# Tag Info

### Closed formula for number of ones in a proper factor tree

A simple generating function, though not a closed formula, for $\gamma(N)$ is given by $\sum_{N\geq 1}\frac{\gamma(N)}{N^s} = \frac{1}{2-\zeta(s)}$, where $\zeta(s)$ is the Riemann zeta function. See ...

### Closed formula for number of ones in a proper factor tree

This seems to be Sloane's A002033, namely the number of perfect partitions of $n-1$. Since the OEIS doesn't give any closed formula, there probably isn't one, but it's probably worth checking the ...

### Closed formula for number of ones in a proper factor tree

Take the formal product $g(x_1,x_2,\ldots)=\prod_{i\ge 1} (1-x_i)$ and define $$f(x_1,x_2,\ldots) = \frac{g(x_1,x_2,\ldots)}{2g(x_1,x_2,\ldots)-1}.$$ Then $\gamma(\prod_i p_i^{\alpha_i})$ is the ...

### Variants of the Bonk-Schramm embedding

Your question (actually, questions) is a bit too vague for my taste. Here is an answer of sorts. If we replace $\mathbb{H}^k$ by a Hadamard manifold with variable (but bounded) curvature, can we ...

### Countable chain condition in topology

A very nice example of a completely regular ccc non-separable space is the Pixley-Roy hyperspace of the reals. While it's not metrizable (and it can't be, as you already know) it is as close as it ...

### Survey of recent developments of the Gelfand-Kirillov dimension

Complementing Manuel Norman's excelent answer, recently I've found a very nice survey about the Gelfand-Kirillov dimension, from 2015, by Jason Bell, called Growth Functions. This survey discusses ...
1 vote
Accepted

### On faces of polytopes

Let $A_0\subset A$ be the set where $\ell$ attains a minimum on $A$. It is a face of some dimension $k<d$. If $k=d-1$, we are done. Assume that $k<d-1$. Without loss of generality, $0\in A_0$ ...
Accepted

### The origin of a planar graph theorem of Steinitz and Rademacher

According to Frank Lutz's article it's in paragraph 46 of the Steinitz-Rademacher book: "every triangulated 2-sphere can be reduced to the boundary of the tetrahedron by a sequence of edge ...

### What does it mean that the Hessian is proportional to the metric?

Since the proof that such a function implies the metric is a warped product metric is fairly simple, I include a complete copy below. 1 Start with $\nabla^2 f = \lambda g$. For any vector field it ...

### Abel–Plana formula with fractional offset

I eventually did find a published derivation of the fractional-offset (2) of the Abel-Plana formula, in A Generalized Mode Summation Formula of the Zero-Point Energy in a Cavity by Norio Inui (2003): ...

### Abel–Plana formula with fractional offset

https://www.semanticscholar.org/paper/The-Euler-Maclaurin-formula-revisited-Elliott/43a86582da7acc57944cee260d1654bfbc5f251c A while ago I was looking at using similar formulas for numerical ...
Accepted

### Are “most” bounded derivatives not Riemann integrable?

In 1977 Clifford E. Weil showed that $A$ is a first Baire category set (i.e. a meager set) in $X$ (sup norm) -- see The space of bounded derivatives. So the situation, at least with respect to one ...
Accepted

### A result on symmetric closed monoidal categories

This is a special case of: Let $F\dashv G$ be an adjunction. If there exists an isomorphism $id \cong GF$, then the unit $id \to GF$ is an isomorphism. Indeed, $[-,A]$ is left adjoint to itself as ...

1 vote
Accepted

1 vote