# All Questions

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### Math word problem for 4th grade [on hold]

Adrian wants his lawn to be mown. Three men apply for the task. The first man can mow the lawn in 6 hours; the second man can mow the lawn in 4 hours; and the third man can mow the lawn in 3 hours. ...
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### Vanishing cech cohomology of a concratible space.

Let $C$ be a concratible space. Is it possible to find a sheaf $F$ such that two Cech cohomology groups don't vanish? E.g. Can I find a sheaf (or a local system) and two integers $i,j$ such that ...
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### How to prove the equivalence relation $A\cup B=B \Leftrightarrow A\cap B = A$? [on hold]

How to prove the equivalence relation? $$A\cup B=B \Leftrightarrow A\cap B = A$$
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### Projective Geometry in $\mathbb{R}^{3}$: “Lonely lines” in source/image planes

I am reading some lecture slides about projective geometry in $\mathbb{R}^{3}$. In particular, given a source plane, $S$, an image plane, $I$, and a focal point, $f$, the issue at hand is the ...
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### Combinatorics project ideas for high school students

It's that time again! Last year I asked for high school project ideas in the area of algebraic geometry, this year it's combinatorics (you can include graph theory and combinatorial game theory if you ...
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### Etymology of normal extensions and subgroups

According to wikipedia, a normal extension is a splitting field of a family of polynomials, and a normal subgroup is one that is invariant under conjugation. Why are normal extensions and normal ...
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### Is a dense and co-dense subset $G_\delta$ or co-$G_\delta$

Let $A \subset \mathbb{R}$ such that $A$ and $A^C$ are both dense. By Baire's Theorem at most one of $A$ and $A^C$ is $G_\delta$ (i.e. a countable intersection of open sets) I couldn't think of an ...
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### On a congruence for the number of finite topologies

I am making search about "On a congruence for the number of finite topologies". I have found a paper. I guess it is written in Russian. How can I find English version of this paper ? I am also ...
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### Complex hyperbolic Trigonometry

When faced with the equation $\cos{z}=\sqrt{2}$ I want to solve for z so I break it up into a sum $z=x+iy$ and get: $\cos{z}=\cos{x}\cosh{y}-i \sin{x} \sinh{y}$ equating real and imaginary parts I ...
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### Commutative matrix question

I was doing my HW, and I am confused with one thing. To show that a matrix is commutative, do we need to show both $x+y = y+x$ and $xy=yx$? Or just by showing $xy=yx$ would suffice?
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### For what natural number $n$ is the following inequality true: $2^n \geq 2\cdot n^2$?

Can you solve this by using induction? The inequality is true for $n = 1$, but is false until $n = 7$. After the induction step I got $$2^n \geq n^2 + 2n + 1.$$ If you take the limit as $n$ ...
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### Origin of period function model of primes

There is a web page attributed to Omar Pol, "Sobre el patrón de los números primos: Determinación geométrica de los números primos y perfectos." ("On the pattern of primes: Geometric Determination of ...
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### Need help with a conic tangent question? (Hyperbolics)

I need to find the equation of the tangent to the hyperbola $$\frac{x^2}{6}-\frac{y^2}{8}=1$$ at the point $(3,2)$. I tried doing it by substituting for $y$ but the algebra is not nice at all and I ...
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### Solving a ln divided by a ln.

I am having trouble figuring out how to calculate this. Thank you for your help. $$.926 = \frac{ln(1+.8u)}{ln(1+u)}$$ What does $u$ equal?
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### Automorphisms of Abelian groups

Let $A$ be a free Abelian group and $N$ a characteristic subgroup of $A$ such that $A/N$ is finite. I also know that $Aut(A/N)$ and $Aut(N)$ are both finite. I have to prove that $Aut(A)$ is finite. ...
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### Geometry of Spans in $\Bbb{R}^2$ and $\Bbb{R}^3$

I'm having difficulty figuring out how to approach the following Geometry of Spans questions. I only seem to understand the "span of a single vector" ones. How would I go about explaining the others? ...
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### How can one know every Cauchy sequence in a complete metric space converges?

I am new to Cauchy sequences. I stumbled onto them in the process of learning what a Hilbert space is. As I understand it, a Cauchy sequence is a sequence whose elements become arbitrarily close to ...
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### How is the Set of all Polynomials Equal to the Following Union?

Given that $P(F)$ is the set containing all polynomials with coefficients from field $F$, I am given the following: $W_1$ is the set of all polynomials $f(x)$ in $P(F)$ such that for: ...
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### Determine if there is a node in a binary postorder anti-sorted tree with key $k$

A binary postorder anti-sorted tree is a binary tree for which the post-order traversal gives the keys that are saved at the nodes of the tree in descending order. Present a pseudocode for the most ...
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### Founding maxima or minima to a function

$g(x)=e^{x-1}+x^{2}-3+2x$ How can I find when this function has maxima and minima? I found the derivative but I can't understand how find the solution when $g'(x)=0$. It's high school material.
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### Prove or disprove $\nu(E)=\lambda(f(E))$ is a measure provided that $f$ is nondecreasing and satisfies the N-condition.

