11
votes
2answers
269 views

Value of $\frac{1}{\sqrt{3}+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{11}}+\frac{1}{\sqrt{11}+\sqrt{15}}+\cdots$ ($n$ terms)

Sum $$\frac{1}{\sqrt{3}+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{11}}+\frac{1}{\sqrt{11}+\sqrt{15}}+\cdots \text{ ($n$ terms)}$$ I know how to use the telescoping series method when the terms are in ...
2
votes
1answer
27 views

Understanding what a statement means

I'm trying review for the finals by reading the text but I ran into an exercise with a sentence that I can't understand. Draw the network associated with the bipartite graph. What does that ...
1
vote
1answer
47 views

A property of polynomials in a paper by Rice

It has been suggested that a mind-reading tag be added. This is, unfortunately, a good candidate for such a tag... I was reading a paper of Rice relating to a property of integer polynomials. The ...
0
votes
1answer
124 views

Order of element in group given order of conjugacy class

What is the order of $x$ in a group of order 21 that contains a conjugacy class of order 3? I know the answer is 7 because the size of the conjugacy class of x equals the index $[G:Z(x)]$ of its ...
3
votes
1answer
83 views

Cannot understand some parts of proof for R be a countable union of closed sets

Im reading chapter9 Category, Real Analysis, Carothers, 1ed, talking about discontinuous functions of metric space. Here is a proof for a theorem that R be a countable union of closed sets,: Baire's ...
2
votes
0answers
42 views

Slope of tangent line using Wallis method

Consider the Wallis method of finding the slope of tangent lines where $x = a$. Use the method to find the slope of the line tangent to the graph of $y = x^2 + 3x + 7$. Use the method to find the ...
1
vote
1answer
32 views

True/False: Differentiablity #2

If the function $f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable and $f(x)\le f(0)$ for all $x\in[-1,1]$, then $f'(0)=0$. I think this one is false. By the definition of differentiable, ...
0
votes
1answer
38 views

How does this random vector indicate that my two random variables are independent?

I've got independent random variables $X$ and $Y$, and a random vector $\vec{V}=(h_1(X,Y),h_2(X,Y))$ where $h_1$ and $h_2$ are (specified) functions of $X$ and $Y$. I found $\vec{V}$'s density ...
0
votes
1answer
77 views

Counting/ probability

Im stuck on this question: "At a certain college, the housing office has decided to appoint, for each floor, one male and one female residential advisor. How many different pairs of advisors can be ...
2
votes
1answer
58 views

Creating a lift chart for a classification tree

This is likely a simple question but I'm new to data mining techniques and am trying to compare two different predictive models. I've created a logistic regression and a classification tree and would ...
3
votes
0answers
26 views

How does this polar function behave?

I came across this question in my textbook for Nonlinear Optimisation and I don't know what to do: Consider the function: $$ f(x_1,x_2)=(r-1)^2-\frac{1}{2}(r-1)^2\cos \left( \frac{1}{r-1}-\phi ...
2
votes
3answers
79 views

Prove by induction $\vphantom{\Large A}3\mid\left(n^{3} - n\right)$

So I'm just studying for my midterm and I came across this exercise: Prove by mathematical induction that $\vphantom{\Large A}3\mid\left(n^{3} - n\right)$ for every positive integer $n$. What ...
2
votes
1answer
41 views

Equation of a plane (not sure if I got this)

When I was doing my calculus midterm, I came across a question that I didn't really know how to solve, I think I skipped over these problems in my studies. The question is: Find an equation of the ...
3
votes
0answers
103 views

Question about complete DVR.

Is there a simple proof that you know to the following statement: If the residue field $k$ of a complete DVR $R$ has the same characteristic as $R$, then $R$ contains a subfield isomorphic to $k$. ...
1
vote
1answer
40 views

Relations $R^2, R^3, R^i and R^*$

Consider the relation on R on the reals where $xRy$ iff $xy=1$ I need to find $R^2, R^3, R^i $ and $R^*$ Ok, so I first started off with the following: $$xR^2z \equiv \exists y: xRy\land yRz \\ ...
0
votes
1answer
386 views

Proving $\left(\tan^2x +\frac{1}{\tan x}\right) \left(\sin x+\cos x\right)=\left(\frac{1}{\cos x}\right)+\left(\frac{1}{sin x}\right)$

$$ \left(\tan^2x +\frac{1}{\tan x}\right) \left(\sin x+\cos x\right)=\left(\frac{1}{\cos x}\right)+\left(\frac{1}{sin x}\right) $$ Can someone show me how to solve this identity and also explain the ...
3
votes
1answer
81 views

True/False: Differentiation

If the differentiable function $f:\mathbb{R}\rightarrow\mathbb{R}$ is monotonically increasing, then $f'(x)\ge0$ for all $x$. I feel like this is true by the Mean Value Theorem, but I'm not ...
0
votes
1answer
38 views

If $U = (1, 2)$, $V = (3, -4)$, is the answer to $2U + \frac{1}{2}V$ the vector $(3.5, 2)$? Check my answer.

