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

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1answer
48 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
221 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
44 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
155 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 ...
2
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1answer
516 views

What sets are Lebesgue measurable?

I cannot detect the fallacy in the set of the following statements in my inconsistent notes: A sigma algebra is a set of the sets in the generating set closed under the set operations countable ...
2
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2answers
103 views

definition of rectangle

I am wondering whether it is fine to define a rectangle with right angles like Wikipedia's page of rectangle. In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. ...
2
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1answer
66 views

What is the function that is not a binary function called?

A binary operation is a calculation involving two elements of the set and returning another element of the set. Suppose it doesn't return an element of the set. What is the function called? For ...
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1answer
86 views

Group of finite ideles

A simple question: If $\overline{\mathbb{Q}}$ denotes the algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$ then what is the definition of the group of finite ideles of $\overline{\mathbb{Q}}$? ...
2
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1answer
63 views

Is Wikipedia's definition of $\omega$-inconsistency problematic in this way?

I could be wrong, but the definition of $\omega$-inconsistency given at over Wikipedia seems slightly problematic. In particular, Wikipedia claims that $\omega$-inconsistency is a property of a theory ...
2
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2answers
180 views

Apostol question on alternative definition of dot product

The problem says: Suppose we define the dot product by $A\cdot B = \sum_{k=1}^n |a_kb_k|$. Which of the following properties hold with this new defition? Does the Cauchy-Schwarz inequality still ...
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2answers
222 views

A definition of algebraic expression

Definition of algebraic expression An algebraic expression is a collection of symbols; it may consist of one or more than one terms separated by either a $+$ or $-$ sign. If by symbol we only mean ...
2
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1answer
63 views

If for every $x_n$ such that $x_n \rightarrow x$, there exists a $x_{n_k}$ such that $Tx_{n_k} = Tx$, is $T$ continuous?

Let $X$ and $Y$ be Banach spaces and $T$ be the (possibly nonlinear) map $T\colon X \rightarrow Y$. $T$ is continuous if for every $x_n \in X$ such that $x_n \rightarrow x$, then $Tx_n \rightarrow ...
2
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1answer
215 views

On the definition of divisors in Riemann Surfaces

The sum notation for a Divisor $D$ in a Riemann Surface $X$ (as in Miranda's "Algebraic Curves and Riemann Surfaces") is $$ D=\sum_{p\in X} D(p)\cdot p $$ That is, $D$ assumes the value $D(p)$ at $p$. ...
2
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2answers
157 views

In a groupoid, do any two objects have a morphism between them?

This is a question about the definition of a groupoid (in category theory). I've read the Wikipedia article, and it says that a groupoid is a small category in which every morphism is an isomorphism. ...
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3answers
1k views

Positive constant scalar definition

In French when we say "$k$ est une constante positive", that means $k\geq 0$. But I remarked that using the same sentence in English, "$k$ is a positive constant", means that $k>0$. Can one explain ...
2
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1answer
242 views

Limit point definition

I have read the definition of a limit point of a set in Real Analysis. The definition goes like: A number $p$ is said to be a limit point of a set of reals, $S$, if every neigbourhood of $p$ has at ...
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2answers
200 views

What is the meaning of the expression $\liminf f_n$?

I am a little confused as to what $\liminf f_n$ means for a sequence $f_n$ of functions converging to $f$. I can not locate a definition anywhere.
2
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1answer
87 views

A question on linear ordered space

A space $X$ is called left-separated if it can be well-ordered in such a way that every initial segment is closed in $X$. And we know every space contains a dense left-separated subspace. My question ...
2
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1answer
74 views

Using the notation $m^{O(1)}$

$m^{O(1)}$ denotes the set of functions $\mathbb{N} \to \mathbb{N}$ which are polynomially bounded. (Is that what is usually means?) Now it used as follows: $$ f(m) \leq m^{O(1)}$$ To express that ...
2
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1answer
425 views

Meaning of no explicit time dependence

What does "no explicit time dependence" mean in this context? : A symmetry of the KdV is given by $$\tilde x=x, \tilde t=t+\epsilon, \tilde u =u$$ as there is no explicit time dependence in the ...
2
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1answer
82 views

What is the name of the group linear functions on a finite field?

More precisely what is the name for the group $$\{ X\mapsto \alpha^2X+\beta : \alpha,\beta \in GF(q), \alpha \neq 0\}$$ I've always called it the special affine group, but I see that can mean ...
2
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1answer
151 views

Is the “binary operation” in the definition of a group always deterministic?

The introduction to group theory that I'm reading requires that the actions of a group are "deterministic"; but the formal definition given makes no mention of this property: A set G is a group if ...
2
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4answers
127 views

What are possible variations of the definition of $\sigma$-additivity?

From what I've read in Wikipedia, $\sigma$-additivity is defined in the following way: Let $\mathcal{A}$ be a $\sigma$-algebra of some underlying set $X$ and let $f$ be a mapping ...
2
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1answer
6k views

What does it mean for a functional equation to have a unique solution?

It is my thinking that unique conventionally means special or one of its kind. But in the context of solving functional equations*, I am confused what it means to have a unique solution... *e.g. Find ...
2
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1answer
452 views

Partial derivative notation: is that a projection function?

