# All Questions

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### Solving a Calculus velocity question using Precalc.

Decide whether on not Calculus is needed for the problem. If it is, use Pre-Calculus to approximate an answer. Find the distance traveled in 10 seconds by an object traveling at a velocity of v(t) = ...
188 views

### Which methods different than the natural one can one devise to confirm that the limit is $\;2/\pi\;$?

Good evening, I have found this exercise (Which methods different than the natural $\lim_{n\to\infty}\frac{|\cos{1}|+|\cos{2}|+|\cos{3}|+\cdots+|\cos{n}|}{n}$) What is the limit of: ...
147 views

### Mismatching Results - Keno and Probability

In Keno, a player picks from 1 to 70 (at least in this version), 20 of these numbers are drawn, and the payouts are based on the number of matches. What I have tried to do is to check that the Swedish ...
101 views

### Problem with understanding generating functions.

I am given generating functions $f(x)= \frac{x}{1-x}$ or $f(x)=\frac{1}{1+x^{2}}$ or $f(x)=\frac{1}{x^2-5x+6}$ and I am obliged to write sequence which are generated by this functions. What is the ...
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### Confidence interval on the difference of probabilities

Suppose you have two coins 1 and 2 with unknown probabilities $p_1$ and $p_2$ of coming up heads. You then flip coin 1 $n_1$ times and coin 2 $n_2$ times. I would like to then be able to say that ...
442 views

### Can I solve an integral (or other tough problem) by playing with knots?

I've seen that in calculating things in knot theory that involves a lot of hard looking integrals and matrices, even though the knots themselves appear fairly simple. So is there some way in which ...
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### divergence problem with polar coordinates

How can I show that $\nabla \cdot (\phi \nabla )=0$. when $\phi (r, \theta)=(r+\frac{R^2}{r})\cos\theta, r> R$?
399 views

### Empty functions are not injective?

Many sources say that empty functions such as $f:\emptyset \rightarrow S$ are injective because it is a vacuous truth. But currently I am reading a book on axiomatic set by Patrick Suppes, and he ...
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### Variance deduction

the definition of variance is $V(X) = E((X-E(X))^2 )$ For a discrete random variable: if we have put $Y = g(X)$ , where $g$ is a real function $E(Y) = E(g(X)) = \sum\limits_{k} g(k)p_X(k)$ , ...
187 views

### Unitary similarity transformation

I have a matrix: $A= \dfrac{i}{3} \begin{bmatrix} 1&-2&1\\-2&1&1\\1&1&-2\end{bmatrix}$ Could someone explain me how to find a corresponding diagonal matrix for a ...
866 views

### Why is an orthogonal matrix called orthogonal?

I know a square matrix is called orthogonal if its rows (and columns) are pairwise orthonormal But is there a deeper reason for this, or is it only an historical reason? I find it is very confusing ...
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### If $x$ is integral over $A_m$ for all maximal ideals $m$, then $x$ is integral over $A$

I am going over an old exam, and there is this question that I am stuck: Given $A$ a commutative ring with unity, show that if $x\in\operatorname{Frac}(A)$ is integral over $A_m$ for all maximal ...
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### Is there a name for this theorem?

Theorem: If you have a number $x$ with $n$ number of decimal places and another number $y$ with $l$ number of decimal places, then $x \cdot y$ will never have more than $n + l$ decimal places. For ...
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### Is this function well-defined or just an abuse of notation?

Let's say we have a real-valued function $f$, perhaps something very simple. I define $g$ to be $g(x) = \int_0^xf(x)dx$. This looks like total nonsense to me. But I can't satisfactorily prove to ...
92 views

### Is this $\epsilon-\delta-$proof correct?

I have to Show that $$\mathbb{C} \rightarrow\mathbb{R}; z \rightarrow \Re z$$ is a continuous function using the $\epsilon-\delta-$criteria. So what I did is the following: I have to Show that ...
207 views

### Prove trigonometric relation

Recently, I found this identities in a sheet of paper I was given as studying material: $$\prod^n_{k=1}\sin\left(\frac{k\pi}{2n+1}\right)=\frac{\sqrt{2n+1}}{2^n}\tag1$$ ...
155 views

### Reducing quartic equations to quadratic

I'm trying to re-learn basic math/algebra, and I can't get passed one question concerned with reducing quartic equation to a quadratic: Find, correct to 3 significant figures, all the roots of the ...
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### Prove that $f(n)$ is in $\Theta(g(n))$

Suppose $f(n) = 1^k + 2^k + \ldots + n^k \;$ and $\; g(n) = n^{k+1}.\;$ Prove that $\;f(n)\in \Theta(n^{k+1})$. My understanding is that we have to find $C_1, C_2 \gt 0$ such that: ...
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### Trick to diagonalize symmetric matrices?

