0
votes
1answer
58 views

What does $\bigcap_{m = 1}^\infty ( \bigcup_{n = m}^\infty A_n)$ mean?

What does $\bigcap_{m = 1}^\infty ( \bigcup_{n = m}^\infty A_n)$ mean? I'm not getting it. ${A_n}$ is a sequence of set in $S$.
-1
votes
2answers
38 views

Prove $f(A_1 \cap A_2 \cap A_3 \cap \dotsb \cap A_n)=f(A_1) \cap f(A_2) \cap f(A_3) \cap \dotsb \cap f(A_n)$

Let $f: R \to R$ be a one to one function. For any collection of subsets $A_1, A_2, A_3, \dotsc A_n$ of $R$, prove that $$f(A_1 \cap A_2 \cap A_3 \cap \dotsb \cap A_n)=f(A_1) \cap f(A_2) \cap f(A_3) ...
0
votes
2answers
31 views

Can one prove divergence by showing a series has at most one solution for an=0?

Say I have any series, I would think it was enough to show that this series equals 0 at most once to prove it diverges. My logic is, For a series: $\sum a_n →∞$, and diverges, if $a_n≠0$ for $n→∞$ ...
2
votes
0answers
17 views

Trigonometric equation using angles

For $ \pi<x<\frac{3\pi}{2}$ with $\cos(x)=-\sin(\frac{\pi}{6})$ find the value of $x$ without actually evaluating $\sin(\frac{\pi}{6})$ using complementary angles.
0
votes
2answers
39 views

can this decimal number be converted into a fraction?

Can $$ 0.45647456647456664745666647456666647456666664745666666647456666666647\dots $$ be converted into a fraction of $\frac{N}{M}$ where $N$ and $M$ are integers? I know there is an algorithm ...
2
votes
5answers
39 views

Prove that $X \subset Y \implies f^{-1}(X) \subset f^{-1}(Y)$

Let E and F be two sets and $f: E \to F $ be a function, and $X, Y \subset F$. Prove that $X \subset Y \implies f^{-1}(X) \subset f^{-1}(Y)$ My answer: Let $y \in X$, then $f^{-1}(y) \in ...
2
votes
3answers
58 views

Math about Geometric series

In a geometric series, the sum of $1^{st}$ term $+$ $2^{nd}$ term $+$ $3^{rd}$ term $= 38$, the sum of $2^{nd}$ term $+ 4^{th}$ term $= 17 \frac{1}{3}$; how to calculate the common ratio? ( it is ...
2
votes
1answer
20 views

Checking psd-ness of matrix

I have the following problem and don't know how to proceed... I want to check if \begin{equation} \frac{1}{2}(B^\top A^\top A + A^\top A B) - \frac{1}{4}B^\top A^\top A A^\top (AA^\top ...
0
votes
2answers
33 views

Prove that if A △ C = B △ C, then A = B

I know what I'm supposed to do. Since A △ C = B △ C -> (A-C) U (C - A) = (B- C) U (C - B) ...
3
votes
1answer
50 views

Sophomore's dream changing “x”

"Sophomore's Dream" says $\sum_{n=1}^{\infty}n^{-n}=\int_0^1x^{-x}$ Can you replace the $x$ and $n$ with $2x$ or $x^3$ (and $2n$ or $n^3$) or something? I would guess not, because replacing $x$ with ...
2
votes
3answers
36 views

Does there exist a surjective continuous map $D^2 \to S^1$?

By considering the induced homomorphism on the fundamental groups, we know that there is no retract $D^2 \to S^1$. But is there any continuous surjection from $D^2$ to its boundary? It seems unlikely ...
0
votes
0answers
8 views

Morita equivalence between right and left ideals of a ring

I would like to know whether Morita equivalence is a useful tool when dealing with right and left ideals of a ring. If so, could someone illustrate it on the example of $2\times 2$ matrices? Thanks
0
votes
0answers
21 views

Generalize a result to any category.

Consider two categories $\mathscr{C}$ and $\mathscr{D}$ where $\mathscr{C}= Grp$ and $\mathscr{D}= \textbf{Set}$, then we are taking the forgetful and faithful functor $p$ (this is, we have a group ...
3
votes
0answers
10 views

Which properties determine the uniqueness of the local Artin map?

Any abelian extension of local fields can be realized as the completion of a global abelian extension. So let $L/K$ be abelian, $w/v$ an extension of places. From the global Artin map on ideles we ...
2
votes
1answer
43 views

Linear Algebra Basis and Dimension

Find a basis for the given subspace by deleting linearly dependent vectors. $S = \text{span}\{(0, 0, 0), (9, 0, 0), (8, 1, 0), (1, 8, 9)\}$ I do not understand how to "delete linearly independent ...
3
votes
1answer
32 views

Characterize magic matrices in terms of their eigenvalues. A Magic Matrix over a field $F$ is a square matrix whose row and colums sums $c\in F$.

