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

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2
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1answer
31 views

$\omega$-limit set of a point $x \in X$

I would like to verify whether the following definition of the $\omega$-limit set of a point $x \in X$ is correct: $$\omega(x) = \{ y \in M : \exists \text{ sequence }\{t_j\}, \text{ where } t_j \...
2
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2answers
38 views

Definition of interior

An interior point is defined as the following in the Euclidean space. If $S$ is a subset of a Euclidean space, then $x$ is an interior point of $S$ if there exists an open ball centered at $x$ ...
2
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2answers
49 views

Marsden's definition of Taylor Series

How does the following definition of Taylor polynomials: $f(x_0 + h)= f(x_0) + f'(x_0)\cdot h + \frac{f''(x)}{2!}h^2+ ... +\frac{f^(k)(x_0)}{k!}\cdot h^k+R_k(x_0,h),$ where $R_k(x_0,h)=\int^{x_0+h}...
2
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1answer
43 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 $A=...
2
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2answers
36 views

Definitional question: difference between a correspondence and a function

Is there a difference between a correspondence and a function? For example, in game theory I am told that for a given strategy set, $\Sigma_i$, the best response given by $BR_i(\sigma_{-i})=\text{...
2
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1answer
54 views

Does the sum of exterior angles of a simple, convex polygon truly = 360°?

The question being asked of me is the following: What is the sum of a polygon's exterior angles? Assuming, again, that the polygon is simple and convex, the answer I see repeatedly given is 360°....
2
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1answer
29 views

Clarification of term in graph theory - about star polygon graphs

I was reading about star polygon graphs from the following link: http://mathworld.wolfram.com/StarPolygon.html. As far as I noticed I felt that whenever $d$ is a proper divisor of $n$, then we get $...
2
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1answer
38 views

What is the definition of $\ell^2(G)$ where $G$ is a group?

First I'll give some context for my question. I'm learning about crossed products of dynamical systems involving $C^*$-algebras and I've just seen the definition of a covariant representation. I have ...
2
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1answer
43 views

Skeptical about (my understanding of) wikipedia's definition of a reflective subcategory.

I am self-learning category theory (though, at this point, I no longer remember what got me started), and I have encountered a troubling definition on wikipedia. The formal definitions of (full) ...
2
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1answer
55 views

Does 'connected surface' in differential geometry actually mean 'path-connected surface'?

While studying differential geometry I often come across propositions with $M$ being a connected surface as their hypothesis. They then often take paths between arbitrary points, which to me suggests ...
2
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1answer
30 views

two definition in automorphism of group

Let $G$ be finite p-group and $\sigma \in Aut(G)$ (automorphism group). what does below symbols mean 1. $[G,\sigma]$ (commutator) 2. $C_G(\sigma)$ (centralizer)
2
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1answer
114 views

Alternate limit definition

I came up with an alternate definition of limit, which I would like to verify it is equivalent to the usual one. A sequence $(a_n)$ has a limit $L$ if, for any $a<L<b$, there are only ...
2
votes
2answers
78 views

Is “probability distribution function” a distribution?

I can understand the definition of distribution as written in https://en.wikipedia.org/wiki/Distribution_(mathematics) On the other hand there are three different terms in the definition of ...
2
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3answers
45 views

A more formal but intuitive understanding (on a definition) of a group action

We know that a symmetric group $S_n$ acts on the set $\{1, 2,\ldots, n\}$. The definition of an action of a group $G$ on a set $S$ is a function $G\times S\to S$ such that: 1) $e\ast s=s$ 2) $g'\ast(...
2
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1answer
50 views

Definition of differentiability at the point in multivariable calculus.

I'm self-studying the analysis from Zorich and the next definition of differentiability is given: $f:E\to \mathbb{R}^n$ is differentiable at the point $x$, which is a limit point of $E\subset \mathbb{...
2
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2answers
55 views

Mathematical formalism for the “dot product” of three vectors

I know that the dot product of two vectors is the sum of element-wise multiplication. Using pseudo-MATLAB notation: (x,y) = sum(x.*y). I'm interested in ...
2
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1answer
27 views

Softball here… what is the precise definition of a proper modification, in the context of complex geometry/manifolds?

It's a classic case of most papers using a concept and proving high-level mathematics about it, without ever stating the definition or some simple properties of the object. Which is just peachy for ...
2
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1answer
49 views

Math notation: What does $I$ mean in this context?

