Questions about studying mathematics without formal instruction.

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2answers
98 views

Elementary properties of closure

Hi everyone I'd like to know if the following is correct. I really appreciate any suggestion. (Honestly the only one that matters me is the second property the others are easy, I think) Thanks. ...
2
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1answer
124 views

Proving that $G$ is a group if $a*x=b$ and $y*a=b$ have solutions.

(Reference : Fraleigh, A first course in abstract algebra) Prove that a nonempty set $G$, together with an associative binary operation * on $G$ such that $a*x=b$ and $y*a=b$ have solutions in G, ...
2
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1answer
302 views

Expected value of sample median given the sample mean.

Let $Y$ denote the median and let $\bar{X}$ denote the mean of a random sample of size $n=2k+1$ from a distribution that is $N(\mu,\sigma^2)$. How can I compute $E(Y|\bar{X}=\bar{x})$? Intuitively, ...
3
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1answer
62 views

Figuring out deficiencies in math education.

I'm mostly self-taught and while I know (and use) many advanced mathematical topics, I often enough find holes in my understanding of lower level math. Is there an exam (or series of exams) I could ...
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1answer
39 views

problem with isometry that doesn't preserves rays, i just don't understand

I try to teach myself a bit of non-euclidean geometry And I am a bit stumped by the following remark in George E. Martins "The foundations of Geometry and the Non-Euclidean Plane" page 217. extra ...
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1answer
278 views

“Easy” (maybe not) question about dual spaces (Lineal Algebra).

Hi everyone is my first time reading about dual spaces and in one part of the notes that I read, says: The dual of the quotient space $V/U$ is naturally a subspace of $V$, namely the annihilators of ...
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2answers
40 views

A question on conditional probability

Question: Let $X$ and $Y$ be two random variables. The relationship between the two is as follows. If $Y$ is less than or equal to $1$, then $X$ is equal to $Y$; If $Y$ is more than $1$, then $X$ ...
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3answers
838 views

Difference between continuity and uniform continuity

I understand the geometric differences between continuity and uniform continuity, but I don't quite see how the differences between those two are apparent from their definitions. For example, my book ...
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2answers
33 views

Absolute value of the difference between two sequences

Consider two sequences $(a_n)$ and $(b_n)$ both contained in $\mathbb{R}$ and assume that these two sequences satisfy $|a_n - b_n| \rightarrow 0$, then this does imply that both $a_n$ and $b_n$ are ...
2
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1answer
71 views

Source for Differential Manifolds/Geometry Questions?

I'm looking for a good source for a large collection of Differential Manifolds/Geometry questions covering a subset of the following topics: inverse function theorem, local coordinates, induced ...
0
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1answer
623 views

Brouwer's Fixed Point theorem proof for 2-dimension

I am trying to find a elementary proof of the Brouwer's fixed point theorem only using basics of point set topology and real analysis. In the one of the textbooks I read, they were proving Brouwer's ...
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2answers
54 views

The mean of a geometric random variable

The mean of a geometric random variable is $$ \frac{1-p}{p} $$ What would be the mean of x for this density function? $$ f_{x}(x) = \sum_{k=0}^{\infty} p(1-p)^{k} \delta (x-k) $$ I got confused ...
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1answer
98 views

Group with an even number of elements.

If $G$ is a group such that $|G|=2n$. Prove that there's an odd number of elements of order 2, and then there's an element which is its own inverse, besides of the identity. If we consider all the ...
3
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0answers
56 views

Prime divisor of the form $2kp+1$ that divides $2^p-1$

The book that I'm reading (Elementary Number Theory by Underwood Dudley) gives a Theorem: If $p$ and $q$ are odd primes and $q|a^p-1$, then either $q|a-1$ or $q=2kp+1$, for some integer $k$. Then it ...
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1answer
382 views

What pure mathematics foundations should an applied mathematician have?

I'm studying mathematics, with some statistics also, and I've always chosen applied courses. I'm getting to the point where I'm studying 3rd year undergraduate to graduate level material. My first ...
3
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0answers
51 views

Totient Function problem

Suppose we know that $(m,n)=2$. Show that this implies that $\phi{(mn)}=2\phi{(m)}\phi{(n)}$. My attempt: So let $m=p_1^{r_1}p_2^{r_2}...p_k^{r_k}, n=p_1^{s_1}p_2^{s_2}...p_k^{s_k}$. Then ...
2
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2answers
77 views

Proving that this is not a group.

