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# Groups with exponentially growing centre

Given a finitely generated group $$G$$ and a finite subset $$S$$, let $$\omega_G(S) = \inf_{n\geq1} |S^n|^{1/n}$$ (this is the exponential growth rate of $$S$$).

1. Is there a finitely generated group $$G$$ with centre $$Z$$ and symmetric generating set $$S$$ such that $$\omega_{G/Z}(SZ/Z) < \omega_G(S)$$?

Here is my idea for a construction. Let $$H$$ be the discrete Heisenberg group, and let $$\varphi$$ be an automorphism of $$H$$ that acts on $$H^\text{ab} \cong \mathbf{Z}^2$$ as a matrix with eigenvalue $$\lambda > 1$$, and let $$G = H \rtimes_\varphi \mathbf{Z}$$. Let $$S$$ be the union of the standard generating sets for $$H$$ and $$\mathbf{Z}$$. Note that $$\varphi$$ preserves the centre of $$H$$, so $$Z(G) = Z(H) \cong \mathbf{Z}$$, and $$G/Z(G) \cong \mathbf{Z}^2 \rtimes_\varphi \mathbf{Z}$$. Since $$\lambda > 1$$ we have exponential growth in $$G/Z(G)$$, and my intuition is that there is enough extra space in $$H$$ that we should have even greater growth in $$G$$. The details elude me, however, and maybe somebody has a reference or a simpler example.

Assuming that the answer to question 1 is "yes", it follows that $$|S^n \cap Z|$$ grows exponentially. Indeed since $$S^n$$ is growing much faster than $$S^n Z/Z$$, the max fibre-size $$\max_g |S^n \cap gZ|$$ must grow exponentially, and $$|S^n \cap gZ| \leq |S^{2n} \cap Z|$$.

1. How fast can $$|S^n \cap Z|$$ grow compared to $$|S^n|$$? In other words, what is the supremum of $$\inf_{n\geq1} |S^n \cap Z|^{1/n} / \omega_G(S)$$ over all $$G$$ and $$S$$? Is it $$1$$?

At the moment I cannot even convince myself that you could not have even $$|S^n \cap Z| \gg |S^n|$$ for all $$n$$.

Finally, a technical question.

1. Do we always have $$|S^n \cap Z|^{1/n} \to \omega_G(S) / \omega_{GZ/Z}(SZ/Z)$$?

By the argument before quetion 2, the exponential growth rate of $$|S^n \cap Z|$$ is at least $$(\omega_G(S) / \omega_{G/Z}(SZ/Z))^{1/2}$$. It's plausible but not obvious that the largest fibre over $$Z$$ is $$S^n \cap Z$$ itself, so it's plausible that we can remove the square-root. The other inequality is more suspect, because we might have lots of small fibres.

• Probably. I'd try Hall's group $G(A)$ of matrices $$\begin{pmatrix}1 & x & z\\0 & t^n & y\\ 0 & 0 & 1\end{pmatrix}$$ with $x,y,z\in A$, $n\in\mathbf{Z}$; here $A$ is the ring generated by $t,t^{-1}$, which leaves several choices, for instance $A=\mathbf{F}_p[t^{\pm}]$ (central extension of the lamplighter) or $A=\mathbf{Z}[1/p]$, $t=p$ (related to Baumslag-Solitar group $\mathrm{BS}(1,p)$). – YCor Sep 11 at 9:18
• Yes, that's simpler. It's not unlike my suggestion, but with the advantage that by taking a different ring than $\mathbf{Z}$ you can take the automorphism $\varphi$ to be diagonal, which makes it easier on the brain. – Sean Eberhard Sep 11 at 9:32
• With $A = \mathbf{F}_p[t^{\pm1}]$ and the generating set consisting of $\varphi^{\pm1}$ times all matrices with entries of degree $0$, I think you have $|S^n Z/Z| \approx p^{2n}$, $|S^n| \approx p^{4n}$, $|S^n \cap Z| \approx p^{2n}$. – Sean Eberhard Sep 11 at 9:35
• My previous comment was hasty, and the situation is actually rather delicate. It turns out that for most points of $S^n$ the coordinate $z$ is determined by $x$ and $y$, basically because of the geometry of the lamplighter: for most points of $S^n$, the lamplighter made an injective walk, and if you only allow $\pm1$ steps then the walk must be unidirectional. I think you can get around this by allowing bigger steps or by walking in $\mathbf{Z}^2$ instead of $\mathbf{Z}$, but I'm still thinking about it... – Sean Eberhard Sep 12 at 13:55
• I have however managed to convince myself that $|S^n \cap Z|$ grows exponentially, so this probably answers question 3 negatively. – Sean Eberhard Sep 12 at 14:00

In fact $$\omega_{G/Z}(SZ/Z) = \omega_G(S)$$, despite the plausible-sounding examples in the comments. The argument is actually embarassingly simple. Let $$S_1$$ be a lift of $$SZ/Z$$ to $$G$$, so that $$|S_1| = |SZ/Z|$$ and $$S \subset S_1 Z_1$$ for some $$Z_1 \subset Z$$. Then $$S^n = S_1^n Z_1^n$$, and $$Z_1^n$$ grows polynomially, so $$\omega_G(S) \leq |S_1| = |SZ/Z|$$. Thus also $$\omega_G(S^n) \leq |S^nZ/Z|$$, but $$\omega_G(S^n) = \omega_G(S)^n$$ and $$|S^nZ/Z|^{1/n} \to \omega_{G/Z}(SZ/Z)$$.
The examples in the comments do however have the property that $$|S^n \cap Z|$$ grows exponentially, but this is not obvious. Let $$G$$ be Hall's group (as suggested by YCor) of matrices of the form $$\left( \begin{array}{ccc} 1 & x & z \\ 0 & t^k & y \\ 0 & 0 & 1 \end{array} \right),$$ where $$k \in \mathbf{Z}$$ and $$x,y,z\in A = \mathbf{F}_p[t^{\pm 1}]$$. Let $$H$$ be the subgroup defined by $$k=0$$ and $$x,y,z\in\mathbf{F}_p$$, let $$\varphi$$ be the diagonal matrix $$(1, t, 1)$$. Let $$S$$ be the symmetric generating set $$H \varphi^{\pm1} H$$. Modulo $$Z$$, this group is just the lamplighter group $$\mathbf{Z}^2 \wr \mathbf{Z}$$, and the generating set $$S$$ represents "change a lamp, take a step, change a lamp". The $$(x, y)$$ coordinates therefore record the lamp configuration at the end of the walk, and the $$z$$ coodinate records second-order information. To be precise, we have $$S^n = \bigcup H^{\varphi^{i_0}} H^{\varphi^{i_1}} \cdots H^{\varphi^{i_n}} \varphi^{-i_n},$$ where the union is over all $$n$$-step walks $$i_0, \dots, i_n$$ in $$\mathbf{Z}$$, by which I mean $$i_0=0$$ and $$|i_t - i_{t-1}| = 1$$ for each $$t > 0$$. We have $$H^{\varphi^{i_0}} H^{\varphi^{i_1}} \cdots H^{\varphi^{i_n}} = \left\{\left( \begin{array}{ccc} 1 & x_0 t^{i_0} + \cdots + x_n t^{i_n} & \sum_{s < t} x_s y_t t^{i_s - i_t} + z_0 + \cdots + z_n \\ 0 & 1 & y_0 t^{-i_0} + \cdots + y_n t^{-i_n} \\ 0 & 0 & 1 \end{array} \right) \colon x_i, y_j, z_k \in \mathbf{F}_p \right\}.$$ Thus the $$z$$ coordinate counts differences $$i_s - i_t$$ between lamp positions where the first lamp was lit before the second lamp. Now consider a walk of the following form: first light an $$x$$ lamp, then walk around lighting $$y$$ lamps however you like, then return to the origin and extinguish the $$x$$ lamp, then go extinguish all the $$y$$ lamps, and finally return to the origin. If you do this then your final $$x$$ and $$y$$ coordinates are zero, since they just record the final lamp configuration, but your $$z$$ coordinate records the configuration of $$y$$ lamps between the two $$x$$ lightings. This argument proves that $$|S^n \cap Z|$$ grows at least as fast as $$p^{n/4}$$.


