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

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3
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3answers
74 views

How to abbreviate $a_0=a_1=\cdots=a_n$?

We write $\sum_{j=0}^n a_j =a_0+a_1+\cdots+a_n$ or $\prod_{j=0}^n a_j=a_0\cdot a_1 \cdots a_n$. How to abbreviate $a_0=a_1=\cdots=a_n$? Maybe ${\Large =}| _{j=0}^n a_j$? Or $\{\forall j,k| ...
1
vote
1answer
34 views

Definition of a pushout of a short exact sequence

What is the definition of pushout of a short exact sequence? In this paper1, page $126$, under the proof of Proposition $2.8$, I don't understand how the author justify the existence of the operator ...
5
votes
1answer
54 views

Is the width of a poset well-defined?

According to the Wikipedia definition (current revision), the width of a poset is the cardinality of any maximum antichain, where "maximum antichain" here means an antichain of maximal cardinality. ...
0
votes
0answers
62 views

What's the meaning or definition of $\frac{\bigtriangleup {x}}{x}$

My teacher circled $\frac{\bigtriangleup{x}}{x}$ today on my answer and wrote "you should know that". I'm trying to figure out what it could mean but I can't come up with anything. ...
1
vote
1answer
29 views

Definition of space $L_f^2$ where $f$ is a function?

http://it.tinypic.com/r/2iqjvbl/9 Hi guys! I'm writing my thesis for my degree and it's about Sturm-Liouville theory applications. I'm using the book "Al-Gwaiz M.A. Sturm-Liouville Theory and Its ...
11
votes
3answers
2k views

Is there an example for an undefinable number?

This question is motivated by a comment of Robert on the question Can any Real number be typed in a computer? : Can you "think of" an undefinable number? – Robert Israel I would like to ...
1
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0answers
53 views

Definition: is a graph allowed to have a “dangling” edge without a vertex at its end(s)?

My textbook gives the following definition "a graph $G=(V,E)$ consisting of $V$, a nonempty set of vertices and $E$, a set of edges. Each edge has either one or two vertices associated with it." Now ...
0
votes
1answer
44 views

Generalization of asymptotic notations like big O or little o notation

Context and question For a thesis I need to prove all the arithmetic rules for asymptotic notations – such as $O(\cdot)$ or $\omega(\cdot)$ – I have used (which are a lot). Currently I look for a ...
1
vote
1answer
45 views

Definition of equivalence of quadratic forms.

I am reading A Course in Arithmetic by J-P Serre. Definition $7$ on page $32$ says Two quadratic forms $f$ and $f'$ are called equivalent if the corresponding modules are isomophic. I am not ...
0
votes
1answer
21 views

Do I Understand Closed Versus Complete in Metric, Normed and Inner Product Spaces?

I've looked at a number of references on this including some questions on stack exchange. Am I correct if I summarize by stating the following ? (1) A space C (metric, normed, or inner product) is ...
2
votes
0answers
17 views

Measure on $S^1$

My book gives the following definition of Lebesgue measure on $S^1$: For each $B\subseteq [0,1]$ in the Borel sigma-algebra on $\mathbb{R}$, define $\mu(B)=\lambda(B)$. This defines a finite measure ...
1
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2answers
30 views

Definitions of a Factor Surrounding $0 | 0$

As far as I can tell, there seems to be some controversy surrounding whether $0 | 0$. Is this partly due to different definitions of what it means for, say, $a$ to be a factor of $b$? [Def 1]: ...
0
votes
1answer
34 views

What does it mean to be less than a constant + $o(1)$?

Suppose $f:(0, \infty)\to\mathbf{R}$ is bounded (by $B$, say) and non-negative. Then $$l := \limsup_{x\to\infty} f(x)$$ exists. The proof I'm reading now claims that $f(x) \le l + o(1)$, which I'm ...
2
votes
3answers
36 views

Tensors as geometric objects

Wikipedia's article on tensors starts with: "Tensors are geometric objects..." https://en.wikipedia.org/wiki/Tensor However there is no definition of "geometric object" in Wikipedia. To my amateur ...
1
vote
1answer
51 views

Definition of weakly continuous map from one Banach space to another

What is the definition of a weakly continuous function from a Banach space to a Banach space? Suppose $X$ and $Y$ are Banach spaces. Define $f : X \rightarrow Y$ as a function. Am I right to say that ...
0
votes
2answers
48 views

What does the notation of $X_n:=n1_{(0,\frac1n)}$ in the following example mean?

