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Questions tagged [interval-arithmetic]

Interval arithmetic is the arithmetic of quantities that lie within specified ranges (i.e., intervals) instead of having definite known values.

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Determine the number of triplets $(A,B,C)$ of non-degenerate intervals that simultaneously fulfill some conditions:

the problem Determine the number of triplets $(A,B,C)$ of non-degenerate intervals that simultaneously fulfill the conditions: one is closed, one open and one semi-open $A\subset (B \cap C)$ $B\cup C=...
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Is it common to use an inverted square bracket for a half-open interval? [duplicate]

I found something like $[a,b[$ in someone's mathematical writing. I wondered if it was a typo − that one meant to write $[a,b]$ − or that one perhaps wanted it to mean the same thing as the more ...
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Properties of disjoint n-intervals whose finite union is n-interval

First let's define what an n-interval is: $$I = \left[a_1, ..., b_1\right] \times ... \times \left[a_n, ..., b_n\right] \subseteq \mathbb{R}^n$$ Now suppose we have $I_1, \dots, I_m$ n-intervals with ...
A.Lugini's user avatar
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When is the quantile function of a gamma distribution concave?

I am thinking about the consequences of adding prediction intervals and the consequence it has on the resulting interval. For example, I am considering when to expect the sum of two such intervals to ...
Galen's user avatar
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Under what assumptions on f can I calculate f([a, b])?

I know that: If $f$ is continuous and monotonous on $[a, b]$, then $$f([a, b]) = [\min(f(a), f(b)), \max(f(a), f(b))]$$ If $f$ is continuous and unimodal on $[a, b]$ with an extremum at $x_0 \in [a, ...
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Determine the minimum value of $S=p+q$

question For a fixed natural number $ n \geq 1$ we consider all rational numbers of the form $\frac{p}{q}$, with $p,q \in N*$ and $\frac{n - 1}{n } < \frac{p}{q }< \frac{n}{ n+1}$ Determine the ...
IONELA BUCIU's user avatar
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Refinement of Lipschitz Continuous Functions on Interval Subdivisions

Hi I am working my way through this interval analysis textbook but I am confused about one of the properties (Theorem 6.1 on page 55-56) http://interval.ict.nsc.ru/Library/InteBooks/IntroIntervAnal-...
Connor's user avatar
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Proving that if $x, y$ are in an open set $S$, then there exists some $x < z < y$ where $z \in S$

Consider a set $S \subseteq \mathbb{R}$. I have learned that $S$ is an open set if $(\forall a \in S)(\exists \epsilon \in \mathbb{R}^+)(\forall b \in \mathbb{R})(|a - b| < \epsilon \implies b \in ...
Christopher Miller's user avatar
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Correctness of multiplication in ball arithmetic

We define $B(c,r) = \{x \in \mathbb{R} \ : \ |x - c| \le r\}$ with $r \ge 0$. Given two balls $b_1 = B(c_1, r_1)$ and $b_2 = B(c_2, r_2)$, multiplication is defined as follows: $$ b_1 * b_2 = B(c_1 ...
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Why if f(x) = 2x - x on [-1, 1] results in [-1, 2] in Interval Arithmetic?

From the Section 3.1 of this paper The Dependency Problem. The main downside of interval arithmetic is that the computed bounds may be extremely pessimistic. As an example, consider the simple ...
Cedric Martens's user avatar
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3 answers
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Why do you get zero by squaring [-1, 1] in interval arithmetic?

In the Wikipedia page on Interval Arithmetic, the example for the Dependency problem is that $f(x) = x^2 + x$ on the interval $\left[-1, 1\right]$ is $\left[-1, 2\right]$. I don't understand why this ...
Cedric Martens's user avatar
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complement of the given interval

A set $M \subseteq \mathbb{R}$ is called open if $\mathbb{R} \setminus M$ is closed. How do you prove that $(2n -1/2, 2n + 1/2)$ is open? I‘m confused because it means $2n - 1/2 < 2n + 1/2$ but how ...
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Differentiable formula that computes rotations in interval arithmetic (bounding box of a family of rotated rectangles)

I want to know how to perform a rotation in 2D interval arithmetic. That amounts to computing the tightest interval containing $$ x\ \mathrm{cos}(\varphi) + y\ \mathrm{sin}(\varphi), \tag{1}$$ where $...
Will's user avatar
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Can functions increase on critical points that are not turning points?

