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Questions tagged [absolute-value]

For questions about or involving the absolute value function.

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Absolute value of tensor product of fields

Suppose that we have the Laurent series fields $F_1:=\mathbb F_p((X))$ and $F_2=\mathbb F_p((Y))$. Equip $F_1$ with the $X$-adic multiplicative absolute value $|\cdot|_1$, i.e. define $|X|_1=\dfrac{...
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124 views

How do you prove this implication using its contrapositive?

$∀x∈R,|2x-1|≤5$ and $|2x-1|>3⇒(x^4+7≤7x^2 )$ or $(2x^3≥8x+5)$ This is what I got for the contrapositive: $∀x∈R,(x^4+7>7x^2 )$ and $(2x^3<8x+5)⇒|2x-1|>5$ or $|2x-1|≤3$ Where would I ...
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Find $x \in \mathbb{Q}(i)$ with $ |x - 1|_{2+i} < \frac{1}{\sqrt{5}} $, $|x+1|_{2-i} < \frac{1}{\sqrt{5}}$ and $|x|_{7} < \frac{1}{7} $

I wanted to try some examples with adeles and strong aproximation. Let $\mathfrak{p}_1 = 2+i$ and $\mathfrak{p}_2 = 2-i$ and $\mathfrak{p}_3 = 7$. Can we a single number $x \in \mathbb{Q}(i)$ that's ...
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110 views

Undo a convolution involving an inverse Laplace transform and definite integral

While asking my previous question , I wanted to solve $(1)$ that is stated below this sentence: $$\mathcal{L}_\text{s}^{-1}\left[\frac{1}{1+\text{L}\cdot\text{C}_2\cdot\text{s}^2}\cdot\mathcal{L}_t\...
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48 views

Is this triangle inequality correct?

I'm reading a book of Calculus and in the proof of the sum law of Limits I came across this: "By the triangle inequality, we have $|f(x)-L_1| + |g(x)-L_2| \le |f(x) - L_1 + g(x) - L_2| = |f(x) + g(x) -...
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92 views

What is the Name of the Function $f(x) = \frac{x + |x|}{2}$

I'm dealing with a function that can be written in all of the following forms: $$f(x) = \frac{x + |x|}{2} \\ = x\ \Theta(x) \\ = \int_{-\infty}^x \Theta(y) \operatorname{d}y,$$ where $\Theta(x)$ is ...
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50 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| +...
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101 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 =...
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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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54 views

Absolute Value of a Integral, separation

This is a homework exercise, and ''Thomas early trascendentals'' book vol.4 says in property that : Def: To integrate an absolute value function, we have to look for specific cases when ; $ v(t)\...
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25 views

Max function with absolute values for arbitrary number of arguments

The maximum between two numbers $x$ and $y$ can be easily written as $$ max(x,y) = \frac12\left(x+y +|x-y|\right). $$ We can obviously generalize this to any number of arguments as $$ max(x_1,\dots,...
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On the maximal unramified extension of $\mathbb{Q}_{p}$ of a given degree.

I'm stucked with a theorem withouth proof saw in a Book about G-Functions by Dwork, I will appreciate any hint, also I provide a ''proof'' of that theorem, but a feel that is too ''bla bla'' and I ...
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57 views

Calculating integral of absolute integrand

I want to calculate the integral $$ \int_0^1\int_0^1\int_0^1 {\rm d}r_1{\rm d}r_2{\rm d}r_3 \, r_1 r_2 r_3\int_0^\pi \int_0^\pi \int_0^\pi {\rm d}\phi_1{\rm d}\phi_2{\rm d}\phi_3 \\ \left| r_1r_2\sin(\...
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50 views

Getting rid of the absolute value in the resolution of a differential equation

I have this differential equation as an exercise: $y' = y^2 - uy$, where u is a real parameter. While trying to resolve it, I got to a point where I have: $|\frac{y-u}{y}| = Ce^{ux}$, where $C$ ...
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36 views

Squeeze Rule Proof variant

The usual proof for the squeeze rule for sequences is the following: However I did a different one but it looks strange that it is normally not used as it's way shorter. First of all I write down ...
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307 views

Minimization of a sum of absolute values

Let $X_{1},...X_{n}$ be a random sample with some known pdf. I found the log likelihood. Now I would like to know how to minimise the following expression in order to minimise this likelihood and ...
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74 views

Definite integral of absolute values over a simplex

I'm attempting to evaluate the following integral (note that $v_n=1-v_1-v_2-\dots -v_{n-1}$, and assume $n$ is even): $$ I_n=(n-1)! \int_{v_1=0}^1 \int_{v_2=0}^{1-v_1} \cdots \int_{v_{n-1}=0}^{1-v_1-\...
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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 ...
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215 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 ...
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63 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: $...
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64 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 ...
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47 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 ...
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47 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}}$...
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84 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\...
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120 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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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 ...
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643 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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123 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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106 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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Integrating $\left|f(x)\right|$ by pulling out $\mathrm{sgn}(f(x))$ from the integral

I tried doing the following integral: $\int_{0}^{\pi/4}\sqrt{1-\sin2x}\mathrm dx$. Firstly I completed the square by rewriting $1$ as $\sin^2x+\cos^2x$ to get the integral revised to this form: $$I=\...
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Rephrasing what this proof is asking

I am new to proofs and am still struggling to parse them. I am not looking for a proof to the following statement; just guidance as to where to start or what the shape of a proof for it looks like. ...
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Is there a symbol that takes the absolute value of each component of a matrix or vector in the linear algebra?

