For questions about or involving the absolute value function.

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4
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117 views

Dealing with absolute values after trigonometric substitution in $\int \frac{\sqrt{1+x^2}}{x} \text{ d}x$.

I was doing this integral and wondered if the signum function would be a viable method for approaching such an integral. I can't seem to find any other way to help integrate the $|\sec \theta|$ term ...
3
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45 views

Nested absolute operations

The question is: are the following two functions equivalent? And if yes, what properties of the absolute value should I use to prove it? $f_1(x,y,z)$ = $|\, x + |y+z| \,|$ $f_2(x,y,z)$ = $| \,|x+y| +...
3
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93 views

Phrase and symbol for “geometric absolute value”$ e^{|\ln(x)|}?$

I'm calculate the median fractional difference between two vectors (to characterise the error in a quantity with a high dynamic range). If $a/b = 0.1$, the fractional difference is $10$, and if $a/b =...
3
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216 views

Proof that there's a unique division quaternion algebra over a locally compact field?

There are many proofs that there is a unique division quaternion algebra over a locally compact field that is not $\mathbb{C}$. For instance this set of notes/book by John Voight contains two proofs: ...
2
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33 views

Proving that $\mathbb{Q}_p$ is not formally real.

I've been looking for a concrete proof without results. The only hint that I have found says: 1) $\mathbb{Q}_2$ contains a square root of $-7$. 2) $\mathbb{Q}_p$ ($p>2$) contains a square root of ...
2
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38 views

Cases in which a certain inequality holds true.

I previously asked this question on this forum, and have been demonstrated counterexamples to the claim that $|a| > |b|$ implies $\big|\frac{b+b^{2}}{a+a^{2}}\big| < 1$, which I had previously ...
2
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133 views

Summation of the absolute value of the variable

The summation of cosine $\sum_{k=1}^N \cos (k x)$ is well known (for example, see the previous question here) and is called Lagrange's trigonometric identity. Is it possible to construct a similar ...
2
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46 views

Reformulate absolute value as quadratic problem

I'm looking for standard approach to reformulate this objective function. The aim is to find values of $x_i$ that are close to either $y_i$ or $-y_i$ ($y_i$s are known) in a least-squares sense: $...
2
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36 views

absolute values and integals

I have the following integral $$\int_{- \infty}^\infty e^{-|x|} dx$$ and the following two questions (1) Since the preimages $x$ determine the the images $e^{-|x|}$ for nonnegative and negative ...
2
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38 views

Around an inequality

I have a very general question, hopefully not too general. Assume that we have real numbers $a_{ij}, b_{ij}$ $(1 \leq i, \: j \leq n)$ such that $-1 \leq a_{ij}, b_{ij} \leq 1$ for all $i,j,$ for ...
2
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40 views

Integral of $|\cos(ax))|\times e^{-x^2/b}$

I can compute the following integral very easily ($a$ and $b$ are real and positive): $$\int_{-\infty}^{\infty} \cos(ax)\times \frac{1}{\sqrt{\pi b}}\cdot e^{-\frac{x^2}{b}}\,dx = e^{-\frac{a^2b}{4}}$...
2
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56 views

Simplifying an expression with absolute values

I am trying to simplify the function $D(\alpha,\beta)$ shown below (with $\alpha,\beta>0):$ $$ D(\alpha,\beta)=\frac{1+\alpha+2\beta}{2} + \frac{|\alpha-1|}{2} - 2 \left(\frac{\frac{1}{2}+\frac{|2\...
2
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37 views

Solving $n \int_{\mathbb{R}}{\left|\frac{1}{n}\sum_{j=1}^n{e^{(itY_j)}}-e^{-\frac{1}{2}t^2}\right|^2}\psi(t)dt$

I have to solve $$ n \int_{\mathbb{R}}{\left|\frac{1}{n}\sum_{j=1}^n{e^{(itY_j)}}-e^{-\frac{1}{2}t^2}\right|^2}\psi(t)dt $$ where $\psi(t)=(2\pi)^{-\frac{1}{2}}e^{-\frac{1}{2}t^2}$ is the density ...
2
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215 views

Fourier Transform of inverse powers of the absolute value

I don't think this question has been asked previously, so here goes. I need to evaluate the following integrals - $$ \displaystyle{{\int\limits_{-\infty}^{\infty}}}\mathrm{d}x\dfrac{e^{-ikx}}{|x|}\...
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35 views

Find the product all real numbers in an equation

What is an easy and fast way to solve the problem without going through all these possibilities: a) $n^2-9n+20>0, 16-n^2>0$, b) $n^2-9n+20>0, 16-n^2<0$, c) $n^2-9n+20<0, 16-n^2>0$, ...
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40 views

Uniform convergence fourier series |sin(x)|

As an exercise we have to calculate the fourier series of |sin(x)| (was no problem) and after that we are meant to show that this series converges uniformly towards |sin(x)|. After thinking about it ...
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33 views

Usefulness of absolute value in optimization algorithms

In a course of Optimization Algorithms at university, professor said that in every algorithm the objective/object function/function cost is defined as: $$f(\bar x)=\lvert x_0 - g(\bar x)\rvert^{2}$$ ...
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31 views

Derivative of the maximum of a function on a interval

My question is as follows: Given a function $f: [-h,\infty) \rightarrow \mathcal{R}$ and the maximum function given by \begin{equation} \max_{s\in [-h,0]} |f(t+s)| \end{equation} for $t\geq0$. Then ...
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29 views

Is this a misprint or am I missing something?

