text:
## Research team

• Guillermo Incatasciato
• Martín Moroni
• Mauricio Tellechea (Assistant Professor)
• Pablo Ventura
• Gonzalo Zigarán

## Recent journal articles

1. X. Caicedo, M. Campercholi, K. A. Kearnes, P. Sánchez Terraf, Á. Szendrei, and D. Vaggione, “Every minimal dual discriminator variety is minimal as a quasivariety,” Algebra universalis 82 (2) p. 36 (2021).

Let $\dagger$ denote the following property of a variety $\mathcal{V}$: \emph{Every subquasivariety of $\mathcal{V}$ is a variety}. In this paper, we prove that every idempotent dual discriminator variety has property $\dagger$ . Property $\dagger$ need not hold for nonidempotent dual discriminator varieties, but $\dagger$ does hold for \emph{minimal} nonidempotent dual discriminator varieties. Combining the results for the idempotent and nonidempotent cases, we obtain that every minimal dual discriminator variety is minimal as a quasivariety

@article{minimal-dual-quasi,
author = {Caicedo, Xavier and Campercholi, Miguel and Kearnes, Keith A. and S{\'a}nchez Terraf, Pedro and Szendrei, {\'A}gnes and Vaggione, Diego},
year = 2021,
title = {Every minimal dual discriminator variety is minimal as a quasivariety},
journal = {Algebra universalis},
month = {Apr},
day = 29,
volume = 82,
number = 2,
pages = 36,
abstract = {Let $\dagger$ denote the following property of a variety $\mathcal{V}$: \emph{Every subquasivariety of $\mathcal{V}$ is a variety}. In this paper, we prove that every idempotent dual discriminator variety has property $\dagger$ . Property $\dagger$ need not hold for nonidempotent dual discriminator varieties, but $\dagger$ does hold for \emph{minimal} nonidempotent dual discriminator varieties. Combining the results for the idempotent and nonidempotent cases, we obtain that every minimal dual discriminator variety is minimal as a quasivariety},
issn = {1420-8911},
doi = {10.1007/s00012-021-00715-8},
url = {https://doi.org/10.1007/s00012-021-00715-8}
}

2. J. Pachl and P. Sánchez Terraf, “Semipullbacks of labelled Markov processes,” Logical Methods in Computer Science 17 (2) (2021).
[ Abstract | BibTeX ]

A \emph{labelled Markov process (LMP)} consists of a measurable space $S$ together with an indexed family of Markov kernels from $S$ to itself. This structure has been used to model probabilistic computations in Computer Science, and one of the main problems in the area is to define and decide whether two LMP $S$ and $S’$ “behave the same”. There are two natural categorical definitions of sameness of behavior: $S$ and $S’$ are \emph{bisimilar} if there exist an LMP $T$ and measure preserving maps forming a diagram of the shape $S\leftarrow T \rightarrow{S’}$; and they are \emph{behaviorally equivalent} if there exist some $U$ and maps forming a dual diagram $S\rightarrow U \leftarrow{S’}$. These two notions differ for general measurable spaces but Edalat proved that they coincide for analytic Borel spaces, showing that from every diagram $S\rightarrow U \leftarrow{S’}$ one can obtain a bisimilarity diagram as above. Moreover, the resulting square of measure preserving maps is commutative (a \emph{semipullback}). In this paper, we extend Edalat’s result to measurable spaces $S$ isomorphic to a universally measurable subset of a Polish space with the trace of the Borel $\sigma$-algebra, using a version of Strassen’s theorem on common extensions of finitely additive measures.

@article{2017arXiv170602801P,
author = {Pachl, Jan and S{\'a}nchez Terraf, Pedro},
title = {Semipullbacks of labelled {M}arkov processes},
journal = {Logical Methods in Computer Science},
volume = 17,
number = 2,
year = 2021,
month = apr,
archiveprefix = {arXiv},
eprint = {1706.02801},
primaryclass = {math.PR},
keywords = {Mathematics - Probability, Computer Science - Logic in Computer Science, 28A35, 28A60, 68Q85, F.4.1, F.1.2},
adsnote = {Provided by the SAO/NASA Astrophysics Data System},
abstract = {A \emph{labelled Markov process (LMP)} consists of a measurable
space $S$ together with an indexed family of Markov kernels from $S$
to itself. This structure has been used to model probabilistic
computations in Computer Science, and one of the main problems in
the area is to define and decide whether two LMP $S$ and $S'$ behave
the same''. There are two natural categorical definitions of
sameness of behavior: $S$ and $S'$ are \emph{bisimilar}
if there exist an LMP $T$ and measure preserving maps
forming a diagram of the shape
$S\leftarrow T \rightarrow{S'}$; and they
are \emph{behaviorally equivalent}
if there exist some $U$ and maps forming a dual diagram
$S\rightarrow U \leftarrow{S'}$.
These two notions differ for general measurable spaces but Edalat
proved that they coincide for analytic Borel spaces, showing that
from every
diagram $S\rightarrow U \leftarrow{S'}$ one can obtain a
bisimilarity diagram as above. Moreover, the resulting square of
measure preserving maps is commutative (a \emph{semipullback}).
In this paper, we extend Edalat's result to measurable spaces $S$
isomorphic to a universally measurable subset of
a Polish space with the trace of the Borel $\sigma$-algebra, using a
version of Strassen's theorem on
common extensions of finitely additive measures.}
}

3. M. Campercholi, M. Tellechea, and P. Ventura, “Deciding Quantifier-free Definability in Finite Algebraic Structures,” Electronic Notes in Theoretical Computer Science 348: 23-41 (2020). 14th International Workshop on Logical and Semantic Frameworks, with Applications (LSFA 2019).

This work deals with the definability problem by quantifier-free first-order formulas over a finite algebraic structure. We show the problem to be coNP-complete and present a decision algorithm based on a semantical characterization of definable relations as those preserved by isomorphisms of substructures. Our approach also includes the design of an algorithm that computes the isomorphism type of a tuple in a finite algebraic structure. Proofs of soundness and completeness of the algorithms are presented, as well as empirical tests assessing their performances.

@article{CAMPERCHOLI202023,
title = {Deciding Quantifier-free Definability in Finite Algebraic Structures},
journal = {Electronic Notes in Theoretical Computer Science},
volume = {348},
pages = {23 - 41},
year = {2020},
note = {14th International Workshop on Logical and Semantic Frameworks, with Applications (LSFA 2019)},
issn = {1571-0661},
doi = {https://doi.org/10.1016/j.entcs.2020.02.003},
url = {http://www.sciencedirect.com/science/article/pii/S1571066120300037},
author = {Miguel Campercholi and Mauricio Tellechea and Pablo Ventura},
keywords = {Definability, logic, decision algorithm, complexity},
abstract = {This work deals with the definability problem by quantifier-free first-order formulas over a finite algebraic structure. We show the problem to be coNP-complete and present a decision algorithm based on a semantical characterization of definable relations as those preserved by isomorphisms of substructures. Our approach also includes the design of an algorithm that computes the isomorphism type of a tuple in a finite algebraic structure. Proofs of soundness and completeness of the algorithms are presented, as well as empirical tests assessing their performances.}
}

4. M. S. Moroni and P. Sánchez Terraf, “The Zhou Ordinal of Labelled Markov Processes over Separable Spaces,” arXiv e-prints 2005.03630 (2020). Accepted for publication at the Review of Symbolic Logic.
[ Abstract | BibTeX ]

There exist two notions of equivalence of behavior between states of a Labelled Markov Process (LMP): state bisimilarity and event bisimilarity. The first one can be considered as an appropriate generalization to continuous spaces of Larsen and Skou’s probabilistic bisimilarity, while the second one is characterized by a natural logic. C. Zhou expressed state bisimilarity as the greatest fixed point of an operator $\mathcal{O}$, and thus introduced an ordinal measure of the discrepancy between it and event bisimilarity. We call this ordinal the “Zhou ordinal” of $\mathbb{S}$, $\mathfrak{Z}(\mathbb{S})$. When $\mathfrak{Z}(\mathbb{S})=0$, $\mathbb{S}$ satisfies the Hennessy-Milner property. The second author proved the existence of an LMP $\mathbb{S}$ with $\mathfrak{Z}(\mathbb{S}) \geq 1$ and Zhou showed that there are LMPs having an infinite Zhou ordinal. In this paper we show that there are LMPs $\mathbb{S}$ over separable metrizable spaces having arbitrary large countable $\mathfrak{Z}(\mathbb{S})$ and that it is consistent with the axioms of $\mathit{ZFC}$ that there is such a process with an uncountable Zhou ordinal.

