Questions tagged [definition]

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

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18 views

What does "stabilize" mean in Conway's game of life?

In wikipedia's article about Conway's game of Life, it often talks about a pattern eventually stabilizing, there's even a page about a type of seed called Methuselah which is "defined" as a ...
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31 views

How can I improve my definition?

I've been trying to write a formal definition for a $k$-involutible function in that the function has to satisfy the following properties: $k$ is a positive integer. $f \in \mathbb{R}(x)$ (as in, $f$ ...
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40 views

What is $\mathbb{E}^d$?

In a paper (chapter 3, the paper is in Italian) I'm reading I found: A Bézier curve of degree $n$ is a parametric polynomial curve $X:[0;1]\to\mathbb{E}^d$ defined as follows: I'm not an expert in ...
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38 views

If $\langle x \rangle =_{df} H(\{x\})$ with $H(S) , (S $ a set $)$ as defined below , how can $\langle x \rangle$ have more than $3$ elements?

Def. of $H(S) :$ $H(S) = \{g\in G \mid \exists n\in \mathbb N , \exists \{g_1, g_2,... g_n\}\subseteq S\cup S^{-1} , g = g_1 ...g_n\}$. Note : it can be shown that $H(S)$ is simply the same set as $\...
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Proof of existence of tensor product of modules.

I’m reading a construction/proof of existence of the tensor product $T$ of $A$-modules $M$ and $N$, and I’d like to ask some clarifying questions. Here’s the construction. Let $C$ denote the free $A$-...
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1answer
29 views

Why do we restrict unbounded linear operator to Banach spaces?

Let $(E, | \cdot|_E)$ and $(F, | \cdot|_F)$ be Banach spaces. An unbounded linear operator $A$ from $E$ to $F$ is a linear map of the form $A: D(A) \to F$ where $D(A)$ is a subspace of $E$. Let $A$ be ...
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2answers
50 views

What is " the equivalence relation associated to $f$" , $f$ being a surjective mapping from a group $G$ and a set $E$.

Context : Equivalence relations and quotient groups. Source : Reversat & Bigonnet, Algèbre pour la licence ( Undergraduate abstract algebra), 1997, p. $31$ . The notion refered to in the title ...
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1answer
45 views

Multiset definition of "all" [closed]

The multiset $\{n,n,n\}$ is a multiset containing elements that are all $n$. It does not contain non-$n$ elements. The empty multiset $\{\}$ also does not contain non-$n$ elements. Is it also a ...
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1answer
36 views

Understanding the definition of path connected subspace

If $X $ is a topological space and $ Y $ is a subspace of $ X$, then what does it mean when we say $Y$ is a path connected subspace of $X$? Does it mean that any two points $ x $ and $ y $ in $ Y $ ...
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2answers
38 views

Meaning of convergence for a double infinite serie. How do we sum two index "at the same time"

In this document is expressed the Fubini theorem for series. It is said that if $a_{jk}$ is a doubly indexed finite sequence that verifies: $$\sum_{j,k} |a_{jk}|<\infty$$ Then, $\sum_{j,k} a_{jk}$ ...
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75 views

Formal definition of union of arbitrary sets

Usually one sees something like this: Finite case Let $n \in \mathbf{N}$ and $A_1,...,A_n$ be sets. Then $\bigcup_{i=1}^n A_i$ exists and is defined to be the object (set) satisfying $$\bigcup_{i=1}^n ...
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1answer
28 views

The boundary of a $singular\; 0-simplex$

In Singular homology, Given a $singular\; n-simplex$, $\varphi$, we define the $singular\; (n-1)-simplex$, $\partial_i{\varphi}$, $$\partial_i{\varphi}(x_0,x_1,\dots,x_{n-1})=\varphi(x_0,x_1,\dots,x_{...
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40 views

Absolute value function definition

The standard definition of is $f(x)=\begin{cases}x,& x\geq 0\\-x,&x<0\end{cases}$. I am wondering what the problem is with the definition $f(x)=\begin{cases}x,& x,\geq 0\\-x,&x\leq0....
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5answers
215 views

(Disagreement among reputable users) Indefinite integral vs. Definite integral vs. Anti-derivative

Suppose, I have a function $\cos(x)$. Now, $$\int{\cos(x)dx}$$ $$\sin(x)+c\\ {\text{[c is a constant]}}$$ Now, there could be an infinite number of values for $c$. For example, $c=1,2,-2,\pi,-\pi, 0, \...
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82 views

Are {$a {\cdot} b\ \ |\ \ a \in A, b \in B$} and {$a {\cdot} b\ \ |\ \ (a, b) \in A \times B$} the same thing? [closed]

Given two sets $A$ and $B$ are the following two sets equivalent? {$a {\cdot} b\ \ |\ \ a \in A, b \in B$}, and {$a {\cdot} b\ \ |\ \ (a, b) \in A \times B$}
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1answer
28 views

Characterization of continuity with open subsets.