What I'm given is this: Evaluate: x = 5, |x| -2 I'm thinking they probably mean |x|=-2, in which case the evaluation would be false. But then again I second ...
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31 views

Show that $|ax^2+bx+c|<1$ gives us $|c|<1$

$a,b,c \in \mathbb{R}$ and for all $-1<x<1$ Show that if $|ax^2+bx+c|=<1$ So : 1) $|c|=<1$ 2) $|a+c|=<1$ 3) $a^2+b^2+c^2=<5$ For the first one ; if I choose x=0 So |c|=<1 ...
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47 views

Compare absolute values of two expressions!

I have two sets of numbers as follows: $$X = \{x_1, x_2, ..., x_n\}\\ Y = \{y_1, y_2, ..., y_n\}$$ And a number $r$. Let $x^\ast$ and $y^\ast$ is average values of set $X$ and $Y$ respectively. ...
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61 views

Least Square Approximation Using Legendre Polynomials

Obtain a fourth degree least squares polynomial for $f(x) = 1/|x|$ over $[-1,1]$ by means of Legendre Polynomials I got stuck when trying the integral over the given interval. Is there another way ...
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65 views

How $|x|<a\implies a>0$

The title is not exactly what I'm asking, so sorry for that. I was doing a problem in my mathematics text book. It is given that $|x|<a$, I thought if $a=2$ then we can put $x=1$ but what if ...
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27 views

Question on valuation axioms - Relating to $\mathbb{R}$

In an earlier thread, I asked if there was a standard generalization of the absolute value of $\mathbb{C}$ that could be placed on a field, but might not take values in $R_{\geq 0}$. What somebody ...
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36 views

Simplifying a function with absolute value and two variables

I am trying to simplify the following $ f(\alpha, \beta)= \big|1-(\frac{\alpha+\beta}{2}+\frac{|\beta-\alpha|}{2})\big|-2 \big|1-\alpha-(\frac{\alpha+\beta}{2}+\frac{|\beta-\alpha|}{2})\big|+ |\...
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38 views

Solve this inequality for B

I am working on a program that is supposed to qualify a value as "in range" and I have come up with the expression: $$\lvert a-b\rvert \leq c$$ to determine the value. Plugging in test numbers ...
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34 views

why and when t0 use norm instead of abs and vice versa

What is the difference between the norm and abs of an expression.. as far i understand does ||a - z|| mean norm and |a-z| abs , but what is the difference?
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37 views

Sudoku and absolute value equation

I know there is many mathematical way to reformulate the Sudoku problem. I'm wondering if there is a way to reformulate this problem as an absolute value equation : \begin{equation} Ax + B|x|=b \end{...
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20 views

X numbers that when subtracted will produce the same absolute value

Let's say I have X unique numbers and I choose one number y out of this set. Is it possible to create these X numbers such that the absolute difference between y and any other number in X will always ...
1
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0answers
38 views

Trivial absolute value

Let $K/L$ be a algebraic extension. Suppose that $\left|\cdot\right|$ is a absolute value in $K$ such that is trivially in $L$. Then is trivially in $K$. Thanks for anny suggestion. If is trivially ...
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35 views

Signal-extraction knowing both the sum and the sum of the absolute values of normally distributed variables

I have two normally distributed variables $X∼N(μ_{x},σ_{x}²)$ and $Y∼N(μ_{y},σ_{y}²)$. I can observe both the sum of their values and the sum of their absolute values, i.e. $Z₁=X+Y$ and $Z₂=|X|+|Y|$. ...
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42 views

How to to minimize a sum by changing summation order

I have two vectors $(x_1,\dots,x_n),(y_1,\dots,y_n) \in \mathbb{R}^{n}$. I want to find a permutation $\sigma$ such that $$ \sum_{i=1}^n |x_i -y_{\sigma(i)}|^2$$ is minimized. Is there a better way ...
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0answers
176 views

Comparing function to parent function without graphing

How can I compare this function to the parent function without graphing? Where did the 5/4 come from and what steps do I need to take to solve this?
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0answers
21 views

Cumulative distribution function of a model similar to the multinominal distribution

I would like to solve a problem similar to the multinominal distribution (http://en.wikipedia.org/wiki/Multinomial_distribution): For k independent trials each of which leads to a success for ...
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156 views

Absolute values nested multiple times

Is there any algorithm to quickly determine "zero points" (i.e. points with undefined derivation) of absolute values functions which are nested multiple times? I do know, that any part of this ...
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0answers
52 views

Zeta function universality: How to compute the shift parameter for simple functions?