I got the set: $G=\{p/q\in \Bbb Q : (p,q)=1$, with $q$ odd number $\}$ and the binary operation $a*b:=a+b$. And I say that $(G.*)$ isn't a group because it doesn't have an identity. My proof is: We ...
3
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2answers
65 views

Proving uniqueness (basics of group theory)

If $(G,*)$ is a group, prove that the identity and the inverse elements are unique. What I did for the first one is: Suppose $\exists e,g\in G$ such that $\forall a\in G a*e=e*a=a$ and also that ...
3
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0answers
108 views

Examples of Talagrand's inequality

I am trying to understand Talagrand's inequality and when it gives better results than Markov/Chebyshev/Chernoff. However I find the formal definition hard to understand. Are there any nice simple ...
3
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1answer
273 views

Mathematical logic book with answers to exercises

I'm sure a question similar to mine has been asked before, but I am looking for a mathematical logic book with answers to the exercises. I am studying independently and although I have good logic ...
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3answers
252 views

Good textbook for learning Sequent Calculus

There are many modern text books teaching logic using Natural Deduction. There are no books teaching logic using the axiomatic method (see Good book for learning and practising axiomatic logic ) Now ...
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1answer
53 views

Shooting Star (Probability)

Assume that a random experiment consists in centering a telescopic sight on a random star. Let $A_{n}$ ($n \in \mathbb{Z}^{+}$) denote the event that the telescopic sight spots exactly $n$ stars. ...
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1answer
74 views

Correspondence between an order of rational function field and a Dedekind cut

In the first page of Real Algebraic Geometry by Jacek Bochnak, Michel Coste, Marie-Francoise Roy, they briefly connect an order of the rational function field $R(X)$ with a Dedekind cut ...
2
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1answer
70 views

Integrating the Fourier Transform

I am trying to show that $$\mathcal{F}\left\{ \frac{f(t)}{t} \right\} = - i \int_{- \infty}^w \hat{f}(w') \, d w'.$$ Shouldn't it be $$\mathcal{F}\left\{ \frac{f(t)}{t} \right\} = - i \int_{w}^{+ ...
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1answer
50 views

why cannot prove convergence of a serie with it's limit

i have the following serie $$ \sum_{k=1}^{\infty} \frac{{\sqrt{k} + k^3}}{{k^4+k^2}} $$ is it enough to calculate the limit to prove that it converges ? so that would be $$\lim_{k \to \infty } ...
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1answer
152 views

Verification and help to simplify an argument about closure of some sets.

Hi everyone I'd like to know if what I have so far is correct, I think is much work for something which is too simple I would appreciate any advice or whatever. Moreover, I have doubt in (3) and (4), ...
2
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1answer
273 views

Would you provide a study routine for Spivak's Calculus? [closed]

I've been working on Spivak's Calculus for the past few days and although I can manage to solve most problems, they take a lot of time. Some chapters have over 20 exercises and it can take several ...
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6answers
180 views

Any ideas how I can rewire my brain such that $\varphi \leq \psi$ “obviously” means that $\varphi$ implies $\psi$?

The Boolean domain $B=\{\mathrm{False},\mathrm{True}\},$ can be viewed as a partially ordered set in two different ways. In the best approach, $\mathrm{False}$ is the least element and $\mathrm{True}$ ...
1
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2answers
309 views

Endomorphisms on finite dimensional vector spaces $f: V \to V$ are surjective $\iff $ injective

Similar questions: Linear map $f:V\rightarrow V$ injective $\Longleftrightarrow$ surjective Proposition: Let $V$ be a finite dimensional vector space over an arbitrary field $\mathbb{K}$. If $f: ...
3
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6answers
278 views

I want a good dictionary of mathematics/ geometry

I noticed I a made a mistake in some geometrical terminology and wanted to better my life by buying a new dictionary of mathematics or more specialised Geometry. (okay I am just a shopaholic for ...
1
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1answer
72 views

rigorous path to statistical mechanics

Is there a path to thermodynamics which is not governed by intuitive and fuzzy postulates but by lemmas and axioms? I also asked this question on physics.stackexchange in order to get different ...
1
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1answer
35 views

Proving this simple equality

$${\sum}_{t=1}^T(t-\bar{t})^2=\sum_{t=1}^T t^2 - T\bar{t}^2$$ Is there any special property going on here? I cant seem to work around it. $\bar{t}$ is the mean $T$ is the last $t$ in sequence ...
2
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1answer
136 views

Is this an abuse of notation?

