For posts looking for feedback or verification of a proposed solution.

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1answer
23 views

Dijkstra's Algorithm for Negative Weights.

Now the problem states that their is a graph $ G = (V,E) $ where some of the edges have negative weights while some of the edges have positive edges. Now the question is why won't Dijkstra's algorithm ...
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2answers
42 views

Let $R$ be a finite ring with unity. Prove that $x$ is a LZD $\iff$ x is a RZD

Let $R$ be a finite ring with unity. Let $x \in R$. Prove that $x$ is a Left Zero Divisor $\iff$ x is a Right Zero Divisor. My attempt Suppose $x$ is a LZD. Then, $\exists y \in R$ such that $xy = ...
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0answers
38 views

A ring with a left cancellable element and a right identity always has an identity.

Let $R$ be a ring with $a, e \in R$ such that $a$ is not a left zero-divisor and $be=b, \forall b \in R$. Prove that $R$ has an identity. My attempt Let, $aeb = ab \Rightarrow aeb - ab = 0 ...
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23 views

Verification: Let $R$ be a unique factorization domain and let $a,b \in R$ be nonzero, non-unit elements. Prove that $a$ and $b$ have a LCM

Let $R$ be a unique factorization domain and let $a,b \in R$ be nonzero, non-unit elements. Prove that $a$ and $b$ have a least common multiple, and describe one such multiple in terms of the ...
2
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1answer
19 views

Finding a sequence of step functions converging uniformly to $f(x)=\sum_{n=0}^\infty 2^{-n}\mathbb{1}(x>q^n)$

As I was revising my Real Analysis course, I came across this strange problem on series of functions. If anybody can verify that the first and last parts of the proof are correct and give me hints ...
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2answers
149 views

Proving $x^{2}+1 \neq n! $,using Gaussian Integer.

I want to show that $$x^{2}+1 \neq n! $$ for $n>3$ where $x,n$ are both integers. Since $$x^{2}+1=(x+i)(x-i) $$ it follows that $x^{2}+1$ has only prime factors on the form ($4k+1$), whereas $n!$ ...
2
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1answer
45 views

Show that $\det \rho(g)=1$ for elements g of odd order and $\det \rho(g)=-1$ for elements g of even order.

$G$ is a finite group. $\rho$ is a representation of $G$, then $\rho \mapsto \rho(g)$ is a $1$-dimensional representation. Show that if $\det \rho(g)=-1$ for some $g \in G$ then $G$ has a ...
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1answer
27 views

For which $p$ the sequence $x^n$ converges in the Sobolev space $W^{1,p}(I)$?

I would like to know for which $p$ the sequence $u(n)=x^n$ converges in the Sobolev space $W^{1,p}(I)$. Is it true that converges only for $p=1$? I find out this looking for which $p$ the Sobolev ...
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1answer
25 views

Finding concavity using a second derivative that is never greater than zero.

When is $\frac{3x-8}{4(2-x)^{3/2}} > 0$? The equation above is the second derivative of the function: $$f(x) = x\sqrt{2-x}$$ I am wanting to find the concavity of the original function. I know ...
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1answer
56 views

calculating the limit of $\int_0^1 \frac{n \cos(x)}{1+x^2n^{1.5}}dx$

i want to calculate $$\lim_{n \to \infty} \int_0^1 \frac{n \cos(x)}{1+x^2n^{1.5}}dx$$ I think this integral is bounded from below by $$n\int_0^1\frac{\cos(1)}{1+x^2n^{1.5}}dx$$ and this equals ...
2
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1answer
59 views

What did I do wrong trying to find this limit?

In another question, a user asked to find: $$\lim_{x\to 0} \frac{\exp(x^2)-\cos(x)}{\sin(x)^2}$$ I thought I could use pure trigonometric identities to find the limit. Apparently I was mistaken, but I ...
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1answer
34 views

If $|S|>1$, then $M(S)$ is not a group with respect to composition. Why?

