# All Questions

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### Probability for slot overlapping for 4 employees

There are 4-shifts in a 24-hour day distributed by 4 employees. If the slot of an employee crosses the 24 hour mark it extends into the next day. What is the probability that all the four employees ...
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### Group of order $1 5$ is cyclic.

Prove that any group of order $1 5$ is cyclic. I am looking at a solution here and I am confused why "there must be one orbit with five elements and three orbits with three elements" and "fixed ...
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### Basic Vector magnitudes and Quadrants Question

Express in unit vector notation each has magnitude of 17.0cm A) Vector E is directed 27 deg counterclockwise from positive x-axis B) Vector F is directed 27 deg counterclockwise from positive y-axis ...
58 views

### The determinant of a linear transformation on a finite vector space

Given a finite vector space $V$ and a linear transformation $f : V \rightarrow V,$ is it true that for any two ordered bases of $V$, call them $a$ and $b$, the determinant of the matrix of $f$ with ...
122 views

### exponential generating function of $\frac{1}{(1-x)^2}$

Hey just had a quick question related to coming up with an explicit formula for $a_n$ with relation to $F(x)=\frac{1}{(1-x)^2}$. I know to compute such a function I have to use the derivative of ...
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### Prove by induction $n^3 < 3^n$. What is the value of $n_0$? [closed]

Prove by induction for $n \geq n_0$, $n^3 < 3^n$. What is the value of $n_0$?
192 views

### Can we determine $A= 1!+2!+3!+…$'s digits starting from last?

After reading a bit about p-adic numbers, I came up with an idea. We know that for every natural number $k$, there exists a natural number $n$ so that for every $m>n$, there are at least $k$ zero ...
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### Solving ODE by contraction mapping

Let a and c be real numbers. Solve the initial value problem y'(x) = ay(x), y(0) = c on the interval [0, 1/2a] with the help of the contraction mapping theorem. I understand that solving this ODE is ...
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### Subgroup of a nilpotent group

Let $G$ be nilpotent and $H \le G$. Let $P_1,P_2,\ldots,P_k$ be the Sylow subgroups of $H$. Is it true that $H = P_1 P_2 \cdots P_k$? I know that when $G$ is nilpotent, it is the direct product of ...
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### Prove if the groups are isomorphic

Prove if the groups are isomorphic. The multiplicative group of all 2x2 matrices with determinant $1$ and $\operatorname{Sym}(\mathbb{Z})$ I have done questions similar to this, but struggle with ...
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### Initial value problem $x'=e^{-|x|}$, $x(0)=0$

Consider $x=x(t)$ and $\frac{dx}{dt}=e^{-|x|}$, $x(0)=0$. Now I want to find the solution of this initial value problem. I want to solve it for $t\geq0$ and $t\leq0$. If $t\geq0$: ...
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### The limit of sequence $(\frac{n \sin(2n)}{n^2 + \cos(n) + 4})$

I have trouble evaluating the limit of the sequence $(\frac{n \sin(2n)}{n^2 + \cos(n) + 4})$. Could anyone help me? Thank you!
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### Local minima of a trivial sin(x) function

The paper on which it came had $4$ choices of answer, out of which $2$ options are creating confusion. For the function $f(x) = sin(x)$ in the interval $x\in[\frac{\pi}{4},\frac{7\pi}{4}]$, the ...
319 views

### a function is continuous iff the graph is pathwise connected

Let f : [a, b] → R. By the graph of f we mean the set F = {(x, f(x)) : a ≤ x ≤ b} ⊆ R Prove that f is continuous if and only if the graph of f is a pathwise connected subset of the plane. I ...
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1k views

### How to determine the equivalence classes of a relation?

I don't fully understand how to find the equivalence classes of a relation. Over $\mathcal P(E)$, where $E = \{1,2,3,4,5,6\}$, $ARB \iff |A\cap\{1,2\}| = |B\cap\{1,2\}|$ From what I've seen, ...
138 views

