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Questions tagged [domain-theory]

Domain theory is a branch of order theory that studies partially ordered sets which are called domains. Do not use this tag for questions about the domain of a function.

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Domain $C^{2+\alpha}$ proprieties

Let $0<\alpha<1$. Hi! I am reading the paper: How to appoximate the heat equation with Neumann Boundary Condition by nonLocal Diffusion Problem (https://link.springer.com/content/pdf/10.1007/...
Luiza Camile's user avatar
2 votes
0 answers
55 views

The subspace topology of $Y_u$($Y$ with upper topology) is strictly coarser than the one induced from $X$?

Let $(X,\leqslant )$ be a poset, we define the upper topology has the principle upper sets, that is upper sets of the form $\left \{ \uparrow x:x\in P\right \} $, as the subbase. We can define ...
Peter's user avatar
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1 answer
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Is there a situation where any intersection of upward-closed subset is not upward-closed(can be infinite)?

In any quasi-ordered set X, we say that a subset A is upward-closed iff whenever $x\in A \ $and$\ x \leqslant y$, then $y\in A$. Consider the collection $\mathcal{O}$ of all upward closed subsets of $...
Peter's user avatar
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2 answers
130 views

Finding Range and Domain of the function $g(x,y) = \sqrt{x - y} + \sqrt{x^{2} + y^{2} - 6}$

I am trying to find domain and range of a function with equation $$g(x, y) = \sqrt{x − y} + \sqrt {x^2 + y^2 − 6}$$ I have reduced this statement, when $g(x, y) = c = 0 $ $$ g(x, y) = (x+0.5)^2+(y-...
Govt_employee's user avatar
2 votes
2 answers
640 views

Why can't a base be negative in an exponential function?

The function $f(x) = a^x$ is generally taught to only allow $a > 0$. This is usually justified by giving a few examples of complex points in cases where $a < 0$. For example: $f(x) = (-2)^x$, if ...
fedsavi's user avatar
  • 35
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0 answers
79 views

Domain of the biggest value for $abcd\dots$ given $a+b+c+d\cdots=10$

I've seen this puzzle on Flammable Maths new video (https://www.youtube.com/watch?v=vW4TjU4IoPY)(optional to watch). It is as follows: "Given a+b+c+d+e...=10, what is the biggest value for abcde.....
Marvin's user avatar
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0 answers
50 views

How to find the domain and range of a 2 variable function

One very simple example is $$f(x,y) = x^2 - y^2$$ I know that the domain is $\Bbb{R}^2 $ but I don't know a general way to find it in any equation. Also I don't know how to find the range of this ...
Chris Costa's user avatar
1 vote
0 answers
27 views

Is the set of sub-dcpos a dcpo?

Given a dcpo $\mathcal{X} = (\le, X)$, consider the set $\mathcal{X}^{sub}$ of all sub-dcpos of $\mathcal{X}$. Can one define a partial order $\le_{sub}$ on $\mathcal{X}^{sub}$ such that $( \le_{sub}...
mathlete42's user avatar
1 vote
0 answers
21 views

When the functions $f:\mathbb{N}\to \mathbb{N}$ are extended to $f:\mathscr{P\omega\to P\omega}$, do they comprise a subdomain of $\mathscr{P\omega}$?

Functions $f:\mathbb{N}\to \mathbb{N}$ are easily extended to corresponding functions $f:\mathscr{P\omega\to P\omega}$ by $\forall x\in\mathscr{P\omega}:\ f(x)=\bigcup\limits_{i\in x}f(i)$. But, of ...
John Forkosh's user avatar
2 votes
1 answer
56 views

Does $f \leq g$ imply $\text{lfp}\ f \leqslant \text{lfp}\ g$?

If I have two functions $f$ and $g$ and $f$ is pointwise smaller or equal to $g$, i.e. $f(x) \leqslant g(x)$, does this imply that $\text{lfp}\ f \leqslant \text{lfp}\ g$ (provided that their least ...
Tom's user avatar
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0 answers
283 views

Smooth domain $\Omega$

A smooth domain $\Omega$ is an open and connected subset of the whole domain, say $\mathbb{R}^n$, of which the boundary $\partial \Omega$ is "smooth". The smoothness of the boundary, ...
sara's user avatar
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6 votes
1 answer
335 views

Laplace Transform for Linear ODEs with Variable Coefficients

If you take the Laplace Transform of the time-domain ODE $x^2 y'' + xy' - 9y = 0$ and do some algebra, you get the new frequency-domain ODE $s^2 Y'' + 3sY' - 8Y = 0$. If you then apply the same ...
user10478's user avatar
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1 vote
1 answer
340 views

The domain of the functions f(x)=√log1/2(2x-x^2) is?

