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

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0
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2answers
44 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
votes
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, $\mathcal{A}$ ...
1
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2answers
30 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 ...
1
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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
votes
4answers
414 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
vote
1answer
30 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 ...
2
votes
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 ...
0
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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
21 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
54 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 ...
0
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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
votes
4answers
967 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
77 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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0answers
58 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 ...
1
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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
vote
1answer
288 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
votes
1answer
38 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
42 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
votes
1answer
22 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
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2answers
94 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
votes
2answers
51 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
61 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. ...
1
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0answers
38 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
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2answers
57 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
32 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
30 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
votes
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 ...
0
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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
votes
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
447 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
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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. ...
1
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0answers
15 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 ...
0
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0answers
38 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
18 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 ...
1
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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 ...
1
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0answers
43 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
vote
1answer
27 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
vote
1answer
47 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
votes
1answer
78 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 $P$ ...
0
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0answers
21 views

How to define a non Triangle Free Graph

Consider a Regular Connected Graph $G= \bigcup\limits_{i=1}^{m} U_i$ where- $U_i$ is a sub-graph of $r$ vertices $\forall i$.(i.e. $G$ is a $r$ regular graph) $\forall i$, $U_i$, is not a ...
0
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0answers
51 views

Definitions of Weil Algebras

I am confused by several definitions of Weil Algebras and their connection to each other. Kock's book on synthetic differential geometry defines a Weil algebra over a ring $R$ as an $R$-algebra of ...
1
vote
1answer
49 views

What is the standard definition of a “periodic” function?

This question is admittedly pedantic, but I like my definitions precise. Tom Apostol, in his calculus book, defines a periodic function as follows. A function f is said to be periodic with ...
2
votes
1answer
35 views

How to prove equivalence of definitions for matrix similarity

It seems the most usual way to define matrix similarity is as follows: "Let $A$ and $A'$ be two n-by-n matrices, we say they are similar if there exist some invertible n-by-n matrix $P$ such that ...
0
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0answers
33 views

Symplectic matrix over $\mathbf{H}$ has determinant $1$ as well?

According to this question, symplectic matrices over $\mathbf{R}$ have determinant $1$. Does the equivalence carry over if we go back to the quaternion definition for $Sp(n)$? In the original ...