Sorry if this is a noobish question, but I don't know what $I$ means in this context: $$\hat{f}(X) = \sum_{m=1}^5 c_m I\{(X_1, X_2) \in R_m\}$$ I am reading about Decision Trees in The Elements of ...
2
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1answer
38 views

Can we state the triangle inequality as $|\int_D f(x) dx| \leq \int_D |f(x)| dx$

$|\int_D f(x) dx| \leq \int_D |f(x)| dx$ is just the infinitestimal version of the triangle inequality commonly presented in any book on vector spaces Can we replace the definition of triangle ...
2
votes
1answer
68 views

Epsilon Delta Limit Intuition

I am having a hard time grasping the definition of a limit. I initially learned this loose definition of a limit: $\lim\limits_{x \to a}f(x)=L$ iff $f(x)$ approaches L when $x$ approaches $a$ from ...
2
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1answer
331 views

Well-posed vs Well-conditioned

What's the difference between a well-posed (ill-posed) and well-conditioned (ill-conditioned) problem ?` Here is my finding up to now: "Even if a problem is well-posed, it may still be ill-...
2
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2answers
176 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 ...
2
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1answer
22 views

Is it okay to define k-th symmetric power of $M$ in this way?

I want to define the tensor algebra and related algebras in a very formal way. I will illustrate how I tried to define algebras below. Let $R$ be a commutative ring and $M$ be an $R$-module. ...
2
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2answers
185 views

What's a local angle?

When I was trying to understand the definition of conformal map I got confused. A conformal map is a function $f: U \to \mathbb C$ where $U \subset \mathbb C$ such that $f$ preserves local angles. ...
2
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1answer
84 views

What is an adjoint operator?

The following conjeture is stated here: Every adjoint operator has a non-trivial closed invariant subspace. Reference 11 where adjoint is supposedly defined can be found here. But I don't have ...
2
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2answers
83 views

What is the definition of closed subspace?

I am trying to understand what is intended with closed subspace, I took the following guess: A closed subspace $M$ of a Hilbert space $H$ is a subspace of $H$ s.t. any sequence $\{x_n\}$ of elements ...
2
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1answer
67 views

Is it possible to have an $a \times b \times c$ matrix?

The book Artificial Intelligence: A Modern Approach states that a certain variable is a $2 \times 2 \times 2$ matrix", but I thought that matrices could only be rectangular (i.e. $a \times b$). Is it ...
2
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2answers
615 views

Defining median for discrete distribution

In probability theory, a median of a probability distribution is a number $M$ such that the CDF of this distribution $F_\xi(x)$ satisfies $F_\xi(M)=\frac{1}{2} \tag1$ This works for continuous ...
2
votes
1answer
53 views

What is a linear equation?

How do we define the linear equation? I mean, it looks like a polynomials with degree one but I'm not sure if $ax+by+c=0$ is a linear equation if $a=b=0$?
2
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2answers
115 views

Is the angle between a vector and a line defined?

Is the angle between a vector and a line defined? The angle between two lines $a,b$ is defined as the smallest of the angles created. The angle between two vectors $\vec{a},\vec{b}$ is the smallest ...
2
votes
1answer
54 views

Family of “something very close to be a curve” over a curve $C$

Hartshorne (IMHO restrictive) definition of a curve: Definition of (complex) curve: A curve is an integral separated scheme of finite type over $\mathbb C$ of dimension $1$. (The definition of a ...
2
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2answers
64 views

Cauchy property of a series

Are these two definitions equivalent, even though the first one has an extra term: If we consider the series $\sum_{n=1}^{\infty}x_{n}$ and the formal definition of a Cauchy property defined in terms ...
2
votes
1answer
680 views

Exact definition of a circular helix

I know that a cylindrical helix is a curve $\alpha: I \to \Bbb R^3$, such that exists a constant vector $\bf u$ which makes a constant angle with the tangent vector $\bf T$ to the curve, at every ...
2
votes
1answer
34 views

Concise description of Lebesgue measure in $\mathbb{R}^{n}$

I would like to confirm that the following is an acceptable description of the Lebesgue measure in $\mathbb{R}^{n}$. The outer Lebesgue measure $E \subset \mathbb{R}^{n}$: $$\lambda^{*}(E) = \inf\...
2
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1answer
95 views

A better definition of Polynomial

Usually, we define a polynomial as $a_n x^n + \cdots + a_1 x + a_0$ where $x$ is called indeterminate. Would it be better to define it as $a_n x^n + \cdots + a_1 x + a_0 x^0$ where $x^0$ means ...
2
votes
1answer
2k views

What's the difference between a cyclic and periodic function?

