text:
# Symmetry transformations on a quantum system; Definitions

We define a symmetry transformation of a system to be any transformation that, when performed, does not change the outcome of a measurement. Wigner's symmetry theorem says that any symmetry of a quantum system is represented by a linear and unitary operator which acts on the Hilbert space of physical states $\mathscr{H}$. So for any symmetry $\mathscr{E}$ there corresponds a unitary transformation $\mathcal{U}(\mathscr{E})$ acting on $\mathscr{H}$.

Suppose $\hat{A}$ is the Hermitian operator corresponding to some observable $A$ which has eigenvalue $\lambda\in\mathbb{R}$ with eigenvector $|\Phi\rangle$. Then, if the system is in a state $|\Psi\rangle$, the probability of measuring the value $\lambda$ of the observable $A$ is given by the Born rule; \begin{align} \text{Prob}(\lambda,\hat{A},\Psi)=\frac{|\langle\Phi|\Psi\rangle|^2}{\langle\Psi|\Psi\rangle\langle\Phi|\Phi\rangle}~. \end{align} We define a symmetry of a quantum system to be one that preserves the above probabilities. However, any unitary operator $\tilde{\mathcal{U}}$ acting on $\mathscr{H}$ will preserve these probabilities; \begin{align} \text{Prob}(\lambda,\hat{A'},\Psi')=\frac{|\langle\tilde{\mathcal{U}}\Phi|\tilde{\mathcal{U}}\Psi\rangle|^2}{\langle\tilde{\mathcal{U}}\Psi|\tilde{\mathcal{U}}\Psi\rangle\langle\tilde{\mathcal{U}}\Phi|\tilde{\mathcal{U}}\Phi\rangle}= \frac{|\langle\Phi|\tilde{\mathcal{U}}^{\dagger}\tilde{\mathcal{U}}\Psi\rangle|^2}{\langle\Psi|\tilde{\mathcal{U}}^{\dagger}\tilde{\mathcal{U}}\Psi\rangle\langle\Phi|\tilde{\mathcal{U}}^{\dagger}\tilde{\mathcal{U}}\Phi\rangle}=\text{Prob}(\lambda,\hat{A},\Psi). \end{align} This would imply that any unitary transformation acting on $\mathscr{H}$ corresponds to a symmetry transformation of the system. I doubt this is true, so where did I go wrong in my definitions and how do I rectify it?

There are many definitions of symmetry in Quantum Theory. However your idea is not correct: symmetries, acting on states, do change outcomes of measurements, at least for one observable. Otherwise we are not speaking of a symmetry but of a gauge transformation.

In the Hilbert space formulation, and I will stick to this case here, and in the absence of superselection rules, a symmetry is a bijective operation that preserves some structure of the space of the states or of the space of observables. As far as I know there are four notions and they are those listed below.

1. Wigner symmetry: a bijective map from the space of rays (unit vectors up to phases) to the space of rays that preserves the probability transitions.
2. Kadison symmetry: a bijective map from the space of states (positive trace-class operators with unit trace) that preserves the convex linear combinations of states, i.e. it preserves the weights of the mixtures of pure states.
3. Kadison symmetry in dual formulation: a bijective map from the lattice of elementary observables (orthogonal projectors in the Hilbert space) into itself preserving the orthocomplemented $\sigma$-complete lattice structure. I.e. it preserves the logical structure of quantum theory.
4. Segal symmetry: a bijective map from the set of bounded everywhere defined self-adjoint operators into the same set that preserves the structure of Jordan algebra of that set, i.e. it preserves the structure of the set of observables.

Each of these definitions can be physically motivated. All definitions lead to the same characterization theorem (for 3 and 4 the Hilbert space must be separable with dimension $\neq 2$ to take advantage of Gleason's theorem).

Theorem [Wigner-Kadison-Segal]. A symmetry of type 1-4 is described by a unitary or antiunitary (depending on the symmetry if the Hilbert space has dimension $>1$ otherwise both possibilities are allowed). Every unitary or anti unitary operator defines a symmetry of type 1-4 simultaneously.

A fundamental symmetry depending on time and continuous with respect to the natural topology (depending on the type 1-4), is the temporal evolution.

A symmetry of type 1-4 (also depending on time) is said to be a dynamical symmetry if it commutes with the temporal evolution.

(At this point, as more or less everybody knows, a theorem similar to Noether's one arises as a consequence of the Stone's theorem provided, as I supposed, the Hilbert space is complex).

When superselection rules enter the game (namely, the center of the von Neumann algebra of observables is non-trivial), the picture becomes more delicate and, for instance, Wigner's notion ceases to be a good notion because the same notion of transition probability becomes ambiguous (pure states and rays are not one-to-one). The same problem arises when the von Neumann algebra of observables admits a non Abelian commutant, as in chromodynamics (even if the center is trivial).

• See Chapter 7 in "Fundamental Mathematical Structures of Quantum Theory" by Valter Moretti (Springer, 2019), and a reference inside: B. Simon, Quantum Dynamics: From Automorphism to Hamiltonian. Studies in Mathematical Physics, Essays in Honor of Valentine Bargmann, ed. by E.H. Lieb, B. Simon, A.S. Wightman (Princeton University Press, Princeton, 1976), pp. 327–349
– mma
Feb 20, 2020 at 5:30


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# Symbolics.jl and Latexify.jl: customize latex representations?

(Julia 1.6)
I deal with equations from publications and the variables there have lots of subscripts and superscripts. They cannot be represented as Julia variable names and I would prefer not to compromise. For example, I might want something like this:

using LaTeXStrings,Latexify, Symbolics
@variables x_plus  # this is a variable
# this is what I would like the variable to be written as
repr_map = Dict(x_plus => L"x^{+}_{A,\alpha_{1}}")

Now I want to do something like

@latexify x_plus + x_plus^2 + factorial(x_plus)
# and also give the macro my repr_map, tell it somehow that I want factorial written as !...

and get x^{+}_{A,\alpha_{1}} + {(x^{+}_{A,\alpha_{1}})}^2 + (x^{+}_{A,\alpha_{1}})!.

Is there a way to do this? One thing I tried was modifying Latexify.jl (this file: Latexify.jl/…/function2latex.jl) to add my own symbols. This doesn’t really work for everything and of course having to modify the library isn’t a good solution. Is there a way to add “custom rules” to Latexify?

If there is no way currently, would such an idea be consistent with what Latexify.jl is trying to do? If so, I would be willing to help implement it.

Perhaps something like this (does not work) would be a good API:

x_plus = L"x^{+}_{A,\alpha_{1}}"
@latexify $x_plus +$x_plus^2 + factorial(\$x_plus)

Something like this might be possible too

@variables var"x^{+}_{A,\alpha_{1}}"
x_plus =  var"x^{+}_{A,\alpha_{1}}"
# continue to use x_plus.

It partially works already. However, when using this, the latex representation crashes (doesn’t use enough parentheses or braces).

julia> latexify(x_plus + x_plus^2 + factorial(x_plus))
L"$$x^{+}_{{A,\alpha\_{1}}} + x^{+}_{{A,\alpha\_{1}}}^{2} + \mathrm{factorial}\left( x^{+}_{{A,\alpha\_{1}}} \right)$$
"

(latex rendering error: “Double exponent: use braces to clarify”)

Also, this won’t help with the factorial to ! issue.

The problem with the “latex rendering error” in my last post can be partially solved by adding even more brackets to the variable name (symbol). Of course, this does not help with things like factorial.

@variables var"{x^{+}_{A,\alpha_{1}}}"

Note that in the output, the second-order-subscript “1” is not represented properly because Latexify escapes the underscore. I don’t think there is a way to mitigate this right now, short of editing Latexify.jl sources.

format: plain


text:
## What happens when a character fails to reject influence that would put a label above +3 or below -2?

The Labels in Masks are fixed between -2 and +3. From the rulebook:

If you ever need to shift a Label and can’t (because the Label is at +3 and would shift up, or is at -2 and would shift down), you must mark a condition, GM’s choice.

When a character with Influence over you tries to shift your labels, this can be prevented with the reject someone’s influence move, but:

On a miss, their words hit you hard. Mark a condition, and the GM adjusts your Labels.

What happens if someone would shift your labels outside the allowed range and you fail to reject their influence? Do you take two conditions?

## John the Ripper / Hashcat rule, reject candidate if char at position X is the same as character at position Y

I’m using John to generate some word lists and I’m trying to figure out the most optimized way to do the next step. What I want to do is add ever possible 3 digit number to a set where the first digit of the number is not the same as the first digit in the set

Set example 123ABC

+ 213 = 123ABC213 OK

+ 131 = 123ABC131 REJECT

I see rules that reject unless a string includes, but not a comparison function like this.

I could make the whole list and prune it after with a python script, but it would be way bigger than needed.

Thank you!

## Enforcing DMARC policy (reject) on an Office 365 tenant

The domain & tenant has SPF and DKIM properly configured and DMARC policy set to p=reject. Still, emails spoofed with the domain in the From header aren’t rejected, but appear in the Junk Email folder on Office 365. People do check their Junk Email for false positives, and are still reading all the CEO frauds, sextortion letters etc.

This seems a feature instead of a bug, as described in Microsoft’s documentation:

How Office 365 handles inbound email that fails DMARC

If the DMARC policy of the sending server is p=reject, EOP marks the message as spam instead of rejecting it. In other words, for inbound email, Office 365 treats p=reject and p=quarantine the same way.

Office 365 is configured like this because some legitimate email may fail DMARC. For example, a message might fail DMARC if it is sent to a mailing list that then relays the message to all list participants. If Office 365 rejected these messages, people could lose legitimate email and have no way to retrieve it. Instead, these messages will still fail DMARC but they will be marked as spam and not rejected.

However, this reasoning has some flaws:

• DKIM protects legitimate mail; DKIM signed messages do pass with the DMARC policy even if it fails to align with the SPF when forwarded on a mailing list. (Mailing lists should change the envelope sender to pass SPF checks, anyway, so the SPF checks are probably passed, but not aligned.)

• By implementing p=reject instead of p=quarantine the owner of the domain has stated that the emails should be rejected. Therefore, Microsoft’s implementation is against RFC 7489, 6.3:

p: Requested Mail Receiver policy ...     reject:  The Domain Owner wishes for Mail Receivers to reject       email that fails the DMARC mechanism check.  Rejection SHOULD       occur during the SMTP transaction.

Is there any setting on Office 365 to alter this behaviour and reject these messages?

## If I “swipe down to reject”, will caller get a chance to leave voicemail?

A few times a year someone who isn’t in my address book might call me, but every day I get several spam/scam calls. So basically anytime I see a plain number, I swipe down to make my phone stop ringing.

I’ve been assuming this sends the call straight to voicemail (so that if it is legit they can leave a message) but it strikes me that “rejecting” a call might hang up completely on the caller.

What happens when I “swipe down to reject” a call from the homescreen? Is there a way I can quickly ignore a call while still leaving the other party a chance to wait for it to go through to voicemail?

This is a fairly stock phone, an “Android One” Moto x4 bought from Google Fi but it does have some extra Motorola tentacles on it.

(I’m also interested in doing this automatically, but only if it uses a built-in feature and not some random app that you happen to trust, sorry.)

## Postfix reject SPAM using FROM in header_check

I’m using postfix (3.1.12) and receiving spam with prevents to be sent from my email address to me. The trick the spammer does is to differ the FROM mail address and the from address in the DATA header.

Jun 13 01:48:01 localhost postfix/cleanup[57940]: AD3DDE06BE: message-id=<3284866135.1622311732354561837738079581085.JavaMail.app@jno.rqlr> Jun 13 01:48:01 localhost postfix/qmgr[31745]: AD3DDE06BE: from=<prvs=60660779a3=gpstemmet@telkomsa.net>, size=241607, nrcpt=1 (queue active) Jun 13 01:48:01 localhost postfix/local[57941]: AD3DDE06BE: to=<xxx@yyy.tld>, relay=local, delay=1.8, delays=1.8/0.02/0/0.02, dsn=2.0.0, status=sent (delivered to maildir)

Date: Thu, 13 Jun 2019 01:47:22 +0200 Message-ID: <3284866135.1622311732354561837738079581085.JavaMail.app@jno.rqlr> From: <xxx@yyy.tld> Errors-To: no-reply@mail.telkomsa.net Subject: xxx Feedback-ID: 5:hc_gfxoi_okpbsv:laceq X-Sender: gpstemmet@telkomsa.net List-Unsubscribe: <http://telkomsa.net/sm/unsubscribe/do?hash=5248854> To: xxx@yyy.tld

I founded many scripts and ways to block a list of specific FROM addresses with header_check using regular expressions.

