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

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1answer
23 views

Are the implicitly definable sets of a second-order theory the sets the second-order quantifiers range over?

I know that in a second-order setting, due to the failure of the Beth definability theorem, implicit and explicit definition come apart (i.e., there are predicates which can be implicitly, but not ...
6
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2answers
659 views

Are ideals also rings?

I am learning about rings and ideals. But I am confused about something. My book (Gallian) says that an ideal of a ring by definition is a subring. But I have talked to other people who insist that an ...
0
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1answer
9 views

What is an example of a non-negative Hermitian form which is still not an inner product?

I was reading the definitions: Let $X$ be a vector space and $f: X \times X \longrightarrow \mathbb K$, where $\mathbb K = \mathbb R$ or $\mathbb C$. $f$ is said to be a Hermitian form on $X$ if: ...
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0answers
15 views

Definition of convergence rate of random variables

What is the definition of rate of convergence of a sequence of random variables? i.e. what does it mean that the convergence rate of the sequence of random variable $X_1,X_2,\ldots X_n,\ldots$ to $X$ ...
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0answers
6 views

What does it mean 'not containing $l^{\infty}(2)$ isometrically'

What does it mean 'not containing $l^{\infty}(2)$ isometrically'? The following is the context: Suppose $X,Y$ are sets and $E,F$ are normed spaces not containing $l^{\infty}(2)$ isometrically. Can ...
2
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3answers
80 views

What does set $\mathbb W$ denote?

What does set $\mathbb W$ denote? I know this may horribly lack context, but I've seen multiple times on M.SE $\mathbb W$ used in some fairly elementary context I think.
0
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1answer
12 views

What is the connection between $V(\vec x) = \frac{1}{2}(x_1^2+x_1x_2+x_2^2)$ and $V(\vec x) = \frac {1}{2} x^TPx $?

For example, $V(\vec x) = \frac{1}{2}(x_1^2+x_1x_2+x_2^2)$ has an equivalent representation $V(\vec x) = \frac {1}{2} x^TPx $ where $P$ is some matrix Can someone make this connection clearer for me ...
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0answers
33 views

What is 'free algebra'?

I've been googling the definition of it, and it seems like somehow it's related to a polynomial ring. But I still quite don't get it. Is a free algebra just a free group with additional operation ...
0
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1answer
27 views

Interpret a relation definition, it's meaning and and how it should be applied

I'm unsure of the definition below : $\mathbb{Z} = (\dots,-2,-1,0,1,2,\dots)$ Take a relation $\rightarrow$ on $\mathbb{Z}^2$ define as $(x,y)$ $\rightarrow$ $(x',y')$ only when: $(x',y')= ...
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2answers
19 views

Is transitive closure defined uniquely?

I'm encountering questions where I'm required to find a transitive closure (and the questions seem to suggest that there is only one), but I probably don't understand something in the definition, ...
1
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1answer
16 views

Help understanding cardinal multiplication and infinite Cartesian products

The cardinal product of two sets is defined to be the cardinality of the Cartesian product. The Cartesian product is: $$\prod_{\alpha \lt\beta}\kappa_{\alpha}=\{f\mid f\colon\beta\rightarrow ...
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1answer
11 views

Are these two definitions of separable extension equivalent?

Definition1 - wikipedia Let $E/F$ be an algebraic extension. Then $E/F$ is separable iff for each $\alpha\in E$, the minimal polynomial of $\alpha$ over $F$ is separable. Definition 2 - ...
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2answers
46 views

Questions about the definition of convergence

I am having difficulty understanding the definition of convergence. I've been rereading and looking at examples during this past week and I haven't made any progress. Definition: We say that ...
3
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1answer
38 views

What does it mean for pullbacks to preserve monomorphisms?

If two arrows $f_A : A \to C$, $f_B : B \to C$ are monomorphisms, then their pullback arrows $p_A : P \to A$, $p_B : P \to B$ are monomorhisms too. Is that what is meant by pullbacks preserving ...
4
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3answers
59 views

What are some illustrative examples that demonstrate how $\succ$ can differ in behavior from $>$ and/or $\geq$?

