text:
# Passing an Object

#### Ryan$Joined Dec 14, 2018 178 I'm primarily programming in c /c++ and using sometimes libraries which already written in java. I have faced something like this Book b= new Book(); for me it's understandable and I know what new Book() mean, what I don't understand is, how an object stored to one variable b? I mean the object might be several memory blocks locations right? so when we write new Book() it's allocating memory location, and what's returning to b? the start address of the object? or it returns the *(start address of the first object) to b and then(analog to c/c++) I write "b." and not "b->" ? what I've understood from java that object means something "concrete" in memory, like kind of data .. ** I know there's no * and pointers in java, I'm just making a visualizing to understand the issue very well because I program in c/c++ ** Another question, will the class itself occupy memory in RAM(not meaning on its objects)? thanks in advance. Last edited: #### 402DF855 Joined Feb 9, 2013 171 Book *b = new Book; // dynamically allocated Book bb; // stack allocated, disappears when function exits I think a class ordinarily will not occupy memory unless it has static members which do not belong to an instance of the class (has no this pointer) but belongs to the class itself. #### Ian Rogers Joined Dec 12, 2012 699 I'm not sure I have ever liked the word "object"... Some numpty always comes on with "It's like a real world object" ie.. concrete as you have just said.. Its a variable!! In another thread we talked about structures... The struct is the fore runner to objects... Objects are structures and the functions that operate on those structures bundled into a single variable... This gives more structured code as each structure can be treated as one single entity!!! Book myBook = new(Book); Book is a structure.. BUT! the member variables and functions are managed... Simply a function to turn a page will hide the page index variable from you to a point you just type myBook.turnpage(); and the structure looks after itself..( assuming that is the function ).. Java has pointers they are just not as apparent as other languages... The only time a class takes up ram is when you initialise them... They have constructors to make an instance from the definition.. #### 402DF855 Joined Feb 9, 2013 171 C: Book b= new Book(); is wrong. "b" must be declared as a pointer! C: Book *b= new Book(); Book b2; Book is a class; "b" is a pointer to an object of type (class) Book. "b2" is an object of type (class) Book. "&b2" is a pointer to an object of type (class) Book. An object is a region of storage. #### WBahn Joined Mar 31, 2012 25,223 I'm primarily programming in c /c++ and using sometimes libraries which already written in java. I have faced something like this Book b= new Book(); for me it's understandable and I know what new Book() mean, what I don't understand is, how an object stored to one variable b? I mean the object might be several memory blocks locations right? so when we write new Book() it's allocating memory location, and what's returning to b? the start address of the object? or it returns the *(start address of the first object) to b and then(analog to c/c++) I write "b." and not "b->" ? what I've understood from java that object means something "concrete" in memory, like kind of data .. ** I know there's no * and pointers in java, I'm just making a visualizing to understand the issue very well because I program in c/c++ ** Another question, will the class itself occupy memory in RAM(not meaning on its objects)? thanks in advance. An object, like a structure or like an array, is always a single contiguous block of memory and its address is always the address of the first byte in the block. How classes are implemented is implementation dependent -- the compiler writer only has to make the behavior conform to the language standard. The class functions and methods are often stored just like any other functions. Some classes have data that belong to the entire class (called static in Java, don't know about C++) and so that requires memory even if there are no objects of that class. In fact, a class doesn't have to be able to instantiate objects at all -- it may only exist in order to provide static methods and data. Many math classes are like that. If you want to get a good feel for how much of this is done, from the bottom up, then work your way through the NAND-to-Tetris project. #### bogosort Joined Sep 24, 2011 441 I'm primarily programming in c /c++ and using sometimes libraries which already written in java. I have faced something like this Book b= new Book(); for me it's understandable and I know what new Book() mean, what I don't understand is, how an object stored to one variable b? I mean the object might be several memory blocks locations right? so when we write new Book() it's allocating memory location, and what's returning to b? the start address of the object? or it returns the *(start address of the first object) to b and then(analog to c/c++) I write "b." and not "b->" ? In Java, the "b" variable in your example is a reference to a newly-created Book object. References act very much like pointers in C: they point to a memory location. The statement "Book b2 = b;" would create another reference to the same object -- note that the object is not copied, just the reference. Likewise, calling some method with "b" as its argument passes the reference to the object, not the entire object. Another question, will the class itself occupy memory in RAM(not meaning on its objects)? I'm not by an means an expert on Java internals, but I would certainly think so. Objects are created at runtime, and so would need runtime access to the class definition. Furthermore, Java allows class variables (declared as static within the class), which belong to the class and not any particular instance of the class. If your Book class had a class variable named "totalBooks", you could (and should) access it without an object reference, e.g.: Book.totalBooks++;. Thread Starter #### Ryan$

Joined Dec 14, 2018
178
thanks to you all!
I have another question which I will not open a new thread for it; in java at main method why it's not allowed to write there a method which doing for me something like printing? For example
Code:
Public class main
{
Public int main()
{
Public void displayMessage()
{
System.out.println("hello");
}
}
}
Why I need to do a new class in order to declare there the function displayMessage? I can inside main itself doing that and the pc will print hello world .. I dont know where's the problem with that? .. I know about objects on java .. But why cant immediately in main class write a function and simple pc will read it and execute it without doing another class for ..

Thanks you guys very much!

#### WBahn

Joined Mar 31, 2012
25,223
What "another class"? Your code above has one class.

Please take at least enough time to TRY to present code that is SOMEWHAT close to being valid and then present it here reasonably formatted.

1) The "public" keyword has to be all lower case.
2) You can't define a method within a method (what you are calling "functions" are called "methods" in Java).
3) Since main() is defined to return a value of type int, it must return a value of type int.
4) Java generally expects the main() method to be type void and to take one argument which is an array of String objects.

Java:
public class main
{
static public void main(String args[])
{
System.out.println("hello");
}
}


text:
Categories

Don’t Dunk the Gunk

How to Avoid Accidentally Amplifying Assholes on Twitter

It happened again. You were doom-scrolling on the hellsite (a.k.a. browsing Twitter) and stumbled upon some stranger’s tweet that was so stunningly ignorant that it almost seems artificial.

But sometimes it’s more than just stupid, it’s also harmful in some tangible way. Homophobic, racist, etc. nonsense is posted to social media almost as frequently as the comments section on YouTube.

This serves to amplify messages of hatred, bigotry, and intolerance, to anyone who follows your friends and peers. Eventually, the bad tweet reaches enough engagement to reach escape velocity, and unsuspecting people find it recommended by the Twitter Algorithm. Again.

Stop Directly Interacting With Bad Takes

No matter how tempting it may be to dunk on a hater, you’re going to ultimately help amplify the idiot’s message if you interact with it directly.

To be clear: I am not saying don’t dunk on them. Just be smarter about it.

What Does Clever Dunking Look Like?

The standard advice you’ll hear from activists is an excellent start.

First, always take a screenshot of the offending content, and tweet that instead of the original tweet. This uncouples your message from the original tweet, and provides you the opportunity to censor anything offensive or inappropriate to your followers.

Next, always archive the original tweet to preserve evidence. There are plenty of web page archiving services online. The Internet Archive and Archive.Today are two. Screenshots can be faked, but third-party archiving services are much more difficult to falsify.

What’s written above is the normal advice you’re probably used to.

But that isn’t clever enough for us. Let’s take it a step further.

Introducing: Dunktags!

A dunktag is a (relatively) unique hashtag, generated from the URL of the original tweet, that we can use to standardize, and thus coordinate, our dunks without having to ever comment on, or reply to, an original bad tweet.

Update: CatmanJon created a dunktag web app for your convenience.

```const { dunktag } = require('dunktags');

// "#dunk_3e45fd4499"
```

The dunktags algorithm, as well as a TypeScript implementation, is open source on GitHub.

How Can Dunktags Help?

With dunktags, our dunking procedure now looks like this.

1. Take a screenshot of the offending tweet.
2. Archive the original tweet.
3. Use my library to generate a dunktag from the original tweet URL.
4. When you choose to dunk on this clown, instead of replying or quote-tweeting them, simply post your screenshot (and archive URLs) with the hashtag generated by the `dunktag()` function.

If you’d like to see what Twitter users have already said about a dumb hot take, you can just take the tweet URL and run it into the `dunktag()` function, and then hit up Twitter Search to enjoy others’ sarcastic one-liners.

What Is A Dunktag, Under the Hood?

Dunktags use BLAKE2b under the hood, with an output length of 5 bytes (10 hex characters), and a constant BLAKE2b key of `0x572a7ff65e3969e1d1f43911cc07ff82463e5ae113f248dc9035d5e21aabad85` (for domain separation).

The key is the BLAKE2b hash of the string, `Soatok Dreamseeker's dunktag function`.

First, the Twitter user’s tweet ID is extracted from the tweet .