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# How to verify a number encrypted with an unknown key

Are there any crypto-constructions to do the following kind of "zero-knowledge proof":

1. Alice encrypts a set of numbers using some cryptosystem and a secret key (e.g. Paillier crypto-system).
2. Bob multiplies the ciphertexts to obtain the sum of numbers.
3. Bob sends to Alice the ciphertext for the sum. Alice decrypts the sum with her secret key and then encrypts the sum with a key known by Bob.
4. Bob receives the sum encrypted with his key and decrypts it successfully.

Question: how can Bob verify that Alice didn't change the sum to a different number before re-encrypting?

-
 If Bob already knows the sum at step 2, what's the point of steps 3 and 4? – David Schwartz Mar 18 at 10:51 @DavidSchwartz Bob doesn't know. The Paillier crypto-sytem is such that by multiplying the ciphertexts you obtain the sum of the plaintexts in encrypted form. – Eugen Mar 18 at 10:58 So the point of this scheme is to communicate to Bob the sum of the plaintexts? In that case, it's pretty simple. The ciphertext in step one can include an HMAC of the sum and the plaintext in step 4 can include the key to the HMAC. In other words, instead of giving Bob the actual length, give Bob what he needs to confirm the length as encoded in the original message. – David Schwartz Mar 18 at 11:02 @Eugen: $\:$ Should 2 say "Bob multiplies the ciphertexts to ..."? $\;\;$ – Ricky Demer Mar 18 at 11:38 @RickyDemer You are right. – Eugen Mar 18 at 12:32

You probably don't need to re-encrypt using the Paillier crypto system.

1) Alice encrypts $c_1=g^{m_1} r_1^n$ und $c_2=g^{m_2} r_2^n$ and computes $r_3=r_1 \cdot r_2$ and $m_3=m_1+m_2$, then sends $c_1$, $c_2$, $m_3$ and $r_3$ to Bob

2) Bob computes $c_3=c_1 \cdot c_2=g^{m_3} r_3^n$ - If the homomorphically computed sum matches the re-encryption Bob will believe that $c_3$ encrypts $m_3$

The basic idea is that $g^{m_1} r_1^n \cdot g^{m_2} r_2^n = g^{m_1+m_2} (r_1 \cdot r_2)^n$

All cipher text operations are executed modulo $n^2$

This proof is backed up by Paillier's original paper. Unfortunately the proof is not absolutely Zero Knowledge as the sum may give you information about the possible summands. However you may choose large similarly sized primes for the $r$s so an attacker would need an efficient solution for prime factorization (see other question here).

-

Yes. This can be solved through standard methods. Alice can prove she decrypted the ciphertext correctly by revealing the decrypted message and the random coins that would be used in encryption to obtain this ciphertext from this message.

Suppose we have a ciphertext $c$, and Alice decrypts it to obtain the message $m$. It follows that $c = g^m r^n \bmod n^2$, where Alice can recover both $m$ and $r$. Alice can send $m$ and $r$ to Bob over a secure channel. Now this information is sufficient that Bob can check that these values satisfy the relationship $c = g^m r^n \bmod n^2$. If they do, this is a proof that Alice correctly decrypted $c$ and $m$ is the correct decryption of $c$.

Revealing $r$ does not endanger secrecy. This is not a zero-knowledge proof, but the knowledge revealed seems inessential.

If you wanted a zero-knowledge proof, Alice could prove she decrypted the ciphertext correctly using a zero-knowledge proof of knowledge of a $n$th root of $c/g^m \bmod n^2$. However, this seems unnecessary.

-
 Do you have a source for the nth root ZKP? – Thomas Lieven Mar 19 at 7:31

format: plain


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# Thread: How to prove that something is an orthonormal basis of L2(N)?

1. ## How to prove that something is an orthonormal basis of L2(N)?

Hi

I am trying to prove the following, but I am getting absolutely nowhere:

{δ_k }_(k=1)^∞ is an orthonormal basis for L^2 (N),where δ_k is the sequence in L^2 (N) whose kth entry is 1.

[/FONT]

2. Originally Posted by Ase
Hi

I am trying to prove the following, but I am getting absolutely nowhere:

{δ_k }_(k=1)^∞ is an orthonormal basis for L^2 (N),where δ_k is the sequence in L^2 (N) whose kth entry is 1.

[/FONT]
and all other entries are 0.

The inner product defined on $L^2(N)$, the space of square summable sequences (also called $l^2$), is $<{a_n},{b_n}>= \sum a_nb_n$ so you need to show three things:

1) That $\{\delta_k\}_{k=1}^\infty$ is a basis for $L^2(N)$

2) That $<\delta_k, \delta_k>= 1$ for all k

3) That $<\delta_k, \delta_j>= 0$ if $k\ne j$

3. $u(i).u(j)=0$ if $i<>j$
$u(i).u(j)=1$ if $i=j$
$||u(i)||=1$
$x.u(i)=x(i)$
$
x.u(i)=0 => x=0$

GTM 233 page 332

4. Thanks a lot HallsofIvy. I will get at it right away and hopefully crack it.

5. ## Basis

Hey,

The sequence

$\{\delta_{k}\}_{k=1}^{\infty}=(0,0,0,\ldots,1,0,0, \ldots)$

is also a basis for the normed vector space $l^{1}(\mathbb{N})$.

Couldn't you show that for any $u \in l^{1}(\mathbb{N})$ there exists unique scalars $c_{k} \in \mathbb{C}$ such that

$u=\sum_{k=1}^{\infty} c_{k}\delta_{k}$ ?


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Definition:Line

Definition

In the words of Euclid:

This can be interpreted to mean that a line is a construct that has no thickness.

This mathematical abstraction can not of course be actualised in reality because however thin you make your line, it will have some finite width.

It can be considered as a continuous succession of points.

The word line is frequently used to mean infinite straight line. The context ought to make it clear if this is the case.

Straight Line

In the words of Euclid:

A straight line is a line which lies evenly with the points on itself.

Curve

A curve is a line which may or may not be straight.

Line Segment

A line segment is any line (straight or not) which terminates at two points.

Straight Line Segment

A straight line segment is a line segment which is straight.

In the words of Euclid:

A straight line segment can be drawn joining any two points.

Endpoint

Each of the points at either end of a line segment is called an endpoint of that line segment.

Similarly, the point at which an infinite half-line terminates is called the endpoint of that line.

In the words of Euclid:

The extremities of a line are points.

Midpoint

Let $L = AB$ be a line segment whose endpoints are $A$ and $B$.

Let $M$ be a point on $L$ such that the line segment $AM$ is equal to the line segment $MB$.