### Proof for hamiltonian cycle in grids having even no. of nodes

How can I go about proving that an undirected graph having even no. of nodes (at least one of the rows or columns are even - excluding line graphs of course) have a hamiltonian cycle? I have managed ...
100 views

### Radon measure not locally integrable

I need some help with this exercise: If we have $N=2$, $\Omega=\mathbb{R}^2$ and $T:\cal{D}(\Omega)\to\mathbb{C}$, with $$\langle T,\phi\rangle=\phi(0,1)-\phi(1,0)$$ I have to show that it is a ...
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### Metric space question

Let (M,p) be a metric space and suppose that ${x_n}$ is a sequence in (M,p) so that $x_n -> x$ and $x_n->y$. prove x=y Let $E>0$. then, $p(x_n,x)->0$ $lim$ $n->inf$ ...
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### Definition of an active hyperplane

We are learning about the Geometry of Duality in Linear Programming, and my prof uses the terminology active hyperplane. I'm wondering what the formal definition of this is. I can't seem to find any ...
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### How to find this Linear Transformation

Q. Find the Linear Transformation $T:V_3\rightarrow V_3$ , such that $T(0,1,2)=(3,1,2)$ $T(1,1,1)=(2,2,2)$ I tried considering $(0,1,2),(1,1,1)$ as basis, it doesnt seem to work that way. Just need ...
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### Proving various properties of metric spaces

Suppose that $p_1$ and $p_2$ are metrics on $M$. Prove that the following are also metrics: (a) $p = p_1 + p_2$ define $p_1(x,y) = |x-y|$ define $p_2(a,b) = |a-b|. p = p_1+p_2 = |x-y|+|a-b|$. But I ...
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### Convergence to Gaussian with infinitesimal condition with relation to Levy Triple.

I have posted this question before under the title "Uniform infinitesimality Condition and convergence in distribution to Gaussian distribution". I managed to find something related to this question ...
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### Almost sure convergence of sample mean

I have a sequence of events $(A_n : n \in \mathbb{N})$ with $\mathbb{P} (A_n) = 1/n^2$ for all $n$. We have $X_n = n^2 1_{A_n} - 1$, and $m_n = (X_1 + ... + X_n )/n$ is the sample mean. I would like ...
263 views

### $L^p$ Spaces, Young's Theorem, Convolutions, and Minkowski's Inequality

I need to show \begin{align} \|f*g\|_p \le \|f\|_p\|g\|_1 \end{align} By using the generalized Minkowski inequality instead of just Young's Theorem. I have spent a lot of time, but I keep hitting a ...
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### Find all real zeros of $f(x)=2x^3+10x^2+5x-12$

Hey guys I'm having a little trouble with one problem: Find all real zeros of $$f(x)=2x^3+10x^2+5x-12.$$ I got $x=-4,(2x^2+2x-3)$. I'm just having trouble using the quadratic formula to get ...
108 views

### Dirac Delta — Symmetry

I had a curiosity question rise up in the middle of the night regarding the behavior of the Dirac Delta. Because it's not a function per-se, I am not sure how a concept like "integration" symmetry ...
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### If $f$ is continuous, so is $f(x^3)$

Suppose $f:R\rightarrow R$ is differentiable and define $g(x)=x^2 f(x^3)$. Show $g$ is differentiable and compute $g'$. So I know how to do the proof, I just want to know that even though $f$ is ...
301 views

### Successive parabolic interpolator for sub pixel interpolation

Is it possible to use successive parabolic interpolator for doing sub pixel interpolation. In the case of non sub pixel interpolation it is very easy to apply successive parabolic interpolation as ...
373 views

### linearization topological sorting which repeatedly removes source nodes from the graph

The chapter suggests an alternative algorithm for linearization (topological sorting), which repeatedly removes source nodes from the graph (page 101). Show that this algorithm can be implemented in ...
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### Initial Segments of Modular Arithmetic

Modular arithmetic (MA) has the same axioms as first order Peano arithmetic (PA) except $\forall x(Sx \neq 0)$ is replaced with $\exists x(Sx=0)$. MA is $\omega$-inconsistent and all infinite models ...
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### Any predetermined sequence in the decimal expansion of an irrational number

I came up with this question in a random math discussion with my friend. I am wondering if one can always find a predetermined sequence of numbers, such as 123456, 33333, in the decimal expansion of a ...
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### What are the steps to take the derivative of this function?