@article{moroni2020zhou,
title = {The {Z}hou Ordinal of Labelled {M}arkov Processes over Separable Spaces},
author = {Moroni, Martín Santiago and S\'anchez Terraf, Pedro},
journal = {arXiv e-prints},
month = may,
year = 2020,
eprint = {2005.03630},
volume = {2005.03630},
archiveprefix = {arXiv},
primaryclass = {cs.LO},
note = {Accepted for publication at the Review of Symbolic Logic},
abstract = {There exist two notions of equivalence of behavior between states of a
Labelled Markov Process (LMP): state bisimilarity and event bisimilarity. The
first one can be considered as an appropriate generalization to continuous
spaces of Larsen and Skou's probabilistic bisimilarity, while the second one is
characterized by a natural logic. C. Zhou expressed state bisimilarity as the
greatest fixed point of an operator $\mathcal{O}$, and thus introduced an
ordinal measure of the discrepancy between it and event bisimilarity. We call
this ordinal the "Zhou ordinal" of $\mathbb{S}$, $\mathfrak{Z}(\mathbb{S})$.
When $\mathfrak{Z}(\mathbb{S})=0$, $\mathbb{S}$ satisfies the Hennessy-Milner
property. The second author proved the existence of an LMP $\mathbb{S}$ with
$\mathfrak{Z}(\mathbb{S}) \geq 1$ and Zhou showed that there are LMPs having an
infinite Zhou ordinal. In this paper we show that there are LMPs $\mathbb{S}$
over separable metrizable spaces having arbitrary large countable
$\mathfrak{Z}(\mathbb{S})$ and that it is consistent with the axioms of
$\mathit{ZFC}$ that there is such a process with an uncountable Zhou ordinal.}
}

## Recent conference papers

1. E. Gunther, M. Pagano, and P. Sánchez Terraf, “Formalization of Forcing in Isabelle/ZF,” in Automated Reasoning. 10th International Joint Conference, IJCAR 2020, Paris, France, July 1–4, 2020, Proceedings, Part II, 2020, p. 221–235.
[ Abstract | BibTeX ]

We formalize the theory of forcing in the set theory framework of Isabelle/ZF. Under the assumption of the existence of a countable transitive model of $\mathit{ZFC}$, we construct a proper generic extension and show that the latter also satisfies $\mathit{ZFC}$. In doing so, we remodularized Paulson’s ZF-Constructibility library.

@inproceedings{2020arXiv200109715G,
author = {Gunther, Emmanuel and Pagano, Miguel and S{\'a}nchez Terraf, Pedro},
title = {{Formalization of Forcing in Isabelle/ZF}},
isbn = {978-3-662-45488-6},
booktitle = {Automated Reasoning. 10th International Joint Conference, IJCAR 2020, Paris, France, July 1--4, 2020, Proceedings, Part II},
volume = 12167,
series = {Lecture Notes in Artificial Intelligence},
editor = {Peltier, Nicolas and Sofronie-Stokkermans, Viorica},
publisher = {Springer International Publishing},
doi = {10.1007/978-3-030-51054-1},
pages = {221--235},
journal = {arXiv e-prints},
keywords = {Computer Science - Logic in Computer Science, Mathematics - Logic, 03B35 (Primary) 03E40, 03B70, 68T15 (Secondary), F.4.1},
year = 2020,
eid = {arXiv:2001.09715},
archiveprefix = {arXiv},
eprint = {2001.09715},
primaryclass = {cs.LO},
abstract = {We formalize the theory of forcing in the set theory framework of
Isabelle/ZF. Under the assumption of the existence of a countable
transitive model of $\mathit{ZFC}$, we construct a proper generic extension and show
that the latter also satisfies $\mathit{ZFC}$. In doing so, we remodularized
Paulson's ZF-Constructibility library.},
adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

2. E. Gunther, M. Pagano, and P. Sánchez Terraf, “First steps towards a formalization of Forcing,” Electronic Notes in Theoretical Computer Science 344: 119-136 (2019). The proceedings of LSFA 2018, the 13th Workshop on Logical and Semantic Frameworks with Applications (LSFA’18).

We lay the ground for an Isabelle/ZF formalization of Cohen’s technique of \emph{forcing}. We formalize the definition of forcing notions as preorders with top, dense subsets, and generic filters. We formalize a version of the principle of Dependent Choices and using it we prove the Rasiowa-Sikorski lemma on the existence of generic filters. Given a transitive set $M$, we define its generic extension $M[G]$, the canonical names for elements of $M$, and finally show that if $M$ satisfies the axiom of pairing, then $M[G]$ also does.

@article{2018arXiv180705174G,
author = {Gunther, Emmanuel and Pagano, Miguel and S{\'a}nchez Terraf, Pedro},
title = {{First steps towards a formalization of Forcing}},
journal = {Electronic Notes in Theoretical Computer Science},
archiveprefix = {arXiv},
eprint = {1807.05174},
primaryclass = {cs.LO},
keywords = {Computer Science - Logic in Computer Science, 03B35 (Primary) 03E40, 03B70, 68T15 (Secondary), F.4.1},
month = jul,
adsnote = {Provided by the SAO/NASA Astrophysics Data System},
abstract = {We lay the ground for an Isabelle/ZF formalization of Cohen's
technique of \emph{forcing}. We formalize the definition of forcing notions as
preorders with top, dense subsets, and generic filters. We formalize
a version of the principle of Dependent Choices and using it
we prove the Rasiowa-Sikorski lemma on the existence of generic filters.
Given a transitive set $M$, we define its generic extension $M[G]$,
the canonical names for elements of $M$, and finally show that if $M$
satisfies the axiom of pairing, then $M[G]$ also does.},
volume = {344},
pages = {119 - 136},
year = {2019},
issn = {1571-0661},
doi = {10.1016/j.entcs.2019.07.008},
url = {http://www.sciencedirect.com/science/article/pii/S157106611930026X},
note = {The proceedings of LSFA 2018, the 13th Workshop on Logical and Semantic Frameworks with Applications (LSFA’18)}
}


text:
# Handling Backslash in File Names for Scripts

Ok need little help here.

I have a String Variable that contains a Filename & Path and this is passed to a Script as a parameter:
“D:\SomeDir\Filename.csv”

However I cannot find a way to escape the "\" or replace them to "/" for use in a file.ReadFromFile(datafile); command. It is driving me crazy

Have tried:

stringvar.replace(/\\/g,"\\\\")
stringvar.replace(/\\/g, "/")
stringvar.split('\').join('/')
stringvar.split('\').join('\\')
stringvar.replace("\\", "\\\\")

All attempts return “D:SomeDirFilename.csv”

arrhhh!

Another failure:

var newstr = 'D:\pathname\Zilename.csv'.replace(new RegExp("\\\\","g"), "/")

Note: The Filename is passed to Samba as a Variable so I have no control over its construction but was hoping to deconstruct it.

This is tricky apparently, because your input string may not really be what you think it is due to JScript doing some pre-escaping itself.

You might believe your are processing c:\path\to\file.csv when really (and hopefully) you are processing this instead: c:\\path\\to\\file.csv

How is the path being passed in, I mean, from what? The FileSystemWatcher?

Because “magically” this works …

function doslashes()
{
var str = 'D:\\pathname\\Zilename.csv';
newstr = str.replace(/\\/g,"/");
return 'in: '+str + "\r\nout: " + newstr;
}

If you run the FSW in Debug, you can see that the path is already escaped (the FSW class does this I believe), so what is really coming into your JScript is probably literally c:\\path\\to\\file.csv

… or at least we can hope that is the case. So try your testing in actual runtime with the code I posted above to see what you get.

EDIT: hmm… not much luck in runtime either. This will be tricky indeed…

Scary… here is the start of what appearz to work… need to replace every “special” escape sequence …

So in this example, \p is not special, but \t and \f are special sequences. We did “ok” so far, but because \p is not “special” it is interpreted simply as p, and so the replacement with / does not occur.

Simplest method would be to replace the entire alphabet !!!

2 Likes

Correct it is the “[:Path]” variable returned.

Well I “sort of glad” you labelled it tricky as well as I was well and truly over it! I saw some long regEx strings to cover a whole heap of characters. I going to just use the [:Filename] part and pass that in with a {SETTING:x} for location so I do not need to deal with a ‘’ again!

