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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3
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4answers
428 views

what exactly is an open set?

Many, infact all the books on topology I have come across define open sets in the following way: "A set $A$ is said to be open if by moving in small amounts in any direction about any point we ...
6
votes
1answer
624 views

Why does the definition of an open subscheme / open immersion of schemes allow for an “extra” isomorphism?

After taking an algebraic geometry course last year, I've been reviewing the material this year, and I remembered something that struck me as odd, but which I'd neglected to ask about at the time: ...
5
votes
2answers
915 views

Embedding, immersion

Could someone please explain what "embedding" means? (Maybe a more intuitive definition) I read that the Klein bottle and real projective plane cannot be embedded in ${\mathbb R}^3$ but is embedded in ...
3
votes
2answers
820 views

Definition of a sphere

If there is no further specification (such as solid or hollow), does a "sphere" refer to the solid/filled form or the hollow shell? Thanks.
2
votes
2answers
2k views

What is the “Donkey Theorem”?

I was watching the Turkish version of Who Wants to Be a Millionaire? and they asked this question: What field is the Donkey Case (or I guess it can be translated as Donkey Theorem) related to? ...
2
votes
1answer
213 views

$A_{i,j}B_{i,j}$ is matrix dot-product in Einstein Notation?

Skim-read an engineering book stopped to this assertion "matrix dot product is $\sum_{i,j \in I} A_{i,j} B_{i,j} := A_{i,j} B_{i,j}$ in Einstein Notation". Sorry but what does it really mean? I have ...
1
vote
2answers
1k views

Interpolation, Extrapolation and Approximations rigorously

A foreign book mentioned that "when the Lagrange's interpolation formula fails (for example with large sample due to Runge's phenomenon), you should use approximation methods such as ...
1
vote
1answer
370 views

What is empirical accuracy?

Can anyone give a definition on empirical accuracy? I Googled this keyword but cannot find a satisfactory definition. It seems to be the accuracy rate of a specific sample. Any link or typed text ...
1
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2answers
800 views

Subexponential growing functions

What is the most common definition of a subexponential growing function ? It seems there are different notions in literature.
1
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3answers
222 views

How to define $-\infty$?

I think I understand the fundamental concept of infinity. Elementary mathematics define $\infty := \frac{x}{0}$, for every $x$. And also $\infty := \frac{-x}{0}$ for every $x$. I know only one ...
1
vote
0answers
165 views

Is the extra condition in this definition superfluous?

I am learning Differential Geometry and someone told me that the second condition of a definition provided in books follows from the first and is hence superfluous. I cannot dispute it, so convince me ...
1
vote
1answer
192 views

“There is a natural [way/map/etc.]…” [duplicate]

Possible Duplicate: What is a natural isomorphism? I have often encountered the phrase "There is a natural [way/map/etc.]..." when describing say isomorphisms, maps, etc. What exactly does ...
8
votes
3answers
348 views

Why is closure omitted in some group definitions?

In some texts, there are three group axioms and in some there are four. The difference is that one of the axioms, the closure ($a,b\in G$ then $a*b \in G$) is omitted. Why is this so?
1
vote
1answer
248 views

Partial derivative notation: is that a projection function?

Consider the following definition: Let $(U,\phi)$ be a chart and $f$ a $C^\infty$ function on a manifold $M$ of dimension $n$. As a function into $\mathbb{R}^n$, $\phi$ has $n$ components ...
18
votes
5answers
2k views

What is a universal property?

Sorry, but I do not understand the formal definition of "universal property" as given at Wikipedia. To make the following summary more readable I do equate "universal" with "initial" and omit the ...
11
votes
3answers
2k views

How can an ordered pair be expressed as a set?

