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7

Although there's a distinction between the noun and adjective abbreviations of monomorphism and epimorphism (mono vs. monic and epi vs. epic), it's fairly common to use iso for both the noun and adjective abbreviation of isomorphism. For example, both of these seem normal to me: $\mathcal{C}$ is a balanced category if every morphism $f$ of $\mathcal{C}$ ...


5

The names you listed in the first paragraph are built from an object of investigation. You study objects named X and prove a bunch of things about them, so you call it X theory. So, it's "group theory" because it concerns the properties of groups. We would not say "real analysis theory" because "real analysis" is not a name of the object under ...


4

In first-order logic, equality is a binary relation for which these two statements are true: Any variable or constant is equal to itself. We call this the Reflexive property, and it can be written $$\text{For all $x$, }x=x$$ or, more formally, $$\forall x(x=x)$$ If two items are equal, anything we can say about the first item in our logical system we ...


3

Discriminants have their roots in abstract algebra and are used to determine the nature of zeroes of polynomials. From Wolfram Mathworld, if $p(z)$ is an $n$th degree polynomial with leading coefficient $a_n$ and roots $r_i$, the discriminant $\Delta$ is $$\Delta(p) = a_n^{2n-2} \prod_{i > j} (r_i - r_j)^2.$$ Discriminants are defined up to a constant ...


2

I'd say: $$r=z^{\frac{1}{n}}e^{\frac{2i\pi k}{n}}$$ It is a multivalued function with $k=0,\dots,n-1$


2

In the semigroup theory S is called a left ideal in the semigroup R under multiplication. See any semigroup theory book(J.M.Howe: Semigroup theory for example)


2

I just call it unit conversion or sometimes unit analysis in my classes, especially in Chemistry and Physics. Both those terms are used in the textbooks I use, though "unit analysis" can also be used for checking that the unit (not the number) is correct, and it is also used for a particular technique to convert units.


2

Calling the plane/vector "perpendicular" to another vector is common and perfectly acceptable. Also common is the word "normal" (e.g. "the vector $(1, -1, 1)$ is normal to the plane $x - y + z = 3$.")


2

Like @Chilango commented, the vector $f_x(x_0, y_0)i+f_y(x_0,y_0)j$ exists even if $f$ is not differentiable at $(x_0, y_0)$. The link you gave conflates differentiability with existence of the gradient vector, but both my calculus textbook and my real analysis textbook give a different definition for differentiability: A function $f$ is differentiable at ...


2

In my experience the expression "real coordinate space" emphasizes that we are not working over the complex numbers, i.e. the space is $\mathbb R^n$, not $\mathbb C^n$. You can use Cartesian coordinates (and a whole bunch of other coordinate systems) on these spaces. The spaces $\mathbb R^n$ are called Euclidean spaces, so they are the same as real ...


1

I've just decided to call them "uniands", but I came here hoping there was something standard. I believe you could get away with calling them "summands" and "factors" by analogy of $\cup$ with $+$ and $\cap$ with $\times$ (the first being the sum and product of the Boolean ring of subsets of a set, and the second being the generic terms for sum and product ...


1

I am partial to what Omnomnomnom mentioned in a comment: it is a proportion, and the authoritative Oxford English Dictionary supports this terminology (snippet produced below):


1

In this context, the term "under the operation" means that $B_n$ is a group when we consider it with the given operation. More generally, if $X$ is a set and we say that we have the group (or some other structure) $X$ under the operation $*$ (or some collection of operations), then we mean that we are putting the operation $*$ (or the collection of ...


1

The Encyclopedia of Math calls it isoperimetric ratio, and Google finds quite a few articles using that phrase for it.


1

Call the vector $\mathbf{n}$, say. Then you could refer to the plane as a "normal plane to $\mathbf{n}$", or "normal plane to $\mathbf{n}$ through $O$" if you wish to specify the location of the plane relative to the origin as in your post.


1

$$z^n=c\implies z=\omega^k\sqrt[n]c$$ Where $\omega$ is a primitive $n^{th}$ root of unity, and $0\le k\in\Bbb{Z}\le n-1$


1

For a non-negative real number $x$ there is always a unique choice of non-negative real $n$-th root, which is usually denoted by $\sqrt[n]{x}$. Furthermore, if $x$ is negative there is a unique choice of $n$-th root if $n$ is odd and none if $n$ is even. In short, if $x \geq 0$ is real and $n$ is even, then the only real $n$-th roots of $x$ are $\pm ...


1

In complex analysis, $\sqrt[n]{x}$ is regarded as a multivalued function. Or you can write it as $$\sqrt[n]{x}=\exp{\frac{\operatorname{Log}(x)}{n}},\space x\ne0.$$ $\operatorname{Log}(x)$ is the inverse function of $\exp(x)$, see here.


1

I think you got the rough idea. We have symbolic expressions and objects, which are two different things. We cannot take the objects themselves and put them on paper, but we can write symbolic expressions that refer to objects, and the objects they refer to are called their values. We might have multiple symbolic expressions that refer to the same object, in ...


1

These are quotient maps. Proof: We want to show the equivalence of these two conditions: For all $A,A'\subseteq X'$, $\operatorname{cl}(f^{-1}(A)) = f^{-1}(A') \implies \operatorname{cl} A = A'$. For all $A\subseteq X'$, $A$ is closed iff $f^{-1}(A)$ is closed. (1 $\Rightarrow$ 2) Suppose (1). Let $A\subseteq X'$. The implication $$ \text{$A$ is ...



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