This calculus derivation is giving me an extremely difficult time. I am having trouble understanding how to manipulate the $e^t$. The function is $$P*{e}^t/[(1-q)*{e}^t]$$ I want to take the ...
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### Finding the Integer solution to $a^3+b^2+c^2=2013$

this is my first time posting and I hope someone can help because this question has been driving me crazy... For a bit of background I am a sophomore math major in college, I have taken math up ...
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### Find image under Riemann map without the explicit map

How can I find the image of the line segments $(0,1)$ and $(0,1+i)$ under the unique Riemann map $f$ from the square with vertices at $\pm1\pm i$ onto the unit disc satisfying $f(0)=0$ and ...
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### Friend Group and Hater Group

Consider a set $S$ of $n$ people such that, for all distinct $x$ and $y$ in $S$, it is the case that either $x$ and $y$ like each other or $x$ and $y$ hate each other. Let us call $S' \subseteq S$ a ...
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### Prove that a finite complete graph can be embedded in $\mathbb{R}^3$

I've actually found a few intuitive examples where edges are taken to a twisted cubic and and the vertices are arranged in a certain way and that's very nice, but I'm actually more interested in a ...
909 views

### Find the power series representation for arctan.

Find a power series representation for the following function and determine the interval of convergence: $f(x)=\arctan\left(\frac{3x}{2}\right)$ So I found its derivative and used U-subsitution and ...
321 views

### How prove this inequality $\frac{x}{x+yz}+\frac{y}{y+zx}+\frac{z}{z+xy}\ge \frac{3}{2}$

let $x,y,z>0$,and such $x^n+y^n+z^n=3(n\ge 1),n\in N^*$, show that: $$\dfrac{x}{x+yz}+\dfrac{y}{y+zx}+\dfrac{z}{z+xy}\ge \dfrac{3}{2}$$ My try: if $n=1$ , since $x+y+z=3$,then use Cauchy-Schwarz ...
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### Group where you can see all the finite subgroups.

Is there any group $G$ for which every finite group is a subgroup ? I thing we can consider $S_\mathbb{N}$, because $S_n$ is subgroup of $S_\mathbb{N}$ for all $n$. so by Caylay's theorem we are ...
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### Calculate the error given a tolerance

I have a noob statistics question. Is there a function, such that given the residuals from the line of best fit, and a probability, A, it will return B such that there is an A probability of being ...
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### Greatest common denominator

My problem is figuring out how to express the GCD as a linear combination of $(9,11)$. So far, I have: $$11 = 9 + 2$$ $$9 = 4 \cdot 2 + 1$$ From here, I'm not sure if I put $2 = 2 \cdot 1$ As for ...
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### Conjugate class in the dihedral group

List all the conjugate classes in the dihedral group of order $2n$ and verify the class equation. The dihedral group is generated by two elements $r$ and $s$. The order of $r$ is two since $r^2=e$ ...
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### infinite limit cycles

Can we find differential equations in the real plane and class $C^{k}(\mathbb{R})\ k\geq 1$" that have an infinity of limit cycles accumulating in the origin with the origin as singular point? I ...
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### Computational complexity of Newtons Method

I'm trying to do a worst case complexity analsis of another algorithm that involves computing an nth root of a real number at each step. I have a bound B on the size of this number also n is fixed and ...
### If $M$ is positive definite, then $\operatorname{det}{(M)}\leq \prod_i m_{ii}$
In the Wikipedia article on positive definite matrices they claim that if $M$ is positive definite, then the determinant of $M$ is bounded by the product of its diagonal entries. How might we show ...