My book says \begin{equation} (a,b)=\{\{a\},\{a,b\}\} \end{equation} I have been staring at this for a bit and it is not making since to me. I have read several others posts on this, but none made ...
1
vote
1answer
519 views

The definition of submanifold of a manifold with boundary

What is the exact definition of submanifold of a manifold with boundary? For example, when $H$ is the half space of the plane and S is a cycle which intersects with the origin in the half plane. Then ...
16
votes
5answers
1k views

Chain Rule Intuition

We know that the chain rule is used to differentiate a composite function ,say $$f(x) = h(g(x))$$ It's defined as the derivative of the outside function times the derivative of the inner function or ...
0
votes
1answer
216 views

Principle of substitution — fields

Could someone possibly explain this definition (applied to fields) to me? The principle of substitution: In a field F, we can, in any formula involving an element $\alpha\in F$, replace $\alpha$ ...
21
votes
3answers
1k views

Definition of convergence of a nested radical $\sqrt{a_1 + \sqrt{a_2 + \sqrt{a_3 + \sqrt{a_4+\cdots}}}}$?

In my answer to the recent question Nested Square Roots, @GEdgar correctly raised the issue that the proof is incomplete unless I show that the intermediate expressions do converge to a (finite) ...
1
vote
1answer
358 views

What does “a set of things” mean?

Suppose we defined some mathematical object $P$, where $P$ is natural number, polynomial, endofunction, geometric figure, etc. What does the expression “$A$ is a set of $P$s” mean: Set inclusion) ...
0
votes
1answer
352 views

Need help understanding Hessian matrix for direction estimation

Additional context: $H = |δ^2f / δx_iδx_j|$ is the Hessian matrix. $(3)$ From my previous question: What are the functionality of δ symbol and $δr^T$?, I got a few questions: I have read more ...
1
vote
1answer
62 views

What are the functionality of δ symbol and $δr^T$?

I got two questions here: Does anybody know what is the functionality of the small delta letter δ here? Is it simply the same as the rate of change just like the big delta letter Δ? And for the ...
4
votes
3answers
1k views

Definitions of direct product and of direct sum

I was wondering if there are some general definitions for direct product and for direct sum, for example in category theory or in set theory, so that the concepts for vector spaces, Abelian groups, ...
1
vote
1answer
150 views

Rephrasing Munkres' Theorem Re: Inverses of Jacobians

Below are theorems from Munkres' "Analysis on Manifolds". The proof of Theorem 7.4 on the right invokes the chain rule, stated on the left. The conditions of Theorem are somewhat strange and appear ...
5
votes
1answer
149 views

What motivates discrepancies between the definitions of “continuous” and “limit”?

I am working from Munkres' Analysis, and I've converted his definitions slightly to make them easier to compare. In the table below, you can fill in the blanks in the top row with words from the ...
1
vote
1answer
332 views

What is a “mixed graph”?

I'm working on a digraph problem in which bidirectional edges need to be treated separately. As such, we could consider them as undirected edges. Clearly, if I replace bidirectional edges with ...
0
votes
1answer
535 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 $B$?
6
votes
2answers
802 views

Precise definition of “weaker” and “stronger”?

If I say that $A$ is stronger than $B$, do I mean that $A \Rightarrow B$, or that $B \Rightarrow A$? (Or something else?) I feel like I have seen both usages in literature, which is confusing. ...
6
votes
5answers
551 views

Definition of Ring

I'm studying Abstract Algebra right now, currently covering rings. In the introduction of the subject, I am curious as to why there is no need for there to be a multiplicative identity. I understand ...
1
vote
3answers
103 views

Should one think of a network as a connected graph ? (Or: What is the right way to think of a network?)

In the definition of a network, are we only considering connected graphs ? Because I keep encountering definitions that don't assume explicitly that we deal with connected graphs, but which would be ...
0
votes
2answers
233 views

Why people have to find quadratic formula,isn't that the formula cannot solve a polynomial with 2 and 1/2 degree?

Why people have to find quadratic formula,isn't that the formula cannot solve a polynomial with 2 and 1/2 degree? and just curious, how many roots does a polynomial with 2 and 1/2 degree have and how ...
1
vote
2answers
151 views

exterior product definition

i have question from vector mathematics,i know that if there is given two vector, for instance $a=\{a_1,a_2,a_3\}$,$b={b_1,b_2,b_3}$; then so called exterior product is determined as $a\wedge ...
1
vote
1answer
211 views

Expressing P = NP as a first order formula

I want to express P = NP in a completely formal way. My first try: There exists an algorithm A and a polynomial bound p such that for all input i, A(i) = true iff i is a satisfiable formula and ...
15
votes
5answers
667 views

Definition of definition

I was wondering if there is a good way to "define" what definition means exactly in mathematics. Since the answers may be subjective or philosophical, I want to ask only for references on this topic. ...
8
votes
3answers
866 views

Is the empty graph connected?