I am reading this lecture. Therefore, I am asking the following question in terms of what the lecture states, and not in terms of what any other individual or group advocates, because I need to answer ...
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Evaluating a 2D cubic Bézier curve with interval coefficients with interval arithmetic

I would like to know how to evaluate 2D cubic Bézier curves at an interval when the Bézier coefficients themselves are intervals. If the coefficients are not intervals, evaluating a Bézier curve on an ...
Will's user avatar
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4 votes
2 answers
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Definition of an Interval

I want to define both real intervals and integer intervals, and I want to avoid duplication of effort. As such, I put together the following definitions. I wanted to share and get feedback/critiques ...
Ryan Pierce Williams's user avatar
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Specific integral $u$ to create the same minimizaton problem

I have the problem A: $$\min (-x_{k\{ n \}}+u*(\sum_{k=1}^{K}x_{k\{ n \}})+\sum_{k=1}^{K}y_{k\{ n \}}))$$ $$\text{where} \quad 0<x_{k\{ n \}}<l, 0<y_{k\{ n \}}<m$$ The problem is that it ...
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How can interval arithmetic be used to determine absence of roots on an interval?

There exist libraries that claim to use interval arithmetic for conservative function graphing, specifically recursive algorithms where interval arithmetic is used to prove absence of roots on a given ...
Ocelot's user avatar
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What does "interval straddling another interval mean?"

I'm looking at the comments in a code base that looks for section of time intervals, and there's this comment saying "If the intervals from the source are straddling the destination interval ...
24n8's user avatar
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Interval quasi-Newton methods?

I am looking to find all zeroes of a smooth function $ f \colon [0,1]^2 \to \mathbb{R}^2, $ using interval arithmetic. The standard way to do this seems to be to use the interval Newton method $$ x_{n+...
Will's user avatar
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How to write (half)-open intervals? [duplicate]

In this Stackoverflow post, the author writes about the interval (26,100]. I believe he means that 100 is included while 26 is excluded. In my country (I'm from ...
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The number of disjoint intervals over which the function $f(x) = |0.5x^2−| x | |$ is decreasing is

The number of disjoint intervals over which the function $f(x) = |0.5x^ 2−| x | |$ is decreasing is A)one B)two C)three D) none of these I actually solved it and the answer is three but I want to ...
sachin's user avatar
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Is there any notation that can get the left boundary value of an real interval?

I have an sequence of half-open interval named with $$\langle\tau_i\rangle_{i=1}^3 = \langle [0, 1),[1,2),[2,3)\rangle.$$ I want to get the left and right boundary value of $\tau_2$, which is 2 and 3, ...
yuki amezaki's user avatar
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In this function, will we include end points for decreasing interval?

Which of these statements is/are correct? The values are correct but am not sure about the interval. Are both correct, or is only one correct? f(x)= sinx+cosx , 0 ≤ x ≤ 2π f(x) is decreasing on ...
Purab Bajaj's user avatar
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Solve ODE IVP with different conditions in finite Real range

How to I approach the following problem? Suppose $\phi (x) : [0,5] \rightarrow [0,1] $ is an IVP given by: $$\phi (x) := \left \{ \begin{array}{l} x, \forall x \in [0,1] \\ 1, \forall x \in (1,5] \end{...
DrThirsty's user avatar
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Nested intervals lemma

In a proof of a theorem about nested intervals (see https://en.wikipedia.org/wiki/Nested_intervals#Theorem), I extracted following lemma, which was used without any proof. Therefore I tried to prove ...
Jo123's user avatar
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The Lebesgue Measure of a subset that contains no intervals

I have taken out a book to help me understand measure theory better, one of the examples (that one has to do alone) asks: 'Show there is a closed subset of [0,1] that contains no interval but it's ...
murpw2011's user avatar
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2 answers
58 views

Can we predict the behavior of a linear composition of known functions?

