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

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3
votes
0answers
9 views

Characterize in terms of fibre

I am not familiar with the notion "characterize" in the following context. Does this mean to redefine or?.... Any help would be appreciated. Thank you. For a function $f:X\to Y$, and y an element of ...
3
votes
2answers
63 views

Are all operations functions?

I have looked at Wikipedia(I know it's not completely reliable) but on it an operation is formally defined as: "A function ω is a function of the form $ω : V → Y$, where $V ⊂ X_1 × … × X_k$." and I ...
1
vote
2answers
25 views

Fibers of an Element

Prepping myself for a graduate abstract course and we are using Dummit and Foote's Text. We are starting on Chapter 1, so I thought it would be a good idea to go over chapter 0. There is a term that ...
18
votes
3answers
401 views

Is every axiom in the definition of a vector space necessary?

Definition: A vector space over a field $K$ consists of a set $V$ and two binary operations $+: V \times V \to V$ and $\cdot: K \times V \to V$ satisfying the following axioms: ...
5
votes
1answer
39 views

Does it matter if you use big $L$ or little $l$ when talking about $L$-norms?

I was reading a post on Quora regarding the application of "$l_1$", "$l_2$" norms for convex linear programming when I became very confused at which $L$-norm the posters are actually referring to. I ...
0
votes
0answers
5 views

Relation between Gâteaux derivatives and partial derivatives

Definition Let $V_1,...,V_n,W$ be nonzero normed spaces over $\mathbb{K}$ and $E$ be open in $ \prod_{i=1}^n V_i$ and $p\in E$. Define $U_i=\{a\in V_i : ...
1
vote
1answer
15 views

Question about usage of $\leq$ in definition of Nash equilibrium

Quick definition: Given $g$, a strategy N-tuple $u^* = (u_1^*,...,u^*_N)$ is said to be a Nash equilibrium if: $$J_i(u_i^*, u^*_{-i}) \leq J_i(u_i, u^*_{-i}), i \in N$$ where $J$ is ...
14
votes
6answers
1k views

How to think of a set?

I am doing self study for the last two months on functional analysis. As I get a bit used to the terms like space, topology, manifold, etc, etc, I realized that everything is defined in terms of set. ...
0
votes
0answers
25 views

Name for kind of big O notation with leading coefficient

Context: As known the big O notation $O(f(n))$ describes a function $g(n)$ such that there is a constant $C \ge 0$ with $\limsup_{n\to\infty} \left|\frac{g(n)}{f(n)}\right| \le C$ (I assume that ...
4
votes
0answers
31 views

Interval arithmetic with different definitions of intervals

Interval arithmetic normally deals with intervals defined as $[a,b]$ with rules like $$[a,b]\cdot[c,d]=[a+c,b+d]$$ I am interested in interval arithmetic with different interval definitions such as ...
2
votes
1answer
33 views

Name for some kind of logarithmic norm/error

As known $(\mathbb R, +)$ and $(\mathbb R^{+}, \cdot)$ are isomorphic with $\exp:\mathbb R\to\mathbb R^{+}$ as an isomorphism. When I transfer the absolute value $|\cdot|$ on $(\mathbb R, +)$ via ...
0
votes
1answer
35 views

Is there a name for operators of the type $A: M \to M$

In some theorem in functional analysis I have noticed that it is important to assume that an operator $A: M \to M$ where $M$ is some set plus conditions, as opposed to $A: M \to N, M \neq N$ Is ...
3
votes
1answer
45 views

Why is the solution of an ordinary differential equation required to be defined on an interval?

I am reading A First Course in Differential Equations with Modeling Applications (10th Edition) and here is a definition: Any function $\phi$, defined on an interval $I$ and possessing at least ...
1
vote
1answer
20 views

Can someone please help me understand what a “player set” is in extensive form game

my text defines player set as: In N-player game $g$, each non-terminating node is partitioned into $N+1$ sets $g^0, ... g^N$. These are player sets. However it makes no attempt to identify ...
1
vote
0answers
21 views

“Second kind” orthogonal polynomials and functions

Recently I've been doing reading in the subject of orthogonal polynomials on the real line (OPRL). Such OPs arise in solving the three-term recurrence relation $$x ...
0
votes
1answer
46 views

Where does this definition of convex set come from?

My textbook says the following. We call a set $E\subset R^k$ convex if $$\lambda x+(1-\lambda ) y\in E$$ whenever $x\in E, y\in E, 0<\lambda <1.$ I'm totally at a loss as to what this ...
7
votes
2answers
66 views

How are asymptotes actually defined in rigorous mathematics?