Given that $$f(x)=\sqrt{\log(2x-x^2)}$$ then find the domain of $f(x)$ My solution: $\log(2x-x^2)\geq0$ $\implies 2x-x^2\geq1 \implies x\leq1$ Also, $2x-x^2\gt0$ $\implies x\gt0$ ,$\ x\lt2$ On finding ...
user281708's user avatar
0 votes
2 answers
438 views

Given $y=\frac{\sqrt{\cos(x)-1/2}}{\sqrt{6+35x-6x^2}}$ find the domain of the function

Find the domain of the function $$y=\frac{\sqrt{\cos(x)-1/2}}{\sqrt{6+35x-6x^2}}$$ I'm unable to find the values for which $\cos(x)\geq 1/2$. PS: This is not a homework question.
user281708's user avatar
2 votes
2 answers
2k views

Are endpoints critical points?

In the function $f(x)=\max\{\sin (x),\cos (x)\}$ for all $x$ belonging to $(0,2π)$ , can we count end points of the domain as critical points, since a function is not differentiable at endpoints?
DarkMan Unknowns's user avatar
8 votes
4 answers
1k views

Understanding Fraleigh's proof of: Every finite integral domain is a field

Here's how Fraleigh proves: Every finite integral domain is a field in his book: Let \begin{equation*} 0, 1, a_1, \dots, a_n \end{equation*} be all the elements of the ...
User31415's user avatar
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0 votes
1 answer
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Construct a discrete subset such that every point of $\partial D$ is a limit point of discrete subset

$D$ is a domain in $\Bbb C$, please construct a discrete subset $E$ of $D$ such that every point of $\partial D\subset \Bbb C$ is a limit point of $E$ $D$ is a unbound domain in $\Bbb C$, please ...
ziang chen's user avatar
  • 7,791
0 votes
1 answer
37 views

Find the domain of the function $f+g$, where $f(x)=\sqrt{4-x}$ and $g(x)=x^2$ for $x\in\mathbb{R}$? [closed]

I know that Dom$ (f) = (-\infty,4]$ Dom $(g) = \mathbb{R}$, but the problem is Dom$(f+g)$. Please also tell Dom$(f-g)$
Rajat Dash's user avatar
-1 votes
2 answers
498 views

Determine the set of all complex numbers satisfying the relation $|z|<4$, here $z$ is a complex number.

Please, solve this above question. Please, tell me how to find domain of a function in terms of complex numbers. I can find domain in terms of real no, but facing issue with complex ones. If I ...
Rajat Dash's user avatar
1 vote
1 answer
52 views

Does anyone know an example of a bounded domain, not L-domain, in which every principal ideal is a join semilattice [closed]

I'm looking for an example of a bounded domain P in which every principal ideal is a Join semilattice but P isn't a L-domain. I have doubts If this kind of poset existe.
Fagner Santana's user avatar
1 vote
1 answer
54 views

Domain not L-domain

I'm looking for an example of a domain(continuos dcpo) which isn't an L-domain(every principal ideal is a complete lattice)
Fagner Santana's user avatar
5 votes
0 answers
106 views

How do we know whether all elements of $[A\to B]$ can be represented as computable functions?

While working through Barendreght's book on the Lambda Calculus, and Abramsky's notes on Domain Theory, I had the following realization: It's often stated that Domain Theory provides a semantics for ...
Alex Appel's user avatar
0 votes
2 answers
173 views

Find the domain of the simplified absolute value expression and state whether it is equal to the original expression

Question: Simplify the expression |x^7+5x|/x where x is negative. Then let h(x) represent the simplified expression and determine its domain, and then decide whether this simplified expression is ...
user avatar
0 votes
1 answer
67 views

Reading a mathametical notation

I am reading a paper on task allocation ...
GENIVI-LEARNER's user avatar
1 vote
0 answers
58 views

Prove that a function defined recursively exists.