Suppose $f$ is a non-decreasing continuous function from $[a,b]$ to $\mathbb{R}$, and $\lambda$ is the Lebesgue measure in $\mathbb{R^1}$. Also, $f$ satisfies the property that $f$ maps Lebesgue ...
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### Solve this problem involving Geometric Brownian Process

The price of a stock follows a geometric Brownian process with annual expected return rate of 20% and volatility 50%. The initial stock price is 10 euros. Compute the probability that the stock price ...
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### Why we throw away the units in the definition of irreducible elements?

In the book "Abstract Algebra" by Dummit, the definition of irreducible element in an integral domain $R$ goes like this. Suppose $r\in R$ is nonzero and is not a unit. Then $r$ is called irreducible ...
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### Universe as a finite 3-manifold without boundary

My question is soft and imprecise, as I know very little differential topology. Nevertheless, I hope it makes some $\epsilon>0$ of sense. Assume the Universe is a 3-manifold without boundary, ...
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### Level curves for “unsolvable” integral

Problem: Sketch the level curves of g defined by $$g(x,y)=\int_x^y{e^{-t^2}dt}$$ (The error function does not need to be used here). Attempts at solution: (1) Apparently we could take $y=x$, then ...
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### There exist uncomputable integer numbers?

This question came from the answer I've given to the question An easy example of a non-constructive proof without an obvious "fix"?. Rereading my answer I had some doubt about the ...
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### What variables is $\delta$ dependent on in the epsilon-delta definition of continuity?

The definition of continuity is: $f$ is continuous at $a$ if: Given any $\epsilon>0$, $\exists \delta > 0$ st. $|x-a|<\delta \implies |f(x)-f(a)|< \epsilon$ $\delta$ obviously depends ...
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### What books do you recommend on mathematics behind cryptography?

I am currently reading the Book Understanding Cryptography from Cristof Paar. I am enjoying the book but i don't like to scratch the surface when it comes to cryptography. I would like do dig a little ...
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### Regular expression building: Comment delimited strings

I'm attempting to build a regular expression that will accept only strings of the form: ...
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### Reduce this third order ordinary differential equation to first order to use Runge Kutta

The ODE I'm workin with is $$\dddot{x} + t^2\ddot{x} + 4x = 0$$ with $$x(0)=1, \dot{x}(0)=0, \ddot{x}=-1$$ I've written a very basic program in C++ to use the RK4 method to approximate a solution to ...
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### Abelian torsion group of rational points of an elliptic curve

I want to find the abelian group of rational points $E(\mathbb{Q})_{\text{torsion}}$ of the elliptic curve $y^2=x^3+8$. E(\mathbb{Q})_{\text{torsion}}=\{P \in E(\mathbb{Q}) | P \text{ of finite ...
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### Computing $\mathrm{gcd} (100!, 3^{100})$

I am trying to compute $\mathrm{gcd}(100!,3^{100})$. I am still not sure how to reach an answer but I feel that Wilson's Theorem (i.e., $(p-1)!\equiv -1 \bmod p, p$ prime) and Fermat's Little theorem ...
I'm trying to solve $\theta_t - D\theta_{xx} = f(x,t)$ on the half-line $0 < x < \infty$ for $0< t < \infty$ given boundary and initial conditions $\theta(0,t) = h(t)$, $\theta(x,0) = ... 3answers 15 views ### Propositional logic problem: Sales, expenses and happiness of the boss Either sales will go up and the boss will be happy, or expenses will go up and the boss won’t be happy. Therefore, sales and expenses will not both go up. I know the solution is that the ... 0answers 13 views ### How do I prove that a given probability distribution is Gaussian I am trying to plot the distribution of a random variable$x$. I got this distribution by marginalising a wishart distribution. When I plot the distribution curve of$x$, it looks like bell shaped ... 2answers 22 views ### Can someone help me prove that$\tau(n)$is odd [duplicate] Can someone help me prove that$\tau(n)$is odd if and only if$n$is a perfect square. So basically I have to prove that$\tau(n)$is odd iff$n = k^2$for some integer$k$. 0answers 15 views ### Equivalence of Definitions of completion of metric space I've come across two different definitions for a completion of a metric space and am trying to figure out why they are equivalent. The definitions are: 1) Let$(X,d)$be a metric space. Then ... 0answers 12 views ### Practicing mathematical proofs in preparation for another course and could use some help [on hold] I'm starting a course on Algorithms and the professor wants to test our induction and proof knowledge. Problem is, our prerequisite courses never focused on such material. I'm hoping someone could ... 1answer 20 views ### No generic is definable in a perfect notion of forcing of a model of Peano Arithmetic I would like to prove Lemma 6.1.2.2 from The Structure of Models of Peano Arithmetic by Kossak and Schmerl. Let$\mathcal{M}$be a countable model of Peano Arithmetic and$\mathbb{P}=\langle P, \le ...
A function $f$ with the argument $x$ is commonly written $f_x : A\to B, x\mapsto f(x)$, or $f_x : \mathbb{R} \to \mathbb{R}, x\mapsto x^2$, but say I don't want to specify the argument, how would I ...