But is there a way to use header_check verifying that the FROM and the DATA FROM are equal ?

## Exim – How to reject mail to RCPT ${run In reference to the recently publicized Exim vulnerability CVE-2019-10149, I am running supposedly patched Exim v. 4.90_1 (built June 4th, 2019) on Ubuntu 18.04.2 LTS. Although it is supposedly patched, according to Canonical, I’m getting a lot of exploit attempt that are ending up as Frozen messages. This is somewhat worrysome. Is there a way to reject messages for recipients starting with a ‘$ ‘?

And how could I test to be sure these exploits aren’t working?

## Turing machine loop and reject example

I’m getting confused on these both. Reject the string does a stop while loop the machine goes on and on.

My textbook has one example on a reject state and no physical one for loop:

Assume that no reject state was given. And I input a string

$$1^{q_{1}}011$$

After all the transitions the turing machine is now on the empty string $$1011 \epsilon^{q_{8}}$$ with no where else to go and its not on a accept state, so it rejects.

Now consider if its on q2 at string $$0$$ so $$10^{q2}11$$

And it cant go anywhere. Would that also reject? Or only if its on the empty string?

Now what is a loop? My idea is

$$1^{q_{1}}011$$

$$1^{q_{2}}011$$

$$1^{q_{1}}011$$

$$\dots$$

forever

## Recognizably turing machine question (reject / loop)

The definition of proving recognizability using dove tailing is below. However I’m wondering if we can also prove loop or reject in the same way?

Give a deterministic TM D that recognizes L such that if $$w \in L$$ then D accepts w. or if $$w \notin L$$, then D does not accept w. I realized this definition on states if D accepts a w

like

$$L = \{\langle M \rangle \mid M$$ is a TM, and $$M$$ accepts some string that ends with 111} is recognizably and easy to prove through dove tailing

$$L = \{\langle M \rangle \mid M$$ is a TM, and $$M$$ reject some string that ends with 111} is this also recognizable and would the proof be the same as the one above ? I think this will work but

$$L = \{\langle M \rangle \mid M$$ is a TM, and $$M$$ loops on some string that ends with 111}

How do you check if something loops?

## Reject in ansible list variable

I have such a variable:

apps:  - {name: kapitalism, extension: .war}  - {name: socialism, extension: .war}  - {name: somethingelse, extension: .ear}

And I need to make another one based on it, which will not include some of the list. I’m trying this:

- name: Reject this please   set_fact:     apps: "{{ apps | map(attribute='name') | reject('search',item.name) | list }}"   when: "item.name.find('socialism') != -1"   with_items: "{{ apps }}"

But here I get such a problem that the extension parameter is lost and I get the variable:

apps: [u'kapitalism', u'somethingelse']

How to make a variable like this:

apps:  - {name: kapitalism, extension: .war}  - {name: somethingelse, extension: .ear}

?


text:
# Economics 2019 Past Questions | WAEC

Study the following Economics past questions and answers for JAMBWAEC  NECO and Post-JAMB. Get prepared with official past questions and answers for upcoming examinations.

46. Which of the following institutions is Concerned with expanding developing countries’ commodity trade

• A. World Trade Organization (WTO)
• C. Economic Commission for Africa (ECA)
• D. African Development Bank (AfDB)

47. The major achievement of the Economic Community of West African States (ECOWAS) is that it has

• A. made capital more mobile
• B. made possible the use of common currency
• C. increased members’ allegiance to former colonial masters
• D. widened the market for goods produced

48. One measure for financing a country’s balance of payments deficit is through

• A. export diversification
• B. import substitution
• C. short-term borrowing from IMF
• D. internal borrowing from commercial banks

49.  Exploitation of forest resources becomes a major problem when

• A. communities fight over ownership
• B. they are not renewed
• C. they are neglected
• D. the supply is fixed in the long-run

50. Natural resources that are renewable are found in the

• A. mining sector
• C. agricultural sector
• D. secondary sector

## ECONOMICS THEORY PAST QUESTIONS 2019 WITH SOLUTION

1. The diagram above represents the equilibrium position of a firm in a perfectly competitive industry. Study it carefully and answer the questions that follow.

1. (a) i. At what level of output and prices is the firm in equilibrium?
2. Calculate the firm’s profit in equilibrium

iii. What type of profits is it?

(b) i. Why is the average revenue (AR) function horizontal?

(c) State any two ways in which marginal cost (MC) and average total cost (ATC) are related.

2.

1. The extract from a country’s balance of payments account is shown below
 Item Imports ($million) Export ($) Agricultural products – 200 Mineral products – 300 Consumer goods 250 – Capital goods 400 – Insurance 50 25 banking 75 30 Transportation 85 25 Loans 150 60
• Using the table above, calculate the:
• (c) balance on current account

3.  Define distribution of goods.

Illustrate the normal chain of distribution of goods.

Describe a consumers’ cooperative society.

Outline any four roles performed by a consumers’ cooperative society

4.  What is an industry?

Explain the following:
i. division of labor
ii. economies of scale

Outline any four internal economies of scale.

5. Define a joint venture.

identify any three merits of a private company over a partnership.

State any three sources of finance to a public enterprise

6.  Distinguish between the following pairs of concepts

elastic demand and inelastic demand

Income elasticity of demand and cross elasticity of demand.

Using diagrams, explain how an increase in price will affect the total revenue of a producer if demand for his product is:
i. price elastic
ii. price inelastic

7. What is public debt?

Outline any three reasons why counties borrow.

Highlight any three effects of a huge national debt on the economy of a country.

8. Distinguish between:
Growing population and a declining population

overpopulation and under population.

Explain any four disadvantages of a rapidly growing population in an economy

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Zbl 1200.42002
Koumandos, Stamatis; Lamprecht, Martin
On a conjecture for trigonometric sums and starlike functions. II.
(English)
[J] J. Approx. Theory 162, No. 5, 1068-1084 (2010). ISSN 0021-9045

For $\mu>0$ let $s_n^\mu(z)=\sum_{k=0}^n (\mu)_k z^k/k!$, where $(\mu)_k=\mu(\mu+1)\dots (\mu+k-1)$. Denote by $\mathcal{A}$ the class of analytic functions in the unit disk $\Bbb{D}=\{z\in\Bbb{C}:|z|<1\}$. For $f,g\in\mathcal{A}$ we write $f\prec g$ ($f$ is subordinate to $g$ in $\Bbb{D}$) if there exists $w\in\mathcal{A}$ with $|w(z)|\leq z$ for $z\in\Bbb{D}$, such that $f=g\circ w$. Also, for $\rho\in (0,1]$ we let $\mu^*(\rho)$ be the unique solution of the integral equation $\int_0^{(\rho+1)\pi}\sin(t-\rho\pi)t^{\mu-1}dt=0$.\par The paper under review is a continuation of the work of {\it S. Koumandos} and {\it S. Ruscheweyh} [J. Approximation Theory 149, No. 1, 42--58 (2007; Zbl 1135.42001)], where its authors guessed that:\par Conjecture. For $\rho\in(0,1]$ the number $\mu^*(\rho)$ is indeed maximal number $\mu(\rho)$ such that for all $n\in\Bbb{N}$ and $\mu\in (0,\mu(\rho)]$ we have $$(1-z)^\rho s_n^\mu(z)\prec \Big(\frac{1+z}{1-z}\Big)^\rho.$$\par In the present paper, the authors give a proof of this conjecture in case $\rho=1/4$. To do this, they prove (based on the properties of completely monotonic functions) a sharp trigonometric inequality. Main result of paper has some applications concerning starlike functions, as more as a corollary, it gives an inequality concerning Gegenbauer polynomials.
[Mehdi Hassani (Zanjan)]
MSC 2000:
*42A05 Trigonometric polynomials
30A10 Inequalities in the complex domain
26D05 Inequalities for trigonometric functions and polynomials
42A32 Trigonometric series of special types
26D20 Analytical inequalities involving real functions

Keywords: positive trigonometric inequality; completely monotonic function; starlike function; Gegenbauer polynomial

Citations: Zbl 1135.42001

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text:
Regarding general understanding of DiffBind
2
0
Entering edit mode
@shubhibartaria-10946
Last seen 5.3 years ago

Hello,

This question is from a rookie in terms of wet lab techniques and new student into bioinformatics. I am working in a lab in Johns Hopkins wherein I am given a few BED and BAM files to analyse and see how similar they are. I am using the diffbind package to do so. The first correlation map(heat map), when I run

>>>MB <- dba(sampleSheet="MBdataframe.csv")

>>>MB

has values corresponding to 0 and below 0 (but not -1)

When I run it using

>>>MB = dba.count(MB)

I get a heatmap with values of 0.9 and 0.955 approx.

1) What exactly do these values signify in both the maps. How are they different?

2) How are they calculated? Do we use the Pearson Correlation Coeff and this is the p-value? If they are high, do they mean that the samples are highly correlated?

3) Can I use something to integrate the values numerically in the heatmap too so that it can make more sense?

I went through this by doing it in a very simple manner using 3 BED and corresponding BAM files and not complicating it by Conditions and replicates etc for now(I don't have that data as well).

I want to understand this in layman terms. I did go through few of your other questions but again ended up getting confused. Hopefully you understand where I am coming from.

Regards

Shubhi Bartaria

diffbind heatmap • 691 views
0
Entering edit mode

Hi,

Can you pass along the output of sessionInfo() please?  The values are correlations, and shouldn't be negative. Higher numbers (darker colours) are higher correlations, i.e. more similar.  (Pearson correlation by default.)

The difference between the two plots is that the first uses the scores in the peak files, while the post-counting heat map uses the actual counts from the BAM files.

There's no capability to incorporate the numbers into the plot, as far as I know (Rory will know for sure).

Cheers,

- Gord

0
Entering edit mode

I hope Rory responds with a way to incorporate exact values into the heatmap.

And my BED files doesn't have a score column. It just has 4 columns

0
Entering edit mode
@shubhibartaria-10946
Last seen 5.3 years ago

This is sessionInfo() output

R version 3.2.2 (2015-08-14)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X 10.11.4 (El Capitan)

locale:

[1] C

attached base packages:
[1] stats4    parallel  stats     graphics  grDevices utils     datasets
[8] methods   base

other attached packages:
[1] DiffBind_1.16.3            RSQLite_1.0.0
[3] DBI_0.4-1                  locfit_1.5-9.1
[5] GenomicAlignments_1.6.3    Rsamtools_1.22.0
[7] Biostrings_2.38.4          XVector_0.10.0
[9] limma_3.26.9               SummarizedExperiment_1.0.2
[11] Biobase_2.30.0             GenomicRanges_1.22.4
[13] GenomeInfoDb_1.6.3         IRanges_2.4.8
[15] S4Vectors_0.8.11           BiocGenerics_0.16.1

loaded via a namespace (and not attached):
[1] Rcpp_0.12.5             lattice_0.20-33
[3] GO.db_3.2.2             gtools_3.5.0
[5] digest_0.6.9            plyr_1.8.3
[7] futile.options_1.0.0    BatchJobs_1.6
[11] ggplot2_2.1.0           gplots_3.0.1
[13] zlibbioc_1.16.0         GenomicFeatures_1.22.13
[15] annotate_1.48.0         gdata_2.17.0
[17] Matrix_1.2-6            checkmate_1.7.4
[19] systemPipeR_1.4.8       GOstats_2.36.0
[21] splines_3.2.2           BiocParallel_1.4.3
[23] stringr_1.0.0           pheatmap_1.0.8
[25] RCurl_1.95-4.8          biomaRt_2.26.1
[27] munsell_0.4.3           sendmailR_1.2-1
[29] rtracklayer_1.30.4      base64enc_0.1-3
[31] BBmisc_1.9              fail_1.3
[33] edgeR_3.12.1            XML_3.98-1.4
[35] AnnotationForge_1.12.2  bitops_1.0-6
[37] grid_3.2.2              RBGL_1.46.0
[39] xtable_1.8-2            GSEABase_1.32.0
[41] gtable_0.2.0            magrittr_1.5
[43] scales_0.4.0            graph_1.48.0
[45] KernSmooth_2.23-15      amap_0.8-14
[47] stringi_1.1.1           hwriter_1.3.2
[49] genefilter_1.52.1       latticeExtra_0.6-28
[51] futile.logger_1.4.1     brew_1.0-6
[53] rjson_0.2.15            lambda.r_1.1.7
[55] RColorBrewer_1.1-2      tools_3.2.2
[57] Category_2.36.0         survival_2.39-4
[59] AnnotationDbi_1.32.3    colorspace_1.2-6
[61] caTools_1.17.1          knitr_1.13

I could attach the heatmaps too but don't know how.