Next, the BLAKE2b hash of this tweet ID is calculated, using the above parameters.

This hex-encoded hash is returned with a constant prefix (`#dunk_`).

That’s it. That’s the entire algorithm. I kept it boring and obviously secure.

Why Only 5 Bytes?

I figure billions of tweets are posted per day, but a dunk would only occur for some very small fraction of those tweets, so the risk of collisions isn’t significant.

You’d expect a 50% chance of collision after $2^{20}$ dunktags, which is a little more than 1 million.

If your tweet and someone else’s bad tweet collide many years later, it’s obvious that they’re not dunking on you.

5 bytes hex-encodes to 10 hexits, which combined with the `#` character and the `dunk_` prefix rounds off nicely to 16 characters of overhead. I think that’s sufficient for most of the Internet’s needs today.

format: plain


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# 16 Experimental design

Experimental design techniques are useful to assess what type of experiments or studies should be conducted to ensure an adequate level of information in the observed data to address the question(s) of interest. In this chapter we discuss and exemplify some of these techniques.

## 16.1 Use of PopED

PopED is a well known optimal experimental design software that can be used both for population and individual studies based on nonlinear mixed-effect models. Methods, underlying PopED are described in Nyberg et al. (2012) and Foracchia et al. (2004). PopED is available as an R package on CRAN. The installer of IQR Tools will install the correct version automatically.

PopED comes with great flexibility for defining models and design specifications. However, the use is made overly complicated by a not well thought through R based interface to functions. The user either needs to be a PopED specialist or he/she will most likely make more or less critical mistakes, leading to wrong results and interpretations.

In order to make the important functionality of this optimal experiment design package available in a much more seamless manner, an interface between IQR Tools and PopED was developed. It benefits from the already available, intuitive, description of models and from the intuitive and easy to setup NLME model specification in IQR Tools. The user is not required to re-code models or NLME model specifications to use PopED. The same objects are used as if a user would like to perform a parameter estimation task.

The approach employed is to convert the IQR Tools model and design specifications to a PopED database. In the current version then standard PopED functions can be used on this database object. In a future version of IQR Tools dedicated functions will be provided to encapsulate the PopED functions, ensuring the easy creation of powerful and report-ready output in terms of tables and graphics.

Below an example for an IQR Tools run with PopED is given, which is based on an example model in the PopED R package. The same example is repeated in native PopED syntax. The results of both approaches are identical. The reader of this documentation can decide freely which syntax is easier to understand, review, validate, and re-use.

### 16.1.1 PopED / IQR Toolsinterface

We will consider a one compartment first order absorption PK model with a direct response, inhibitory IMAX, model as PD. This model in the IQRmodel syntax is available here. In a first step this model is loaded into R:

# Load the model into R
model <- IQRmodel("material/01-08-X1-popED/model.txt")

The study design to be evaluated is defined as a list with the named elements time, dosing, groupsize. The example implements a parallel design with three arms/groups with same observation times between groups and observables. N=4 subjects per group. Each group obtains a different dose (20, 40, and placebo).

# Define study design to be evaluated
design <- list(

# The time element defines the sampling times for the observables.
# Here we have two observables (PK as OUTPUT1 and PD as OUTPUT2).
# It is easily possible to define different observation times for the different outputs.
# In this example we chose the same sampling times as in the original PopED example (even if the final
# sampling time appears twice ... just as in the PopED example).
time = list(
g1 = list(OUTPUT1 = c(1, 2, 8, 240, 240),
OUTPUT2 = c(1, 2, 8, 240, 240)),
g2 = list(OUTPUT1 = c(1, 2, 8, 240, 240),
OUTPUT2 = c(1, 2, 8, 240, 240)),
g3 = list(OUTPUT1 = c(1, 2, 8, 240, 240),
OUTPUT2 = c(1, 2, 8, 240, 240))
),

# Define the dosing schedule for the different groups.
dosing = list(
g1 = IQRdosing(TIME = 0, ADM = 1, AMT = 20, ADDL = 10, II = 24),
g2 = IQRdosing(TIME = 0, ADM = 1, AMT = 40, ADDL = 10, II = 24),
g3 = IQRdosing(TIME = 0, ADM = 1, AMT =  0, ADDL = 10, II = 24)
),

# Define the size of the groups
groupsize = list(
g1 = 4,
g2 = 4,
g3 = 4
)
)

The model specification for the NLME model is defined in an identical fashion as if the user would like to generate a NONMEM, MONOLIX, or NLMIXR model with IQRtools. The modelSpec variable defined below is a list whose construction is aided by the function modelSpec_IQRest():

# Define model specifications related to the parameter, statistical, covariate model
modelSpec     <- modelSpec_IQRest(
# Define initial guesses for the fixed effect parameters
POPvalues0  = c(V=72.8, KA=0.25, CL=3.75,   Favail=0.9, E0=1120, IMAX=0.807, IC50=0.0993),
# Define if a fixed effect is to be estimated (1) or not (0)
POPestimate = c(V=1,    KA=1,    CL=1,      Favail=0,   E0=1,    IMAX=1,     IC50=1),
# Define initial guesses for the random effects (standard deviations used - while PopED uses variances)
IIVvalues0  = sqrt(c(V=0.09, KA=0.09, CL=0.25^2, Favail=0,   E0=0.09, IMAX=0,     IC50=0)),
# Define if random effects are present and estimated (1), not present (0),
# or fixed to a specified value and not estimated (2)
IIVestimate = c(V=1,    KA=1,    CL=1,      Favail=0,   E0=1,    IMAX=0,     IC50=0),

# By default lognormal distribution of individual parameters is assumed.
# There are a multitude of additional optional settings which are not needed for
# the present example.

# Define error model and the initial guesses. In this example additive-proportional error models
# are used for both observables.
errorModel = list(
OUTPUT1 = c("absrel", c(abs0=5e-6, rel0=0.04)),
OUTPUT2 = c("absrel", c(abs0=100,  rel0=0.09))
)
)

The IQR Tools function IQRpopEDdb() takes as required input arguments the structural model, the desired design, and the NLME model specification modelSpec and converts this information to a PopED database object.

# Run the PopED interface to establish the poped database
poped.db <- IQRpopEDdb(model = model,
design = design,
modelSpec = modelSpec)

This PopED database object then can be used with standard PopED functions. To perform model predictions, design evaluations, etc. In a soon to be available IQR Tools version proper IQR Tools functionality will be provided to simplify the evaluation of designs, etc.

# Execute standard PopED functions
out1_IQR <- PopED::model_prediction(poped.db)
out2_IQR <- PopED::evaluate_design(poped.db)

The main result of the above design evaluation are the predicted relative standard errors for the different parameters. Results for this example are shown below. The output is still fairly cryptic, as based on PopE functionality and also might require some further post-processing.

