4

Boundaries "at infinity" of spaces and groups had already been in the air for a while, perhaps going back to the boundary of the hyperbolic plane. Then in the 1940's we had the Freudenthal-Hopf Theorem in group theory, in which the concept of the ends of a (locally compact, locally connected) topological space and the ends of a finitely generated group were ...


3

For Euclid, an "angle" is formed by two rays which are not part of the same line (see Book I Definition 8). So, to Euclid, a "straight angle" is not an angle at all, and so Proposition 31 is not a special case of Proposition 20 since Proposition 20 only applies when you have an angle at the center.


3

Equivalently, the problem is to compute $L=\sum_{s\in S}(-1)^s/(s-1)$ where $S$ is the set of all perfect powers. A similar problem of computing $\sum_{s\in S}1/(s-1)\color{gray}{[{}=1]}$ goes back to Goldbach as far as I know. The idea is pretty much the same. Let $N$ be the set of all integers (strictly) greater than $1$. Then each $s\in S$ has a ...


3

What follows is taken directly from Borweins' Pi and the AGM. Let $N$ be a positive number and $q_N=e^{-\pi\sqrt{N}}$ and $$k_N=\frac{\vartheta_{2}^{2}(q_N)}{\vartheta_{3}^{2}(q_N)},k'_N=\sqrt{1-k_N^2},G_N=(2k_Nk'_N)^{-1/12},g_N=\left(\frac{2k_N}{{k'} _N^{2}}\right)^{-1/12}\tag{1}$$ where $\vartheta _2,\vartheta_3$ are theta functions of Jacobi defined by $$...


2

In more familiar terms, by the formula for the square of a sum, $$\frac1n\sum_{i=1}^n x_i^2-\left(\frac1n\sum_{i=1}^n x_i\right)^2=\frac1n\sum_{i=1}^n x_i^2-\frac1{n^2}\sum_{i=1}^n x_i^2-\frac2{n^2}\sum_{i=1}^n\sum_{j=i+1}^n x_ix_j.$$ The notation $S(x_1x_2)$ is questionable, because it doesn't clearly shows that the summation indexes do not cover the ...


2

First, here's the geometric example illustrated: Assume that on the circumference of a circle there is a fixed point $A$ about which a ray revolves. When this ray passes through the center of the circle, we call the other point at which it intersects the circle the point B associated with this position of the ray. Now imagine that the ray “revolves ...


1

Statements that are undecidable in Peano arithmetic, but can be stated in it and are provable in something larger such as ZF or second-order arithmetic, include $\varepsilon_0$ induction, Goodstein's theorem and the Paris–Harrington theorem. I'm not sure, though, if they can be formatted as Gödel sentences. See also @DanielWainfleet's comment.


1

Try reading $S(x_1)$ as $\sum\limits_i x_i$, then $S(x_1^2)$ as $\sum\limits_i x_i^2$ and $S(x_1 x_2)$ as $\sum\limits_{i,j} x_ix_j$


1

There is a lot of analytic number theory even in Landau's other books on algebraic number theory. A standard reference before Landau is Paul Bachmann's book on analytic number theory from the 1890s. Then there are several lecture notes that Siegel made available (Lectures on analytic number theory, and several on "Funktionentheorie") the university library ...


1

It (or, rather, its determinant) appears in the second volume of Cauchy's "Exercices d'analyse et de physique mathematique", as part of "Memoir on alternating functions and alternating sums" (https://books.google.se/books?id=DRg-AQAAIAAJ, pages 151-159, see particularly formula (10).) Reading just that section, the calculation of the determinant is presented ...


1

The persistence algorithm was first described in: Barannikov, S. (1994). "Framed Morse complex and its invariants". Advances in Soviet Mathematics. 21: 93–115.


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