# Tag Info

5

Your difficulty stems from the use of the letter $y$ for two different purposes: (a) as coordinate variable in the $(x,y)$-plane, and (b) as variable for (unknown) functions $x\mapsto y(x)$ whose graphs are lying in the $(x,y)$-plane. When dealing with ODEs for the first time we are given a function $f:\ (x,y)\mapsto f(x,y)$ defined in some region $\Omega$ ...

3

The notation $\dfrac{\mathrm dy(x)}{\mathrm dx}$ is short for $\dfrac{\mathrm dy}{\mathrm dx}(x)$ or $y'(x)$, if you prefer. In this context, the equality $\dfrac{\mathrm{d}y(x)}{\mathrm{d}x} = f(x,y)$ should be read as $\dfrac{\mathrm dy}{\mathrm dx}(x)=f(x,y(x))$. As for your last example, you got the wrong idea, $y$ is a function whose domain is a ...

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(Too long for a comment:) I can offer an explanation showing that dividing by $n$ would give an underestimation of the variance. The sum of squares $\sum (X_i - \overline{X})^2$, where $\overline{X}$ is the sample mean, is smaller than the sum $\sum (X_i - \mu)^2$ where $\mu$ is the true mean. This is the case since $\overline{X}$ is expected to be ...

3

$\dfrac{dt}{t}$ is the (well, unique up to a constant factor, so "a", strictly) Haar measure on the topological group $(0,+\infty)$ (with multiplication). That makes some transformations particularly nice when written in that style, $$\int_0^\infty f(at)\,\frac{dt}{t} = \int_0^\infty f(t)\,\frac{dt}{t}$$ for all $a > 0$, so $$\int_0^\infty t^s ... 2 In the book Handbook of Product Graphs 2nd Edition - Hammack et al. you can find the following definition: The strong product of G and H is the graph denoted as G \boxtimes H, and defined by$$ V(G\boxtimes H) = \{(g,h) | g \in V(G) \text{ and } \in V(H) \}.  E(G\boxtimes H) = E(G\square H) \cup E(G\times H). $$2 For anything you write that you want other people to read, do not use Cyrillic letters if you don't know what you're doing. If there were any trend to use Cyrillic in math then the Russians would use them, and they don't. They write almost everything with Latin and Greek letters like everyone else. And they write "sin" for the sine function, "lim" for limit, ... 1 The first variant. Everything is in one real variable, so you do not get Jacobian matrices to compute determinants. The best way to understand that identity is to think of a delta-approximating sequence with compact support, for instance based on the quadratic or cubic B-Spline. Then consider small disjoint intervals around the roots of g, make the index ... 1 The Cyrillic Л, which is analogous to L, is the first letter in the name Lobachevsky and has been used in hyperbolic geometry for the Lobachevsky function$$ Л(\theta) = -\int_0^\theta \log|2\sin t|\,dt. $$This notation was introduced by Milnor. See 1) chapter 7 of Thurston's "Geometry and Topology of 3-manifolds" (written by Milnor), 2) the appendix ... 1 too long for a comment: Greek letters are precariously overloaded. There is both the Dirichlet eta \eta(s) and the Dedekind eta function \eta(\tau). "\pi" can occasionally mean a permutation or a prime in a field of characteristic k as well as the ratio of the circumference to the diameter of a circle and the prime counting function \pi(n) ... 1 Well, the idea is to assume that for a ring R and this special subset H, we want H to be an ideal of R, that is,$$ \forall x,y \in H, \forall r \in R, \quad x+y, rx \in H.  Then, for each element $r \in R$, we group together all the elements $\{ r+h \, | \, h \in H \}$. Note that $H$ doesn't have to be finite as in the notation you suggested by ...

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