I don't understand the notation of the following term $$ X_n:=n1_{(0,\frac1n)} $$ It comes from this specific problem: Let $P$ be uniform distribution on $[0,1]$ and let $X_n:=n1_{(0,\frac1n)}$ ...
1
vote
2answers
32 views

A question concerning mathematical nomenclature - formal and informal, rigorous and non-rigorous

I've never been quite sure of the exact meanings of the terms formal, informal, rigorous and non-rigorous in mathematics. For example, I've read a set of notes in which the author speaks of a ...
0
votes
1answer
44 views

Why does a subgroup not require the inverse of $y$ as well?

Here is the definition of a subgroup that I found in a book: If $H$ is a subgroup of $G$ then $H$ is a subset of $G$ such that it contains the unit element $e$ as well as satisfying the following ...
10
votes
3answers
1k views

Has my understanding of a generating set been wrong this entire time?

My understanding of the generating set (informally) was the following Pick any number of elements, say $g_1, g_2, \dotso, g_n$, in a finite group $G$. The set of all elements produced by a finite ...
0
votes
0answers
34 views

Definition of renorming of a space

The following is Proposition $4.5$ from Kalton's paper. Let $X$ and $Y$ be Banach spaces such that there exists a Lipschitz embedding $L:X \rightarrow Y$ such that $L(0)=0$ and ...
0
votes
3answers
31 views

Subtle difference between convex and strictly convex, why?

A function is convex if $f(\theta x + (1-\theta) y) \leq \theta f(x) + (1-\theta) f(y)$, $\theta \in [0,1]$ A function is strictly convex if $f(\theta x + (1-\theta) y) < \theta f(x) + ...
0
votes
1answer
21 views

how to define a subset of a population which give the max values for a set of functions

I am a computer scientist, struggling with writing a mathematical definition for a subset of a population which contains individuals that are the best scoring in the population for a set of functions. ...
0
votes
0answers
47 views

Definition of $\nabla f(ux_n - S(t))$ on a Banach space

Suppose $X$ is a Banach space and $f$ is a Lipschitz Gateaux differentiable function on $X$. Let $S : [0,1]^{\mathbb{N} \backslash \{ n \}} \rightarrow X$. Then we have $$f(x_n + S(t)) - f(S(t)) ...
1
vote
1answer
18 views

To define Dehn surgery, should one allow orientation reversing diffeomorphisms or arbitrary ones?

So for the definition of Dehn surgery (also called rational/integer surgery), what is correct: Definition Let $K$ be a knot in an oriented $3$-manifold with a regular neighbourhood $N(K)\simeq ...
2
votes
2answers
37 views

For completeness, does the limit of the Cauchy sequence need to be in the same space as the sequence?

I know $\mathbb{R}$ is complete since every Cauchy sequence of numbers has a limit. But does this limit need to be in the same metric space as the sequence. For example is $\mathbb{Q}$ complete? Every ...
3
votes
3answers
67 views

Meaning of $\int_E {f(x) \mu(dx)}?$

Suppose $f$ is a measurable real-valued function defined on a measure space $(E, X , \mu)$. What is the meaning of the RHS of the following integral $$\int_E{f d\mu} = \int_E {f(x) \mu(dx)}?$$ I ...
1
vote
0answers
31 views

Can the parameter of prior probability depends on data?

In Bayseian approach https://en.wikipedia.org/wiki/Prior_probability we often use prior probability. Can we have a prior probability distribution with parameters and while estimating the posterior ...
3
votes
1answer
91 views

Why are these two definitions of differentiability identical?

Recently, I have learned the following as the definition of multivariable differentiability. Assume that one can express $f(x, y)$ in the following form: $$f\left(x, y\right) = f\left(x_0, ...
2
votes
1answer
36 views

First version of Strong Law - finite vs bounded

Iirc, bounded functions are finite and not all finite functions are bounded. From Williams' Probability w/ Martingales: Why 'finite 4th moment' ? It's technically right I guess, but why not ...
0
votes
1answer
37 views

What is the point of having a special integer $N$ in the definition of a limit

I don't quite understand what we are trying to show when we find an $N$ such that $\forall n\geq N\in\mathbb{R}$ we have $|a_n-L|< \varepsilon$. For example I have been experimenting a bit with the ...
1
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0answers
24 views

Different notions of Submanifold

There are three types of submanifolds discussed in my book. Let $M$ be a smooth manifold. Then 1.) An immersed submanifold of $M$ is a set $S\subseteq M$ such that $S=F(S)$, where $F:N\to M$ is an ...
0
votes
0answers
22 views

How to make sense of the δ and ε limit proof and get used to it?

So when you are proving using δ and ε definition, is it necessary to create the image in your mind or you just examine the inequality equations and find a δ or ε ?
1
vote
1answer
22 views

Confusion Concerning Arbitrary Neighborhoods, Boundary Points, and Isolated Points

I've been using Steven R. Lay's book, Analysis with an Introduction to Proof as a self-study for real analysis. I thought I understood the definitions of a neighborhood (that contains its center), a ...
2
votes
2answers
22 views

The definition of a subspace in linear algebra

I'm trying to learn linear algebra on my own but I am stuck on the definition of a linear subspace. Let's assume I want to find out if $S$ is a subspace of $\mathbb{R}^2$, where $ S = [X_1 , X_2] $ ...
2
votes
1answer
40 views

Can integers be divisible by real numbers?