1 Like

I was beating on this issue for a few hours yesterday, and I still cannot figure out a good working solution.

I think the problem with detection is that escaped characters, when special, are multi-byte. But an escaped regular character (non-special) are not. For example, the length of this string, according to JS is 16 … but it should be 19

C:\path\to\file.csv
^
012345678901234567890
11111111112

Every instance of a backslash is not counted and does not make up part of the string length. It is a difficult problem to solve.

I think the solution probably involves converting the string into byte form. I was playing around with the encodeURI() function which is able to detect escaped special characters, but it does not detect escaped non-special characters.

C%3Apath%09o%0Cile.csv
|||    ||| |||
:     \t  \f

The thing is, it misses is \p because it is not special. I do think that is the “right approach”, but we need a different conversion.

JS ECMA6 has String.raw which I think might work, but alas, we are working with ECMA3 I believe.

P.S. I wold like to figure out a good working solution, but in the meantime, there is a very simple workaround to the problem as it pertains to your implementation:

Do not use backslashes in your File Monitor Path. Use Forward slashes.

It still works, and all your problems go away.

This guy has done it …

Wow @QMcKay - it like a whole project!

I appreciate you spending time on this but please do not spend anymore I was a bit like you thinking surely it cannot be that hard? <- Understatement.

So I am going to just create a {SETTING:x} that will hold the PATH as a D:\\PATH\\ or as D:/PATH/ and then I will just append to [:Filename] which is outputted from File Monitor. Would like to use just 1 so need to find the compatible specification for both File Monitor and JScript.

PS: Of course I have not studied the code but there may be a possibility that the DLL (FM) could be changed to return file spec in '/' or '\\' but in any case best practises would require a configurable Directory Path so point is moot.

I was thinking the same thing, but then I thought to myself “how hard could it possibly be to get proper output from JScript?”.

Well, I implemented the library above and I still cannot get it to work… not sure why. But I cannot figure out how to “uncheck” the last option on that webpage in the code, and that is the important option to “turn off” to make it work as seen in the screenshot (“treat as JavaScript string body” option needs to be “off”).

In any case, because of the inherent “issues” with backslashes being an escape character, I would recommend you don’t use them in paths at all; instead use the forward slash / everywhere. Nothing will break AFAIK, and you won’t need to deal with this “issue” ever again.

I remember when putting together the Browser Printer Customer Display, I was fighting a similar problem when trying to display images for the Order Items via the Printer Template. In the end, I flipped all the slashes to / and all the problems were gone. No workarounds necessary.

Windows and Programs understand the forward slash anyway …

1 Like

Ok @QMcKay I had to come back here because I cannot use {SETTING:x} as a Path Spec in File Monitor as I wanted to just pass the correct usage of “/” to both File Monitor & JScript

But I have a Solution!
Apply as a Path specification "D:/SomePath/" in File Monitor which will return:

Then in your JScript you can use something like:

var fContent = file.ReadFromFile(location.substr(0,location.lastIndexOf("/")+1)+datafile);

Location = [:Path]
Datafile = [:FileName]

Just saved a heap of Rules, Actions and Screens …

1 Like

LOL, nice. I noticed that as well and was wondering how you might handle it.

format: plain


text:
# Ngô Quốc Anh

## June 5, 2010

### Why the conformal method is useful in studying the Einstein equations?

Filed under: Nghiên Cứu Khoa Học, PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 19:20

I presume you have some notions about general relativity, especially the Einstein equations

${\rm Eins}_{\alpha\beta}=T_{\alpha\beta}$.

As these equations are basically hyperbolic for a suitable metric, it is reasonable to study the Cauchy problems for them. Under the Gauss and Codazzi conditions, we have two constraints called Hamiltonian and Momentum constrains. Cauchy problem is to determine the solvable of these constrains of variables $K$-the extrinsic curvature and $g$-the spatial metric. Interestingly, the conformal method says that we can start with an arbitrary metric then we recast the constrain equations into a form which is more amenable to analysis by splitting the Cauchy data. In this method, we try to solve $\gamma$ within the conformal class represented by the initial metric. So, in general, the conformal factor is chosen so that we eventually have a simplest model.

This idea is given via the following theorem.

Theorem. Let $\mathcal D =(\gamma, \sigma, \tau,\psi,\pi)$ be a conformal initial data set for the Einstein-scalar field constraint equations on $\Sigma$. If

$\displaystyle \widetilde \gamma =\theta^\frac{4}{n-2}\gamma$

for a smooth positive function $\theta$, then we define the corresponding conformally transformed initial data set by

$\displaystyle\widetilde{\mathcal D} =(\widetilde\gamma, \widetilde \sigma, \widetilde \tau,\widetilde\psi,\widetilde \pi)=(\theta^\frac{4}{n-2}\gamma, \theta^{-2}\sigma, \tau,\psi,\theta^\frac{-2n}{n-2}\pi)$.

Let $W$ be the solution to the conformal form of the momentum constrain equation w.r.t. the conformal initial data set $\mathcal D$ and let $\widetilde W$ be the solution to the conformal form of the momentum constrain equation w.r.t. the conformal initial data set $\widetilde{\mathcal D}$ (we just assume both exist). Then $\varphi$ is a solution to the Einstein scalar field Lichnerowicz equation for the conformal data $\mathcal D$ with $W$

$\displaystyle \Delta_\gamma \varphi - \mathcal R_{\gamma, \psi}\varphi +\mathcal A_{\gamma, W, \pi}\varphi^{-\frac{3n-2}{n-2}}-\mathcal B_{\tau, \psi}\varphi^\frac{n+2}{n-2}=0$

if and only if $\theta^{-1}\varphi$ is a solution to the Einstein scalar field Lichnerowicz equation for the conformal data $\widetilde{\mathcal D}$ with $\widetilde W$

$\displaystyle \Delta_{\widetilde\gamma} (\theta^{-1}\varphi) - \mathcal R_{\widetilde\gamma, \widetilde\psi}(\theta^{-1}\varphi) +\mathcal A_{\widetilde\gamma, \widetilde W, \widetilde\pi}(\theta^{-1}\varphi)^{-\frac{3n-2}{n-2}}-\mathcal B_{\widetilde\tau, \widetilde\psi}(\theta^{-1}\varphi)^\frac{n+2}{n-2}=0$.

We refer the reader to a paper due to Yvonne Choquet-Bruhat et al. [here] published in Class. Quantum Grav. in 2007 for details. We adopt this theorem from that paper, however, there is no proof there.

Outline of the proof. Let

$\overline \gamma=\varphi^\frac{4}{n-2}\gamma$.

Obviously we have (go to this entry for further information)

$\displaystyle \frac{n-2}{4(n-1)} \overline{{\rm Scal}}_{\overline\gamma}=\varphi^{-\frac{n+2}{n-2}}\left(-\Delta_\gamma\varphi + \frac{n-2}{4(n-1)}{\rm Scal}_\gamma \varphi\right)$.

However, it is clear to see that

$\overline \gamma=(\theta^{-1}\varphi)^\frac{4}{n-2}\widetilde \gamma$

which also yields

$\displaystyle \frac{n-2}{4(n-1)} \overline{{\rm Scal}}_{\overline\gamma}=(\theta^{-1}\varphi)^{-\frac{n+2}{n-2}}\left(-\Delta_{\widetilde\gamma}(\theta^{-1}\varphi) + \frac{n-2}{4(n-1)}{\rm Scal}_{\widetilde\gamma} (\theta^{-1}\varphi)\right)$.

Thus,

$\displaystyle \boxed{-\Delta_{\widetilde\gamma}(\theta^{-1}\varphi) + \frac{n-2}{4(n-2)}{\rm Scal}_{\widetilde\gamma} (\theta^{-1}\varphi)=\theta^{-\frac{n+2}{n-2}}\left(-\Delta_\gamma\varphi + \frac{n-2}{4(n-1)}{\rm Scal}_\gamma \varphi\right)}$.

Keep in mind that

$\displaystyle \mathcal R_{\gamma, \psi}=\frac{n-2}{4(n-1)}\left({\rm Scal}_\gamma - |\nabla \psi|_\gamma^2\right)$.

Next we have

$\displaystyle |\nabla \psi|_\gamma^2 =\gamma^{ij}\nabla_i \psi\nabla_j\psi=\theta^{\frac{4}{n-2}}{\widetilde\gamma}_{ij}\nabla_i \psi\nabla_j\psi=\theta^{\frac{4}{n-2}}|\nabla \widetilde\psi|_{\widetilde\gamma}^2$,

i.e.