This question is coming from the analytic geometry viewpoint. Please ignore the viewpoint of algebraic geometry here, unless that viewpoint is somehow able to handle non-algebraic curves like $x ...
1
vote
1answer
303 views

What is this No thing?

What exactly is this No? Is there any other use of it other than graphs? Thank you so much. I am not trying to cram or anything it's just that I took a course online and a lot of the time it focused ...
2
votes
1answer
38 views

Orthogonality v. Perpendicularity

In Intro to Linear Algebra (my class) two vectors are defined to be orthogonal if their dot product is zero. And the dot product of two $n$-vectors $\vec a\cdot\vec b=0$ means that the two vectors are ...
9
votes
9answers
391 views

Motivation for the Definition of Compact Space

A compact topological space is defined as a space, $C$, such that for any set $\mathcal{A}$ of open sets such that $C \subseteq \bigcup_{U\in \mathcal{A}} U$, there is finite set $\mathcal{A'} ...
4
votes
3answers
394 views

Definition of homeomorphic?

I am looking up the definition of "homeomorphic" and the source I am looking at says there are two different definitions: Possessing similarity of form, Continuous, one-to-one, in surjection, ...
1
vote
1answer
19 views

Meaning of 3-disjoint

Definition: Two edges $\{x, y\}$ and $\{w, z\}$ of $G$ are said to be 3-disjoint if the induced subgraph of $G$ on $\{x, y, w, z\}$ consists of exactly two disjoint edges. (See page 5 of this file.) ...
-2
votes
2answers
46 views

What is meant by “functional analysis is the study of vector spaces endowed with a topology” [closed]

Lecture notes on Functional Analysis by Razvan Gelca open with the definition: Functional analysis is the study of vector spaces endowed with a topology, and of the maps between such spaces. ...
0
votes
1answer
20 views

A conditioned on B is independent of C

Let $A,B,C$ be measurable sets on a probability space. I'm trying to understand the meaning of the sentence: A conditioned on B is independent of C. The conditional probability was defined as: $$ ...
5
votes
2answers
499 views

Definition of a bounded sequence

My professor gave the following definition: A sequence $\{x_n \}$ is said to be bounded if $\exists M > 0$ such that $|x_n| \le M$ for all $n \in \mathbb N^+.$ But then what about the sequence ...
0
votes
0answers
31 views

Notation/definition problem for commutative binary operation

I'm trying to describe/define the commutative binary operation on a three-element set which when the operands are the same, gives the same element and when they are different gives the element which ...
1
vote
1answer
39 views

Definition in Operator Theory

I have started learning some Operator Theory. I encountered the following definition. I would like to know why it is that the $f(z)$ in the integrand and the $f(a)$ are both labelled as $f$ where it ...
7
votes
1answer
114 views

What is $\varphi(0)$? [duplicate]

$\varphi$ is Euler's totient function. My question is: When/if $\varphi$ is defined at $0$, what is it usually defined as? Is there a "most natural" or more commonly accepted definition of ...
2
votes
1answer
57 views

Is it a definition of cotangent?

$\cot(\alpha)$ - a value of $x$ coordinate of a point of intersection of line $y=1$ and a line passing through $(0,0)$ and a point rotated $\alpha$ radians counterclockwise on the unit circle from ...
2
votes
1answer
49 views

Can anyone please help to clarify the sentences “ into a fat tail part in L2 plus a fat body part in L1.”

In the link https://en.wikipedia.org/wiki/Fourier_transform#On_Lp_spaces what does this sentences mean: into a fat tail part in L2 plus a fat body part in L1? Would anyone please help?
0
votes
0answers
34 views

Definition of triangle

If a polygon has 3 sides, but one side has zero length (or one angle is zero degree), is it still a triangle by definition of triangle? and how about if it has 2 sides, if not 3 sides are zero?
0
votes
3answers
37 views

Axioms for Group Action

Reading the Wikipedia article on group action, I am wondering, why are the axioms stipulating that a group action obey both "compatibility" and "identity"? If a group action is merely a group ...
1
vote
0answers
41 views

Can we have different methods to estimate elements from Lp spaces?

Sorry if my question is vague. Consider I have some time samples and it is known to be summation of sinusoidal. Problem is to estimates the frequencies. Generally, Fast Fourier transform (FFT) is the ...
1
vote
2answers
57 views

Why is it called a “multiset”?