I have searched for many definitions of divisibility and they all seem to go like this: Let $a, b \in \mathbb{Z}$ then $b$ is divisible by $a$ if there exists $c \in \mathbb{Z} : b = ac$. Is ...
4
votes
3answers
36 views

Absolute Extrema at Infinity

Define $f(x)=1+\dfrac{1}{x}$ on the interval $[0,+∞)$. How can you find the absolute extrema of the function on this interval? My first step was to take the first derivative, which gave me ...
3
votes
1answer
66 views

Is there a rigorous mathematical definition of the Koch curve?

Is there a rigorous mathematical definition of the Koch curve? Wikipedia says that mathematics is not given a rigorous formal definition of a fractal in general. And also I have not found a strict ...
6
votes
2answers
349 views

We can define the derivative of a function whose domain is a subset of rational numbers?

Usually the derivative is defined for a function $f:A\to \mathbb{R}$ where $A \subset \mathbb{R}$, and the usual definition of the derivative at a point $a$ require the existence of an open ...
0
votes
2answers
30 views

Confusion between vector space, field and sets

From my understanding, a vector space is a set that is closed under addition and multiplication, so let $A$ denote a set, a vector space $V = (A, +, \times)$ But whenever you read the definition of ...
1
vote
1answer
22 views

Is it true you use adjoint for linear transforms while you use transpose for matrix representation of linear transforms?

In one lectures on linear algebra, my professor wrote on the first day: Let $L$ be in a linear map $L: X \to Y$, then $Im(L) \oplus Ker(L^*) = Y$ On the second day, he wrote: $L: X \to Y$, then ...
0
votes
0answers
22 views

Definition of Poincare Section

Def: Let $X\subset Y$ with $\phi_t:Y\to Y$ a semiflow. $X$ is called a poincare section of $\phi_t$ if the first return time $\tau(x):=$inf$\{t>0:\phi_t(x)\in X\}\in\mathbb{R}^+$ for every $x\in ...
1
vote
1answer
21 views

Definition of determinant of a derivative.

Can someone please help me with the following definition: $B$ is a bounded open set in $R^n$ and $g$ : $\bar{B} \rightarrow R^n$ is $C^1$ . We say $a$ is a regular value of $g$ if ...
1
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0answers
32 views

Why are these definitions equivalent for $\alpha=1$?

I am getting confused over what I can only assume is a technicality of a limit in the following definitions; Definition 1: Let $I=(a,b)\subset \mathbb{R}$ be an interval and $f, F:I\rightarrow ...
0
votes
2answers
27 views

Sequence definition

Is it correct to say: "E is an infinite sequence that maps each time instant in T to a system state." More precisely my problem is to know if I can use the term "map" for a sequence here, since we ...
0
votes
0answers
15 views

Definition of geodesic closure

What is the definition of the "geodesic closure" of an open subset $U\subsetneqq M$?, where $M$ is a compact riemannian manifold with boundary.
1
vote
1answer
16 views

How to inductively define a set?

I am trying to define a set inductively. Suppose the set I want to define is: S = {(a, b) | a, b ∈ Z,(a − b) mod 3 = 0}. I know that to define this inductively I need a basis, some rule to make a ...
1
vote
1answer
28 views

Proving A is a subset of S by mathematical induction?

Suppose I have a question similar to: Let $S$ be defined recursively by (1) $5 ∈ S$ and (2) if $s ∈ S$ and $t ∈ S$, then $st ∈ S$. Let $A = \{5^i \mid i ∈ Z+\}$. Prove that $A ⊆ S$ by ...
1
vote
1answer
76 views

Examples of tangent cone

In http://users.ece.utexas.edu/~cmcaram/EE381V_2012F/Lecture_2_Scribe_Notes.final.pdf The definition of a tangent cone is defined as the closure of the feasible directions. Definition 9. (Tangent ...
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vote
3answers
64 views

How does the following definition of isomorphism between vector space imply “structure preserving bijection”?

I am really confused by this concept of isomorphism, it seems to be a new name for something that is already well understood. Every time I look up the definition for isomorphism, the definition ...
1
vote
4answers
62 views

What is $X$? ${}{}$

Polynomials are things of the form: $$ \sum_{i\geq0} a_i {X}^i= a_0+a_1X^1+\dots $$ Where only finite $a_i$'s are non-zero. My question is, what kind of object is $X$? We call it an ...