$\displaystyle\boxed{|\nabla \widetilde\psi|_{\widetilde\gamma}^2=\theta^{-\frac{4}{n-2}}|\nabla \psi|_\gamma^2}$.

We refer the reader to this entry showing that how to calculate the norm of tensors. It is also clear to see that

$\displaystyle \boxed{\mathcal B_{\widetilde\tau, \widetilde\psi} = \mathcal B_{\tau, \psi}}$

since $\tau$ and $\psi$ remain unchanged under the conformal change. Concerning $\mathcal A$, from its definition

$\displaystyle \mathcal A_{\gamma, W, \pi}=\frac{n-2}{4(n-1)}\left(|\sigma+\mathbb DW|_\gamma^2+\pi^2\right)$

we see that

$\displaystyle \boxed{\mathcal A_{\widetilde\gamma, \widetilde W, \widetilde\pi}=\theta^{-\frac{4n}{n-2}}\mathcal A_{\gamma, W, \pi}}$.

$\displaystyle\begin{gathered} - {\Delta _{\widetilde\gamma }}({\theta ^{ - 1}}\varphi ) + \frac{{n - 2}}{{4(n - 2)}}{\text{Sca}}{{\text{l}}_{\widetilde\gamma }}({\theta ^{ - 1}}\varphi ) \hfill \\ \qquad\qquad- \frac{{n - 2}}{{4(n - 2)}}|\nabla \widetilde\psi |_{\widetilde\gamma }^2({\theta ^{ - 1}}\varphi ) + {\mathcal{A}_{\widetilde\gamma ,\widetilde W,\widetilde\pi }}{({\theta ^{ - 1}}\varphi )^{ - \frac{{3n - 2}}{{n - 2}}}} + {\mathcal{B}_{\widetilde\tau ,\widetilde\psi }}{({\theta ^{ - 1}}\varphi )^{\frac{{n + 2}}{{n - 2}}}} \hfill \\ \qquad= {\theta ^{ - \frac{{n + 2}}{{n - 2}}}}\left( { - {\Delta _\gamma }\varphi + \frac{{n - 2}}{{4(n - 1)}}{\text{Sca}}{{\text{l}}_\gamma }\varphi } \right) \hfill \\ \qquad\qquad- \frac{{n - 2}}{{4(n - 2)}}{\theta ^{ - \frac{4}{{n - 2}}}}|\nabla \psi |_\gamma ^2({\theta ^{ - 1}}\varphi ) + {\theta ^{ - \frac{{4n}}{{n - 2}}}}{\mathcal{A}_{\gamma ,W,\pi }}{({\theta ^{ - 1}}\varphi )^{ - \frac{{3n - 2}}{{n - 2}}}} + {\mathcal{B}_{\tau ,\psi }}{({\theta ^{ - 1}}\varphi )^{\frac{{n + 2}}{{n - 2}}}} \hfill \\ \qquad= {\theta ^{ - \frac{{n + 2}}{{n - 2}}}}\left[ { - {\Delta _\gamma }\varphi + \frac{{n - 2}}{{4(n - 1)}}{\text{Sca}}{{\text{l}}_\gamma }\varphi - \frac{{n - 2}}{{4(n - 2)}}|\nabla \psi |_\gamma ^2 + {\mathcal{A}_{\gamma ,W,\pi }}{\varphi ^{ - \frac{{3n - 2}}{{n - 2}}}} + {\mathcal{B}_{\tau ,\psi }}{\varphi ^{\frac{{n + 2}}{{n - 2}}}}} \right]. \hfill \\ \end{gathered}$


text:
# It's Now or never -- Question about "universal" time

1. Sep 12, 2014

### johniha

I am not sure whether this is a 'relativity' query as it may not be concerned with the order of events nor simultaneity in the usual relativistic sense. So whether it remains, is moved, or is deleted I'll leave to the mercy of the Mentors. Very likely I've made a simple blunder or else the topic has already been done to death, in either case, please enlighten me... thanks!

At any one moment in time the whole universe exists

There cannot be a moment when the universe does not exist

There cannot be a moment when the universe only partially exists.

At any one moment in time the whole universe exists regardless of the relative time-frame of any object within it.

Therefore the concept of a universal 'now' holds true regardless of relativistic concepts. So... there must be a fundamental and universal principle orchestrating the whole shebang... ???

2. Sep 12, 2014

### Staff: Mentor

Hi, johniha, and welcome to PF!

You're assuming that the concept of "any one moment in time" is meaningful. In relativity, it isn't. I realize that seems highly counterintuitive, but that's the way it is.

3. Sep 12, 2014

### ghwellsjr

If there is, then you neglected to mention one other attribute of this principle:

It has the uncanny ability to remain illusive.

So why bother speculating along these lines?

4. Sep 12, 2014

### CKH

This counterintuitive problem is eliminated in (neo-)Lorentz Ether Theory (LET) which is equivalent to SR (in it's experimental effects on measurements). In this theory there is absolute time, space and simultaneity.

The theory explains precisely why moving clocks run slower and lengths contract in classical terms. It does not require that time and space themselves are relative to the observer. In LET there is one and only one now in the Universe.

5. Sep 12, 2014

### ghwellsjr

Yes, and it's illusive. No one can identify the one and only state in which light propagates at c and in which time, space and simultaneity are absolute. So why bother?

6. Sep 12, 2014

### Staff: Mentor

This is called "petitio principii" or "begging the question". Your premise "At any one moment in time the whole universe exists" contains the conclusion that there is a universal now.

http://www.fallacyfiles.org/begquest.html

7. Sep 12, 2014

### bahamagreen

Plain language is inherently full of assumptions and causes no end of problems in science questions and answers. For example, someone is informed that there is "no absolute space and therefore no absolute motion, only relative motion".

He may think about that and talk to himself "logically" saying that if two objects are in relative motion then at least one of them must be in absolute motion even if we can't determine or assign motionlessness to any particular object... because there are two objects in relative motion he could only assign motionlessness to at most only one of them. They both can't be motionless if in relative motion; they both might be in absolute motion, or only one of them... but at least one of them has to be in motion. So if relative motion is happening with two objects, that must mean that at least one of the motions must be "real motion" with respect to an absolute space.

This has the appearance of a logical argument, like your idea about absolute time, but it also is full of dozens of assumptions in the clumsy plain language that need to be identified and examined...

8. Sep 12, 2014

### CKH

Isn't that argument a little like the argument "I cannot see it, therefore it doesn't exist"?

SR postulates that c is constant in all IRF. In order to explain such a strange phenomenon, SR concludes that space and time themselves must be frame dependent. The reason SR is forced to mess with space and time is the additional postulate that the speed of light is isotropic in all frames (and the implicit postulate that all clocks "measure time correctly").

It has been since been pointed out by many physicists that these last postulates assume more than is necessary to explain the experimental results (in which the Lorentz transformations are validated). Furthermore, isotropy cannot be proved or disproved using these experimental results.

In LET a mechanism is provided that explains why measurement of c is constant in all frames. Why should we abandon a physical mechanism to explain things when such a mechanism is viable? Why should we resort to an entirely different idea about time and space than the classical one when there is no need? Why should we abandon the notion that waves are an effect that occurs in some medium?

To be clear I'm a novice in SR. However, it was only when I was shown how to account for the Lorentz transformations in classical physics with an aether (mechanistic physics) that I could say "It makes perfect sense". Perhaps this is because I was not indoctrinated through SR training to accept the mainstream interpretation of the physics of the Lorentz transform.

9. Sep 12, 2014

### Staff: Mentor

Because we *observe*, experimentally, that $c$ is constant in all IRF. We didn't just make up the postulate from thin air.

Which we also observe, experimentally. If you don't like this postulate, take it up with Nature.

No. First of all, we don't assume that all clocks measure time correctly; some clocks do it better than others. Second, we don't postulate that clocks measure time correctly; we postulate that "time" is what clocks measure. In other words, we have a theoretical model in which "time" appears, and we link this model to what we actually observe by linking the "time" in the model to the observations we make of clocks. This has to be done for any physical theory.

Really? Why not? Can't we just observe light moving in different directions?

Because this "mechanism" is inherently unobservable.

Because there still is a need. Nobody actually observes this "absolute time" or "absolute space"; the "different idea about time and space" is about the time and space we actually *observe*.

Because we can't observe the medium, even in principle.

