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

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1answer
16 views

Writing order of geospacial coordinates

Is there a defined order in which to write geospacial coordinates? When looking at GPX files generated by my GPS, latitude is named first but when looking at a KML file generated by google earth ...
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1answer
339 views

Definition of Derivative for $sgn(x)$

When using the definition of the second derivative for $sgn(x)$, I'm a little confused on evaluating something like $sgn(x+h)$. Since $h\rightarrow 0$ does that mean that I should treating $sgn(x+h)$ ...
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2answers
51 views

What is the definition of 'line' in $\hat{\mathbb{C}}$?

What is the definition of straight line in $\hat{\mathbb{C}}$? Is it defined as $\{x\in\mathbb{C}: \frac{Re(x-a)}{Re(b)} = \frac{Im(x-a)}{Im(b)}\}\cup \{\infty\}$? ($a,b$ are complex numbers and ...
60
votes
14answers
4k views

Are “if” and “iff” interchangeable in definitions?

In some books the word "if" is used in definitions and it is not clear if they actually mean "iff" (i.e "if and only if"). I'd like to know if in mathematical literature in general "if" in definitions ...
2
votes
2answers
97 views

Odd and even functions.

I have a book which says: If a function $f$ satisfies $f(-x)=f(x)$ for all $x$ in its domain, then $f$ is called an even function. However, if $f(-x)=-f(x)$ for every $x$ in the domain of $f$, ...
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3answers
100 views

Orthonormal Bases

I am struggling to get my head around orthonormal bases, this is the defintion in my course notes: If anyone could clarify/explain the concept to me, it would be much appreciated. I am a university ...
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1answer
840 views

normalized subgroup by another subgroup

Let $A$ and $B$ be two subgroups of the same group $G$. What does it mean for the subgroup $A$ to be normalized by the subgroup $B$?
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0answers
38 views

Computing the differential (not the Jacobi-Matrix) independent of a basis choice

I am eager to learn more about the differential. I was told that it is a good practice to compute the differential independent of a choice of a basis using the following definition Definition: ...
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1answer
34 views

Derivatives from the left and right

I need to give a definition of the derivative from right and from left. (I know it doesn't make a whole lot of sense, but it is supposed to be similar to the same thing as the limits from the right ...
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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 ...
2
votes
1answer
85 views

The differential is NOT the Jacobi Matrix?

In the book Analysis II by C.T. Michaels the differential is introduced as the Jacobi-Matrix. In class we had the following definition: Definition: Let $U \subset \mathbb{R}^m$ be open, $f: U \to ...
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2answers
34 views

Homomorphisms, Linear Transformations, and Distributivity

What are the differences between homomorphisms, linear transformations, and distributive operations? To me, they all seem essentially the same, they just are different names for the same phenomenon ...
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1answer
28 views

A question about the span of functions.

I am having trouble understanding the span of functions, my problem is: What is $\operatorname{span}\{a\sin(x),b\}$, and what is its dimension? I understand this in terms of vectors, but not in ...
1
vote
1answer
23 views

Can't remember definition of $\lvert G \rvert_{p'}$

For $G$ a finite group, I know that $\lvert G\rvert$ denotes the order of the group. My question is: What is $\lvert G\rvert_{p'}$? Also is this the same as $\lvert G\rvert_p$ (without the prime on ...
5
votes
2answers
181 views

If a finite poset has greatest and least elements, is it a lattice?

Let $P$ be a finite partially ordered set with elements $0$ and $1$ such that $0 \le x \le 1$ for any $x \in P$. Does it follow that $P$ is a lattice? If not, what is a counterexample? I believe this ...
3
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1answer
80 views

Equivalent definitions for strictly positive elements

We have two usual definitions for strictly positive elements in C*-algebras: Let $A$ be a C*-algebra Definition (a) [MURPHY, C$^*$-algebras and Operator Theory] An element $a\in A_+$ is said to be ...
4
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3answers
479 views

Definition of set.

A set is defined as a collection of distinct objects. Why have we defined a set to contain only distinct objects? Why is a collection of objects which may have identical objects not called a set? ...
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2answers
78 views

What is the definition for holomorphic functions on the Riemann sphere?