I've seen the words "cyclic" and "periodic" used to describe characteristics of a given function. What do they mean? I can't seem to find a difference. Wikipedia says a periodic function is one that ...
2
votes
1answer
125 views

Definition of disc and open ball

I have the following definitions in my notes for arbitrary discs and open balls - $$D^n = \{x \in \mathbb{R^{n+1}}: ||x|| \le 1\}$$ $$B^n = \{x \in \mathbb{R^{n+1}}: ||x|| < 1\}$$ The $\mathbb{R^...
2
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1answer
69 views

formalize definition of topology

In my studies I used this definition of topology, but I am reading on wikipedia a different definition... I thought to formalize: Def. let be $A$ a set and $B \in \mathcal{P}(\mathcal{P}(A))$, $(A,B)...
2
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2answers
72 views

A problem with the domain of function in the defintion of limits

My Stewart's Calculus gives the following definition of limit: $f(x)$ is defined on some open interval containing $a$, except at possibly $a$. So, $\lim_{x\to a} f(x) = L $ if and only if for ...
2
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1answer
67 views

“internal” definition of complete regularity?

There is something strange (I think) about the complete regularity separation axiom. Consider the definitions. T0 means for every two distinct points there is an open set containing exactly one of ...
2
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1answer
583 views

Clear definition of degeneracy of a graph.

There are at least two questions on this topic but the answers are not clear to me and WiKi link didn't make it any clear either. Could someone please clarify is the degeneracy of a graph $G$ ...
2
votes
1answer
109 views

Definition of an $n$-tuple agreeing with the Kuratowski's definition of an ordered pair

Is there a nice and elegant definition of an $n$-tuple ($n$ is a nonnegative integer) in ZFC, which will at the same time agree with the Kuratowski's definition of an ordered pair, i.e. $\left ( a,b \...
2
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2answers
44 views

Does the definition of “mean”/“average” require the result to be in the domain set?

If I have a function that calculates the mean value of a set of elements that is an arbitrary subset of some set $X$, does the mean, by definition, have to also be in $X$? (In other words, if the mean ...
2
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3answers
65 views

I'm confused about the definition of poset.

The definition of poset : $\qquad$A set with a partial ordering. Partial ordering is a binary relation $\preceq$ over a set ($P$). If I understood the definition of relation correctly, then ${\preceq}...
2
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1answer
85 views

Why do people not use “partially directed” graphs?

Are there structures in use which are a mix of directed and undirected graphs? I.e. the effective edge-set consists of both directed and undirected vertex pairs. In the case that the graph is simple, ...
2
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1answer
50 views

Why do we want to define a $k$-scheme to be birational if the rational map (and its inverse) to $\Bbb A_k^n$ is over $k$?

Two varieties $X,Y$ are said to be birational if there exist rational maps in each direction such that either composition is the identity on a open dense subset. Note that here the morphisms aren't ...
2
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1answer
226 views

Subspace with different vector space operations

Let $A,B$ be vector spaces such that $A\subseteq B$. Is it true that $A$ is a subspace of $B$? I claim that the answer is no, because it is possible that $A$ and $B$ might be equipped with different ...
2
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1answer
45 views

What is a cofactor in this case?

Im looking at a homework problem I have and I am a bit confused. The first part of the question is to show that 8128 is a perfect number. This is simple enough: $1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + ...
2
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1answer
160 views

Definition of pull-back analogous to push-forward

Let $U\subseteq\mathbb{R}^n, V\subseteq\mathbb{R}^m$ be open subsets, and $f:U\rightarrow V$ a $C^1$ map. Let $u,v$ be vector fields on $U,V$ respectively. Then the push-forward $f_*u$ is equal to $v$ ...
2
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1answer
46 views

definition clarification in graph theory

I was studying about Almost Self-Centered Graphs (ASC). ASC graphs are introduced as the graphs with exactly two non-central vertices. Of course, the remaining two vertices are diametrical. My doubt ...