# All Questions

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### Triangle Inequalities

Anybody have a hint on how to begin to Prove that $\lvert x-y \rvert \lvert z-w \rvert \leq \lvert x-z \rvert \lvert y-w \rvert + \lvert x-w \rvert \lvert y-z \rvert$ for any $w,x,y,z \in \Bbb R$?
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### How do I group unique pairs of sequential numbers in a grid?

Not sure if this SE site the best place to help find a solution for this problem. I am open to suggestions! It basically boils down to this: Given a grid of numbers where the numbers are ordered ...
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### integration limit

How the integration, \begin{align}I= \Im\int_{-\infty}^{\infty} {\rm e}^{-\left(r - {\rm i}kR/2\right)^{2}} \,r\,{\rm d}r \\[3mm]\end{align} can be written as \begin{align}I= \Im\int_{-\infty - {\rm ...
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### Finding a non-recursive formula for a recursively defined sequence

So I have a recursive definition for a sequence, which goes as follows: $$s_0 = 1$$ $$s_1 = 2$$ $$s_n = 2s_{n-1} - s_{n-2} + 1$$ and I have to prove the following proposition: The $n$th term of the ...
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### Financial Linear Programming Problem

I'm very new at linear programming and I'm trying to figure out a way to approach this problem below: ...
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### Help Getting a Zero out of the Denominator of a Limit

Why is the following limit equal to $1/2$. I get undefined. :-( $$\lim_{x\to 0}\frac{(x+1)^{1/2}+1}x$$ this should $= 1/2$. When I multiply the top and bottom by $(x+1)^{1/2} - 1$, I end up with ...
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### Automorphisms of free groups

Suppose $U$ is a subgroup of finite index in the free group on $k$ generators $F_k$. Suppose $\sigma$ is an automorphism of $F_k$ such that $\sigma|_U = \text{id}$, then must $\sigma = \text{id}$?
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### let $B=(f:\mathbb{C} \rightarrow \mathbb{C}/f(x)=x^ke^{cx};\mbox{ }k\in \mathbb{Z},\mbox{ }c\in \mathbb{C})$

Let $\mathbb{Z}^*=\mathbb{Z}^+\bigcup \left \{ 0\right \}$ and let $B=(f:\mathbb{C} \rightarrow \mathbb{C}/f(x)=x^ke^{cx};\mbox{ }k\in \mathbb{Z^*},\mbox{ }c\in \mathbb{C})$ Prove that $B$ is a ...
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### What is the most motivating way to introduce modular arithmetic?

What the best way to introduce congruences in a number theory course? I am looking for something which will have an impact. What are the really interesting applications of congruent mathematics?
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### Can the limit of a product exist even if the limit of one of the factors doesn't?

Show an example where $\lim_{x\to c} f(x)$ does not exist and $\lim_{x\to c} g(x)$ exists but $\lim_{x\to c} f(x)g(x)$ exists.
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### What are the details behind structures such as $F[X]$?

I see the notation $F[X]$ often, where $F$ is an algebraic structure (usually a field). The notation $\Bbb R[X]$ has never been explained to us in class, other than that it refers to the polynomials ...
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### What is the principal branch of $f(z)=\sqrt{1-z}$, $z\in\mathbb{C}$?

My question is, essencially, about the definition of the principal branch of a function that is not a Logarithm. In this case, $f(z)=\sqrt{1-z}$.
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### books on fundamentals on statistics

I have mathematical backgroud but I am new to Statistics. So, could you please advice any books on Statistics and/or Biostatistics for beginners?
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### Expressing elementary matrices in terms of each another

How can I express an elementary matrix of type 2 in terms of the product of elementary matrices of types 1 and 3? Just for clarity, here are the types: Type 1: \begin{bmatrix}1&a\\0&1\\ ...
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### Are ideals of an sup semilattice always non-empty?

I am trying to do an exercise from the book A Compendium of Continuous Lattices. Exercise: Let $L$ be a set with a transitive relation, and let $A,B$ be ideals of $L$. (i) $A\cap B$ is an ideal of ...
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### Using Janet Basis to solve a nonlinear polynomial system

I am trying to solve a nonlinear polynomial equation system using Janet basis, when they have finite many solutions. For example the solution of the system: $$xy^2-y^3-3x^2=0,x^2+y^2+xy=0.$$ There ...
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### If X is Geometric(p), show that $\displaystyle E\left[\frac{1}{1+X}\right]=\log\left( (1-p)^{\frac{p}{p-1}}\right)$

