + y + z)^{10}$

I need to find the number of coefficients in the expansion $(x + y + z)^{10}$. I had this exercise on a recent assignment. The answer I gave is: $3^{10} = \binom {3 + 10 - 1}{10} = \binom{12}{10} = ...
1
vote
1answer
41 views

A question on the morphism of projective varieties

The continuation of this, my question I want to show that $X$ and $Y$ are smooth and irreducible curves then $f(X)$ is either $Y$ or a point. Note that I know the proof of this ...
3
votes
2answers
40 views

$L^1$ is complete in its metric

Theorem: The vector space $L^1$ is complete in its metric. The following proof is from Princeton Lectures in Analysis book $3$ page $70$. Some of my questions about the proof of this theorem are as ...
2
votes
1answer
67 views

the diagonal $\Delta (Y) =\{(y,y)\in Y \times Y\}$ is closed in $\Bbb P^m \times \Bbb P^m $

Asumme that $\Pi : \Bbb P^n \times \Bbb P^m \rightarrow \Bbb P^m $is a closed map. $X\subset \Bbb P^n$ And $Y\subset P^m$ $f: X\rightarrow Y$ be a morphism of projective varieties. ...
0
votes
2answers
80 views

Every projective algebraic set can be written as the zero set of finitely many homogeneous polynomials of the same degree.

Definition: Let $I \subset k[x_0,\ldots,x_n]$ be a homogeneous ideal (or a set of homogeneous polynomials). The set $Z(I) := \{(a_0 : \cdots : a_n)\in P^n ; f(a_0,\ldots,a_n) = 0 \ \ \forall f \in ...
0
votes
0answers
83 views

Rings having the same characters but not isomorphic.

I want to show that these two rings have the same characters but they are not isomorphic for $\nu>2$ Thank you for helping. $$H=k+kt^{4\nu}(1+t)+kt^{6\nu}(1+t)+kt^{7\nu}(1+t)+k[[t]]t^{8\nu}$$ ...
38
votes
7answers
2k views

When working proof exercises from a textbook with no solutions manual, how do you know when your proof is sound/acceptable?

When working proof exercises from a textbook with no solutions manual, how do you know when your proof is sound/acceptable? Often times I "feel" as if I can write a proof to an exercise but most ...
7
votes
3answers
294 views

Is formal logic necessary for pure/“higher” mathematics?

I'm asking this as an autodidact who wants to learn math rigorously for its own sake. And I was just wondering if understanding proofs could be achieved without a formal grounding in symbolic logic. I ...
2
votes
0answers
63 views

“Teach yourself” guides [closed]

I really liked Teach Yourself Logic: A Study Guide by the user Peter Smith. It is a thorough guide how to teach yourself logic and set-theory from scratch up to any level with book recommendations for ...
2
votes
2answers
95 views

Second Course in Number Theory - Self Study

I just finished a first course in number theory using Dudley's Elementary Number Theory. This was by far my favorite math course and I want to learn more number theory this summer. As far as ...
1
vote
1answer
49 views

Differential forms and wedge product exercise.

Show that $$\omega \wedge v(\left <a_1,a_2,a_3 \right>,\left <b_1,b_2,b_3 \right>) = c_1 dx\wedge dy + c_2 dx\wedge dz + c_3 dy \wedge dz.$$ I wasn't given the form of $\omega$ or ...
10
votes
0answers
134 views

Soft question: How does basic differential geometry “fit together”?

I'm self-studying diff geom from Lee's Introduction to Smooth Manifolds. He warns the reader that there's a lot of machinery to construct, which is fine, and he explains things with wonderful clarity. ...
1
vote
1answer
18 views

“And”/“or” in solutions of inequalities

I'm working on introductory chapters of Lang's "A First Course in Calculus", and in the part on inequalities I didn't quite understand why, when the solution of an inequality consists of more than one ...
2
votes
1answer
25 views

Minimum distance from origin in $\mathbb{R}^3$ but Hessian is indefinite

Problem: Consider the Set $M= \lbrace (x,y,z) \in \mathbb{R}^3 \mid x^2+2y^2-z^2=1 \rbrace$ and find all Points on $M$ which have minimal euclidian distance from the origin. My approach: I ...
1
vote
1answer
21 views

Bayes with conditional independence

I have a problem that I can't work out I've two conditional independent A,B such as $P(A,B|C) = P(A|C)P(B|C)$ Now I've to find posterior formula for: $P(C | A,B)$, now what I got was pretty ...
2
votes
1answer
91 views

Prove the Gaussian curvature $K=0$

If two families of a geodesics on a surface intersect at a constant angle $\theta$, prove that the Gaussian curvature of the surface is zero, i.e. $K=0$. Please explain how to show the question. ...
2
votes
3answers
26 views

Find a vector orthogonal to other two given and ends at a plane

I am reviewing Calculus III using Mahavier, W. Ted's material and get stuck on one question in chapter 1. Here is the problem: Assume $\vec{u},\vec{v}\in \mathbb{R}^3$. Find a vector ...
4
votes
0answers
68 views

Strategy for self-studying after M.S.

