For questions about or involving the absolute value function.

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2
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1answer
89 views

Does “Expected Absolute Deviation” or “Expected Absolute Deviation Range” exist in stats and have another name?

So everyone is familiar with Variance and Standard Deviation from high school, but it seems no one has any familiarity with a philosophical justification for such weird, seemingly arbitrary measures. ...
1
vote
1answer
21 views

What is this equal to? : $|A+B|^2$ where $A = P e^{ia}$ and $B = Q e^{ib}$

$A$ and $B$ are two complex numbers: $A = P e^{ia}$ $B = Q e^{ib}$ I would like to know what is this equal to? : $|A+B|^2$ Please also give a small proof if possible.
1
vote
1answer
34 views

If $|\alpha|\leq 1$ and $|\beta|\leq 1$, prove that $|\alpha+\beta|\leq |1+\overline{\alpha}\beta|$

Note $\alpha$ and $\beta$ are complex numbers and $\overline{\alpha}$ is the conjugate of $\alpha$. I've tried using variations of the triangle inequality and I couldn't find anything to work.
1
vote
1answer
66 views

Getting rid of absolute value in integrating factor

If I have this equation $$|I|=e^C |x^3|$$ where $C$ is a constant, yet to be determined. Is it allowed to say: let $A$ be a constant such that $$\begin{cases} A=-e^C \space\space\space ...
1
vote
1answer
30 views

Absolute values and inequalities

So I've been trying to solve this one for a few hours and am now out of ideas on how to approach this problem. Here are the inequalities: $$\text{show that if}$$ $$z,w \in \Bbb C$$ $$|z| < ...
1
vote
1answer
55 views

Rearranging absolute values (limit proof)

My textbook ends a proof with the following: $|x-9| \over \sqrt(x) + 3$ < $\epsilon$ can be rearranged to conclude: |$x-9 \over \sqrt(x) -3$ - 6| < $\epsilon$ However, I don't understand ...
1
vote
1answer
52 views

Evaluating Absolute Value Expression Within Ranges

I am trying to evaluate an absolute value expression but I am struggling to know whether to place a (+) or a (-) on each expression when evaluating each interval. For example, is there a quick ...
1
vote
1answer
246 views

Intersection of a point and absolute value function contained within a circle

I'm attempting some crazy ideas while programming a game and ran into the following math problem that has been bugging me for a few days: Given a unit circle and a random point $P$ within the circle, ...
3
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0answers
39 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
votes
0answers
89 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
votes
0answers
89 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 ...
2
votes
0answers
44 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 ...
2
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0answers
34 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
votes
0answers
97 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 - $$ ...
2
votes
0answers
186 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: ...
1
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0answers
25 views

Consider fourier transformations of $|p(\mathbf{r})|^2$

If we have $\mathbf{k}=(k_x,k_y,k_z)=\frac{2\pi}{L}(j,s,l)$ with $j,s,l \in \mathbb{Z}$ and we have $$p(\mathbf{r})=\sum_{\mathbf{k}}\tilde{p}(\mathbf{k})e^{-i\mathbf{k}\cdot \mathbf{r}} \implies ...
1
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0answers
33 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|+ ...
1
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0answers
30 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 ...
1
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0answers
26 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?
1
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0answers
23 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 ...
1
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0answers
14 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
31 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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0answers
22 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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0answers
25 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 ...
1
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0answers
82 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?
1
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0answers
17 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 ...
1
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0answers
80 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 ...
1
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0answers
42 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 ...
1
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0answers
695 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 ...
1
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0answers
142 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 ≤ ...
1
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0answers
73 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 ...
1
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0answers
78 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 ...
1
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0answers
90 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 ...
0
votes
0answers
49 views

two dimensional Gaussian integral with complex exponent of an absolute value

I am trying to solve the following two dimensional integral: $$\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}{e^{ia(\left|x\right|-\left|y\right|)} \frac{1}{2\pi\sqrt{1-\rho^2}}e^{-\frac{x^2+y^2-2\rho ...
0
votes
0answers
26 views

How to solve an equation with absolute value and x as exponent.

The inequation is: I thought that the solution would be the same as if there was no x as exponent, but in Microsoft Math it says it has no solution and about the equation it said it has solutions. ...
0
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0answers
26 views

equivalence of norms vs equivalence of absolute values

Consider a field, $k$. A norm on $k$ is a map, $|-| : k \to \mathbb{R}$ such that $(1)$ $|v|= \iff v=0$ $(2)$ $|vw| = |v||w|$ $(3)$ $|v+w| \leq |v|+|w|$ and we say two norms $|-|_1, |-|_2$ are ...
0
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0answers
18 views

When does $\overline{U(0,1)}=B(0,1)$ hold?