If X is Geometric(p), show that $\displaystyle E\left[\frac{1}{1+X}\right]=\log\left( (1-p)^{\frac{p}{p-1}}\right)$ I found that $E\left[\frac{1}{1+X}\right]=\log p^{\frac{p}{p-1}}$ instead using ...
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### Finite ring has zero divisors

”If the nonzero element $a$ of the ring $\mathbb{D}$ does not have a multiplicative inverse, then $a$ must be a zero divisor”. If $\mathbb{D}$ has finitely many elements, then the statement is true. ...
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### Epsilon-Delta questions

Prove the following limits using only the epsilon delta definition: Q1: $$\lim_{x\to2^-} \sqrt{4-x^2}= 0$$ and Q2: $$\lim_{x\to\infty}\dfrac{x^2+2x}{x^2+1} = 1$$ For 1, I got stuck at the ...
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### Why does the Method of Successive Approximations for a Differential Equation work?

Time dependent perturbation theory in quantum mechanics is often derived using the Method of Successive Approximations for a Differential Equation. I have not seen an explanation or a more rigorous ...
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### Describing multilinear maps as linear operators

Let a finite dimensional complex vector space $V$ be given. Let $T^k(V)$ denote the vector space of multilinear maps $V^k\to\mathbb C$. My original question was going to be as follows: Does there ...
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### Is open induction as strong as bounded induction without free bounds?

As was established in my question here, one reason that $Q$ + induction on formulas with bounded quantifiers is stronger than $Q$ + induction on quantifier-free formulas is that the variable that ...
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### How to find the following Integral

I am unable to get the following integral. I know the basics of integration. I have tried looking it up but to no avail. $\int_0^\infty x^{-\frac{1}{2}}e^{-\frac{x}{2}}\,dx$ Thanks for the help.
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### Problem involving trigonometric equality

Hello I am having some problems trying to solve the following trigonometric equation when is $$-\tan(x) +3\sin(x) = \cos(x)$$ on the interval $0$ to $2\pi$.
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### Does $xe^{1/\log(x)}$ have an oblique asymptote when x tends to infinity?

Does $xe^{1/\log(x)}$ have an oblique asymptote when $x$ tends to infinity? If so, what is the equation of this asymptote and how can we find it?
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### Olympiad inequality: is this reasoning sound?

I am trying to show that for $a,b,c>0,\;abc=1:$ $$\underbrace{\frac{1}{b(a+b)}+\frac{1}{c(b+c)}+\frac{1}{a(c+a)}}_{X}\geq \frac{3}{2}$$ This problem is from the Zhautykov Olympiad of 2008. ...
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### I need help in limits

Find the values of real constants $a$ and $b$ such that $$\lim_{x\to \infty} \sqrt{ax^2 + bx} - \sqrt{x^2 + x + 1} = 1$$ If I obtained $a = 1$ and $b=0$, is that the correct answer? Or is b ...
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### Distance between a point and a plane

I was just working on some review textbook problems in James Stewart's Multivariable Calculus when I encountered a problem that looked like the following: Find the distance between the point ...
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### Prove that if $\sum{a_n}$ converges absolutely, then $\sum{a_n^2}$ converges absolutely

I'm trying to re-learn my undergrad math, and I'm using Stephen Abbot's Understanding Analysis. In section 2.7, he has the following exercise: Exercise 2.7.5 (a) Show that if $\sum{a_n}$ converges ...
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### Why is echelon form important?

My professor gave us this definition for a system of equations in echelon form: A system of m linear equations in n variables is called an echelon system if m ≤ n. Every variable is the ...
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### Will the rules of calculus stay the same when a real-valued function is defined over infinite number of variables?

So the question would be: Can we ever talk about a real-valued function that is defined over infinite number of variables? Will the rules of calculus remain the same for such functions described in ...
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### Proving closed sets [duplicate]

Please may you help with the following question: let E be a non-empty subset of R. Let E' be its derived set(the set of all the limit points of E). How to prove that E' is a closed set. closed set ...
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### Understanding quaternions & gradient descent in a paper on inertial / magnetic sensor arrays

I hope this question is appropriate here! I and a friend at work are trying to understand Sebastian Madgwick's paper, "An efficient orientation for inertial and inertial/magnetic sensor arrays" ...
### Characteristic of a finite ring with $34$ units
Let $R$ be a finite ring such that the group of units of $R$, $U(R)$, has $34$ elements. I would like to find the characteristic of $R$. Let $k:= \mathrm{Char}(R)$. If $\varphi$ denotes the ...
Can we have distinct positive real $x,y,z \neq 1$ with $$x^{\left( y^z \right)} = y^{\left( z^x \right)} = z^{\left( x^y \right)}$$ in cyclic permutaion? It does not work well if any ...