That is, let $M$ be the bisector of $L$.

Then $M$ is the midpoint of $L$.

Infinite Line

An infinite line is a line which has no endpoints.

Infinite Half-Line

An infinite half-line is a line which terminates at an endpoint at one end, but has no such endpoint at the other.

Infinite Straight Line

An infinite straight line is a straight line which has no endpoints, or equally, a straight line which is infinite.

Also see

In the words of Euclid:

The extremities of a surface are lines.

• Results about lines can be found here.


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Exporting UV coordinates by face elements

I'm trying to write my own export script, and I'm not sure how to reference the uv coordinates as an element array, similar to how .obj format works (ie. you have a list of all uv coordinates "vt s t" and each face selects the correct index). I looked at previous threads on similar things, but there doesn't seem to be anything similar to my issue. Do I need to hash all uv coordinates and compute the index for each face myself? Or is there a built-in way to do this? How does the .obj exporter work? Also, for further reference, is it possible to see the python source for the .obj exporter?

• It would be nice if you could post your edit as an answer. – stacker Mar 22 '14 at 10:28
• @stacker And done! – spearmunkie Mar 22 '14 at 10:40

I found the .obj export source in /usr/share/blender It's probably different on your computer, the best place to see where you can find it is to go into user preference under Addons. It seems like the official exporter just creates a hash map of uv coordinates. This seems to be where the magic happens.

# Make our own list so it can be sorted to reduce context switching
face_index_pairs = [(face, index) for index, face in enumerate(me.polygons)]
uv = f_index = uv_index = uv_key = uv_val = uv_ls = None

uv_face_mapping = [None] * len(face_index_pairs)

uv_dict = {}
uv_get = uv_dict.get
for f, f_index in face_index_pairs:
uv_ls = uv_face_mapping[f_index] = []
for uv_index, l_index in enumerate(f.loop_indices):
uv = uv_layer[l_index].uv
uv_key = veckey2d(uv)
uv_val = uv_get(uv_key)
if uv_val is None:
uv_val = uv_dict[uv_key] = uv_unique_count
fw('vt %.6f %.6f\n' % uv[:])
uv_unique_count += 1
uv_ls.append(uv_val)

Since I answered my own question, should this be deleted? I feel like this has some value to other people who may want to do something similar, as this is a specific and non-trivial task.

• Answering your own questions is fine, you even have the option to answer as you post the question, it is a way to share your knowledge with others. – sambler Mar 23 '14 at 1:48


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# Intuition Behind Scale Invariance Power Spectrum

1. May 22, 2015

### center o bass

In the book "Statistical physics for cosmic structures" at p. 171 a read a definition of scale invariance (leading to the so called scale invariant power spectrum) given as the requirement that $\sigma^2_M(R=R_H(t)) = constant$, where $R_H(t)$ is the horizon, i.e. the maximal distance that light could have traveled in cosmological time $t$.

In other words, the normalised mass variance over a sphere of radius the horizon distance should be independent of time. So if we computed the mass variance at some time $t_0$ when the horizon was $R_H(t_0)$ this should be the same as if we computed it at any other $t_1$ when the horizon was $R_H(t_1)$ even though $R_H(t_0)$ might be much smaller than $R_H(t_1)$.

I am trying to get intuition for why one would believe such a requirement to be true. Does anyone have some enlightening explanations/insights?

2. May 22, 2015

### bapowell

Scale invariance is only true for de Sitter expansion in which $\dot{\rho} = 0$. The background is steady state, so there are no time-dependent dynamics that would impart a scale dependence on the perturbations.


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# Two Period Kyle (1985) Model

1. Motivation This post shows how to solve for the equilibrium price impact and demand coefficients in a $2$ period Kyle (1985)-type model where informed traders see a noisy signal about the fundamental value of a single asset. There are various … [Continue reading]

# Fano’s Inequality and Resource Allocation

1. Motivation This post describes Fano's inequality. It's not a particularly complicated result. After all, it first shows up on page 33 of Cover and Thomas (1991). However, I recently ran across the result again for the first time in a while, and … [Continue reading]

# Wavelet Variance

1. Motivation Imagine you're a trader who's about to put on a position for the next month. You want to hedge away the risk in this position associated with daily fluctuations in market returns. One way that you might do this would be to short the … [Continue reading]

# WSJ Article Subject Tags

1. Motivation This post investigates the distribution of subject tags for Wall Street Journal articles that mention S&P 500 companies. e.g., a December 2009 article entitled, When Even Your Phone Tells You You're Drunk, It's Time to Call a … [Continue reading]

# Randomized Market Trials

1. Motivation How much can traders learn from past price signals? It depends on what kind of assets sell. Suppose that returns are (in part) a function of $K = \Vert {\boldsymbol \alpha} \Vert_{\ell_0}$ different feature-specific … [Continue reading]


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# Our picks for first quarter 2010 MIP (most important QG paper)

## Which papers (see titles below) will prove most valuable to future research?

46.2%

7.7%

0 vote(s)
0.0%

7.7%

15.4%

0 vote(s)
0.0%

15.4%

15.4%

0 vote(s)
0.0%

23.1%

7.7%

0 vote(s)
0.0%

7.7%

7.7%
15. ### Verlinde

38.5%
1. Mar 30, 2010

### marcus

Of these fifteen candidates, please indicate the paper or papers which you think will contribute most significantly to future quantum gravity research. Multiple choice is possible in the poll, so please select several if you wish. Newest papers are listed first. In some cases, a pair of related papers has been combined to make one choice.

Surface Density of Spacetime Degrees of Freedom from Equipartition Law in theories of Gravity
http://arxiv.org/abs/1003.5665

Bonder Sudarsky
Searching for spacetime granularity: analyzing a concrete experimental setup
http://arxiv.org/abs/1003.5245

Afshordi
Dark Energy, Black Hole Entropy, and the First Precision Measurement in Quantum Gravity
http://arxiv.org/abs/1003.4811

Mielczarek Cailleteau Grain Barrau
Inflation in loop quantum cosmology: dynamics and spectrum of gravitational waves
http://arxiv.org/abs/1003.4660

Bianchi Rovelli Vidotto
Towards Spinfoam Cosmology
http://arxiv.org/abs/1003.3483

Durka Kowalski-Glikman
Hamiltonian analysis of SO(4,1) constrained BF theory
http://arxiv.org/abs/1003.2412

Modesto Randono
Entropic corrections to Newton's law
http://arxiv.org/abs/1003.1998

Barrett Dowdall Fairbairn Gomes Hellmann Pereira
Asymptotics of 4d spin foam models
http://arxiv.org/abs/1003.1886

Hossenfelder
Comments on and Comments on Comments on Verlinde's paper "On the Origin of Gravity and the Laws of Newton"
http://arxiv.org/abs/1003.1015

Easson Frampton Smoot
Entropic Accelerating Universe
http://arxiv.org/abs/1002.4278
and followup:
Entropic Inflation
http://arxiv.org/abs/1003.1528

A spin foam model for general Lorentzian 4-geometries
http://arxiv.org/abs/1002.1959
and followup:
Spin foams with timelike surfaces
http://arxiv.org/abs/1003.5652

Ashtekar Campiglia Henderson
Casting Loop Quantum Cosmology in the Spin Foam Paradigm
http://arxiv.org/abs/1001.5147

Smolin
Newtonian gravity in loop quantum gravity
http://arxiv.org/abs/1001.3668

Freidel Speziale
Twisted geometries: A geometric parametrisation of SU(2) phase space
http://arxiv.org/abs/1001.2748

Verlinde
On the Origin of Gravity and the Laws of Newton
http://arxiv.org/abs/1001.0785

Last edited: Mar 30, 2010
2. Mar 30, 2010

### marcus

Since multiple choice is possible in the poll, you may wish to check off your top 5. There were a lot of potentially important papers this quarter making it difficult to narrow down to a short list. I'll fetch the abstracts, as a reminder of what each paper is about. BTW many of these were spotted by other PF members and called to our attention either as an item added to the bibliography or in a discussion thread. Special thanks to MTd2, Francesca, and Atyy for keeping us alert to new research output.