Both the heatmaps use pearson correlation?

0
Entering edit mode

And when you say score in the peak files, does that mean an explicit column in the BED files or it calculates it on its own?

Does the first one only use the BED files and the second uses both BED and BAM files?

0
Entering edit mode

Initially (before counting) the scores are from the bed file.  If there are no scores, they will (if I recall correctly) default to 1.  After counting, the bed file(s) are used to form a consensus peak set, but then the score itself is calculated from the reads in the bam files.

0
Entering edit mode
@shubhibartaria-10946
Last seen 5.3 years ago

I just faced another dilemma. When I used the dba.count after adding one more bed and bam file which was a combination of all the other 3 files(Mid Brain 1,2,3), I got a plot which was a little different than the one I plotted without the Joint file. Now, if comparison was a pairwise comparison, how did the values change in the heatmap? It was 0.84 between MB2 and MB3 and 0.85 between MB3 and MB1. Without the Joint file, it was 0.9 between MB3 and MB1 and approx 0.89 between MB2 and MB3 (can we not get exact values when plotting heatmaps?)

Could you explain to me how this would have happened.

And I just wanted to inform that my BED files didn't have any score column. It just has chr #, start, end and name. Would that affect the occupancy plot or the DiffBind package calculates it on its own?

I hope I am being clear regarding my queries, because I am very new to this. Anyway I can better articulate my query, do kindly let me know

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The numbers could change for a couple of reasons:

1) The normalization can (will) change when you add new data, so the numbers could shift a bit.

2) The addition of the new data will change the consensus peak set, so the calculation of the correlation will change.

I'm not sure what you mean by a "joint" file... you shouldn't be merging bed files and then adding that as a new peak set.  What is it you're really trying to achieve here?

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A: DiffBind affinity vs occupancy Heatmaps

-Rory


text:
desctable {desctable} R Documentation

## Generate a statistics table

### Description

Generate a statistics table with the chosen statistical functions, and tests if given a "grouped" dataframe.

### Usage

desctable(data, stats, tests, labels)

## Default S3 method:
desctable(data, stats = stats_auto, tests, labels = NULL)

## S3 method for class 'grouped_df'
desctable(data, stats = stats_auto, tests = tests_auto, labels = NULL)

### Arguments

 data The dataframe to analyze stats A list of named statistics to apply to each element of the dataframe, or a function returning a list of named statistics tests A list of statistical tests to use when calling desctable with a grouped_df labels A named character vector of labels to use instead of variable names

### Value

A desctable object, which prints to a table of statistics for all variables

### Labels

labels is an option named character vector used to make the table prettier.

If given, the variable names for which there is a label will be replaced by their corresponding label.

Not all variables need to have a label, and labels for non-existing variables are ignored.

labels must be given in the form c(unquoted_variable_name = "label")

### Stats

The stats can be a function which takes a dataframe and returns a list of statistical functions to use.

stats can also be a named list of statistical functions, or purrr::map like formulas.

The names will be used as column names in the resulting table. If an element of the list is a function, it will be used as-is for the stats.

### Tests

The tests can be a function which takes a variable and a grouping variable, and returns an appropriate statistical test to use in that case.

tests can also be a named list of statistical test functions, associating the name of a variable in the data and a test to use specifically for that variable.

That test name must be expressed as a single-term formula (e.g. ~t.test), or a purrr::map like formula (e.g. ~t.test(., var.equal = T)). You don't have to specify tests for all the variables: a default test for all other variables can be defined with the name .default, and an automatic test can be defined with the name .auto.

If data is a grouped dataframe (using group_by), subtables are created and statistic tests are performed over each sub-group.

### Output

The output is a desctable object, which is a list of named dataframes that can be further manipulated. Methods for printing, using in pander and DT are present. Printing reduces the object to a dataframe.

### See Also

stats_auto

tests_auto

print.desctable

pander.desctable

datatable.desctable

### Examples

iris %>%
desctable()

# Does the same as stats_auto here
iris %>%
desctable(stats = list("N"      = length,
"Mean"   = ~ if (is.normal(.)) mean(.),
"sd"     = ~ if (is.normal(.)) sd(.),
"Med"    = stats::median,
"IQR"    = ~ if(!is.factor(.)) IQR(.)))

# With labels
mtcars %>% desctable(labels = c(hp  = "Horse Power",
cyl = "Cylinders",
mpg = "Miles per gallon"))

# With grouping on a factor
iris %>%
group_by(Species) %>%
desctable(stats = stats_default)

# With nested grouping, on arbitrary variables
mtcars %>%
group_by(vs, cyl) %>%
desctable()

# With grouping on a condition, and choice of tests
iris %>%
group_by(Petal.Length > 5) %>%
desctable(tests = list(.auto = tests_auto, Species = ~chisq.test))

[Package desctable version 0.3.0 Index]


text:
# Phoebe

Phoebe has been discovered in 1899; this was the first discovery of an irregular satellite in the solar system and the first discovery of a moon through photography. With a size of 219 km × 217 km × 204 km and containing >98% of the mass of Saturn’s irregular-moon system, it is the giant of Saturn’s outer moons. Phoebe’s mean distance to Saturn is ∼13 million kilometers; this is among the smallest of all known Saturnian irregular moons, with one revolution around the planet on a retrograde orbit requiring 1 year and 6 months.
By synchronizing the shape model of Phoebe with ground-based stellar occultations and lightcurves, the rotation period was determined to 9 h 16 min 25.14 sec ± 0.07 sec. Different to the orbit direction, the spin direction is prograde. Since the north pole of Phoebe’s rotation axis points quite close to Saturn’s rotation north pole, the seasonal variations on Phoebe’s surface are very similar to the planet’s and the inner moons’s seasons; the sub-solar latitude varies by about ±25° over the course of one Saturn revolution about the Sun. On 11 June 2004, the Cassini spacecraft made a close flyby three weeks before arrival at Saturn.

Fig. (left): Short animation of Cassini images of Phoebe while moving through constellation Virgo on 07 Jan 2015 (21 frames; time span: 2:04 h; exposure times: 1.5 sec; range: 12.7 million km; Cassini orbit: rev 211). In the first image, the object is marked by a black cross. The background stars move through the field of view because the camera of the fast moving Cassini spacecraft was tracking Phoebe. The slight horizontal jitter of Phoebe is due to improper tracking at the arcseconds level. Flickering bright spots stem from cosmic-ray hits on the camera’s CCD detector, fixed bright or dark pixels are incorrectly calibrated hot or cold pixels on the CCD, respectively. The space background, in reality pitch black, is displayed in dark gray because this makes visibility of the object easier. Image IDs: N1799321159 to N1799328599.

Fig. (right): Phoebe landscape — high-resolution view from Cassini’s targeted flyby on 11 Jun 2004. Image resolution is ∼65 meters per pixel, phase angle ∼83°.

This page is intended to compile (parts of) our knowledge of Phoebe in compact form. Its focus is on the documentation of my Cassini-ISS work (observation planning and data analysis), but it also provides general information obtained from other work, like discovery circumstances and orbital and physical parameters (including Phoebe’s rotation period). It does not include the raw data (images or spectra) taken by the Cassini spacecraft, these are available at NASA’s Planetary Data System (PDS). For further reading on Phoebe and on irregular moons of Saturn in general, see the reference list at my outer-Saturnian moons page or the reference list below.

This website is still under development, and more content will be added in the forseeable future. I will remove this note when the page will be close to completion. In the meantime, for example the Phoebe entry in Wikipedia is a rich first-order source of information and references.

Last update: 05 Jan 2021 — page content is best displayed on a screen at least 1024 pixels wide

### (1) Astronomical and physical properties

 Moon name Saturn range Orbit period Orbit direction Size Rotation period Discovery year Phoebe million km years retrograde km h min 1899

Sheet (1) offers basic information about Phoebe in tabular form:
(1A) Designations and discovery circumstances
(1B) Orbit parameters
(1C) Physical parameters (body properties)
← Tables (1A) to (1C) in text format (last updated: 31 Jul 2019)
← Screenshot of Table 4 in Denk et al. (2018) (many values of the tables here in sheet (1) refer to it)

Most fundamental values are highlighted in red. The notes offer explanations, calculations, accuracies, references, etc. The data were obtained from spacecraft as well as from ground-based observations.

###### (1A) Designations and discovery circumstances
 Moon name(1) Phoebe IAU number(3) Saturn IX First observation date(7) 16 Aug 1898 Moon abbrev. (TD)(2) Pho Provisional desig.(4) — Announcement date(7) 17 Mar 1899 SPICE ID(5) 609 Harvard Obs. announcement(7) Bull. no. 49 Also-used label(6) S9 Discoverers(8) W.H. Pickering, Stewart

Notes for Table 1A:

(1) Phoebe’s name was suggested by W.H. Pickering about a month after discovery in a note from E.C. Pickering published in the Harvard College Observatory Circular and in identical letters to the Journal of Popular Astronomy, the Astronomical Journal, and the Astronomische Nachrichten. It is named after Phoebe, a sister of Saturn and a Titaness in greek mythology, and follows the scheme of naming Saturnian moons by his siblings, introduced by John Herschel in his work Results of Astronomical Observations made at the Cape of Good Hope from 1847 (in paragraph (398) of chapter VI on page 415).

(2) I use this 3-letter abbreviation in the diagrams of my publications simply for practicability reasons. These have no offcial character.

(3) Moon numbers are assigned by the International Astronomical Union (IAU)’s Committee for Planetary System Nomenclature. For satellites, roman numeral designations are used.

(4) Designation given to the object in the first announcement; the guidelines are explained here and have not been invented at the time of Phoebe’s discovery.

(5) SPICE is a commonly-used information system of NASA’s Navigation and Ancillary Information Facility (NAIF). It assists engineers in modeling, planning, and executing planetary-exploration missions, and supports observation interpretation for scientists. Each planet and moon obtained a unique SPICE number.

(6) ‘S’ for ‘Saturnian moon’ plus the roman numeral designation in arabic numbers are often-used labels for satellites. Not sure how official that is.

(7) Given are the date of the photography wherein Phoebe was spotted for the first time, and the discovery-announcement date. The Phoebe discovery was reported in a hand-written Bulletin of the Harvard College Observatory in Cambridge, MA. Phoebe was the first ever satellite in the solar system detected photographically.

(8) The discoverer team included: William H. Pickering, DeLisle Stewart. The discovery announcement was made by Edward C. Pickering, older brother of W.H.P. and director of Harvard College Observatory at that time.

###### (1B) Orbit parameters
 Orbit direction(1) retrograde Group member(2) Norse Dynamical family(3) ? Periapsis range(4) 10.83 ⋅ 106 km Semi-major axis(5) 12.948 ⋅ 106 km Apoapsis range(6) 15.06 ⋅ 106 km Semi-major axis(7) 215 R♄ Semi-major axis(8) 0.087 au Semi-major axis(9) 0.20 RHill Orbit eccentricity(10) 0.163 Orbit inclination(11) 175.24° Inclination supplemental angle(12) 4.76° Orbital period(13) 548 d Orbital period(14) 1 y 6 m Mean orbit velocity(15) 1.71 km/s

Notes for Table 1B:

(1) Prograde (counterclockwise as seen from north) or retrograde (clockwise as seen from north) — this difficult problem was solved by Pickering (1905).