For someone who has finished their M.S. degree in pure mathematics, what is a good way to keep learning mathematics within your specialization? Would you suggest reading research articles from ...
1
vote
2answers
37 views

Problem of continuous function

Define the function $g(x) = x^2\cos\frac1x$ for $x\ne 0$. What should be the value of $g(0)$ if $g(x)$ is a continuous function? Explain your work and justify your answer. Frankly, I have no ...
1
vote
2answers
70 views

Third and average price auction

Third price auction: the winner is the highst bidder but this time instead of paying the second highst bid, he would pay the third highst bid. -assume there are at least 3 bidders. - Average price ...
0
votes
1answer
49 views

Prove the assertion on the game theory.

if a dominant strategy for player1 is added to finite normal form game then the payoff to player1 at any equlibrium of the new game must be at least as great as any nash equlibrium payoff ...
0
votes
3answers
63 views

How do I find $\lim_{x\rightarrow \infty} x\sin \frac {c}{x}$?

How do I find the following limit for some real $ c $? $$\lim_{x\rightarrow \infty} x\sin \frac {c}{x} $$
0
votes
0answers
20 views

Polar coordinates: Slope of tangent

Would anyone mind telling me how to solve this problem? It seems strange as my answer is $-1$. Do I have to apply this formula, $(r'sinθ +rcosθ)/(r'cosθ -rsinθ)$ ?
0
votes
2answers
31 views

Need help with this absolute value equation

I need to solve the following equation involving absolute value: $$|x-1| = 1-x$$ Looking at the term $x-1$, I thought I'd divide the interval into parts: $x < 1$ and $x \geq 1$. Now, when ...
2
votes
1answer
60 views

Misuse of Tangent Vector

I am quite confused with the term tangent used in differential geometry books. It seems to me that people use this word quite loosely. For example, one definition about tangent space in my book is as ...
0
votes
1answer
30 views

Series representation of cosh or the hyperbolic cosine. [closed]

What is the Series representation of cosh or so-called the hyperbolic cosine? Can you help me guys?
2
votes
2answers
35 views

Question in relation to Fundamental Theorem of Calculus

I obtain two answers, one is $\dfrac1{\sqrt{1+x^6}}$ and another one is $\dfrac{2x}{\sqrt{1+x^{12}}}$ by using Fundamental Theorem of Calculus, but I am not so sure. Would anyone help me?
1
vote
1answer
46 views

Washer method and shell method

(1) Sketch the region enclosed between the curve $y=sin^2x$ and the straight line $y =2x/π$ (2) Find the volume of the solid $S$ obtained by revolving the region in part (1) about the $y$-axis by ...
1
vote
2answers
192 views

Partitions of Natural Numbers [duplicate]

This is a question from Complex analysis by Stein. The question is Prove that it is not possible to partition $\mathbb N$ into finitely many infinite AP's with distinct common differences.(other ...
1
vote
3answers
49 views

Definite integrals: Calculating Volume

Suppose $D$ is the region in the $xy$-plane bounded by the parabola $y=x^2$ and the line $y=2x$. Find the volume of the solid generated by rotating $D$ about 1) $x$-axis 2) $y$-axis. Are the ...
1
vote
1answer
53 views

Calculus: Application of definite integrals

Suppose $a>0$ is a constant. Let $C$ be the curve $y=\cosh x$, for $-a \leq x\leq a$. Let $D$ be the region bounded by $C$, $|x| = a$ and the $x$-axis. 1) Find the length of $C$ 2) Find the area ...
1
vote
2answers
36 views

Proving this binomial identity

I'm required to prove the following binomial identity: $$\sum\limits_{k=0}^l {n \choose k} {m \choose l-k} = {n+m \choose l}$$ I tried various arrangements but reached nowhere. Finally I turned to ...
0
votes
1answer
24 views

Every metrizable space with a countable dense subset has a countable basis

I'm working on this problem from Munkres: Show that every metrizable space with a countable dense subset has a countable basis. Here's my attempt at a proof. Let $X$ be a metrizable space with ...
0
votes
0answers
43 views

Full Course in Mathematics

I'm in a bit of a bind. I've recently taken up a degree in Computer Science, and it bypasses much of the mathematics. I picked this course as it's been years since I've exercised any math beyond basic ...
5
votes
3answers
237 views

Laplace transforms: Convolution

Find $$1*1*1*\cdots*1\quad n\,\,\text{ factors}$$ that is, a function $f(t)=1$ convolution with itself for a total of $n$ factors. Would anyone mind helping me? I have no idea what I should do. ...
2
votes
2answers
30 views

Two sequences are equivalent. Prove that one is Cauchy iff the other is Cauchy.

This question has already been asked and answered here Let $ϵ>0$ be given. With loss of generality, we may assume $ϵ$ is rational. Suppose $a_n$ is a Cauchy sequence and $b_n, a_n$ are ...
2
votes
0answers
49 views

Continuity of the right-hand derivative of a Convex function (help with the proof)

Hi everyone I have some trouble with one point in the following proof. Let $f$ be a convex function (strict convex function) on a real interval. If $f'_-(a)=f'_+(a)$ where $f'_-$ and $f'_+$ are ...