Consider the following definition: Let $(U,\phi)$ be a chart and $f$ a $C^\infty$ function on a manifold $M$ of dimension $n$. As a function into $\mathbb{R}^n$, $\phi$ has $n$ components ...
2
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1answer
35 views

Mathematical definition of the word “generic” as in “generic” singularity or “generic” map?

I've been trying to work out what generic means but I'm not making much progress. You can find an example of the usage of the word generic for example here: "School on Generic Singularities in ...
2
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1answer
43 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 ...
2
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1answer
22 views

Understanding the definition of the direct sum of subspaces of a vector space

I have a question regarding the definition of direct sum of a vector space in relation to subspaces. Definition: A vector space $V$ is called the direct sum of $W_1$ and $W_2$ if $W_1$ and $W_2$ ...
2
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1answer
40 views

What is the precise relationship between vector space and vecctor field?

I have looked up precise definition of a vector field and a vector space but I could not understand the relationship between them. On wikipedia A vector field is: Given a subset $S$ in $R^n$, a ...
2
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1answer
99 views

Lost in terminology: What is the meaning of the words “Constraint” and “Parameter” in a goodness of fit?

This is somewhat related to this previous question of mine. I need a clear distinction and/or definition of the words 'parameter' and 'constraint' in the following context which is the the only source ...
2
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1answer
100 views

Weak Gâteaux Derivative

Suppose $X$ and $Y$ are Banach spaces. Let $F:X \rightarrow Y$ be a function and $U \subset X$ be an open set. The Gâteaux derivative of $F$ at $u \in U$ in the direction $\phi \in X$ is given by ...
2
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2answers
39 views

Different definitions for the complex inner product.

I have asked this question on P.S.E. and have gotten some nice answers, but I felt I might get even more satisfactory answers if I post it here. "I have just now noticed that Griffiths (in his book ...
2
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1answer
32 views

Understanding Homotopy Definition

I'm having trouble with part of the definition of homotopy, but I may just have a misunderstanding about continuous maps from product spaces. This is the definition my book uses: Two continuous ...
2
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3answers
57 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 ...
2
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2answers
41 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 ...
2
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2answers
33 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
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1answer
44 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 ...
2
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1answer
90 views

On the terms “nowhere dense” and “dense-in-itself”

Are there literal interpretations of the terms "nowhere dense" and "dense-in-itself" from which these terms' definitions follow? If I were to guess what it means for a subset $A$ of a topological ...
2
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1answer
56 views

What is Convolution?

The definition of convolution is known as the integral of the product of two functions $$(f*g)(t)\int_{-\infty}^{\infty} f(t -\tau)g(\tau)\,\mathrm d\tau$$ But what does the product of the functions ...
2
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1answer
48 views

Orthogonality v. Perpendicularity

In Intro to Linear Algebra (my class) two vectors are defined to be orthogonal if their dot product is zero. And the dot product of two $n$-vectors $\vec a\cdot\vec b=0$ means that the two vectors are ...
2
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1answer
64 views

Confusion regarding the $\omega$-limit of a set in a flow

In Salamon's Connected Simple Systems, p.8, the author writes that the $\omega$-limit of a set $Y$ inside a flow $\Gamma$ has the two equivalent descriptions $$ \omega(Y) = I(\overline{Y \cdot ...
2
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1answer
35 views

Pulling back a connection to a curve

What does it mean to pull a connection back to a curve? For example, if I take the connection $\nabla s = ds$ on the trivial bundle $\mathbb R^2 \times \mathbb R^2$ over $\mathbb R^2$, and the curve ...
2
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1answer
89 views

When do you call something “a calculus” vs. “a logic”?

Similar to What do Algebra and Calculus mean?, what is the difference between a logic and a calculus? I am learning about the different kinds of logics, and often when I look them up in a different ...
2
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1answer
18 views

Simplicial homology: chain group with basis open n-simplices vs. chain group with basis closed n-simplices

In his Algebraic Topology book, Hatcher defines the chain groups for simplicial homology as free abelian groups with basis the open $n$-simplices of some simplicial complex X. Is there any ...
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1answer
51 views

Uniqueness of a number $area(A)$

I have the following definition of area: Let $A$ be a bounded set from $\mathbb{R}^2$. We say that $A$ has area if there exist two sequences $(E_n)_{n\in \mathbb{N}}, (F_n)_{n\in \mathbb{N}}$ of ...
2
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1answer
72 views

Meaning and Underlying idea of a definition or a theorem

What does it mean by 'explain the meaning and underlying idea of a definition or a theorem'? For example, if we are asked to explain the Fundamental Theorem of Algebra, how should we explain its ...
2
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1answer
36 views

Rouches theorem, $|f(z)|\gt |g(z)|$ at each point on $C$. $\color{red}{\text{On or In?}}$

Rouche's theorem from both of my resources say the following: Let $C$ denote a simple closed contour and suppose that: Two functions $f(z)$ and $g(z)$ are analytic inside and on $C$ $|f(z)|\gt ...
2
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1answer
36 views

Definition of “vector fields never have opposite direction”

Good day! As in my other question I am referring to the book "Differential Equations and Dynamical Systems" by Lawrence Perko, chapter 3.12. I have a question regarding Lemma 2: Lemma 2. If $v$ ...
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1answer
40 views

X - Y in a finite set

Question: The domain is the power set of a finite set. $X$ is related to $Y$ if $X - Y$ is not empty. Is it a partial or strict order? If it is a partial order or strict order, is it also a total ...