Questions about studying mathematics without formal instruction.

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1answer
25 views

Proving the Boolean expressions

Are these two Boolean expressions the same? *$co$ is the carry out while $ci$ is the carry in.
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1answer
34 views

How to prove there exists a bijection between the ideals of $A/a$ and the ideals of $A$ containing $a$

original proposition is there is a one-to-one order-presserving correspondence between the ideals of $A$ which contain $a$ and the ideals of $A/a$. I think one-to-one correspondence mean ...
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1answer
104 views

help and verification of 3 short exercises

I'm reading an old book and find the following three question. I'd like to know two things: if my attempts are correct and also it would be great if someone could give suggestions in more detail. ...
14
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2answers
977 views

Show that these two numbers have the same number of digits

I want to show that for $n>0$, $2^n$ and $2^n + 1$ have the same number of digits. What I did was I found that the formula for the number of digits of a number $x$ is $\left ...
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2answers
56 views

How to prove “ideal $I$ is prime iff $A/I$ is a integral domain ”?

$A$ is a commutative ring with identity. $I$ is a ideal of $A$. then ideal $I$ is prime iff $A/I$ is a integral domain. here is what I thought $(\Rightarrow)$ We want to prove $A/I$ is a integral ...
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2answers
27 views

Qualification of a Universal Quantification

Let us say I have a predicate, $P(n)$, and I want to say that it holds for every integer greater than $2$ (an example would be $P(n) = 2n>2+n$). Let us furthermore say that the UOD (universe of ...
2
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1answer
33 views

Correct Way to Write a Statement in First-Order Logic

I am teaching myself set theory. I am at a point where the set of rationals, $\mathbb{Q}$, has been defined, along with its ordering relation, $<_\mathbb{Q}$. Now, working towards a definition of a ...
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2answers
60 views

Question about some algebra theorem

1. order-presserving = monotonic But we haven't define order structure on the ring. 2. I try to prove x is a unit <=> (x)=A ,and fail. That's what I think: => we want to prove (x)=A. ...
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3answers
67 views

How to find the sum of series $\sum_{i=1}^{\infty}\frac{i}{2^i}$? [duplicate]

I am learning about series of numbers at the moment. In the book there is an exercise in which I need to find the sum of : $$\sum_{i=1}^{\infty}\frac{i}{2^i}$$ I know it is equal to $2$. But how do I ...
0
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1answer
17 views

Question regarding dimension of linear transformation.

I saw in an exercise that if $T$ is a linear transformation $T: V\rightarrow W$ and $T_2: W\rightarrow Z$ and $T_1: X\rightarrow V$ are invertible then the rank of the composition doesn't change. So, ...
1
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1answer
47 views

How is an open set defined without referring to any distance function in a topology?

I am currently studying general topology. The definition given by Royden looks very confusing to me. It says that the elements of a topology is called open sets without actually defining what exactly ...
0
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1answer
34 views

A proof problem about congruence relation

Given a ring R and a two-sided ideal I in R, we may define an equivalence relation ~ on R as follows: a ~ b if and only if a − b is in I. Using the ideal properties, it is not difficult to check ...
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1answer
41 views

Checking if systems of linear equations are equivalent

I'm going through Hoffman and Kunze's Linear Algebra on my own and can't for the life of me figure out part of Exercise 3 of Section 1.2: Are the following two systems of linear equations equivalent? ...
4
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0answers
86 views

Prove that a series is $O(t^a)$.

Consider the series $$ u(t,x) = \sum_{i \geq 1} {u_i(x) t_1^i } + \sum_{i+2j \geq k+2, j\geq 1} {\varphi_{i,j,k}(x) t_1^i t_2^j y^k} $$ where $t \in \tilde{\mathbb{C} \setminus \{ 0 \}}$, $x$ is ...
2
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2answers
27 views

Prove $ab + ab\overline{c} + bcd = b(a+c)(a+d)$

Do I need to use absorbtion law to prove them? $ab + ab\overline{c} + bcd = b(a+c)(a+d)$ $ab + cd = (a+c)(a+d)(b+c)(b+d)$. For 1), I simplified $ab+ ab\overline{c} + bcd$ into $b(a\overline{c} + ...
2
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2answers
113 views

$A \times B$ is an open set in $\Bbb R^2 \implies A$ and $B$ are both open in $\Bbb R$; $A,B \neq \emptyset$