Let $$f(x) = a_1f_1(x)+...+a_Nf_N(x)$$ be a function composed of known continuous and smooth functions $f_1...f_N$ and $a_1...a_N$ some constants. the term "critical point" is defined as the ...
Hosein Javanmardi's user avatar
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1 answer
62 views

If the intervals converge, do the members also converge under the same function? [closed]

If $S$ is is a sequence of intervals over the reals starting with $S_0 = (a,b)$ where $a,b\in\mathbb{R}$ and the rest are generated by some function $f : S_n → S_{n+1}$ If I can show that: $S_{n+1} \...
LeiMagnus's user avatar
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Why is $[-1,1]^2 = [0,1]$?

I have an example of a function $f(x)=x^2+x$, this is specifically for an example explaining the dependency problem. In it we are interested the values over the interval $[-1,1]$, with them being $[-\...
Ellis Thompson's user avatar
1 vote
1 answer
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How do we use the epsilon-delta notations in the context of a limit of a sequence of sets?

This is in the context of "set theory" and "sequence of sets" and their corresponding notations. I was trying to solve a question which says: $C_1, C_2, C_3 \dots$ is a non-...
koustav_ch's user avatar
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2 answers
73 views

We can show bijection between [a,b] and [c,d] in $\mathbb(R)$. But how can one establish an bijection between $[a,b] and (c,d)$. [duplicate]

We can show bijection between [a,b] and [c,d] in $\mathbb(R)$ by send $x \in [a,b]$ to $(d-b)/(c-a)x + bc-da$. But how can one establish an bijection between $[a,b]$ and $(c,d)$ can we explicitly ...
Samrudhi Thakar's user avatar
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Generalization of boolean logic on intervals

I'm working on a generalization of Boolean algebra in a way that it can encompass 'series' of true/false values. My motivation is to introduce logic gates which are mapping from $\{0, 1\}^n$ to $\{0, ...
Mark Lumar's user avatar
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1 answer
41 views

Find the intervals to which $p$ belong for these $4$ conditions

Suppose there is a quadratic equation $$f(y)=y^2-(p+1)y+p^2+p-8=0$$ then find the intervals in which $p$ should belong to fulfill the following conditions $1.$ both the roots are less than $2$ $2.$ ...
Vanessa's user avatar
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How to replace two binary additions by one ternary

Consider the "sum" of two intervals of natural numbers $\sigma_i = \langle n_i, k_i \rangle := \{ n_i,n_i+1,\dots,n_i + k_i\} $ being the set of numbers $\sigma = \sigma_1 \oplus \sigma_2 $ ...
Hans-Peter Stricker's user avatar
3 votes
2 answers
69 views

Which interval is correct here$?$

The equation $$2\textrm{sin}^2\theta x^2-3\textrm{sin}\theta x+1=0$$ where $\theta \in \left(\frac{\pi}{4},\frac{\pi}{2}\right)$ has one root lying in the interval $(0,1)$ $(1,2)$ $(2,3)$ $(-1,0)$ ...
user avatar
2 votes
2 answers
124 views

$x^2-px+p^2-4<0$ for at least one $x<0$

Find the set of all values of $p$, for which $x^2-px+p^2-4<0$ for at least one $x<0$. My work: Let $x=k<0$ be a root of this inequality. So our inequality becomes $$k^2-pk+p^2-4<0$$ Here $...
user avatar
1 vote
1 answer
43 views

Integer Inequality for Repeating Intervals

I've tried poking around a little bit for a more formal way of asking this question, but can't seem to find anything. So you'll forgive my ignorance and (undoubtedly) sloppy notation/ terminology. Let'...
Sav's user avatar
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1 answer
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Trouble understanding indexed sets