Surface Density of Spacetime Degrees of Freedom from Equipartition Law in theories of Gravity
http://arxiv.org/abs/1003.5665
20 pages
(Submitted on 29 Mar 2010)
"I show that the principle of equipartition, applied to area elements of a surface which are in equilibrium at the local Davies-Unruh temperature, allows one to determine the surface number density of the microscopic spacetime degrees of freedom in any diffeomorphism invariant theory of gravity. The entropy associated with these degrees of freedom matches with the Wald entropy for the theory. This result also allows one to attribute an entropy density to the spacetime in a natural manner. The field equations of the theory can then be obtained by extremising this entropy. Moreover, when the microscopic degrees of freedom are in local thermal equilibrium, the spacetime entropy of a bulk region resides on its boundary."

Bonder Sudarsky
Searching for spacetime granularity: analyzing a concrete experimental setup
http://arxiv.org/abs/1003.5245
9 pages
"In this work we show that the spin pendulum techniques developed by the Eöt-Wash group could be used to put very stringent bounds on the free parameters of a Lorentz invariant phenomenological model of quantum gravity. The model is briefly described as well as the experimental setup that we have in mind."

Afshordi
Dark Energy, Black Hole Entropy, and the First Precision Measurement in Quantum Gravity
http://arxiv.org/abs/1003.4811
4 pages
(Submitted on 25 Mar 2010)
"The two apparently distinct phenomena of dark energy (or late-time cosmic acceleration) and quantum gravity dominate physics on extremely low, and extremely high energies, but do not seem to have any apparent empirical connection. Nevertheless, the two have a theoretical connection, through the cosmological constant problem. I argue that the finite temperature quantum gravitational corrections to black hole entropy yields a pressure for the gravitational vacuum (or gravitational aether). Assuming that the relative corrections are linear in horizon temperature (i.e. are suppressed by one power of Planck energy), the pressure is comparable to that of dark energy for astrophysical black holes. This implies that the observation of late-time cosmic acceleration may have provided us with the first precision measurement of quantum gravity, i.e. that of black hole entropy."

Mielczarek Cailleteau Grain Barrau
Inflation in loop quantum cosmology: dynamics and spectrum of gravitational waves
http://arxiv.org/abs/1003.4660
11 pages, 14 figures
(Submitted on 24 Mar 2010)
"Loop quantum cosmology provides an efficient framework to study the evolution of the Universe beyond the classical Big Bang paradigm. Due to holonomy corrections, the singularity is replaced by a "bounce". The dynamics of the background is investigated into the details, as a function of the parameters of the model. In particular, the conditions required for inflation to occur are carefully considered and are shown to be generically met. The propagation of gravitational waves is then investigated in this framework. By both numerical and analytical approaches, the primordial tensor power spectrum is computed for a wide range of parameters. Several interesting features could be observationally probed."

Bianchi Rovelli Vidotto
Towards Spinfoam Cosmology
http://arxiv.org/abs/1003.3483
8 pages
"We compute the transition amplitude between coherent quantum-states of geometry peaked on homogeneous isotropic metrics. We use the holomorphic representations of loop quantum gravity and the Kaminski-Kisielowski-Lewandowski generalization of the new vertex, and work at first order in the vertex expansion, second order in the graph (multipole) expansion, and first order in 1/volume. We show that the resulting amplitude is in the kernel of a differential operator whose classical limit is the canonical hamiltonian of a Friedmann-Robertson-Walker cosmology. This result is an indication that the dynamics of loop quantum gravity defined by the new vertex yields the Friedmann equation in the appropriate limit."

Durka Kowalski-Glikman
Hamiltonian analysis of SO(4,1) constrained BF theory
http://arxiv.org/abs/1003.2412
9 pages
"In this paper we discuss canonical analysis of SO(4,1) constrained BF theory. The action of this theory contains topological terms appended by a term that breaks the gauge symmetry down to the Lorentz subgroup SO(3,1). The equations of motion of this theory turn out to be the vacuum Einstein equations. By solving the B field equations one finds that the action of this theory contains not only the standard Einstein-Cartan term, but also the Holst term proportional to the inverse of the Immirzi parameter, as well as a combination of topological invariants. We show that the structure of the constraints of a SO(4,1) constrained BF theory is exactly that of gravity in Holst formulation. We also briefly discuss quantization of the theory."

Modesto Randono
Entropic corrections to Newton's law
http://arxiv.org/abs/1003.1998
7 pages, 2 figures
"It has been known for some time that there is a deep connection between thermodynamics and gravity, with perhaps the most dramatic implication that the Einstein equations can be viewed as a thermodynamic equation of state. Recently Verlinde has proposed a model for gravity with a simple statistical mechanical interpretation that is applicable in the non-relatvistic regime. After critically analyzing the construction, we present a strong consistency check of the model. Specifically, we consider two well-motivated corrections to the area-entropy relation, the log correction and the volume correction, and follow Verlinde's construction to derive corrections to Newton's law of gravitation. We show that the deviations from Newton's law stemming from the log correction have the same form as the lowest order quantum effects of perturbative quantum gravity, and the deviations stemming from the volume correction have the same form as some modified Newtonian gravity models designed to explain the anomalous galactic rotation curves."

Barrett Dowdall Fairbairn Gomes Hellmann Pereira
Asymptotics of 4d spin foam models
http://arxiv.org/abs/1003.1886
10 pages
"We study the asymptotic properties of four-simplex amplitudes for various four-dimensional spin foam models. We investigate the semi-classical limit of the Ooguri, Euclidean and Lorentzian EPRL models using coherent states for the boundary data. For some classes of geometrical boundary data, the asymptotic formulae are given, in all three cases, by simple functions of the Regge action for the four-simplex geometry."

Hossenfelder
Comments on and Comments on Comments on Verlinde's paper "On the Origin of Gravity and the Laws of Newton"
http://arxiv.org/abs/1003.1015
10 pages
"We offer some, hopefully clarifying, comments on Verlinde's recent claim that gravity is an entropic force. A suitable identification of quantities shows that both formulations of Newtonian gravity, the classical and the thermodynamical one, are actually equivalent. It turns out that some additional assumptions made by Verlinde are unnecessary. However, when it comes to General Relativity there remain some gaps in the argument. We comment on whether this identification can be done also for electrostatics. Finally, some thoughts on the use of this reinterpretation are offered."