Question: If $|S|>1$, then $M(S)$, the set of all mappings from $S$ to $S$, is not a group with respect to composition. Why? My answer: Any mapping that maps more than one element from the domain ...
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2answers
59 views

Finding an operation on $G^S$ that yields a group

Problem: Assume $S$ is a nonempty set and $G$ is a group. Let $G^S$ denote the set of all mappings from $S$ to $G$. Find an operation on $G^S$ that will yield a group. Update (full attempted ...
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0answers
8 views

Correct mathematical notation for showing the selection procedure for multiple equation solution

I have an equation that solving it results in 4 solutions for $\theta$. Only one of these solutions is correct. So, in order to select the correct $\theta$, I use a function that I already know the ...
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2answers
17 views

Picking Unique Balls from a Bin

Problem: We have a bin with 5 red balls, 7 green balls, and 9 blue balls. We draw 3 balls out of the bin, without replacement. What is the probability that no two of the three balls have the ...
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0answers
22 views

Problem with the primary decomposition theorem

I need to use the primary decomposition theorem in the linear transformation $T:\mathbb R^3 \to \mathbb R^3$ defined by the matrix $\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -6 & ...
2
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1answer
59 views

Question on Graph Connectivity

Now this question is on graph connectivity and I still can't get my head around these graph theory questions. Now we've been given an n-node graph which is represented by $G=(V,E)$ and two nodes $s$ ...
3
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1answer
25 views

Number of Possible Pairs from Two Sets

The Question: We have 18 distinct gadgets and 22 distinct widgets. We want to pick five pairs, each with one gadget and with one widget. In how many different ways can the five pairs be ...
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1answer
30 views

Minimize the cost of a 3 cubic unit volume box, given the price of each of six sides per unit area

I was wondering if anyone could help verify my answer of a question, or if it is incorrect to maybe let me know my mistake? The questions asks to minimize the cost of a 3 cubic unit volume rectangle ...
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1answer
64 views

Question related to Directed Acyclic Graphs

In an assignment I got a question, "Show that the strongly connected component of a DAG is also a DAG." Now I wasn't able to solve this. The problem I faced with this question was that the DAG is ...
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1answer
33 views

Topology, Showing that two metric spaces are topologically equivalent

Can someone verify if this is true? $X=\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ . $d$ is the standard metric on $\mathbb{R}$ and $d'(x,y)=d\left(\tan(x),\tan(y)\right)$ . We want to show that ...
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1answer
26 views

How to find periodic continued fraction expansion of $\frac{\sqrt{7}}3$

How to find periodic continued fraction expansion of $\frac{\sqrt{7}}3$ Using this formula here (it begins in the middle of the page), I obtained $\frac{\sqrt{7}}3=[0;1,\overline{7,2}]$ but ...
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0answers
36 views

roots of $z^7 − 2z^5 + 6z^3 − z + 1 = 0$ in the disc $|z| < 1$?

The Q is How many roots does the equation $z^7 − 2z^5 + 6z^3 − z + 1 = 0$ have in the disc $|z| < 1$? My approach : To count the roots of $f$ in the unit disk $D =\lbrace \vert z \vert < 1 ...
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0answers
37 views

Prove/disprove that if $(n a_n)$ is bounded then $(a_n)$ converges to zero

I need to prove or disprove the following statement: If $(n a_n)$ is a bounded sequence, then $\lim_{n\to\infty}(a_n)=0$ I think the statement is true but I'm not sure about my proof: Because ...
1
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1answer
18 views

Is my reasoning correct for the following linear congruence?

