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0
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1answer
21 views

Find a matrix whose column space contains the column space of the given matrix.

Let $$A = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix}\text{.}$$ $C(A)$ denotes the column ...
0
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1answer
25 views

Do the spaces spanned by the columns of a matrix and by the columns of a set of matrices coincide?

As in Do the spaces spanned by the columns of the given matrices coincide?, let $$A = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ ...
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0answers
72 views

verify that the set $\{0,1,2,3\}$ is not a group under multiplication modulo $4$

Given the set $\{0,1,2,3\}$: -Associativity holds for this set -Closure holds for this set (constructing the Cayley table, all entries in the tables are in this set). -there is an identity element ...
0
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1answer
40 views

Check My Work on a Poisson Process/Distribution Question

I'm just curious if my work is correct, and if not, where I made a mistake. My Task: Cars arrive according to a Poisson process with a rate of 12 per hour. (1) What is the probability that the ...
2
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2answers
29 views

Do the spaces spanned by the columns of the given matrices coincide?

Reviewing linear algebra here. Let $$A = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix} \qquad ...
1
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2answers
25 views

Show that properties of norm are satisfied

Show that \begin{align} & \|y\|_M= \max_{a \leq x \leq b} |y(x)| \tag 1 \\[8pt] & \|y\|_1=\int_a^b |y(x)|\, dx \tag 2 \end{align} satify the properties of a norm in $C[a,b]$. That's what I ...
1
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2answers
38 views

Does $(f_n)$ converge pointwise/uniformly on $I$?

Does $(f_n)$ converge pointwise/uniformly on $I$ if $$f_n(x) = \frac{x^n}{1+x^n} ~~~~~~ I=[0,1]$$ My attempt: If $x \in [0,1): \displaystyle \lim_{n \to \infty}f_n(x) = 0$ If $x=1: ...
2
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1answer
26 views

How to answer this graph theory question?

Okay so let me define some terms before I ask my problem: Let $K_n$ denote the complete graph on $n$ vertices with $n\geq 2$ and let $C_3$ be a cycle of length $3$ (a triangle). Suppose $x,y,z$ ...
3
votes
1answer
38 views

Laurent Series Expansion of $\frac{-3z^2+8z+1}{(z-2)(z^2+1)}$

Laurent Series Expansion of $\frac{-3z^2+8z+1}{(z-2)(z^2+1)}$ on the annulus $A(1,2)$ I think $A(1,2)$ denotes the set $\{z:1<|z-0|<2\}$, so it excludes the poles. using partial fraction ...
1
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0answers
32 views

What are the solutions to this equation? (hyperboloid)

Equation: $$(x - y - z ) A - (x^2 - y^2 - z^2)=0$$ I am trying to find all the possible solutions for the equation above. $A$ is a real strictly positive constant , $A>0$. $x,y,z$ are non ...
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2answers
25 views

evaluating the double integral

I tried to calculate $\int _0^9 dx\:\int _{-\sqrt{x}}^{\sqrt{x}}\:y^2dy$ which yielded $c$ as in this integral has no particular value...when I plot the graphs for it's D however, a certain area does ...
3
votes
2answers
27 views

Computing line integral using Stokes´theorem

Use Stokes´ theorem to show that $$\int_C ydx+zdy+xdz=\pi a^2\sqrt{3}$$ where $C$ is the curve of intersection of the sphere $x^2+y^2+z^2=a^2$ and the plane $x+y+z=0$ My attempt: By Stokes´ theorem ...
3
votes
1answer
49 views

Is it necessary for $A$ to be symmetric with non-zero determinant?

Today, in a Differential Geometry test, I was asked to prove that: $$S:=\{x \in \mathbb{R}^3: x^TAx+c=0\}$$ where $A$ is a symmetric $3\times 3$ matrix and $c \in \mathbb{R}$ is a regular surface ...
1
vote
1answer
14 views

Probability of event in join sample space of X & Y

From what I understand, the answer should be $(0.1+0.35+0.05)$, since the given points have probability summation $1$. Am I correct? By the way, the correct answer unknown.
1
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4answers
60 views

Solve a linear equation .

The equation I'm trying to solve is $x+2=x$. Here's what I tried: $(x+2)^2=x^2$ $x^2+4x+4=x^2$ $4x=-4$ so $x=-1$
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2answers
37 views

Derive the solution to $\frac{dQ}{dt} = kQ$

Derive the solution to $\frac{dQ}{dt} = kQ$ in terms of $Q_0$ Here is my work: $\frac{dQ}{dt} = kQ$ $\frac{dQ}{Q} = kdt$ $\int\frac{dQ}{Q} = \int kdt$ $lnQ = kt + C$ $Q = e^{kt}e^{C}$ Did I ...
0
votes
1answer
17 views

Population Growth Word Problem Using the Law of Natural Growth

The problem is included in the image below. There are three parts to the problem, and all three are on the same page. I am looking for solution verification on all three parts, but I have a specific ...
0
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1answer
13 views

Law of Natural Growth World Problem: How many years will it take to sell 100 franchises?

