For requesting, clarifying, and comparing definitions of mathematical terms.

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2answers
49 views

If the empty set is a subset of every set, why isn't $\{\emptyset,\{a\}\}=\{\{a\}\}$?

If the empty set is a subset of every set, why it isn't written with the elements of a set? like so $\{1,2,3,\emptyset\}$ Or why isn't $\{\emptyset,\{a\}\}=\{\{a\}\}$? I know one has two elements ...
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2answers
53 views

Find the right cosets of $H$ in $G$ simple example

Question: Let $G$ be a group and $H<G$ a subgroup with $|G:H|=2$ Show that the right cosets of $H$ in $G$ are $H$ and $G\backslash H$ Answer given: There are two right cosets, they are disjoint ...
0
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2answers
33 views

Give an example of a set which is not transitive

Transitive set: set $x$ is transitive if $\forall y\in x(y\subseteq x)$ I think $\{\varnothing\}$ is not transitive since $\varnothing\in\{\varnothing\}$ but $\varnothing\not\subseteq\{\varnothing\}$ ...
3
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1answer
37 views

slightly different definition of an ordered pair

In a paper I was reading an ordered pair had a slightly different definition $\langle a,b \rangle = \{a,\{a,b\}\}$ instead the normal Kuratowski definition which is that $\langle a,b \rangle = ...
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0answers
20 views

The maximum notation as regards the absolute value?

We know that $\max(\textbf{A})$ gives the maximum element of the array $\textbf{A}$. What is the notation, or a short formula, if we seek the element having the largest absolute value? e.g., ...
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2answers
24 views

Why can we write a uncontinual function continual?

Let's consider $$f(x) = \frac{(x-1)(x-2)(x-3)}{x^2-3x+2}$$ with definition $D_{f} = \mathbb{R} \setminus \lbrace 1, 2 \rbrace$. This means we are allowed to set $x$ to every value of $\mathbb{R}$ ...
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1answer
28 views

Local Homeomorphisms: Characterization

Problem Consider for simplicity a surjection $F:X\to Y$. Are these characterizations of local homeomorphisms equivalent: $$\forall x\in X:\exists U_x\in\mathcal{T}_X, V_y\in\mathcal{T}_Y:\quad ...
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1answer
31 views

Can every function be a composite to itself and how to know if a composite between two functions is defined?

Can every function be a composite to itself? like we have $f:A\to B$ is $f \circ f$ always defined? Can we say that if $f$ is a injection/surjection/bijection then so is $f\circ f$? Also, how do ...
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2answers
33 views

Determinant of Polynomial

I was reading some paper and it says 'Let $\Delta$ denote the determinant of the polynomials $P,Q$ and $R$ with respect to the basis $1,X,X^2$' ($P,Q$ and $R$ are degree 2 polynomials). And then I ...
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1answer
47 views

definition of a $\Delta$ - complex

I've been given this image from hatchers algebraic topology as an example of a $\Delta$ complex but with the explicit definition as follows A $\Delta$-complex structure on a space X is a collection ...
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1answer
72 views

Question about Definition of Boundary in Stokes' Theorem

I was wondering if what my teacher said was correct and complete in that in Stokes' Theorem the "boundary curve" of a surface can be defined as the mapping along the boundary of the two-dimensional ...
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0answers
23 views

Holomorphs and split extensions

The notion holomorph was introduced in Maria S. Voloshina's Ph.D. thesis On the Holomorph of a Discrete Group. It is defined as follows: Let $G$ be a group and let $\mathrm{Aut}(G)$ be the ...
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0answers
27 views

Difference between upper bound, maximal element, and maximum

In order to learn the difference between upper bound, maximum, and maximal element (of a set), I wrote down the following. Is it correct? Upper bound not necessarily element (of set) greater than ...
0
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0answers
15 views

What is meant by “commutes with spatial translations”?

Let $f\colon(X,\mathcal{B},m_1)\to(X,\mathcal{B},m_2)$, where $(X,\mathcal{B},m_i), i=1,2$ are measure spaces. What is meant when saying that f commutes with all spatial translations?
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0answers
14 views

$k$-cube definition clarrification

If $c:I^k\rightarrow\mathbb{R}^n$ is a singular $k$-cube. What will $\partial c$ be? Will it be a singular $(k-1)$-cube or a $(k+1)$-cube or something else? Definition - A singular $k$-cube on $U$ ...
6
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1answer
88 views

Is there a name for a regular semigroup with zero in which the product of any two different idempotents is zero?

