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2
votes
1answer
16 views

Determining which vectors are solutions of a given system of equations.

Determine which vectors are solutions of the system. \begin{align*} & \hphantom{+}3x-2y-5z = \hphantom{+}4 \\ & \hphantom{+}2x+4y-\hphantom{1}z = \hphantom{+\llap{$0$}}2 \\ & {-}4x-8y+9z ...
3
votes
0answers
56 views

A quick Galois Theory question

Let $\mathbb{F}_q$ be a finite field of order $q$ where $q$ is a prime power. For any $d \in \mathbb{N},$ we have an inclusion $\mathbb{F}_{q^d} \subseteq \overline{\mathbb{F}}_q.$ Both ...
1
vote
0answers
28 views

Checking if this set is a vector sp.

Question: Define a set $V=\{(x,y):x,y\in\Bbb R\}$. For any two elements $u=(u_1,u_2),v=(v_1,v_2)$ in $V$ and $t\in\Bbb R$, addition and scalar multiplication as, ...
0
votes
1answer
24 views

What is the quickest way to find Nash equilibria in two player bimatrix game?

Suppose the cost/penalty matrix of a game is given as: $$M = \begin{bmatrix} (-5,-5) & (0,0) \\ (0,0) & (-3,-3) \end{bmatrix}$$ Then the game as two equilibria $(u_{11},u_{21})$ and ...
3
votes
4answers
82 views

Proof of $(A\cup B)-(A\cap B)=(A-B)\cup(B-A)$

I was trying to prove $(A\cup B)-(A\cap B)=(A-B)\cup(B-A)$ and came across issues in translating (pertaining to what I did with $\emptyset$) and got through the proof but was doubting its accuracy so ...
0
votes
1answer
27 views

Check if algebraic structure is a field

Check if algebraic structure $(\mathbb{R^2},+,\cdot)$ is a field where binary operations $(+)$ and $(\cdot)$ are given by $$(x,y)+(u,v)=(x+u,y+v)$$ $$(x,y)\cdot(u,v)=(xu-2yv,xv+yu)$$ Structure ...
0
votes
0answers
26 views

Rate of change problem. Over which interval is the function increasing? Over which intervals the function decreasing?

I would just like to verify that this solution is correct. Thanks. The table shows the number of animals killed by planes in West Africa from 1986 through 1990. Suppose the number of animals ...
0
votes
0answers
21 views

Convergence of a sequence using Banach contraction principle: $x_{n+1}=\frac{1}{x_n+a},x_1>0,a \in \mathbb{R},n=1,2,…$

$$f(x)=\frac{1}{x+a}$$ $$f:\mathbb{R}_{\le0}\rightarrow \mathbb{R}_{\le0},a<0$$ $$f:\mathbb{R}_{=0}\rightarrow \mathbb{R}_{=0},a=0$$ $$f:\mathbb{R}_{\ge0}\rightarrow \mathbb{R}_{\ge0},a>0$$ ...
1
vote
1answer
80 views

I just need your approval

Verify by susbtitution if the given functions are a solution to the next differential equations. a) $$x^2y''+xy'-y=\ln x \quad ,\quad y_p=x^{-1}-\ln x $$ simplifying: $$ x^2y''+xy'-y=\ln x $$ $$ ...
0
votes
0answers
35 views

Ball shape on different metric spaces and interior set

I am starting to study topology, and to assess if I am on the right track, I kindly ask if someone can check my reasoning below. Let the metric space $(\mathbb{R}^2,d)$, where $d$ is the Euclidean ...
2
votes
0answers
48 views

Artin's Algebra, Exercise 2.4.11. (1st edition)

I have been working through Artin's algebra book. Here's a simple exercise, but I want to make sure I am not missing anything important. Let $(G,\cdot)$ be a group and let $x,y \in G$ with orders ...
0
votes
0answers
39 views

Are these Finance-related calculations correctly solved?

1)A car costs R130 000 is advertised:'no deposit necessary and first payment due three month after date of purchase'.The interest rate is 18% p.a compounded monthly. 1.1)Calculate the amount owing ...
1
vote
1answer
10 views

Investigate the limiting behavior at the origin on the three lines $x = 0 , y = 0$ and $y = x$.

