history-and-theory-of-mathematics

Hello Physics Forums Community, I would like to introduce an open-source infrastructure designed for high-precision numerical analysis and verification of infinite-dimensional dynamic systems: SO-HMNS (Sovereign Absolute Invariant Truth Infrastructure). * Official Repository... Read more

It is a well known fact that the Galois group $G$ of a Galois extension $K\subseteq L$ is a profinite group, as $G$ is equal to the inverse limit of the Galois groups of the finite subextensions of ...

Let $p_1, p_2, p_3, p_4, p_5 ... = 2, 3, 5, 7, 11, ...$ be the prime numbers in increasing order. Let $k > 5$ and $A_k = {1, 4, 5, 6, 7, 9, 11, p_6, ..., p_k}$. In $A_k$ all least common multiples ...

My question is really just about one-dimensional spaces. I am primarily using covering dimension but I'd be sufficiently happy with an answer for small or large inductive dimension. Question: Suppose ...

The n queens problem is to place on an n × n chessboard n queens so that none attacks any other. This means there is only one queen on every horizontal, vertical, and diagonal line. When n is a prime number ≥ 5, it is sufficient to place the queens on a line that has slope 2, 3, 4, …, […] The post Queens on a prime order board first appeared on John D. Cook .

I am having troubles on the spectral sequence of a double complex, as treated in Weibel's AITHA. In Example 1.2.4 it is defined a double complex in an abelian category $\sf A$: a double complex ...

A reflective subcategory is a full subcategory such that objects and morphisms in have “reflections” and in . Every object in looks at its own reflection via a morphism and the reflection of an object is equipped with an isomorphism . A canonical example is the inclusion of the category of abelian groups into the category of groups, whose reflector is the operation of abelianization. A useful pro…

Everything Is Logarithms Some connections between things, which I have not seen elsewhere. Maybe they mean something? 1. The Baseless Logarithm Normally one writes a logarithm with a base, \(\log_b (x)\), to mean \[y = \log_b (x) \Lra b^y = x\]And then you can change the base of the logarithm with \[\log_b (x) = \frac{\log_a (x)}{\log_a(b)}\]Which follows from rearranging \(\log_a (x) = \log_a (b…

Let $f(z)$ be an entire function of exponential type $1$, and let $s,t>0$, $s+t=1$. Can we always find entire functions $g,h$ of exponential types $s,t$ such that $f=gh$ ?

superalgebra and (synthetic ) supergeometry Supergeometry is the (higher) geometry over the base topos on superpoints modeled on the canonical line object in there. As ordinary differential geometry studies spaces – smooth manifolds – that locally look like vector spaces, supergeometry studies spaces – supermanifolds – that locally look like super vector spaces. As ordinary algebraic geometry stu…

One variant of the traditional sphere packing problem in $\mathbb R^3$ asks what the density $\rho_r$ of spheres of equal radii $r$ one can fit in a unit cube without exceeding the boundary ...

I am looking for a reference—either a book or a specific paper—that contains the actual proof of the result that no three consecutive positive integers are perfect powers. While reading Wacław ...

A semifunctor is a homomorphism between semicategories, like a functor is a homomorphism between categories. A semifunctor from a semicategory to a semicategory is a map sending each object to an object and each morphism in to morphism in , such that If is a category, then need not preserve its identity morphisms, but the composition axiom does require that it send them to idempotents in . In Rel…

A variation of Abel's summation formula is $$\sum_{n = 1}^N f(n)g(n) = f(N)G(N) - \sum_{n = 1}^{N - 1} (f(n + 1) - f(n))G(n),$$ where $$G(n) = \sum_{k = 1}^n g(k).$$ Suppose that $\alpha \in ...

In Mandelbrot(1968)'s paper, the fractional brownian motion, denoted by $B_{H}(t,\omega)$,(t>0) is defined by $$B_{H}(0,\omega)=b_{0}$$ $$B_{H}(t,\omega)-B_{H}(0,\omega)=\frac{1}{\Gamma(H+\frac{1}{2})}\{\int^{0}_{-\infty}[(t-s)^{H-1/2}-(-s)^{H-1/2}]dB(s,\omega)+\int^{t}_{0}(t-s)^{H-1/2}dB(s,\omega)\}$$ I have difficulty understanding fractional brownian motion by self study.Is there an intuitive…

This is a submission for the June Solstice Game Jam What I Built The Longest Night is a browser game about codes, daylight, and one impossible question. It's June 21 — the solstice — and you're the night-shift cryptanalyst at a remote listening station. Four encrypted transmissions arrived at noon. Command wants them broken before the sun goes down, and the sun is going down: an animated sky drai…

In a project on Quantum Mechanics, we ran into the following identities with Bessel functions and their derivatives: ...

The lower corner of the page of a book is folded over so as to just reach the inner edge of the page. Find the width of the part folded over, measured along the bottom of the page, when the length of ...

When trying to design asymmetric smooth-step functions $f_{\lbrack 0,1\rbrack}(x)$ that for $0<a<1$ satisfy: $f(0)=0,\ f'(0)=0,\kappa\lbrack f\rbrack(0)=0$ $\kappa\lbrack f\rbrack(a)=0$ ...

Wikipedia gives the following proof that the Cauchy-completeness of the real numbers implies Dedekind-completeness: Let $S$ be a nonempty set of real numbers. If $S$ has exactly one element, then its ...