# All Questions

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### Integral Power Rule Step-by-step

Real quick, two things: I'm sorry if my notation or terminology is incorrect, and I know what I'm asking isn't strictly necessary for my studies, but writing something out step by step helps me to ...
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### How to show that for $a_n := \left( \frac{(2n)!}{n!n^n} \right)$, $a_n \rightarrow 4/e$

I'm asked [in a homework problem set] to show that for $a_n := \left( \frac{(2n)!}{n!n^n} \right)$, $a_n \rightarrow 4/e$. I'm told to show that this function is Riemann integrable for $ln(a_n)$. I ...
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### Why continuous growth (based on e) is being simply scaled to match non-limit cases (limit of the (1+1/n)**n formula)?

The constant $e$ is the maximum exponential growth that is possible when it is done continuously, i.e. when what can be called "continuous breeding" occurs. That is the meaning of the ...
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### How to find the largest n for which one can solve a problem with a given algorithm?

This is from Discrete Mathematics and its Applications I am working on 15b and 15f. First I converted the rate to say that 10^9 operations are carried out in one second. so for 15b. In one ...
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### Vector Fields on Real Numbers

I'm looking at vector fields on the manifold $\mathbb{R}$, in the sense that a vector field $v$ is a smooth function from $\mathbb{R}$ to $\mathbb{R}\times T_p\mathbb{R}$. These seem so simple that ...
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### T/F question on free groups

Is this statement True or False- If F is a free group with basis {$x,y$} and H is the subgroup generated by {$x^2,y^2,xy,yx$} then H is a free group of rank $3$. What should be my approach to solve ...
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### Surface of revolution and curvatures

Let $f(x)$ be a smooth function. Consider a surface of revolution, $$M(u, v) = (f(v) \cos(u), f(v) \sin(u), v).$$ (a) Calculate coefficients of the first and second ...
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### what is the area of a symmetric lens with the circles having a radius of 35' being 60' apart

I need this answer for work. Ive tried for hours to figure this out. If anyone could help it would be much appreciated. I need to find the area of a symmetrical lens of two circles who's radius is ...
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### Growth faster than polynomial, slower than exponential.

Assume $F(n)$ is a positive function. If $F$ is growing faster than a polynomial then is it growing exponentially fast? Is this statement true? Can we find a function $F(n)$ such that ...
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### Determine the boundedness of a complex function

How can I determine whether {$\frac{z}{1+z^2}$; z $\in$ $\mathbb{C}$ \ {-i, i}} is bounded? My textbook is very poor at describing boundedness for complex functions. Thanks for the help!
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### Maximum value of a given function.

The question is to find the maximum value of the function: $f(x)= \frac{|x|-2 -x^2}{|x| +1}$. What I tried to do was maximise the numerator and minimise the denominator to obtain the maximum value ...
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### First examples for topology of non-Hausdorff spaces

I have absolutely no intuition about non-Hausdorff spaces. I would like to understand the topology of non-Hausdorff spaces (in particular spaces obtained by "bad" group actions). As a first example, ...
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### Proving Decimal Representation

Prove that if the decimal representation of a nonnegative integer n ends in 5 or 0 then 5 | n. (Hint: As a first step show that if the decimal representation of a nonnegative n integer ends in d0 then ...
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### Decreasing family of path connected open sets containing two points has a common path.