Investigate the limiting behavior at the origin on the three lines $x = 0 , y = 0$ and $y = x$. For the function $f :\mathbb R^2 \to \mathbb R$ given by $$f(x,y)=\begin{Bmatrix} 0, \quad ...
1
vote
0answers
40 views

If $f$ is one to one show that $f(a) \in \partial \Omega$

Let $G$ be a region. Let $a \in G$. Suppose that $f:(G-{a}) \to \mathbb{C}$ is an analytic function such that $f(G-{a})=\Omega$ is bounded. i) Show that $f$ has a removable singularity at $z=a$ ii) ...
1
vote
1answer
49 views

Solve complex equation $5|z|^3+2+3 (\bar z) ^6=0$

I'm stuck in trying to solve this complex equation $$ 5|z|^3+2+3 (\bar z)^6=0$$ where $\bar z$ is the complex conjugate. Here's my reasoning: using $z= \rho e^{i \theta}$ I would write $$ ...
1
vote
1answer
151 views

Can a method related to “Weisfeiler-Lehman Method” provide better time complexity for Graph Isomorphism than existing result?

Cai-Furer-Immerman showed that the W-L(Weisfeiler-Lehman ) hierarchy cannot distinguish general graphs except at linear dimension. Even besides CFI's result, there is good reason to believe that ...
2
votes
1answer
38 views

Complex irreducible representations of the Klein 4 group

I wrote an answer to the following question. Can someone please verify it? Completely and explicitly describe, up to isomorphism, the set of all complex irreducible representations of the Klein 4 ...
2
votes
1answer
40 views

Show that $f\equiv 0$ in $|z|<1$.

Let $f$ be analytic in $|z|<1$ and $f\left(\frac{1}{n^2}\right)=\frac{1}{n}$ , for all $n>2$. Show that $f\equiv 0$ in $|z|<1$. Since $f$ is analytic so Taylor's series expansion of $f$ ...
3
votes
1answer
78 views

Question about Meijer-G definition and identity

I'm trying to wrap my mind around computation involving the Meijer $G$ function, as defined here. (Edit: I'm actually using a somewhat mixed notation using the definitions from MathWorld and the ...
0
votes
1answer
78 views

Taking Limits of Sets

I know this sounds like a ridiculous idea- but it's the only one I can think of for this radius of convergence problem for a power series involving sine. I want to let $P:= \{k:|sin(k)| \geq \delta ...
3
votes
0answers
49 views

Determining prime ideals lying above a given ideal

Let $R=\mathbb{Z}[x]/(f)$, where $$f(x)=x^4+42x^3-11x^2+22x-2002002002002002.$$ Let $I=3R$, the ideal generated by $3$ in $R$. Find all prime ideals of $R$ that contain $I$. I am hoping to ...
3
votes
3answers
62 views

How to solve the trigonometric equation $\sin x-\cos x-2(2)^{\frac 1 2}\sin x\cos x=0$

the question is: Find the solutions of the equation: $\sin x-\cos x-2(2)^{\frac 1 2}\sin x\cos x=0$. Let $\sin x+\cos x=u \text{ and } \sin x \cos x=v \implies \sin^2x+\cos^2x+2\sin x\cos x=u^2 ...
0
votes
2answers
53 views

Convergence of a sequence by induction: $x_{n+1}=\frac{x_n+1}{x_n+a},x_1>0,a>0,n=1,2,…$

Assume that $x_n>0$ and prove $x_{n+1}>0$ $x_{n+1}=1-\frac{a-1}{x_n+a}$ $x_n+a>a$ $-\frac{a-1}{x_n+a}>-\frac{a-1}{a}\Rightarrow x_{n+1}>0$ Is it necessary to find upper bound to ...
2
votes
0answers
16 views

Step response for different definitions of step function

I was thinking about the solution of the known problem of determining the step response for the concentration leaving a CSTR tank. The differential equation is: ...
1
vote
2answers
37 views

Broken Pens: combinations and probabilities

A container hold 50 pens. Exactly 10 pens are broken. What is the chance of finding: a) In a random sample of 10 drawn from the container, 2 or more are broken? b) The last broken pen to ...
0
votes
1answer
40 views

