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9

If $x^2=x$, then $$ (2x-1)^2=4x^2-4x+1=4x-4x+1=1 $$ So you actually have the split-complex numbers in disguise.


7

In mathematics, $x$ usually denotes a variable and $a$ denotes a (fixed) constant (however, any constant). The idea that the author wanted to give is that if you can calculate the derivative at any point, then you can consider the function that sends each point $x$ to the derivative of $f$ at $x$ (function known precisely as the derivative of $f$).


6

Aha, I found it! The $f$ function I'm using is Cantor's pairing function.


5

There's no real difference between $x$ and $a$ used as variable names. We distinguish between variables, parameters, constants - and yet this distinction does not really exist. This hierachy is just a customary order among variables, as it is customary to denote "more variable" variables with $x$ and "more constant" variables with $a$, say. Or to use $n,m,k$ ...


3

Similarly to Micah's answer, but showing an ismorphism to $\Bbb{R}\times \Bbb{R}$, note that \begin{align} (a(1-x)+bx)(c(1-x)+dx) &= ac(1^2-2x+x^2)+(ad+bc)(x-x^2)+bdx^2 \\ &= ac(1-2x+x)+(ad+bc)(x-x)+bdx \\ &=ac(1-x)+bdx. \end{align} Therefore the mapping to $\Bbb{R}\times \Bbb{R}$: $\varphi(a(1-x)+bx)=(a,b)$ preserves multiplication. Showing ...


3

Let $ \Bbb{F} $ denote the base field of $ V $. If we assume that $ \star $ is a binary operation on $ V $ that turns $ V $ into an $ \Bbb{F} $-algebra, i.e., Left distributivity: $ x \star (y + z) = x \star y + x \star z $ for every $ x,y,z \in V $, Right distributivity: $ (x + y) \star z = x \star z + y \star z $ for every $ x,y,z \in V $, and ...


2

There are two things to point out here. $2^{\aleph_0}=\aleph_1$ is an assertion called "The Continuum Hypothesis". It cannot be proved, nor disproved, from the standard axioms of set theory. Note that in $\Bbb N$ not every number has a logarithm. For this you need to extend to the real numbers. For example $\log_25$ is not a natural number at all. So the ...


2

A constant is something like a "number". It doesn't change as variables change. For example $3$ is a constant as is $\pi$. A parameter is a constant that defines a class of equations. $$\left(\frac xa\right)^2 + \left(\frac yb\right)^2 = 1$$ is the general equation for an ellipse. $a$ and $b$ are constants in this equation, but if we want to talk about ...


2

I haven't heard "cumulative density" used before. People are saying that it is a contradiction. To see why, it helps to look at the definitions of cumulative distribution function (CDF) and probability density function (PDF). Assume $X$ is a continuous random variable. The CDF is $$F(x) = P(X\leq x) = \int_{-\infty}^x f(x) \, dx.$$ It is the integral of ...


2

The terms diagonal and anti-diagonal are descriptive and "culturally apt". The first is standard in geometry and topology (the diagonal embedding of a topological space $X$ is the inclusion $X \hookrightarrow X \times X$ defined by $x \mapsto (x, x)$), and I'm almost positive I've seen the second in connection with the normal bundle of a diagonal embedding ...


2

I don't think there is a generally-accepted name for such sets. Sometimes they are written as dilations of $\mathbb{Z}$: $\dfrac12\mathbb{Z}$ or the like. Do not confuse the set $\dfrac12\mathbb{Z}=\{\dfrac n2:\ n\in\mathbb{Z}\}$ with the half-integers $\{n+\dfrac12:\ n\in\mathbb{Z}\}$.


1

Notations include $q^{-1}\mathbb{Z}$,$\frac{1}{q}$,$\mathbb{Z}/p$. I would probably call it "Integers over p" or the likes.


1

$r$ is the base of a Cunningham chain of the second kind. A Sophie-Germain prime is the base of a Cunningham chain of the first kind.


1

I think the second definition is better. $\frac{dy}{dx}=0$ would be considered a differential equation. A differential equation doesn't necessarily need to involve the function itself.


1

I think this paper gives the definition you want, if I understand you correctly: Definition 2.9. A poset $P$ will be called locally ranked if all its principal lower ideals are ranked.


1

I think it must mean that although the assertion might not be true of the sequence in question, it is true of some subsequence of it. For an example where the phrase is clearly used this way, see this answer. This usage is not quite the same as the one described in the Wikipedia article: any equivalence relation that considered all sequences equivalent to ...


1

The decomposition with the stronger condition is called effective. It appears in connection with pseudo-Riemannian symmetric spaces, where one deals with a special case of the symmetric decomposition $\mathfrak{g}=\mathfrak{h}\oplus \mathfrak{m}$, namely that the decomposition is effective, i.e., that it is minimal, which precisely means that ...


1

The logarithm of an infinite cardinal number $\alpha$ is defined as follows: $$\log\alpha=\min\{\beta\ |\ 2^\beta\ge\alpha\}.$$ This definition is given on p. 74 of Cardinal Functions in Topology by István Juhász, and I am not aware of any competing definitions. According to this definition, $$\log\aleph_0=\log\aleph_1=\aleph_0.$$ It should be noted that ...


1

In principle, "$F$ is a left adjoint" is fine because the adjunction is determined uniquely up to unique isomorphism by $F$. This seems to contradict your intuition that "$F$ is a left adjoint" sounds wrong, but it's perfectly consistent with standard usage such as talking about "the" limit of $G$: this sort of usage works fine except in contexts where it's ...



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