Given $R$ an absolute valued ring (with unit), sometimes $\overline{U(0,1)}=B(0,1)$ (for example, $\mathbb{Q},\mathbb{R},\mathbb{C},\mathbb{H}$) and sometimes $\overline{U(0,1)}\neq B(0,1)$ (for ...
0
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0answers
20 views

Proving a Comparison

If $\varrho(x,y)=min\{|x-y|,p-|x-y|\}$ ($p>0$ and $x,y\in[0,p)$) then prove that: $\varrho(x,y)\leq\varrho(x,z)+\varrho(z,y)$ This is how far I've gotten: If ...
0
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0answers
23 views

Time derivative of a function involving absolute value

I have a functional looking like this $$ L[u] = \int \int_{\Omega} k \left| \nabla u \right|^2$$ , for which I want to take the time derivative $\dot{L}$. I am not sure that I handled the time ...
0
votes
0answers
7 views

Derivative gradient power metric

I use the the following definition of gradient power metric of an image $I$ $M(I)=\sum_{i,j} \left|\frac{||I|*[-1, 1]|}{\sum_{i,j} ||I|*[-1, 1]|} \right|$ (I take $|I|$ bacause $I$ may have complex ...
0
votes
0answers
34 views

integrate an absolute value periodic function

$$ \int_{-\frac{\pi}{2}}^{t} |\cos{t}|dt = \sin(t-\pi\lfloor(\frac{t}{\pi}+\frac{1}{2})\rfloor)+ 2\lfloor(\frac{t}{\pi}+\frac{1}{2})\rfloor $$ In know that this integral holds. It can be obtained by ...
0
votes
0answers
32 views

Why the plus-minus sign within a pseudo-Riemannian-manifold arc length integral?

Deep with the Wikipedia page on arc length, there exists the following puzzling excerpt (mathematics further marked up by yours truly for readability): Generalization to (pseudo-)Riemannian ...
0
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0answers
51 views

Division Algorithm With Negative and Absolute Value

(a) Prove that $d \, |\, a$ implies that $d \,| (−a)$. (b) Prove that $d\, |\, a$ if and only if $d \,| (−a)$. (c) Prove that $d \,|\, a$ if and only if $d\, \Big|\, |a|$. I can see why these ...
0
votes
0answers
35 views

Definition of the absolute value of a polynomial

I am having a hard time verifying that this is the definition of the absolute value of a polynomial: Given a polynomial with (possibly) complex coefficients: $p(z) = a_0 + a_1 z + a_2 z^2 + ... + ...
0
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0answers
23 views

A system of absolute value equalities

Background: I'm trying to show that the transformation $T:\Bbb R^n\to\Bbb R^n$ defined by $T(x_1,\dots,x_n) := (|x_2-x_1|,|x_3-x_2|,\dots,|x_1-x_n|)$ is (or is not, this is out of curiosity only) ...
0
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0answers
62 views

Simple proof explanation - Possibly triangle inequality involved

I'd like some help with understanding the following statements...I saw it on the internet while searching for a proof, and I'd like to understand why its true: let $A$ be a diagonally dominant matrix ...
0
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0answers
32 views

Formula to convert value to absolute value

This is probably a 'dumb' question (it's a while since I studied maths) but is there a way to convert a value to an absolute value using only the +,-,x,/ symbols? I'm pretty certain that the only way ...
0
votes
0answers
29 views

Questions about $|f(1+a+bi)|<|f(1+a)|$

Let $a,b >0$ and $|*|$ denote the absolute value. Let $f(z)$ be a realvalued analytic function defined for $Re(z)>1.$ For any $a,b$ we have $|f(1+a+bi)|<|f(1+a)|$. Some questions : $1)$ If ...
0
votes
0answers
115 views

Calculation of the sub gradient of the first norm of a matrix

Lets say I have a matrix X and its first norm $||X||_1$. How do I calculate the subgradient of this norm with respect to matrix X itself.
-2
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0answers
20 views

Adjustment of a percentage

Suppose my formula for something used to always be $$\frac{85}{1000} \frac{val}{1-P}$$ It turns out that $1000$ should really be $900$. Rather than changing the formula to $\frac{80}{900}$, I want ...