Check if $(S,+)$ is a group where $S=\{\frac{4n+1}{4m+1}:m,n\in\mathbb{Z}\}$

Addition is closed and associative in $S$.There is no neutral element for addition in $S$ because if $e=0\Rightarrow n=\frac{-1}{4} \in \mathbb{Q}$ and also there is no inverse. $(S,+)$ is a ...
3
votes
2answers
42 views

Largest domain where the function $e^z/(\sin z+\cos z)$ is analytic

I have a function $f(z) = \frac{\exp{z}}{\sin z+\cos z}$ and I need to show the region where $f(z)$ is analytic. My work so far :- As the function is the sum and product of holomorphic functions, I ...
0
votes
2answers
53 views

Showing a function space is infinite dimensional

Let $C_\infty (\mathbb R) = \{ f \in C(\mathbb R) \mid \lim_{|x|\to \infty} f(x) = 0 \}$ and Let $C_0(\mathbb R) = \{ f \in C(\mathbb R) \mid f \text{ has compact support} \}$. I want to show that ...
2
votes
4answers
34 views

How do you solve for θ in the equation $\tan \frac{\theta}{5} + \sqrt{3} = 0$

$$\tan \frac{\theta}{5} + \sqrt{3} = 0$$ Alright so the $\frac{\theta}{5}$ is confusing me. Would it be wrong to do \begin{eqnarray} \tan \frac{\theta}{5}&=&-\sqrt{3}\\ ...
4
votes
1answer
54 views

Some questions about an exercise about $C^\infty \subset L^\infty$

Let $$ L^\infty (\mathbb R) = \{f : \mathbb R \to \mathbb C\mid \text{essential sup of } f < \infty \text{ and } f \text{ Borel measurable} \}$$ and $$ C^\infty (\mathbb R ) = \{ f: \mathbb R ...
1
vote
1answer
69 views

Finding Automorphisms of Irregular graph through Regular Sub-Graphs.

Objective : To find a set of permutations for a irregular graph which is also a set of automorphism. This finding process uses permutations of 2 regular subgraphs of the given graph. Description and ...
0
votes
2answers
67 views

How many permutations of the word TOMORROW can be made if the O's can't be together?

I'm trying to answer this question. This is my attempt of solution: First we distiguish the O's and R's, then we have the word: $TO_1MO_2R_1R_2O_3W$. We have $8!-7!\cdot3!-6!\cdot 3!$ different ...
1
vote
1answer
31 views

Probability of 2 students being chosen the both have under 100 books at home

Suppose we select two students at random from the class of fifteen. What is the probability that both students chosen have less then 100 books at home? Data provided is the amount of books each ...
1
vote
0answers
27 views

Remainder Estimate for Integral test

I have the following question, it is a fill in the blank type question, however when I submit my answer, the system which verifies it say it is incorrect. I believe I am right, so I was hoping for ...
1
vote
3answers
58 views

Right answer, wrong explanation, probability of grids?

Two unit squares are selected at random without replacement from an $n\times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected squares are ...
3
votes
4answers
91 views

$f(x) = \frac{2}{x}$ is not uniformly continuous on $(0,1]$

Show that $f(x) = \frac{2}{x}$ is not uniformly continuous on $(0,1]$ WLOG Suppose, $0< \delta \leq 1.$ Let, $\epsilon = 1$ and $x = \frac{\delta}{2}, y = x + \frac{\delta}{3}, x,y \in (0,1]$ ...
0
votes
1answer
37 views

Show that $y=\frac{4\sin\theta}{2+\cos\theta}-\theta$ is increasing function when $\theta \in [0,\frac\pi2]$

Show that $$y=\dfrac{4\sin\theta}{2+\cos\theta}-\theta$$ is increasing function when $\theta \in [0,\frac\pi2]$ What I have done If $\theta_1,\theta_2\in[0,\frac\pi2]$ then $$\sin\theta_1 < ...
0
votes
1answer
21 views

Finding a power series solution for a given differential equation and identifying the function represented by the power series.

