Questions tagged [definition]

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

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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...
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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 ...
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\...