Questions about studying mathematics without formal instruction.

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50 views

Can I start apostol (vol 2) with no background in multivariable calculus?

I searched a lot in web and almost everyone says if you want to read spivak or apostol, you should first read an introductory book on calculus like stewart. I didn't read stewart but I studied single ...
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1answer
101 views

Learning math vs problem solving

Ok so I am about to start my final year in high school we will be learning calculus this year, but I already know single- and multi-variable calculus and linear algebra so I want to spend my final ...
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1answer
92 views

Question about limit points in relation with continuity and functional limits

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I have the feeling that the author is being careless about limit points in his theorems or I am not understanding ...
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2answers
183 views

Proof of Divergence Criterion for Functional Limits

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I don't understand corollary 4.2.5 on page 107. To be more specific, let me first write down the theorem that precedes ...
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4answers
41 views

Find $n(A \cap B)$

Question: In group of people, 60% like coffee and 70% like tea. How many people like both of them.? My Effort: We have to find how many people like both the items that means we have to find $n(A ...
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1answer
50 views

on limits of cumulative density functions

Theorem 1.5.3 of Statistical Inference by Casella and Berger States that the function $F(x)$ is a cdf if and only if the following three conditions hold: 1) $\lim_{x \to - \infty} F(x) = 0 \text{ and ...
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5answers
1k views

Self-learning mathematics - help needed!

First, I apologise for the nebulous nature of my title but I can't adequately explain myself concisely. I am about to start an MSc in pure maths after a fairly shaky undergraduate degree. I am very ...
3
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0answers
58 views

How to approach real analaysis

I'm just starting first year in university in Europe and here there there is no Calculus, instead you jump right into Analysis. The trouble is, for some time I self-studied through US style books and ...
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1answer
14 views

Find catesian coordinate of T-point $P(-\frac{65\pi}{2}) $

Find the Cartesian coordinates of T-point $P(-\frac{65\pi}{2}) $. It is easy when there is no negative sign. I don' t know how to do with negative sign.
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1answer
32 views

Paradox in connection with definition of limit points and order limit theorem?

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I come across something that appears (to me) as a paradox. Let me first write down one definition and two theorems that ...
3
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2answers
141 views

Is there any closed form for the finite sum $1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+…+\dfrac{1}{n}?$ [duplicate]

I know that the infinite summation $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}+...$$ is divergent and also the sequence $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}-\ln ...
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2answers
24 views

Finding Principale period of $\cos$ function

Find principle period of $3\cos (2x-3)$. Today I have learned about principle period of various trigonometric function. I know that principle period of cos is $2 \pi$. Please someone can help me ...
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1answer
23 views

prove that $g$ is a function of $(x_1-x_2,x_2-x_3,\dots,x_{n-1}-x_n)$

$g$ is a function of $(x_1-x_2,x_2-x_3,\dots,x_{n-1}-x_n)$ iff $$g(x_1+a,x_2+a,\dots,x_n+a)=g(x_1,x_2,\dots,x_n)\forall a \in \mathbb R$$ Trial: Only if part : Consider $g$ is a function of ...
0
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1answer
35 views

How to get this inequality

Let $c>0$, $n \in \mathbb N$ and $q>1$. How to get the following approximating inequality when $n$ is large, please? To be more specific, I cannot see how to get rid of the square root. $$ ...
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1answer
68 views

Exercise 3.3.8 from Understanding Analysis by Stephen Abbott

Motivation: trying to prove that if $K \subseteq \mathbb{R}$ is compact (and thus, by the Heine-Borel theorem, closed and bounded), then this implies that any open cover for $K$ has a finite subcover. ...
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2answers
56 views

exercise on pointwise convergence of an (easy) function.

Exercise 6.2.5. Taken from understanding analysis of Stephen Abbott For each n $\in N$, define $f_n on \ R$ by $$f_n(x) = \begin{cases} 1, & \mbox{if} \ |x| \ge 1/n \\ n|x|, & \mbox{if} \ ...
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3answers
101 views

If $G$ is a simple $f$ an homomorphism, and $A\lhd H$ is such that $[H:A]=2$, show $f(G) \subset A$

I'm stuck with the following problem. Can someone help me by providing a hint? Suppose $G$ is simple and let $f$ be an homomorphism between $G \to H$. If $\#G\ne2$, $A\lhd H$, and $[H:A]=2$. Then ...
4
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1answer
63 views

hint with an exercise algebra

I'm stuck with the following I hope someone could help me. Let $N$ a normal subgroup of $G$. Show that if $[G:N]=4$, exists a normal subgroup $M$ of $G$ s.t. $[G:M]=2$. My idea: Since $G/N$ has ...
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1answer
92 views

Prove that line segments are parallel.

Prove using slope of lines that line segment joining the midpoint of $\overline { AB}$ and $\overline{AC}$ in $\Delta ABC$ is parallel to $\overline {BC.}$ Need to prove using slope of lines means I ...
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6answers
1k views

How should I self-study calculus?

So I already took Pre-Calc, and ended up with a B both semesters. I am an incoming senior in high school. My special-ed case manager won't let me take it because she doesn't want to see me panic ...
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0answers
35 views

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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2answers
138 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
48 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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1answer
81 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 ...
3
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3answers
124 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 ...
2
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1answer
44 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 ...
4
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2answers
170 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 ...
2
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1answer
31 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 ...
2
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1answer
38 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
62 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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0answers
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
41 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
30 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 ...
2
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1answer
59 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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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 ...
8
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1answer
124 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
40 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
48 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
56 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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4answers
264 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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1answer
151 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
85 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
41 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
24 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
71 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 ...
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0answers
84 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
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
96 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
41 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 ...