Lets say I am given " find the set" $$\bigcup_{k∈N}B_k$$ $$ B_k = \left[ \frac{3}{k}, \frac{5k+2}{k} \right) \cup \{10+k\} $$ I understand that $k$ is an argument and $N$ is a set however ...
ramanujans alkhazarim's user avatar
2 votes
1 answer
204 views

Let $K_n:=(n,\infty)$ for $n\in\mathbb{N}.$ Prove that $\bigcap_{n=1}^\infty K_n=\emptyset.$

I did a proof for exercise 2.5.9 of Introduction to Real Analysis: 4th edition by Bartle and Sherbert. The exercise is: Let $K_n:=(n,\infty)$ for $n\in\mathbb{N}.$ Prove that $\bigcap_{n=1}^\infty ...
blakedylanmusic's user avatar
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Show that $\mathbb{R}$ and the open interval $(-1, 1)$ have the same cardinality.

I am a little confused about using functions to show that two sets of intervals have the same cardinality. I believe that if we can find a bijective function $f$ such that $f: \mathbb{R} \to (-1, 1)$, ...
Jason Chiu's user avatar
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1 answer
75 views

Evaluating sets and interval union, intersection and exclusion [closed]

I am given the sets A = {2,4,6,8}, B = [2,6)andC = (3,8)`. Calculate each of the following. ...
yowhatsup123's user avatar
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1 answer
48 views

Show/"sketch" which area f is strictly positive.

I have the following question: Let $c>0$ and consider the following function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ given by: $$f(x,y)=\begin{cases} c y e^{-x} & \text{0 < x < $\infty$...
Mugge513's user avatar
1 vote
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34 views

Let `x` be a real number between two values, let `y` be a real number between two other values. What's the range of `x/y`?

Like the title says: $$ x \in \Bbb R : x \in [x_{min}, x_{max}] $$ $$ y \in \Bbb R : y \in [y_{min}, y_{max}] $$ What is the range of: $$ x \over y $$ ? I find this hard to reason about because the ...
Helloer's user avatar
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When given an interval and asked to operate on it do we consider intersection or union?

I had a question: Let $A=[-1,4]$. Find $A^2$ and $\frac{1}{|A|}$. Someone told me that the answers are $A^2=[0,16]$ and $\frac{1}{|A|}=[1,\infty]$. However, if you notice in the first part we are ...
This is not me.'s user avatar
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187 views

Constant function if the derivative is null

Let $f$ be a function defined on $(a,b)$. I know that $$f'(x)=0,\ \forall x\in(a,b)\iff f\, \text{constant in}\, (a,b)$$ I know that if the domain of the function is not an interval then this result ...
pawel's user avatar
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1 vote
1 answer
100 views

Confused about interval notation

I have a function $f$ and a subset $A$ of its domain, $f(A)$ representing its range. Letting $f(x) = x^2$ and $A = [0,2]$, is it correct to calculate range $f(A)$ like this: $A = [0,2] \implies f(A) = ...
GMoss's user avatar
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Equivalent definition for an open interval around a point in $\mathbb{R}$

Let $I$ be an open interval containing $a\in\mathbb{R}$. I know that $I=(\inf(I),\sup(I))$. Is it possible to find $\delta>0$ such that $(a-\delta,a+\delta)=I?$ This would imply that $\sup(I)-\...
Vab22's user avatar
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2 votes
1 answer
149 views

Defining an interval as a subset of the naturals

$x \in (a,b) \subset \Bbb N$ as a way to say "$x$ is any natural number in the interval $a < x < b$." I like this expression better than $x \in \Bbb N, x \in (a,b)$, but I'm not sure ...
user110391's user avatar
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1 vote
1 answer
249 views

Length of symmetric difference of two intervals.

I have two intervals characterized by the pairs $a,b$ and $c,d$, all numbers larger than zero: $$S_1 = (\min(a,b), \max(a,b))$$ $$S_2 = (\min(c,d), \max(c,d))$$ The length $s$ of an interval is ...
user196574's user avatar
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