Easson Frampton Smoot
Entropic Accelerating Universe
http://arxiv.org/abs/1002.4278
10 pages, 1 figure
"To accommodate the observed accelerated expansion of the universe, one popular idea is to invoke a driving term in the Friedmann-Lemaître equation of dark energy which must then comprise 70% of the present cosmological energy density. We propose an alternative interpretation which takes into account the temperature intrinsic to the information holographically stored on the screen which is the surface of the universe. Dark energy is thereby obviated and the acceleration is due to an entropic force naturally arising from the information storage on a surface screen. We consider an additional quantitative approach based upon the entropy and surface terms usually neglected in General Relativity and show that this leads to the entropic accelerating universe."
and followup:
Entropic Inflation
http://arxiv.org/abs/1003.1528
14 pages
"One of the major pillars of modern cosmology is a period of accelerating expansion in the early universe. This accelerating expansion, or inflation, must be sustained for at least 30 e-foldings. One mechanism, used to drive the acceleration, is the addition of a new energy field, called the Inflaton; often this is a scalar field. We propose an alternative mechanism which, like our approach to explain the late-time accelerating universe, uses the entropy and temperature intrinsic to information holographically stored on a screen enclosing the observed space. The acceleration is due in both cases to an emergent entropic force, naturally arising from the information storage on the horizon."

A spin foam model for general Lorentzian 4-geometries
http://arxiv.org/abs/1002.1959
27 pages, 1 figure
"We derive simplicity constraints for the quantization of general Lorentzian 4-geometries. Our method is based on the correspondence between coherent states and classical bivectors and the minimization of associated uncertainties. For spacelike geometries, this scheme agrees with the master constraint method of the model by Engle, Pereira, Rovelli and Livine (EPRL). When it is applied to general Lorentzian geometries, we obtain new constraints that include the EPRL constraints as a special case. They imply a discrete area spectrum for both spacelike and timelike surfaces. We use these constraints to define a spin foam model for general Lorentzian 4-geometries. "
and followup:
Spin foams with timelike surfaces
http://arxiv.org/abs/1003.5652
22 pages
"Spin foams of 4d gravity were recently extended from complexes with purely spacelike surfaces to complexes that also contain timelike surfaces. In this article, we express the associated partition function in terms of vertex amplitudes and integrals over coherent states. The coherent states are characterized by unit 3-vectors which represent normals to surfaces and lie either in the 2--sphere or the 2d hyperboloids. In the case of timelike surfaces, a new type of coherent state is used and the associated completeness relation is derived. It is also shown that the quantum simplicity constraints can be deduced by three different methods: by weak imposition of the constraints, by restriction of coherent state bases and by the master constraint."

Ashtekar Campiglia Henderson
Casting Loop Quantum Cosmology in the Spin Foam Paradigm
http://arxiv.org/abs/1001.5147
36 pages
"The goal of spin foam models is to provide a viable path integral formulation of quantum gravity. Because of background independence, their underlying framework has certain novel features that are not shared by path integral formulations of familiar field theories in Minkowski space. As a simple viability test, these features were recently examined through the lens of loop quantum cosmology (LQC). Results of that analysis, reported in a brief communication [1], turned out to provide concrete arguments in support of the spin foam paradigm. We now present detailed proofs of those results. Since the quantum theory of LQC models is well understood, this analysis also serves to shed new light on some long standing issues in the spin foam and group field theory literature. In particular, it suggests an intriguing possibility for addressing the question of why the cosmological constant is positive and small."

Smolin
Newtonian gravity in loop quantum gravity
http://arxiv.org/abs/1001.3668
16 pages
"We apply a recent argument of Verlinde to loop quantum gravity, to conclude that Newton's law of gravity emerges in an appropriate limit and setting. This is possible because the relationship between area and entropy is realized in loop quantum gravity when boundaries are imposed on a quantum spacetime."

Freidel Speziale
Twisted geometries: A geometric parametrisation of SU(2) phase space
http://arxiv.org/abs/1001.2748
28 pages
"A cornerstone of the loop quantum gravity program is the fact that the phase space of general relativity on a fixed graph can be described by a product of SU(2) cotangent bundles per edge. In this paper we show how to parametrize this phase space in terms of quantities describing the intrinsic and extrinsic geometry of the triangulation dual to the graph. These are defined by the assignment to each triangle of its area, the two unit normals as seen from the two polyhedra sharing it, and an additional angle related to the extrinsic curvature. These quantities do not define a Regge geometry, since they include extrinsic data, but a looser notion of discrete geometry which is twisted in the sense that it is locally well-defined, but the local patches lack a consistent gluing among each other. We give the Poisson brackets among the new variables, and exhibit a symplectomorphism which maps them into the Poisson brackets of loop gravity. The new parametrization has the advantage of a simple description of the gauge-invariant reduced phase space, which is given by a product of phase spaces associated to edges and vertices, and it also provides an abelianisation of the SU(2) connection. The results are relevant for the construction of coherent states, and as a byproduct, contribute to clarify the connection between loop gravity and its subset corresponding to Regge geometries."

Verlinde
On the Origin of Gravity and the Laws of Newton
http://arxiv.org/abs/1001.0785
29 pages, 6 figures
"Starting from first principles and general assumptions Newton's law of gravitation is shown to arise naturally and unavoidably in a theory in which space is emergent through a holographic scenario. Gravity is explained as an entropic force caused by changes in the information associated with the positions of material bodies. A relativistic generalization of the presented arguments directly leads to the Einstein equations. When space is emergent even Newton's law of inertia needs to be explained. The equivalence principle leads us to conclude that it is actually this law of inertia whose origin is entropic."

Last edited: Mar 30, 2010
3. Mar 31, 2010

### Demystifier

The poll question reads - which papers will prove most valuable to future research?
My question reads - what does it mean to be "the most valuable"?

If valuableness is measured by the number of citations, then there is almost no doubt that it is the Verlinde's paper. That's why I voted for Verlinde. But obviously, some voters here have a different definition of " valuableness".

4. Mar 31, 2010

### marcus

I believe the value of fundamental research is something decided collectively by mankind over the long haul. Citation counts are helpful, but they are not a perfect measure.

How do you value understanding the various aspects of nature and the universe?
I suppose for each of us this rests on subjective grounds. However, citation counts, if you watch them long enough (for enough years), are one possible indicator. Also the scientific community itself ultimately sorts these things out and rates relative merit in various never-quite-consistent ways. That too finally rests on individual subjective judgements. What it comes down to is no automatic criterion, no monolithic correct "definition" of value. But if you Demy, personally like the citation count definition, that's fine! Let's look at things in those terms, for the moment.

Demy, you seem confident that Verlinde's paper will be far more cited than, for example, Padmanabhan's. But maybe Padmanabhan's is deeper and more enlightening---couldn't it turn out to be more cited in the long run?
Verlinde has made a splash, but has he said anything new?

Last edited: Mar 31, 2010
5. Mar 31, 2010

### atyy

In spacetime there is a preferred tensor we call the metric. On the value of papers manifold maybe there isn't - perhaps Hossenfelder's bimetric theory is right :tongue:

6. Mar 31, 2010

### marcus

I found your choice intriguing, Atyy. Modesto and Randono have a surprising nontrivial message. Maybe Verlinde was assuming too simple an area-entropy relation. Is S = A/4 exactly? If not, there should be corrections to Newton's law! And Modesto Randono point to two well-motivated corrections to S = A/4.

One of them, the log A correction, I've seen arising in LQG context. Several papers on this by Corichi and his group, including some numerical work. The socalled volume correction is a term in A3/2. They cite several QG papers for that.