Recently I learned about the following characterization for continuity: A function $f: X \to Y$ with $X,Y \subseteq \mathbb{R}$ is continuous if for every open subset $U \subseteq Y$ the inverse image ...
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1answer
67 views

Standard Tic Tac Toe is -- Impartial or Partisan?

I am currently studying basic game theory (combinatorial) and was introduced to impartial games. The "definition" of impartial games I saw was: "An impartial game is a two-player game ...
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1answer
62 views

Is this modification of connexity necessary, or redundant in the definition of partial ordering?

My question pertains to Fundamentals of Mathematics, Volume 1 Foundations of Mathematics: The Real Number System and Algebra, Edited by H. Behnke, F. Bachmann, K. Fladt, W. Süss and H. Kunle Vol-1 A8....
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1answer
27 views

Neighborhood base generated by uniformity

In Topology for analysis by A. Wilansky the theorem 11.1.2 states: Let $(X,\mathcal{U})$ be a uniform space, let $\mathcal{B}$ be [uniformity] base for $\mathcal{U}$ and let $x \in X$. Then $\{U(x) :...
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1answer
70 views

Understanding the definition of Sylow $p$-subgroups

Here is the definition of Sylow $p$-group (source: wikipedia) For a prime number $p$, a Sylow $p$-subgroup of a group $G$ is a maximal $p$-subgroup of $G$, i.e. a subgroup of $G$ that is a $p$-group (...
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54 views

When $p\in X\times Y$, is there a rule that allows us to infer $p=(p_x,p_y)$?

For $p\in X \times Y$, is there a inference rule that allows us to say that $p=(p_x,p_y)$ for some $p_x\in X, p_y\in Y$? For context, I am reading Pinter's "A Book of Set Theory" and couldn'...
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1answer
51 views

Are transition maps diffeomorphisms?

For two charts $(U, \phi)$ and $(V,\psi)$ on a topological manifold that are $C^\infty$-compatible, the transition maps $$\phi \circ \psi^{-1}:\psi(U \cap V)\rightarrow \phi(U \cap V)$$ $$\psi\circ\...
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1answer
125 views

Exotic Definitions of Groups

Inspired by this question I was wondering, whether there are alternative definitions of groups, namely ones different from the usual 4 axioms. I already suspected that the category theorists have one ...
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1answer
36 views

definition of $f(n)$ diverging to infinity for $n\to\infty$ [closed]

Can someone maybe help me, and tell me if the following is a valid defintion for divergence to infinty? Thanks in advance $f(x)$ diverges to $\infty$ for $x \to \infty$, if for each $\epsilon > 0$...
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54 views

Intuition for a uniform space?

Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and uniform convergence. According to Wikipedia, ...
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69 views

About a definition for an inner product in linear algebra books.

I am reading a linear algebra book. The typical definition of an inner product is the following: Let $F=\mathbb{R}$ or $\mathbb{C}$. Let $V$ be a vector space over $F$. An inner product on $V$ is a ...
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1answer
64 views

It seems that the limit function is "cherry-picking" as values approach $0$

A rule that I've been taught over and over is that dividing by zero gives an undefined value. Another fact is that multiplying by zero, equals zero. Also, zero is nothing. Now, when I'm learning ...
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2answers
55 views

Is there other function $f$ such that $f(x, y) + f(y, z) \geq 2f(x,z)$? What is it name?

Let function $f: \mathbb R^d\times \mathbb R^d \rightarrow \mathbb R$ satisfy the following conditions: For all $x, y, z\in \mathbb R^d$, Non-negativity: $f(x, y)\geq 0$ Identity: $f(x, x)=0$ "...
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0answers
52 views

when two curves are **transverse**?

in the "A Primer on Mapping Class Groups by Benson Farb and Dan Margalit" we define intersection number as follow : Let $\alpha$ and $\beta$ be a pair of transverse, oriented, simple closed ...
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27 views

Definition $f(x)$ diverges to negative infinity for $x \to a$

Can someone tell me, if the following is a valid definition of $f(x)$ diverging to $-\infty$ for $x\to a$ $f(x)$ diverges to $-\infty$ for $x\to a$, if and only if for all $\epsilon\in\mathbb{R}$, an ...
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1answer
63 views

What else is a neighbourhood?

I need to settle an argument. Yes or no: If $(X,\tau)$ is a topological space ($X$ is an arbitrary - and I mean arbitrary - set, $\tau\subseteq\mathcal P(X)$ is the collection of open sets), then a ...
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1answer
45 views

How do you read the definitions of range and domain of a relation aloud?

