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

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0
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1answer
40 views

Why do we need $\sigma$-algebra in the definition of a measurable space?

A measurable space is a triple $(M, \Sigma, \mu)$. I am confused by one thing, to my knowledge almost everything you think of is measurable. So why do we need $\Sigma$? Why is that a measurable set ...
1
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1answer
41 views

What is the group $O(2)$?

I have learnt about the group $$ SO(2) = \left\{\pmatrix{\cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta)} : \theta \in \mathbb{R} \right\}. $$ This group has do with rotations. ...
0
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1answer
17 views

What is “analytic vector for closed operator”?

I need the defenition of "analytic vector of closed operator that acts on Hilbert space". I cant find it in google and in my textbooks (Khelemsky "Lectures And Exercises on Functional Analysis"), I ...
0
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1answer
38 views

Matching the definition of hom-functor with how these are used when defining adjuncts

I have a problem matching the definition of a hom-functor (from nlab) with how this concept it used in the definition of adjunction (from nlab): The hom-functor is defined on $C^{\text{op}}\times C$, ...
1
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1answer
51 views

Defining adjoint functors: What does “natural bijection” mean?

Take the following definition of adjunction from the nlab This definition can be found in numerous other places. My brain parses this definition perfectly up till the point where it says that for ...
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2answers
48 views

Is $f(x) = 2x+1$ injective? Is it surjective? [closed]

How would I answer this? I know what it means to be surjective and injective. Is the function $f(x)=2x+1$ injective? Is it surjective? Give reasons for your answers. I assume they are both because ...
0
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1answer
25 views

Meaning of denseness in a $L^p$ spaces?

I am currently studyind Density theorems in $L^p$ - spaces. In that, I have encountered a theorem which goes like this - The space of integrable simple functions is dense in $L^p $(E, $\mathcal{A}$ ...
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2answers
32 views

Problem in understanding definition of absolutely continuous?

Suppose $(E, \mathcal{A})$ is a measurable space. Let $\mu$ and $\gamma$ be two distinct measures of this space. Now we say that $\gamma$ is absolutely continuous with respect to $\mu$ if for every $A ...
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0answers
69 views

What is the name of the property $x^m=x$ when $x$ is in a ring?

I have been doing problems in Atiyah & MacDonald's Introduction to Commutative Algebra, and in problem 1.6 it asks to assume the existence of an idempotent element in an ideal whenever the ideal ...
5
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4answers
416 views

Definition of the total derivative.

I am trying to understand the following definiton. $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$ . The total derivative of $f$ in point $a$ is the unique linear map $Df|_a$ such that $$\lim_{h ...
1
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1answer
32 views

Definition request: explicit definition of covering compactness in terms of set notation

Part of my confusion with covering compactness stems from the fact that it is a definition given almost completely in a high level manner (in English no less). When I look at: A set $A \subset ...
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4answers
66 views

What is the exact definition of a metric?

In some book I found, a metric on a non-empty set $X$ defined as a map $$X\times X\to \Bbb R^{+}$$ and some other place as $$X\times X\to \Bbb R$$ So, is a metric a real valued function or a ...
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0answers
21 views

Formal definition of a face of a polyhedron

Given an $n$-dimensional convex polyhedron, an $(n-1)$-dimensional face of it can be defined as an intersection of the polyhedron with a supporting hyperplane. What is the formal definition in the ...
1
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1answer
22 views

If $f:\mathbb{R}^2\to X$, how do we call $f_{x_0}(y) =f(x_0,y)$ as a function of one variable?

For obvious reasons coming from probability theory I have been calling the function $f_{x_0}(y)$ ($x_0$ fixed) "marginal function". However, reviewing some literature I've noticed that the word ...
1
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0answers
55 views

Ignoring the lack of rigor, is this a fair argument to make when considering if 0^0 should be equivalent to 1? [closed]

The Professor of Mathematics argued that 0^0 is undefined because the limits $0^x$ and $x^0$ as x approaches 0 don't agree. That seemed logical to me, but then Scott pointed out in the comments that ...
4
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4answers
164 views

Iff Interpretation

I understand that (1) "$A$ if and only if $B$" ($A\iff B)$ means that (2) "$A$ implies $B$ and $B$ implies $A$" $(A\implies B)\land (B\implies A)$. The phrase "$A$ if and only if $B$" sounds as ...
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2answers
51 views

What does “closed” mean in Heine Borel for $C^0$?