I really, really want to understand the generalization of metric spaces known as continuity spaces. Unfortunately, I always get tripped up right at the beginning. The problem is that I have little or ...
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3answers
33 views

Difference between generators and basis

What is the difference between the terms "generator set" and "basis"? Don't they both just mean a set of objects that you can use to obtain all of the objects in a larger set under some operations? ...
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0answers
23 views

Are there symbols for the map function or the filter function?

I'm going to define two notations, because I'd like to use them, but I'm wondering whether someone has done this before in serious mathematics. The idea comes from functional programming. Definition: ...
0
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1answer
25 views

Alternate definition limit

The definition of the limit of a sequence is: $L=\lim\limits_{n\to\infty}f(n)\Leftrightarrow\forall\epsilon>0:\exists N\in\mathbb{N}:\forall n\in\mathbb{N}:\left(n>N\Rightarrow ...
1
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1answer
24 views

Question on definition of group acting on a topological space.

I know that a group can act on a graph by acting on the set of vertices on a graph. I also know that a graph can be viewed as a CW complex and therefore a topological space and i am trying to bridge ...
0
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1answer
14 views

Definition of Cofinal Segment

Recently I encountered the term 'cofinal segment' in the paper 'The Point of Continuity Property, Neighbourhood Assignments and Filter Convergences' by Ahmed Bouziad, example $2.3$. Question: What ...
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3answers
36 views

What does inversion mean?

I am in highschool taking some advanced math courses and I have some questions about terminology. There appears to be more definitions to the meaning of inversion in math than I can count. I'm ...
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0answers
10 views

If $C=M \times (0,1)$, what is an integral over $\partial C$?

Let $M$ be a compact Riemannian manifold. Define $C=M \times (0,1)$ so that $\partial C = M \times \{0\} \cup M \times \{1\}$. If $f:\partial C \to \mathbb{R}$, is then the integral over $\partial C$ ...
2
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2answers
295 views

When is it allowed to take a constant out of a series?

When is it allowed to take a constant out of a series? Suppose we have a series $\sum ca_n$, when can we write it as $c\sum a_n$? It's pretty obvious when we know beforehand the series converges ...
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0answers
26 views

Why is delta used to describe difference between two entities

I hear a lot about extracting "delta" between two properties in my current job. I come from a User Interface programming background and I do not really have much math background. I looked up delta ...
0
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1answer
21 views

How to interpret the definition of inductive set?

I can't understand the sentence below: "A subset Y ⊂ X will be called inductive if, for every x ∈ X such that y ∈ Y for all y ∈ X such that y < x, we have x ∈ Y." please tell me what's the meaning, ...
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0answers
31 views

Does this violate the notion of positive definiteness?

From a previous course in linear algebra, I was taught that a function is positive definite if it satisfies $$\left< \vec x \mid v(\vec x) \mid \vec x\right> > 0 $$ In simpler notation, ...
2
votes
1answer
114 views

I need help understanding Frege's definition of number

I have really been trying to understand Frege's definition of a number or at least gain a strong intuition of it. However, my attempts have not been fruitful. If someone could help me it would be much ...
17
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2answers
220 views

To whom do we owe this construction of angles and trigonometry?

I've come across what is, to me, the most precise, beautiful and thorough definition of what we know of as the angle between two vectors. I say this because most literature either skims over things ...
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2answers
42 views

What exactly is the meaning of the following $\inf\{ s_n : n > N\}$ and $\sup\{ s_n : n > N\}$

What exactly is the meaning of the following $$u_N = \inf\{ s_n : n > N\} \ \ \text{ and} \ \ v_N = \sup\{ s_n : n > N\}$$ This might seem a stupid question, but I am not understanding the ...
0
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1answer
14 views

Definition of normal extension.

Let $E/F$ be an algebraic extension. Then, $E/F$ is normal iff $E$ is a splitting field of a family of polynomials in $F[X]$. So does this mean that if $E$ is a splliting field of a given ...
4
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2answers
130 views

Evaluating a definite integral by definition

I have an area function $A(x)$ defined as $$A(x) = \int_{-1}^{x} (t^2 + 1)\space dt$$ ... and I would like to use the definition of definite integral to evaluate it. I started this way $$A(x) = ...
1
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1answer
61 views

Why does this author define cardinality indirectly?

I'm studying Enderton's Elements of Set Theory and in the page 129 he defines what it means two sets being equinumerous: After that in the page 136 he defines cardinality: Why doesn't he define ...
0
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1answer
48 views

What does the notation $\mathbb R[x]$ mean?

