Questions about studying mathematics without formal instruction.

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1answer
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Convergence of expecations implies convergence of positive and negative parts?

If we have $E|X_n| \rightarrow E|X|$ does that imply \begin{equation} \lim_{n\rightarrow\infty} E X_n^\pm = X^\pm \end{equation} How about if we only have $EX_n \rightarrow EX$? Is this true in ...
2
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1answer
41 views

Complete list of math topics to study up to college level math?

Aspiring mathematician here. I have always been fascinated by math, physics, and just logic in general. I have noticed that I generally grasp topics and ideas quite quickly, but I am being hindered ...
2
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0answers
39 views

Equillibrium between Programming and Math Skills? [closed]

So I enjoy recreationally doing math and programming and am now at a stage where I will be pursuing them in University but I have found myself in a bit of a bind. My programming ability seems to lag ...
40
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6answers
4k views

What should an amateur do with a proof of an open problem?

Assuming that somebody is not an employee of a university, just a math amateur, and makes a proper proof of some well known open math problem, what should he do with it? Publish on the internet for ...
7
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1answer
102 views

Math competitions for hobbyists?

Are there any math competitions for hobbyist / amateur mathematicians? Something like the Putnam or the International Mathematical Olympiad, but open to regular people who are not full-time students?
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0answers
38 views

How do you look for classic/normative/standard books about an established branch of mathematics?

If you want to immerse yourself in a branch of mathematics (e.g. linear algebra and linear optimisation) which is new to you, then you often look for standard books which you can rely on. You could ...
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15answers
6k views

Why learn to solve differential equations when computers can do it?

I'm getting started learning engineering math. I'm really interested in physics especially quantum mechanics, and I'm coming from a strong CS background. One question is haunting me. Why do I need ...
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2answers
46 views

How do you move from just plugging in variables to an equation to actually understanding the equation?

I really can't say I focused on math when in school (didn't focus much ) , but now much later in life, I am trying to learn math. While I can sometime just plug and chug variables into an equation ...
2
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2answers
54 views

Sandwiching Limsups & liminfs of expectations

Why is it that if we sandwich a liminf of an expectation between two equal quantities we get that the limit exists? Can we somehow deduce the limsup from that and conclude that it's the same or am I ...
5
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4answers
234 views

What sequence should I study these topics in?

I will shortly list a series of topics that amount to what is essentially the first two years of an undergraduate degree. I'd like to know what is considered best order in which to study these ...
6
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1answer
146 views

Modeling Rain on a Windshield for various Speeds using Calculus

A question was recently posed to Click & Clack Talk Cars (http://www.greatfallstribune.com/story/life/2014/08/07/click-clack-rainy-day-raises-physics-question/13750681/). The topic is rain hitting ...
8
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1answer
63 views

Changing the order of integration without sketching?

When changing the order of double integrals, I have always relied on sketching the region. I have recently come across this example on MSE by @FelixMartin which seems to avoid visual-based reasoning, ...
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2answers
36 views

How to find a basis of an image of a linear transformation?

I apologize for asking a question though there are pretty much questions on math.stackexchange with the same title, but the answers on them are still not clear for me. I have this linear operator: ...
0
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0answers
21 views

Distance between points and parametric equations of line.

Find the distance between the line $x=3t-1$, $y = 2-t$, $z=t$, and each of the following points: a) $(0,0,0)$ b $(2,0,-5)$ c) $(2,1,1)$ Here is how I proceeded: Find v of the line: (3,-1,1) Find ...
1
vote
1answer
59 views

Sufficient requirements for graduate school in Mathematics?

I am 24 years old, and I will be completing a degree in Computer Science in May 2015. Over the years, I have taken the Calculus series (up to Multivariable Calculus), Intro to Linear Algebra, Intro ...
3
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0answers
67 views

In what order should I study?

I would like to study the basic fundamentals of mathematics from the beginning and move on from there as my understanding of the subject is lacking. In what order should I study? Arithmetic ...
0
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1answer
31 views

induction exercise doubt

the exercise states: Let $x_1 , ...,x_n$ be strictly positive numbers such that their product is equal to 1. Show then that $\sum_{k=1}^{n} {x_k} \ge n $, for every $n \ge 2$. My solution: for the ...
3
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2answers
90 views

If $X$ and $Y$ are uniform$(-1,1)$, how can I find the distribution of $W=X^2+Y^2$?