(2) Norse, Inuit, or Gallic

(3) Phoebe might be the parent body of several small irregular moons (esp. the Mundilfari group), but this is completely unclear so far; see also Fig. 1 and Table 2 in Denk et al. (2018)

(4) $r_{Peri}=a\cdot(1-e)$

(5) Orbit semi-major axis a, from Table 4 in Denk et al. (2018)

(6) $r_{Apo}=a\cdot(1+e)$

(7) Saturn radius R = 60330 km (100 mbar level)

(8) Astronomical Unit 1 au = 149 597 870.7 km

(9) Saturn’s Hill sphere radius $R_{Hill}=\sqrt[3]{m_♄/3m_☉}\cdot r_{♄↔☉}$= ∼65 ⋅ 106 km = ∼1085 R♄ = ∼3° as seen from Earth at opposition (with mass of Saturn m = 5.6836 ⋅ 1026 kg and perihel range Saturn↔Sun r♄↔ = 1.353 ⋅ 109 km)

(10) Orbit eccentricity e, from Table 4 in Denk et al. (2018)

(11) Orbit inclination i, from Table 4 in Denk et al. (2018)

(12) Orbit “tilt” or inclination supplemental angle i’ = i for prograde moons; i’ = 180°−i for retrograde moons

(13) From Table 4 in Denk et al. (2018)

(14) Value from (13) in units of years, months, weeks

(15) From Table 4 in Denk et al. (2018)

###### (1C) Physical parameters
 Mean size(1) 213 $^{+1}_{−1}$ km Min. equatorial axes ratio(4) 1.01 Mass(6) 8.29 ⋅ 1018 kg Mean radius(2) 106.4 km Axes radii (a × b × c)(5) ∼ 109 × 108 × 102 km Mean density(7) 1.64 g/cm3 Equatorial circumference(3) ∼ 685 km Surface escape velocity(8) ∼ 320 km/h Rotation period (sidereal)(9) 9.27365 h +/- (9) 0.07 s Spin rate(9) 2.588 d−1 Spin direction(10) prograde Pole dir. (ecliptic longitude λ)(12) 61.2° ± 0.3° Pole direction (geocentric, RA)(13) 356.6° ± 0.3° Seasons(11) none Pole dir. (ecliptic latitude β)(12) +64.5° ± 0.1° Pole direction (geocentric, Dec)(13) +78.0° ± 0.1° Absolute visual magnitude(14) 6.6 mag Apparent vis. mag. from Earth(15) 16 mag Best apparent mag. for Cassini(16) 5.1 mag Spectral slope(17) ∼ −2.5 %/100nm B−R color index(17) ∼ 0.98 / ∼ 0.91 Albedo (V-band)(18) 0.086 Hill sphere radius(19) ∼ 18000 km Hill sphere radius(20) ∼ 170 rPho

Notes for Table 1C:

(1) From Thomas et al. (2018); see also Table 4 in Denk et al. (2018) for a tabular compilation of many Phoebe properties.

(2) Half the diameter value. While the diameter is the intuitive size number, the radius r is mainly used in formulas to calculate other quantities.

(3) Estimated from the equatorial axes radii, see note (5).

(4) From Castillo-Rogez et al. (2012) (see also Table 3 in Denk et al. (2018)).

(5) Here, a is the long equatorial, b the short equatorial, and c the polar axis dimension of the reference ellipsoid (see Table 4 in Denk et al. (2018)).

(6) From Jacobson et al. (2006); see also Table 4 in Denk et al. (2018).

(7) From Thomas et al. (2018); see also Table 4 in Denk et al. (2018).

(8) From Porco et al. (2005); see also Table 4 in Denk et al. (2018).

(9) From Gomes-Júnior et al. (2020), determined through synchronization of the Gaskell et al. (2013) shape model with six stellar occultations. The “older”, ∼30x less accurate value from Bauer et al. (2004) (9.2735 ± 0.0006 h) was used in Tables 3 and 4 in Denk et al. (2018). The spin rate is 24/P, measured in units of one per day.

(10) Valid entries: Prograde (counterclockwise as seen from north), retrograde (clockwise as seen from north), ‘lying on the side’ (pole direction almost perpenticular to ecliptic pole), or ‘unknown’.

(11) Valid entries: “None” (rotation axis points close to one of the ecliptic poles), “moderate” (rotation axis is moderately tilted), or “extreme” (rotation axis is highly tilted, points somewhere close to the ecliptic equator), or ‘unknown’.

(12) From Giese et al. 2006; see also the pole-direction sheet below for details.

(13) As (12), but in geocentric coordinates (Right Ascension and Declination).

(14) From Miller et al. 2011; see also Tables 2 and 4 in Denk et al. (2018). The absolute visual magnitude H is the magnitude (brightness) of an object (in the visible wavelength range) if located 1 au away from the sun and observed at 0° phase angle (i.e., in this definition, the observer virtually sits at the center of the sun). The magnitude scale is logarithmic, with an object of 6th mag being 100x darker than a 1st mag object.

(15) Apparent visual magnitude V; from Table 2 in Denk et al. (2018).

(16) From Table 3 in Denk et al. (2018). Given is the best apparent magnitude as seen from Cassini at a time when a low-resolution lightcurve observation took place (i.e., not during the targeted flyby in June 2004 where Phoebe became much brighter and bigger).

(17) Color information: Spectral slope S’2  and BR color index; from Table 3 in Grav and Bauer (2007). The higher the value, the “redder” the color of the object. Mean wavelengths: 445 nm for B (“blue”), 658 nm for R (“red”) filters. BR of the Sun is 1.01 (Ramírez et al. 2012).

(18) From Miller et al. 2011; see also Table 4 in Denk et al. (2018).

(19) Hill radius at periapsis under the assumption of the given density (see note (7)).

(20) Hill radius at periapsis in Phoebe-radius units. With $R_{Hill}=\sqrt[3]{4\pi\rho_{Pho}/9m_♄}\cdot r_{Pho↔♄}$, this number only depends on the object’s distance to the central body (Saturn; linear dependency) and on the object’s density (proportional to the cubic root; see also note (7)).

### (3) Images

Phoebe as seen with Cassini’s narrow-angle camera within ±1 hour of closest approach in June 2004.

Left: Before; right: after closest approach. The spatial resolution is ~80 m/pixel (left) and ~65 m/pixel (right). See NASA’s planetary photojournal (PIA18411) for more details on these two images.

### (5) Shape model

General:
– Available
– References: Porco et al. (2005); Giese et al. (2006); Gaskell (2013)

Shape model of Phoebe from PDS Small Bodies Node (Gaskell 2013). Link to original pdf file →here

Shape model of Phoebe with relative color-coded heights relative to an equipotential surface; adapted from Fig. S1 from Porco et al. (2005).

The calculation assumed homogeneous mass distribution, and accounts for rotational accelerations. Equatorial views with north at the top. Viewpoint longitudes are noted.

### (9) References for Phoebe

Discovery: Harvard College Observatory Bulletin no. 49 (17 Mar 1899)
Wikipedia:  Phoebe (moon) Phoebe (Mond)
Planetary Photojournal (NASA/JPL): Phoebe images and maps
My ‘Outer Moons of Saturn’ website: Sheet ‘links and references’
JPL’s Icy Moon Treks page: Phoebe Trek

References (that include my work) (time-ordered; newest first)

 Denk, T., Mottola, S. (2019): Studies of Irregular Satellites: I. Lightcurves and Rotation Periods of 25 Saturnian Moons from Cassini Observations. Icarus 322, 80-103. doi:10.1016/j.icarus.2018.12.040 Denk, T., Mottola, S., Tosi, F., Bottke, W.F., Hamilton, D.P. (2018): The Irregular Satellites of Saturn. In: Enceladus and the Icy Moons of Saturn (Schenk, P.M., Clark, R.N., Howett, C.J.A., Verbiscer, A.J., Waite, J.H., editors), Space Science Series, The University of Arizona Press, pp. 409-434. Jaumann, R., Clark, R.N., Nimmo, F., Hendrix, A.R., Buratti, B.J., Denk, T., Moore, J.M., Schenk, P.M., Ostro, S.J., Srama, R. (2009): Icy Satellites: Geological Evolution and Surface Processes. In: Saturn after Cassini-Huygens, Dougherty, M., Esposito, L., Krimigis, S. (eds.), Springer-Verlag, chapter 20, 637-681. Roatsch, Th., M. Wählisch, F. Scholten, A. Hoffmeister, K.-D. Matz, T. Denk, G. Neukum, P.C. Thomas, P. Helfenstein, C.C. Porco (2006): Mapping of the icy Saturnian satellites: First results from Cassini-ISS. Planetary and Space Science 54, 1137-1145. Giese, B., G. Neukum, Th. Roatsch, T. Denk, C.C. Porco (2006): Topographic modeling of Phoebe using Cassini images. Planetary and Space Science 54, 1156-1166. Porco, C.C., Baker, E., Barbara, J., Beurle, K., Brahic, A., Burns, J.A., Charnoz, S., Cooper, N., Dawson, D.D., DelGenio, A.D., Denk, T., Dones, L., Dyudina, U., Evans, M.W., Giese, B., Grazier, K., Helfenstein, P., Ingersoll, A.P., Jacobson, R.A., Johnson, T.V., McEwen, A.S., Murray, C.D., Neukum, G., Owen, W.M., Perry, J., Roatsch, T., Spitale, J., Squyres, S.W., Thomas, P.C., Tiscareno, M., Turtle, E.P., Vasavada, A.R., Veverka, J., Wagner, R., West, R. (2005): Cassini Imaging Science: Initial Results on Phoebe and Iapetus. Science 307, 1237-1242.

♦ References (work from colleagues) (alphabet-ordered)

 Bauer, J.M., Buratti, B.J., Simonelli, D.P., Owen, W.M. (2004): Recovering the rotational light curve of Phoebe. Astrophys. J. 610, L57-L60. Bauer, J.M., Grav, T., Buratti, B.J., Hicks, M.D. (2006): The phase curve survey of the irregular saturnian satellites: A possible method of physical classification. Icarus 184, 181-197. Bottke W.F., Nesvorný D., Vokrouhlický D., Morbidelli A. (2010): The irregular satellites. The most collisionally evolved population in the solar system. Astron. J. 139, 994-1014. Buratti, B.J., Hicks, M.D., Davies, A. (2005): Spectrophotometry of the small satellites of Saturn and their relationship to Iapetus, Phoebe, and Hyperion. Icarus 175, 490-495. [ Word of caution: Nomenclature is erroneously mixed in this paper. S/2000 S3 is (S29) Siarnaq, not (S21) Tarvos. S/2000 S11 is (S26) Albiorix, not (S29) Siarnaq. The four irregulars observed for this work were Ymir, Paaliaq, Siarnaq, and Albiorix — but not Tarvos. ] Castillo-Rogez, J.C., Johnson, T.V., Thomas, P.C., Choukroun, M., Matson, D.L., Lunine, J.I. (2012): Geophysical evolution of Saturn’s satellite Phoebe, a large planetesimal in the outer Solar System. Icarus 219, 86-109. Gaskell, R.W. (2013): Gaskell Phoebe Shape Model V2.0. CO-SA-ISSNA-5-PHOEBESHAPE-V2.0. NASA Planetary Data System. Gomes-Júnior, A.R., Assafin, M., Braga-Ribas, F., Benedetti-Rossi, G., Morgado, E., Camargo, J.I.B., Vieira-Martins, R., Desmars, J., Sicardy, B., Barry, T., Campbell-White, J., Fernández-Lajús, E., Giles, D., Hanna, W., Hayamizu, T., Hirose, T., De Horta, A., Horvat, R., Hosoi, K., Jehin, E., Kerr, S., Machado, D.I., Mammana, L.A., Maybour, D., Owada, M., Rahvar, S., Snodgrass, C. (2020): The first observed stellar occultations by the irregular satellite Phoebe (Saturn IX) and improved rotational period. Month. Not. Roy. Astron. Soc. 492, 770-781. Grav, T., Bauer, J.M. (2007): A deeper look at the colors of the saturnian irregular satellites. Icarus 191, 267-285. Grav, T., Bauer, J.M., Mainzer, A.K., Masiero, J.R., Nugent, C.R., Cutri, R.M., Sonnett, S., Kramer, E. (2015): NEOWISE: Observations of the irregular satellites of Jupiter and Saturn. Astrophys. J. 809(3), 9 pp. Graykowski, A., Jewitt, D. (2018): Colors and Shapes of the Irregular Planetary Satellites. Astron. J. 155(184), 10 pp. Jacobson, R.A., Antreasian, P.G., Bordi, J.J., Criddle, K.E., Ionasescu, R., Jones, J.B., Mackenzie, R.A., Meek, M.C., Parcher, D., Pelletier, F.J., Owen, W.M., Roth, D.C., Roundhill, I.M., Stauch, J.R. (2006): The Gravity Field of the Saturnian System from Satellite Observations and Spacecraft Tracking Data. Astron. J. 132, 2520-2526. Miller, C., Verbiscer, A.J., Chanover, N.J., Holtzman, J.A., Helfenstein, P. (2011): Comparing Phoebe’s 2005 opposition surge in four visible light filters. Icarus 212, 819-834. Pickering, W.H. (1905): The ninth satellite of Saturn. Annals of Harvard College Observatory 53, 45-74. Thomas, P.C., Tiscareno, M.S., Helfenstein, P. (2018): The Inner Small Satellites of Saturn and Hyperion. In: Enceladus and the Icy Moons of Saturn (Schenk, P.M., Clark, R.N., Howett, C.J.A., Verbiscer, A.J., Waite, J.H., editors), Space Science Series, The University of Arizona Press, 387-408.


text:
# Tag Info

1

This is the classical sampling theorem. When you sample a continuous signal $x(t)$, you are basically multiplying it by a sample train $s(t) = \sum_{-\infty}^{+\infty}\delta(t-kT_s)$, the value at consecutive $T_s$ being your samples. In frequency domain, it effect is to convolve $X(\omega)$ and $S(\omega)$ S(\omega) = \sum_{-\infty}^{+\infty}\delta(\...