I'm trying to study complex analysis in general setting, but i have troubles with defining things in this general setting. I have skimmed definitions in Ahlfors, Conway and wikipedia, but these all ...
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1answer
74 views

What is this metric called?

Ahlfors -complex analysis p.20 Consider a stereographic projection between the 2-sphere and $\overline{\mathbb{C}}$ (i.e. one-point compactification of $\mathbb{C}$) Let $z,w$ be complex numbers. ...
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1answer
57 views

A question on Holder spaces

A function $f$ is said to belong to the Holder space if Holder condition is satisfied, i.e. $\exists \beta,L\geq0$ such that $$|f(x)-f(x')|\leq L|x-x'|^\beta$$ for all $x,x'$ in the domain of $f$. ...
2
votes
1answer
29 views

Why do we define product of morphisms in this way

I'm always forget the definition of product of morphisms in a category, maybe the main reason is because I don't know the motivation beyond the definition: I need help to see this abstract ...
2
votes
1answer
157 views

Help with definition: partition mod ultrafilter.

I do not see how the partition is well-defined. By definition $A\neq\varnothing\mbox{ mod }D\iff A\notin I_{D}$. Since D is a maximal filter $A\notin I_{D}\iff A\in D$ . So ...
2
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3answers
31 views

Can someone explain me how to read members of a set to prove uncountability?

I have trouble in understanding what the elements in the following set are: $$ V = \{f:\mathbb{N}\to \mathbb{N} \mid \text{there is $N_f \in \mathbb{N}$ so that $f(x) \le N_f$ for all ...
2
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1answer
147 views

Embedding and monomorphism

What is the difference between an embedding and a monomorphism? As far as I can see, most introductory abstract algebra texts treat them as if they are the same, i.e. an injective function from one ...
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4answers
284 views

What is the *exact* definition of a bounded subset in a metric space (in relation with the Heine-Borel Theorem)?

I see quite a lot of different definitions of a bounded space. For instance, from nLab: Let $E$ be a metric space. A subset $B⊆E$ is bounded if there is some real number $r$ such that ...
4
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1answer
82 views

How to understand structure groups?

I'm studying fiber bundles and I'm somewhat confused on how Structure Groups appears. The definition of fiber bundle I have is the following: A bundle is a tuple $(E,B,\pi)$ where $E,B$ are ...
8
votes
2answers
987 views

Definition of a monoid: clarification needed

I'm only in high school, so excuse my lack of familiarity with most of these terms! A monoid is defined as "an algebraic structure with a single associative binary operation and identity element." ...
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2answers
88 views

Limits to infinity?

As a part of homework, I was asked What does $\lim_{x\to a} f(x)=\infty$ mean? In an earlier calculus class I was taught that in order for $L=\lim_{x\to a}f(x)$ to exist, we need that ...
1
vote
1answer
92 views

A set with a supremum and an infinum inside

What is the name of a set $A$ that has its infimum $i \in A$ and a supremum $s \in A$? Examples: $[0,1]$ has a supremum $1$ and an infimum $0$ which are inside $[0,1]$. $\{0,1,2,3\}$ has a supremum ...
3
votes
3answers
291 views

Equality of positive rational numbers.

I am reading the second article Rational Numbers of the book "A Treatise on Advanced Calculus" by Philip Franklin. I have mainly 3 questions regarding this article. I am writing all these $3$ ...
0
votes
1answer
55 views

$\epsilon$ rules for uniform and pointwise convergence.

Could someone please provide with the $\epsilon$ definitions of uniform and pointwise convergence. I'm trying to really get my head around the differences between them (I do know the differences, but ...
3
votes
3answers
131 views

Can I create my own function like Trigonometric or Exponential

When I want to solve mathematical problems, most of the time I meet the following functions Algebraic like polynomials. Trigonometric like sin(), cos(), tan(), cot(). Logarithmic like log(). ...
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2answers
132 views

How does the epsilon-delta definition define a limit?

I understand what the epsilon-delta definition is saying in regards to the distance from a point c and the distance from your limit, but I don't understand how this defines a limit. Any help is ...
1
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4answers
155 views

Vertical bar notation: $\frac{d}{dt}|_{t=0}f(a+tv)=$?