Is the empty graph always connected ? I've looked through some sources (for example Diestels "Graph theory") and this special case seems to be ommited. What is the general opinion for this case ? As ...
4
votes
2answers
428 views

True, false, or meaningless?

Are the following two assertions always true, always false or meaningless? $\exists i \in \emptyset$ $\forall i \in \emptyset$ Because it seems that one encounters expressions of this kind fairly ...
7
votes
3answers
703 views

What is the operation $\boxtimes$?

Reading papers about $p$-adic analysis and Galois representations, I have found objects like this $D \boxtimes \mathbb{Q}_p$. So my question is what is $\boxtimes$ and how do we read it ?
4
votes
3answers
661 views

Definition of Ring Vs Rng

When I took abstract algebra I learned that a ring was a set that is an abelian group under addition and monoid under multiplication (along with the distributive property). In preperation to tutor ...
3
votes
1answer
1k views

What is Hermite data?

Using fairly simple language, what is Hermite data? I encountered it here, http://www.frankpetterson.com/publications/dualcontour/dualcontour.pdf and could not get an answer on standard StackExchange, ...
4
votes
1answer
375 views

What is this R-like symbol power 2?

I found this in a computer medical research text. What is the meaning of this R-like letter? S, in this context is an iso-intensity surface. [edit] Since context is not sufficient, I think it is ...
2
votes
3answers
195 views

“Defined as” versus “Equivalent to”

This is a lazy question, but very often textbooks use the "$\equiv$" (equivalent to) sign and the "$:=$" (defined as) sign in the same places from book to book. I suppose equivalence to a previously ...
2
votes
1answer
283 views

Set-theoretical definitions of the notion of “structure”

What general set-theoretical definitions of the notion of "structure" are there? By general definition of "structure" I mean a formula $\Phi(x)$ in the first-order language of set theory such that ...
19
votes
3answers
2k views

Is a line parallel with itself?

Simple Question, but I'm finding a lot of dispute on the "lesser" internet. Basically, given a line, is it parallel with itself?
1
vote
2answers
111 views

Definition of identity in a monoid

I'm having trouble understanding the way the identity element is defined in Lang's Algebra. Below is the relevant information. Suppose we have a monoid G with elements $x_{1},...,x_{n}$. We can define ...
4
votes
2answers
309 views

equivalent definitions of orientation

I know two definitions of an orientation of a smooth n-manifold $M$: 1) A continuous pointwise orientation for $M$. 2) A continuous choice of generators for the groups $H_n(M,M-\{x\})=\mathbb{Z}$. ...
8
votes
4answers
610 views

If a function can only be defined implicitly does it have to be multivalued?

What is the general reason for functions which can only be defined implicitly? Is this because they are multivalued (in which case they aren't strictly functions at all)? Is there a proof? ...
2
votes
1answer
122 views

Free presentations of $\mathbb{Z}G$-modules

Dear All, I have a doubt about a specific definition, but I cannot find any help on the web or on the books that I have. Talking about $\mathbb{Z}G$-modules, what does one intend saying "take a free ...
0
votes
1answer
259 views

Definition of a subcomplex of a $\Delta$-complex

I am taking the following as the definition of a $\Delta$-complex. (i) one starts with an indexing set $I_n$ for each $n \in \mathbb{Z}_{\ge 0}$. (ii) for each $\alpha \in I_n$, one takes a ...
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2answers
152 views

Informal Equivalents of Mathematica “Set” and “SetDelayed”

How would one distinguish between what is meant by Mathematica's "Set" and "SetDelayed" functions in informal mathematical notation? Is there a way to make this distinction any any reasonably standard ...