What does the notation $\mathbb R[x]$ mean? I thought it was just the set $\mathbb R^n$ but then I read somewhere that my lecturer wrote $\mathbb R[x] = ${$\alpha_0 + \alpha_1x + \alpha_2x^2 + ... + ...
0
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1answer
32 views

Is this alternate definition of Limit correct?

Would it be correct to define the limit of a series as the smallest number that no number in the series is greater than (for an increasing series, the other way around for a decreasing series, i.e the ...
0
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1answer
35 views

How do I prove cardinality is well-defined?

I define equinumerous and cardinality in this way: $A$ and $B$ are equinumerous (written $A\sim B$) if there is a bijection between them. We say $card(X)=card(Y)$ if $X\sim Y$. I would ...
0
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0answers
10 views

Defining the rational function field in n variables.

Reading over an editing my dissertation "Elementary functions" and i am having trouble with my definition of a rational functions in n variables, this is what i have written but its missing one part: ...
1
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1answer
41 views

Vector Space: is there a general definition of annihilator beyond the dual space?

The definitions I see for the annihilator of a subset S of a vector space V over a field F is the subset $S^0$ of the dual space $V^*$given by $S^0 = \{\varphi\in V^\star:\varphi(S)=0\}$, and the ...
0
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1answer
32 views

Does the line y=mx have equal intercepts?

This line passes through the origin and has zero intercepts. Can this be called as a line having equal intercepts? What is the definition of intercepts of a line then?
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1answer
108 views

Why does 2+2 equal to 4? [duplicate]

The question is in the title. I am very appreciative of any time and concern put into belaboring this relatively little problem.
0
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1answer
15 views

Riemann integrable function over bounded set

In my calculus course we extended the definition of riemann integrable to functions whose domain are jordan-measurable sets, but can we extend the definition if we just ask for the domain to be ...
2
votes
2answers
54 views

Definition of Aut(G) in the graph theory and group theory

For a fixed group G, we define the collection of group automorphisms is the automorphism group Aut(G) in the group theory. (An automorphism: a permutation on the set G) In the graph theory, on the ...
3
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0answers
77 views

Funny translations of mathematical words [closed]

As already noticed in this question there are some mathematical words that literally translated from a language to english (or from english to this language) means something totally different. A few ...
0
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1answer
69 views

How to translate math technical terms?

What is a good way to translate mathematical technical terms? This can sometimes be hard because some words have different meanings in some language. For example: "eigenwert" (= "eigenvalue" ...
3
votes
1answer
140 views

Is my attempt to define the concept “smooth manifold” as a structure satisfying certain axioms correct?

In the lecture notes for a class I'm currently taking, smooth manifold structures are defined as equivalence classes of atlases. However, the issue I'm having is that its not entirely clear (to me) ...
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1answer
18 views

Radical function in two (or more) variables

I'm reading a paper where the author uses the word radical function for a function $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$. I understand the definition of a radical function if $n=1$, but what if ...
0
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1answer
72 views

Complex number field: ''essentially'' unique?

I solved the following exercise but have trouble making sense of the result: If $\widetilde{\mathbb C}$ is another field of complex numbers and $\varphi : \mathbb C \to \widetilde{\mathbb C}$ is a ...
0
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1answer
38 views

Useful definition of limit?

Is the following definition of a limit a useful one? Does it make sense? Why/Why not? $lim_{x\rightarrow a} f(x) = b \Leftrightarrow \forall \varepsilon > 0 : 0<|x-a|<\varepsilon ...
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0answers
19 views

Finding the Partial Derivative of this function (Chain Rule)

This problem is not part of the assigned problem set, but I would really like to figure it out. I spend the last 2 days thinking about it and stuck on the chain rule part. I know how to do this using ...
1
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2answers
29 views

Is the empty family of sets pairwise disjoint?

„A family of sets is pairwise disjoint or mutually disjoint if every two different sets in the family are disjoint.“ – from Wikipedia article "Disjoint sets" What about the empty family of sets? Is ...
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0answers
13 views

Checking the definition of absolutely irreducible representations

The definition of an irreducible representation $(\rho, V) $ is one with no subrepresentations. Am I correct in saying that a absolutely irreducible means "it is irreducible over the algebraic ...