10. Sep 12, 2014

### Staff: Mentor

I don't know why you would equate "why bother" with "does not exist". "Why bother" means that it does not matter if it exists or not, it doesn't affect anything measurable.

11. Sep 12, 2014

### johniha

Thank you guys.

Well it certainly is from my perspective, and I refuse to believe anyone who tells me that the universe is not out there, and not out there at this moment!

Would it be better to say that there is a different universe for each point in space-time, with neighboring points having fairly similar universes? (Not that points could have universes of course)

12. Sep 12, 2014

### Staff: Mentor

You might spend some time thinking about what "at this moment" means experimentally and why you think that the universe must not only be "out there" but must be "out there" specifically "at this moment" which is a distinct concept. How would one prove it experimentally?

13. Sep 12, 2014

### Staff: Mentor

Really? Suppose you and I are on opposite sides of Earth. The information you are getting from me is, at best, about 70 milliseconds old. The information you are getting from the Moon at the same instant (meaning, that arrives at your eyes and other sensory apparatus at the same event) is about 1.25 seconds old. The information you are getting from the Sun at the same instant is about 500 seconds old. And from Jupiter, about 45 minutes old. And from Alpha Centauri, about 4.3 years old. And from the Andromeda Galaxy, about 2 million years old. And...

Your brain presents to you the illusion that all of these objects exist in a "now", even though the information you get from them is never fully up to date; there is always some delay. Before relativity was discovered, we could tell ourselves that the delay didn't matter, that there was still a single "now" despite the delay, because *something* in the universe could connect things without the delay (perhaps gravity, which according to Newton's theory acted instantaneously).

But now we know that the delay is fundamental: *all* information transfer in the universe is limited to the speed of light. There is *no* way around it. And that means there is no way around the fact that the "now" that you are so sure is "out there" is an illusion. That doesn't mean the universe doesn't exist; it means that the simple deduction you think you are drawing from your immediate perceptions is in fact a huge leap, which your perceptions by themselves do not justify. The only thing your immediate perceptions justify is belief in the existence of someone on the other side of the Earth 70 milliseconds ago, or the Moon 1.25 seconds ago, or the Sun 500 seconds ago, etc. But these, put all together, do not form a single "now". (What they form, in relativity, is called your past light cone.)

No; that just compounds the problem, because you would have to explain why there are all these different universes, instead of just all these different "nows". It's much simpler just to admit that "now" is a construction, that it's relative.

14. Sep 12, 2014

### johniha

I'm far from sure what "this moment" means and was hoping for some straightforward pointers, but instead I'm getting elusive 'Zen' type responses , without anyone actually closing the door on the question. The implication being that the question isn't totally without merit, but so far as experimental physics is concerned answers remain elusive?

15. Sep 12, 2014

### johniha

OK belay my previous post... you've been at some pains to address my misapprehension Peter, thank you, I shall try to absorb...

16. Sep 12, 2014

### johniha

OK, you've dumbed it down enough for me... I think I got it... thank you all! - Our intuitive understandings do not correspond to the physical reality of the world. It's safe to say there is a universe out there, but my everyday understanding of what "this moment" is simply isn't up to the job, according to SR....

...nonetheless I cannot resist a peek at LET.

17. Sep 12, 2014

### CKH

I am not contradicting that postulate; the two-way speed of light is supported by hard evidence. It is not that postulate that is unnecessary. It's the others (isotropy of c and that clocks measure "real time" regardless of motion) that force us to an explanation in which time and space themselves are only relative things.

The postulate that moving clocks measure "actual time" is unnecessary to explain the Lorentz transformations, as is the postulate that the speed of light is the same in all directions.

You cannot prove the isotropy of c with experiments.

That moving clocks measure actual time is a definition of time in SR. That definition is unnecessary to explain the LTs.

Experiments are easily explained using wave physics in absolute space and time. In LET, moving clocks are demonstrated to tick slower than those at rest in the ether. The Lorentz transformation for time as measured by clocks is a direct consequence of a simple analysis of waves in a medium.

I can make a similar argument to yours. SR assumes some things that you cannot detect so why believe that it is the correct interpretation of nature?

You can attempt to measure the one-way speed with a ruler and clocks, but you have to choose a convention for clock synchronization to measure one-way velocity. There are many choices. Einstein's choice implicitly assumes that the speed of light is isotropic. That choice is not required to explain experimental results.

I would agree that the absolute frame is undetectable in the context of the Lorentz transformations. Beyond that, who knows?

I think you will agree that isotropy of the speed of light is undetectable if you study some papers on the subject of clock synchronization.

You can define "space" as what is measured by similar rods in any frame and "time" as what is measured by a similar clocks in any frame regardless of motion, and assert that time and space are frame dependent i.e. relative. You can apply Minkowsky space and all the things that have already been done using the concept of relativity. That's perfectly correct in terms of results. You can do the same thing in LET.

What is lacking in SR is a physical theory of why "space' and "time" (as defined above) behave the way they do. This view that time and space are not absolute is forced by the "extra" postulates of SR.

On the other hand, it is easily demonstrated with a simple application of classical physics to waves in a medium that moving clocks tick slower and consequently that length is measured as contracted in a frame moving with respect to the medium. That the speed of light is an upper bound is obvious for waves in a isotropic medium from classical physics. In LET nothing can travel faster. Tachyons are impossible.

In SR you cannot observe the one-way speed of light, even in principle. SR assumes that c is isotropic.

There is nothing that you can observe "directly" in principle. All you ever get to do is measure quantities and infer or theorize what is actually there. These "things that you observe" have the same level of reality as ether.

In LET, waves in the ether is "all" that is observable. Light and matter are waves. Because we are waves we cannot directly perceive the ether but we can measure it's properties by seeing how the waves behave.

In summary, SR is forced to assume that actual time and space are relative by certain postulates that cannot be demonstrated.

SR deprives light waves and matter waves of a medium in which to propagate. All the other types of waves that we study in physics are the result of changes in a medium.

Many physicists have also pointed out that motion cannot be entirely relative because we can detect absolute rotation using the Sagnac effect (laser gyros). Also, acceleration is absolute.

Last edited by a moderator: Sep 13, 2014
18. Sep 12, 2014

### phinds

Well, no, that's not quite right. Our "common sense" / "intuition", and so forth DO correspond to an EXTREMELY limited range of things in the universe. Things that are close to us and are not moving at relativistic speeds, are a of macro size, and ... and ... and ...

We evolve with survival mechanisms that require our ability to make sound judgments based on an awareness of the world the including those constrictions without being aware of, or even NEEDING to be aware of, their existence. They just don't matter to how individuals have survived since living beings came into existence on this planet.

Consequently, when you start to study cosmology (the very large) and quantum mechanics (the very small) and black holes (the very heavy/dense) you are going to have moments when your brain just flat out says NO WAY !!!

Consequently, to be successful in such studies you have to be willing to suspend disbelief long enough to work things out with mathematics, NOT English language, and see that your initial disbelief was wrong.

19. Sep 12, 2014

### johniha

When I consider that I only exist in the current 'now' - the future has no existence, neither does the past - my existence is limited to an infinitesimal wave of time, likewise the rest of the universe has no temporally sustained existence... that idea is very much at odds with the way I view the world which is more 'me passing through a landscape were objects are fairly permanent temporal fixtures'... So it seems to me my mind has embellished reality. It has created a world distinctly different to the actual reality

20. Sep 12, 2014

### CKH

While I agree with what you are saying about experience being limited, this does not imply that we should not employ our experience to explain things when we can.

SR is not intuitive. In fact it is so counter-intuitive that complicated explanations are required for untrained people to "get it". That people keep popping up with apparent paradoxes illustrates the difficulty of the concepts.

In the LET interpretation, we do not need the counter-intuitive concepts that space, time and simultaneity are just relative to explain nature. So why adopt them, as in SR?

With the lack of any "reasonable" interpretation in Quantum Mechanics, it seems that most physicist have abandoned any attempt to try. Thus it's become popular to say that the equations are physics and leave any deeper explanation as impossible or unnecessary.

Now, the idea that understanding nature is impossible because of some limitation of our imaginations has become common. I have to agree with Einstein and not accept that as true and instead believe that our understanding is as yet far from complete.


text:
The goal of Automated Theorem Proving is to automatically generate a proof, given a conjecture (the target theorem) and a knowledge base of known facts, all expressed in a formal language. Automated Theorem Proving is useful in a wide range of applications, including the verification and synthesis of software and hardware systems.