0

We can do it in 2 simple steps: You can actually estimate $\omega$ of your input signal easily by taking FFT of the measured n samples. I am suggesting this only because offline processing is an option, otherwise this would be pretty computationally costly. Once you have a close estimate of $\omega$, then you can use Weiner filtering to predict $x[n]$ from ...

0

On a simpler note, if you take WLAN systems, the preamble has initial sequence called STF which is comprised of repetitions of known sequences. Let us say the length of basic sequence is $L$. If there are 4 repetitions (like 802.11n), total length of STF will be $5/4L$ including Cyclic Prefix. For now, we will ignore cyclic prefix part and let us consider ...

0

The noise of interest would ultimately be over the bandwidth of your signal and specifically all filtering done prior to ultimately making a decision over which symbol was transmitted- which would typically be the bandwidth of your matched filter. So given a noise density in $dBm/Hz$ you can translate that to the total noise in the signal to noise ratio of ...

1

Yes this is very common to have a dynamic loop bandwidth such that during acquisition the loop bandwidth is wider, and then once acquired to tighten it up for better noise performance. A typical loop will have an error signal determined which is presented to the input of the loop filter. The filtered version of this error signal can be thresholded and used ...

2

Negative frequencies exist both mathematically and logically and you could probably accomplish the logical demonstration yourself if you want but I'll try. The mathematical demonstration is much more straightforward. OK so the logical approach would be this. Consider the energy flow in a tank circuit in a problem you are analysing. When the energy flows from ...

2

There are two numbers that square to be $-1$. Pick either one of those two numbers and call that "$j$". Then the other one is "$-j$". Doesn't matter which one is picked. The difference between $+j$ and $-j$ is only an arbitrary choice. A convention. Now multiply that $+j$ and $-j$ by a single non-zero real number. Doesn't matter which sign but let's ...

3

Consider a wheel rotating counter-clockwise at one revolution per second. Its frequency of rotation is 1 Hz. If it rotates clockwise, its frequency of rotation is -1 Hz. It's as simple as that.

1

What does it mean exactly to "exist" vs "just theoretical"? Do we for some reason think that $cos(\omega t)$ exists while $e^{j\omega t}$ does not? Both are equally mathematical constructions that describe our physical world. We somehow conclude that the latter as a complex quantity does not exist but the former as a real quantity does, but I don't see a ...

-1

With respect to a known given fixed point in time (and space if you assume a time-space coordinate system), a negative frequency sine wave is simply the negative of a sine wave or sine() function basis vector that starts with a phase of exactly zero at that same exact point in time. The fixed point in absolute time can also be defined to be periodic with ...

0

Can someone please kindly explain this idea of representation of real sinusoidal signals in terms of positive and negative frequency components? Here is my view. As you can see in most books related to DSP, a complex exponential $e^{j\omega_0 t + \phi}$ is a fundamental signal representing a point on unit circle on complex plane at the angle $\omega_0 t ... 0 You can have the channel of 200kHz split into orthogonal sub-carriers. But for this you need to meet certain requirements to eliminate ICI (Inter-Carrier Interference, Not ISI Inter Symbol Interference). If you lose orthogonality, the data in one sub-carrier will be affected by data in another sub-carrier (hence inter-carrier interference). In order to ... 0 In addition to the distinction that frequency is an objective measure and pitch is a subjective one, it's also useful to note that the pitch of a note may not be directly related to any easy measure of frequency. Case in point: a bell, tuned to A440, will generally emit sound energy at roughly 880Hz, 1320Hz, 1760Hz, etc. -- in other words, for a pitch ... 0 In simple terms your regular frequency is how many times in a second, the signal goes through the same cycle. This is not restricted to a sinusoid alone (because any real world signal can be represented by sum of sinusoids). When you say regular frequency is 60Hz, the signal cycles itself 60 times in a second. Each period is 1/60s. A sinewave is a ... 3 Frequency is mathematically defined as the number of cycles per second. So it is a more strict word mathematically. It is represented numerically by the unit called Hertz.$f=1/T$, where$T$represents the one-period length of a waveform. This makes frequency quantifiable. Pitch on the other hand, is a perceptual characteristic of a sound frequency, so it'... 0 It's a matter of the units used to measure frequency as events which happen per unit of time. For the regular frequency f, we ask: How any wavelengths pass a specific point per second?\ For the angular frequency$\omega$, we ask: How many radians are traversed per second? The easiest way to visualise the angular frequency is by understanding the ... 3 @Man. Hi. Measuring frequency in terms of$\omega$(radians/second) or in terms of$f$(cycles/second, Hz) is the same as measuring speed in miles/hour or kilometers/hour. People measure frequency in terms that are convenient for them. In algebra equations it's convenient to represent frequency in terms of$\omega$because it's easier to write the single$\...

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Many signals we observe are functions of time, i.e., their value changes with an independent time variable, which can be measured in seconds. Frequency is a measure of how fast a signal changes and thus measured in "per second" which is the same as Hertz: 1 Hz = 1/s. An oscillation is an excellent example: a harmonic wave can be described as \$e^{\jmath 2\...

1

Frequency is a mathematical/physical concept while pitch is a perceptual concept that correlates with frequency. Edit: or in wikipedias words: « Frequency is an objective, scientific attribute that can be measured. Pitch is each person's subjective perception of a sound wave, which cannot be directly measured.» https://en.m.wikipedia.org/wiki/Pitch_(music)

1

I'm not sure but there's something called the time-frequency resolution limit, basically the shorter your windowing interval the broader your spectrum is going to be. The frequency resolution in the low frequency area may be poor because of this. Things like wavlets attempt to resolve this problem but i dont know anything about them.

Top 50 recent answers are included


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# Splitting a multiplication into multiple smaller steps, reaching the same result

Suppose I have a number, x, which should be doubled every second.

If one had a function which is called exactly once every second, the solution would be simple: All you would have to do was multiplying x by 2.

What, however, can you do to achieve the same results when working not in one-second but very tiny steps?

The target function should, given a small delta time value, still multiply x by 2 every second.

An example:

1. x == 3

2. Apply the function every millisecond for one second, therefore passing 1/1000 as parameter to it

3. Now x == 6

• How can I solve this problem?
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The function should multiply by $2^{1/1000}$ (the 1000th root of 2). – Florian Jul 25 '11 at 12:42
@Florian, that's certainly the right answer from the point of view of pure mathematics, but I suspect that if you tried to implement it in floating point, it wouldn't do what you want it to do. – Gerry Myerson Jul 25 '11 at 12:47
Perhaps this isn't quite what you want, but... have the function store an internal counter. If the counter is less than 1000, the function returns its input. If the counter equals 1000, then the function sets the counter to 0 and returns twice its input. – Chris Taylor Jul 25 '11 at 12:59

You want a function of time that multiplies by a fixed coefficient every $\delta$ time step, and which doubles every second. In other words, $x(t+\delta) = a x(t)$. Assuming $\delta = 1/n \text{ sec}$ for some integer $n$, you get $x(t+1)=x(t+n\delta) = a^nx(t)$, and hence you have to solve $a^n=2$, which results in $a=2^{1/n}$.

The only problem is that the smaller you make $\delta = 1/n$, the faster this procedure will accumulate error from what you would get just multiplying by $2$ every second, though I can't say off the top of my head whether or not the deviation is negligible for your purposes. (You wouldn't have to worry at all though if you didn't bother with successive iteration and simply calculated the value of $x(t)$ explicitly as $2^tx(0)$ - accurate and much less resource-intensive.)

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The mathematically correct answer has been given a couple of times, so allow me to suggest another method that produces similar results without accumulating error.

Create an array $A$ whose indexes range from $A[0]$ to some big number—preferably, a power of two, like $A[1024]$. Set $A[0] = 1$ and $A[1024] = 2$. Now, to calculate the values in between, find the midpoint of this array ($A[512]$) and set it to the geometric mean of $A[0]$ and $A[1024]$; that is, $\sqrt{A[0] \cdot A[1024]} = \sqrt{2}$. Again, find the midpoint of two numbers and take their geometric mean, so that, for example:

$$\begin{array}\\A[256] = \sqrt{A[0] \cdot A[512]}\\ A[384] = \sqrt{A[256] \cdot A[512]}\\ A[448] = \sqrt{A[384] \cdot A[512]}\\ A[416] = \sqrt{A[384] \cdot A[448]}\end{array}$$

And so on. After you've calculated this array, your "slow doubling" function can grab successive values out of this array, since those values gradually increase from 1 to 2.

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You need an internal counter to implement this. On the other hand, if you can use an internal counter, you actually do not need this array: just double the number once in 1000 times, and do nothing for the rest. – Tsuyoshi Ito Jul 25 '11 at 15:27

You do not say if the the function can use an internal counter, nor the desired intermediate results. Moreover you did not even specify which set x belongs to: integers? rationals? complex numbers?

If x is a real number and you want a linear function xi+1 <- f(xi), of course the answer is: use A radix of 2. (There can be two of them if the number of steps is even, or plenty of them if you work on complex numbers, or ...)

On the other side, if you do not necessarily need a linear function, there are lots of awful tricks to obtain function giving random-looking results but ensuring fn(x) = 2x.

A simple example working with small integers (but this is linear in the GF) is the following:

x <- (x*11) % 2601127

i.e. multiply by 11 modulo 2601127. Iterate the function above 1000 times and it will double the initial (integer and smaller than 1300000) value.

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# Kaldi* Statistical Language Model Conversion Tool¶

The Kaldi* Statistical Language Model (SLM) Conversion Tool is a command-line tool that converts Kaldi language model resources to the format supported by the OpenVINO Speech Recognition Demos.

## Command Line¶

kaldi_slm_convertion_tool HCLG.const.fst transitions.txt words.txt slm.fst labels.bin

## Input Parameters¶

HCLG.fst

The HCLG.const.fst parameter is the input weighted finite-state transducer (WFST) file in the OpenFST const format.

Most example scripts create a language model file in that format. If you have a WFST in a different OpenFST format, convert it with the following command:

$KALDI_ROOT/tools/openfst/bin/fstconvert --fst_type=const HCLG.fst HCLG.const.fst The source Kaldi language model file HCLG.fst can be found in directories like exp/tri2b/graph_xyz, where tri2b is the name of the model used for speech recognition tests. transitions.txt The WFST transitions file describes the relations between WFST transitions and neural acoustic model outputs. This file is usually not generated by Kaldi example scripts, so you have to create it with the following command: $KALDI_ROOT/src/bin/show-transitions phones.txt final.mdl > transitions.txt

For this call, the phones.txt file is the phoneme description file, which can often be found in data/lang/phones.txt. The final.mdl file is the neural acoustic model that is used for speech recognition.

words.txt

The words.txt file defines the mappings from word IDs used internally to their text representation. For many Kaldi example scripts, the file can be found in the same directory as HCLG.fst.

## Output Parameters¶

slm.fst

The output file slm.fst is generated by the SLM Conversion Tool. It contains the information needed for the OpenVINO speech recognition demos for decoding.

labels.bin

The labels.bin file defines mappings from word IDs to word strings, like the words.txt file, but in the binary format. The OpenVINO speech recognition example needs the labels.bin file to convert recognized words into a human-readable format.