Heine Borel for $C^0$: A set $\mathcal{E} \subseteq C^0([a,b], \mathbb{R})$ is compact if it is closed, bounded and equicontinuous. I don't really understand what closed mean in the ...
7
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4answers
980 views

What is wrong with this argument that closed interval [0, 1] is not compact?

I am a student majoring engineering. I am studying real analysis with textbook 'Measure and Integral' by Wheeden and Zygmund. This book defined compact like the following: $E$ is compact if ...
4
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0answers
82 views

Why is cos at $\pi/2$ not undefined?

If the $\cos$ function is based off of the ratio of the adjacent side of Euclidean, right triangle, with fixed hypotenuse length (such as the unit circle), then how does this correspond to a defined ...
1
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1answer
96 views

How does one “join” two graphs in graph theory?

I am asked to find the join of two graphs in graph theory. But I cannot find the exact definition! I know that in lattice theory, we join every vertex of a graph to every vertex of another graph to ...
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2answers
48 views

$\nabla ^2 G$ meaning

If $G$ is a function with three components. $G$ is actually a Green's function in my case, like $G( \textbf{x} , \xi)$ with $\textbf{x} = (x,y)$ and $\xi = (\xi _x, \xi _y)$ so I am guessing it is ...
1
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1answer
23 views

Series representation of a differential form

I have a problem understanding the general series representation of a p-form. For 1-form things are pretty clear to me: For $h = (h_1, \dots, h_n)^T \in \mathbb{R}^n $ and $ h = \sum\limits_{i = ...
2
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3answers
102 views

Understanding iff [duplicate]

I'm having difficulty understanding why it is appropriate to use if and only if, something I thought I had a firm grasp on. From Lara Alcock's book, How to Study as a Mathematics Major: ...
1
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1answer
290 views

A proof that $|x+yi|=\sqrt{x^2+y^2}$, based on the given the conditions

If we attempt to define $|x+yi|$ by following conditions: $|x|=|xi|=x\operatorname{sgn}(x)$ (implicitly meaning the result will always be $\ge 0$) $|xz|=|x||z|$ $|z^x|=|z|^x$ for $x \in ...
0
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1answer
39 views

I know what a Complex Plane is, but what is a complex $k$-plane?

This may be somewhat related to physics, but I saw in a non-English paper (which I googled "complex $k$-plane" for some constant real $k$) that mentioned a complex $k$-plane. $k$ in its context was ...
2
votes
1answer
58 views

What's the difference between a Nash, Correlated, and Extreme equilibrium?

As the title states, what's the difference? As I understand it: The Nash Equilbirum (NE) is a solution concept in non-cooperative games where no player has incentive to unilaterally deviate from a ...
21
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7answers
2k views

Proof that the empty set is a relation

In the book Naive Set Theory, Halmos mentions that the "The least exciting relation is the empty one." and proves that the empty set is a set of ordered pairs because there is no element of the empty ...
0
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1answer
24 views

Simpler way to define convex functions

I find the standard way of defining convex and concave functions slightly tricky. For me, it is the introduction of the new variable $0<λ<1$. However, I understand it intuitively. I was ...
0
votes
2answers
96 views

In definition of a category , what is the meaning of 'consists of'

A category $\mathsf C$ consists of the following three mathematical entities: A class $\operatorname{ob}(\mathsf{C})$, whose elements are called objects; A class $\hom(\mathsf{C})$, whose ...
5
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2answers
53 views

Is there a name for a group having a normal subgroup for every divisor of the order?

Suppose, G is a group of order $n$. Is there a name (or an easy criterion) for the property that for every divisor $d|n$, there is a normal subgroup of order $d$ ? The abelian groups and the ...
1
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0answers
62 views

How are non-associative groups called?

Is there a name for the algebraic objects that have all the properties of groups expect associativity? For example, the unit octonions have this property. They satisfy the following definition. ...
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0answers
43 views

Does there exist a derivative of the Golden Ratio (equation)?

On my calculus exam, there was a question asking "Is the Golden Ratio differentiable? If so, use the definition of a derivative to show it." The "definition of a derivative" is ...
3
votes
2answers
59 views

Definition of convergence of $\sum_{i=-\infty}^\infty a_i$

This is a really basic question, but I'm unsure about the definition for convergence of $$\sum_{i=-\infty}^\infty a_i$$ The definition $$\sum_{i=-\infty}^\infty a_i=\lim_{n\to \infty}\sum_{i=-n}^n ...
0
votes
1answer
10 views

$\alpha$ in a normal confidence interval

Related to: Deriving the confidence interval $P(-\Phi^{-1}_\alpha < X < \Phi^{-1}_\alpha) = 1-2\alpha$ I'm not sure why, but I'm having some trouble with the definitions here. ...
1
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1answer
33 views

Why does an injective resolution give a complex?