# Display the predicted relative standard errors for the parameters based on the chosen study design
print(out2_IQR$rse) ## V KA CL E0 IMAX IC50 d_V ## 18.144697 22.274804 9.626962 15.824337 22.126141 88.275210 87.071162 ## d_KA d_CL d_E0 ## 130.774210 57.766256 49.274222 PopED results can also be plotted, using PopED functionality. PopED::plot_model_prediction(poped.db, facet_scales = "free") This conlcudes the example for the current IQR Tools / PopED interface. We still have some work to do to provide adequate functionality to post-process PopED output to produce reasonably nice tables and graphics that are more informative. Stay tuned ... it will come in one of the next IQR Tools versions! In the following section the same example as the one above is shown in native PopED syntax. ### 16.1.2 Same example in PopED In this section, the example from the previous section, conducted with the IQR Tools interface to PopED, is performed in the native PopED syntax. The example code was taken from the examples in the PopED package on CRAN. In a first step the model prediction function needs to be defined. This can be done using ODEs or in this case by the analytic solution. Compare this model specification to that one here. It is far easier to make a mistake in the approach below: # Model prediction function ff <- function(model_switch,xt,parameters,poped.db){ with(as.list(parameters),{ y=xt MS <- model_switch # PK model N = floor(xt/TAU)+1 CONC=(DOSE*Favail/V)*(KA/(KA - CL/V)) * (exp(-CL/V * (xt - (N - 1) * TAU)) * (1 - exp(-N * CL/V * TAU))/(1 - exp(-CL/V * TAU)) - exp(-KA * (xt - (N - 1) * TAU)) * (1 - exp(-N * KA * TAU))/(1 - exp(-KA * TAU))) # PD model EFF = E0*(1 - CONC*IMAX/(IC50 + CONC)) y[MS==1] = CONC[MS==1] y[MS==2] = EFF[MS==2] return(list( y= y,poped.db=poped.db)) }) } In a second step parameters and their transformation needs to be defined. For more information, please see the PopED documentation. # Parameter definition function sfg <- function(x,a,bpop,b,bocc){ parameters=c( V=bpop[1]*exp(b[1]), KA=bpop[2]*exp(b[2]), CL=bpop[3]*exp(b[3]), Favail=bpop[4], DOSE=a[1], TAU = a[2], E0=bpop[5]*exp(b[4]), IMAX=bpop[6], IC50=bpop[7]) return( parameters ) } In a third step the residual error model function is defined. Again, the interested reader is referred to the PopED documentation for further clarification. # Residual Error function feps <- function(model_switch,xt,parameters,epsi,poped.db){ returnArgs <- ff(model_switch,xt,parameters,poped.db) y <- returnArgs[[1]] poped.db <- returnArgs[[2]] MS <- model_switch pk.dv <- y*(1+epsi[,1])+epsi[,2] pd.dv <- y*(1+epsi[,3])+epsi[,4] y[MS==1] = pk.dv[MS==1] y[MS==2] = pd.dv[MS==2] return(list( y= y,poped.db =poped.db )) } The create.poped.database() function in PopED then is used to combine the above information and to define the design to be evaluated. # Create PopED database poped.db <- PopED::create.poped.database(ff_fun="ff", fError_fun="feps", fg_fun="sfg", groupsize=4, m=3, bpop=c(V=72.8,KA=0.25,CL=3.75,Favail=0.9,E0=1120,IMAX=0.807, IC50=0.0993), notfixed_bpop=c(1,1,1,0,1,1,1), d=c(V=0.09,KA=0.09,CL=0.25^2,E0=0.09), sigma=c(0.04,5e-6,0.09,100), notfixed_sigma=c(0,0,0,0), xt=c( 1,2,8,240,240,1,2,8,240,240), minxt=c(0,0,0,240,240,0,0,0,240,240), maxxt=c(10,10,10,248,248,10,10,10,248,248), discrete_xt = list(0:248), G_xt=c(1,2,3,4,5,1,2,3,4,5), bUseGrouped_xt=1, model_switch=c(1,1,1,1,1,2,2,2,2,2), a=list(c(DOSE=20,TAU=24),c(DOSE=40, TAU=24),c(DOSE=0, TAU=24)), maxa=c(DOSE=200,TAU=40), mina=c(DOSE=0,TAU=2), ourzero=0) Once this is completed, we execute the same functions as in the previous IQR Tools / PopED example. # Execute standard PopED functions out1_IQR <- PopED::model_prediction(poped.db) out2_IQR <- PopED::evaluate_design(poped.db) The results are basically identical to the previous example. The observed small differences stem from the fact that in this native PopED example an analytic solution was used, while in the other example the model was implemented using ordinary differential equations and the solutions to the model were integrated numerically. # Display the predicted relative standard errors for the parameters based on the chosen study design print(out2_IQR$rse)
##          V         KA         CL         E0       IMAX       IC50        d_V
##  18.156519  22.290494   9.625457  15.824391  22.126653  88.276094  87.119440
##       d_KA       d_CL       d_E0
## 130.861826  57.763841  49.274240

Same plot as in previous example:

PopED::plot_model_prediction(poped.db, facet_scales = "free")

## 16.2 Use of profile likelihood

Example to be added ... soon!


text:
## Transits Across g-Darkened Stars

Titan transiting its oblate fluid planet (Credit: HST)

We have previously looked at modelling the transit of an extrasolar planet incorporating limb darkening. How can we build upon this? For starters, it would be nice to actually take into consideration more physical phenomena – nature is not always simple. Take for example the case of a non-spherical star. Consider the case of Saturn, whose rapid rotation distorts the planet away from a spherical shape and more into an oblate shape. Stars, being fluid bodies like gas giant planets, are subject to similar physics, and have measurable effects on the transit light curve.

How does this affect a system? A rotating fluid body like a gas giant planet or a star is subject to centrifugal force which causes the equatorial radius to be larger than the polar radius. This effect, called oblateness, can be measured with

$\displaystyle \text{oblateness} = \frac{a-b}{a}$

Where a is the equatorial radius and b is the polar radius. In the case of Saturn, the oblateness is about 0.1. Despite a higher rotational velocity, Jupiter’s higher density and surface gravity are able to keep it’s oblateness lower, but still noticeable, at about 0.06. All of the other planets have very different, non-gas-dominated compositions, and have much a lower oblateness (in all cases, less than 0.025). The sun, however, with it’s high surface gravity (28 times that of Earth) and long rotation period (about a month), is almost perfectly spherical, with an oblateness of 0.000009.

In the case of a self-luminous body like a star, oblateness will cause the surface gravity and therefore surface temperature of the star to be a function of latitude. Parts of the surface closer to the poles will be hotter than those near the equator which are pushed further from the stellar centre due to the centrifugal force imposed by the stellar rotation. The temperature of the surface at a given latitude, θ, is given by

$\displaystyle T_{eff}(\theta)=T_p\left(\frac{g(\theta)}{g_p}\right)^\beta$

Where Tg and Tp are the surface gravity and temperature at the pole, respectively, and β is the gravity darkening coefficient, which much like limb-darkening coefficients, are dependent on the properties of the star and will have to be tinkered with to get a good fit.

An example of this comes from the nearby and well-known rapidly rotating, A-type star Altair, which has been spatially resolved using interferometry, allowing for the construction of a low-resolution temperature map.

Altair (Source)

Notice the oblateness of Altair is greater than even that of Saturn. This has a strong effect on the surface temperature of the star, with the equator being over 1,000 K cooler than the poles!

Much like in the case of limb-darkening, gravity-darkening will cause the luminosity profile of the stellar disc to be concentrated in one particular area. Unlike the case of limb-darkening, however, this area need not be the centre of the disc, but could instead be anywhere on the disc. This allows for transit light curves to actually be asymmetric in the case of a misaligned planetary orbit. A transiting planet in a polar orbit of a gravity-darkened star with, for example, a 45 degree rotation axis inclination, will have its planet occulting the bright polar region before moving over the more equatorial regions, causing the minima of the transit light curve to be displaced from the transit centre over toward the ingress.

Consider for example the case of KOI-13, a rapdily rotating A-type star, much like Altair, which was discovered to have a transiting hot Jupiter by the Kepler mission.

 KOI-13 Light Curve (Source) KOI-13 Light Curve Residuals

You may need to click on the light curve to expand it to full size to see the slight difference between the gravity-darkened and non-gravity-darkened model. If nothing else, the residuals to the non-gravity-darkened model are clear that the star is not uniformly luminous.

The stars where gravity darkening will be most pronounced will of course be the stars which have high absolute rotation rates. While a high v sin Istar implies a rapid rotation, the converse is not necessarily true since the stellar inclination angle is not known a priori. However, because a gravity-darkened star appears rotationally symmetric when viewed pole-on, the effects it has on the transit light curve are greatly diminished for low values for v sin Istar. This conspires to mean that the stars for which gravity-darkening effects on the light curves of transiting planets will be those stars where Doppler spectroscopy is less effective. Thus, while the Rossiter-McLaughlin effect may not be measurable due to the stellar spectral line broadening and corresponding lack of radial velocity precision, one can still in some cases use gravity-darkening to determine the projected stellar obliquity (or misalignment between the planetary orbit axis and stellar spin axis, however one chooses to interpret the system). Your typical system for which this is applicable is a bloated hot Jupiter transiting a rapidly rotating A-type star, and at present, only a couple such such systems are known.

From the perspective of modelling, gravity darkening is essentially asymmetric limb-darkening. This post was designed to give you the idea behind gravity darkening of stars and how it can affect the light curve for a transiting planet. In a future post, we will look at modelling transiting planets orbiting gravity-darkened stars.

## Dancing in the Dark

Epislon Eridani, the nearest known planet-hosting star other than the sun.

The motion of a star around the barycentre of a planetary system is not necessarily confined to detectability through indirect methods only. The barycentric motion of a star in a planetary system can, with sufficient precision, be directly observed. Logically, the larger the star’s barycentric semi-major axis, the more easy it would be to observe such a motion. However very long, multi-decade period orbits are harder to observe simply because they progress at timescales comparable to the human lifespan. As such, astrometry is biased toward intermediate-period, massive planets.

Often, the barycentric motion of a star will be tiny. Our sun being an example rarely diverges from two solar-radii from the Solar System barycenter. Across the extreme distances, these changes are difficult to resolve, leading to large error bars that can swamp out the real orbital motion. For this reason, it is often the case that a large number of measurements are required to build up confidence in the detection of an astrometric signal, much as with the Doppler spectroscopy method.

While the above plot is messy, it makes clear that the stellar orbit is far from face-on. It gives us a rough measurement of the stellar orbit’s inclination.