In my textbook on computability theory there's a Lemma that's a special case of the Knaster-Tarski and Kleene fixpoint theorems. The lemma states: Let $a \in \mathbb{N} ,\ g:\mathbb{N} \rightarrow \...
gjl's user avatar
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3 votes
0 answers
85 views

Constructing a Galois connection

Consider $C, A_1, A_2$ complete lattices and assume they are related by two Galois connections $\alpha_i : C \to A_i$, $\gamma_i: A_i \to C$, for $i\in \{1,2\}$. I would like to construct a Galois ...
Marta Fornasier's user avatar
1 vote
0 answers
47 views

What's the representation of the identity function using Stoy's discussion of $\mathscr{P\omega}\sim[\mathscr{P\omega\to P\omega}]$?

Stoy, pages 117-122, discusses how to represent functions $f:\mathscr{P\omega\to P\omega}$ as elements $\mbox{graph}(f)\in\mathscr{P\omega}$, definition 7.5 page 120. What would $\mbox{graph}(\...
John Forkosh's user avatar
2 votes
1 answer
76 views

Is the down-closure of a sub-dcpo a sub-dcpo?

Recall that a dcpo is a partially ordered set $(X,\leqslant)$ possessing suprema for all directed subsets. The Scott topology of a dcpo $(X,\leqslant)$ has as closed sets those sub-dcpos which are ...
მამუკა ჯიბლაძე's user avatar
0 votes
0 answers
335 views

Is there a term for extensions of functions on discrete sets to continuous sets?

The pi function, $\Pi(z)$ – defined as $\Pi(z) = \Gamma(z+1)$, where $\Gamma(z)$ is the gamma function – extends the factorial in that $$\Pi(n) = (n)!$$ for all positive integers $n$. In other words,...
HelloGoodbye's user avatar
4 votes
1 answer
226 views

Continuous Lattices - Confusing passage in Dana Scott's paper

In Dana Scott's paper called "Continuous Lattices" from 1971, I am lost in the following passage. This passage occurs in some exposition after his Proposition 2.4. Before going any deeper, ...
Patrick Nicodemus's user avatar
1 vote
0 answers
145 views

Scott topology vs the topology generated by the complements of principal ideals

In Scott topology, it is easy to show that principal ideals are closed sets (and their complements are open). I suppose that in general case Scott topology is not necessarily generated by the ...
Alexey's user avatar
  • 2,162
0 votes
0 answers
31 views

What kinds of non-topological continuity concepts exist between partially-ordered sets

Beside Scott-continuity, what other kinds of continuity concepts are there between partially-ordered sets that can be stated without recourse to topological concepts?
Evan Aad's user avatar
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2 votes
2 answers
121 views

A Scott-like continuity between partially-ordered sets

Given two partially ordered sets $P$ and $Q$, a function $f : P → Q$ between them is Scott-continuous if it preserves all directed suprema, i.e. if for every directed subset $D$ of $P$ with supremum ...
Evan Aad's user avatar
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1 vote
2 answers
44 views

Maximal domain of function and its derivative if both have Lebesgue > 0

Consider $f(x) = \ln(x), \sqrt{x}, |x|$ and $\lfloor x \rfloor$. Based on these, I have the following conjecture. Please prove/disprove. The maximal domain of $f$ and its derivative differ only by a ...
BCLC's user avatar
  • 13.6k
1 vote
0 answers
92 views

The Domain (Co)Monad

Many different kinds of data structures can be captured as Monads. Lists and trees are two good examples. A domain (dcpo) is like a tree, with extra axioms. Definition. A directed subset of a ...
Ben Sprott's user avatar
  • 1,281
0 votes
1 answer
438 views

how to translate from one "domain of a function" to anothe

I have these values, all these values are in the range between $0$ and $1 (0, +1)$. $0.3$ $0.5$ $0.8$ $0.9$ I want to change (translate) these values to a new range between $-1$ and $+1$ $(-1, +1)$, ...
nimo23's user avatar
  • 103
7 votes
0 answers
229 views

How is a Scott domain Cartesian-closed?

I read the following excerpt from nLab but I need further explanation: The problem Scott solved is to faithfully model untyped lambda calculus; in categorical terms, the basic problem is to construct ...
Yan King Yin's user avatar
  • 1,219
1 vote
2 answers
26 views

Fractional ideals: Is $D$ included in $I'I$?