I am studying Analysis on my own. Reading The Elements of Real Analysis by Bartle. Came across the above problem and I came up with the following solution but am very unsure about it. Would be very ...
2
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1answer
59 views

Bridge the gap to university mathematics [closed]

Can anyone suggest some good books to help an high school student to "bridge the gap" to university math? I've heard of http://www.amazon.com/How-Study-as-Mathematics-Major/dp/0199661316 and ...
2
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2answers
79 views

Example for a set in $\Bbb R^p$ whose interior is $\emptyset$ and closure is $\Bbb R^p$

The following exercise was in the Elements of Real Analysis by Bartle. Give an example of a set $A$ in $\Bbb R^p$ such that $A^{\circ} = \emptyset$ and $A ^ - = \Bbb R^p$. Can such a set be ...
2
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0answers
60 views

Proof involving prime factorization

I'm beginning some self-study in Number Theory and have come across a problem that I'm not really sure how to solve. Here's the problem: Prove that, if, $$ a=q_{1}^{e_{1}}q_{2}^{e_{2}} . . . ...
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2answers
36 views

Regular or normal topological space

How do we define an open neighborhood around a closed set? My question is with respect to a normal or regular topological space where we use the concept of an open neighborhood around a closed set.
2
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0answers
72 views

Difference between Metric Space and Topological Space

I am reading Chapter 11 of Real Analysis written by Royden and Patrick (4th). It says "The concept of uniform convergence of a sequence of functions is a metric concept. The concept of pointwise ...
25
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7answers
2k views

Active learning vs Passive learning in Math

I am trying to improve how I learn in general but specifically in math and a common suggestion I keep coming across is the difference between active learning and passive learning. The problem is, most ...
1
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1answer
24 views

Prove the following property of holomorphic functions.

Let $\rho(x)$ be a holomorphic function on a disk $D \subseteq \mathbb{C}$ with the property that $\rho(x) \notin \mathbb{N^*} = \{1,2,\dots\}$ on $D$. Prove the following: There exists an $R$ ...
2
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0answers
109 views

Unordered summation

I have a question regard unordered sumation. Definitions: Let $X$ be a set (possible uncountable) and let $f: X\rightarrow \mathbb{R}$ be a function. We say that $\sum _{x\in X}f(x)$ converges ...
3
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1answer
159 views

A subset of $\Bbb R^p$ is open iff it is the union of a countable collection of open balls

I am studying analysis on my own and need some help verifying the solution to the above exercise found in Bartle's Elements of Real Analysis. I know there are other posts answering the same question ...
4
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1answer
53 views

Show that $T\neq{T^*}$

Let $V=P_2(\mathbb{R}), T\in \mathcal{L}(P_2(\mathbb{R})),$ where $T(p)=(a_1x)$. Make $V$ an inner product space by defining $$\langle p,q\rangle=\int_0^1{p(x)q(x)\,dx}$$ So I calculate $$\langle ...
3
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0answers
53 views

What else can I do to learn math better? [closed]

I feel like I am not learning math well or efficiently enough. I read the textbook, do the exercises but I still don't get the marks I would like. For the time I put in, it seems like I should be ...
1
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1answer
21 views

Reference request to study Borel summation

Could someone recommend sources to learn about Borel summation procedure? Books, articles or reviews? I have a background in basic analysis.
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1answer
112 views

Book on discrete mathematics for self study

I am searching for book on discrete mathematics which is suitable for self study. This mean I want it to have exercises with answers (It would be ideal if it had solutions). I have already read ...
1
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1answer
31 views

How to evaluate: $\int_{0}^{1-z} y^{j-1} (1-z-y)^{n-k} dy$

How can I compute the integral: $$\int_{0}^{1-z} y^{j-1} (1-z-y)^{n-k} dy\quad\text{where}\ z \in (0,1) $$ Had it not been for $z$ , the integral would look like an incomplete beta function but what ...
2
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1answer
81 views

An Increasing Function Discontinuous at All Rational Numbers

Let $q: \mathbb{N}\rightarrow \mathbb{Q}$ be a bijective map and let $g: \mathbb{Q}\rightarrow \mathbb{R}$ define $g(q(n))=2^{-n}$. Show that $\sum_{r\in \mathbb{Q}}g(r)$ is absolutely convergent. ...
1
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1answer
25 views

Find the point of inflection

Will there be an inflection point if there is no solution for $x$ when $f ''(x) = 0$? For example, $$ f(x)=\frac{x^2-x+1}{x-1} $$ with domain $\mathbb{R}-\{1\}$ Also, is that when $x$ is smaller than ...
0
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0answers
44 views

Generating an open set using basis of standard topology.