If $Y$ and $X$ are independent uniform (-1,1) random variables, I would like to derive the distribution of $W=X^2+Y^2$. At first I thought that I could use the CDF technique and a geometric ...
1
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1answer
30 views

Bounded Almost Sure convergence implies convergence in pth mean

A book I'm reading gave the following result. If $X_n \to X $ a.s. and $|X_n|^p \le Z$ for some random variable $Z$ with finite expectation, then we have convergence in $p$th mean. I was wondering, if ...
1
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1answer
24 views

How to prove the second inequality

This might be very trivial to show. But I still cannot figure it out. Let $a \in [-1, 1]$ and $b_i, c_i \in \mathbb R$ with $i \in \mathbb N$. Show that $$\sum_i ab_ic_i \leq |\sum_i ab_ic_i| \leq ...
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0answers
43 views

Inequality among trigonometric sums of normal random variables

This is an inequality used in a proof which I do not know how to prove. $$\left(\sum_{k = 2^j +1}^{2^{j+1}} \frac{\sin(k\pi t)}{k}G_k\right)^2 \leq \left|\sum_{k = 2^j +1}^{2^{j+1}} \frac{e^{ik\pi ...
2
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1answer
46 views

Topology of a nested sequence of subsets

Hi everyone I'd like to know if the following proof is correct, I think so. And also if there is a more direct approach without the many subcases. Thanks in advance Let $X$ be an infinite set, and ...
2
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1answer
39 views

Uniform convergence of $xe^{-nx}$

Does the sequence $(f_n)$ on $[0, \infty)$ given by $ f_n(x) = > xe^{-nx} $ converge uniformly? This is from Bartle's Elements of Real Analysis. I've already proven that the sequence is ...
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3answers
41 views

Prove that $\frac{\tan^2\theta(\csc\theta-1)}{1+\cos\theta}=\frac{(1-\cos\theta)\csc^2\theta}{\csc\theta+1}$

Question: $$\frac{\tan^2\theta(\csc\theta-1)}{1+\cos\theta}=\frac{(1-\cos\theta)\csc^2\theta}{\csc\theta+1}$$ Prove that L.H.S.=R.H.S. My Efforts: ...
2
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1answer
33 views

Find the value of $27\csc^2\theta+8\sec^2\theta$

$10\sin^4\theta+15\cos^4\theta=6$, then find the value of $27\csc^2\theta+8\sec^2\theta$ I don't know how to do it have just tried by converting sin and cos into csc and sec. But can't get the ...
0
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1answer
42 views

Completeness and closedness

I got confused with these two concepts when consider the set $\Omega$ of real valued continuous functions defined on $[0, 1]$. By definition, $\Omega$ is certainly closed since every set is a closed ...
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1answer
34 views

Covariant and contravariant bases on a diffeomorphism

If we allow two domains $\Omega, \bar{\Omega}\in \mathbb{R}^3$, allow $\mathbf{\Theta}: \Omega \to \mathbf{E}^3$ and $\mathbf{\bar \Theta}: \bar \Omega \to \mathbf{E}^3$ to be two ...
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3answers
36 views

Trigonometric Proof:

Question: If $m\cos\alpha-n\sin\alpha=p$ then prove that $m\sin\alpha+n\cos\alpha=\pm \sqrt{m^2+n^2-p^2}$ My Efforts: $(m\cos\alpha-n\sin\alpha)^{2}=p^2$ ...
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3answers
110 views

Trigonometric proof [L.H.S.=R.H.S]

Question: $$\frac{2-3\sin\theta+\sin^3\theta}{\sin\theta+2}=2\sin\theta (\sin\theta-1)+\cos^2\theta$$ I don't know how to start with these problem. Normally these type of proof confuse me. In my ...
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0answers
22 views

Probability Distribution in Cumulative Follow-Up Study

Data layout for a cumulative type of follow-up study is : $$\text{table 01. Data layout for a cumulative follow-up study}$$ $$ \begin{array}{l|cc|l} & \text{Exposed}(E) & ...
2
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0answers
43 views

Rotate the unit circle by a fixed angle, what does happen is $\alpha/\pi$ is rational? and irrational?

Hi everyone I´d like if someone could say me if the following is correct. Thanks in advance Rotate the unit circle by a fixed angle $\alpha$, say $R: C \rightarrow C$; $(1,\theta)\mapsto ...
3
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3answers
142 views

Convergent or divergent $\sum_{n=1}^{\infty} \frac{e^nn!}{n^n}$?