The advantages of astrometry justify the difficulty in performing it for planetary systems. By directly observing the barycentric motion of a star, it is possible to reconstruct the orbit of the orbiting planet fully in 3D (while distinguishing which node is the ascending node, $\it{\Omega}$, will require a combination of astrometric and radial velocity data). Because astrometry provides the full 3D orbit, the inclination may be measured and thus the true mass of the planet is found, resolving the inclination degeneracy that plagues Doppler spectroscopy.

Because the star is not the only thing in motion, some work is needed before jumping into searching for exoplanet-induced astrometric signals. Firstly, the proper motion (the natural drift of the position of the star in the sky as the result of each star’s independent galactocentric orbit) of the star must be modelled out. Additionally, the orbital motion of Earth around the Sun will cause another signal in the astrometric data that must be modelled out. After this is done, whatever remains must be the intrinsic motion of the star under the influence of other bodies.

To give some idea of comparison, from “above” the solar system, the sun’s astrometric signal would look like this.

The large yellow circle represents the diameter of the sun, and the black line is the path it takes, due to the influence of the planets in our solar system. Jupiter and Saturn dominate the astrometric signal. The other planets aren’t detectable in this timespan. Notice that the amplitude of the astrometric variation is comparable to the diameter of the star itself. More specifically, for a star with a distance$D$and a single orbiting companion, the astrometric amplitude can be estimated as

$\displaystyle \alpha = \arctan{\left( \frac{a * q}{D(q+1)} \right)}$

where q is the mass ratio between the planet and the star$M_p / M_*$, and a is given as

$\displaystyle a = (G(M_* + M_p))^{1/3} (P/2 \pi)^{2/3}$

The need for precision thus makes itself apparent. So far, very few extrasolar planets have been detected this way. On the positive side, radial velocity candidates whose true masses are significantly higher than their RV-derived minimum mass have larger than expected amplitudes. This makes astrometry quite effective at determining if exoplanet candidates are actually low mass or even main sequence stars. In most cases, astrometry can set an upper limit to the astrometric amplitude, which translates directly into an upper limit for the planet’s mass. This can permit even a non-detection to secure the planetary nature of a candidate extrasolar planet.

## Bulk Properties of the Planets

Even without going outside our solar system, we can take a look at the eight that are well known to even our children and get a preliminary grasp on the types of the planets that we can expect to find throughout the Universe. Looking at the graphic above, which has the size of our planetary system’s members to scale (but not their separations from each other), we readily note that the planets available to use for close study assume a wide variety of appearances and sizes. Furthermore, they visually seem to fall into two or three different classifications. Let’s take a more quantitative approach to comparing these planets.

The planets that we are most intuitively familiar with are the smallest members of the planetary zoo. They are the terrestrial planets, and are composed of rock and metal, silicates and iron. They have high densities and sudden phase transition boundaries between states of matter separating their solid and gaseous components (which we might call a “surface”). For our solar system, this corresponds to the planets Mercury, Venus, Earth and Mars.

Because Earth is the most massive representative of this type of planet that we have available to us for study, terrestrial planets of ~Earth-mass are generally called “Earth-like,” but this similarity is in bulk composition alone, and is not to imply habitability. The term is loose enough that even the infernally hellish Venus would fit the description. It’s important to remember that the differences between Earth and Venus are “skin-deep” so to speak. That which makes the two planets so different is confined to a thin layer of gases clinging onto their surfaces, and is a negligible fraction of the planet’s total mass. Planets that are more massive than Earth, typically about twice the mass of Earth and above, are called “super-Earths,” but the term is a loose one.

Being the prototypical terrestrial planet, the mass of Earth defines one of the more commonly used mass measurements. One “earth-mass,” $M_\oplus$, is a mass unit of mass equivalent to the mass of Earth.

Moving further out in the Solar System, we find the gas giant planets Jupiter and Saturn.

The Bulk Composition of the Gas Giants by Mass

These planets are those who have the majority of their mass in the form of “gases,” or at least those elements labelled so on the Periodic Table. As the depth into the planet increases, the atmospheric pressure increases considerably. Deep within the planet, the pressure is great enough that the gas slowly transitions into a liquid, with exotic properties that generate enormously powerful magnetospheres. At their centre may exist a core of solid material of several Earth-masses.

As a planet-forming disk evolves, it’s composition will change such that the gas-to-dust ratio drops until all of the gas has either fallen into the star, been accreted by planets, or pushed out of the planetary system through the radiation pressure of sunlight from the star. Therefore, planets that form first will more closely resemble the composition of the disk where they were formed. We can tell therefore that Jupiter formed first as its composition most closely resembles not only the sun, but that of the protoplanetary disk out of which it formed as well. In the case of gas giants, large solid cores of several Earth-mass grow in size until they begin runaway accretion of hydrogen from the disk (this typically occurs at a mass of ~10$M_\oplus$). Furthermore, Saturn formed later, as is evidenced by the lower gas-to-rock fraction.

Jupiter is the most massive example of this class of planets in our Solar system and therefore sets another unit of mass, the “Jupiter-mass,”$M_J$, roughly equal to 318$M_\oplus$,

Further out in the Solar system lie the planets that formed later, and had access to only the scraps of what was left of the hydrogen and helium in the disk. Their compositions are dominated by the ices that condensed out of the volatiles in the disk. These planets are called the “ice giants,” and are represented by Uranus and Neptune.

Bulk Composition of the Ice Giants by Mass

Like the gas giants, their atmospheres get thicker with depth but they reach a boundary that is more compositional than arising from a state of matter, where the composition goes from an H-He envelope to an icy mantle, likely dominated by water ice with other volatiles (ammonia, methane, etc). There is likely not a quick phase transition between corresponding to a surface on such planets.

Neptune is the most massive of these planets in our Solar System, and is thus the prototypical “Neptune-like planet.” A unit of mass based off Neptune can obviously be conceived, but it is not used much in practice. With Neptune having a mass of ~17$M_\oplus$, the masses for these planets are usually expressed in terms of$M_\oplus$or$M_J$.

When the masses and radii of the Solar System planets are plotted, they group fairly nicely sorted by their type.

Mass-Radius Diagram for the Solar System

In an upcoming post, we’ll look at doing this for extrasolar planets to see how our solar system fits into the range of extrasolar planet masses and radii and how well these neat cookie-cutter categories hold up in the wealth of examples of extrasolar planets with measured masses and radii to date.


text:
# On the Potts Antiferromagnet on Random Graphs

@article{CojaOghlan2016OnTP,
title={On the Potts Antiferromagnet on Random Graphs},
author={Amin Coja-Oghlan and Nor Jaafari},
journal={Electron. J. Comb.},
year={2016},
volume={23},
pages={P4.3}
}
• Published 29 February 2016
• Mathematics, Computer Science
• Electron. J. Comb.
Extending a prior result of Contucci et al (Comm. Math. Phys. 2013), we determine the free energy of the Potts antiferromagnet on the Erdos-Renyi random graph at all temperatures for average degrees $d \le (2k-1)\ln k - 2 - k^{-1/2}$. In particular, we show that for this regime of $d$ there does not occur a phase transition.
6 Citations

#### Topics from this paper

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• Mathematics, Computer Science
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• 2017
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• 2019
We verify a key component of the replica symmetry breaking hypothesis put forward in the physics literature (Mézard and Montanari in Information, physics and computation. Oxford University Press,Expand
Information-theoretic thresholds from the cavity method
• Computer Science, Mathematics
• 2018
A formula for the mutual information in statistical inference problems induced by random graphs is established and it is shown that the Mutual information holds the key to understanding certain important phase transitions in random graph models. Expand
Continuous spin models on annealed generalized random graphs
• Mathematics, Physics
• 2016
Abstract We study Gibbs distributions of spins taking values in a general compact Polish space, interacting via a pair potential along the edges of a generalized random graph with a given asymptoticExpand
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This paper proves physics predictions on the precise location and nature of phase transitions in random constraint satisfaction problems that predicted that their satisfiability thresholds are quite generally preceded by several other thresholds that have a substantial impact both combinatorially and computationally. Expand
Sidorenko's Conjecture, Colorings and Independent Sets
• Mathematics, Computer Science
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• 2017
This paper proves Sidorenko's conjecture for certain special graphs G, which states that in the last two cases (independent sets and Widom-Rowlinson colorings) the graph $H$ does not need to be bipartite. Expand