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# Inequalities with fractions

## Summary and examples

Roughly speaking, inequalities with fractions are inequalities with one or more fractions in which the unknown variable (e.g. $x$) occurs in the denominator or in both the numerator and denominator.
First study the topic Equations with fractions before continuing this topic about inequalities.
The following examples show how you can solve these types of inequalities.

##### Example 1

Solve:

$\displaystyle\frac{1}{x-1}>0$

This is a simple example. The numerator in the left-hand side is always positive, then the fraction will be positive if the denominator is positive as well.

So the fraction is greater than $0$ if:

$x-1>0$

so if:

$x>1$

##### Example 2

Solve:

$\displaystyle\frac{x-1}{x+2}<0$

The solution of this inequality is a bit more difficult to find. The fraction is less than $0$ for those values of $x$ for which the numerator is positive and the denominator is negative, or vice versa. But we have to do more.
A convenient approach is the following. For each factor in both the numerator and denominator draw a so-called number line. For each factor find the values of $x$ for which it is positive, equal to $0$ or negative.

In this case we do this for the factors $x-1$ and $x+2$.

$x-1$ ----------------------------(1)+++++++

$x+2$ ---------------(-2) ++++++++++++++

quotiënt ++++++++(-2) ---------(1)+++++++

The first number line shows that $x-1=0$ for $x=1$, $x-1>0$ for $x>1$ and $x-1<0$ for $x<1$; the second number line shows that  $x+2=0$ for $x=-2$, $x+2>0$ for $x>-2$ and $x+2<0$ for$x<-2$.

The third line is the result of the other two. We are dealing with fractions and therefore we look at the results of divisions in certain intervals: $+/+\rightarrow+$, $+/-\rightarrow-$, $-/+\rightarrow-$ and $-/-\rightarrow+$ . In this case we get: for $x<-2$ the fraction is positive $(-/-)$; for $-2 the fraction is negative $(-/+)$ and for $x>1$ the fraction is positive again $(+/+)$.
This method is quite simple and can also be applied if more factors are involved as we can see in next example.
The inequality above is satisfied for $-2.

##### Example 3

Solve:

$\displaystyle\frac{x^2+x-2}{x+1}>0$

We notice that the numerator can be written as follows:

$x^2+x-2=(x+2)(x-1)$

and thus we have to solve:

$\displaystyle\frac{(x+2)(x-1)}{x+1}>0$

Again we draw a number line for each factor. Make sure that the numbers $-2$, $-1$, $+1$ have the right positions.

x+2 ----------(-2)++++++++++++++++++

x-1 ------------------------------------(1)+++++

x+1 ---------------------(-1)+++++++++++++

result --------(-2)+++(-1)----------(1)++++++

The inequality is satisfied for:

$-2 or $x>1$

##### Example 4

Solve:

$\displaystyle\frac{1}{x}>\displaystyle\frac{x-1}{x+1}$

This example is somewhat more complicated because fractions have to be added or subtracted before we get the right form of inequality.
We move the fractions to the left-hand side:

$\displaystyle\frac{1}{x}-\displaystyle\frac{x-1}{x+1}>0$

We can subtract the fractions after having made the denominators equal:

$\displaystyle\frac{1}{x}.\displaystyle\frac{x+1}{x+1}-\displaystyle\frac{x-1}{x+1}.\displaystyle\frac{x}{x}>0$

Now we can write the inequality as:

$\displaystyle\frac{x+1-x(x-1)}{x(x+1)}>0$

or:

$\displaystyle\frac{-x^2+2x+1}{x(x+1)}>0$

or:

$\displaystyle\frac{x^2-2x-1}{x(x+1)}<0$

The numerator can be factorized:

$x^2-2x-1=(x-1+\sqrt{2})(x-1-\sqrt{2})$

by applying the $abc$-formula.
As a result we have to solve the following inequality:

$\displaystyle\frac{(x-1+\sqrt{2})(x-1-\sqrt{2})}{x(x+1)}<0$

We draw number lines for each of the four factors (-0.4 and 2.4 are approximations):

$x-1+\sqrt{2}$    ----------(-0.4)++++++++++++++++

$x-1-\sqrt{2}$    ------------------------------------------(2.4)+++

$x$   --------------------------------------(0)+++++++++++

$x+1$   -----------(-1)++++++++++++++++++++++

result        +++++++(-1)------(-0.4)++++(0)--------------(2.4)+++

This inequality is satisfied if:

$-1 or $0

We mentioned before that cross-multiplication yields incorrect solutions. We will show this below.

Cross-multiplication would give:

$x+1>x(x-1)$

$x+1>x^2-x$

$x^2-2x-1<0$

We can immediately see that this is not the correct answer.

0


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# Symbols:Greek/Eta

Jump to navigation Jump to search

Previous  ... Next

## Eta

The $7$th letter of the Greek alphabet.

Minuscule: $\eta$
Majuscule: $\Eta$

The $\LaTeX$ code for $\eta$ is \eta .

The $\LaTeX$ code for $\Eta$ is \Eta .

### Order Type of Rational Numbers

$\eta$

The order type of $\struct {\Q, \le}$ is denoted $\eta$ (eta).

Previous  ... Next


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# Frederick S. Woods

Frederick Shenstone Woods (1864–1950) was an American mathematician.

He was a part of the mathematics faculty of the Massachusetts Institute of Technology from 1895 to 1934,[1] being head of the department of mathematics from 1930 to 1934[2] and chairman of the MIT faculty from 1931 to 1933.[3]

In 1901 and 1903 he published two papers on non-Euclidean geometry.[4][5] He also wrote several textbooks.[6]

Following Wilhelm Killing (1885) and others (see History of Lorentz transformations#Woods), Woods described motions in spaces of non-Euclidean geometry in the form:[7]

${\displaystyle x_{1}^{\prime }=x_{1}\cos kl+x_{0}{\frac {\sin kl}{k}},\quad x_{2}^{\prime }=x_{2},\quad x_{2}^{\prime }=x_{3},\quad x_{0}^{\prime }=-x_{1}k\sin kl+x_{0}\cos kl}$

which becomes a Lorentz boost by setting ${\displaystyle k^{2}=-1}$, as well as general motions in hyperbolic space[8]

{\displaystyle {\begin{matrix}{\begin{aligned}x_{1}^{\prime }&=x_{1}\cosh \alpha -x_{0}\sinh \alpha \\x_{2}^{\prime }&=x_{2}\cos \beta -x_{3}\sin \beta \\x_{3}^{\prime }&=x_{2}\sin \beta +x_{3}\cos \beta \\x_{0}^{\prime }&=-x_{1}\sinh \alpha +x_{0}\cosh \alpha \end{aligned}}\end{matrix}}}

## Notes

1. "Faculty - MIT Mathematics". math.mit.edu.
2. "Facts - MIT Mathematics". math.mit.edu.
3. "MIT History - MIT Faculty". libraries.mit.edu.
4. Woods, F. S. (1901). "Space of constant curvature". The Annals of Mathematics. 3 (1/4): 71–112. JSTOR 1967636.
5. Woods, F. S. (1905) [1903]. "Forms of non-Euclidean space". The Boston Colloquium: Lectures on Mathematics for the year 1903: 31–74.
6. A course in mathematics (1907), Analytic geometry and calculus (1917), Elementary calculus (1922), Higher geometry (1922)
7. Woods (1903/05), p. 55
8. Woods (1903/05), p. 72


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# Symmetric and Cyclic Assumptions

It is quite noticeable than this word – “WLOG” (=Without Loss Of Generality) is strewn across the net and books …. This post shall convey its real meaning.

(1) If a function – $f(x,y,z)$ is symmetric

$\implies f(x,y,z) = f(x,z,y) = f(y,x,z) = f(y,z,x) = f(z,x,y) = f(z,y,x)$.

If for some permutation – $f(a,b,c)$ with $a \ge b \ge c$

Since the function is symmetric,

$f(a,b,c) = f(x,y,z)$

Therefore , Proving,

$f(x,y,z) \ge P \Leftrightarrow f(a,b,c) \ge P$ with $a \ge b \ge c$

Thus a pairwise order can be assumed in symmetric functions..

(2) If a function – $f(x,y,z)$ is cyclic

Since the function is only cyclic , we only have –

$f(x,y,z) = f(y,z,x) = f(z,x,y)$

We can either have –

$x \ge y,z$

\$latex y \ge z,x\$

$z\ge x,y$

Let $(x,y,z) = (a,b,c)$ with $a \ge b,c$

$(y,z,x) = (a',b',c')$ with $a' \ge b',c'$

$(z,x,y) = (a'',b'',c'')$ with $a'' \ge b'',c''$

Thus , proving ,

$f(x,y,z) \ge P \Leftrightarrow f(a,b,c) \ge P$

where $(a,b,c)$ is any cyclic permutation of $(x,y,z)$ with $a \ge b,c$

P.S. Thanks to Tejs Mathias for making me understand this.


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# Constrained optimization - quantitative finance

I am trying to perform constrained opmitization for portfolio performance attribution analysis. Specifically, I am trying to determine the impact of sectors performance on the S&P 500 index.

Min y-(b1x1+b2x2+...+bnxn+bn+1xn+1+....bpxp)

subject to b1+b2+...+bp = 1,

0 <= bi <= 1 for i=1,2,...,p

yis the time-series return of S&P500 and xi is the time-series return of a sector for all i = 1, 2, ..., p.

The idea of the betas is that the higher the beta, the greater the impact of the variable (sector) is on the portfolio.

The problem with this optimization is that, sometimes, the sector and market are negatively correlated so beta is negative. But the constraint forces it to be between 0 and 1, so beta becomes 0, which suggests that the sector had no impact on the S&P 500 index (even though it DID have an impact, negatively).

What's the better way of solving this?

Thank you very much in advance!

• If beta can be legitimately negative (can it?), why is is constrained to be $\ge 0$? – Mark L. Stone Jul 11 '18 at 13:10
• @MarkL.Stone Good question. The constraint that beta >=0 exists because I want the betas to represent the impact of a sector on the index in "percentage". – Jun Jang Jul 11 '18 at 13:14
• Say you have two sectors which have equal impact the S&P 500. Assume one is positively correlated, and the other is negatively correlated. Are you saying the desired output of the algorithm in this case is $\beta_1=\beta_2=0.5$? – scherm Jul 11 '18 at 13:41
• @scherm Yes exactly, because I am concerned about the impact of a sector on S&P 500. In your opinion, is that a good analytical method? – Jun Jang Jul 11 '18 at 13:43
• Sorry, I'm not an economist, so I can't comment on the validity of your model. You could try removing the constraints on $\beta$, solve the optimization problem, and then normalize the $\beta_i$'s to sum to 1 afterward to try and understand their relative contributions, but I would be cautious about drawing conclusions that way. As a more general note, it does seem strange to assume that the time-series $y$ can be approximated by a convex combination of other time-series $x_i$. Again, I am not an expert, but this seems like a very simple model for something as complex as market trend analysis. – scherm Jul 11 '18 at 16:46


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I had an interesting idea this morning. At least I find it interesting, but I haven’t slept much lately. Either way, here it is: in Markdown, if you misname a reference link, forget to fill one in or have a malformed URL, your broken Markdown shows up in your output. Wouldn’t it be nice if your preview highlighted those for you before you went to publish?

Marked 1.3, which is coming along very nicely, has a few JavaScripts built in to the preview (which can be turned off in preferences). It provides a table of contents based on headers in your document, smooth scrolling, tool tips to show you where external links will go and, as of this morning, highlighting of broken links.

The code is simple. This version is jQuery, just because that was convenient, but you can pull this off in plain old JavaScript with barely any extra code:

It just scans for Markdown-style links ([text][title] or [text](link)) that are still in your text after converting to HTML. If it finds them, it highlights them in red. Just thought I’d share.

By the way, in addition to the JavaScript fun, the next version of Marked has multiple custom styles (unlimited), can open any text file with any extension, can load MathJax for MathML rendering and much more. I’ll keep you posted on its release, but it should be soon.


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# Noetherian and local Laurent polynomials

Let $R$ be a commutative ring. I am interested in the following two questions:

1) If $R$ is Noetherian, then is $R[u,u^{-1}]$ Noetherian? In fact, I know that this is true (the Hilbert basis theorem tells us $R[u]$ is Noetherian and then $R[u,u^{-1}]$ is a localisation of $R[u]$, which preserves the Noetherian property).