A Magic Matrix over a field $F$ is a square matrix whose row and colums sums $c\in F$. Characterize magic matrices in terms of their eigenvalues. I know that $c$ is an egenvalue and $[1,...,1]^T$ is ...
-5
votes
1answer
55 views

Assume that $f(t)$ is a known continuous function on $[0,\infty)$and $\lim_{t\to\infty} f(t)=2005$ [on hold]

Assume that $f(t)$ is a known continuous function on $[0,\infty)$and $\lim_{t\to\infty} f(t)=2005$ Consider a 1st order differential equation $dy/dt + 409y = f(t)$ a)Solve and write the general ...
1
vote
1answer
37 views

Cauchy Sequences To Prove $f(z)$ is not continous [on hold]

I've learn ways to prove the discontinouity of a complex function. I have not learn Cauchy Sequences however. I cannot find useful information on the subject. Please explain
2
votes
0answers
12 views

Conversion from 2-dimensional parabolic coordinates to cartesian and cylindrical

I have been looking at the Wolfram Mathworld page on parabolic coordinates here: http://mathworld.wolfram.com/ParabolicCoordinates.html and I'm having trouble grasping how to convert between parabolic ...
4
votes
3answers
88 views

Find the probability that the final score is 4 in a dice game with two throws

A game uses an unbiased die with faces numbered 1 to 6. The die is thrown once. If it shows 4 or 5 or 6 then this number is the final score. If it shows 1 or 2 or 3 then die is thrown again and the ...
11
votes
4answers
374 views

I think I see mysterious lines inside triangles—how to prove their existence?

Lately I've been fooling around with points inside a triangle and the sum of their distances from all sides. This was when I noticed a weird behaviour: For each point I chose there always seemed to ...
0
votes
2answers
25 views

rotation matrix and vector - understand step calculation

I have an extremely equation, but I just don't understand which step they made to get to the last line. ${\bf W}$ and ${\bf V}$ are all 3d vectors. A is a rotation matrix. How did they get that ...
1
vote
0answers
36 views

Uniformly convergence question

I have the following exercise: Let $\varphi:[0,+\infty)\to \mathbb{R}$ an increasing continuous function that satisfies that $1/2\leq \varphi(x) <1$ for all $x>0$. Let $f_0:[0,+\infty)\to ...
0
votes
2answers
22 views

Fourier series coefficient problem

I am having trouble calculating the $a_n$ coefficient for when $n=1$ for the following function. The function $f(x)$ is periodic with period 2 pi, and is defined on the interval $-\pi<x<\pi$ by ...
0
votes
0answers
9 views

Is there a classification of ideals of $\mathcal O_K$ ($K$ quadratic) over ramified and split primes depending on $d \pmod 4$?

I am unsure if the following argument is correct. I have not been taught to use something like this by my lecturer, so I'm a bit skeptical, since this seems like a very simple way of computing norms ...
0
votes
0answers
7 views

Intersection of decomposition groups in abelian extensions

Let $L/K$ be a normal number field extension such that $G = \operatorname{Gal}(L/K)$ is abelian. Further let $P$ be a prime in $\mathcal O_K$ and $Q_1,Q_2$ distinct primes of $\mathcal O_L$ that lie ...
3
votes
0answers
30 views

Positivity of alternating series

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $\limsup_n \frac{1}{n}\log a_n=-\infty$. Then $$ f(x)=\sum_{n\geq 0}a_n x^n $$ converges absolutely for all $x$. Under ...
1
vote
1answer
26 views

How can I find how many unique strings there are with an equal numbers of elements, given a string length and number of elements to choose from?

The question is all in the title. Here's an example: Elements: A, B; Length: 4: AABB ABAB ABBA BABA BBAA BAAB There are 6 such unique strings for 2 elements and ...
1
vote
0answers
13 views

Analytic function with alternating taylor series

Is there a characterization of the sequences of non-negative real numbers $\{a_n\}_{n=0}^{\infty}$ such that $$ f(x)=\sum_{n\geq 0}a_n (-x)^n $$ converges absolutely for all $x\geq 0$? It's not hard ...
4
votes
1answer
69 views

Prove $ \lim\limits_{n\to\infty}\int_0^1 f(x)g(nx)\,dx=\int_0^1 f(x)\,dx\int_0^1 g(x) \, dx $ [duplicate]

Let $f$ and $g$ be a real valued continuous functions on $\mathbb{R}$ such that $f(x+1)=f(x)$ and $g(x+1)=g(x)$ for all $x\in \mathbb{R}$. Prove that $$ \lim_{n\to\infty}\int_0^1 ...
1
vote
0answers
24 views

Calculating a formula for variables with multiple values equaling the same total

I'm having a bit of trouble puzzling a formula for some code I'm using to develop a piece of software. I'm not very savvy with what the technical terms for all of what I'm describing are, but I'll try ...
-1
votes
0answers
12 views

Ratio and Prop0rtion [on hold]

4 skilled workers can do a job in 5 days. 5 Sami-Skilled workers can do the same job in 6 days. How long does it take 1 Sami-Skilled and 2 skilled workers to do the job working together?
3
votes
1answer
29 views

About negligible terms in a limit

When is it valid to deal with a term as a "negligible" one in a limit? I am asking this question because I usually do not take limits very seriously, and I can do a lot of "illegal" moves just to ...
3
votes
1answer
55 views

Has this approximation $0.41468250985111166$ a name?