If $U = (1, 2)$, $V = (3, -4)$, is the answer to $2U + \frac{1}{2}V$ the vector $(3.5, 2)$? I did the following: \begin{align*} U : ( ( 2 * 1 ), ( 2 * 2 ) ) &= ( 2 , 4 )\\ V : ( ( 0.5 ...
3
votes
1answer
144 views

Generating and solving recurrence relations

I am trying to do this question but don't know where to go from here: The question: For $n\ge1$ let $t_n$ be the number of ways to tile the squares of a 2xn checkerboard using 1x2(which can be rotated ...
3
votes
1answer
79 views

Möbius Transformation of Triangles

I understand that Möbius transformations are angle preserving transformations. Knowing this, my professor asked us to think about how the image of equilateral triangle is not an equilateral triangle ...
3
votes
1answer
388 views

Optimization Word Problem, revenue

A sorority plans a bus trip to the Great Mall of America during Thanksgiving break. The bus they charter seats 44 and charge a flat rate of 350 dollars plus 35 dollars per person. However, for every ...
0
votes
1answer
30 views

A jump process as an integrand in Itô integral with respect to an Itô process

So, $X_1(s)$ is a jump process, $X_2(s)$ is another jump process, $X_2^c(s)$ is the continuous part of $X_2(s)$. And $\int_0^tX_1(s-)dX_2^c(s) = \int_0^tX_1(s)dX_2^c(s)$, is it because the ...
1
vote
2answers
99 views

Limit $\lim_{n \to \infty} n (1 - \mathrm{e}^{t/n})$

Can someone help how to compute (step-by-step) the following limit? When I use the online calculator, the answer is -t. However, I do not know how to get that answer. $$ \lim_{n \to \infty} n (1 - ...
0
votes
1answer
189 views

Prove a commutative ring with characteristic n has a subring isomorphic to $\mathbb{Z}_n$

Let $R$ be a commutative ring with identity such that the characteristic of $R$ is $n$, char$R=n$. Prove that is $n>0$ then $R$ contains a subring isomorphic to $\mathbb{Z}$$_n$, the additive ...
2
votes
0answers
56 views

A probability inequality

Let $(\Omega,\mathscr{F},P)$ be a probability space. Assume $X_1,X_2,Y_1,Y_2$ are four random variables. Assume $X_1$ and $X_2$ are independent. Is it necessarily true that ...
6
votes
2answers
89 views

Has the idea of generalizing the codomain of a metric been seriously considered?

The long line is much longer than $\mathbb{R}$, and indeed many chains have this property. Thus, since metrics are usually assumed to be real-valued, this can be understood as an assumption that ...
1
vote
0answers
95 views

Prove that a linear operator is indecomposable

Let $V$ be a fi nite-dimensional vector space over $F$, and let $T: V \rightarrow V$ be a linear operator. Prove that $T$ is indecomposable if and only if there is a unique maximal T-invariant proper ...
0
votes
1answer
210 views

Area using Fermat's method of quadratures [closed]

Use Fermat's method of quadratures to find the following: a. The area between the curve $y = x^5$ and the x-axis over the interval $(0,a)$. b. The area between the curve $y = 1/x^2$ and the x-axis ...
1
vote
1answer
90 views

Equivalent Definitions of Divergence

I am having a hard time wrapping my head around the equivalence of two definitions of diverging ($+ \infty$). The first definition, which I asked a previous question about, was purely negating the ...
2
votes
1answer
121 views

Integral mean value and value of second derivative

Let $f:[a,b] \rightarrow \mathbb{R}$ be a twice differentiable function such that $f(a) = f(b) = \frac{1}{b-a} \int_a^b f(x)\,dx$. Show there exists $c \in (a,b)$ such that $f''(c) = 0$. Attempts: ...
1
vote
0answers
66 views

Weierstrass and Borel summation

In the Wikipedia article on Borel summation, there is the following quote attributed to Gösta Mittag-Leffler: Borel, then an unknown young man, discovered that his summation method gave the ...
3
votes
1answer
770 views

Integrating Dirichlet's Kernel

Determine $\frac{1}{\pi}\int_{-\pi}^{\pi}\left[D_{m}(t)\right]^{2}dt$ for $m=100$ where $D_{m}(t)=\frac{1}{2}+\sum_{n=1}^{m}\cos{nt}$ (Dirichlet's kernel). Initially, I thought of using the ...
0
votes
2answers
69 views

Finding the inverse of a natural log

How would I find the inverse of $$\ln(8x-64)?$$ I've tried put $8x-64$ as the power to the base of $e$, I don't know what to do from there on, thanks in advance
5
votes
4answers
99 views