As the title says, my question is: is there a name for a regular semigroup with zero in which the product of any two different idempotents is zero? Note that any such semigroup is necessarily an ...
0
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0answers
37 views

Is there a name for a function that returns only binary values?

Is there a name for a function that returns only binary values (e.g., $f(x) : X \rightarrow \{0,1\}$)?
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1answer
14 views

Is $(-\infty,0)\times S$ for a compact closed manifold $S$ a “manifold with boundary and cylindrical ends”?

I read the following definition from this paper. Definition: Let $N$ be a Riemannian manifold with boundary $\partial N$. We say $N$ is a manifold with boundary and cylindrical ends if there ...
2
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1answer
34 views

What does “be the inclusion” mean?

Can anyone explain what the phrase means? To be specific, my notes has the phrase "let $f:A \rightarrow B $ be the inclusion". Does this mean the identity map?
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2answers
48 views

Power set of $\{\{\varnothing\}\}$

$$\mathcal{P}(x)=\{y\mid y\subseteq x\}$$ $$\mathcal{P}(\varnothing)=\{\varnothing\}$$ $$\mathcal{P}(\{\varnothing\})=\{\varnothing,\{\varnothing\}\}$$ ...
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0answers
6 views

What does bimeasurable mean? Is an invertible transformation bimeasurable?

What exactly is the meaning of a bimeasurable transformation? I did not find a very clear answer to that. As far as I see it means that Borelsets are maped to Borelsets. So an invertible ...
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2answers
23 views

Corresponding partition in equivalence relation

The relation $R$ on the set $A=\{2,4,6,8,10\}$ is defined by $$R=\{(2,2),(2,6),(2,10),(4,4),(4,8),(6,2),(6,6),(6,10),(8,4),(8,8),(10,2),(10,6),(10,10)\}$$ Question 1 Verify if $R$ is an ...
6
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2answers
172 views

Is a function defined at a single point continuous?

Is a function defined at a single point continuous? For example $f:\{0\}\to\{0\}$ defined by $f(x)=\sqrt{x}+\sqrt{-x}$ is a sum of two continuous functions and is therefore continuous, however for ...
2
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1answer
46 views

What is a regular homotopy?

The definition of regular homotopy from Wikipedia says that two immersion $f,g:M\to N$ are regularly homotopic if they represent points in the same path-component of $\text{Imm}(M,N)$. What does ...
2
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2answers
69 views

Questions which have false conditions

There are many "questions" on the internet like If $$1=5$$ $$2=6$$ $$3=7$$ $$4=8$$ then how many is $5$? With one "logic" answer is $9$ because $n=n+4$, then $5=9$. With other "logic" ...
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1answer
44 views

Is “algebraic system” the same as “algebraic structure”?

1. algebraic system | planetmath.org: http://planetmath.org/algebraicsystem It seems that algebraic system is only a set on which some operations are defined. Is it necessary that some additional ...
0
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1answer
47 views

The Sobolev Space $H^{1/2}$

this is a very stupid question. In my course of linear PDEs, the professor used $H^{1/2}$ without defining, and I have looking on google to find a definition, but the only related thing I found was ...
2
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1answer
57 views

Show a function is well defined

My understanding of a function being well defined: Let $(G,\circ)$ be a group and let $H\unlhd G$ be a normal subgroup. Let $G/H=\{ah\mid a\in G\}$. For a group $(G/H,\star)$(the quotient ...
2
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2answers
42 views

A question about two common definitions

Two definitions make me puzzled ! 1. The definition of $\textbf{Functions Differentiable at a Point}$: A function $f$ defined in a neighborhood $(x_{0}-\delta,x_{0}+\delta)$of a point $x_{0}$, ...
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0answers
13 views

Word for two objects with coplanar axes.

Is there a suitable word for describing two objects with coplanar axes (e.g. cylinders)? The word parallel springs to mind, but I wondered if there was anything more specific.
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3answers
64 views

Which quadrant is the “first quadrant”?

In the coordinate plane split into four quadrants by the $x$- and $y$-axes, I learned (educated in a public school in the U.S.) that the "first quadrant" was the one with both $x$ and $y$ positive, ...
0
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1answer
39 views

What is the definition of a properly-immersed arc?

I am going over a paper in which a theorem includes a condition that certain arcs are "properly-immersed geodesic arcs" in a surface $\Sigma$ with nonempty boundary. I have done a search, but I have ...
0
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0answers
34 views

Are my definitions correct? (Formal language)

I will describe how I understand below. Please tell me where I'm thinking wrong or correct. Symbol is an undefined term just like a set. Symbol can be ragarded as a object we consider. ...
2
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1answer
60 views

What is an adjoint operator?