I was trying to use van Kampen theorem on fundamental groupoids to prove the van Kampen theorem on fundamental groups. I got stuck at the following technicality. Let $\{U_i\}$ be a decreasing ...
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### Homework : Anti log expression

I have this expression $x(r) = y(a)r^a$ where $r$ is a random variable and I want to express the expression in terms of $r$. The objective is to substitute the variable $r$ into the pdf of $r$, ...
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### Efficient algorithm for calculating hypervolume

Given a $d$-dimensional hyperrectangle that spans from the origin to the integer coordinates $l_1,l_2,l_3,\cdots,l_d$. If $V$ is the hypervolume of the solid formed by all points in the ...
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### Question about permutations and cycles (Groups and Symmetries)

Let $a = (123)(456)$ be in $S_{10}$. Find a permutation $b$ in $A_{10}$ such that the disjoint cycle form of does not contain any $3$ cycles and $a$ is in $\langle b \rangle$. Attempt: I'm assuming ...
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### Examples of forcings which add no “definable” Aronjain tree

Maybe a bit board question but: Fixing a regular cardinal $\kappa$ in the ground model, I am looking for examples of set forcing notions which preserve regularity of $\kappa$ and add no new $\kappa$ ...
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### Find $\int_C ydx+zdy+xdz~$ where $C$ is the curve of the intersection of the two surfaces $z=xy$ and $x^2+y^2=1$

Find $\int_c ydx+zdy+xdz~$ where $C$ is the curve of the intersection of the two surfaces $z=xy$ and $x^2+y^2=1$ , traversed once in a direction that appears counterclockwise when viewed from high ...
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### Eigenvector of canonical form matrix

So if you have the canonical matrix: $$\begin{pmatrix} \lambda &1\\ 0 & \lambda \end{pmatrix}$$ the eigenvector is $(1,0)$. I've found this from multiple sources. But where does that come ...
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### Equivalence proof

Let $(f_n)$ and $(g_n)$ be sequences with e.g.f $F(t)$ and $G(t)$ respectively. Show the equivalence of the following: $$g_n=\sum^n_{k=0}{n\choose k}\ f_k$$ $$G(t)=F(t)e^{t}.$$ ...
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### Consider this primality test: Fix an initial segment of primes (e.g. 2,3,5,7), and combine a $b$-pseudoprime test for each b in that list…

Consider this primality test: Fix an initial segment of primes (e.g. 2,3,5,7), and combine a $b$-pseudoprime test for each b in that list. For several such initial segments, find the first $n$ for ...
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### a characterization of $L^p$ space

The following question should be part of the questions I recently asked here Prove or disprove a claim related to $L^p$ space If $g \in L^p(\Omega, \lambda)$ where $\Omega$ is a bounded subset of ...
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### For the following functions, the domain and codomain is {a,b,c,d} which ones are one to one and which are onto? Give reasons for each

For the following functions, the domain and codomain is {a,b,c,d} which ones are one to one and which are onto? Give reasons for each. a) f(a) = b, f(b) = a , f(c) = c, f(d) = d b) f(a) = b, f(b) = ...
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### Is it possible to construct a 1-D linear differential operator with given spectrum $0\leq\lambda_0\leq \lambda_1\leq\dots\leq\lambda_n\le\dots$?

Suppose one is given with a sequence $S$ of non-negative real numbers $0\leq\lambda_0\leq \lambda_1\leq\dots\leq \lambda_n\leq\dots$. Under what conditions on $S$, is it possible to construct a Linear ...
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### Determine if $h(x)=\sin(x^2)$ is uniformly continuous on $(-\infty,\infty)$ [duplicate]

Normally I know how to do these kind of questions using different kind of methods, but on this specific one I have no idea what to do: Determine if $f(x)=\sin(x^2)$ is uniformly continuous on ...
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### The two planes 2x-y+3z=2 and -4x+2y-6z=3 are parallel. Let V= the set of P1P2: P1 is an element of equations one and P2 is an element of equation 2.

The two planes 2x-y+3z=2 and -4x+2y-6z=3 are parallel. Let V= the set of P1P2: P1 is an element of equations one and P2 is an element of equation 2. Is this a vector space? P1P2 should have a line ...
Assume that we take an $n$-simplex (with side length of 2 units) and place a unit $(n-1)$-sphere at each vertex. For $n=2$, half of a circle is enclosed inside the $2$-simplex. For $n=3$, solid angle ...