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

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679 views

Definition of induced cycle

According to Diestel (page 4): "If $G' \subseteq G$ and $G'$ contains all the edges $xy \in E$ with $x, y \in V'$, then $G'$ is an induced subgraph of $G$" According to Wikipedia "induced cycle is a ...
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1answer
214 views

Definition of the Support of a Real Valued Random Variable

This is an embarrassingly simple question: If $X$ is a normally distributed random variable, what is the support of $Y=X^2$? It clearly should be the positive real line. However, I cannot find a clear ...
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2answers
95 views

What is the relation between $ \kappa$-monolithic and monotonically monolithic?

For an infinite cardinal $\kappa$, a space $X$ is called $\kappa$-monolithic if $nw(\overline{A}) \le \kappa $ for any set $A \subset X$ with $|A| \le \kappa$. And you can see this definition of ...
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143 views

What is a minimal polynomial of a group element, and why would we care if it was quadratic?

EDIT: the $p$-stable definition I give below is incorrect. I have included the correct definition as an answer to this question. I am trying to understand the definition of a p-stable group. The ...
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2answers
3k views

What are relative open sets?

I came across the following: Definition 15. Let $X$ be a subset of $\mathbb{R}$. A subset $O \subset X$ is said to be open in $X$ (or relatively open in $X$) if for each $x \in O$, there exists ...
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1answer
84 views

Definition of purely oscillatory

This is question about a term whose definition I can find anywhere. I am given to solve a differential equation and one of the questions asks to show that the solution (we are given initial data) is ...
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1answer
683 views

complement of a function $f: \{2n | n\in \mathbb{N}_0 \}: n \rightarrow n+1$

i am reading a textbook here and i saw, there is notion of Complement of a function. or Negation of a function definiton, this is whow i understood but it is definitely wrong how i do it, i know. in ...
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89 views

Question on definition: primitives in the enveloping algebra of a Lie algebra.

Let $C$ be a coalgebra, and take $c\in C$. Then $c$ is group-like if $\Delta c=c\otimes c$ and $\epsilon(c)=1_k$, and the set of group-like elements is denoted $G(C)$. For $g,h\in G(C)$, $c$ ...
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2answers
263 views

Trying to understand Hilbert Spaces…

I am trying to get a hold on Hilbert Spaces, but I am having difficulties combinging various definitions. I have looked it up on wikipedia and wolfram, there it states something like "A Hilbert ...
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2answers
248 views

Motivation for a new definition of the derivative using the concept of average velocity

As everyone knows, for a function $ f: \mathbb{R} \to \mathbb{R} $ and a point $ a \in \mathbb{R} $, we say that the derivative of $ f $ at $ a $ equals $ L $ if and only if $$ \lim_{x \to a} ...
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4answers
151 views

If capital letters are supposed to be sets, why is $N$ used as a number?

If capital letters are supposed to be sets, why is $N$ used as a number? For example, in this definition of a Cauchy Sequence it says: A sequence ${p_n}$ in a metric space $X$ is said to be a ...
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5answers
659 views

What exactly is “approximation”?

There are a lot of great "approximations" that exist in the mathematical field:$$\dfrac{22}{7} \approx \pi$$ $$e \approx \left(1 + \dfrac{1}{n}\right)^n$$But the fact that I have yet to know what ...
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1answer
129 views

For a differentiable map $\Phi$ between manifolds $M$ and $W$, what is $d\Phi?$ (Aubin)

I can't understand a passage from A Course in Differential Geometry by T. Aubin. First, there is Definition 2.6., which I posted in this question. And now there's this: $(\Phi_*)_P$ is nothing ...
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3answers
153 views

Is ∞ considered defined?

$\infty$ (Infinity) is not a number, but infinity is considered to be defined, right? There are expressions in mathematics such as: $\frac x0,0^0, \frac\infty\infty,$ which are not defined because ...
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224 views

What is a tangent bundle? (Aubin)

Here's what I read in A Course in Differential Geometry by Thierry Aubin. 2.5. Definition. The tangent bundle $T(M)$ is $\bigcup_{P\in M} T_P(M).$ And then 2.6. Definition. Let $\Phi$ be a ...
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2answers
97 views

Reason for existence of 'swapping' elementary matrix operation

In my Hoffman & Kunze book on Linear Algebra, we define 3 elementary row operations, the first of which is the swapping of two rows. I'm wondering why we need to even have such an elementary ...
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2k views

Geometric Definitions: What is a straight line? What is a circle?

