Questions about studying mathematics without formal instruction.

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What would be the way to learn algorithms?

I want to know the mathematical theory associated algorithms. Story with a base of calculation, analytical geometry and linear algebra. What would be the previous topics to begin the study of ...
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52 views

Is there any proof for this formula $\lim_{n \to ∞} \prod_{k=1}^n \left (1+\dfrac {kx}{n^2} \right) =e^{x⁄2}$

Some times ago, In a mathematical problem book I sow that this formula. I don't no whether it is true or not. But now I'm try to prove it. I have no idea how to begin it. Any hint or reference would ...
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1answer
41 views

General formula for $\frac{a^{2n+1}-a^{2n-1}}{a^n-a^{n-1}}$. Is such proof correct?

I have a very simple case: Find general formula for $\frac{a^{2n+1}-a^{2n-1}}{a^n-a^{n-1}}$. Of course dividing one by another was quite simple with outcome: $a^n(a+1)$. However I would like to prove ...
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0answers
18 views

Monotonic Functions and Uniform Convergence

The following is a proof from "Heavy-Tail Phenomena" by Resnick (2007). I have some questions about the proof. (2.3) seems to be an identity. The left side the global sup over $[a, b]$ and hence ...
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3answers
84 views

How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$?

I am reading a chapter about mathematical proofs. As an example there is: Prove that: $$(1) \space\space\space\space\space\space\space\space\space\space\space \frac{a+b}{2} \geq \sqrt{ab}$$ for ...
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1answer
25 views

Invariance Properties of Brownian Motion

I am trying to make sense of the Scaling-Invariance and Time-Inversion properties of Brownian motion by producing a sample path. For the record, I am using the following definitions. Let $B(t)$ be the ...
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2answers
127 views

A counter example of Brownian Motion

Here is an example in my textbook to illustrate why we need the continuous sample path in the definition of Brownian motion. Let $(B_t)$ be a Brownian motion and $U$ be a uniform random variable on ...
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1answer
21 views

Filtration from a Brownian Motion

The textbook I am reading defines the filtration induced from a Brownian Motion as follows. Let $\{B(t): t \geq 0\}$ be a Brownian Motion defined on some probability space, then we can define a ...
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1answer
33 views

Using Taylors to show convergence in probability

I'd like to show that \begin{equation} \sqrt{n} \left( (1-\frac{1}{n})^{n\bar{X}} - e^{-\bar{X}} \right) \to 0 \end{equation} in probability for a random variable with mean $\mu$ and finite variance ...
2
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1answer
51 views

$\mathbb E[\mathbb E(X|Y, Z)|Y]$ or $\mathbb E\{\mathbb E[(X|Y)|Z]\}$?

To begin with, the standard iterated law of probability is as follows. $$ \mathbb E X = \mathbb E [\mathbb E(X|Y)]. (1) $$ I am perfectly happy with $(1)$ and there is also some quite good ...
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20 views

Formula given 1 number how do I get the other two numbers

I am doing some programming which involves traversing through dataset rows. I am returning data in rows of 10 per page. So the first page is rows 1-10. The second page is 11-20. The third page is ...
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0answers
27 views

Asymptotic Equivalence implies same asymptotic distribution?

A book I'm reading stated that if we have nonnegative random variables, and if $X_n\to X > 0$ in distribution and $\frac{Y_n}{X_n} \to 1$ in probability then $Y_n \to X$ in distribution. However, ...
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1answer
23 views

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 ...
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0answers
54 views

How do I gain a comprehensive background on a mathematical topic without taking a course? [closed]

Here is my dilemma: I am really interested in mathematics and plan to take BC next year, but in the meantime, I might be interested in some other interesting side topics. First, what do you ...
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1answer
31 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 ...
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36 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 ...
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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
82 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
34 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 ...
37
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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
45 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 ...
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2answers
43 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 ...
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3answers
156 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 ...
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31 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 ...
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1answer
51 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
32 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: ...
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0answers
16 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 ...
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1answer
46 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
61 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 ...
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1answer
29 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
85 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 ...
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1answer
20 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 ...
2
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1answer
19 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
40 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
40 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 ...
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0answers
19 views

Minimal Surface Problem in Gelfand & Fomin

I'm working through Gelfand and Fomin's Calculus of Variations, and I'm stuck on problem 19 in the first chapter. The verbatim text says: "Find all minimal surfaces whose equations have the form ...
2
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1answer
35 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 ...
2
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3answers
36 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: ...
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1answer
26 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 ...
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1answer
35 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
32 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
32 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
50 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) & ...
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0answers
25 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 ...
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3answers
130 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.
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3answers
43 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
135 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 ( ...
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2answers
31 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 ...
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0answers
29 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 ...