TREND DATASET BEST METHOD PAPER TITLE PAPER CODE COMPARE

# HOList: An Environment for Machine Learning of Higher-Order Theorem Proving

5 Apr 2019tensorflow/deepmath

We present an environment, benchmark, and deep learning driven automated theorem prover for higher-order logic.

751

# HolStep: A Machine Learning Dataset for Higher-order Logic Theorem Proving

1 Mar 2017tensorflow/deepmath

We propose various machine learning tasks that can be performed on this dataset, and discuss their significance for theorem proving.

751

# Learning to Prove Theorems via Interacting with Proof Assistants

21 May 2019princeton-vl/CoqGym

Proof assistants offer a formalism that resembles human mathematical reasoning, representing theorems in higher-order logic and proofs as high-level tactics.

195

# On-demand Injection of Lexical Knowledge for Recognising Textual Entailment

We approach the recognition of textual entailment using logical semantic representations and a theorem prover.

165

127

# Learning as Abduction: Trainable Natural Logic Theorem Prover for Natural Language Inference

29 Oct 2020kovvalsky/LangPro

Tackling Natural Language Inference with a logic-based method is becoming less and less common.

77

# LangPro: Natural Language Theorem Prover

LangPro is an automated theorem prover for natural language (https://github. com/kovvalsky/LangPro).

77

# GamePad: A Learning Environment for Theorem Proving

In this paper, we introduce a system called GamePad that can be used to explore the application of machine learning methods to theorem proving in the Coq proof assistant.

65

# G2SAT: Learning to Generate SAT Formulas

The Boolean Satisfiability (SAT) problem is the canonical NP-complete problem and is fundamental to computer science, with a wide array of applications in planning, verification, and theorem proving.

29

# Holophrasm: a neural Automated Theorem Prover for higher-order logic

8 Aug 2016dwhalen/holophrasm

I propose a system for Automated Theorem Proving in higher order logic using deep learning and eschewing hand-constructed features.

26


text:
# Can I use Runge-Kutta to solve these equations?

Edit: I'm going to give some more background and derivation to show how I got to these equations. I am basically following the derivation that is found in the appendix of the following paper:

R.M. Wood and B.E. Watts, "The Flow, Heat, and Mass Transfer Characteristics of Liquid Films on Rotating Disks", Trans. Instn Chem. Engrs., Vol 51, 1973.

The derivation is for the laminar fluid flow on the top of a rotating disk, where the liquid is dispensed on the center only.

Parameters: Fluid flow in the $r$-direction is $u(r,z)$, flow in the $\theta$-direction is $v(r,z)$. $\omega$ is the rotation rate of the disk, $h(r)$ is the height of the fluid film. Liquid flow rate is $Q=\int_0^{2\pi}\int_0^{h(r)}u(r,z)\,dz\,r\,d\theta=2\pi\,r\,h\,u_m$, where $u_m$ is the mean velocity in the $r$-direction, or $u_m(r)=\frac{1}{h(r)}\int_0^{h(r)}u(r,z)\,dz$. Similarly $v_m(r)=\frac{1}{h(r)}\int_0^{h(r)}v(r,z)\,dz$.

Assumptions: There is rotational symmetry such that $\frac{\partial}{\partial \theta}=0$. Flow is only horizontal so that flow in the $z$-direction $w=0$. Similarly $\frac{\partial P}{\partial z}=0$.

Assume that flow in the $r$-direction is parabolic and that $u=0$ at $z=0$ and $\frac{\partial u}{\partial z}=0$ at $z=h$. This gives us $u$ in terms of the local mean $u_m$ and fluid height $h$:

$$u(r,z)=3u_m\left[ \frac{z}{h}-\frac{1}{2}\left(\frac{z}{h} \right)^2 \right]$$

We do a similar assumption for flow in the $\theta$-direction, so that in this case $v=\omega\,r$ at $z=0$ and $\frac{\partial v}{\partial z}=0$ at $z=h$. This gives us:

$$v(r,z)=r\,\omega+\frac{3}{2}(r\,\omega- v_m)\left[\left(\frac{z}{h} \right)^2 -2 \frac{z}{h}\right]$$

## Derivation

At this point we take the steady-state Navier-Stokes equations in cylindrical coordinates, assuming that $\frac{\partial P}{\partial r}=0$, keeping the advection terms for $u$ and $v$, and keeping the $\frac{\partial^2}{\partial z^2}$ viscous terms:

$$u\frac{\partial u}{\partial r}-\frac{v^2}{r}=\nu \left(\frac{\partial^2 u}{\partial z^2} \right)$$

$$u\frac{\partial v}{\partial r}+\frac{vu}{r}=\nu \left(\frac{\partial^2 v}{\partial z^2} \right)$$

At this point you integrate both equations from $0$ to $h$ in the $z$-direction. Using the above assumptions for $u(r,z)$ and $v(r,z)$, you can perform the integration for the viscous term on the RHS of the N-S equations. For the advection terms on the LHS, Wood and Watts then simply replace $u$ with $u_m$ and $v$ with $v_m$ and throw in an $O(1)$ constant for each, eliminating the need to solve the integral for the advection terms. Finally you use the definition of $Q=2\pi\,r\,h\,u_m$ to eliminate $h$ from the equations with $h=Q/(2\pi\,r\,u_m)$. For Wood and Watts' derivation this gives the following ODE's:

$$u_m \frac{du_m}{dr}=\frac{{v_m}^2}{r}-\frac{12 K_1 \nu\, \pi^2\,r^2\,{u_m}^3}{Q^2}$$

$$u_m \frac{dv_m}{dr}=-\frac{u_m v_m}{r}+\frac{12 K_2 \nu\, \pi^2\,r^2\,(r\,\omega-v_m){u_m}^2}{Q^2}$$

My goal however, is to eliminate the need for the $K_1$ and $K_2$ fudge factor constants by using the full assumed forms of $u(r,z)$ and $v(r,z)$ and integrating all the terms on the LHS of the NS equations. Finishing the integration and simplification (done using MATLAB symbolic math package) I have the following two equations:

$$69ru_m\frac{du_m}{dr}=8r^2\omega^2-16r\omega v-21{u_m}^2+48{v_m}^2-\frac{480\pi^2\nu r^3{u_m}^3}{Q^2}$$ $$48ru_m\frac{dv_m}{dr}=-21r(v_m-r\omega)\frac{du_m}{dr}-69u_m v_m+37r\omega u_m+\frac{480\pi^2\nu r^3(v_m-r\omega){u_m}^2}{Q^2}$$

## Problem

I want to solve them simultaneously for $u(r)$ and $v(r)$, and initial conditions of $u(r_0)$ and $v(r_0)$ are known.

How to use the Runge-Kutta method is not my question. I've used it in the past and know how it works. My question/problem comes from the $\frac{du}{dr}$ term in the 2nd equation. Without that term I could just solve the two equations simultaneously using Runge-Kutta quite easily. But with it I'm having trouble understanding the best way to solve it. I could simply solve the first equation for $\frac{du}{dr}$ and substitute it into the second equation, but that would make the formulation really messy and I'd like to avoid that. Is there a way I could introduce a simple 3rd equation to deal with that? I'm thinking of something similar to how when you solve a 2nd degree ODE like $y''=f(x)$ using the Rung-Kutta method you split it into two equations:

$$y_1'=y_2$$

$$y_2'=f(t)$$

However I'm having trouble coming up with a formulation that would let me do that.

## Update

With some help from the folks over at Computational Science Stackexchange, I have solved the system of equations. The solution can be seen here.

• You didn't finish your algebraic derivation yet. Like you said, you can solve the first equation for du/dr and then substitute in the second. Having said that, the coefficients in your equations look very funny. How did you end up with numbers like 69 and 37? Are you sure you didn't make a mistake somewhere along the way? – CuriousOne Dec 22 '14 at 22:19
• Even if you didn't want to do that substitution (and assuming everything else is correct) -- actually write out the discrete form of your equations as they are and it should be pretty easy to formulate the integration. Particularly if you write the left-hand side as a matrix multiplying a vector of {u,v}. – tpg2114 Dec 22 '14 at 22:38
• can you define what $\omega$ is - is it a constant - I would normally guess it is $d\theta /dt$, but here that seems to be $v$ - would help to answer if that was clear – tom Dec 22 '14 at 22:52
• Crossposted to math.stackexchange.com/q/1078136/11127 Would Computational Science be a better home for this question? – Qmechanic Dec 22 '14 at 22:53
• Is $\omega$ the rate of spin of the disk? – tom Dec 22 '14 at 23:12

The way I see it you have two choices.