Find a power series for the solution of the differential equation $y'(t)-2y(t)=0 ,\ y(0)=5$, and then identify the function represented by the power series. (I use the following information ...
1
vote
2answers
127 views

Find $\lim_{n \to \infty} n[(1+\frac{1}{n})^n - e]$

$\lim_{n \to \infty} n[(1+\frac{1}{n})^n - e]$ I let, $x = \frac{1}{n}$, then as $\lim_{x \to 0} \frac{1}{x}[(1+x)^\frac{1}{x} - e] = \infty$ L'hopital's: $\lim_{x \to 0} ...
2
votes
3answers
48 views

Uniform convergence of $f_n(x) = x^n$ on $[0,c]$

Let $c \in (0,1)$ be fixed. Let $$f_n(x) = x^n,\quad x \in [0,1)$$and$$f(x) = 0,\quad x \in [0,1)$$ Show that $f_n$ converges uniformly to $f$ on $[0,c]$. So, we have, $f_n(0) = 0, f_n( c) = ...
0
votes
1answer
15 views

solution verification: find characteristic of integral domain under given conditions

Okay, so this seems an easy problem, but I was having doubts if my solution was correct or not. I would really appreciate if somebody could verify it for me. Suppose $R$ is an integral domain such ...
1
vote
1answer
31 views

Taylor Approximation of $\cos(0.02)$

Use a Maclaurin $(a=0)$ polynomial for $\cos{(x)}$ with $3$ nonzero terms to approximate $\cos{(0.02)}$. Also, use the Taylor Remainder Theorem to find a bound on the error $\left(\displaystyle ...
3
votes
1answer
59 views

Find $U(P,f,\alpha)$ and $L(P,f,\alpha)$, where $f(x) = 1+x^2$

Given $f(x) = 1+x^2, \alpha(x) = x^3, x \in [-1,1], P = \{-1,\frac{-1}{2},0,\frac{1}{2},1\}$. Find $U(P,f,\alpha)$ and $L(P,f,\alpha)$. Ok, for this problem, we have $\alpha (x) = x^3$. And, I am ...
1
vote
0answers
16 views

Please could someone check and help me with my answer to part two of this exercise about vector fields along maps?

I previously solved the following (first half of an) exercise: Let $U$ be an open neighbourhood of $0 \in \mathbb R^2$ and let $f = (f_1, f_2) : U \to \mathbb R$ be a smooth map such that $f(0) = ...
0
votes
0answers
19 views

Please would someone check my answer to this exercise on vector fields along maps?

I believe I solved the following exercise and would appreicate it greatly if someone could check my answer: Let $U$ be an open neighbourhood of $0 \in \mathbb R^2$ and let $f = (f_1, f_2) : U \to ...
2
votes
1answer
52 views

Prove that $a_j=b_j$ for the $2$ sequences $a$ and $b$

Let $n$ is a natural number and $(n,6)=1$. Given $2$ sequences $a$ and $b$ such that $a_1>a_2>\ldots a_n$ and $b_1>b_2>\ldots b_n$. And for all $1 \leq j < k <l \leq n$, it is ...
3
votes
2answers
29 views

Show that $\frac{1}{b-a_{1}} + \frac{1}{b-a_2}+ …+ \frac{1}{b-a_n} \geq \frac{n}{b-\frac{1}{n}(a_1+a_2+…+a_n)}$

Let, $b> \max\{a_1,a_2,...,a_n\}.$ Show that $\frac{1}{b-a_{1}} + \frac{1}{b-a_2}+ ...+ \frac{1}{b-a_n} \geq \frac{n}{b-\frac{1}{n}(a_1+a_2+...+a_n)}$ f is convex if ...
1
vote
2answers
39 views

Upper and lower Riemann sums problem

Let $c > 0$ and $f(x) = x, x\in [0,c].$ Let $P = \{x_0, x_1, x_2,...,x_n\}$ be a partition of $[0,c]$ with $x_i = \frac{i}{n}c, i = 0,1,2,...,n.$ Find $U(P,f).$ Find $\lim_{n \to \infty} U(P,f).$ ...
1
vote
1answer
32 views

Evaluate products and sums

Did I evaluate the following terms correctly? Does the set notation in example b) allow me to chose the order of the terms? $$ a) \sum_{i=1}^6 ix^{i+1} = x^2+2x^3+3x^4+4x^5+5x^6+6x^7 \\ b) \prod_{i ...
2
votes
3answers
30 views

How fast is the village clock.

A man started from home at 14:30 hours and drove to a village, arriving there when the village clock indicated 15:15 hours. After staying 25 minutes, he drove back by a different route of length (5/4) ...