For questions about or involving the absolute value function.

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2
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1answer
68 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
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1answer
33 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
24 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
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1answer
27 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
43 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
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1answer
50 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
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1answer
220 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, ...
0
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1answer
39 views

Inequality which involves complex numbers and absolute values

How can I solve the following inequality: $|\frac{(1+(1-\theta)z)}{1-\theta z}| \leq 1$ ? $z$ is a complex number. I have to find the values of $\theta$ for which the inequality is satisfied.
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1answer
20 views

Absolute value of the sume of two complex number

I have a question about the following. $|A+B|^2$, where $A, B $ is complex number. The question is , when can $|A+B|^2$ be equal to $|A|^2 + |B|^2$?
0
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1answer
30 views

Hassle with Absolute Value and Square Root

Are my questions invalid or difficult cause I'm not getting answers since many days? Question 1:      By definition absolute value gives just no of units and does not indicate any ...
0
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1answer
19 views

Inequalities finding the set of solutions

Find the set of solutions to this inequality? $|x − 3| + |x − 6| < 5$ I have been taught to do it by treating $x$ in $3$ separate cases however I am not getting the correct answer. The answer is ...
0
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1answer
45 views

Determine the symmetry of $y=|x-4|$

Determine whether the graph of $y = |x − 4|$ is symmetric with respect to the origin, the $x$-axis, or the $y$-axis. A. not symmetric with respect to the $x$-axis, not symmetric with respect to the ...
0
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1answer
28 views

Differential Inequalities involving Absolute Values

I have to show that $|f '(x)| \leq 1, \ \forall x\in R$. The information I have been given is $|f(x)-f(y)|\leq |x-y|$ ... cauchy schwarz inequality. This is for calculus. Thanks so much.
0
votes
1answer
32 views

Maclaurin series for $\frac{1}{|1+x|}$

I believe that there is no Maclaurin Series for $\frac{1}{|1+x|}$ as the latter is not differentiable at $x=-1$. However, would it be appropriate for me to refer $\frac{1}{|1+x|}$ as 'not a smooth' ...
0
votes
1answer
22 views

Find the value of parameter $m$ such that the equation has real solutions…

For which values of real parameter "m" the equation:$$\sqrt3*|\tan x+\cot x|=4m$$ has real solutions? My only thought is that $m\gt 0$ because the right part of the equation is an absolute value which ...
0
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1answer
48 views

Solve the following inequality…

Can you please verify if I've done this exercise correctly, and if you have a better solution, please, show it to me. Thank you! (The exercise is in the left top corner.)
3
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0answers
38 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
85 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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0answers
88 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
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0answers
32 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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0answers
62 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
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0answers
176 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: ...
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0answers
10 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 ...
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0answers
25 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
11 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
19 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
29 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
16 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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0answers
72 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
37 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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0answers
625 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
128 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 ≤ ...
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0answers
70 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
71 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
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0answers
39 views

How does this textbook determine when this function diverges?

The question is "state the limits of x for which this binomial expansion is valid." $\frac{1}{(1+2x)^3}=1-6x+24x^2-80x^3+...$ Obviously, it's not valid where $(1+2x)^3=0$, so $(1+2x)^3\ne0$ ...
0
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0answers
48 views

Proof of nearest integer equality

Let $N(n)$ be the nearest-integer function undefined on half-integers. There are many valid ways to define $N(n)$, I like to choose $N(n) =\arg \min_{z \in \mathbb{Z}} |n-z|$. Consider the function ...
0
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0answers
23 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 ...
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0answers
25 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 ...
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0answers
37 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 ...
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0answers
39 views

Using the negation of a statement to disprove original statement

Prove the following statement is false by first writing the negation, then proving the negation is true: For all sets, S, if S ⊆ ℕ, then there exists some t ∈ S such that |t| ≥ 1. So far, I've ...
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0answers
23 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 + ... + ...
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0answers
20 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) ...
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0answers
12 views

Difference in magnitude between two cross-correlations by two different way of calculations.

I think there are two ways of calculating cross-correlations for two difference random variables, X and Y. I am assuming discrete functions. 1) Multiplication $$ \sum_{m=-\infty}^\infty x[m]y[m+n] ...
0
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0answers
48 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
63 views

Looking for a counter example: limit of absolute value of $f(x)$

Consider the following: $$\lim_{x\rightarrow a}f(x)=L\Rightarrow \lim_{x\rightarrow a}|f(x)|=|L|$$ I proved it using the "second triangle inequality", but I tried to think why is the reversed ...
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0answers
28 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 ...
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0answers
17 views

Prove that the following function of binary random variables is monotonic

Consider a binary random variable $y$ over the space $\mathcal{Y} = \{+1, -1\}$ such that $\Pr(y = 1) = q$. Consider also $r$ binary random variables $y^1, \ldots, y^1$ over the space $\mathcal{Y}$ ...
0
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0answers
28 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 ...
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0answers
99 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.