Expected Value of Game

The game is as follows: Start with 1. If you are at $n$, add one with a probability $\frac{1}{n+1}$, otherwise subtract one. End the game if you hit 0. What is the expected number of rounds that this ...
0
votes
0answers
85 views

number of directed acyclic graphs

Given number of vertices $k$, how many DAGs over $k$ are there? I know from here that it can be computed in a recursive manner. I am wondering whether there are other simple formulas.
2
votes
0answers
82 views

Chain homotopy inverse to inclusion

I am currently trying to solve the following problem: Given a simplicial complex $K$ (being the union of simplices such that any face of a simplex in $K$ is also in $K$ and any two simplices intersect ...
2
votes
3answers
68 views

Prove that if $n^{2} - \left(n - 2\right)^{2}$ is not divisible by $8$ then $n$ is even

Let n be an integer. Prove that if $n^{2} -\left(n - 2\right)^{2}$ is not divisible by $8$ then $n$ is even. Can anyone help me step by step to understand this.
1
vote
1answer
77 views

What is the degree of following differential equations

Can you help me find the degrees of the following differential equations? $$\left(\frac{dy}{dx}\right)^2 + \frac{1}{dy/dx}=2$$ $$x+\left(\frac{dy}{dx}\right)= ...
0
votes
0answers
52 views

Quasicompact? Why the distinction?

What is the reason that some topologists use quasicompact? Why is the distinction made? quasi means "not really", so why use this terminology?
1
vote
4answers
232 views

Without using Cayley’s theorem, prove that there are at most $n^{2n−2}$ labelled trees with n vertices.

I am studying for a test I have and I found a past problem which I have no idea how to go about doing.. My thoughts are. I know not to use Cayley's theorem but it says that there are $n^{n-2}$ ...
3
votes
2answers
86 views

Reference request: localisation of categories

I'm trying to track down a result mentioned in Verdier's Categories Derivees apearing in SGA 4.5. In chapter 2, 3.1 (page 280 in SGA 4.5) Verdier mentions that categorical localisations always exist, ...
0
votes
1answer
51 views

Finding Neutral element and symmetric element Of a Group

So i have $(C^{*}, .)$ and its a group. so i have $\alpha= a+b*i$ | $\beta = c+d*i$ $\alpha+\beta = (a+c)+(b+d)*i$ $\alpha*\beta = (ac-bd)+(ad+bc)*i$ how can i find the neutral and the symmetric ...
4
votes
1answer
43 views

Given $c_{i} \rightarrow c$, prove if $c \ge 0$ then $\limsup c_{i} a_{i} = c \limsup a_{i}$

Let $a_{i}$ be a sequence of real numbers and suppose that $\limsup a_{i}$ is finite. Let $c_{i}$ be another sequence and suppose $c_{i}$ converges to $c$. Prove that if $c \ge 0$, then $\limsup ...
1
vote
1answer
138 views

Maximum number of edges in a simple graph?

I found that the maximum number of edges in a simple graph is equal to $$\sum\limits_{i=1}^{n-1} i$$ Where $n$ = # of vertices. For example in a simple graph with 6 vertices, there can be at most 15 ...
1
vote
1answer
166 views

If two sequences converge in a metric space, the sequence of the distances converges to the distance of limits of the sequences.

Let $(p_n)$ and $(q_n)$ be sequences in the metric space $(X, d)$ and assume that $p_n \rightarrow p \in X$ and $q_n \rightarrow q \in X$. Prove that $d(p_n, q_n)$ converges to $d(p, q)$. Ok, so ...
1
vote
2answers
182 views

Reduction Transitive Relation Problem

I have this problem on my homework, it's my last one left but I'm having trouble with it. Any help would be appreciated.
0
votes
1answer
151 views

Prove or disprove the following proposition

Prove or disprove the following proposition: There are no positive integers $x$ and $y$ such that $$x^2 - 3xy + 2y^2 = 10$$
0
votes
1answer
80 views

Holder continuity and Hilber space

Let $\Omega\subset \Re^n$ be an open set and let $u \in H^1_{loc}(\Omega)$ be a weak solution of $\Delta u=f $ in $\Omega$, with $f \in C^{0,\alpha}(\Omega)$. Prove that $u \in C^{2,\alpha}(K)$ for ...
-1
votes
1answer
72 views

Characteristic function say something about the expectation and variance [closed]

Show that if $\lim_{t \downarrow 0} (\varphi(t) -1) / t^2 = c > -\infty$ then $EX = 0$ and $E|X|^2 = -2c < \infty$. In particular, if $\varphi(t) = 1 + o(t^2)$, then $\varphi(t) \equiv 1$. Where ...
1
vote
1answer
66 views

Number of ways to form isosceles triangle by picking points on a circle

Given a circle with 24 evenly spaced points, how would you find the number of possible isosceles triangles (which includes equilateral) that can by drawn using the points? My attempt was to say that ...

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