Is there a symbol that takes the absolute value of each component of a matrix or vector in the linear algebra? For example, Abs$(-1,-2)=(│-1│,│-2│)= (1,2)$
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28 views

On the Newton Polygon for $p-$adic Power series

I'm studyng a Book about $p-$adic numbers, and I have troubles with a "degenerate" case of a Newton polygon. Let $f(X)=\sum a_{i}X^{i}\in\mathbb{Q}_{p}[\![X]\!]$, we define the Newton poligon of $f$ ...
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85 views

On an Inequality for the Riemann Zeta Function

Okay, firstly a bit of background to set the scene. My question comes from the approaches made by R. Spira in his paper, "An inequality for the riemann zeta function," regarding the initial steps he ...
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Multiplicative absolute value

The absolute value of a real number $r$ is defined to be the additive inverse of $r$ is $r < 0$, and $r$ is $r \geq 0$, where $0$ is the additive identity of the commutative group $\mathbb{R}$. ...
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58 views

The image of a valuation is dense in $\mathbb{R}$

I'm reading Dwork's Boook An introduction to G-Functions and I'm stuck in some part of a proposition. We say that $|-|$ is a valuation in the field $K$ (With values in $\mathbb{R}_{\geq 0}$) if 1) $|...
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Is there an extension of the reverse triangle inequality to $n$ variables?

The triangle inequality for real numbers states $|x+y| \leq |x| + |y|$. This is extended easily on induction to the corresponding result for $n$ variables $$|x_1 + x_2 + ... + x_n| \leq |x_1| + |x_2| +...
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How to Solve a System of Absolute Value Equations Algebraically

How can one find the points of intersections of the two graphs algebraically? $$f(x) = |(x-1)^2-4|$$ $$f(x) = |x+3|-2$$ I understand that it could be solved graphically and through some sort of ...
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GRE absolute values multiple answer question

This GRE question is confounding me. The answer given is A, C, D and E. First off, if you plug in zero for all of the variables, all of the options are right. I see nothing wrong with saying $|0| = ...
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How to solve an integration when the increment is in absolute value?

I have seen many solutions of integration of absolute values such as $\int |x| dx$ however how do you proceed when you have something like this $\int x |dx|$. I have been struggling with this for ...
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Is the $\left|\frac{f(x)}{g(x)}\right|$ always equivalent to $\frac{\left|f(x)\right|}{\left|g(x)\right|}$

Can I split the modulus of a rational function into two parts like this $$ \left|\frac{f(x)}{g(x)}\right|=\frac{\left|f(x)\right|}{\left|g(x)\right|} \ $$ Is this statement always true for any ...
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37 views

always get absolute value of unknown input data (sometimes positive/sometimes negative)

I need to find an absolute value for a variable, but it is homomorphic encrypted. So, sometimes my x might be x=2, or other times x=-4. I want my function to always return the positive value even if ...
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116 views

Minimizing sum of absolute values

here is my objective function: \begin{equation*} \begin{aligned} & \underset{}{\text{minimize}} & & \frac{1}{n} \sum_{i=1}^{n} \alpha|x_i| + (1-\alpha)|x_i-c| \\ & \underset{}{\text{s....
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23 views

What should $\mu$ be here in order for the logistic map to be stable?

I think this might be an error in A Survey of Computational Physics: Introductory Computational Science by Landau. On page 292, he mentions that in order for the logistic map to be stable, we must ...
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44 views

Peculiar Integral with Power Laws

I am trying to calculate the potential for a system of two non-linear coupled oscillators. To do that I would have to calculate the following integral: \begin{equation} V(x,y)=\alpha \int_{}|x|^{n-1}x ...
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91 views

Proof of Corollary 2.4 in Stein and Shakarachi Fourier Analysis

The book states that: $$2\pi \hat{f}(n) = \frac{-1}{n^{2}} \int_{0}^{2\pi} {f^{\prime \prime}(\theta) e^{-in\theta} d\theta}$$ Therefore, $$2\pi |n|^{2} |\hat{f}(n)| \leq \left| \int_{0}^{2\pi} {f^{\...
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71 views

For the x=a path, the limit doesn't exist. Does the multivariable limit exist?

I have the following limit:$$L=\lim_{(x,y) \to (1,1)} \dfrac{x^2+y^2-2}{|x-1|+|y-1|}$$ If I take the limit for $x=1$, then we get:$$L_1=\lim_{y \to 1_{+}}\dfrac{y^2-1}{|y-1|}=2$$, and $$L_2=\lim_{y \...
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69 views

Boundaries for the domain of an absolute value function's inverse

I have the function $f(x) = \mid 2x-1 \mid$, and I'm trying to find its inverse function(s). I found an inverse relationship $\frac {1 \pm x}{2}$. In order to get inverse functions out of this ...
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68 views

Absolute value inequality example

Can anyone please help me with this example from Spivak? I am math autodidact. I have to express this without absolute value: $$a-|a-|a||$$ The answer is $$\begin{cases}a&a\ge0\\ 3a&a<0\...
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318 views

Rewrite an expression without absolute value signs

I'm trying to rewrite $2x − |x − |x + 1||$ without the absolute value signs. If I input this into Wolfram Alpha, it returns the alternate form: $2x-\sqrt{\left(x-\sqrt{(x+1)^2}\right)^2}$. Any help on ...