#### References

SHOWING 1-10 OF 38 REFERENCES
Antiferromagnetic Potts Model on the Erdős-Rényi Random Graph
• Mathematics, Physics
• 2013
We study the antiferromagnetic Potts model on the Poissonian Erdős-Rényi random graph. By identifying a suitable interpolation structure and an extended variational principle, together with aExpand
The Condensation Phase Transition in Random Graph Coloring
• Physics, Mathematics
• 2016
Based on a non-rigorous formalism called the “cavity method”, physicists have put forward intriguing predictions on phase transitions in diluted mean-field models, in which the geometry ofExpand
A positive temperature phase transition in random hypergraph 2-coloring
• Mathematics, Computer Science
• ArXiv
• 2014
This paper establishes the existence and asymptotic location of this so-called condensation phase transition in the random hypergraph $2$-coloring problem. Expand
Potts glass on random graphs
• Physics
• 2008
We solve the q-state Potts model with anti-ferromagnetic interactions on large random lattices of finite coordination. Due to the frustration induced by the large loops and to the local tree-likeExpand
The Replica Symmetric Solution for Potts Models on d-Regular Graphs
• Mathematics, Physics
• 2012
We establish an explicit formula for the limiting free energy density (log-partition function divided by the number of vertices) for ferromagnetic Potts models on uniformly sparse graph sequencesExpand
Decay of Correlations for the Hardcore Model on the $d$-regular Random Graph
• Mathematics, Physics
• 2014
A key insight from statistical physics about spin systems on random graphs is the central role played by Gibbs measures on trees. We determine the local weak limit of the hardcore model on randomExpand
MCMC sampling colourings and independent sets of G(n, d/n) near uniqueness threshold
This work focuses on the k-colouring model} and the hard-core model with fugacity \lambda when the underlying graph is an instance of Erdos-Renyi random graph G(n,p), where p=d/n and d is fixed and uses the Markov Chain Monte Carlo method for sampling from Gibbs distribution. Expand
Reconstruction on Trees and Spin Glass Transition
• Physics
• 2006
Consider an information source generating a symbol at the root of a tree network whose links correspond to noisy communication channels, and broadcasting it through the network. We study the problemExpand
On the chromatic number of random d-regular graphs
• Mathematics
• 2008
In this work we show that, for any fixed d, random d-regular graphs asymptotically almost surely can be coloured with k colours, where k is the smallest integer satisfying d (2k−3)log(k−1), then theExpand
On the chromatic number of a random hypergraph
• Computer Science, Mathematics
• J. Comb. Theory, Ser. B
• 2015
We consider the problem of k-colouring a random r-uniform hypergraph with n vertices and cn edges, where k, r, c remain constant as n ? ∞ . Achlioptas and Naor showed that the chromatic number of aExpand


text:
# CLT and Proportions

#### mathematicallyUNinclined

##### New Member
I would like to ask those of you who know their CLTs and Proportions a few questions.
I am completing my HW right now and wanted do ask you guys a few Qs please. :wave:

Example: The mean price of shoes in a certain store is $585 with a standard deviation of$195. I am randomly selecting 35 pairs of shoes in this store. What is the probability that the mean price will be between $500 and$690 ?

I identified this example as CLT.

I started with the x bar being $585, st.dev$195 and n=35. Then wrote down that the mean of my x bar is basically the mean = $585, then calculated the sigma for the x bar and wrote down my probability as P ($500< x bar < $690). then calculated the z score for the$500 and $690 and calculated that my area on diagram based on the z scores from the table. I came up with 99% probability that my shoe prices will fall between$500 and \$690.

Does this sound any good? I would appreciate feedback.


text:
# Expectation Value of Momentum

1. Nov 15, 2012

### Johny18

Good Evening Fellows,
I have the following question,
So far I have learned that the expectation value of momentum is equal the time derivative of the expectation value of position. If the potential only depends upon position and not on time. Then, if we use the time independent schrodinger equation the wavefuntion will be separable into a purely function of x and a function of t. Therefore, is it correct to assert that the expectation value of momentum will always be zero for this case, since the expectation value of position will be a constant?

2. Nov 15, 2012

### Staff: Mentor

For the stationary states (energy eigenstates) that you get by solving the time-independent SE, this is true.

However, it is not true for states that are superpositions of energy eigenstates. Consider for example a superposition of the first two energy eigenstates of the "particle in a box:"

$$\Psi(x,t) = a_1 \psi_1(x)e^{-iE_1 t / \hbar} + a_2 \psi_2(x)e^{-iE_2 t / \hbar}$$

For this wavefunction, the expectation values of position and momentum are not constant.


text:
# 15th Varenna Conference on Nuclear Reaction Mechanisms

Jun 11 – 15, 2018
Villa Monastero
Europe/Zurich timezone

## Nuclear reaction study for high-level radioactive waste: Cross section measurements for proton- and deuteron-induced spallation reactions of long-lived fission products

Jun 14, 2018, 11:30 AM
20m
Villa Monastero

Varenna (Italy)

He Wang

### Description

The disposal of high-level radioactive waste from nuclear power plants is one of the major issues in worldwide. As a promising solution, research and development has been devoted to the partitioning and transmutation technology where long-lived nuclides are converted to stable or short-lived ones. In particular, the transmutation on the long-lived fission products (LLFPs) has received much attention because the LLFP nuclei have large radiotoxicities and they can be produced continuously in the accelerator driven systems and next-generation nuclear reactors. However, experimental reaction data for LLFP nuclei are very limited.

Aiming at bringing a new invention to the nuclear transmutation on LLFP, we have performed a series of systematic studies on the proton- and deuteron-induced spallation for the long-lived fission products ($^{90}$Sr,$^{93}$Zr,$^{107}$Pd,$^{126}$Sn,$^{137}$Cs) at reaction energies ranging from 50 to 200 MeV/nucleon at RIKEN Radioactive Isotope Beam Factory. The inverse kinematics technique was adopted. Namely, LLFP beams were used and proton/deuteron targets were conducted to induce the reactions. Our study on $^{137}$Cs and $^{90}$Sr [1] is the first attempt in the history of nuclear physics to solve the problem of the LLFP transmutation and has triggered the reaction studies for other long-lived fission products.

The present work focuses on $^{137}$Cs, $^{90}$Sr and $^{107}$Pd. Cross sections on proton/deuteron were successfully obtained for these three nuclei and both target and energy dependence of reactions were investigated. In addition, the newly obtained data were compared with the nuclear interaction model including both intra-nuclear cascade and evaporation processes in the frame work of the Particle and Heavy Ion Transport code System (PHITS). In the presentation, the results for LLFP nuclei $^{137}$Cs, $^{90}$Sr [1] and $^{107}$Pd [2] as well as the potential of spallation reaction on the LLFP transmutation will be discussed.

This work was supported and funded by the ImPACT program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan).

[1] H. Wang, H. Otsu, H. Sakurai et al., Phys. Lett. B 754, 104(2016).
[2] H. Wang, H. Otsu, H. Sakurai et al., Phys. Theor. Exp. Phys. 2017, 021D01 (2017).

### Primary authors

He Wang Dr Hideaki Otsu (RIKEN NIshina Center) Hiroyoshi Sakurai (RIKEN) Dr Satoshi Takeuchi (Tokyo Institute of Technology) Yukinobu Watanabe (Kyushu University) Dr Shoichiro Kawase (Kyushu University) Dr Toshiyuki Sumikama (RIKEN Nishina Center) Dr Nobuyuki Chiga (RIKEN Nishina Center)


text:
MathSciNet bibliographic data MR143240 60.30 Baum, Leonard E. On convergence to $+\infty$$+\infty$ in the law of large numbers. Ann. Math. Statist. 34 1963 219–222. Article

For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.


text:
Differentiable Architecture Search (DARTS)
This paper addresses the scalability challenge of architecture search by formulating the task in a differentiable manner. Unlike conventional approaches of applying evolution or reinforcement learning over a discrete and non-differentiable search space, our method is based on the continuous relaxation of the architecture representation, allowing efficient search of the architecture using gradient descent. Extensive experiments on CIFAR-10, ImageNet, Penn Treebank and WikiText-2 show that our algorithm excels in discovering high-performance convolutional architectures for image classification and recurrent architectures for language modeling, while being orders of magnitude faster than state-of-the-art non-differentiable techniques. …

State-Reification Network
Machine learning promises methods that generalize well from finite labeled data. However, the brittleness of existing neural net approaches is revealed by notable failures, such as the existence of adversarial examples that are misclassified despite being nearly identical to a training example, or the inability of recurrent sequence-processing nets to stay on track without teacher forcing. We introduce a method, which we refer to as \emph{state reification}, that involves modeling the distribution of hidden states over the training data and then projecting hidden states observed during testing toward this distribution. Our intuition is that if the network can remain in a familiar manifold of hidden space, subsequent layers of the net should be well trained to respond appropriately. We show that this state-reification method helps neural nets to generalize better, especially when labeled data are sparse, and also helps overcome the challenge of achieving robust generalization with adversarial training. …