Is there a more insightful proof? I feel like just the above two facts, whilst telling us it is true, don't really offer much insight into why this is true.

2) If $R$ is local, then is $R[u,u^{-1}]$ still local? We are localising $R[u]$ at the multiplicative subset $S=\{ u^i \}_{i \ge 0}.$ The result would follow if I knew $S$ was a maximal multiplicative subset of $R[u]$, but is this true?

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"Insightful proof" is too subjective for me but I have answered question 2). –  Georges Elencwajg Jan 29 '12 at 22:53
Dear Juan, The Hilbert Basis Theorem is enough in this case (see the answer by anonymous below); you don't need to think about localizations if you don't want to. (Although you can if you want to; your argument is perfectly correct.) The Hilbert Basis Theorem is a basic source for Noetherianness of rings; it's not clear what additional insight you want. Regards, –  Matt E Jan 30 '12 at 1:13
@Matt - thanks for the comments. I see now that HBT is indeed required for this! –  Juan S Jan 30 '12 at 1:28

1) I think the best way to think of this ring is as $R[u,v] / (uv-1)$. With this presentation it is clear that it is noetherian.
@Juan: Dear Juan, No, because of the Hilbert Basis Theorem (it is a quotient of $R[u,v]$, which is Noetherian by HBT). Regards, –  Matt E Jan 30 '12 at 1:04
2) No: $\mathbb C$ is local but $\mathbb C[u,u^{-1}]$ isn't.
Indeed, its maximal ideals are the principal ideals $(u-a)$'s with $a\in \mathbb C^*$.


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## G = Dic9.Q8order 288 = 25·32

### The non-split extension by Dic9 of Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — Dic9.Q8
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C2×Dic9 — C4×Dic9 — Dic9.Q8
 Lower central C9 — C2×C18 — Dic9.Q8
 Upper central C1 — C22 — C4⋊C4

Generators and relations for Dic9.Q8
G = < a,b,c,d | a18=c4=1, b2=a9, d2=c2, bab-1=dad-1=a-1, ac=ca, cbc-1=a9b, bd=db, dcd-1=a9c-1 >

Subgroups: 292 in 84 conjugacy classes, 40 normal (38 characteristic)
C1, C2 [×3], C3, C4 [×8], C22, C6 [×3], C2×C4 [×3], C2×C4 [×4], C9, Dic3 [×5], C12 [×3], C2×C6, C42, C4⋊C4, C4⋊C4 [×5], C18 [×3], C2×Dic3 [×4], C2×C12 [×3], C42.C2, Dic9 [×2], Dic9 [×3], C36 [×3], C2×C18, C4×Dic3, Dic3⋊C4 [×4], C4⋊Dic3, C3×C4⋊C4, C2×Dic9 [×4], C2×C36 [×3], Dic3.Q8, C4×Dic9, Dic9⋊C4 [×4], C4⋊Dic9, C9×C4⋊C4, Dic9.Q8
Quotients: C1, C2 [×7], C22 [×7], S3, Q8 [×2], C23, D6 [×3], C2×Q8, C4○D4 [×2], D9, C22×S3, C42.C2, D18 [×3], C4○D12, D42S3, S3×Q8, C22×D9, Dic3.Q8, D365C2, D42D9, Q8×D9, Dic9.Q8

Smallest permutation representation of Dic9.Q8
Regular action on 288 points
Generators in S288
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126)(127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162)(163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198)(199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216)(217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234)(235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252)(253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270)(271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288)
(1 111 10 120)(2 110 11 119)(3 109 12 118)(4 126 13 117)(5 125 14 116)(6 124 15 115)(7 123 16 114)(8 122 17 113)(9 121 18 112)(19 227 28 218)(20 226 29 217)(21 225 30 234)(22 224 31 233)(23 223 32 232)(24 222 33 231)(25 221 34 230)(26 220 35 229)(27 219 36 228)(37 264 46 255)(38 263 47 254)(39 262 48 253)(40 261 49 270)(41 260 50 269)(42 259 51 268)(43 258 52 267)(44 257 53 266)(45 256 54 265)(55 191 64 182)(56 190 65 181)(57 189 66 198)(58 188 67 197)(59 187 68 196)(60 186 69 195)(61 185 70 194)(62 184 71 193)(63 183 72 192)(73 241 82 250)(74 240 83 249)(75 239 84 248)(76 238 85 247)(77 237 86 246)(78 236 87 245)(79 235 88 244)(80 252 89 243)(81 251 90 242)(91 130 100 139)(92 129 101 138)(93 128 102 137)(94 127 103 136)(95 144 104 135)(96 143 105 134)(97 142 106 133)(98 141 107 132)(99 140 108 131)(145 278 154 287)(146 277 155 286)(147 276 156 285)(148 275 157 284)(149 274 158 283)(150 273 159 282)(151 272 160 281)(152 271 161 280)(153 288 162 279)(163 205 172 214)(164 204 173 213)(165 203 174 212)(166 202 175 211)(167 201 176 210)(168 200 177 209)(169 199 178 208)(170 216 179 207)(171 215 180 206)
(1 219 216 160)(2 220 199 161)(3 221 200 162)(4 222 201 145)(5 223 202 146)(6 224 203 147)(7 225 204 148)(8 226 205 149)(9 227 206 150)(10 228 207 151)(11 229 208 152)(12 230 209 153)(13 231 210 154)(14 232 211 155)(15 233 212 156)(16 234 213 157)(17 217 214 158)(18 218 215 159)(19 171 282 121)(20 172 283 122)(21 173 284 123)(22 174 285 124)(23 175 286 125)(24 176 287 126)(25 177 288 109)(26 178 271 110)(27 179 272 111)(28 180 273 112)(29 163 274 113)(30 164 275 114)(31 165 276 115)(32 166 277 116)(33 167 278 117)(34 168 279 118)(35 169 280 119)(36 170 281 120)(37 252 97 197)(38 235 98 198)(39 236 99 181)(40 237 100 182)(41 238 101 183)(42 239 102 184)(43 240 103 185)(44 241 104 186)(45 242 105 187)(46 243 106 188)(47 244 107 189)(48 245 108 190)(49 246 91 191)(50 247 92 192)(51 248 93 193)(52 249 94 194)(53 250 95 195)(54 251 96 196)(55 270 86 130)(56 253 87 131)(57 254 88 132)(58 255 89 133)(59 256 90 134)(60 257 73 135)(61 258 74 136)(62 259 75 137)(63 260 76 138)(64 261 77 139)(65 262 78 140)(66 263 79 141)(67 264 80 142)(68 265 81 143)(69 266 82 144)(70 267 83 127)(71 268 84 128)(72 269 85 129)
(1 75 216 62)(2 74 199 61)(3 73 200 60)(4 90 201 59)(5 89 202 58)(6 88 203 57)(7 87 204 56)(8 86 205 55)(9 85 206 72)(10 84 207 71)(11 83 208 70)(12 82 209 69)(13 81 210 68)(14 80 211 67)(15 79 212 66)(16 78 213 65)(17 77 214 64)(18 76 215 63)(19 41 282 101)(20 40 283 100)(21 39 284 99)(22 38 285 98)(23 37 286 97)(24 54 287 96)(25 53 288 95)(26 52 271 94)(27 51 272 93)(28 50 273 92)(29 49 274 91)(30 48 275 108)(31 47 276 107)(32 46 277 106)(33 45 278 105)(34 44 279 104)(35 43 280 103)(36 42 281 102)(109 241 177 186)(110 240 178 185)(111 239 179 184)(112 238 180 183)(113 237 163 182)(114 236 164 181)(115 235 165 198)(116 252 166 197)(117 251 167 196)(118 250 168 195)(119 249 169 194)(120 248 170 193)(121 247 171 192)(122 246 172 191)(123 245 173 190)(124 244 174 189)(125 243 175 188)(126 242 176 187)(127 220 267 161)(128 219 268 160)(129 218 269 159)(130 217 270 158)(131 234 253 157)(132 233 254 156)(133 232 255 155)(134 231 256 154)(135 230 257 153)(136 229 258 152)(137 228 259 151)(138 227 260 150)(139 226 261 149)(140 225 262 148)(141 224 263 147)(142 223 264 146)(143 222 265 145)(144 221 266 162)

