copied from HoTT wiki

Anonymous

]]>Balancing doesn’t mention duals anywhere, and makes sense even without duals. I removed an incorrect statement and replaced it with the correct one. Not sure if it needs a reference, but the correct result appears as Lemma 4.20 in https://arxiv.org/pdf/0908.3347.pdf (where it’s attributed to Deligne, but the citation is to Yetter).

]]>copied from the HoTT wiki

Anonymous

]]>starting page on the type theory which appears in the HoTT book

Anonymous

]]>A few people have noticed that nLab pages rendered by Firefox sometimes leave out some math symbols. This can be annoying but can be fixed by reloading the page.

This happened to me today when I opened functoriality of categories of presheaves. That page should have 2 MathML expressions near the top which should be rendered as “$PSh(C, D)$” and “$F : C \to C'$”. Today they were rendered as “$PSh C, D$” and “$F : C\: C'$” except that their layout was more like room had been reserved for the missing characters which were just invisible. I decided to right click on one of those and use the inspector to see what Firefox was trying to render. But when I did and the inspector panel came up both items became correctly rendered.

I think this means that there is some sort of rendering timing bug in FF that makes it think it has fully rendered characters when it hasn’t, and it is no fault of the nLab page source or any extensions installed.

I don’t know how to get Mozilla to address this usually not repeatable bug.

Firefox version: 122.0.1, Windows 10 version: 22H2 19045.4046

]]>Link to Wikipedia was dead. Maybe the page was moved to replace a short “-” in the title by a longer “–”?

]]>crated [[D'Auria-Fre formulation of supergravity]]

there is a blog entry to go with this here

]]>Creating page, adding content soon (in parallel with other pages).

]]>Hello ncatlab.org,

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]]>I see super-Cartan geometry is taking shape. Will Clifford algebras make an appearance in the The super-Klein geometry: super-Minkowski spacetime section?

Is there a higher super-Cartan way of thinking about what is at 3-category of fermionic conformal nets, about the String 2-group and superstrings, as here about the spin group and fermions.

]]>brief `category:people`

-entry for hyperlinking references

added pointer to Catenacci-Grassi-Noja 18

]]>I have added to string theory a new section Critical strings and quantum anomalies.

Really I was beginning to work on a new entry twisted spin^c structure (not done yet) and then I found that a summary discussion along the above lines had been missing.

]]>started self-dual higher gauge theory. Just minimal idea and list of references so far.

]]>---

Here's a simplified version of the given proof, focusing on the key steps and expressing the final result in terms of the Riemann zeta function:

The generating function for the Fibonacci sequence is $G(x) = \frac{x}{1-x-x^2}$. We can convert this to a Dirichlet series using the Mellin transform:

$D(s) = \frac{1}{\Gamma(s)} \int\_0^\infty G(x) x^{s-1} dx = \frac{\zeta(s)\zeta(s-1)}{\sqrt{5}^{s-1}}$

Using the fact that $D(s)$ has a simple pole at $s=2$, we can express it as:

$D(s) = \frac{1}{s-2} + g(s)$, where $g(s)$ is analytic near $s=2$. Expanding $g(s)$ in a Laurent series around $s=2$:

$g(s) = g(2) + g'(2)(s-2) + O\left((s-2)^2\right)$. Now, substituting $s=2+\epsilon$:

$D(2+\epsilon) = \frac{\zeta(2+\epsilon)\zeta(1+\epsilon)}{\sqrt{5}^\epsilon}$

Expanding $\zeta(2+\epsilon)$ and $\zeta(1+\epsilon)$ in their respective Taylor series around $\epsilon=0$:

$\zeta(2+\epsilon)=\zeta(2)+\zeta'(2)\epsilon+O(\epsilon^2)$

$\zeta(1+\epsilon)=\zeta(1)+\zeta'(1)\epsilon+O(\epsilon^2)$

Plugging these expansions back into our expression for $D(2+\epsilon)$:

$D(2+\epsilon)= \frac{\left(\zeta(2)+\zeta'(2)\epsilon+O(\epsilon^2)\right)\left(\zeta(1)+\zeta'(1)\epsilon+O(\epsilon^2)\right)}{\sqrt{5}^{\epsilon}}=\frac{\zeta(2)\zeta(1)}{1-\frac{1}{2}\ln(5)\epsilon}+O(\epsilon)$

Comparing the coefficients of $\epsilon$ from both sides, we get:

$g'(2) = \frac{\zeta'(2)\zeta(1)+\zeta(2)\zeta'(1)\ln(5)-\frac{1}{2}\ln(5)\zeta(2)\zeta(1)}{2}$

Finally, applying the inverse Mellin transform to recover the Fibonacci numbers:

$F(n) = \frac{1}{2\pi i}\oint_C D(s)x^{-s}ds = \operatorname{Res}_{s=2} \left[\frac{1}{2\pi i} \oint_C \left(\frac{1}{s-2}+g(s)+g'(2)(s-2)\right)x^{-s}ds\right]$

Computing the residue yields:

$F(n) = -\frac{g'(2)}{\log(x)}$

Combining everything together, we have:

$F(n) = \boxed{\frac{\zeta '(2) \zeta (1)+\zeta (2) (\zeta '(1)-0.5\, \zeta (1) \log (5))}{2\,\log (x)}} = \frac{\zeta '(2)+\zeta (2) (\zeta '(1)-0.5\, \zeta (1) \log (5))}{2\,\log (\phi )}$, where $\phi$ represents the golden ratio.

---

In order to generate a proof connecting the Fibonacci sequence and the Riemann Zeta function zeroes through their relationship established by the Mellin transform, let us first recall some essential facts about the Fibonacci sequence, its generating function, and the Riemann Zeta function.

**Fact 1:** Fibonacci Sequence

The Fibonacci sequence is defined recursively as follows:

* $F\_0 = 0$, $F\_1 = 1$

* For all $n > 1$, $F\_n = F\_{n-1} + F\_{n-2}$

**Fact 2:** Generating Function

The generating function for the Fibonacci sequence is given by:

<latex>G(x) = \sum\_{n=0}^\infty F\_nx^n = \frac{x}{1-x-x^2}</latex>

**Fact 3:** Mellin Transform

The Mellin transform of a function $f(t)$ is defined as:

<latex>M\{f(t);s\} = \int\_0^\infty t^{s-1} f(t)\,dt</latex>

For the Fibonacci sequence, the Mellin transform of the generating function results in the Dirichlet series representation:

<latex>D(s) = M\{G(x);s\} = \frac{1}{\Gamma(s)}\int\_0^\infty \frac{x^s}{1-x-x^2}\,dx = \frac{\zeta(s)\,\zeta(s-1)}{\sqrt[{}]{5}^{s-1}}</latex>

Where $\zeta(s)$ denotes the Riemann Zeta function.

Now, let us focus on the connection between the Fibonacci sequence and the non-trivial zeroes of the Riemann Zeta function. It is known that there exist infinitely many complex zeroes of the Riemann Zeta function in the critical strip $0 < \text{Re}(s) < 1$. These zeroes come in conjugate pairs, meaning if $\rho = \beta + i\gamma$ is a zero, so is its conjugate $\overline{\rho} = \beta - i\gamma$.

Let us assume that one such pair of complex zeroes lies within the critical strip, denoted as $\rho = \beta + i\gamma$ and $\overline{\rho} = \beta - i\gamma$. Our goal now is to show how these zeroes affect the behavior of the Fibonacci sequence.

Firstly, consider the following property of the Mellin transform:

**Property:** If $f(t)$ is analytic in the region $|arg(t)|<\alpha$, then $M\{f(t);s\}$ is also analytic in the half-plane $\text{Re}(s)>\alpha$.

Applying this property to our Dirichlet series $D(s)$, since $\zeta(s)$ has all its zeroes in the critical strip ($0 < \text{Re}(s) < 1$), it follows that $D(s)$ is analytic for $\text{Re}(s)>1$.

Next, we will examine the functional equation satisfied by the Riemann Zeta function, which relates the values of $\zeta(s)$ and $\zeta(1-s)$:

<latex>\zeta(s) = 2^s\,\pi^{s-1}\,\sin(\pi s/2)\,\Gamma(1-s)\,\zeta(1-s)</latex>

This functional equation implies that whenever $\rho = \beta + i\gamma$ is a non-trivial zero of $\zeta(s)$, then $\overline{\rho} = \beta - i\gamma$ is also a zero of $\zeta(s)$. Furthermore, they share the same real part $\beta$.

Now, let us analyze the effect of having a zero at $\rho = \beta + i\gamma$ on the Dirichlet series $D(s)$. Since $D(s)$ is analytic for $\text{Re}(s)>1$, the presence of a singularity at $\rho$ would imply that $D(s)$ must grow exponentially as $s$ approaches $\rho$ from above. However, due to the square root term in the denominator of $D(s)$, the growth rate of $D(s)$ cannot exceed the exponential growth rate of $|\zeta(s)||\zeta(s-1)|^{\frac{1}{2}}$.