Algorithmic Polynomial
The approximate degree of a Boolean function $f(x_{1},x_{2},\ldots,x_{n})$ is the minimum degree of a real polynomial that approximates $f$ pointwise within $1/3$. Upper bounds on approximate degree have a variety of applications in learning theory, differential privacy, and algorithm design in general. Nearly all known upper bounds on approximate degree arise in an existential manner from bounds on quantum query complexity. We develop a first-principles, classical approach to the polynomial approximation of Boolean functions. We use it to give the first constructive upper bounds on the approximate degree of several fundamental problems: – $O\bigl(n^{\frac{3}{4}-\frac{1}{4(2^{k}-1)}}\bigr)$ for the $k$-element distinctness problem; – $O(n^{1-\frac{1}{k+1}})$ for the $k$-subset sum problem; – $O(n^{1-\frac{1}{k+1}})$ for any $k$-DNF or $k$-CNF formula; – $O(n^{3/4})$ for the surjectivity problem. In all cases, we obtain explicit, closed-form approximating polynomials that are unrelated to the quantum arguments from previous work. Our first three results match the bounds from quantum query complexity. Our fourth result improves polynomially on the $\Theta(n)$ quantum query complexity of the problem and refutes the conjecture by several experts that surjectivity has approximate degree $\Omega(n)$. In particular, we exhibit the first natural problem with a polynomial gap between approximate degree and quantum query complexity. …

Dynamic Graph Neural Network (DGNN)
Graphs, which describe pairwise relations between objects, are essential representations of many real-world data such as social networks. In recent years, graph neural networks, which extend the neural network models to graph data, have attracted increasing attention. Graph neural networks have been applied to advance many different graph related tasks such as reasoning dynamics of the physical system, graph classification, and node classification. Most of the existing graph neural network models have been designed for static graphs, while many real-world graphs are inherently dynamic. For example, social networks are naturally evolving as new users joining and new relations being created. Current graph neural network models cannot utilize the dynamic information in dynamic graphs. However, the dynamic information has been proven to enhance the performance of many graph analytical tasks such as community detection and link prediction. Hence, it is necessary to design dedicated graph neural networks for dynamic graphs. In this paper, we propose DGNN, a new {\bf D}ynamic {\bf G}raph {\bf N}eural {\bf N}etwork model, which can model the dynamic information as the graph evolving. In particular, the proposed framework can keep updating node information by capturing the sequential information of edges, the time intervals between edges and information propagation coherently. Experimental results on various dynamic graphs demonstrate the effectiveness of the proposed framework. …


text:
# Taylor Series for $\dfrac{x}{x ^ 2 + x + 1}$.

I'm trying to calculate the taylor series of $$\dfrac{x}{x ^ 2 + x + 1}$$.

Algebraic manipulation didn't get me anywhere, since the roots of $$x ^ 2 + x + 1$$ are complex.

Integrate or derive made the problem worse

Any tips on how to proceed?

• Complex roots shouldn't faze one. But your function equals $\frac{x-x^2}{1-x^3}$. Oct 15, 2019 at 18:26
• @gmn_1450 Observe that $(1+x+x^{2})^{-1}=1-(x+x^2)+(x+x^2)^{2}+...$, if $|x+x^{2}|<1$ Oct 15, 2019 at 18:26
• Partial fractions. Oct 15, 2019 at 18:33
• Taylor series can be centered lots of places. Do you mean a Taylor series centered at $0$? Oct 15, 2019 at 19:38

There seem to be many ways to go about this, so here is one: put $$\dfrac{x}{x ^ 2 + x + 1}=\sum_{n=0}^{\infty} a_nx^n$$, then $$x=\sum_{n=0}^{\infty} a_nx^n(x^2+x+1)=a_0+(a_1+a_0)x+\sum_{n=2}^{\infty}(a_n+a_{n-1}+a_{n-2})x^n,$$ and by comparing the coefficients we get $$a_0=0$$, $$a_1=1$$, and $$a_n+a_{n-1}+a_{n-2}=0$$ for $$n \geq 2$$. You can see the coefficients repeat ($$a_2=-1,a_3=0,a_4=1,\dots$$), so we have $$a_{3k}=0, a_{3k+1}=1, a_{3k+2}=-1$$, or in other words $$\dfrac{x}{x ^ 2 + x + 1}=x-x^2+x^4-x^5+\dots$$

$$\begin{array}\\ \dfrac{x}{x ^ 2 + x + 1} &=\dfrac{x(1-x)}{1-x^3}\\ &=(x-x^2)\sum_{k=0}^{\infty} x^{3k}\\ &=\sum_{k=0}^{\infty} x^{3k+1}-\sum_{k=0}^{\infty} x^{3k+2}\\ \end{array}$$

Suppose we have the recurrence $$q_{n+2}=aq_{n+1}+bq_n\qquad n\ge0$$ where $$q_0,q_1$$ are given. We then define the generating function for this sequence, $$q(x)=\sum_{n\ge0}q_nx^n.$$ We see that $$q_{n+2}x^{n+2}=ax\cdot q_{n+1}x^{n+1}+bx^2\cdot q_nx^n,$$ so that, upon applying $$\sum_{n\ge0}$$ to both sides, $$q(x)-q_1x-q_0=ax(q(x)-q_0)+bx^2q(x)$$ and $$q(x)=\frac{(q_1-aq_0)x+q_0}{1-ax-bx^2}.$$ We apply this to the problem in question ($$a=b=-1$$), and get $$\frac{x}{x^2+x+1}=\sum_{n\ge0}q_nx^n,$$ where $$q_{n+2}+q_{n+1}+q_{n}=0\qquad q_0=0,q_1=1.$$ The first few values for $$q_n$$ are $$0,1,-1,0,1,-1,0,1,-1,0,...$$

Hint

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

For $$-1

$$(1+x+x^2)^{-1}=(1-x)(1-x^3)^{-1}=?$$

Using Binomial Series/ Infinite Geometric Series, $$\dfrac x{1+x+x^2}=x(1-x)(1-x^3)^{-1}=(x-x^2)\sum_{r=0}^\infty(x^3)^r=\sum_{r=0}^\infty(x^{3r+1}- x^{3r+2})$$

• Oct 15, 2019 at 18:37
• For this one, it is not needed. It would be if it had $1/(1-x)^3$. Oct 16, 2019 at 17:28
• @martycohen, Please find the updated post Oct 17, 2019 at 5:31


text:
# Corollary of the Hahn-Banach theorem.

Recall the Hahn-Banach Theorem in Normed Vector Spaces:

"Let $X$ be a NVS, and let $W$ be a linear subspace of $X$. Then, for any $f_W\in W'$ there exists an extension $f_X\in X'$ such that $\|f_X\|=\|f_W\|$."

And consider its corollary,

"Let $W$ be a proper closed subspace of an NVS $X$ and let $x\in X \setminus W$. Set $\inf_{w\in W}\|x-w\|=\delta\gt0.$ Then, there exists $f\in X'$ such that $\|f\|=1,\,f(w)=0,\, \forall\, w\in W$ and $f(x)=\delta.$"

What does this actually tell us about a bounded linear functional $f\in X'$, and its relation to $W\subset X$? In my notes it says that this corollary is an seperation result, whereas the Hahn-Banach theorem is centered on showing when an extension exists.

I want to say that this corollary is almost the opposite of the Hahn-Banach theorem, since the Hahn-Banach theorem allows us to find an $f_X\in X'$ such that $f_X(w)=f_W(w),\,\forall w \in W$, whereas its corollary states that $f\in X'$ is such that $f(w)=0,\,\forall\, w\in W$. I am looking to get an intuitive understanding of what is going on.

• Maybe this is too vague, but I think of it as a relationship between the topological structure of a Banach space and its algebraic structure: If a point can be "topologically distinguished" from $W$ (in that it doesn't lie in the closure of $W$), then it can be "algebraically distinguished" from $W$ (by a continuous linear functional). – Noah Olander Jan 5 '16 at 19:43


text:
# How to install a package from a local repository?

I am unable to connect to the package repository server using MiKTeX due to a proxy error. How would I go about installing a package from a local repository? I am trying to install the package beamer. Where do I actually find the file to download?

You need to get the packages from the miktex folder on ctan:

http://mirrors.ctan.org/systems/win32/miktex/tm/packages/

Put whatever you want to install e.g.

http://mirrors.ctan.org/systems/win32/miktex/tm/packages/beamer.tar.lzma

in some empty folder (outside miktex).

Beside this you need the three database files

Put them also in the folder. Then you can use the folder as local repository.

• Tried that, I'm getting: ! LaTeX Error: File `etoolbox.sty' not found. – John Doe Nov 28 '18 at 17:31
• Well then you will have to install etoolbox in the same way. mirrors.ctan.org/systems/win32/miktex/tm/packages/… – Ulrike Fischer Nov 28 '18 at 17:41
• I'm getting an unending number of missing dependencies. Is there any way to find all of them I need to use beamer? – John Doe Nov 28 '18 at 18:13
• Not really. You can search for some log and guess, but it would be easier to download the whole lot, e.g. with wget. – Ulrike Fischer Nov 28 '18 at 18:15
• There are an enormous number of dependencies. I've manually download around 15 now. No end in sight, – John Doe Nov 28 '18 at 18:28

From https://ctan.org/tex-archive/systems/win32/miktex/tm/packages, download to a directory on your h.d. `beamer.tar.lzma` and the 5 following files:

`pr.ini`, `files.csv.lzma`, `miktex-zzdb1-2.9.tar.lzma`, `miktex-zzdb2-2.9.tar.lzma` and `miktex-zzdb3-2.9.tar.lzma`.