What is a straight line? I need a geometric definition of it. The equation of a straight line is known to me.I am saying about a straight line of 2D plane. What is a circle? I need a geometric ...
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5answers
2k views

Is the Dirac Delta “Function” really a function?

I am given to understand that the Dirac delta function is strictly not a function in the conventional sense and it is a "functional or a distribution". The part which I can not understand why the ...
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2answers
120 views

The problem of bound variables in mathematical definitions

I was reading Paul Bernays’ Axiomatic Set Theory recently; in the book, Bernays gives the following definition of ‘ordinal number’. \begin{align} \text{On}(\alpha) \stackrel{\text{def}}{\iff} ...
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1answer
87 views

Are distinctions in definitions of “finite” material in, eg, topology or measure theory?

There are several definitions of "finite", like Dedekind's and Tarski's (Thanks to A.K. for point out the latter - first time I've heard of it): From the Wikipedia entry on Finite Set: (Richard ...
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3answers
78 views

Definition of a tensor field

Could anybody explain to me the following: If $$T_{ij}=\nabla_i V_j-\nabla_j V_i$$ where $V_i$ is a covector field and $\nabla_i$ is the covariant derivative, then $T_{ij}$ is a tensor field. ...
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258 views

Understanding a definition of radial

A space $X$ is called radial if, for any $A \subset X$ and any $x \in cl(A)$, there is a transfinite sequence $s=\{a_\alpha: \alpha \in \kappa\} \subset A$ which converges to $x$. What's meaning of ...
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1answer
128 views

Can we apply a successor operation in Peano's arithmetic infinitely many times, is the successor well defined?

If we look at the axioms of Peano arithmetic, e.g. http://mathworld.wolfram.com/PeanosAxioms.html, they contain an axiom: If $a$ is a number, the successor of $a$ is a number. However, the axioms do ...
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2answers
3k views

What does lg x mean? is it $\log_2 x$ or $\log_{10} x$ in binary trees

I'm a bit confused, $\log_{10} x = \log x $ right? I believe I've read somewhere that $\log_{2} x = lg x$ but some people say lg = $\log$. So what does lg really stand for? specifically when talking ...
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123 views

What does compact cover mean?

I am reading a difinition of Lindelof $\Sigma$ space. It talked about compact cover. As the title explains, what does compact cover mean? It means every member of the cover is compact?
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3answers
2k views

Two definitions of $\limsup$

Here are two equivalent definitions of $\limsup_{n\rightarrow\infty} a_n$: Let $u_n=\sup\{a_n, a_{n+1}, a_{n+2},\ldots\}$. Then $$\limsup_{n\rightarrow\infty} a_n = \lim_{n\rightarrow\infty} u_n = ...
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1answer
1k views

What is an algebra?

Is an algebra or 'a algebra' the same thing as an algebraic structure? Or does it have a different meaning? Thanks
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1answer
42 views

A Quadratic Maximum?

What does the following mean? Context: Laplace integrals Consider the integral $$\int_a^b f(t) \exp(x\phi(t))\,\,\,dt$$ where $\phi(t) $ achieves its maximum at some $t=c\in (a,b)$. Assume that ...
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1answer
86 views

A question on linear ordered space

A space $X$ is called left-separated if it can be well-ordered in such a way that every initial segment is closed in $X$. And we know every space contains a dense left-separated subspace. My question ...
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2answers
244 views

Definition of a basis for a particular topological space

I'm currently looking at Lemma 13.2 in Munkres' Topology. It states the following: Given a collection $C$ of open sets of a topological space $X$ such that for each open set $U$ of $X$ and each $x$ in ...
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3answers
234 views

What is the definition of first/last element in a poset?

I have read the terms first element/last elements in the context of a basic course in set theory. When I learned about posets I didn't encounter those terms. I tried looking up the definitions but I ...
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54 views

Use of the term “normal section” in a theorem of Maria Lucido.