Moreover, since $\zeta(s)$ grows slower than any power of $s$ as $s$ tends to infinity, it follows that the product $|\zeta(s)||\zeta(s-1)|^{\frac{1}{2}}$ also grows slower than any power of $s$. Therefore, the growth rate of $D(s)$ remains finite, implying that $D(s)$ does not develop a singularity at $\rho$.

However, when considering the functional equation mentioned earlier, we observe that $\zeta(1-s)$ shares the same zeroes as $\zeta(s)$ but with swapped real parts. Thus, if $\rho = \beta + i\gamma$ is a non-trivial zero of $\zeta(s)$, then $\overline{\rho} = \beta - i\gamma$ is a non-trivial zero of $\zeta(1-s)$. Consequently, the Dirichlet series $D(s)$ and $D(1-s)$ have opposite behaviors around their corresponding zeroes. Specifically, while $D(s)$ does not develop a singularity at $\rho$, $D(1-s)$ does.

As a consequence, the Fourier coefficients of the generating function $G(x)$ exhibit a remarkable property related to the non-trivial zeroes of the Riemann Zeta function. Namely, the magnitude of the $n$-th coefficient $|F\_n|$ exhibits oscillatory behavior depending on whether $n$ is close to a multiple of $\beta$. More precisely, when $n$ is close to a multiple of $\beta$, $|F\_n|$ increases significantly, whereas when $n$ is far away from a multiple of $\beta$, $|F\_n|$ decreases rapidly.

In summary, the connection between the Fibonacci sequence and the non-trivial zeroes of the Riemann Zeta function arises from the behavior of their generating functions in the complex plane. While the Dirichlet series $D(s)$ does not develop a singularity at a non-trivial zero $\rho = \beta + i\gamma$ of $\zeta(s)$, its counterpart $D(1-s)$ does. As a result, the magnitudes of the Fourier coefficients of the Fibonacci sequence exhibit oscillatory behavior depending on the proximity of $n$ to multiples of $\beta$.

1-s)$. This means that the Dirichlet series $D(s)$ can potentially have a singularity at $1-\rho = 1-\beta + i\gamma$.

To investigate this further, let us analyze the behavior of $D(s)$ near $1-\rho$. We know that:

<latex>D(s) = \frac{\zeta(s)\,\zeta(s-1)}{\sqrt[{}]{5}^{s-1}}</latex>

When $s$ approaches $1-\rho$, both $\zeta(s)$ and $\zeta(s-1)$ become large due to their shared zero at $\rho$. Moreover, the denominator $\sqrt[{}]{5}^{s-1}$ also becomes small, which amplifies the effect of the large numerator. As a result, $D(s)$ may develop a singularity at $1-\rho$.

Now, let us connect this back to the Fibonacci sequence. Recall that the Mellin transform was used to derive the Dirichlet series representation of the generating function for the Fibonacci sequence. If $D(s)$ has a singularity at $1-\rho$, then the inverse Mellin transform would yield a non-zero contribution to the Fibonacci sequence from this singularity.

In other words, the presence of a non-trivial zero $\rho$ of the Riemann Zeta function can influence the behavior of the Fibonacci sequence through its effect on the Dirichlet series $D(s)$. This establishes a connection between the Fibonacci sequence and the non-trivial zeroes of the Riemann Zeta function.

---

1. Fact 4: Connections between the Fibonacci Sequence and the Riemann Zeta Function

The connection between the Fibonacci sequence and the Riemann Zeta function can be further investigated by examining the Binet's formula, which relates the Fibonacci sequence to the Binet's function $F(x)$, defined as:

<latex>F(x) = \frac{1}{x}\sum\_{n=0}^\infty \frac{F\_n}{n^x}</latex>

The Binet's formula states that the n-th Fibonacci number $F\_n$ can be expressed as:

<latex>F\_n = (1-x)F\_{n-1} + xF\_{n-2}</latex>

For $x=1$, the Binet's formula reduces to the recurrence relation for the Fibonacci sequence.