In the book I'm reading, I'm given two definition that pertain to relations. Suppose R is a relation from A to B. Then the domain of R is the set $$ Dom(R) = \{a \in A \vert \exists b \in B((a,b) \...
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17 views

Formal definition of being in the top n elements

Let $n$ be a positive integer, let $P$ be a partial order, and let $x$ be an element of $P$. How does one formally define the relation "$x$ is in the top $n$ elements of $P$"? A prerequisite ...
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5answers
355 views

Extending $f: (0,1]\mapsto\mathbb{R}$ to a continuous function from $[0,1]$ to $\mathbb R$

Theorem Consider the continuous function $f: (0,1]\mapsto\mathbb{R}$ defined by $f(x)=\sin(\frac{1}{x}).$ I have to answer the following question : show that it is impossible to extend this function ...
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0answers
35 views

What is the topology on continuous groups?

I was look at this page where it says that A group having continuous group operations. A continuous group is necessarily infinite, since an infinite group just has to contain an infinite number of ...
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1answer
66 views

How to turn elements of a ring $A$ into functions on $\text{Spec}A$?

Let $A$ be a commutative ring with $1$, and $a \in A$. In our class, we’ve just introduced a construction that aims to turn $a$ into a function on $\text{Spec}A$. There are some points I’m not clear ...
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1answer
62 views

Difference between $\bigcap F$ and $F \cap$...

I don't understand what something like $\bigcap F$ means. Here's an example of where it's used. Theorem. Suppose $F$ and $G$ are families of sets, and $F \cap G \neq \varnothing$. Then $\bigcap F \...
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1answer
58 views

Definition of Unitary Operators: Why do we need surjectivity or boundedness?

Consider some Hilbert space $(\mathcal{H},\Vert\cdot\Vert)$. Now, there are several equivalent definitions of unitary operator, i.e. see for example on wikipedia. All of them assume either that $U$ ...
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49 views

Can the median and the interquartile range ($IQR=Q_3-Q_1$) be defined if the data consists of complex numbers?

If we have a collection of data consisting of complex numbers $a_1,\dots, a_n$. Technically the average can still be defined using $$\frac{1}{n}\sum_{i=1}^n a_i.$$ How about the median and the ...
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4answers
206 views

Formal definition of $n$ by $0$ and $0$ by $n$ matrices

A matrix is usually informally defined as a rectangular array of numbers. To make this definition formal, we can define a matrix as a map from $\{1,...,m\} \times \{1,...,n\}$ to the underlying field ...
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2answers
54 views

Derivatives and integral of distribution valued functions

Given a $C^1$ distribution valued function $f: \mathbb{R}^l \to \mathcal{D}'(\mathbb{R}^n)$ how does one defined its, say, j-th partial derivative? How does one define its integral over a compact set $...
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Equivalent definition of term

This problem deals with sequences. I have found many definitions for cluster point and there equivalences on MSE and beyond. Here is the one my text uses: Let X be a topological space and A $\subset$ ...
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2answers
120 views

Why do you calculate the dot product that way?

I am learning dot product these days. I understand the geometric meaning of one vector's interpretation in the same direction of the other to calculate the work in terms of force and distance in the ...
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1answer
71 views

What is a "positive statement" in mathematical proof?

I'm going through How to Prove It: A Structured Approach by Daniel J. Velleman and some terms that I frequently see are "positive statement" and "negative statement". I'm not sure ...
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1answer
36 views

Convertibility of Two Lambda Expressions Equivalent to Existence of a Common Reduct

Suppose $\rightarrow$ is $\rm{\beta}$ reduction and $\twoheadrightarrow$ denotes a reduction sequence from $\rm{\beta}$ reductions. Convertibility of two lambda expressions is defined as follows: two ...
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201 views

Is there a name for this property of functions on groups?

Let $G$ be a group and $F:G^n \to G$ with the following property: If $x_1,…,x_n,h \in G$, then $F(hx_1,…,hx_n)=hF(x_1,…,x_n)$. Is there a name for this type of function property? It is something I’ve ...
4
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2answers
115 views

Putting "$\forall y(y \in x \to \exists A \in F(y \in A))$" into words

I'm new to mathematical proof and I struggle sometimes with putting definitions into words. If I had one like this: $$\forall y(y \in x \to \exists A \in F(y \in A))$$ Would it be correct to read this ...
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1answer
34 views

Discrete random variables definition

A random variable $X : \Omega \to X(\Omega)$ is said to be discrete, when there is a finite or countable set of values $Y \subseteq X(\Omega)$ such that $P(X \in Y ) = 1$. The function $p : Y \to [0, ...
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0answers
50 views

Let $E$ be a Banach space, $x \in E$, and $f \in E^\star$. What does $\langle f, x\rangle$ mean?

I'm reading Part 3.2 of textbook Functional Analysis, Sobolev Spaces and Partial Differential Equations: Let $E$ be a Banach space and let $f \in E^{\star}$. We denote by $\varphi_{f}: E \rightarrow \...
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59 views

Is it true that $\left(-\frac{1}{64}\right)^{-\frac 43}=256$? [duplicate]

I have included this picture in its original form from my textbook here. I think this is wrong because it contradicts the definition of $a^x$. Because we define $$a^x=e^{x\ln a}$$ where $a>0,x\in\...

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