The definition of the directional derivative is $D_vf(a)$=$\left.\dfrac{d}{dt}\right|_{t=0}f(a+tv)=\lim \limits_{t\to0}\dfrac{f(a+tv)-f(a)}{t}$. However, I do not understand how this bar notation is ...
3
votes
1answer
46 views

Linear function definition

I'm trying to figure out what is linearity and what is a linear function. But the wikipedia page confused me. Firstly it defines as polynomial : $f(x) = ax +c$ Than it defines as linear map: ...
0
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4answers
309 views

Definition of Normalized Number

Which is correct? Are they both correct? Definition 1 A floating point number is called normalized if the leading digit of the fraction is nonzero. for example $(0.10101)_{2}\times 2^{3}$ is ...
0
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4answers
104 views

How is the word “contains” defined in set theory? (In relation with neighborhoods in topology).

From Wiki: Some basic sets of central importance are the empty set (the unique set containing no elements) Thus, this make me think that "contained" is equivalent to the $\in$, as in: if $a$ is ...
2
votes
1answer
121 views

Clear definition of degeneracy of a graph.

There are at least two questions on this topic but the answers are not clear to me and WiKi link didn't make it any clear either. Could someone please clarify is the degeneracy of a graph $G$ ...
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2answers
40 views

Confused about class definition

I find this in my Set theory material: [0] = {x:0==x(mod2)} = {x:2|0-x} , where I'm replacing equivalence sign ("=" with extra horizontal line) with double ...
10
votes
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 ...
1
vote
1answer
145 views

What is the definition of “Winning Strategy” in an Ehrenfeucht-Fraïssé game?

I've read many descriptions and applications of a Winning Strategy, as much as many for a Strategy tout court, but when a formal, algebraic definition is called upon, I've found close to no input. I ...
5
votes
0answers
87 views

Are the words “function”, “map”, and “mapping” synonymous? [duplicate]

Is it correct to say that "A function or a map or a mapping is a binary relation such that ..."
1
vote
1answer
53 views

What can you tell me about integrable functions and riemann integrals?

Define the concepts of integrable function and Riemann integral for functions of two variables (across a rectangle and over an arbitrary area). I know how to define for a rectangle but not an ...
0
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0answers
40 views

About definition of “extended absolute value” (in $\overline{\Bbb{R}}=\Bbb{R}\cup\{+\infty,-\infty\}$)

Is correct following definition? Def.: Let be $a \in \overline{\Bbb{R}}=\Bbb{R}\cup\{+\infty,-\infty\}$, $$|a|=\begin{cases} |a|& \text{ if } a\in \Bbb{R} \\ +\infty & \text{ if } a \in ...
4
votes
0answers
92 views

Defining “Penon Infinitesimals”.

In this lecture (which is accompanied by these slides), right near the end (so page 9 in the pdf of the slides; I don't think you have to watch the lecture), P. Johnstone refers to the "Penon ...
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2answers
53 views

Limit Point of a Set

Definition. A point $x$ is a limit point of a set A if every $\varepsilon$-neighborhood of $x$ intersects the set A in some point other than $x$. I understand the definition in that $x$ is our limit ...
0
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1answer
33 views

Analytic, never zero on domain D graph and its local max. min

If $f$ is analytic and never zero on domain D, then $|f(z)|$ has no local minimum or maximum. Now, If I use the fact that $|f(z)|\leq$ max $|f(z+re^{it})|$ ($0\leq t\leq 2\pi$), then I can show ...
1
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2answers
182 views

Definition of subalgebra

Is it generally implicitly assumed that if $B$ is a subalgebra of a unital Banach algebra $A$ then $1 \in B$? I tried to find a definition of subalgebra but the only definition I found was in Murphy ...
1
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1answer
21 views

'Union' of maps

Let $f : A \to Y$, $g : B \to Y$. Suppose that $f(x) = g(x)$ whenever $x \in A \cap B$. Define $$ h : A \cup B \to Y, \\ h(x) = \begin{cases} f(x) & \text{ if $x \in A$} \\ g(x) & \text{ if ...
0
votes
2answers
54 views

What is the easiest way to prove by induction?

Is there any easy way to do this? I get the basic step.. where you prove it for some number.. but I don't get the induction step. Do you literally take the given equation that you just proved with ...