The following conjeture is stated here: Every adjoint operator has a non-trivial closed invariant subspace. Reference 11 where adjoint is supposedly defined can be found here. But I don't have ...
5
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5answers
413 views

Is $1234567891011121314151617181920212223…$ an integer?

This question came from that one and from that talk where it's noted that "integers have a finite count of digits", so that the "number" in the title is not at all a number (not integer nor rational ...
4
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2answers
72 views

Multiplication Operation

I am a father of two young boys and I looks forward to exploring mathematics with them for as long as they will let me :-). I would really like for them to have a deeper understanding of mathematics ...
8
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2answers
101 views

Is function $f:\mathbb C-\{0\}\rightarrow\mathbb C$ prescribed by $z\rightarrow \large \frac{1}{z}$ by definition discontinuous at $0$?

Is function $f:\mathbb C-\{0\}\rightarrow\mathbb C$ prescribed by $z\rightarrow \large{\frac{1}{z}}$ by definition discontinuous at $0$? Personally I would say: "no". In my view a function can only ...
0
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1answer
24 views

What is the definition of $I=(f(X,Y),g(X,Y))$?

What is the definition of this ideal in $\mathbb C[X,Y]\ I=(f(X,Y),g(X,Y))$ for some polynomials $f,g \in \mathbb C[X,Y]$
0
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1answer
39 views

definition of product of modules

I have been given this definition of a product between modules: If $I$ is an indexing set with $M_i$ as an $R$-Module then the product $\prod \limits_{i \in I} M_i$ is defined as the set consisting ...
0
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0answers
38 views

About a variation of the primitive root idea.

Let $n $ be called a half primitive root mod ($m$) if $n^{\phi(m)/2} = 1$ mod($m$) and for any $t $ with $1 < t < \phi(m)/2$, $m$ does not divide $(n^t - 1)$. So the order of $n$ mod($m$) is ...
1
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1answer
34 views

Why are the summands $-1,0,1$?

I have some problem to understand the following: Let $X=\left\{0,1,2\right\}$ and consider $X^{\mathbb{Z}^d}, d\geq 2$ as being the set of all function from $Z^d$ to $X$. So for $\eta\in ...
10
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3answers
140 views

Is it possible to define $\lceil x\rceil$ in terms of $\lfloor\ldots\rfloor$?

I have tried to define the ceiling function of $x$ in terms of its floor function; I thought this might be easy, but it isn't. I can easily do this with a piecewise equation, but I need to do without ...
0
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0answers
33 views

Well defined uncomputable numbers.

For any prefix-free universal computable function $F$ with domain $P_F$, the Chaitin’s constant $$ \Omega_F=\sum_{p\in P_F}2^{-|p|} $$ is a number $\in [0,1]$ and seems "well defined". But this ...
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0answers
14 views

What is the $\bar{d}$-metric for translation-invariant measures?

I've often heard of the so called $\bar{d}$-metric for translation-invariant measures. I found something like $$ \bar{d}(m_1,m_2)=\inf\text{Prob}^m\left\{\eta(0)\neq\delta(0)\right\}, $$ where $m_1$ ...
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0answers
18 views

What is the coupling of two measures?

I know what means coupling for random variables, as explained here. But what is a coupling of measures?
3
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0answers
56 views

Functors that preserve the subset relation on constructs

Is there a name for functors between concrete categories over Set that preserve '$\subset$'? Example, for a functor $F:\mathbf {Top}\to \mathbf {Grp}$, if $Y$ is a subspace of $X$ then $F(Y)$ ...
2
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2answers
46 views

What is the definition of closed subspace?

I am trying to understand what is intended with closed subspace, I took the following guess: A closed subspace $M$ of a Hilbert space $H$ is a subspace of $H$ s.t. any sequence $\{x_n\}$ of elements ...
0
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2answers
68 views

Do all the properties of exponents work for every real exponent? [closed]

I am writing an essay, but i doubt at typing that the properties of exponents works for every Real, I know it works for every Rational number. $$\forall a,n,m\in\mathbb R$$ or $$\forall a\in\mathbb ...
2
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1answer
56 views

Is it possible to have an $a \times b \times c$ matrix?

The book Artificial Intelligence: A Modern Approach states that a certain variable is a $2 \times 2 \times 2$ matrix", but I thought that matrices could only be rectangular (i.e. $a \times b$). Is it ...
0
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0answers
28 views

Set of functions and sequences

By $A^G$ I mean $\left\{x\colon G\to A\right\}$. Is it then to same to write $$ A^G=\left\{x=(x_g)_{g\in G}, x_g\in A\right\}? $$