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

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2
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1answer
62 views

Closure of a pre compact subspace is compact?

Suppose we have a metric space $(X,d)$ We say that $Y\subset X$ is pre compact if every sequence in $Y$ has a convergent subsequence. Notice that generally the subsequence might not converge in $Y$. ...
1
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2answers
62 views

Definition of “in terms of” for a constant vs a variable

Say I have a question that says: "answer the question in terms of x" where x is a variable. vs. "answer the questions in terms of n" where n is any constant. What is the difference between ...
3
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2answers
103 views

Strange definition of real differentiable function

I am currently refreshing my complex analysis knowledge (I had a course many years ago, but I've forgotten almost all of it). The German textbook I'm using, which is Funktionentheorie by Fischer & ...
1
vote
1answer
52 views

What is Convolution?

The definition of convolution is known as the integral of the product of two functions $$(f*g)(t)\int_{-\infty}^{\infty} f(t -\tau)g(\tau)\,\mathrm d\tau$$ But what does the product of the functions ...
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1answer
43 views

Alternative definition of differential

I've been thinking about idea of derivative, in particular about multivariable case. It seems that it is be pretty well-defined geometrically, so I thought maybe there is another definition, possibly ...
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2answers
27 views

Does the arccos itself contain all solutions or just one solution?

For the equation $$\cos(x)=\frac{1}{2}$$ All solutions are: $$x=\pm\frac{\pi}{3}+p2\pi,\quad p\in\mathbb Z\:.$$ To find these solutions, I use the inverse cosine ($\arccos$ or $\cos^{-1}$). Is the ...
2
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1answer
52 views

Definition of adjoint functor similar to the definition of homotopy equivalence?

I am new at category theory and I haven't get the definition of adjoint functor. I have seen the another definition of natural transformation much similar to the definition of homotopy. Other ...
3
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0answers
80 views

What's the derivative of a map defined on manifolds?

I'm going through Warner's book on differentiable manifolds. On page 8 he defines what it means for a map $f: U \subset M \to \mathbb R$ to be differentiable: $f$ is differentiable iff $f \circ ...
11
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5answers
892 views

Why countable unions, intersections etc.?

I was just wondering why one always insists on countability when it comes to the definition of a $\sigma$-algebra in measure theory. I mean, measure theory works as it does, but is there a deeper ...
1
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1answer
44 views

The (matrix) definition of a positive-definite function

In the definition of a positive-definite function, why are the elements of the n$\times$n matrix chosen as $f(x_i-x_j)$ for $i, j = 1,...,n$? Also, it says that "for any real numbers $x_i$". Does ...
2
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3answers
43 views

A more formal but intuitive understanding (on a definition) of a group action

We know that a symmetric group $S_n$ acts on the set $\{1, 2,\ldots, n\}$. The definition of an action of a group $G$ on a set $S$ is a function $G\times S\to S$ such that: 1) $e\ast s=s$ 2) ...
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4answers
1k views

Why is it necessary for a ring to have multiplicative identity?

I have read earlier that in a ring $(R,+,.)$ the following needs to hold: $(R,+)$ is an abelian group multiplication is associative and closed left and right distribution laws hold. However, I ...
2
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0answers
49 views

Is it possible to define submanifold like this

Wikipedia offers the following definition for an (embedded) submanifold: An embedded submanifold (also called a regular submanifold), is an immersed submanifold for which the inclusion map is a ...
2
votes
2answers
71 views

What is a primitive root?

So I'm trying to learn about RSA and have come across various subtopics, including the discrete logarithm problem. This mentions primitive roots, which I do not understand. Essentially all I want ...
0
votes
1answer
85 views

What to $t\wedge s$ mean when $t$ and $s$ are just scalars?

In my experience $\wedge$ has something to do with the outer product, but I am not sure what it means when $t$ and $s$ are not vectors and the book I am reading does not explain it. I thought maybe it ...
7
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6answers
222 views

How is $\mathbb N$ actually defined?

I know perfectly well the Peano axioms, but if they were sufficient for defining $\mathbb N$, there would be no controversy whether $0$ is a member of $\mathbb N$ or not because $\mathbb N$ is ...
2
votes
1answer
35 views

Definition/Clarification of Graph Embeddings

Recently I started reading about graph embeddings, but I am unable to grasp its definition from Wikipedia. Can anyone explain this term with an example.
2
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1answer
42 views

Definition of differentiability at the point in multivariable calculus.

I'm self-studying the analysis from Zorich and the next definition of differentiability is given: $f:E\to \mathbb{R}^n$ is differentiable at the point $x$, which is a limit point of $E\subset ...
3
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3answers
34 views

A question about the formulation of the definition of a limit for sequences

So I know the definition of a limit of a the sequence is: $a$ is a limit of a sequence $\{x_n\}$ if given $\epsilon>0$ there exists a positive integer $N$ such that $|x_n-a|<\epsilon$ for all ...
1
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1answer
20 views

Real analysis: Characteristic property for unconditional divergence

A convergent series $\sum_{k=1}^\infty a_k$ is called unconditional convergent, when it's value is invariant under any permutation $\sigma:\mathbb N\to\mathbb N$ of it's summands, i.e. ...
0
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0answers
58 views

Definition of an exponential polynomial

From Wikipedia: For formal exponential polynomials over a field $K$ we proceed as follows. Let $W$ be a finitely generated $Z$-submodule of $K$ and consider finite sums of the form ...
0
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1answer
40 views

Incompatible results with double factorial different definitions

Maybe this is a stupid question but I'm lost. The double factorial is defined as: $$n!!=\prod_{k=1}^\frac n2 2k=n\times(n-2)\times(n-4)\times\dots\times2$$ For $n$ even. By definition $0!!=1$ as an ...
7
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3answers
199 views

In the definition of a functor, why is it necessary that $F(id_{A})=id_{F(A)}$?

A functor $F$ is defined to be a mapping from category $\mathcal{C}$ to $\mathcal{D}$ such that: (1) $F(f\circ_{\mathcal{C}} g)=F(f)\circ_{\mathcal{D}} F(g)$ (say, for a covariant functor). (2) ...
0
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3answers
149 views

Give a good reason to define a function from A to B as a triple (F, A, B) rather than a functional set of pairs with domain A and image included in B.

The operative part of this question is "good reason": either an example or an argument, without preconceptions or fallacies. The object is comparing two definitions for "a function $f$ from $A$ to ...
3
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0answers
26 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
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2answers
87 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 ...
2
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2answers
95 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 ...
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0answers
9 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 : ...
5
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1answer
54 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 ...
1
vote
1answer
26 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 ...
6
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2answers
131 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 ...
25
votes
3answers
582 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: ...
0
votes
1answer
38 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 ...
1
vote
1answer
33 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 ...
2
votes
1answer
63 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 ...
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0answers
30 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
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1answer
57 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 ...
1
vote
1answer
388 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 ...
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0answers
39 views

Is there anything called kernel space?

Here I am referring kernel as an integral operation.The wikipedia link is this https://en.wikipedia.org/wiki/Integral_transform My question is: consider the function insider the integral $f(t)$ is ...
3
votes
1answer
66 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 ...
2
votes
1answer
46 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 ...
4
votes
3answers
586 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, ...
10
votes
9answers
552 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'} ...
0
votes
1answer
52 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: $$ ...
3
votes
1answer
38 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.) ...
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2answers
85 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 ...
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0answers
45 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?
2
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1answer
57 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?
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0answers
40 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
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1answer
46 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 ...