I am given 2x = (1+3i) mod (3+8i). Here are my steps: I found that 1 = (-1)(3+8i) + 2(2+4i). I took mod (3+8i) and got 2(2+4i) = 1 mod (3+8i). Then I took the product (2+4i)(1+3i) and got -10 + 10i. ...
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2answers
148 views

Fundamental unit in the ring of integers $\mathbb Z[\frac{1+\sqrt{141}}{2}]$

Find a fundamental unit in the ring of integers $\mathbb Z[\frac{1+\sqrt{141}}{2}]$ of $\mathbb Q(\sqrt{141})$ I have different corollaries for different numbers, the most appropriate for $141$ ...
2
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1answer
17 views

Prove $U = \{u(x) \in P_4 | u(0) + u(1) = 0\}$ is a subspace of $P_4$

I am trying to prove $U = \{u(x) \in P_4 | u(0) + u(1) = 0\}$ is a subspace of $P_4$ For U is nonempty I have: Let $u(x) = 0x^4 + 0x^3 + 0x^2 + 0x + 0$ For U is closed under $+$ I have: Let $x, y ...
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0answers
186 views

$\pi(z)-\omega(z-1)-\{-1,0,1\}= \pi(2z-1)- \pi(z)$ when $z(z-1)$ is divisible by all primes ${<}\sqrt{z}$

I have encountered the below problem: Given $z(z-1)$ divisible by all primes ${<}\sqrt{z}$ (and the prime factors of $z(z-1)$ are consecutive primes), prove (or disprove) ...
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0answers
22 views

Is this method correct for working out these probabilities?

I'm doing this question and wanted to see if this was the correct method. My answer a) $X \sim N(8,16) \rightarrow Z=\frac{X-\mu}{\sigma}\sim N(0,1)$ then manipulating what you want you find ...
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0answers
85 views

$(z-k)$ is composite then $(z-1)+(k-1)$ is also composite(A proof for composite number).

Given $z(z-1)$ is divisible by all prime $< n$ where $ n>\sqrt z$ $(z+k)$ is prime. Prove or disprove if $(z-k)$ is composite then $(z-1)+(k-1)$ is also composite. ...
1
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1answer
32 views

Higher Order Derivatives problem involving the position of a particle

Given the position of a particle is $$s(t)=t^3-12t^2+36t-20$$ a. Find the velocity and acceleration functions b. When is the particle moving to the right? c. When is the particle ...
2
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3answers
64 views

Solve $(y')^2=(y/c)^2-1$

Can someone help me solve $(y')^2=(y/c)^2-1$? WolframAlpha is giving me $\frac 12(c^2 e^{(x/c)-k}+e^{k-(x/c)})$. One book I have is giving me $y=c\cdot \cosh(\frac {x+b}c)$ -- but that one won't ...
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1answer
30 views

Why am I getting two different answers to this simple calculus of variations problem?

A worker is disposing of radioactive material of mass $\mu$ and needs to minimize her exposure. Being near the radioactive material exposures her to radiation at a rate of $\frac {dE_n}{dt}=c\mu$, ...
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0answers
25 views

Help on Proof By Induction

We had to prove the following algorithm by induction: $ a^n = a^{n/2*2} = a^{n/2}*a^{n/2} $ if $n$ is even $ a^n = a^{\frac {n-1}2*2}*a = a^{\frac {n-1}2} * a^{\frac {n-1}2} * a $ if $n$ is ...
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2answers
61 views

Find $y'$ for $ln(x+y)=arctan(xy)$

Find $y'$ for $ln(x+y)=arctan(xy)$ Here is my attempt at a solution. Is this correct? Any hints or advice would be appreciated.
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1answer
28 views

Find the slope of the tangent line for $y^4+3y-4x^3=5x+1$ at the point P(1,-2)

Find the slope of the tangent line for $y^4+3y-4x^3=5x+1$ at the point $P(1,-2)$ Here is what I have tried. Does this look correct? Any hints or advice would be appreciated. EDIT 1: Here is ...
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0answers
31 views

fundamental theorem of calculus and chain rule

Let $f:[0,1] \to \mathbb{R}$ be a differentiable function with $f(0) = 0$ and $f'(x) \in (0,1)$ for every $x \in (0,1)$. I want to take the first and second derivative of $F(x) = ...
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0answers
35 views