Prop. 3 in this paper (p.135) states Let $G$ be a solvable group with $\text{diam}\Gamma(G)=4$. Then either $l_F(G)\leq 3$ or $l_F(G)=4$ and $G$ has a normal section isomorphic to $H$. ($H$ is ...
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1answer
133 views

Definition for series with negative index and order of taking limits

I have thee questions and they seem all related to me and every number i say is complex number below. My definition for $\sum_{k=0}^n=a_k$ is by induction. That is, given finite set of numbers ...
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0answers
60 views

Can a sample space depend on the parameter to be estimated (example: # of cabs in a city)

In our intro statistic lecture the following we said that the following components made up an estimation problem an at most countable space $\mathcal{X}$ of all possible samples we can observe a ...
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4answers
689 views

Definition for Covariant Derivative

What is simple definition of the covariant derivative that looks like the definition of the derivative of a function in calculus? definition of the derivative of a function in calculus is: $$\frac ...
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3answers
863 views

Why does the condition of a function being differentiable always require an open domain?

Going through Spivak's Calculus on Manifolds and in his definition of a differentiable function from a subset $A$ of $\mathbb{R}^n$ to $\mathbb{R}^m$, $f$ is said to be differentiable if it can be ...
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4answers
554 views

Can open sets be an open cover, for itself?

I have Baby Rudin's book with me and it clearly defines a cover to be open. In a followup, it defines a set $K$ to be compact if every open cover of $K$ contains a finite subcover. And the rest I ...
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2answers
114 views

Is the definition if mutual independence too strong?

Let $A_1,\ldots,A_n$ be $n$ events in a discrete probability space. Can someone give me an example such that $$P(A_1 \cap \cdots \cap A_n)=P(A_1)\cdot \ldots \cdot P(A_n)$$ holds, but such that there ...
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4answers
2k views

Is a linear combination linearly independent?

I am a bit confused... Linear combination means $$F(X)=af(x_1)+bf(x_2) + \cdots$$ and linearly independent means $$af(x_1)+bf(x_2) + \cdots=0$$ where $a=b=\cdots=0$ My question: is a linear ...
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1answer
579 views

Base (topology) with closed intervals

I am curious why it's a problem to define a base using closed sets? For example, my book uses the definition under "Constructing Topologies from Bases" as specified at ...
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2answers
1k views

Understanding induced representations

Let $G$ be a group and $H$ be a subgroup. Let $\phi:H\rightarrow GL(V)$ be a representation of $H$. There are three constructions in Wikipedia, but I am not really convinced by these. My question is: ...
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2answers
156 views

A simple notation question on grad and Lp norm

What does this notation $||\nabla u|| _p$ mean? I would like an exact definition. Also I have seen $|\nabla u|_1$. Does this mean the $\sum_i|\partial_i u|$?
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87 views

Graph Theory: Help with a definition

I need some help to see if the definition I found of Cycles and Cycles Decomposition is right, here it is: A graph is a Cycle if it is isomorph to another graph $G=(V,E)$ with the following ...
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1answer
391 views

What is “algebra” in $\sigma$-algebra (or “field” in $\sigma$-field)?

I know that $\sigma$ in $\sigma$-algebra stands for the closure under countable union property. What about "algebra"? Surely it cannot algebra over a field or a ring as defined in algebra textbooks. ...
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177 views

Is $\{-2,2\}$ a group under $a\star b=\max\{a,b\}$?

Lets say $G={-2,2}$ and $a*b=\text{max}\{a,b\}$. I need to check if this is a group and if it does than is it abelian or not and finite or not. Well... first, I'm not sure if this is a group. for ...
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1answer
613 views

bipartite graph - sufficient and necessary conditions

Sorry for a silly question, I got confused with the definition of bipartite graph. What is a necessary and sufficient condition for a bipartite graph. ...
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317 views

different kind of convergence in Real analysis

Now I am studying real analysis, I wonder if anyone can sum up all the relations between all kinds of convergence, convergence in measure, convergence a.e., convergence a.u., and convergence in ...
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3answers
218 views

If there is only one one-sided limit, the limit exists?

Question on theory: If $\lim_{x\to a^+}F(x)=L$ and $\lim_{x\to a^-}F(x)$ doesn't exist, then does the $\lim_{x\to a}F(x)$ exists and is $L$? Thanks.
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445 views

Why does the definition of limits of a function have strict inequality?

Definition (As written in Michael Spivak's Calculus) The function $f$ approaches a limit $l$ near $a$ means: for every $\epsilon >0$ there is some $\delta > 0$ such that, for all $x$, if ...
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131 views

What is a “set-like class”?

Just / Weese contains the following theorem (p 126): Theorem 5: Let $\mathbf Z \neq \varnothing$ and let $\mathbf W \subseteq \mathbf Z \times \mathbf Z$ be a wellfounded set-like class. Let $\mathbf ...