Let $(\mathbb R, \mathscr T$) be a topological space where $\mathscr T$ be a standard topology. Let $K = \{ \frac1n | \; n \in \mathbb N \}$. How can I generate $K$ from the basis elements of this ...
2
votes
1answer
21 views

Example of an equivlance relation that is transitive

I am just tying to figure our this example but am having difficulty understand the math being used. The example state: Let R be a relation on the set $\mathbb{Z}$ defined as $(m,n)\in$ R if and only ...
0
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2answers
38 views

Trying to understand an example of an equivlance relation that is symmetric

I am just tying to figure our this example but am having difficulty understand the math being used. The example state: Let R be a relation on the set $\mathbb{Z}$ defined as (m,n)$\in$ R if and only ...
0
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2answers
38 views

How to integrate absolute function

I have this absolute e-function, but I don't know how to calculate the integration $$ \int_{-2}^{2} e^{\frac{1}{2}j\omega |x|}dx $$ Any idea?
1
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1answer
119 views

How to prove the Jordan's Inequality?

Can anyone tell me how to prove it by using the concepts related to limit?
-4
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1answer
46 views

What is a right way to think when going about trying to solve a math problem? [closed]

What is good step-by-step method to deconstruct a math problem.
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0answers
19 views

Inverse Fourier Transform of $S_Y(f)$

I have this power spectral density $$ S_Y(f) =\frac{N_0}{4 \pi ^{2} f^{2}}\left [ 1- \cos(2\pi f T) \right ] $$ Can any one help me how to find the Inverse Fourier transform?
3
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1answer
81 views

Exponentiation of Real Numbers?

I'm looking to learn Real Analysis on my own. Am reading Elements of Real Analysis by Bartle. I came across this project which defines the powers of real numbers i.e. exponentiation. Firstly I am ...
5
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2answers
131 views

What can I do with measure theory that I can't with probability and statistics

I've studied mathematics and statistics at undergraduate level and am pretty happy with the main concepts. However, I've come across measure theory several times, and I know it is a basis for ...
3
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2answers
44 views

one-to-one and onto functions help

I am trying to understand this exercise. Define $S : \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ by the rule: For all integers $n$, $S(n) =$ the sum of the positive divisors of $n$. a. Is $S$ one-to-one? ...
5
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2answers
138 views

What all maths do I need to know to become good at machine learning.

I am a computer science engineer and I took a couple of maths classes in my first year they were on Fourier series(not transform) partial differential equations, vector calculus, infinite series ...
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3answers
70 views

Calculus: Limit and continuity

Would anyone mind telling me how to solve these two questions? I know it sounds silly but I really have no idea.
0
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1answer
28 views

PMF of Two Random Variables

X and Y are independent and geometrically distributed random variables with $$ P(X = m) = p(1-p)^{m}, m=0,1,2... $$ $$ P(Y = n) = p(1-p)^{n}, n=0,1,2... $$ To find the probability mass function ...
0
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2answers
42 views

one-to-one and onto function question

I am trying to understand this exercise: Let S be the set of all strings of 0's and 1's, and define f: S -> $Z^{nonneg}$ by f(s) = the length of s, for all string in S. a. Is f one-to-one? The ...
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0answers
201 views

Modern research into Grassman's “theory of forms”?

I quote from Petsche's Hermann Graßmann: Biography (emphasis mine): The mathematical part of the book begins with the conception of the “General Theory of Forms”. Starting with a perspective on ...
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0answers
77 views

$(x^r)^s=x^{rs}$ for the real case

Hi everyone I'd like to know if the following is correct and if someone knows a better way to do it. Definition Let $x>0$ be a real, and $\alpha$ be a real number. We define the quantity ...
3
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1answer
28 views

Determining quadratic residues quickly

Let's say that I'm looking for all quadratic residues of a number. THe example from my book is 31. So I can just evaluate $i^2\equiv{a}\pmod{31}$, for $i=1..15$. While not a terribly difficult ...
0
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1answer
25 views

Most powerful size $\alpha$ test

Someone can help me to check this answer? How to find the Most Powerful Test size $\alpha$ and Power of Test, Since I have $H_0 : X \thicksim f_{\theta 0}= (1/\sqrt(2\pi) \exp^{(-x^2/2)}$ and $H_1 : ...