Find the slope at $t=16$ for $s(t) = $arctan$(\sqrt{t})$

A particle moves along the x axis so that its position at any time when t is greater than or equals zero is $s(t) = $arctan$(\sqrt{t})$. Find the velocity of the particle at $t=16$. The point of ...
1
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1answer
25 views

Critical points of a function $f(x) = x\sqrt{x-a}$

Find the critical points of a function $f(x) = x\sqrt{x-a}$. A function $f(x)$ is said to have critical points at points $c$ such that $f^\prime(c)$ is $0$ or undefined. For a function $f(x) = ...
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0answers
24 views

How can I verify the result of modular exponentiation

I ask a computer to calculate $x^y \pmod z$, where $x,y,z$ are all large numbers. How can I verify the correctness of the result returned by the computer. I assume that I myself cannot afford ...
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0answers
25 views

Prove that the given linear transformation is bijective

Suppose that $T:P_2(\mathbb{R})\to P_2(\mathbb{R})$ given by $T(p(x))=p(x+1)$ Let $p(x)$ and $q(x)$ be elements in $P_2(\mathbb{R})$ So if $T(p(x)) = T(q(x))$ then $p(x+1) = q(x+1)$ then $p(x) = ...
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2answers
27 views

Related Trigonometric Rates

A port and a radar station are $3 \text{ mi}$ apart on a straight shore running east and west. A ship leaves the port at noon traveling at a rate of $13 \frac{\text{mi}}{\text{hr}}$. Find the rate of ...
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1answer
23 views

Verification of a certain identity in wavelet basis lemma.

This is from Lemma 7.1 in Mallat's Wavelet Tour 2nd edition. I am trying to show that $$ b(2x)h(x) + c(2x)g(x) = a(x) $$ when \begin{align*} b(2x) &= \frac{1}{2}\left[ a(x)h(x)^* + ...
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0answers
9 views

Conical Related Rates

Sand falls from a conveyor belt at a rate of 11 $\frac{\text{m}^3}{\text{min}}$ onto the top of a conical pile. The height of the pile is always three-eights of the diameter of the base. Give the rate ...
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0answers
21 views

Related rates with similar triangles

A $6 \text{ ft}$ tall man walks at $6.00 \text{ ft/s}$ toward a street light that is $16.0 \text{ ft}$ above the ground. Find at what rate the end of the man's shadow moving when he is $6.0 \text{ ...
2
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2answers
49 views

$T$ linear operator s.t. $\lim\limits_{n\to\infty}x_n{=}0_X$ $\Longrightarrow$ $\lim\limits_{n\to\infty}T(x_n){=}0_Y$ then $T$ is bounded

Let $X$ and $Y$ be two normed spaces, with $X$ a reflexive space. I suppone that $T:X\rightarrow Y$ is an operator such that: $\lim\limits_{n\to\infty}x_n{=}0_X$ $\Longrightarrow$ ...
1
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0answers
19 views

Differentiability of Function of Two Variables

Define $g(x, y) := (|x| + |y|)^{1/2}$. Find those points in $\Bbb R^{2}$ at which $g$ is differentiable. My Idea of a Solution: In the $1^{st}$ Quadrant, $g(x,y)= (x + y)^{1/2}$, which is a ...
4
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1answer
27 views

Determining te probability that a message can not be corrected

A bit error occurs with probability $10^{-7}$ . A message consists of 8000 bits. Upto three bit errors can be corrected at the receiver with FEC (Forward Error Correction) code in the message. ...
0
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0answers
19 views

Trying to determine the relationship of m and n in a Casting Out m under base n

While exploring $\mathbb{Z/n}$ I stumbled upon this It explains that Casting Out Nines works because our common counting system is decimal and thus there exist a congruence relation as follows ...
1
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3answers
49 views

Evaluating $\int_0^3|3x-1|\,dx$.

The definite integral I have to evaluate is $$ \int_0^3|3x-1|\,dx. $$ The answer I got was $13.5$ only because I solved for the area after where the graph of the function starts to increase after ...