I've come across Zeta function universality. For a nice function $f$ in a nice subset $U$ of the complex strip between real $0$ and $1$, one can find a real $t$, such the zeta function $\zeta$ shifted ...
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981 views

Properly Solving Absolute Value Inequality and Quadratic Inequality Problems

How do I solve the following absolute value inequality and inequality problems properly? 1) $\newcommand\abs[1]{|#1|}\abs{2x+9}>x$ Solving this problem algebraically, I get When $x > 0, x >...
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177 views

Integration of the absolute value of an unknown function

I'm doing a vector arclength problem, and have gotten to the part where I have $\int | r'(t) | dt. $ Both $r(t)$ and $r'(t)$ are unknown functions, though I do know that $0 \leq r(t)$ for $a ≤ t ≤ b$...
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82 views

Banach spaces over complete fields with their own absolute value

Let $F\hspace{.03 in}$ be a field, and let $E\hspace{.03 in}$ be an ordered subfield of $F$. Let $\;\; |\hspace{-0.03 in}\cdot\hspace{-0.03 in}| \: : \: F \: \to \: E \;\;$ be such that for all ...
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93 views

Basic question about $p$-adic expansions

I was recently introduced to the $p$-adic numbers, and have been asked to show that for $n > 0$, the $p$-adic expansion of $\frac{1}{1 - p^n}$ is $\sum_{i=0}^\infty p^{in}$. Could someone tell me ...
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92 views

Fields with their own absolute value

Let $F\hspace{.02 in}$ be a field. $\:$ Let $E\hspace{.02 in}$ be a non-zero subring of $F$. Let $\hspace{.03 in}\leq\hspace{.03 in}$ be a total order on $E\hspace{.02 in}$ that makes $E\hspace{.02 ...
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0answers
24 views

On the Arrangement of Intermediate Subgroups

I am trying to find a journal paper 'On the Arrangement of Intermediate Subgroups' by M.S. Bah and Z.I. Borevich appearing Rings and Linear Groups, Krasnodar (1988), 14-41. This is a Russian text and ...
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0answers
26 views

Is complex multiplication the only multiplication operation on $\mathbb{R}^2$ that works with the Euclidean norm?

What I'm asking is: viewing complex multiplication as binary operation on $\mathbb{R}^2$, is usual multiplication of complex numbers the only operation $\otimes$ on two vectors $\vec{u}$ and $\vec{v} \...
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0answers
62 views

Compare difference between mean and actual

My problem is: I have two sets of numbers as follows: $X = {x_1, x_2, ..., x_n}; Y = {y_1, y_2, ..., y_m}$. Where $r$ is the actual value. $x^*$ is the mean of set X, $y^*$ is the mean of set Y, (n!=m)...
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0answers
14 views

Automorphisms of local field

(1)Suppose that $K$ is a local field but not $\mathbb C$. Then show that any automorphism of $K$ is continuous. (We can assume that $K$ is $\mathbb R$ with classical absolute value or $K$ is a finite ...
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19 views

$\sqrt[n]{x}$ as a power series in a complete field with absolute value.

Let K be a field complete with respect to a non-archimedean absolute value $| \cdot|:K^*\rightarrow \mathbb{R}$ and $char(K)=0$ or $char(K)\nmid n$. I want to prove that if $|x|\leq 1$ (i,e $x$ is in ...
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28 views

The maximum of the absolute value of a real-valued function

For a continuous real-valued function $f$ on a compact set $X\neq \emptyset$, we have \begin{equation*} \max_{x\in X}|f(x)| = \max \{\max_{x\in X} f(x), \max_{x\in X} -f(x)\}. \end{...
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0answers
11 views

generalised eigenvalue problem with absolute value

Problem: $\max_w |w^t A w|-|w^t B w|$ s.t $w^t C w=1$ If there was no absolute values, i.e. if the problem was $\max_w w^t A w-w^t B w$ s.t $w^t C w=1$ this would, by using the appropriate Lagrange ...
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31 views

Sketch subset of $\mathbb{C}$ which satisfies $|z-3-4i|=5$

I proceeded by plotting $z$ on the complex plane, and the modulus of $z-3-4i$: From this I deduced that: $$\begin{align}&\mathrm{Re}(z)=5\cos\theta+3\\&\mathrm{Im}(z)=(5\sin\theta+4)i\end{...
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0answers
15 views

determine the absolute difference in bits between using an indexed colour

Given the following 4 x 4 image, determine the absolute difference in bits between using an indexed colour (where the index uses the lowest number of bits possible, but the colour is represented as ...