Pizza Unlimited is a national pizza firm and is selling franchises throughout the country. The president estimates that the number of franchises N will increase at a rate of $15$% a year, that is, ...
10
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1answer
92 views

Something Isn't Right With My Parking

A few days ago in my Calculus BC class we were given a page of 6 challenging end of the year problems. That was a refreshing change from the drudgery we usually do (WebAssign). One of them went like ...
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3answers
29 views

Solution check: Let $f_n$ be fibonacci numbers. Prove: $\sum_{k=0}^{n-1} \binom{n+k}{2k+1} = f_{2n-1}$ and $\sum_{k=0}^n \binom{n+k}{2k} = f_{2n}$

The question: Let $f_n$ be fibonacci numbers. Prove: $\sum_{k=0}^{n-1} \binom{n+k}{2k+1} = f_{2n-1}$ and $\sum_{k=0}^n \binom{n+k}{2k} = f_{2n}$ For every $n\in N$. $f_0=f_1=1$, ...
0
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1answer
30 views

Equivalence relations regarding binary relations

Let $R \subseteq X \times X$ be a binary relation for $X = \{a, b, c, d\}$. $R = \{(a, a), (b, c), (c, d), (b, d)\}$. Is the relation an equivalence relation? I don't know if I am proving it correctly ...
3
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1answer
47 views

Two chess players, A and B, are going to play 7 games. There are three possible outcomes for each game, A wins, A loses, or Tie

Two chess players, A and B, are going to play 7 games. There are three possible outcomes for each game, A wins, A loses, or Tie Addtionally, a win is worth 1 point, draw 0.5 points and loss 0 points. ...
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0answers
26 views

Storing k people's phone numbers in n locations using hash function.

Assuming no two people have the same name and locations are chosen at random, what is the probability that at least one location has more than one phone number stored there? (From Introduction to ...
2
votes
2answers
49 views

Probability over an infinite set. Of two numbers.

Suppose there is a man, who chooses two completely random numbers. So they can be equal too. And they can be only positive real numbers. One of them can even be $285294.38285967281$ or anything ...
2
votes
1answer
20 views

Which of the following collections are topologies for $\mathbb R$? Am I correct?

Which of the following collections are topologies for $\mathbb R$? I think I have these correct I just want to double check my answers. (a) $\{\mathbb R, \emptyset, (-\infty, 0), (0,\infty)\}$- Yes, ...
1
vote
1answer
42 views

Let $f:\mathbb R \rightarrow \mathbb R$ be given by $ f(x) = x^2 -3$

Let $f:\mathbb R \rightarrow \mathbb R$ be given by $ f(x) = x^2 -3$ Find $f([-2,1])$= $[-3,1]$ Find $f^{-1}([-2,1])$= $[-1,1]$ I am not wonderful at these types of problems and I seem to make ...
1
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0answers
12 views

Proving that a certain function is an integral of motion for a Hamiltonian

Let $H=q_1p_1-q_2p_2-aq_1^2+bq_2^2$ (with $a,b$ constant) be a Hamiltionian. Show that $G=\dfrac{p_1-aq_1}{q_2}$ is a first integral (integral of motion) of this system. According to the ...
2
votes
2answers
56 views

Solve the equation -

Solve $$ 3-\frac{4}{9^x}-\frac{4}{81^x}=0 $$ I had this question for an exam today and I want to find out if my answer was correct.
2
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2answers
33 views

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. Then 1. $Cl(A) = Cl(Int(A))$ 2. $Int(A) = Int(Cl(A))$

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. Then $Cl(A) = Cl(Int(A))$ $Int(A) = Int(Cl(A))$ I believe both of these statements are false and I think I ...
0
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0answers
51 views

Finite solution of Power Diophantione Equation.

Given an equation $x^2+k=y^3$ where k is a constant and $y=f(x)$,$f(x)$ is differentiable and algebraic. for which- $$\frac{d}{dx}x^{2} \neq\frac{d}{dx} f(x)^3$$ 1. Can I infer that the ...
0
votes
2answers
47 views

There is an element, whose order is the exponent of $H$

If $H$ is a subgroup of $K^*$, where $K$ is an arbitrary field, then there is an element $h\in H$, whose order is the exponent of $H$, that is the least common multiple of the elements of $H$ I ...
1
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0answers
25 views

Evaluate this integral containing a piecewise function.

Please see the image for the problem. I am unfamiliar with integrating piecewise functions correctly, so I would like verification for this problem. Did I get the correct answer?
0
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1answer
45 views

Integral involving trigonometric functions and a second variable

Integrate the following: $$\int_0^{\pi/2}(1+\sin\theta)^5\cos\theta dx$$ To solve this, I integrated as per usual, as shown in my work below. However, I'm now doubting myself- I'm concerned ...
0
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2answers
57 views

Proving something it NOT and integral domain

Let $R$ and $S$ be two commutative rings with unity. Prove that $R\times S$ is NOT an integral domain. This is the best I could think of so far, please give me a push in the right direction and ...
0
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2answers
34 views

Probability Urn

Suppose that an urn has 1 green ball, 1 yellow ball, 1 blue ball and 1 red ball. You draw 4 with replacement. What is the probability that you draw exactly two are exactly the same color? So I ...
1
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2answers
42 views

Finding a nullspace of a matrix.