According to Wolfram MathWorld, "A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is generally also ignored ..." and A multiset is "A ...
5
votes
4answers
243 views

Are all groups algebraic?

I know the definition of a group as a set with an operation that satisfies certain axioms. I have heard that there is something called an algebraic group and that this is a group with a topology such ...
2
votes
1answer
21 views

question about the definition of Embedding

Suppose that we have two finite geberated vector spaces $V$ and $S$ over a field $\mathbb{K}$. Let $\phi:V \rightarrow S$ a function. What does it mean that $\phi$ is an embedding?
0
votes
1answer
316 views

what is the argument of 0?

When $z\neq 0$, $\arg z$ is defined to be the set $\{\theta \in \mathbb{R} : z=|z|e^{i\theta}\}$. What if $z=0$? Usually does one leave the argument of $0$ undefined? Or is $\arg 0 = \mathbb{R}$?
8
votes
4answers
434 views

Definition of General Associativity for binary operations

Let's say we are talking about addition defined in the real numbers. Then, by induction we define $\sum_{i=0}^{0}a_i=a_0$ and $\sum_{i=0}^{n}a_i=\sum_{i=0}^{n-1}a_i+a_n$ for $n> 1.\:$ Now, how do ...
-5
votes
3answers
68 views

In the real domain, are there any theorems or definitions that state all functions are differentiable? [closed]

I want to ask about basic theory of calculus, say differentiation. We know that not every function can be integrable, but as far as I know all functions are differentiable in the real domain. My ...
2
votes
1answer
36 views

Confusion regarding the $\omega$-limit of a set in a flow

In Salamon's Connected Simple Systems, p.8, the author writes that the $\omega$-limit of a set $Y$ inside a flow $\Gamma$ has the two equivalent descriptions $$ \omega(Y) = I(\overline{Y \cdot ...
3
votes
0answers
67 views

“Identify” terminology

What does it mean to "identify" two objects in algebra, as distinct from having an isomorphism or automorphism between them? I was led to think about this while reading Stewart's Galois Theory, where ...
0
votes
0answers
15 views

Probability Law of Stochastic Process Definition

I am reading Probability and Stochastics by Çınlar, and am confused by the following definition in it: I must be missing something because this definition does not seem correct to me. For ...
1
vote
1answer
206 views

$P \cap N$ is a Sylow $p$-subgroup of $N$, where $N$ is normal in $G$ and $P$ is a Sylow $p$-subgroup of $G$?

In 'A Course in Group Theory' by Humphreys, Proposition 11.14 says that if $G$ is a finite group, $P$ is a Sylow $p$-subgroup of $G$ and $N$ is a normal subgroup of $G$, then $P \cap N$ is a Sylow ...
0
votes
1answer
24 views

The definition of a functional structure on a topological space

I am reading Glen E. Bredon's Topology and Geometry and trying to solve the following problem in the book. But I can't understand what the blue g means. Could anyone explain please? Show that a ...
2
votes
2answers
42 views

Mathematical formalism for the “dot product” of three vectors

I know that the dot product of two vectors is the sum of element-wise multiplication. Using pseudo-MATLAB notation: (x,y) = sum(x.*y). I'm interested in ...
49
votes
16answers
5k views

Why do we not have to prove definitions?

I am a beginning level math student and I read recently (in a book written by a Ph. D in Mathematical Education) that mathematical definitions do not get "proven." As in they can't be proven. Why not? ...
6
votes
2answers
373 views

Is 2+2=4 an identity?

I know this seems like a silly question, but someone was trying to debate with me about how $2+2=4$ should be called an identity and not an equation. I mentioned how it has no variables and isn't true ...
3
votes
2answers
59 views

Name of the mathematical term $\frac{a-b}{a+b}$

I calculated a kind of proportion between $a$ and $b$, not $\frac{a}{b}$ but $$\frac{a-b}{a+b}$$ Do this mathematical term has a name? If so, it would help me to explain my calculations...
0
votes
0answers
29 views

Trouble With the Definition of $\Delta$-Complex in Hatcher's Book

On Pg. 103 of Hatcher's Algebraic Topology the author has defined what a $\Delta$-complex is: This is what I have gathered from what the author writes: A $\Delta$-complex is a collection of ...
1
vote
0answers
32 views

Open balls in the definition of a Euclidean submanifold

Stroock (Essentials of Integration Theory For Analysis, $\S8.3.4$) defines a $k$-dimensional submanifold of $\mathbb{R}^n$ to be a set $M \subseteq \mathbb{R}^n$ such that for all $x \in M$, there ...