I will write an exam on Quantum Mechanics soon. I was wondering whether there is any smart and fast way to determine the eigenvalues/eigenvectors of a symmetric 3x3 matrix other than by calculating ...
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### Let $N$ be the product of the first $m$ primes and $M$ be the product of the first $n-m$ primes.

Could someone tell me if this proof is correct? Suppose there are finitely many primes: $p_1,p_2,\cdots,p_n$ primes. Let $m<n$. Then let $N=p_1p_2p_3\cdots p_m$ and $M=p_{m+1}\cdots p_{n-1}p_n$. ...
129 views

### Axiom of Completeness for set of integers

If $A$ is a subset of the integers $\mathbb{Z}$, and is bounded above, then A has a supremum $\alpha$ that is an element of the integers $\mathbb{Z}$. Is this statement true?
137 views

### How many trailing zeroes does 1000! have in base 8

First I checked this for powers of two: $$\sum\limits_{i=1}^{9}\left\lfloor\frac{1000}{2^i}\right\rfloor=994$$ I was told the answer to this is $331$ since $994=331\cdot 3+1$. I'm wondering why its ...
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### Why are these two series identical?

Could someone please show/explain to me explicitly why this is true (from wiki): $S = 1 - 1 + 1 - 1 + 1 - 1 + \cdots$ The series can be rearranged as: $S = 1 - (1 - 1 + 1 - 1 + 1 - 1 + \cdots)$ I ...
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### directional derivatives at (0,0) vanish

Is the statment that all directional dervatives vanish at (0,0) really true, it seems to me the last equation states the opposite. The example is from: Mathematical Analysis: An Introduction to ...
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### Finding general solution for a nonhomogeneous system of equations

I have a system of differential equations: $\begin{cases} x_1'=x_2+2e^t \\ x_2'=x_1+t^2 \end{cases}$ And I want to find the general solution for it. I started by finding the general solution for the ...
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### Inverse Laplace of $F(s) = \frac{3s}{(s^2+9)^2}$

Can somebody please show how to go about answering the following; ${\scr L^{-1}}(F(s))$ where $F(s) = \dfrac{3s}{(s^2+9)^2}$ I know the ${\scr L}\left(\dfrac{3}{s^2+9}\right)=\sin(3t)$ and that ...
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### Solve the following differential equation: $y\frac{dx}{dy}-x=2y^2$, with the initial condition $y(1)=5$.

Solve the following differential equation: $y\dfrac{dx}{dy}-x=2y^2$, with the initial condition $y(1)=5$. The thing that is throwing me off (I think) is the $\dfrac{dx}{dy}$ instead of what I am ...
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### almost sure convergence. martingale

I have a simple question regarding almost sure convergence. Assume a sequence $X_n = a_1 + ... + a_n$. and $P(a_n \neq -1 \space i.o) = 0$ This means $X_n/n \rightarrow -1 a.s$. Does it also mean ...
512 views

### What pure mathematics foundations should an applied mathematician have?

I'm studying mathematics, with some statistics also, and I've always chosen applied courses. I'm getting to the point where I'm studying 3rd year undergraduate to graduate level material. My first ...
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### How do I convince someone that $\mathbb{R}^2$ and its copy inside $\mathbb{R}^3$ are different?

One of my friends is taking a first course in linear algebra now, and one of the problems on his latest homework was to explain why $\mathbb{R}^2$ and $\{(a_1,a_2,a_3) \in \mathbb{R}^3 \mid a_3 = 0\}$ ...
181 views

### Having trouble with a combinatorics question.

I'm not so good at combinatorics, but I want to know if my answer for this question is right. Originally this question is written in spanish and it says: Se dispone de una colección de 30 pelotas ...
181 views

### Does the distributive law for dot products go for both addition and subtraction?

I know that $\vec a\cdot(\vec b+\vec c)=\vec a\cdot\vec b+\vec a\cdot\vec c$, but is it also true that $\vec a\cdot(\vec b-\vec c)=\vec a\cdot\vec b-\vec a\cdot\vec c$?
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### expectation conditioned on sigma fields

could anyone please explain to me a simple question regarding expectations conditioned on sigma fields? Consider a sample space {a, b, c} and $F_1$ = $\sigma$(a) and $F_2$ = $\sigma$(b) and a random ...
257 views

### Pushout from initial object isomorphic to coproduct

Let $C$ be a category with an initial object $0$, pushouts, and coproducts. If $G$ and $H$ are two objects, I want to know whether $G\sqcup H$ and $G\sqcup_0 H$ are isomorphic. Letting $i_1$ and $i_2$ ...
196 views

### Determine a sequence of random variables is a martingale

I'm trying to solve a problem from an old exam. This is an easy but a bit lengthy exercise, divided into subproblems. Since they are based on each other and probably are quite short, I was hoping that ...
OK, I am looking at Cartan and Eilenberg Homological Algebra book (1956, 1973 printing). Chapter V.9, p97 they define functors T(-,-) of type L$\Sigma$ and R$\prod$. T is of type L$\Sigma$, if T(A,C) ...