You have to declare the local repository from MiKTeX console, Settings, like this:


text:
icalp15_lra.pdf (323.22 kB)

# On Temporal Graph Exploration

conference contribution
posted on 07.05.2015, 09:24 by Thomas Erlebach, Michael Hoffmann, F. Kammer
A temporal graph is a graph in which the edge set can change from step to step. The temporal graph exploration problem TEXP is the problem of computing a foremost exploration schedule for a temporal graph, i.e., a temporal walk that starts at a given start node, visits all nodes of the graph, and has the smallest arrival time. We consider only temporal graphs that are connected at each step. For such temporal graphs with $n$ nodes, we show that it is $\NP$-hard to approximate TEXP with ratio $O(n^{1-\epsilon})$ for any $\epsilon>0$. We also provide an explicit construction of temporal graphs that require $\Theta(n^2)$ steps to be explored. We then consider TEXP under the assumption that the underlying graph (i.e. the graph that contains all edges that are present in the temporal graph in at least one step) belongs to a specific class of graphs. Among other results, we show that temporal graphs can be explored in $O(n^{1.5}k^2\log n)$ steps if the underlying graph has treewidth $k$ and in $O(n\log^3 n)$ steps if the underlying graph is a $2\times n$ grid. Finally, we show that sparse temporal graphs with regularly present edges can always be explored in $O(n)$ steps.

## Citation

Proceedings of the 42nd International Colloquium on Automata, Languages, and Programming, Lecture Notes in Computer Science Volume 9134, 2015, pp. 444-455 (12)

## Author affiliation

/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of Computer Science

ICALP, Kyoto

## Version

AM (Accepted Manuscript)

## Published in

Proceedings of the 42nd International Colloquium on Automata

## Publisher

Springer Berlin Heidelberg

0302-9743

## isbn

978-3-662-47671-0;978-3-662-47672-7

1007

20/06/2016

06/07/2015

10/07/2015

en


text:
Question

# Find the smallest possible number by which 8788 must be multiplied to obtain a perfect cube.

To write the prime factorization of any number, start by dividing the number by the first prime number, which is 2 and then continue to divide by 2 until you get a number which is not divisible by 2 (which means that you get a decimal or remainder on dividing the number by 2). Then start dividing the number by the next prime number which is 3. Continue dividing the number by 3 until you get a number which is not divisible by 3. Thus, continuing the process, keep dividing the numbers by series of prime numbers $5,7,...$until the only numbers left are prime numbers. Write the given number as a product of all the prime numbers (considering the fact to count each prime number as many times as it divides the given number) to get the prime factorization of the given number.
Thus, the prime factorization of 8788 is $8788={{2}^{2}}\times {{13}^{3}}$.
We observe that ${{13}^{3}}$ is already a perfect cube. So, to make 8788 a perfect cube, we need to make ${{2}^{2}}$ a perfect cube. Thus, the smallest number by which we need to multiply ${{2}^{2}}$ is 2.


text:
## Unit4.4.3Parallelizing the packing

Since not parallelizing part of the computation can translate into a slow ramping up of performance, it is worthwhile to consider what else perhaps needs to be parallelized. Packing can contribute significantly to the overhead (although we have not analyzed this, nor measured it), and hence is worth a look. We are going to look at each loop in our "five loops around the micro-kernel" and reason through whether parallelizing the packing of the block of $A$ and/or the packing of the row panel of $B$ should be considered.

### Subsubsection4.4.3.1Loop two and parallelizing the packing

Let's start by considering the case where the second loop around the micro-kernel has been parallelized. Notice that all packing happens before this loop is reached. This means that, unless one explicitly parallelizes the packing of the block of $A$ and/or the packing of the row panel of $B \text{,}$ these components of the computation are performed by a single thread.

###### Homework4.4.3.1.
• Copy PackA.c into MT_PackA.c. Change this file so that packing is parallelized.

• Execute it with

export OMP_NUM_THREADS=4
make MT_Loop2_MT_PackA_8x6Kernel

• Be sure to check if you got the right answer! Parallelizing the packing, the way PackA.c is written, is a bit tricky.

• View the resulting performance with data/Plot_MT_Loop2_MT_Pack_8x6.mlx.

Solution

On Robert's laptop (using 4 threads), the performance is not noticeably changed:

###### Homework4.4.3.2.
• Copy PackB.c into MT_PackB.c. Change this file so that packing is parallelized.

• Execute it with

export OMP_NUM_THREADS=4
make MT_Loop2_MT_PackB_8x6Kernel

• Be sure to check if you got the right answer! Again, parallelizing the packing, the way PackB.c is written, is a bit tricky.

• View the resulting performance with data/Plot_MT_Loop2_MT_Pack_8x6.mlx.

Solution

On Robert's laptop (using 4 threads), the performance is again not noticeably changed:

###### Homework4.4.3.3.

Now that you have parallelized both the packing of the block of $A$ and the packing of the row panel of $B\text{,}$ you are set to check if doing both shows a benefit.

• Execute

export OMP_NUM_THREADS=4
make MT_Loop2_MT_PackAB_8x6Kernel

• Be sure to check if you got the right answer!

• View the resulting performance with data/Plot_MT_Loop2_MT_Pack_8x6.mlx.

Solution

On Robert's laptop (using 4 threads), the performance is still not noticeably changed:

Why don't we see an improvement? Packing is a memory intensive task. Depending on how much bandwidth there is between cores and memory, packing with a single core (thread) may already saturate that bandwidth. In that case, parallelizing the operation so that multiple cores are employed does not actually speed the process. It appears that on Robert's laptop, the bandwidth is indeed saturated. Those with access to beefier processors with more bandwidth to memory may see some benefit from parallelizing the packing, especially when utilizing more cores.

### Subsubsection4.4.3.2Loop three and parallelizing the packing

Next, consider the case where the third loop around the micro-kernel has been parallelized. Now, each thread packs a different block of $A$ and hence there is no point in parallelizing the packing of that block. The packing of the row panel of $B$ happens before the third loop around the micro-kernel is reached, and hence one can consider parallelizing that packing.

###### Homework4.4.3.4.

You already parallelized the packing of the row panel of $B$ in Homework 4.4.3.2

• Execute

export OMP_NUM_THREADS=4
make MT_Loop3_MT_PackB_8x6Kernel

• Be sure to check if you got the right answer!

• View the resulting performance with data/Plot_MT_Loop3_MT_Pack_8x6.mlx.

Solution

On Robert's laptop (using 4 threads), the performance is again not noticeably changed:

### Subsubsection4.4.3.3Loop five and parallelizing the packing

Next, consider the case where the fifth loop around the micro-kernel has been parallelized. Now, each thread packs a different row panel of $B$ and hence there is no point in parallelizing the packing of that block. As each thread executes subsequent loops (loops four through one around the micro-kernel), they pack blocks of $A$ redundantly, since each allocates its own space for the packed block. It should be possible to have them collaborate on packing a block, but that would require considerable synchronization between loops... Details are beyond this course. If you know a bit about OpenMP, you may want to try this idea.


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# Can I recover a group by its homomorphisms?

There is finitely generated group $G$ which I don't know. For every finite group $H$ I can think of, I know the number of homomorphisms $G \to H$ up to conjugation. (By this I mean that two homomorphisms $\phi_1$ and $\phi_2$ are being considered equivalent if there is a $h \in H$ such that $\phi_1(g)h = h\phi_2(g)$ for all $g \in G$.)

Given these numbers, do I have enough information to recover $G$?

Edit: The question is motivated from physics. A flat $H$-connection on a manifold $M$ is a homomorphism $\pi_1(M) \to H$ and a gauge transformation is a conjugation. So I'm interested whether I can recover the fundamental group by counting equivalence classes of connections, for arbitrary finite gauge groups.

Edit 2: It would be interesting as well if we count the number of homomorphisms without taking the equivalence by conjugation into account.

