Definitions are the core of mathematical precision; they come to answer "what is X" in mathematics. Into this category fit questions regarding equivalence of definitions; clarifications regarding complicated definitions; as well questions with purposed definitions for mathematical notions with ...

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1answer
74 views

“[T]ransversely isotropic and mirror-symmetric (space group:$D_{\infty h}$)”, its orbifold notation?

I am trying to understand this frieze pattern $D_{\infty h}$ aka its orbifold notation. This describes spider's silk. The authors call it a space group, some sort of generalization from orbifolds. ...
4
votes
2answers
687 views

Definition of an “Experiment” in Probability

One can define the fundamental concepts of probability theory (such as a probability measure, random variable, etc) in a purely axiomatic manner. However, when we teach probability, we start off with ...
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0answers
62 views

Improper or Undefined

Let $f(x)=0 $ if $x\neq 1 $ and $f(1)=\infty $ then the Riemann integral $\int_{0} ^1 f(x)$ $ dx $ = $ 0 $ or is it undefined? If we take it as a legitimate function for improper Riemann ...
12
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8answers
2k views

Why do we accept Kuratowski's definition of ordered pairs?

I've been struggling understanding Kuratowski's definition of ordered pairs. I understand what it means but I don't see why I should accept it. I've seen this question and this one, most importantly ...
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2answers
327 views

About the definition of n-tuple

I've read from the theory of sets that the definition of ordered pair is foundational. To define formally the term of $n$-tuple I suppose we need to use the concept of ordered pair as well as the ...
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3answers
417 views

Expected Value Function

My text-book defines expected value as $$E(X) = \mu_x = \sum_{x \in D} ~x \cdot p(x)$$ And so, if I was to find the expected value of a random variable $X$, where $X = 1,2,3$, then it would resemble ...
10
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1answer
971 views

Semi-Norms and the Definition of the Weak Topology

When I was searching for the definition of the weak topology, I found two different definitions. One defines the weak topology in terms of a family of semi-norms, while the other defines it in terms ...
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6answers
829 views

How exactly can't $\delta$ depend on $x$ in the definition of uniform continuity?

I'm told that a function defined on an interval $[a,b]$ or $(a,b)$ is uniformly continuous if for each $\epsilon\gt 0$ there exists a $\delta\gt 0$ such that $|x-t|\lt \delta$ implies that ...
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3answers
322 views

How can the real numbers be a field if $0$ has no inverse?

I'm reading a linear algebra book (Linear Algebra by Georgi E. Shilov, Dover Books) and the very start of the book discusses fields. 9 field axioms discussing addition and multiplication are given ...
2
votes
3answers
979 views

Linear independence and dependence of vectors

I am really stuck in this problem, I have only 2 days to learn matrix's base, and its generator. My problem is that I know definitions but I don't understand intuitively what they mean. What I know: ...
10
votes
1answer
331 views

Is there a way to relate convexity to Gaussian curvature?

This is a vague question because I'm not sure what I want to ask. An ellipsoid has positive curvature everywhere, and bounds a convex subset of $\mathbb R^3$. What I want to say now is "It seems as ...
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1answer
544 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
166 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
90 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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4answers
137 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 ...
4
votes
2answers
2k 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 ...
1
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1answer
66 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 ...
1
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1answer
409 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 ...
1
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1answer
78 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 $g,h\in G(C)$, $c$ ...
4
votes
2answers
216 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 ...
4
votes
2answers
200 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
147 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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votes
5answers
531 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 ...
5
votes
1answer
117 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
152 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 ...
4
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3answers
199 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
86 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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2answers
1k 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 ...
8
votes
5answers
1k 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 ...
2
votes
2answers
115 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} ...
3
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1answer
86 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 ...
3
votes
3answers
71 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. ...
5
votes
3answers
202 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 ...
2
votes
1answer
114 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 ...
3
votes
2answers
2k 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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1answer
114 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
1k 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 = ...
4
votes
1answer
836 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
1
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1answer
41 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 ...
2
votes
1answer
82 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
225 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 ...
2
votes
3answers
191 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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1answer
49 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
118 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 ...
2
votes
0answers
56 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 ...
2
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4answers
612 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 ...
6
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2answers
659 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 ...
6
votes
4answers
492 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 ...
2
votes
2answers
101 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 ...