Either you introduce a third variable for $du/dr$ - let us call it $y$ and calculate $dy/dr$ using your first equation. The you should be able to do the Runge Kutta with three variables. and you can set $du/dr = y$

Or you substitute in for $du/dr$ from the first equation into the second.

I think the second method, though messy, is probably the best because it looks as if $y$ is not an indepedent variable as it is completely defined by $u$, $v$, $r$ etc.

• If I did that though, wouldn't I essentially be calculating $d^2u/dr^2$? Since my equations are 1st order I would then have to have some initial condition for $du/dr$, which I don't have. – Derek Dec 23 '14 at 16:33
• @Derek yes - that is why I think option 2 is better (substitute in for $du/dr$ in 2nd equation using information from first equation) - the reason I suggest substituting it with $y$ is that it makes it easier to handle the $du/dr$ term on the right hand side of the second equation. + you can calculate the initial value for $du/dr$ with your first equation. – tom Dec 23 '14 at 16:52
• I could do that with a simple Euler method, but I don't think that would work with an automated Runge-Kutta code, since it needs to evaluate the RHS of the equations at several points for each iteration. I would prefer to be able to just use the Runge-Kutta functions in MATLAB to solve this, but if I can't find another way I can always revert to the Euler method. – Derek Dec 23 '14 at 17:08
• @Derek, if it will work with Euler - it can also be programmed with Runge Kutta, but it is just a bit more difficult to programme in my experience. I'm afraid I am not familiar with the functions in MATLAB. – tom Dec 24 '14 at 23:11
• I think this explanation was good enough to get me to the next step - actually getting it to work. Thanks for the help. – Derek Jan 7 '15 at 22:03


text:
# Can you please explain Boyle's Law Charle's Law in a simlified manner? I would be very thankful to u for this?

Dear student!

Simply, the Boyle's law shows the relation between the pressure and volume of a gas at constant temperature, so as : For a fixed mass of gas at constant temperature, the volume is inversely proportional to the gas.

It means if, the volume is doubled the pressure will be half or vice versa.

It is mathematically expressed as: V ∝ 1/P (at constant temperature and mass)

Or, so we write as , PV = constant

Hence, P1 /P2 = V2 /V1

Similarly, the  Charle's Law explains that: For a fixed mass of gas at constant pressure (P) ,the volume is directly proportional to the kelvin temperature (T). So, if we double the volume, the temperature will also be doubled.

Mathematically, At constant P and Mass (or no. of moles = n)

V ∝ T

Or, T1 /T2 = V1 /V2

In equation of both the laws we obtain, PV = nRT

• -1

Boyle's law gives the relationbetween pressure (p) and volume (V) of a fixed amount of gas. Boyle's law states that at constant temperature, the pressure of a fixed amount (number of moles,n) of a gas is inversely proportional to its volume.

Mathematically, it is represented as

, where= Proportionality constant

From the above equation, it is found that at constant temperature, the product of pressure and volume of a fixed amount of a gas is constant. Also, if at constant temperature,

V1= Volume of a gas at pressurep1

V2= Volume of the same gas at pressurep2

Then,

p1V1=p2V2= Constant

Charles' Law states that at constant pressure, the volume of a fixed amount of gas is directly proportional to its absolute temperature. Mathematically, it is represented as follows

V ∝ T

or V = kT

where V is the volume, T is the temperature and k is the proportionality constant. The above equation also gives the following relation

V2/ V1= T2/ T1

where V1and V2are the initial and the final volume of the gas and T1and T2are the initial and the final temperature of the gas respectively.

• 4

Boyle's Law

To put it in a simple manner, Boyle's Law is a correlatian or it shows the relation between pressure and volume, when pressure acts on a sample of gas.

P∝ 1/V means that pressure is inversely proportional to the volume of the gas and when the pressure increases the volume decreases.(Temperature always has to remain the same.)

A formula to find the change that has occured when more or less pressure is applied is P1*V1= P2*V2 , where P is pressure and V is volume.P1 and V1 are initial pressure and volume respectively, while P2 and V2 are the applied pressure and the resulting volume respectively.

• 0
What are you looking for?


text:
# GNU Radio - General Filtering Questions - Any feedback is appreciated!

This is a fairly general question I feel like.

So, the scenario I am presenting is this:

I am generating random data using the random source module and feeding that data into the Constellation Modulator, thus producing a Basebase modulated signal. This takes us to my first question, why is the signal from (-f to +f) and not just (0 to +f)? Attached is a picture.

Next question is this, if I implement a low pass filter from (0 to f) f being a frequency relatively close to 0 Hz, why do I not see the rejection following with the roll off? Also, why is the negative frequency component still there?

I am teaching myself quite a bit of dsp and gnuradio, so any constructive feedback would be great!

If you look at the picture that actually has some decent pulse shaping, it seems to me that the cutoff frequency is far to high as well, but it's only at this range that I get decent pulse shaping.

Thanks!

-DasBoy

• Why do you expect the spectrum to be zero for $f<0$? – MBaz Oct 9 '19 at 23:19
• If your modulated signal at baseband is a complex signal (as opposed to a real signal) then the negative and positive exponential frequency components will not be the same. For the case of real signals the positive and negative frequencies are complex conjugate symmetric (same magnitude opposite phase) so the redundant negative frequency need not be displayed but they still exist. This more detailed answer may help you better understand the concept of negative frequencies and what you are doing with the Constellation Modulator: dsp.stackexchange.com/questions/41073/ – Dan Boschen Oct 10 '19 at 1:57
• Regardless if the baseband signal is real or complex, by showing the positive and negative frequencies at baseband as you did here, we see what the spectrum will look like when it it up-converted to any particular carrier frequency (assuming a linear distortion-free up-conversion process)- so this is the common approach to plot the spectrum of your baseband waveform even if your signal is completely real (such as a BPSK modulation). – Dan Boschen Oct 10 '19 at 2:00


text:
## College Algebra 7th Edition

Interval: $(- \infty,\ \frac{7}{5}]\cup [\frac{8}{5},\ \infty)$
$\frac{1}{|2x-3|}\leq 5$ We see that $x\ne\frac{3}{2}$ because otherwise we would get division by 0. $1\leq 5|2x-3|$ $\frac{1}{5}\leq|2x-3|$ We split the absolute value into the positive and negative: $\frac{1}{5}\leq 2x-3$ $\frac{16}{5}\leq 2x$ $\frac{8}{5}\leq x$ $2x-3 \leq-\frac{1}{5}$ $2x\leq\frac{14}{5}$ $x\leq\frac{7}{5}$ Interval: $(- \infty,\ \frac{7}{5}]\cup [\frac{8}{5},\ \infty)$


text:
# Jacobian conjecture

In mathematics, the Jacobian conjecture is a famous problem on polynomials in several variables. It was first posed in 1939 by Ott-Heinrich Keller.[citation needed] It was widely publicized by Shreeram Abhyankar, as an example of a question in the area of algebraic geometry that requires little beyond a knowledge of calculus to state.

The Jacobian conjecture is notorious for the large number of attempted proofs that turned out to contain subtle errors. As of 2018, there are no plausible claims to have proved it. Even the two-variable case has resisted all efforts. There are no known compelling reasons for believing it to be true, and according to van den Essen (1997) there are some suspicions that the conjecture is in fact false for large numbers of variables. The Jacobian conjecture is number 16 in Stephen Smale's 1998 list of Mathematical Problems for the Next Century.

## The Jacobian determinant

Let N > 1 be a fixed integer and consider polynomials f1, ..., fN in variables X1, ..., XN with coefficients in a field k. Then we define a vector-valued function F: kNkN by setting:

F(c1, ..., cN) = (f1(c1, ...,cN),..., fN(c1,...,cN)).

(F is a polynomial mapping.)

The Jacobian determinant of F, denoted by JF, is defined as the determinant of the N × N Jacobian matrix consisting of the partial derivatives of fi with respect to Xj:

${\displaystyle J_{F}=\left|{\begin{matrix}{\frac {\partial f_{1}}{\partial X_{1}}}&\cdots &{\frac {\partial f_{1}}{\partial X_{N}}}\\\vdots &\ddots &\vdots \\{\frac {\partial f_{N}}{\partial X_{1}}}&\cdots &{\frac {\partial f_{N}}{\partial X_{N}}}\end{matrix}}\right|,}$

then JF is itself a polynomial function of the N variables X1, ..., XN.