Any suggestion/hint, not the whole solution, how to determine convergence/divergence of $$ \sum_{n=1}^{\infty}\dfrac{e^n \cdot n!}{n^n} $$ I'm currently stuck.
5
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3answers
49 views

Trigonometric identity, simplifying an expression to $(1-\sin^2 a\cos^2a)/(2+\sin^2a\cos^2a)$

Question: $$\left(\frac{1}{\sec^2A-\cos^2A}+\frac{1}{\csc^2A -\sin^2A}\right)\sin^2A\cos^2A=\frac{1-\sin^2A \cos^2A}{2+\sin^2A\ \cos^2A}$$ Prove L.H.S. = R.H.S. My Efforts: ...
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4answers
160 views

Suggestion for a book on Linear Algebra [duplicate]

Please suggest a Linear Algebra book with an introduction and rigorous theory (description) on Eigenvectors , eigen-values , Cayley-Hamilton theorem , Diagonalisation of matrices ; Quadratic forms ( ...
0
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2answers
34 views

Ways to select a hand of 9 cards from a deck of 36

This is a very basic self learning question, the scenario is there are 36 cards of 4 suits from 1 to 9 of each suit. One can pick a hand of 9 cards. My question is how many ways can someone pick a ...
0
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0answers
36 views

Doubts: Proof of Deduction Theorem

I am reading Robert Wolf's A Tour Through Mathematical Logic and am enjoying it. But the author omits proofs for the Deduction and Generalization Theorems. I looked through Intermediate Logic by ...
5
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4answers
206 views

Studying mathematics: Is proving things yourself worth the time?

When studying mathematics, is proving things yourself (before reading the proof given in the text) worth the time? This approach takes significantly longer than simply trying to follow along, but you ...
0
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2answers
79 views

Prove that $(2n+1)+(2n+3)+\dots +(4n-1) = 3n^2$ by induction

Note: This is for self study, the book is Elementary analysis by Kenneth. A. Ross How to prove the following by mathematical induction, I am stuck
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1answer
63 views

Does anyone have any good resources for learning high level maths. [closed]

Does anyone have any good resources for learning higher level maths? The topics I'm considering are algebra, triangles and anything you people think is cool and useful.
1
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0answers
61 views

Baker's transformation: continuity, orbits of irrational and rational points

I've reading the Pugh's Analysis book and I have problems with one exercise. This says: The baker's transformation: a rectangle of dough is stretched to twice its length and folded back on itself. ...
5
votes
1answer
87 views

What math have I missed as an Engineeering graduate? [closed]

To explain, I have a Master's in Engineering from a well known university. We did a wide variety of mathematical topics, vector calc, perturbation methods, numerical methods, linear algebra, discrete ...
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1answer
43 views

Books to get started on mathematics

I'm studying grammar and I feel a based mathematics would help me. What you recommend to start considering I'm not familiar with well developed therms and etc?
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1answer
40 views

Inner product of functions as integration

I am trying to teach my self some linear algebra in preparation for a module in machine learning. I am using Gilbert Strang's text Introduction to Linear Algebra and am having some difficulties. My ...
11
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1answer
132 views

How to Self-Study Higher Math Without Solutions

I've been lurking on this site for several months, and as someone studying higher mathematics independently (i.e., outside of a college/institutional setting), this forum has probably been the best ...
4
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3answers
113 views

reduction of $6xy + 8 y^2 -12x-26y + 11 = 0$ to canonical one of a second-order curve

I have this polynomial $$ 6xy + 8 y^2 -12x-26y + 11 = 0 $$ and I need to reduce it to a canonical equation of a second-order curve. The correct answer from the textbook is that it is a hyperbola ...
2
votes
3answers
61 views

Why $ (1- \sin \alpha + \cos \alpha)^2 = 2 (1 - \sin \alpha)(1+ \cos \alpha)$?

Why $ (1- \sin \alpha + \cos \alpha)^2 = 2 (1 - \sin \alpha)(1+ \cos \alpha)$? I am learning trigonometric identities one identity I have to proof is the next: $ (1- \sin \alpha + \cos \alpha)^2 = ...
1
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1answer
70 views

Typical material covered in Calculus 1 course?

I have a copy of Larson's Calculus: early transcendental functions, 2nd edition. I was wondering what material I would need to cover to have the equivalent of a Calculus 1 course at a University. I ...
1
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3answers
304 views

Have any one studied this binomial like coefficients before?

Consider the following identities. $\dfrac{n}{n-r}\dbinom{n-r}{r}=\dfrac{n}{r}\dbinom{n-r-1}{r-1}$ ...
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2answers
33 views

Could anybody provide a more detailed explanation of a tangent equation in its general form?

In my textbook I'm currently at the topic of a tangent line to an ellipsis and hyperbola. And there I've encountered this statement: If a curve has an equation $$ y = f(x) $$ then an equation of a ...
0
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3answers
67 views

how to prove $ax + by = cx + dy \implies a = c, b = d$?

Actually the question is in the title. I just have saw such a method $$ ax + by = cx + dy \implies a = c, b = d $$ in my textbook, so I can assume it is true, but I'm very interested on proving ...