Here is a proof says that the differential of Gauss map is self-adjoint. But I seems there is an abuse of notation at (1) in it. Since $dN_p$ is linear, it suffices to verify that $\langle ...
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2answers
116 views

Exercise in well-ordered sets

Definitions: Given a subset $A$ of a non-empty well-ordered set $X$ we define the supremum as follows: $\text{sup(A)}:=\text{min}\bigg(\big\{\,y\in X \oplus\{ +\infty\}: \forall x\in A \;(x\le y)\, ...
1
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2answers
277 views

What does the 'triple line equals' sign with a strikethrough mean?

I understand that the triple equal sign can mean "identical to" or "defined to be equal to", so intuitively, I assume that the triple equal sign with a strikethrough means "is not identical to" or ...
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5answers
996 views

Self studying math, how can I learn the most?

I am currently studying Pre-Calculus on my own. I have a few texts I am working with but feel like I could learning a lot more than I am. When people typically ask these kind of questions the common ...
0
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1answer
276 views

Square Root of Random Variables

Question: Suppose that $\displaystyle \frac{2}{\theta_0}\sum_{i=1}^n y_i\sim\displaystyle\chi_{2n}^2$ and $\displaystyle 2\theta_0\sum_{i=1}^n x_i\sim\displaystyle\chi_{2n}^2$. And these two are ...
0
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2answers
82 views

Positive recurrent, zero recurrent and transient states

How do we prove that a state in a Markov chain process is positive recurrent, zero recurrent or transient? For example, if we have a transition matrix $$P=\left(\begin{array}{ccc} \frac{1}{3} ...
2
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1answer
84 views

Evaluate $\iint\limits_{\substack{x<u,y<v, \\ x^2+y^2<1}} dxdy$

How can I evaluate the following double integral: $$\iint\limits_{\substack{x<u,y<v, \\ x^2+y^2<1}} dxdy$$ If we didn't have the restrictions $x<u, y<v$ polar coordinates would have ...
3
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2answers
89 views

The set of all the partial ordering of $X$ is a partially ordered set.

I have trouble with the following exercise. I don't understand it. Also I think that $x,y\in P$ should be $x,y\in X$. Could someone give an insight of the exercise and also some hint of how to solve ...
3
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3answers
68 views

$X$ and $1 - X$ are identically distributed

I have a two-fold goal for this question. First, I'm trying my hand at making hypotheses and proving them as far as I can. I want to understand the limits of proof, not just the techniques. Second, ...
0
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1answer
46 views

polynomial interpolation

I have a function, for example $f(x)=\frac{-x^2}{2}+|x|$, which is divided on $[-1,0)$ and $[0,1]$. How do we interpolate this function with a polynomial $p$ in the maximum degree 4 with ...
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1answer
212 views

Intuition behind proof and verification partially ordered sets

Hi everyone in the book that I read I have trouble to understand the argument of the proof at the below proposition. There is a lot of point which are left as exercises, which is great. One of these ...
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0answers
36 views

Question on numerical quadrature and precision

I am studying numerical analysis and I came across this question: Let $P_{n+1}(x)\in\Pi_{n+1}$ be orthogonal to $\Pi_n$ relative to a weight function $w(x)\geq 0$ on $[a,b].$ Denote by ...
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1answer
278 views

Exercise: Every non-empty subset of $X$ contains a minimum and maximum element and the converse

Hi everyone is my second exercise of posets. And find the following, the first part I think is not difficult at all. But for the converse I have some serious troubles to prove it. I have two ...
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0answers
102 views

Partially ordered set exercise

Hi everyone is my first time working with posets and I'd like to know if the solution of the next exercise is really correct or maybe there is some errors that I cannot see. Thank in advance. ...
2
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2answers
106 views

Calculate the following conditional expectation.

Let $\Theta$ and $R$ be two independent random variables, where $R$ has density $f_{R}(r)=re^{-\frac{1}{2}r^2}$ for $r>0$ (zero otherwise) and $\Theta$ is uniform on $(-\pi,\pi)$. Let $X=R ...
0
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0answers
266 views

Problem with Spivak calculus.What should I do?

For past two weeks I have been working throught the Spivaks calculus book.Needless to say I am very pleased with his writing style,but I have a slight issue. The issue is namely that in those two ...
3
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2answers
124 views

What is the purpose of universal quantifier?

The universal generalization rule of predicate logic says that whenever formula M(x) is valid for its free variable x, we can ...
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2answers
157 views

Pigeonhole Principle and maximum length of the repeating section

The question I have is, when 5 / 20483 is written as a decimal, what is the maximum length of the repeating section of the representation? I believe I need to divide 5 by 20483 which is equal to ...