G:=sub<Sym(288)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162)(163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198)(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234)(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252)(253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270)(271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288), (1,111,10,120)(2,110,11,119)(3,109,12,118)(4,126,13,117)(5,125,14,116)(6,124,15,115)(7,123,16,114)(8,122,17,113)(9,121,18,112)(19,227,28,218)(20,226,29,217)(21,225,30,234)(22,224,31,233)(23,223,32,232)(24,222,33,231)(25,221,34,230)(26,220,35,229)(27,219,36,228)(37,264,46,255)(38,263,47,254)(39,262,48,253)(40,261,49,270)(41,260,50,269)(42,259,51,268)(43,258,52,267)(44,257,53,266)(45,256,54,265)(55,191,64,182)(56,190,65,181)(57,189,66,198)(58,188,67,197)(59,187,68,196)(60,186,69,195)(61,185,70,194)(62,184,71,193)(63,183,72,192)(73,241,82,250)(74,240,83,249)(75,239,84,248)(76,238,85,247)(77,237,86,246)(78,236,87,245)(79,235,88,244)(80,252,89,243)(81,251,90,242)(91,130,100,139)(92,129,101,138)(93,128,102,137)(94,127,103,136)(95,144,104,135)(96,143,105,134)(97,142,106,133)(98,141,107,132)(99,140,108,131)(145,278,154,287)(146,277,155,286)(147,276,156,285)(148,275,157,284)(149,274,158,283)(150,273,159,282)(151,272,160,281)(152,271,161,280)(153,288,162,279)(163,205,172,214)(164,204,173,213)(165,203,174,212)(166,202,175,211)(167,201,176,210)(168,200,177,209)(169,199,178,208)(170,216,179,207)(171,215,180,206), (1,219,216,160)(2,220,199,161)(3,221,200,162)(4,222,201,145)(5,223,202,146)(6,224,203,147)(7,225,204,148)(8,226,205,149)(9,227,206,150)(10,228,207,151)(11,229,208,152)(12,230,209,153)(13,231,210,154)(14,232,211,155)(15,233,212,156)(16,234,213,157)(17,217,214,158)(18,218,215,159)(19,171,282,121)(20,172,283,122)(21,173,284,123)(22,174,285,124)(23,175,286,125)(24,176,287,126)(25,177,288,109)(26,178,271,110)(27,179,272,111)(28,180,273,112)(29,163,274,113)(30,164,275,114)(31,165,276,115)(32,166,277,116)(33,167,278,117)(34,168,279,118)(35,169,280,119)(36,170,281,120)(37,252,97,197)(38,235,98,198)(39,236,99,181)(40,237,100,182)(41,238,101,183)(42,239,102,184)(43,240,103,185)(44,241,104,186)(45,242,105,187)(46,243,106,188)(47,244,107,189)(48,245,108,190)(49,246,91,191)(50,247,92,192)(51,248,93,193)(52,249,94,194)(53,250,95,195)(54,251,96,196)(55,270,86,130)(56,253,87,131)(57,254,88,132)(58,255,89,133)(59,256,90,134)(60,257,73,135)(61,258,74,136)(62,259,75,137)(63,260,76,138)(64,261,77,139)(65,262,78,140)(66,263,79,141)(67,264,80,142)(68,265,81,143)(69,266,82,144)(70,267,83,127)(71,268,84,128)(72,269,85,129), (1,75,216,62)(2,74,199,61)(3,73,200,60)(4,90,201,59)(5,89,202,58)(6,88,203,57)(7,87,204,56)(8,86,205,55)(9,85,206,72)(10,84,207,71)(11,83,208,70)(12,82,209,69)(13,81,210,68)(14,80,211,67)(15,79,212,66)(16,78,213,65)(17,77,214,64)(18,76,215,63)(19,41,282,101)(20,40,283,100)(21,39,284,99)(22,38,285,98)(23,37,286,97)(24,54,287,96)(25,53,288,95)(26,52,271,94)(27,51,272,93)(28,50,273,92)(29,49,274,91)(30,48,275,108)(31,47,276,107)(32,46,277,106)(33,45,278,105)(34,44,279,104)(35,43,280,103)(36,42,281,102)(109,241,177,186)(110,240,178,185)(111,239,179,184)(112,238,180,183)(113,237,163,182)(114,236,164,181)(115,235,165,198)(116,252,166,197)(117,251,167,196)(118,250,168,195)(119,249,169,194)(120,248,170,193)(121,247,171,192)(122,246,172,191)(123,245,173,190)(124,244,174,189)(125,243,175,188)(126,242,176,187)(127,220,267,161)(128,219,268,160)(129,218,269,159)(130,217,270,158)(131,234,253,157)(132,233,254,156)(133,232,255,155)(134,231,256,154)(135,230,257,153)(136,229,258,152)(137,228,259,151)(138,227,260,150)(139,226,261,149)(140,225,262,148)(141,224,263,147)(142,223,264,146)(143,222,265,145)(144,221,266,162)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162)(163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198)(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234)(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252)(253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270)(271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288), (1,111,10,120)(2,110,11,119)(3,109,12,118)(4,126,13,117)(5,125,14,116)(6,124,15,115)(7,123,16,114)(8,122,17,113)(9,121,18,112)(19,227,28,218)(20,226,29,217)(21,225,30,234)(22,224,31,233)(23,223,32,232)(24,222,33,231)(25,221,34,230)(26,220,35,229)(27,219,36,228)(37,264,46,255)(38,263,47,254)(39,262,48,253)(40,261,49,270)(41,260,50,269)(42,259,51,268)(43,258,52,267)(44,257,53,266)(45,256,54,265)(55,191,64,182)(56,190,65,181)(57,189,66,198)(58,188,67,197)(59,187,68,196)(60,186,69,195)(61,185,70,194)(62,184,71,193)(63,183,72,192)(73,241,82,250)(74,240,83,249)(75,239,84,248)(76,238,85,247)(77,237,86,246)(78,236,87,245)(79,235,88,244)(80,252,89,243)(81,251,90,242)(91,130,100,139)(92,129,101,138)(93,128,102,137)(94,127,103,136)(95,144,104,135)(96,143,105,134)(97,142,106,133)(98,141,107,132)(99,140,108,131)(145,278,154,287)(146,277,155,286)(147,276,156,285)(148,275,157,284)(149,274,158,283)(150,273,159,282)(151,272,160,281)(152,271,161,280)(153,288,162,279)(163,205,172,214)(164,204,173,213)(165,203,174,212)(166,202,175,211)(167,201,176,210)(168,200,177,209)(169,199,178,208)(170,216,179,207)(171,215,180,206), (1,219,216,160)(2,220,199,161)(3,221,200,162)(4,222,201,145)(5,223,202,146)(6,224,203,147)(7,225,204,148)(8,226,205,149)(9,227,206,150)(10,228,207,151)(11,229,208,152)(12,230,209,153)(13,231,210,154)(14,232,211,155)(15,233,212,156)(16,234,213,157)(17,217,214,158)(18,218,215,159)(19,171,282,121)(20,172,283,122)(21,173,284,123)(22,174,285,124)(23,175,286,125)(24,176,287,126)(25,177,288,109)(26,178,271,110)(27,179,272,111)(28,180,273,112)(29,163,274,113)(30,164,275,114)(31,165,276,115)(32,166,277,116)(33,167,278,117)(34,168,279,118)(35,169,280,119)(36,170,281,120)(37,252,97,197)(38,235,98,198)(39,236,99,181)(40,237,100,182)(41,238,101,183)(42,239,102,184)(43,240,103,185)(44,241,104,186)(45,242,105,187)(46,243,106,188)(47,244,107,189)(48,245,108,190)(49,246,91,191)(50,247,92,192)(51,248,93,193)(52,249,94,194)(53,250,95,195)(54,251,96,196)(55,270,86,130)(56,253,87,131)(57,254,88,132)(58,255,89,133)(59,256,90,134)(60,257,73,135)(61,258,74,136)(62,259,75,137)(63,260,76,138)(64,261,77,139)(65,262,78,140)(66,263,79,141)(67,264,80,142)(68,265,81,143)(69,266,82,144)(70,267,83,127)(71,268,84,128)(72,269,85,129), (1,75,216,62)(2,74,199,61)(3,73,200,60)(4,90,201,59)(5,89,202,58)(6,88,203,57)(7,87,204,56)(8,86,205,55)(9,85,206,72)(10,84,207,71)(11,83,208,70)(12,82,209,69)(13,81,210,68)(14,80,211,67)(15,79,212,66)(16,78,213,65)(17,77,214,64)(18,76,215,63)(19,41,282,101)(20,40,283,100)(21,39,284,99)(22,38,285,98)(23,37,286,97)(24,54,287,96)(25,53,288,95)(26,52,271,94)(27,51,272,93)(28,50,273,92)(29,49,274,91)(30,48,275,108)(31,47,276,107)(32,46,277,106)(33,45,278,105)(34,44,279,104)(35,43,280,103)(36,42,281,102)(109,241,177,186)(110,240,178,185)(111,239,179,184)(112,238,180,183)(113,237,163,182)(114,236,164,181)(115,235,165,198)(116,252,166,197)(117,251,167,196)(118,250,168,195)(119,249,169,194)(120,248,170,193)(121,247,171,192)(122,246,172,191)(123,245,173,190)(124,244,174,189)(125,243,175,188)(126,242,176,187)(127,220,267,161)(128,219,268,160)(129,218,269,159)(130,217,270,158)(131,234,253,157)(132,233,254,156)(133,232,255,155)(134,231,256,154)(135,230,257,153)(136,229,258,152)(137,228,259,151)(138,227,260,150)(139,226,261,149)(140,225,262,148)(141,224,263,147)(142,223,264,146)(143,222,265,145)(144,221,266,162) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126),(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162),(163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198),(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216),(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234),(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252),(253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270),(271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288)], [(1,111,10,120),(2,110,11,119),(3,109,12,118),(4,126,13,117),(5,125,14,116),(6,124,15,115),(7,123,16,114),(8,122,17,113),(9,121,18,112),(19,227,28,218),(20,226,29,217),(21,225,30,234),(22,224,31,233),(23,223,32,232),(24,222,33,231),(25,221,34,230),(26,220,35,229),(27,219,36,228),(37,264,46,255),(38,263,47,254),(39,262,48,253),(40,261,49,270),(41,260,50,269),(42,259,51,268),(43,258,52,267),(44,257,53,266),(45,256,54,265),(55,191,64,182),(56,190,65,181),(57,189,66,198),(58,188,67,197),(59,187,68,196),(60,186,69,195),(61,185,70,194),(62,184,71,193),(63,183,72,192),(73,241,82,250),(74,240,83,249),(75,239,84,248),(76,238,85,247),(77,237,86,246),(78,236,87,245),(79,235,88,244),(80,252,89,243),(81,251,90,242),(91,130,100,139),(92,129,101,138),(93,128,102,137),(94,127,103,136),(95,144,104,135),(96,143,105,134),(97,142,106,133),(98,141,107,132),(99,140,108,131),(145,278,154,287),(146,277,155,286),(147,276,156,285),(148,275,157,284),(149,274,158,283),(150,273,159,282),(151,272,160,281),(152,271,161,280),(153,288,162,279),(163,205,172,214),(164,204,173,213),(165,203,174,212),(166,202,175,211),(167,201,176,210),(168,200,177,209),(169,199,178,208),(170,216,179,207),(171,215,180,206)], [(1,219,216,160),(2,220,199,161),(3,221,200,162),(4,222,201,145),(5,223,202,146),(6,224,203,147),(7,225,204,148),(8,226,205,149),(9,227,206,150),(10,228,207,151),(11,229,208,152),(12,230,209,153),(13,231,210,154),(14,232,211,155),(15,233,212,156),(16,234,213,157),(17,217,214,158),(18,218,215,159),(19,171,282,121),(20,172,283,122),(21,173,284,123),(22,174,285,124),(23,175,286,125),(24,176,287,126),(25,177,288,109),(26,178,271,110),(27,179,272,111),(28,180,273,112),(29,163,274,113),(30,164,275,114),(31,165,276,115),(32,166,277,116),(33,167,278,117),(34,168,279,118),(35,169,280,119),(36,170,281,120),(37,252,97,197),(38,235,98,198),(39,236,99,181),(40,237,100,182),(41,238,101,183),(42,239,102,184),(43,240,103,185),(44,241,104,186),(45,242,105,187),(46,243,106,188),(47,244,107,189),(48,245,108,190),(49,246,91,191),(50,247,92,192),(51,248,93,193),(52,249,94,194),(53,250,95,195),(54,251,96,196),(55,270,86,130),(56,253,87,131),(57,254,88,132),(58,255,89,133),(59,256,90,134),(60,257,73,135),(61,258,74,136),(62,259,75,137),(63,260,76,138),(64,261,77,139),(65,262,78,140),(66,263,79,141),(67,264,80,142),(68,265,81,143),(69,266,82,144),(70,267,83,127),(71,268,84,128),(72,269,85,129)], [(1,75,216,62),(2,74,199,61),(3,73,200,60),(4,90,201,59),(5,89,202,58),(6,88,203,57),(7,87,204,56),(8,86,205,55),(9,85,206,72),(10,84,207,71),(11,83,208,70),(12,82,209,69),(13,81,210,68),(14,80,211,67),(15,79,212,66),(16,78,213,65),(17,77,214,64),(18,76,215,63),(19,41,282,101),(20,40,283,100),(21,39,284,99),(22,38,285,98),(23,37,286,97),(24,54,287,96),(25,53,288,95),(26,52,271,94),(27,51,272,93),(28,50,273,92),(29,49,274,91),(30,48,275,108),(31,47,276,107),(32,46,277,106),(33,45,278,105),(34,44,279,104),(35,43,280,103),(36,42,281,102),(109,241,177,186),(110,240,178,185),(111,239,179,184),(112,238,180,183),(113,237,163,182),(114,236,164,181),(115,235,165,198),(116,252,166,197),(117,251,167,196),(118,250,168,195),(119,249,169,194),(120,248,170,193),(121,247,171,192),(122,246,172,191),(123,245,173,190),(124,244,174,189),(125,243,175,188),(126,242,176,187),(127,220,267,161),(128,219,268,160),(129,218,269,159),(130,217,270,158),(131,234,253,157),(132,233,254,156),(133,232,255,155),(134,231,256,154),(135,230,257,153),(136,229,258,152),(137,228,259,151),(138,227,260,150),(139,226,261,149),(140,225,262,148),(141,224,263,147),(142,223,264,146),(143,222,265,145),(144,221,266,162)])

54 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 9A 9B 9C 12A ··· 12F 18A ··· 18I 36A ··· 36R order 1 2 2 2 3 4 4 4 4 4 4 4 4 4 4 6 6 6 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 2 4 4 18 18 18 18 36 36 2 2 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + + + - - - - image C1 C2 C2 C2 C2 S3 Q8 D6 C4○D4 D9 D18 C4○D12 D36⋊5C2 D4⋊2S3 S3×Q8 D4⋊2D9 Q8×D9 kernel Dic9.Q8 C4×Dic9 Dic9⋊C4 C4⋊Dic9 C9×C4⋊C4 C3×C4⋊C4 Dic9 C2×C12 C18 C4⋊C4 C2×C4 C6 C2 C6 C6 C2 C2 # reps 1 1 4 1 1 1 2 3 4 3 9 4 12 1 1 3 3

Matrix representation of Dic9.Q8 in GL4(𝔽37) generated by

 20 26 0 0 11 31 0 0 0 0 1 0 0 0 0 1
,
 8 9 0 0 1 29 0 0 0 0 36 0 0 0 0 36
,
 7 14 0 0 23 30 0 0 0 0 0 1 0 0 36 0
,
 26 20 0 0 31 11 0 0 0 0 32 23 0 0 23 5
G:=sub<GL(4,GF(37))| [20,11,0,0,26,31,0,0,0,0,1,0,0,0,0,1],[8,1,0,0,9,29,0,0,0,0,36,0,0,0,0,36],[7,23,0,0,14,30,0,0,0,0,0,36,0,0,1,0],[26,31,0,0,20,11,0,0,0,0,32,23,0,0,23,5] >;

Dic9.Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_9.Q_8
% in TeX

G:=Group("Dic9.Q8");
// GroupNames label

G:=SmallGroup(288,99);
// by ID

G=gap.SmallGroup(288,99);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,254,219,100,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^18=c^4=1,b^2=a^9,d^2=c^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^9*b,b*d=d*b,d*c*d^-1=a^9*c^-1>;
// generators/relations

׿
×
𝔽