## Formulation of the conjecture

It follows from the multivariable chain rule that if F has a polynomial inverse function G: kNkN, then JF has a polynomial reciprocal, so is a nonzero constant. The Jacobian conjecture is the following partial converse:

Jacobian conjecture: If JF is a non-zero constant and k has characteristic 0, then F has an inverse function G: kNkN, and G is regular (in the sense that its components are polynomials).

The obvious analogue of the Jacobian conjecture fails if k has characteristic p > 0 even for 1 variable. The characteristic of a field must be prime, so it is at least 2. The polynomial xxp has derivative 1 − px xp−2 which is 1 (because px is 0) but it has no inverse function. However, Adjamagbo (1995) suggested extending the Jacobian conjecture to characteristic p > 0 by adding the hypothesis that p does not divide the degree of the field extension k(X) / k(F).

For real maps, the condition JF ≠ 0 is related to the inverse function theorem in multivariable calculus. In fact for smooth functions (and so in particular for polynomials) a smooth local inverse function to F exists at every point where JF is non-zero. For example, the map x → x + x3 has a smooth global inverse, but the inverse is not polynomial.

## Results

Wang (1980) proved the Jacobian conjecture for polynomials of degree 2, and Bass, Connell & Wright (1982) showed that the general case follows from the special case where the polynomials are of degree 3, or even more specifically, of cubic homogeneous type, meaning of the form F = (X1 + H1, ..., Xn + Hn), where each Hi is either zero or a homogeneous cubic. Drużkowski (1983) showed that one may further assume that the map is of cubic linear type, meaning that the nonzero Hi are cubes of homogeneous linear polynomials. These reductions introduce additional variables and so are not available for fixed N.

Connell & van den Dries (1983) proved that if the Jacobian conjecture is false, then it has a counterexample with integer coefficients and Jacobian determinant 1. In consequence, the Jacobian conjecture is true either for all fields of characteristic 0 or for none. For fixed N, it is true if it holds for at least one algebraically closed field of characteristic 0.

Let k[X] denote the polynomial ring k[X1, ..., Xn] and k[F] denote the k-subalgebra generated by f1, ..., fn. For a given F, the Jacobian conjecture is true if, and only if, k[X] = k[F]. Keller (1939) proved the birational case, that is, where the two fields k(X) and k(F) are equal. The case where k(X) is a Galois extension of k(F) was proved by Campbell (1973) for complex maps and in general by Razar (1979) and, independently, Wright (1981). Adjamagbo (1995) suggested extending the Jacobian conjecture to characteristic p > 0 by adding the hypothesis that p does not divide the degree of the field extension k(X) / k(F). Moh (1983) checked the conjecture for polynomials of degree at most 100 in two variables.

de Bondt, van den Essen & 2005, 2005 and Drużkowski (2005) independently showed that it is enough to prove the Jacobian Conjecture for complex maps of cubic homogeneous type with a symmetric Jacobian matrix, and further showed that the conjecture holds for maps of cubic linear type with a symmetric Jacobian matrix, over any field of characteristic 0.

The strong real Jacobian conjecture was that a real polynomial map with a nowhere vanishing Jacobian determinant has a smooth global inverse. That is equivalent to asking whether such a map is topologically a proper map, in which case it is a covering map of a simply connected manifold, hence invertible. Sergey Pinchuk (1994) constructed two variable counterexamples of total degree 25 and higher.

It is well-known that the Dixmier conjecture implies the Jacobian conjecture (see Bass et al. 1982). Conversely, it is shown by Yoshifumi Tsuchimoto (2005),[1] and independently by Alexei Belov-Kanel and Maxim Kontsevich (2007), that the Jacobian conjecture for 2N variables implies the Dixmier conjecture for N dimension. A self-contained and purely algebraic proof of the last implication is also given by P. K. Adjamagbo and A. van den Essen (2007) who also proved in the same paper that these two conjectures is equivalent to Poisson conjecture.

## References

1. ^ Tsuchimoto, Yoshifumi (2005). "Endomorphisms of Weyl algebra and $p$-curvatures". Osaka Journal of Mathematics. 42 (2): 435–452. ISSN 0030-6126.


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## 4.12 Histograms

In Section 4.10, we learned how to use geom_bar() to generate bar charts for visualizing the distributions of discrete variables. You may be wondering, how about visualizing continuous variables? One popular plot is called histograms.

Let’s again use the sahp housing price data set.

### 4.12.1 Using the hist() function

To generate a histogram, you can simply use hist() with the variable as the argument.

library(r02pro)

### 4.12.3 Aesthetics in geom_histogram()

Next, we introduce the aesthetics in histograms, which are very similar to those in bar charts. For example, we can map a variable to the fill aesthetic.

ggplot(data = sahp) +
geom_histogram(mapping = aes(x = sale_price, fill = house_style),
bins = 5)

We can see that like the bar chart, the bar for each bin is now divided into sub-bars with different colors. The different colors in each sub-bar correspond to the values of house_style. And the height of each sub-bar represents the count for the cases with the sale_price in this particular bin and the specific value of house_style. Just like geom_bar(), we have a global aesthetic called position, which does position adjustment for different sub-bars. The default position value is again "stack" if you don’t specify it.

a. Stacked bars

ggplot(data = sahp) +
geom_histogram(mapping = aes(x = sale_price, fill = house_style),
position = "stack",
bins = 5)

As expected, we get the same histogram as before.

b. Dodged bars

The second option for position is "dodge", which places the sub-bars beside one another, making it easier to compare individual counts for the combination of a bin of sale_price and house_style.

ggplot(data = sahp) +
geom_histogram(mapping = aes(x = sale_price, fill = house_style),
position = "dodge",
bins = 5)

c. Filled bars

Another option for optional is position = "fill". It works like stacking, but makes each set of stacked bars the same height.

ggplot(data = sahp) +
geom_histogram(mapping = aes(x = sale_price, fill = house_style),
position = "fill",
breaks = seq(from = 0, to = 600, by = 100))

Just like geom_bar(), the y axis may be labeled as “proportion” rather than “count,” to be more precise. This makes it easier to compare proportions of different values of cut for different bins of price. For example, we can see that the proportion of house_style = "1.5Fin" is highest for the houses with sale price less than 100.

### 4.12.4 Density Estimate Using Histograms

In addition to using histograms to visualize the distribution of a discrete variable, you can also construct a density density of variable using a proper normalization. To generate such a density estimate, you can add y = ..density.. as a mapping in the aes() function in geom_historgram(). Let’s see an example as below.

ggplot(data = sahp) +
geom_histogram(aes(x = sale_price, y = ..density..),
breaks = seq(from = 0, to = 6e2, by = 1e2)) +
scale_y_continuous(breaks = seq(from = 0, to = 7e-3, by = 1e-3))

Here, we added the breaks arguments on the bins as well as on the y-axis.

Let’s try to calculate the height of the first bar together. We know that the total area of the bars is 1, agreeing with the definition of density. First, we get the area of the first bar. Then we divide it by the width to get the height.

sale_price_no_na <- na.omit(sahp\$sale_price)
sum(sale_price_no_na < 1e2)/length(sale_price_no_na) #area of the first bar
#> [1] 0.1036585
sum(sale_price_no_na < 1e2)/length(sale_price_no_na) /1e2 #height of the first bar
#> [1] 0.001036585

You can see that the height matches the y axis for the first bar.

### 4.12.5 Exercises

Use the sahp data set to answer the following questions.

1. Create three different histograms on the living area (liv_area) for each of the following settings
• Use 10 bins
• Set the binwidth to be 300
• Se the bins manually to an equally-spaced sequence from 0 to 3500 with increment 500.
1. Create histograms on the living area (liv_area) with 5 bins, and show the information of different kit_qual values in each bar. What conclusions can you draw from this plot?


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# How do Feynman diagrams and loop quantum gravity fit together?

Feynman diagrams are a good way of calculating effects in quantum electrodynamics on a constant background space-time.

Spin foams (LQG) are conjectured to be a good way of calculating quantum effects with gravity.

Is there any recent ideas of how the two ideas will fit together?

Are particles thought to be identified with nodes on the spin-network. Or perhaps they are smeared round loops? Or on the lines? Why is this problem so hard to solve?

In other words, how (if it is known) would one embedd a Feynman graph on a spin-foam?

I would imagine if $$V$$ represents the vertices of a spin foam. Then one should be able to define fields $$\phi(V)$$ which have values for every vertex. Then one should be able to quantize those fields.