Let $D$ be a domain and let $F$ be its field of fractions. If $I$ is fractional, we define $I'$ to be the set of elements $c \in F$ such that $cI$ is included in $D$. It is clear that if $I$ is ...
user404634's user avatar
1 vote
1 answer
88 views

Find the interval of $c$ such that the rational function $\frac{x^2+2x+c}{x^2+4x+3c}$ takes all real values

Find the interval of $c$ such that the rational function $$f(x)=\frac{x^2+2x+c}{x^2+4x+3c}$$ takes all real values $\forall$ $x\in D_f$ I tried in the following way: Let $$y=\frac{x^2+2x+c}{x^2+4x+...
Umesh shankar's user avatar
3 votes
1 answer
144 views

Limits in a dcpo with the Scott topology

This is exercise 4.7.7 in Non-Hausdorff Topology and Domain Theory by Jean Goubault-Larrecq. Consider a dcpo $(X,\leq)$ and equip it with the Scott topology where the opens are the upward-closed sets ...
mrp's user avatar
  • 5,106
1 vote
1 answer
87 views

what is K-completeness? (Kelly completeness)?

I'm studing Domain theory and working on maximal point space problem. I couldn't find any book or sci-text about K-completness and D-completness!Can any body give me some help with this?
Yuval's user avatar
  • 11
4 votes
1 answer
216 views

Scott topology on $\mathcal{O}X$

In Wikipedya Scott-open set is defined as follows: Def. A subset $O$ of a partially ordered set $P$ is called Scott-open if It is an up set. All directed sets $D$ with supremum in $O$ have non-...
Evgeny Kuznetsov's user avatar
0 votes
1 answer
61 views

Inverse domain problem

My task is to find the domain of inverse of : $f(x) = \dfrac{\cos e^x}{1-\cos e^x}$ . Now, if I am correct the inverse should be : $f^{-1}(x) = \ln\Bigl(\arccos\dfrac {x}{x+1}\Bigr)$ . If I solve ...
marco's user avatar
  • 1
-1 votes
1 answer
21 views

Domain definition for $R(a) = \{(s_1,s_2), ...\}$

If I have $a \in A$ and $s \in S$ and different function values $R(a)$ which, for instance, could be $$R(a_1) = \{ (s_1,s_2),(s_2,s_3),\ldots \}$$ What is the definition of the function $R$? I guess ...
Jamgreen's user avatar
  • 819
2 votes
1 answer
87 views

Find $b$ and $c$ for $f(x)=\log(-x^2+bx+c) $

I have a function $f(x)=\log(-x^{2}+bx+c) $ and the domain of $f$ is $(1,3)$. I have to use this fact to find the values of $b$ and $c$. I thought about solving the inequality $1 < \log(-x^2+bx+c) &...
Patrick Robertson's user avatar
0 votes
1 answer
196 views

Monotonicity, Continuity and Least fix point of function

I am learning semantics of language Haskell and there i come around this question. For answering this, I am thinking in this way: a) For a function to be monotonic, let's say I have 2 arguments: $f$ ...
nobe's user avatar
  • 155
2 votes
1 answer
86 views

What is the relationship, if any, between maximal domains of $f$ and $f'(x)$?

(Edited question after some Calculus review) Let $D$ be the maximal domain of function $f$. Let $D^d$ be the largest subset of $D$ where $f'(x)$ is defined. Maximal domain of $f(x) = \ln(x)$ is $D_1 ...
BCLC's user avatar
  • 13.6k
0 votes
2 answers
68 views

How would you sketch this subset of $\mathbb{C}$?

How would you sketch the subset $\{z \in \mathbb{C} : |e^{z}| <1 \}$ of $\mathbb{C}$? Is there a general method for this kind of problem? Also, how is it possible to tell whether a subset of $\...
M Smith's user avatar
  • 2,727
0 votes
2 answers
48 views

What is the domain of $f(x,y) = xy/\sqrt{x^2+y^2}$

I think the domain is all real except $(0,0)$, because this point anulates the denominator, but what about with the fact that the limit exists when $(x,y)$ tends to $(0,0)$?
Ronald Becerra's user avatar
1 vote
1 answer
229 views

Why is the union of two closed sets again closed in the Scott topology?

I want to figure out why the Scott topology really forms a topology. In particular, I want to find out why the union of two closed sets is again closed. So say we have a partial order $P = (P, ≤)$. A ...
k.stm's user avatar
  • 18.6k