William Hughes calculated on WolframAlpha the expression $$ \sum_{n=1}^{\infty} \frac{1}{2^{\operatorname{prime}(n)}} $$ and got the approximate value $0.41468250985111166$. If one enters this value ...
2
votes
3answers
44 views

$G$ a group and $H$,$K$ subgroups, $kHk^{-1} \subseteq H \implies kHk^{-1} = H$?

As post said, if $G$ a group and $H,K \leq G$ and for FIXED $k \in K$ does $kHk^{-1} \subseteq H$ imply that $kHk^{-1} = H$ ?
0
votes
0answers
9 views

finding min and max after removing percentage of num values knowing the standard deviation.

I have a question i have some data and i know it's (number of values, min, max, mean and standard deviation) can I know the minimum after removing x% of the total number of values and the maximum ...
2
votes
4answers
88 views

Compute $\lim_{n \to \infty} \left(\frac{1}{\sqrt{n^3+1}} + \frac{1}{\sqrt{n^3+4}} + \cdots + \frac{1}{\sqrt{n^3+n^2}}\right)$

How do I evaluate the following limit? I guess I should do a comparison, but I've got no clue about what to do. Could you give me a hand? $$\lim_{n \to \infty}\left( \frac{1}{\sqrt{n^3+1}} + ...
1
vote
2answers
28 views

Curve fitting the cross sectional area of a cake.

For my final Calculus project I have to find the area of a Bundt cake through the use of cross sectional areas. (Cakeulus) While most seniors in High School who run into this popular calculus project ...
-1
votes
1answer
38 views

About summer course or online course of Linear algebra and real anyasis

I just looking for the online course for Linear algebra or real analysis but it should be upper level. i saw MIT and another college but our university said it was not upper level its likely ...
2
votes
1answer
17 views

Upper bound for the infimum of $(K-x)^2 + (T/x)^2$

I have a function $$f(x)=(K-x)^2 + \left( \frac{T}{x} \right)^2$$ where $K$ and $T$ are positive constants and $x>0$. The function $f$ (hopefully) has an infimum, in terms of $K$ and $T$. I ...
2
votes
4answers
51 views

Is there a book on proofs with solutions?

I am a biochemistry graduate student who works on cancer. I am interested in learning proofs as a personal interest. I use math as a tool, but would like to start building a deeper understanding on my ...
1
vote
1answer
46 views

Finding the real root of the polynomial $2x^3-3x^2+2 $

I want to get exactly roots of this equation... $2x^3-3x^2+2 = 0$ I try to solve it but can not find the solution. wolframealpha just give me aproximation.. I know the real root is $-1< root ...
2
votes
1answer
24 views

Writing the ideal $m=\langle X, Y \rangle$ in $R=k[X, Y]$ as a countable union of prime ideals

Here's a problem (Exercise 3.21) from "A Term in Commutative Algebra" by Altman & Kleiman: Let $k$ be a field, and $R=k[X, Y]$ be polynomial ring in two variables. Let $\mathfrak{m}=\langle ...
9
votes
3answers
537 views

why is 2.2250738585072014e-308 not a number?

In programming the min value of a float is: $$2.2250738585072014e-308$$ but when I type this into a calculator, it says Not a Number. what I am wondering is why ...
0
votes
1answer
13 views

Boolean Alegebra De morgans rule 2

hi i am told to perform a simplification using demorgans rule 2. Here is the question ' = Equals Not B . (C + B')' I got B' + (C' + B'') then B' + (C' + B) Now i dont know where ...
0
votes
3answers
36 views

Complex number $\tan \alpha+i$

Given that $z=\tan \alpha+i$, where $0<\alpha<\frac{1}{2}\pi$ Find $\left |z \right |$. I've never seen this kind of example in my book. Can anyone guide me? Thanks a lot. How to find $arg ...
0
votes
1answer
23 views

Solving for all equations of x trigonometry

Solve for all the values of $x$. $$\tan^2 x=\tan x $$ I don't know how to do this. I've tried similar examples but have failed to get this one.
2
votes
2answers
38 views

Topological proof that the interval $[a,b)\subset \mathbb{R}$ is not closed

I want to prove that the interval $[a,b)\subset \mathbb{R}$ is not closed using the definition that a set $A$ in a topological space $X$ is closed iff its complement $X-A$ is open. Here, the topology ...
0
votes
1answer
17 views

Quick Factoring/Multiplying with recursion question

I am wondering if anyone can help to shed some light on something that I think should be very easy but I dont quite understand. In my textbook, how does author make this conclusion, From $$ ...
0
votes
2answers
36 views

Calculate: $\lim_{n \to \infty} \frac{2^{\sqrt{(\ln n)^2 + \ln n^2}}}{n^2+1}$ and $\lim_{n \to \infty} \frac{10^{\sqrt{(\ln n)^2 + \ln n^2}}}{n^2+1}$

I have to evaluate the following limits (which are similar). However, I don't know how to evaluate them. Could you give me a hand? $$\lim_{n \to \infty} \frac{2^{\sqrt{(\ln n)^2 + \ln ...

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