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

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2
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1answer
110 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 ...
0
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0answers
24 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
48 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 ...
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0answers
16 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
21 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 $U$ ...
7
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1answer
113 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 ...
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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., $f(x) : X \rightarrow \{0,1\}$)?
1
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1answer
16 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
39 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
50 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
12 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
35 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 ...
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2answers
215 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
votes
1answer
53 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
71 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
50 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 ...
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1answer
53 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
68 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 ...
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2answers
48 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
15 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.
2
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3answers
70 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
46 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 ...
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0answers
37 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
64 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 ...
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5answers
439 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 ...
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2answers
76 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 ...
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2answers
104 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 ...
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0answers
43 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
vote
1answer
36 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 ...
11
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3answers
181 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
46 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
15 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 $m_1$ ...
2
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0answers
22 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?
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0answers
59 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
votes
2answers
51 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 ...
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2answers
70 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
votes
1answer
59 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
29 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 $$ A^G=\left\{x=(x_g)_{g\in G}, x_g\in A\right\}? $$
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0answers
59 views

What is an undefinable ordinal? How large are the least undefinable ordinal and supremum of all undefinable ordinals?

What does it mean to say that a particular ordinal/cardinal number is "definable"? Where and how do we define this "definability"? If there are undefinable ordinals/cardinals, how large are the least ...
4
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3answers
75 views

what is meant by $ f ∈ C^{2}[a, b] ?$

What is the meaning of $ f ∈ C^{2}[a, b] ?$ I think it says that $f$ is twice differential on $[a,b]$, isn't it?
1
vote
1answer
59 views

o-minimal structures and definable functions

Consider the following definition of an o-minimal structure: An o-minimal structure $O=\{O_n\}$ is a sequence of Boolean algebras $O_n$ of subsets of $\mathbb{R}$ which satisfies the following ...
1
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1answer
63 views

What's the exact definition of symmetry?

My question is basically the same as this, but I haven't found the answer given satisfying. The definition of symmetry I've come up with is this: Let $X \subseteq R^2$. A symmetry of $X$ is an ...
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0answers
54 views

What is a coupling argument?

In an article I've read in a proof that distinguishes two cases something like: "the second case can be shown by an easy coupling argument using the first case." What is a coupling argument? Edit ...
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0answers
13 views

Definition of Bernouilli shift

Let $\mathfrak{B}$ denote the Borel field on $X$ generated by its topology and let $\mu_{p_0,p_1,p_2}$ be product measure on $X$ in which the $i$'s have density $p_i$. A translation-invariant ...
0
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1answer
67 views

Is “small disk” well-defined?

I saw the notion "small disk" very frequently used in literature. For example, in Brunnian braids on surfaces by V. G. Bardakov, R. Mikhailov, V. V. Vershinin, J. Wu, one line reads: Let $P_n(M)$ ...
6
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2answers
245 views

Defining sine and cosine

We know the following are true about sine and cosine (and that they can be proven geometrically): $\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$ $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$ ...
7
votes
6answers
333 views

Removable discontinuity question

A quick and easy question on terminology: If we have a function $$f(x) = \frac{x(x-2)}{(x-2)}$$ then clearly the function is not defined at $x = 2$. If we cancel out "$(x-2)$" then the function is ...
0
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1answer
28 views

Definition of rank of an element in tensor algebra

Let $V$ be a vector space over a field $K$ and let $T(V)$ be its tensor algebra; that is, $$T(V)=\displaystyle\bigoplus_{k=0}^{\infty}V^{\otimes k}=K\oplus V\oplus V\oplus(V\otimes V)\oplus(V\otimes ...