Questions tagged [absolute-value]
For questions about or involving the absolute value function also known as modulus function.
3,226 questions
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Linearization Question for max-min|x| Bi-level Optimization Problem
I'm currently working on a bi-level optimization problem with the following structure:
max min |x|
I attempted to linearize this problem using the following approach:
Introduce an auxiliary variable ...
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Evaluating the definite integral of an absolute value involving trigonometric functions
I'm trying to evaluate the following definite integral but am unsure how to handle the absolute value efficiently over the given interval. Any hints on finding the points where the expression inside ...
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Handling the absolute value in this derivation of the quadratic formula
I tried to follow many explanations of how the quadratic formula is obtained, but I get lost at a particular step.
I know we start from $ax^2+bx+c=0$ and then we do some algebraic manipulation in ...
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Solving $|3d-5|+7< 4$. Why is my answer wrong?
Solve the following absolute value inequality
$$|3d-5|+7< 4$$
My Steps:
Subtract 7 from both sides
$$|3d-5|<-3 \tag1$$
Remove the absolute value bars and solve for the variable
$$3d-5<-3 ...
- 137
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Approximate order statistics for folded normal random variables
Assume we have $n$ sampled random variables $X_{1:n}$ from a gaussian distribution with mean $1$ and variance $\sigma^2$.
Problem formulation: After first finding the absolute value and then sorting ...
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How to solve the definite integral $\int_{-1}^{1} \sqrt\frac{|x+1|}{|x|+1} \, {\rm d} x$?
I have the definite integral
$$ I_1 := \int_{-1}^{1} \sqrt\frac{|x+1|}{|x|+1} \, {\rm d} x $$
and I am having trouble solving it. The following is a plot of the integrand, $x \mapsto \sqrt\frac{|x+1|}{...
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Expressing piecewise linear functions using absolute value terms
Given a continuous piecewise linear function from the reals to the reals, can it be expressed as a sum of terms of form $k | ax+b |$ ? [assuming a finite number of pieces]
If some piecewise functions ...
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Concern with extraneous solutions for equation $|f(x)|=g(x)$
Suppose I have the generic equation $|f(x)|=g(x)$. When I solve this using casework, I eliminate the extraneous solutions by checking whether the solutions fit within the domain of the absolute value ...
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Finding the range of $k$ such that $-2x + k = 3|x - 4| + 6$ has no real roots. Why does squaring both sides remove values of $k$?
$$-2x + k = 3|x - 4| + 6$$ has no real roots. Find the range of possible values for the constant $k$.
Isolating the absolute value:
$$-2x + (k - 6) = 3|x - 4|$$
After squaring both sides and ...
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$1,2,\ldots,32$ are written on a board, at each step replace $a$ and $b$ with $|a-b|$ and $|32-a-b|$. What is maximum number of $0$s we can get?
This is a problem from the first stage of Computer Olympiad $(2011)$:
Positive integers from $1$ to $32$ are written on a board. At each step, we select two non-zero numbers (say, $a$ and $b$) and ...
- 8,489
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Calculations involving absolute values - is it necessary to write down all the procedures?
I am trying to show that the operation $\ast$ gives an associative binary operation on $\mathbb{R}^*$, where $\mathbb{R}^*$ is the set of all real numbers except $0$, and $\ast$ on $\mathbb{R}^*$ is ...
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Solving an Exponential-Absolute Value Equation [closed]
3^∣x∣⋅(∣2−∣x∣∣)=1
This is a problem, that involves finding real non-integral solutions, we cant just put abstract values, i tried to convert it into logarithmic equation and ended up
a=-log_3 (2-a)
If ...
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Show that $|S_1|+|S_2|+\dots+|S_m|=[\frac{m+1}{2}][\frac{m+2}{2}]$ for any $m\in\mathbb N^*$
The problem
For each $n \in \mathbb N^*$ we denote $S_n=\{ x\in[0,1)| x\{nx\} = \frac{1}{2} \}$. Let $|S_n |$ be the number of elements of the set $S_n$. Show that $|S_1|+|S_2|+\dots+|S_m|=[\frac{m+1}{...
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Show that $|a-\frac{b+c}{2}| +|b-\frac{a+c}{2}|+ |c-\frac{b+a}{2}| \leq 4$ if $|a|,|b|,|c|\leq 1$. [closed]
The problem
Let $a,b,c\in \mathbb R$ such that $|a|,|b|,|c|\leq 1$. Show that
$$\left|a-\frac{b+c}{2}\right| +\left|b-\frac{a+c}{2}\right|+ \left|c-\frac{b+a}{2}\right| \leq 4.$$
My idea
So I ...
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Does $|(y\cdot \nabla)^N f| \, \le \, C_N \, |(y\cdot \nabla) \, f|$?
This is a problem important to me. But I do not know if it is true. And I do not know if there is some geometric perspectives.
Fix $y\in \mathbb{R}^n$. Let
$$
f(x) = |x-y|-|x|,
$$
we CAN restrict $x$ ...
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