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

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1answer
472 views

Elliptic Operators

I'm studying Elliptics Operators like this: $$Lu=a_{ij}(x)D_{ij}u+b_i(x)D_iu+c(x)u$$ for $u\in C^2(\Omega)\cap C(\overline{\Omega})$. I want to know what the difference when: $L$ is elliptic in $\...
2
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4answers
119 views

Definition of open set/metric space

On Proof Wiki, the definition of an open set is stated as Let $(M,d)$ be a metric space and let $U\subset M$, then $U$ is open iff for all $y\in U$, there exists $\epsilon \in \mathbb{R}_{>0}$ ...
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2answers
222 views

Definition of metastability

I was reading Terence Tao's blog post on analogies between soft and hard analysis when I saw that the soft analysis statement "$x_n$ is convergent" corresponds to the hard analysis statement "$x_n$ is ...
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2answers
73 views

Name of the order with irreflexivity, antisymmetry and transitivity?

I have an order otherwise poset aka partial order but it is irreflexive so relationships such as 1R1 and 2R2 are impossible. What is the name of this order?
2
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1answer
143 views

Property of Division by vector for a field

Serge Lang in "Linear Algebra" on page 2 says that The essential thing about a field is that it is a set of elements which can be added and multiplied, in such a way that additon and ...
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1answer
767 views

Meanings of the terms “conjunct” and “disjunct” in a logic?

This sentential logic problem is stated as: Suppose that $A \models B$, where $A$ is a conjunction of literals and $B$ is a disjunction of literals. Show that $ \models \neg A$, $ \models B$, or a ...
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1answer
104 views

Affine algebra of an algebraic group

From what I understand there are two approaches to defining an algebraic group. One can start talking about varieties and the Zariski topology and such and get to a definition of an algebraic group. ...
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1answer
1k views

Definition of the complement of a set

My book defines the complement of a set as, "Let $U$ be the universal set. The complement of the set $A$, denoted by $\bar{A}$, is the complement of $A$ with respect to $U$. Therefore, the complement ...
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1answer
146 views

Can we apply a successor operation in Peano's arithmetic infinitely many times, is the successor well defined?

If we look at the axioms of Peano arithmetic, e.g. http://mathworld.wolfram.com/PeanosAxioms.html, they contain an axiom: If $a$ is a number, the successor of $a$ is a number. However, the axioms do ...
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1answer
117 views

Definition of a deterministic Pushdown automaton

According to my book the definition of a deterministic Pushdown automaton allows for $\delta(q,\epsilon,Z)$ to be non-empty if $$\forall\sigma\in\Sigma:\,\delta(q,\sigma,Z)\neq\emptyset$$ Can someone ...
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202 views

Question about inverse limit

I'm puzzled by the definition of inverse limit in this Wolfram article. I thought if an object was defined by a universal property it meant that the object is unique up to unique isomorphism. This ...
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1answer
350 views

Free boolean algebra

Consider the following definition: Let $X$ be a set and $e : X \mapsto A$ a mapping to a boolean algebra $A.$ We say that $A$ is free over $X$ (with respect to $e$) if for every mapping $f:X ...
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1answer
24 views

What is actually the standard definition for Radon measure?

I see that there are various definitions for Radon measure and they are NOT equivalent, but they are equivalent on locally compact Hausdorff spaces. I think this is the reason why Radon measure has ...
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3answers
53 views

Why other definitions of convergence fail to be correct?

The following is an exercise from the book Advanced Calculus: Well obviously the b. is not correct since for $\epsilon= \frac19$, ${\{a_n}\}={\{\frac1n}\}$ and $N=2$ it fails to be correct. But the ...
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1answer
22 views

Sublinear functions on a Riemannian manifold

I would like to know if there is any notion of sublinear function or subadditive function for Riemannian manifolds. Thank you!
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2answers
40 views

Some doubts on the definition of minimality of a graph. EDITED

If a graph $G$ is minimal with property $P$, does that specifically mean: Any proper subgraph of $G$ does not have that property $P$? Or, Any graph with less number of vertices or less number of ...
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2answers
36 views

Does “sphere” denote the surface or the entirety of a solid ball?

In everyday English, the word "sphere" denotes a 3-dimensional object, including the points inside the surface and its center. However, I get the sense that in mathematics, the sphere is used ...
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1answer
48 views

If $\lim_{x\to a} g(x) = M$, show that there exists a number $\delta > 0$ such that $0 < |x - a| < \delta \Rightarrow |g(x)| < 1 + |M|.$

The hint given was to take $\epsilon = 1$ for the formal definition of a limit. Doing this means that $|g(x) - M| < 1.$ Using the triangle inequality, you get $$|g(x)| = |(g(x) - M) + M| \le |g(x) -...
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1answer
24 views

Definition of measure-preserving: why inverse image?

In the definition of measure-preserving dynamical system, the crucial equation is $$ \mu \left(T^{-1} \left(A\right)\right) = \mu\left(A\right) . $$ Why is it not the seemingly more natural $$ \...
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 \...
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2answers
40 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 $x$ ...
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2answers
50 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
45 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=...
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2answers
39 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{...
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1answer
56 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°....
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1answer
30 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 $...
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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 ...
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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
57 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)
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1answer
115 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 ...
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2answers
79 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 ...
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3answers
47 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
28 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
51 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 ...
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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
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1answer
69 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
342 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-...
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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. ...
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2answers
187 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. ...
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1answer
85 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 ...
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2answers
84 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 ...
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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 ...
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2answers
640 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 ...
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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 $a=b=0$?
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2answers
117 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 ...
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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 ...