The following is the definition of the right-derived functor from Lang's Algebra - Let $\mathcal A$ and $\mathcal B$ be abelian categories with enough injectives. Consider a covariant functor ...
0
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1answer
35 views

$f$ attains its minimum, delta definition

I just want to check if the following "definition" makes sense and is correct: A function $f: X \rightarrow \mathbb{R}$ attains its minimum if $\exists \delta > 0$ such that $f(x) \geq \delta$, ...
0
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3answers
70 views

If I prove that $|A| > |\mathbb{N}|$, does it mean that $A$ is uncountably infinite?

My intuition is that if something is larger than $\mathbb{N}$ then it follows that it is larger than $\mathbb{Q}$ as there is a bijection from $\mathbb{N}$ to $\mathbb{Q}$. So if a set is larger than ...
0
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0answers
25 views

What are notions of “well-behavedness” for distributions?

When talking about functions, there are many notions that are interpreted as well-behaved, such as increasing, continuous, integrable, and many of these terms can be directly understood in some ...
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2answers
54 views

Show that $\frac{te^t}{e^{2t}-1}$ is integrable?

How to show that the function $f\mapsto\dfrac{te^t}{e^{2t}-1}$ is integrable on $(0,+\infty)$ ? I think it suffices to say that $f\mapsto\dfrac{te^t}{e^{2t}-1}$ is well-defined and continuous on ...
-1
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2answers
69 views

Is there a definition that defines the set of all factors of a natural number? [closed]

So let's say that you have the number $n=12$. The factors of $12$ are $1$, $2$, $3$, $4$, $6$, and $12$. I'm wondering if there's some definition that when you plug in a natural number $n$, it will ...
4
votes
4answers
462 views

Is every Closed set a Perfect set?

From 'baby' Rudin. I've seen that a set is closed iff it contains all of its limit points. In Rudin, $(d)$ says if every limit point of E is a point of E, then $E$ is closed. He also says $(h)$: $E$ ...
4
votes
0answers
30 views

Prove Limit Addition Thm as x approaches infinity

I am supposed to prove prove $\lim_{x\to \infty} [f(x)+g(x)]= L+M$. Starting to realize I don't really understand the formal definition of a limit, although I do understand the general concept. ...
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0answers
16 views

Different meanings of $L_r(P)$: norm, metric, class of functions

I have doubts on the meaning of the symbol $L_r(P)$ for $r\in \mathbb{N}$. Consider a random variable $X: \Omega \rightarrow \mathbb{R}$ with probability distribution $P$ and a random function ...
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0answers
42 views

Definition of q-ary lattices

Lattices defined over the vector space over $\mathbb{R}^n$, whereas q-ary lattices consists of only integers i.e., Let A be a $\mathbb{Z}_q^{n\times m}$ then q-ary lattice is defined as $$\Lambda(A) = ...
2
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0answers
20 views

The different between Non-parametric statistics and Parametric statistics?

I have 1D data. I want to classify the data to $N$ cluster. The two common ways can use Using the mean/average value as a criterion to classify Assumption that the data follows a distribution with ...
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2answers
53 views

Defining Positive integers..

While reading Calculus by Apostol I found the set of positive integers defined as "Set of Real numbers that belong to every Inductive set"... The question is "Why we don't define the set of ...
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0answers
47 views

What does non-degenerate mean?

The context is that f is a real-valued function on a plane such that for every non-degenerate square ABCD in the plane, f(A)+f(B)+f(C)+f(D)=0.... What does this word mean here? I've found that "in ...
1
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1answer
28 views

Formula for Determining Cost

I am trying to put together a formula for determining company costs for certain services in-house. A little background: before I arrived, the company outsourced their IT work. Now that I've been ...
1
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1answer
51 views

Is there any difference between covariant vectors and one-forms?

Just need to have it clarified, are the 2 expressions interchangeable, or is there any difference? I'm trying to learn differential geometry on my own and it is really hard.
7
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1answer
80 views

The category of locally $P$ spaces

Let $P$ be a class of topological spaces (for example, compact spaces). The class of locally $P$ spaces consists of those spaces in which every point has a neighborhood basis consisting of $P$ ...