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7

This is a falling factorial: $$ (n)_k = n^{\underline k} = \underbrace{n\cdot(n-1)\cdot(n-2)\cdots(n-k+1)}_{k\text{ factors}} = \frac{n!}{(n-k)!} $$


6

Because the quotient of $V \times W$ by $W$ is $V$. (more precisely, the quotient of $V \times W$ by the subspace $\{ 0 \} \times W$ is naturally isomorphic to $V$) There are more general situations, but that it covers this particular situation is, in my opinion, a pretty strong motivation. The idea is even clearer in the case of abelian groups; finite ...


5

It's called the canonical map into the double dual. It does not have another name that I know of. In a monoidal category with duals (I'm being vague), the uncurried version of it, $\operatorname{ev} : X \otimes X^* \to I$, is called evaluation.


3

You could say that a digraph has this property if and only if all connected components are strongly connected. I'm not sure if there is one word for it.


3

French prepositions: in and sur (meaning on).


3

Such a morphism $f : a \to b$ is said to be $F$-(hyper)cocartesian. This is in connection to Grothendieck opfibrations. You may like to work out what this means concretely in the case of $\mathrm{dom} : [\mathbf{2}, \mathcal{C}] \to \mathcal{C}$ where $\mathcal{C}$ is a category with pushouts.


2

The most common way to express this is that component $c_1$ is not simply connected (it has a hole). Of course, this doesn't help to give a term for $c_2$, but in analogy with political maps, we might call it an enclave.


2

In linear algebra the double-dual comes to mind, but the general term that's most relevant is "evaluation map". One somewhat common notation is $\mathrm{ev}_{x}(f)=f(x)$. Either ev (your ~) is the evaluation map, or each $\mathrm{ev}_{x}$ is the evaluation map at $x$.


2

In the context of functional analysis, the map $$ \Phi: A \to (\Bbb F)^{\Bbb F^A}\\ \Phi:x \mapsto (f \mapsto f(x)) $$ is called the "evaluation map".


2

$\newcommand{\Basis}{\mathbf{e}}\newcommand{\Brak}[1]{\langle #1\rangle}$Let $A$ be an $n \times n$ orthogonal matrix, and put $\Basis = (1, 1, \dots, 1)$. The sum of the entries of $A$ is the inner product $$ \Brak{\Basis, A\Basis} = n\cos\theta, $$ with $\theta$ the angle between $\Basis$ and $A\Basis$. Since there exists an orthogonal matrix fixing $\...


1

I'm no mathematician, and there may be more concrete definitions, but this is how I think of a function as exponential "growth" and "decay." Theory If an exponential function is "skyrocketing" (for lack of better terminology) and heads towards $\pm\infty$, then it's "growing" (you can think of it as "absolute" growth, and disregard the sign). If an ...


1

Thanks to Zhen Lin for his comment, here is what I believe is the correct interpretation of the words and phrases having been unclear to me. i) The pullback of the morphism $f_A:A \to C$ along the morphism $f_B:B \to C$ is the morphism $p_B:P \to B$ where $\langle P, p_A, p_B \rangle$ is the pullback, i.e. $f_A \circ p_A = f_B \circ p_B$ where $P$ is ...


1

Looking at $x \in X$ as a map $x \colon 1 \to X$, the operator $\tilde x$ is precisely the natural transformation given by co-Yoneda's embedding $$ \mathsf{Set}(X,-) \stackrel{\mathsf{Set}(x,-)} \longrightarrow \mathsf{Set}(1,-) \simeq \mathrm{id}_{\mathsf{Set}} $$ So you could call it the restriction along $x$, or precomposition by $x$, and denote it $x^\...


1

Two possible motivations: For finite-dimensional vector spaces over finite fields, $$ |V/W|=|V|/|W|\;. $$ And for $m\mid n$, $$ n\mathbb Z/m\mathbb Z\sim(n/m)\mathbb Z\;. $$ More generally, see also the Wikipedia article on equivalence classes. Whenever the equivalence classes of an equivalence relation all have the same size $d$, forming the quotient ...


1

Your list includes some subjects, some algebraic structures, and some objects within algebraic structures. I'll separate them to organize the list a little better. Anyone can feel free to contribute to/ edit this list as I'm certainly not an expert in all of this. Subjects: Linear Algebra: the study of vector spaces and the linear transformations ...


1

According to Wikipedia and some papers: Exterior algebra = Grassmann algebra (= differential forms, since they are a construction of the exterior algebra) (=derivations, since derivations are just one possible construction of the dual object to differential forms). EDIT: To make the claim that "exterior algebra=differential forms" precise, since as it ...


1

Short answer to (1): There's no obvious reason why you can't do this, but you have to be really, really careful with this sort of stuff because it's very easy to get it wrong. What you're defining here is not a function, but an operation which acts on an expression (in a way that is not particularly clear, but the idea seems plausible). Short answer to (2):...


1

Since no one immediately answered this question, I suspect that there are no standard terms for the number of descendants or the number of ancestors of a given node in a directed acyclic graph (DAG). But remember that the point of writing is just to clearly convey an idea to the reader, so you should feel free to invent terms as you need them, just so long ...



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