3

At least in the context of elasticity theory, this is known as "injectivity almost everywhere", see, e.g., Ciarlet's "Mathematical Elasticity vol. 1", Problem 5.7 (following the work of Ball) and Section 7.9 (following Ciarlet-Nečas).


2

The following older (mostly before 1900) usage may be of interest. Equations of the form $ax^2 + c = 0$ such that $a \neq 0$ used to be called pure quadratic equations (sometimes incomplete quadratic equations) and equations of the form $ax^2 + bx + c = 0$ such that $a \neq 0$ and $b \neq 0$ used to be called affected quadratic equations (sometimes complete ...


2

The full quote reads: As in the case of finite numbers, the infinitesimal numbers form an isolated subgroup $\mathbf{R}_{\text{inf}}$ of $^*\mathbf{R}$; this implies that the factor group $^*\mathbf{R}/\mathbf{R}_{\text{inf}}$ inherits naturally its order relation from $^*\mathbf{R}$. From this context, I would infer that for an ordered group $(G,\leq)$, a ...


2

I don't know a name for this construction but it is adjoint to the "Cartesian product" of directed graphs (warning: despite the name, this is not the categorical product in the category of directed graphs). That is, writing $[G,H]$ for your graph of homomorphisms from $G$ to $H$, there is a natural bijection between homomorphisms $G\to [H,K]$ and ...


2

The two results above express the basic property of the derivability relation $\vdash$. The symbol $\Gamma \vdash \phi$ means that there is a derivation $\mathcal D$ (in the Natural Deduction proof system) of conclusion $\phi$ from the set $\Gamma$ of assumptions (or premises). In the linked comment, the existence of such a derivation is symbolized with: $\...


2

The term is 'integer division'. The symbol used is $\lfloor \frac{322}{100} \rfloor$ or sometimes $[\frac{322}{100}]$ and $100\,|\,300$ if one is a divisor of the other.


1

This is what your textbook means: Let's throw 2 fair dice: Event $E$: the sum of the "2 dice's face up" is even Event $F$: the sum "2 dice's face up" is 2,3 or 4 Find $$\mathbb{P}[E \cup F]=\mathbb{P}[E ]+\mathbb{P}[ F]-\mathbb{P}[E \cap F]=\frac{1}{2}+\frac{6}{36}-\frac{4}{36}=\frac{5}{9}$$


1

The polynomials over $\Bbb C$ (or more generally any field $\Bbb K$) are expressions of the form $p_0+p_1X+p_2X^2+\cdots+p_{m-1}X^{m-1}+p_mX^m$, where $X$ is an indeterminate and $p_0,p_1,p_2,...,p_{m-1},p_{m}$ are in $\Bbb C$ (or more generally $\Bbb K$).


1

Find the $y$ coordinates of intersection of curves $x=y^2$ and $x = 3y^2$ with line $x+y=2$. $x=2-y = y^2 \implies y = 1$ $x=2-y = 3y^2 \implies y = \frac{2}{3}$ For $0 \leq y \leq \frac{2}{3}, y^2 \leq x \leq 3y^2$ For $\frac{2}{3} \leq y \leq 1, y^2 \leq x \leq 2-y$ So the integral becomes, $\displaystyle \int_0^{2/3} \int_{y^2}^{3y^2} f(x,y) dx \ dy + \...


1

If you were teaching students about this, then perhaps you could mention strategies that ultimately don't work. If we rewrite $$ ax^2+bx+c=0 $$ as $$ x=-\frac{ax^2+c}{b} $$ then $x$ is still written in terms of $x^2$, meaning that we can't get anywhere. Similarly, if we make $x^2$ the subject of the equation, then we don't get anywhere. The crux of the ...


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