For questions about or involving the absolute value function.

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6
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0answers
106 views

Lower bound on absolute value of determinant of sum of matrices

I needed to find a lower bound on $|\det(A+B)|$ where $|.|$ is the absolute value operator. Because I was unable to get such a bound so I was trying to guess a bound and prove it. But ...
3
votes
0answers
43 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
107 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
91 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
votes
0answers
208 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
votes
0answers
119 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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0answers
44 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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0answers
34 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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0answers
37 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
votes
0answers
39 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 = ...
2
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0answers
52 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
votes
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36 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
168 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 - $$ ...
1
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0answers
27 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 ...
1
vote
0answers
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 ...
1
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0answers
42 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. ...
1
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0answers
47 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 ...
1
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0answers
64 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 ...
1
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0answers
23 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 ...
1
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0answers
35 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
34 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
29 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
32 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
19 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
36 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
32 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 $Z₂=|X|+|Y|$. ...
1
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0answers
37 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
144 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
20 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
136 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
vote
0answers
46 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
964 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
168 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
80 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
85 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
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 ...
0
votes
0answers
19 views

Function with absolute value in denominator - limits

f(x)=(x-1)/(|2-x|-1) |2-x|= { |2-x|; x < +2} {-|2-x|; x >= +2} State domain, range and the equations of the asymptotes. D(f)= {x | x > 3 or x < 3} R(f)= {y | y > 1 or y <= -1} ...
0
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0answers
23 views

Continuity properties of an example function $f:\mathbb{R}^n\to\mathbb{R}$

Consider the function $f:\mathbb{R}^n\to\mathbb{R}$ defined as follows: $$ f(x)=\begin{cases} ||x||^2 & \text{if $||x||\le 1$,}\\ 1/||x||^2 & \text{if $||x||> 1$,} \end{cases} $$ where ...
0
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0answers
26 views

Exploring n + abs(x)

In class we are focussing on 'Lattice Theory' at the moment. I have an assignment with instructions saying simply - "Explore the following:" followed by a list of 5 theorems. The purpose of the ...
0
votes
0answers
35 views

How to determine whether expression is positive or negative?

Given expressions $|x - 3 + y|$ and $|x + 3 + y|$ how can I determine, whether are those positive or negative, and determine their value in the intervals of: $y < -x - 3$ $y \in [-x - 3, 3 - x)$ ...
0
votes
0answers
27 views

Cauchy Determinant with Absolute Values

This is perhaps a straightforward question but I'm a little confused. An $n\times n$ Cauchy matrix $A$ is a matrix with entries $$a_{i,j}=\frac{1}{x_i-y_j}$$ for $1\le i,j\le n$, where $x_i$ and $y_j$ ...
0
votes
0answers
56 views

Use axioms to solve inequality $| x-2| +| x-4| < 1$

I have a feeling that the inequality is false for all values of x, but I don't know at which point that should have become clear to me. I am supposed to solve the inequality using $x < a$ ...
0
votes
0answers
25 views

Absolute value with inequalities

Can we solve the following $ |f(x)| + |g(x) | < b$ by taking the intersection of the solutions for $f(x) + g(x) < b$ $-f(x) - g(x) < b$ $f(x) - g(x) < b$ $-f(x) + g(x) < ...
0
votes
0answers
14 views

Graphing an Absolute Value Equation

How would I graph the equation: abs(x)+abs(y)=1+abs(xy). I have tried to consider cases, but am not sure if I need to graph it by considering a piecewise-defined function.
0
votes
0answers
23 views

Turning points of an absolute value equation

Given the equation $$\left| x\; -\; c \right|\; +\; \left| x\; +\; c \right|\; -\; \left| y \right|\; -\; \left| \left| x\; -\; c \right|\; -\; \left| y \right| \right|\; = b \text{,} -c, c, b \in ...
0
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0answers
17 views

Question related to $F(x) = |x-a_1| + |x-a_2|+ … + |x-a_N|?$

Suppose $a_1 < a_2 < \cdots < a_N $. So $F(x) = S_N - 2S_i + (2i-N)x$ if $a_i < x < a_{i+1}$ with $S_i = a_1 + ... + a_i$. Assume $ a_i < u < a_{i+1} < v $, we have: $F(u) = ...
0
votes
0answers
25 views

Absolute Value Inequality Proof - Hint needed

I am having difficulty on a problem. If someone could explain where I should start or what I can use to help solve the proof it would help. Prove that if $|(x+2)| \lt 1$ then $|(x^2 +2)| \gt 3$
0
votes
0answers
18 views

Functional Derivative of Complex Absolutes

It is said, that the lowest order complex amplitude equation shows potential dynamics with the functional \begin{align} V[A] = \int\limits_{a}^{b} \text{d}X\left[- \vert A \vert^2 + \frac{1}{2} \vert ...
0
votes
0answers
23 views

Minimizing the absolute value of the sum, |sum(cx)| ,by linear program

Now I need to use linear program to minimize the absolute value of the sum.The mathematical model of linear program can be expressed as Min |sum(cx)| Subject to |Ax|<=b ...
0
votes
0answers
38 views

Decomposition and valuation rings

I am reading Algebraic Number Theory by A. Fröhlich, M. J. Taylor, it first introduced the theory: $(K,u)$ be a field and its absolute value, $(K_u,\bar u)$ be its completion and absolute value ...