So it's interesting---I agree with your choice, in effect. (BTW you were the first respondent, I was a close second.) Suppose the law is actually NOT as simple as S = A/4, suppose it contains other terms in A. And suppose these people like Verlinde Jacobson Padmanabhan are right and Newton's law derives from the [correct] area-entropy relation. Then we have a consistency check. The correct Newton law should show modifications due to correction in the area-entropy equation.

Last edited: Mar 31, 2010
7. Mar 31, 2010

### MTd2

This time I only voted for Padmanabhan. It seems this time a contestant hold his paper until the last moments just to get my vote, I mean, to take profit from my hype.

The entropic relation he found is very closed to Boussos holographic bound, so I was very impressed by that! Id vote for Verlinde if he also updated his paper with the contents of his blog posts, but he didnt.

Last edited: Mar 31, 2010
8. Mar 31, 2010

### atyy

Some things whose links I wonder about - the log corrections in mentioned by Modesto and Randono and the corrections to the boundary law eg. in http://arxiv.org/abs/1003.2434 by Brian Swingle, who has worked with Wen. Some of Wen's stuff is related to a recent paper by Xu and Horava http://arxiv.org/abs/1003.0009, while other bits of it http://arxiv.org/abs/0809.2821 are related to LQG http://arxiv.org/abs/0907.2994 .

Last edited by a moderator: Apr 24, 2017
9. Apr 1, 2010

### Demystifier

I agree that the Padmanabhan's paper is deeper than the Verlinde's one. Still, I am confident that the Verlinde's paper will be far more cited even in the long run. Simply, that's how the scientific society works. Usually, the most cited papers on a given subject are the first ones, not the best ones. I can give a number of examples of that type, and I'm sure that you can give them too.

So, which paper is more valuable? One can say that more valuable paper is the one that triggered, influenced and motivated more FURTHER research. Sometimes, it is more important (and more difficult) to identify the problem than to solve it. With such a view, it becomes justified to say that the Verlinde's paper is more valuable. But of course, values are subjective things, so one is allowed to have his own measure of valuedness.

10. Jul 2, 2010

### marcus

Getting fifteen candidates for the 2nd quarter poll.

Chamseddine Connes
Noncommutative Geometry as a Framework for Unification of all Fundamental Interactions including Gravity. Part I
http://arxiv.org/abs/1004.0464
http://arxiv.org/cits/1004.0464

Rovelli
A new look at loop quantum gravity
http://arxiv.org/abs/1004.1780
http://arxiv.org/cits/1004.1780

Bianchi Magliaro Perini
Spinfoams in the holomorphic representation
http://arxiv.org/abs/1004.4550
http://arxiv.org/cits/1004.4550

Lisi Smolin Speziale
Unification of gravity, gauge fields, and Higgs bosons
http://arxiv.org/abs/1004.4866
http://arxiv.org/cits/1004.4866

Alesci Rovelli
A regularization of the hamiltonian constraint compatible with the spinfoam dynamics
http://arxiv.org/abs/1005.0817
http://arxiv.org/cits/1005.0817

Denicola Marcolli al-Yasry
Spin Foams and Noncommutative Geometry
http://arxiv.org/abs/1005.1057
http://arxiv.org/cits/1005.1057

Mercuri Randono
The Immirzi Parameter as an Instanton Angle
http://arxiv.org/abs/1005.1291
http://arxiv.org/cits/1005.1291

Randono
Gravity from a fermionic condensate of a gauge theory
http://arxiv.org/abs/1005.1294
http://arxiv.org/cits/1005.1294

Freidel Livine
U(N) Coherent States for Loop Quantum Gravity
http://arxiv.org/abs/1005.2090
http://arxiv.org/cits/1005.2090

Rovelli Smerlak
Thermal time and the Tolman-Ehrenfest effect: temperature as the "speed of time"
http://arxiv.org/abs/1005.2985
http://arxiv.org/cits/1005.2985

Bonanno Contillo Percacci
Inflationary solutions in asymptotically safe f(R) gravity
http://arxiv.org/abs/1006.0192
http://arxiv.org/cits/1006.0192

Freidel Speziale
From twistors to twisted geometries
http://arxiv.org/abs/1006.0199
http://arxiv.org/cits/1006.0199

Borja Diaz-Polo Garay Livine
Dynamics for a 2-vertex Quantum Gravity Model
http://arxiv.org/abs/1006.2451
http://arxiv.org/cits/1006.2451

Dittrich Ryan
Simplicity in simplicial phase space
http://arxiv.org/abs/1006.4295
http://arxiv.org/cits/1006.4295

Lisi
An Explicit Embedding of Gravity and the Standard Model in E8
http://arxiv.org/abs/1006.4908
http://arxiv.org/cits/1006.4908

=========================

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Last edited: Jul 2, 2010
11. Jul 2, 2010

### BenTheMan

12. Jul 4, 2010

### vacuumcell

I think its worth posting why this paper is so important, which is that in it the authors prove that "in ten dimensions, all supersymmetric theories of gravity without known inconsistencies are realized in string theory."

13. Jul 5, 2010

### BenTheMan

He's also almost completed the proof in six dimensions, I think. He's also working on the proof in six dimensions, which is an important step to the four dimensional proof, which I think we'll see in the next few years. This is, in my opinion, one of the most interesting results in quantum gravity this year---the fact that no consistent theories of nature in ten dimensions can be constructed that don't come from string theory is pretty remarkable. The fact that marcus fails to list such an important result in his most influential quantum gravity papers of XXXX'' should be an indication to all that he is not interested in the state of science as a whole, but rather in pimping his own pet ideas (which is ironic, because he doesn't have any of his own).

14. Jul 5, 2010

### MTd2

Interesting:

"In the physically-interesting case of 4 dimensions, the absence of gravitational anomalies means that the anomaly-cancellation does not impose much of a constraint at all on the possible gauge groups and matter content. It’s not clear at all why the 6D case (even if it were true that all anomaly-free 6D supersymmetric theories could be UV-completed in string theory) would be a good guide to 4D."

http://golem.ph.utexas.edu/~distler/blog/archives/001989.html

The rest of the post is rather insightful.

And for such a field with a huge number of researchers, like string theory, 3 citations in one year, in which none of them were citations by any other than the authors and associates, clearly shows how people are hopeless with their paper:

http://www.slac.stanford.edu/spires/find/hep?c=ARXIV:0906.0987 [Broken]

Last edited by a moderator: May 4, 2017
15. Jul 5, 2010

### marcus

This thread is about FIRST quarter papers in (non-string) QG. We've been doing this four times a year for I don't remember how long, over three years I guess. We compare research of every kind that is basically 4D---no extra dimensions---and treats gravity essentially as the geometry of 4D spacetime (the way General Relativity does.)

It's always interesting to look at the citation counts, to see how our picks have been doing by that (imperfect but useful) measure. However, especially after only a few months the citation count can't be taken as indicating the longterm importance or value of a paper. It is understood that always has a subjective component in any case.

That said, I'll put in the links to make it easy to see who and how many other researchers have cited each of these papers from the first quarter. In part this is simply for my own convenience. I may be the only one who wants to keep track of these papers from earlier in the year.