I am given the following matrix $A$ and I need to find a nullspace of this matrix. $$A = \begin{pmatrix} 2&1&4&-1 \\ 1&1&1&1 \\ 1&0&3&-2 \\ ...
1
vote
1answer
26 views

SAT2 Level 2 Book Answer Error

I am currently studying for my SAT2 Subject Test in Mathematics Level 2 and was check my answers to a practice test when I can across this (below) question. Problem: George invests $\$1000$ into ...
0
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1answer
18 views

Proof using smaller step size and increasing step, Euler method tend to exact solution(solution verification)

Please help to verify is the proof below contain any error. I start by considering a differential equation $\frac{dy}{dt}=f(t)$ and using a step size of $\frac{h}{n}$ where n is consider to be a very ...
4
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3answers
39 views

Let $a \leq x_{n} \leq b$ for all n in N. If $x_{n} \rightarrow x$. Then prove that $a \leq x \leq b$

Let $a \leq x_{n} \leq b$ for all n in N. If $x_{n} \rightarrow x$. Then prove that $a \leq x \leq b$ Attempt - If I assume that $x$ is greater than both $a$ and $b$. Then since series is given ...
1
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0answers
24 views

$X + Y \overset{\mathcal{D}}{=} X \Longrightarrow \mathbf{P}[Y = 0] =1$

Let $X$ and $Y$ be independent, real random variables. Show that $X + Y \overset{\mathcal{D}}{=} X$ implies that $\mathbf{P}[Y = 0] =1$. Note: $U \overset{\mathcal{D}}{=} V$ means that the ...
2
votes
1answer
35 views

Show for any prime $p$ and $a \in \mathbb{F}_{p}$ that $x^p-a$ has multiple roots

Show for any prime $p$ and $a \in \mathbb{F}_{p}$ that $x^p-a$ has multiple roots using the derivative of $x^p-a$ which is $px^{p-1}$ if they are relatively prime then $x^p-a$ only has simple roots. ...
1
vote
1answer
24 views

Did I do this Continuous Probability Problem Correctly?

I'm new to evaluating continuous probability density functions. I'd like someone to check my work, please. Problem: Suppose $X$ has density $f(x) = c/x^6$ for $x>1$ and $f(x) = 0$ otherwise, ...
1
vote
3answers
37 views

Finding quadratic factors

Show that $(x-√3)$ and $(x+√3)$ are factors of $x^4+x^3-x^2-3x-6$. Hence write down one quadratic factor of $x^4+x^3-x^2-3x-6$, and find a second quadratic factor of this polynomial. My attempt: ...
0
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3answers
25 views

Find the values of the polynomial equation.

Find the values of A, B and C such that $A(x^2+4)+(x-2)(Bx+C)=7x^2-x+14$ My attempt to solve the question: $Ax^2+4A+Bx^2+Cx-2Bx-2C=7x^2-x+14$ $Ax^2+Bx^2+Cx-2Bx+4A-2C=7x^2-x+14$ ...
1
vote
1answer
100 views

Is this derivation of the Dirichlet Integral using a derivative under the integral sign, incorrect?

To find the integral of the Sinc function: Start with, \begin{equation} I(a)=\int_{-\infty}^{\infty}\frac{\sin\ ax }{x}dx %\hspace{20.0} ; (a>0) \end{equation} \begin{equation} \Longrightarrow ...
0
votes
0answers
35 views

Bond percolation probability on a Bethe lattice

I derived the bond-percolation probability for a Bethe lattice and my result disagrees with a seemingly reputable reference by Albert & Barabasi (see below). I would be happy if somebody could ...
1
vote
2answers
53 views

IVP to $y'=\frac{2xy^2+2x}{x^2+1}$

Find a solution of $y'=\frac{2xy^2+2x}{x^2+1}$ with given IVP $y(0)=\sqrt{3}$. My solution: $\int \frac{1}{y^2+1}dy=\int \frac{2x}{x^2+1}dx$ $\Rightarrow \tan^{-1}(y)=\log(x^2+1)+c, c\in ...
0
votes
1answer
25 views

Second Derivative Test and Hessian for $f(x,y) = x^2 + y^2$.

My task was to find the critical points of the function $f(x,y) = x^2+y^2$, to then compute the Hessian, and to use the second derivative test to determine whether the critical points are local maxima ...
4
votes
1answer
36 views

Proving the uniqueness of the weak limit

In "A First Look at Rigorous Probability Theory" by J. S. Rosenthal there is the following exercise: Prove that weak limits, if they exist are unique. That is, if $\mu, \nu, \mu_1, \mu_2, \ldots$ ...
3
votes
0answers
34 views

Show that the ideal generated by $4$ in $\mathbb Z_{12}$ is not a prime ideal.

Show that the ideal generated by $4$ in $\mathbb Z_{12}$ is not a prime ideal. Hint: Give a counter-example This is my rough proof to this question. I was wondering if anybody can look over it ...