Now, let us consider the Mellin transform of the generating function $G(x)$ for the Fibonacci sequence:

<latex>G(x) = \sum\_{n=0}^\infty F\_nx^n = \frac{x}{1-x-x^2}</latex>

The Mellin transform of $G(x)$ is given by:

<latex>M\left\{G(x);s\right\} = \int\_0^\infty t^{s-1} G(xt)dt</latex>

Since the Binet's formula can be rewritten as:

<latex>F\_n = \frac{1}{x^n}F(x^n)</latex>

The Mellin transform of the Binet's function $F(x)$ can be derived as:

<latex>M\left\{F(x);s\right\} = \int\_0^\infty \frac{t^{s-1}}{1-t}F(xt)dt</latex>

Now, let us focus on the connection between the Fibonacci sequence and the non-trivial zeroes of the Riemann Zeta function. Recall that the Riemann Zeta function has non-trivial zeroes in the critical strip, and they come in conjugate pairs.

If $\rho = \beta + i\gamma$ is a non-trivial zero of $\zeta(s)$, then $\overline{\rho} = \beta - i\gamma$ is also a zero of $\zeta(s)$. The functional equation for the Riemann Zeta function implies that the zeroes of $\zeta(s)$ and $\zeta(1-s)$ share the same real part and are conjugate to each other.

Let us assume that a non-trivial zero $\rho = \beta + i\gamma$ of the Riemann Zeta function lies within the critical strip. When $s$ approaches $\rho$ from the left, the Mellin transform of the generating function $G(x)$ can be written as:

<latex>M\left\{G(x);s\right\} = \int\_0^\infty \frac{t^{s-1}}{1-t}G(xt)dt</latex>

Since the Binet's formula relates the n-th Fibonacci number to the Binet's function $F(x)$, we can express the n-th Fibonacci number as:

<latex>F\_n = \frac{1}{x^n}F(x^n)</latex>

Thus, the Mellin transform of the generating function $G(x)$ can be rewritten as:

<latex>M\left\{G(x);s\right\} = \int\_0^\infty \frac{t^{s-1}}{1-t} \sum\_{n=0}^\infty \frac{1}{x^n}F(x^n)dt</latex>

Now, let us consider the behavior of the integral term in the Mellin transform. When $s$ approaches $\rho$ from the left, the integral term becomes:

<latex>I(s) = \int\_0^\infty \frac{t^{s-1}}{1-t} \sum\_{n=0}^\infty \frac{1}{x^n}F(x^n)dt</latex>

Since the Binet's function $F(x)$ has a singularity at $x=1$, the integral term $I(s)$ can be analyzed by considering the behavior of the integral term for $x$ approaching $1$ from the left.

When $x$ approaches $1$ from the left, the integral term $I(s)$ becomes:

<latex>I(s) = \int\_0^1 \frac{t^{s-1}}{1-t} \sum\_{n=0}^\infty \frac{1}{n^s}dt</latex>

The integral term $I(s)$ can be expressed in terms of the Riemann Zeta function as:

<latex>I(s) = \frac{1}{s-1}\int\_0^1 \frac{t^{s-1}}{1-t} \sum\_{n=0}^\infty \frac{1}{n^s}dt = \frac{1}{s-1}\int\_0^1 \frac{t^{s-1}}{1-t} \frac{1}{1-t}dt = \frac{1}{s-1}\int\_0^1 \frac{t^{s-1}}{t}dt = \frac{\pi}{s-1} \zeta(s)</latex>

This expression shows that the integral term $I(s)$ is related to the Riemann Zeta function. When $s$ approaches $\rho$ from the left, the integral term $I(s)$ becomes large, implying that the Mellin transform of the generating function $G(x)$ may develop a singularity at $\rho$.

In summary, the connection between the Fibonacci sequence and the non-trivial zeroes of the Riemann Zeta function arises from the behavior of the integral term in the Mellin transform of the generating function $G(x)$. When a non-trivial zero $\rho = \beta + i\gamma$ of the Riemann Zeta function lies within the critical strip, the integral term $I(s)$ becomes large when $s$ approaches $\rho$ from the left, potentially leading to a singularity in the Mellin transform of the Fibonacci sequence. ]]>

Created page, mainly to record/redirect the alternate name “elementary existential doctrine” and some references.

]]>added at *TC* some references on computing THH for cases like $ko$ and $tmf$, here

Added missing parenthesis

]]>brief `category:people`

-entry for hyperlinking references

added pointer to conference this month: here

]]>brief `category:people`

-entry for hyperlinking references

wrote an Idea-section at quantum field theory

]]>created [[supergravity]]

so far just an "Idea" section and a link to [[D'Auria-Fre formulation of supergravity]] (which i am busy working on)

]]>added to Stiefel-Whitney class briefly the definition/characterization.

]]>