Surface Density of Spacetime Degrees of Freedom from Equipartition Law in theories of Gravity
http://arxiv.org/abs/1003.5665
http://arxiv.org/cits/1003.5665

Bonder Sudarsky
Searching for spacetime granularity: analyzing a concrete experimental setup
http://arxiv.org/abs/1003.5245
http://arxiv.org/cits/1003.5245

Afshordi
Dark Energy, Black Hole Entropy, and the First Precision Measurement in Quantum Gravity
http://arxiv.org/abs/1003.4811
http://arxiv.org/cits/1003.4811

Mielczarek Cailleteau Grain Barrau
Inflation in loop quantum cosmology: dynamics and spectrum of gravitational waves
http://arxiv.org/abs/1003.4660
http://arxiv.org/cits/1003.4660

Bianchi Rovelli Vidotto
Towards Spinfoam Cosmology
http://arxiv.org/abs/1003.3483
http://arxiv.org/cits/1003.3483

Durka Kowalski-Glikman
Hamiltonian analysis of SO(4,1) constrained BF theory
http://arxiv.org/abs/1003.2412
http://arxiv.org/cits/1003.2412

Modesto Randono
Entropic corrections to Newton's law
http://arxiv.org/abs/1003.1998
http://arxiv.org/cits/1003.1998

Barrett Dowdall Fairbairn Gomes Hellmann Pereira
Asymptotics of 4d spin foam models
http://arxiv.org/abs/1003.1886
http://arxiv.org/cits/1003.1886

Hossenfelder
Comments on and Comments on Comments on Verlinde's paper "On the Origin of Gravity and the Laws of Newton"
http://arxiv.org/abs/1003.1015
http://arxiv.org/cits/1003.1015

Easson Frampton Smoot
Entropic Accelerating Universe
http://arxiv.org/abs/1002.4278
http://arxiv.org/cits/1002.4278
and followup:
Entropic Inflation
http://arxiv.org/abs/1003.1528
http://arxiv.org/cits/1003.1528

A spin foam model for general Lorentzian 4-geometries
http://arxiv.org/abs/1002.1959
http://arxiv.org/cits/1002.1959
and followup:
Spin foams with timelike surfaces
http://arxiv.org/abs/1003.5652
http://arxiv.org/cits/1003.5652

Ashtekar Campiglia Henderson
Casting Loop Quantum Cosmology in the Spin Foam Paradigm
http://arxiv.org/abs/1001.5147
http://arxiv.org/cits/1001.5147

Smolin
Newtonian gravity in loop quantum gravity
http://arxiv.org/abs/1001.3668
http://arxiv.org/cits/1001.3668

Freidel Speziale
Twisted geometries: A geometric parametrisation of SU(2) phase space
http://arxiv.org/abs/1001.2748
http://arxiv.org/cits/1001.2748

Verlinde
On the Origin of Gravity and the Laws of Newton
http://arxiv.org/abs/1001.0785
http://arxiv.org/cits/1001.0785

Last edited: Jul 5, 2010
16. Jul 5, 2010

### marcus

Keeping in mind that in these polls we only compare non-string papers, and that citation counts do not tell the whole story (especially after only a few months), folks might want to see which papers turned out to be the most highly cited.

This is as of 5 July, and I haven't bothered to exclude self-citations.
As of today, the top ten were:

Verlinde 87
Smolin 39
Easson Frampton Smoot(1) 23
Easson Frampton Smoot(2) 20
Modesto Randono 17
Freidel Speziale 12
Hossenfelder 9
Bianchi Rovelli Vidotto 7
Ashtekar Campiglia Henderson 6

Last edited: Jul 5, 2010
17. Jul 5, 2010

### BenTheMan

The bias is pretty unacceptable, as quantum gravity'' is understood by people who actually work in the field to contain string theory as a subset.

18. Jul 5, 2010

### marcus

MTd2, you make an interesting point. Only 3 citations is not a hopeful sign for that paper
One has to bear in mind, though, that it's a string paper, and the point of these MIP polls is that they are always about NON-string QG so it's kind of off-topic. If someone wants to discuss stringy research papers, they can start a separate thread (and even have a poll if they want ) Plus I should repeat that citation counts don't tell the whole story.

Anyway, I just checked citations for the first quarter MIP poll papers, and as you might expect there is a good correlation between the ones we at PF picked three months ago in the poll and the ones that turned out to get cited a lot. I will list them with the titles and maybe put the abstracts, to give an idea of the non-string QG research areas that attracted most attention earlier this year.

Several of the first-quarter papers in the poll (even though only out for a few months) have already garnered 20-or-more cites.
Notably Erik Verlinde's (with 87) and Lee Smolin's (with 39).

It's worth remarking that Leonardo Modesto and Andy Randono are both just postdocs (fairly recent PhDs) while Smoot is a Nobel laureate! They were in the same ballpark, with papers exploring the same (entropic force) idea. I will get the titles of the first 5 or so of these papers.

Verlinde (87 cites)
On the Origin of Gravity and the Laws of Newton
http://arxiv.org/abs/1001.0785
29 pages, 6 figures
"Starting from first principles and general assumptions Newton's law of gravitation is shown to arise naturally and unavoidably in a theory in which space is emergent through a holographic scenario. Gravity is explained as an entropic force caused by changes in the information associated with the positions of material bodies. A relativistic generalization of the presented arguments directly leads to the Einstein equations. When space is emergent even Newton's law of inertia needs to be explained. The equivalence principle leads us to conclude that it is actually this law of inertia whose origin is entropic."

Smolin (39 cites)
Newtonian gravity in loop quantum gravity
http://arxiv.org/abs/1001.3668
16 pages
"We apply a recent argument of Verlinde to loop quantum gravity, to conclude that Newton's law of gravity emerges in an appropriate limit and setting. This is possible because the relationship between area and entropy is realized in loop quantum gravity when boundaries are imposed on a quantum spacetime."

Easson Frampton Smoot (22 cites)
Entropic Accelerating Universe
http://arxiv.org/abs/1002.4278
10 pages, 1 figure
"To accommodate the observed accelerated expansion of the universe, one popular idea is to invoke a driving term in the Friedmann-Lemaître equation of dark energy which must then comprise 70% of the present cosmological energy density. We propose an alternative interpretation which takes into account the temperature intrinsic to the information holographically stored on the screen which is the surface of the universe. Dark energy is thereby obviated and the acceleration is due to an entropic force naturally arising from the information storage on a surface screen. We consider an additional quantitative approach based upon the entropy and surface terms usually neglected in General Relativity and show that this leads to the entropic accelerating universe."

Easson Frampton Smoot (20 cites)
Entropic Inflation
http://arxiv.org/abs/1003.1528
14 pages
"One of the major pillars of modern cosmology is a period of accelerating expansion in the early universe. This accelerating expansion, or inflation, must be sustained for at least 30 e-foldings. One mechanism, used to drive the acceleration, is the addition of a new energy field, called the Inflaton; often this is a scalar field. We propose an alternative mechanism which, like our approach to explain the late-time accelerating universe, uses the entropy and temperature intrinsic to information holographically stored on a screen enclosing the observed space. The acceleration is due in both cases to an emergent entropic force, naturally arising from the information storage on the horizon."

Modesto Randono (17 cites)
Entropic corrections to Newton's law
http://arxiv.org/abs/1003.1998
7 pages, 2 figures
"It has been known for some time that there is a deep connection between thermodynamics and gravity, with perhaps the most dramatic implication that the Einstein equations can be viewed as a thermodynamic equation of state. Recently Verlinde has proposed a model for gravity with a simple statistical mechanical interpretation that is applicable in the non-relatvistic regime. After critically analyzing the construction, we present a strong consistency check of the model. Specifically, we consider two well-motivated corrections to the area-entropy relation, the log correction and the volume correction, and follow Verlinde's construction to derive corrections to Newton's law of gravitation. We show that the deviations from Newton's law stemming from the log correction have the same form as the lowest order quantum effects of perturbative quantum gravity, and the deviations stemming from the volume correction have the same form as some modified Newtonian gravity models designed to explain the anomalous galactic rotation curves."

Surface Density of Spacetime Degrees of Freedom from Equipartition Law in theories of Gravity
http://arxiv.org/abs/1003.5665
20 pages
(Submitted on 29 Mar 2010)
"I show that the principle of equipartition, applied to area elements of a surface which are in equilibrium at the local Davies-Unruh temperature, allows one to determine the surface number density of the microscopic spacetime degrees of freedom in any diffeomorphism invariant theory of gravity. The entropy associated with these degrees of freedom matches with the Wald entropy for the theory. This result also allows one to attribute an entropy density to the spacetime in a natural manner. The field equations of the theory can then be obtained by extremising this entropy. Moreover, when the microscopic degrees of freedom are in local thermal equilibrium, the spacetime entropy of a bulk region resides on its boundary."

This is not to suggest that the more highly cited papers are necessarily better, or that the others are not worth studying. But folks might want to check them out, just in case, to see what they're about. It's a way of benefitting from the collective intelligence of the nonstring research community---see what other professionals are finding useful and want to reference and build on in their own work.

Last edited by a moderator: May 4, 2017
19. Jul 6, 2010

### vacuumcell

My feeling is that MTd2's point is irrelevant since he may have failed to notice that version 1 of this paper was submitted to the archive only four weeks ago. It takes time for papers to acquire citations and some of us can tell when a paper is important before it's received any.

IMHO, it would be helpful if in the future you specified that by papers in "quantum gravity" you mean nonstringy papers, or papers written by people other than those who happen not to do research on your favourite (whatever your reasons) approach to this field. That way you help avoid any appearance of purely personal bias and thus avoid giving people the wrong impression about research in quantum gravity.

Just my two cents.

Last edited: Jul 6, 2010
20. Jul 6, 2010

### MTd2

I referred to last year's paper, not the one submitted last month.


text:
##### 46. Consider the state space model of a system...

46. Consider the state space model of a system, as given below

$\begin{bmatrix}\dot x_1\\\dot x_2\\\dot x_3 \end{bmatrix} = \begin{bmatrix}-1&1&0\\0&-1&0\\0&0&-2\end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix}+ \begin{bmatrix}0\\4\\0 \end{bmatrix}u$ ; $y=\begin{bmatrix}1&1&1\end{bmatrix} \begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}$

The system is

(A)  controllable and observable

(D)  controllable and unobservable

(C)  uncontrollable and unobservable

(B)  uncontrollable and observable

Hint:


text:
# How do the pollutants that cause acid rain get into the air?

Apr 5, 2016

Pollutants that cause acid rain are typically released into the air as gases from industrial processes (factories usually).

#### Explanation:

The pollutants most commonly attributed to acid rain are sulfur dioxide ($S {O}_{2}$) and nitrogen dioxide ($N {O}_{2}$).

$S {O}_{2}$ and $N {O}_{2}$ are released from factories in gaseous form into the atmosphere, where they mix with water (${H}_{2} O$) and hydroxides ($O H \cdot$) which are normally present. When these chemicals mix they cause the neutral pH of atmospheric gases to become slightly acidic (the pH drops). The pH remains acidic as it falls as rain. Learn more about how acid rain forms here.

Infrastructure that is habitually exposed to acid rain will eventually deteriorate or discolor, plants aquatic life can be killed after just one storm. I'll write out the chemical equations below:

Sulfur Dioxide

$S {O}_{2} + O H \cdot \to H O S {O}_{2} \cdot$

$H O S {O}_{2}$ is an intermediary so it's immediate broken down through exposure to oxygen forming a hydroperoxyl ($H {O}_{2} \cdot$) and sulfur trioxide ($S {O}_{3}$):

$H O S {O}_{2} \cdot + {O}_{2} \to H {O}_{2} \cdot + S {O}_{3}$

$S {O}_{3}$ is another intermediary and is immediately broken down into sulfuric acid (${H}_{2} S {O}_{4}$) when ${H}_{2} O$ is present:

$S {O}_{3} \left(g\right) + {H}_{2} O \left(l\right) \to {H}_{2} S {O}_{4} \left(a q\right)$

Nitrogen Dioxide

Nitrogen dioxide ($N {O}_{2}$) and a hydroxide ion ($O H \cdot$) combine to form nitric acid ($H N {O}_{3}$):

$N {O}_{2} + H O \cdot \to H N {O}_{3}$


text:
# Don't think about $$a+b+c+d$$ as a known equation!

Algebra Level 3

$\begin{cases} a+b+c-d = 1 \\ a+b-c+d = 2 \\ a-b+c+d = 3 \\ -a+b+c+d = 4 \end{cases}$

If the system of equations above holds true for the positive real number of $$a, b, c, d$$, then find the value of the expression below.

$\begin{eqnarray} 8a^2 + 9b^2 + 12c^2 + 13d^2 \end{eqnarray}$

×


text:
# Question: What Type Of Solid Is Boron Nitride?

Boron Nitride.

Boron nitride (BN) is a chemical compound that involves equal numbers of boron and nitrogen atoms.

Boron nitrate exists in various crystalline forms that are isoelectronic to carbon.

The hexagonal (h-BN) form, which corresponds to graphite, is the most stable and soft among polymorphs.

## Is boron nitride ionic or covalent?

Cubic Boron Nitride

In the cubic form of boron nitride, alternately linked boron and nitrogen atoms form a tetrahedral bond network, exactly like carbon atoms do in diamond. So it is a 3D giant covalent lattice. In fact it is isoelectronic with the giant covalent lattices of carbon.

covalent bonds

## Is Boron Nitride a network solid?

Boron nitride, BN, is a covalent network solid with a structure similar to that of graphite. Boron nitride is a covalent network solid with a structure similar to that of graphite.

## How hard is Boron Nitride?

But cubic boron nitride is still, at best, just the world’s second hardest material with a Vickers hardness of around 50 GPa. Its hexagonal form (w-BN) was initially reported to be even harder but these results were based upon theoretical simulations that predicted an indentation strength 18% higher than diamond.

## Is boron nitride toxic?

Boron nitride (BN) nanomaterials have been increasingly explored for potential biological applications. However, their toxicity remains poorly understood.

## What is the hardest material?

Scientists have calculated that wurtzite boron nitride and lonsdaleite (hexagonal diamond) both have greater indentation strengths than diamond. Source: English Wikipedia. (PhysOrg.com) — Currently, diamond is regarded to be the hardest known material in the world.

## Does nitrogen bond with boron?

Both Nitrogen and Boron are sp hybridized, making it possible for them to make a triple bond, so that Boron will lose 3 electrons, whilst Nitrogen will gain the lost electrons.

## Which is more electronegative boron or nitrogen?

Here, oxygen is the most electronegative element with electronegativity of 3.44. Then is nitrogen followed by carbon, boron, and aluminium.

## Is Boron Nitride a ceramic?

Boron Nitride (BN), often referred to as ‘white graphite’ due to its similar planar hexagonal structure and lubricious characteristics, is a hard ceramic material with high thermal conductivity and good thermal shock resistance.