• As anomaly's answer shows, you should restrict attention to groups which can be detected based on their finite quotients. These are the residually finite groups (en.wikipedia.org/wiki/Residually_finite_group). It's known, for example, that the fundamental groups of closed manifolds of dimension at most $3$ are residually finite. – Qiaochu Yuan Feb 23 '15 at 20:19
• In addition to @QiaochuYuan's comment, recent work of Bridson and Wilton essentially covers this topic. Specifically, this paper proves that, given two finitely presented, residually finite groups, it is undecidable whether their profinite completions are isomorphic. The "profinite completion" of a group is an object which encodes all its finite quotients. See also this related paper, which proves this it is undecidable whether a finitely presented group has a finite quotient (equivalently, has trivial profinite completion) – user1729 Feb 23 '15 at 21:24
• Well, no, since you might need infinitely many $H$es, but it still suggests that you won't make much progress without further assumptions. – Qiaochu Yuan Feb 23 '15 at 21:48
• Alright, so I should be asking "How much information about $\pi_1$ can I recover from counting $H$-connections?" or so. Possibly the Morita equivalence class or so..? – Turion Feb 23 '15 at 21:53
• @Turion: any finitely presented group is the fundamental group of some smooth closed $4$-manifold, and it's known that there are finitely presented groups that are not residually finite. – Qiaochu Yuan Mar 12 '15 at 21:43

No. If $G$ is infinite and simple, then any homomorphism $f:G \to H$ for $H$ finite must be trivial; otherwise, $\ker f$ would be a proper normal subgroup of $G$. Such (finitely-generated) groups exist, though their construction is nontrivial; look at the Thompson groups, for example, or one of Higman's examples.

• Very beautiful answer, +1. Can one say anything if $G$ is finite? – Alex Wertheim Feb 23 '15 at 14:44
• Thanks. That's a much harder question, and I don't have any ideas offhand. My instinct would be to say that the answer is no, by analogy with the case in representation theory where two groups can have the same character table; but that's nowhere near an actual proof. – anomaly Feb 23 '15 at 14:46
• @AlexWertheim: For $G$ finite the answer should be yes. If you can determine for every finite group $H$ whether it is a quotient of $G$, then the biggest quotient will be $G$ itself. I think you can get the number of surjective homomorphisms $G \to H$ modulo conjugation by induction: the number of all homomorphisms is given and every non-surjective homomorphism has a proper subgroup $U$ of $H$ as image, so you know their number by induction (you'll have to correct this number considering $N_H(U)$ and the conjugates $U^h$). – j.p. Feb 25 '15 at 13:30
• @j.p.: What about maps that are conjugate in some $H > K$ but not in $K$ itself? – anomaly Feb 25 '15 at 15:20
• @Turion: I hope to have some time for it at the weekend. – j.p. Feb 27 '15 at 16:35

For $G$ finite the answer is yes.

It is clearly enough to know which finite groups $H$ are quotients of $G$, as $G$ will the biggest of them. For this we determine the number of surjective homomorphisms $G \to H$ by induction: For $H=1$ there is exactly one. So we may assume that we know the number of surjective homomorphisms $G \to U$ for all proper subgroups $U$ of $H$.

By assumption we are given the number $n_H$ of homomorphisms $\varphi:G \to H$ up to conjugation by elements $h$ of $H$ where $\varphi^h(g) = (\varphi(g))^h$. Let $U = \varphi(G)$ be the image of $\varphi$ in $H$. For $h\in C_H(U)$ centralizing $U\le H$ we have $\varphi^h = \varphi$, and vice versa the equality implies that $h$ centralizes $U$. So counting modulo conjugation is the same as weighing each homomorphism with one over the index of the centralizer of its image. (For $h\in N_H(U)$ normalizing $U\le H$ we know $\varphi^h(G)=\varphi(G)$. For general $h\in H$ we get that that the image of $G$ in $H$ under $\varphi^h$ is a subgroup of $H$ conjugated to $U$: $\varphi^h(G) = U^h$.)

Writing the number $n_H$ of homomorphisms (up to conjugacy) as $$n_H = \sum_{\varphi:G\to H} \frac{|C_H(\mathrm{Im}(\varphi))|}{|H|} = \sum_{U\le H}\sum_{{\varphi:G\to U} \mbox{ surjective}} \frac{|C_H(U)|}{|H|}$$ the left side $n_H$ is known by assumption. By induction all but one term (for $U=H$) of the outer sum on the right side can be calculated, which enables us to determine the missing term $x = \sum_{{\varphi:G\to H} \mbox{ surjective}} \frac{|C_H(H)|}{|H|}$. The number of surjective homomorphisms $G\to H$ is then $x$ times the index of the center of $H$.

The proof can easily be adapted for the case without conjugation mentioned in edit 2.

• Instead of $n_G$ you mean $n_H$ in the big formula? On what exactly do you do induction? Is there some sort of lattice of groups and their subgroups? – Turion Mar 2 '15 at 14:25
• @Turion: Yes, thanks, this should be $n_G$ (also in the line below the formula). Induction is on the order of the group $H$. – j.p. Mar 2 '15 at 14:31
• Shouldn't it be $n_H$? – Turion Mar 11 '15 at 17:53
• @Turion: Do you have a clue about the answer to your question for finitely generated residually finite groups? – j.p. Mar 11 '15 at 18:49
• Given that your proof feels a bit like magic, no I don't :I – Turion Mar 11 '15 at 20:52


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# What is the limit as x turns to 0 of x^sinx?

May 11, 2018

${\lim}_{x \to 0} {x}^{\sin x} = 1$

#### Explanation:

If we try to directly substitute $x = 0$ into ${\lim}_{x \to 0} {x}^{\sin x}$, we end up with ${0}^{0}$, an indeterminate form. Thus, this is a case in which L'Hospital's Rule is useful.

Consider that our function is $y = {x}^{\sin x}$. Taking the log of both sides gives $\ln y = \ln \left({x}^{\sin x}\right) = \sin x \ln x$. L'Hospital's Rule generally works only on quotients, so we note that $\sin x = \frac{1}{\csc x}$.

This gives ${\lim}_{x \to 0} \ln y = {\lim}_{x \to 0} \sin x \ln x = {\lim}_{x \to 0} \ln \frac{x}{\frac{1}{\sin} x}$. At this point we can actually apply L'Hospital's rule, giving: lim_(x->0) lnx/(1/sinx) = lim_(x->0) (1/x)/(-cosx/(sin^2x).

We continue to rearrange and apply L'Hospital's rule as necessary. I will omit the ${\lim}_{x \to 0}$ for now.

$\frac{\frac{1}{x}}{\frac{- \cos x}{{\sin}^{2} x}} = {\sin}^{2} \frac{x}{- x \cos x} \implies \left(L ' H\right) \implies \frac{2 \sin x \cos x}{- \cos x + x \sin x}$

At this point, we have a determinate form.

lim_(x->0) (2sinxcosx)/(xsinx - cosx) = (2sin(0)cos(0))/((0)(sin(0)) - cos(0)
$= \frac{0}{0 - 1} = \frac{0}{- 1} = 0$

So we've found that ${\lim}_{x \to 0} \ln y = 0$. Note that $y = {e}^{\ln y}$. Thus our original limit can be written ${\lim}_{x \to 0} {e}^{\ln y}$. Note that $e$ is just a constant, so we can simply plug in the limit we found for $\ln y$ as $x \to 0$, giving:

${\lim}_{x \to 0} {x}^{\sin x} = {\lim}_{x \to 0} {e}^{\ln y} = {e}^{0} = 1$.


text:
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# Math Help - Linear Algebra: Linear Independence question

1. ## Linear Algebra: Linear Independence question

Hey guys,

Can you tell me if this solution looks alright?

Thanks!
James

2. Sorry the picture didn't come up. Here it is again.

3. The row reduction looks good to me. I'm not sure I would your matrix equation the way you did. I would write it this way:

$\left[\begin{array}{rrr}\mathbf{v}_{1} &\mathbf{v}_{2} &\mathbf{v}_{3}\end{array}\right]\left[\begin{array}{r}c_{1}\\ c_{2}\\ c_{3}\end{array}\right]=\left[\begin{array}{rrr}1 &2 &0\\ \lambda^{2} &\lambda &0\\ 1 &4 &1\\ 2 &8 &2\end{array}\right]\left[\begin{array}{r}c_{1}\\ c_{2}\\ c_{3}\end{array}\right].$

$\left[\begin{array}{r}c_{1}\\ c_{2}\\ c_{3}\end{array}\right]$

disappeared on the RHS for some reason. Other than that, as I say, it looks good to me!

[EDIT] See Deveno's post for a correction.

4. one must be extremely careful in accounting for each step of the row reduction.

i found that to eliminate λ from the row-reduced matrix, it was necessary to divide by λ. this suggests that λ = 0 is a special case.

and indeed, when λ = 0, we have 2(1,0,1,2) + 2(0,0,1,2) = (2,0,4,8), so that {(1,0,1,2), (2,0,4,8), (0,0,1,2)} is a linearly dependent set.

thus the statement: "regardless of the parameter λ, the only scalar values of c1, c2, c3 such that c1v1 + c2v2 + c3v3 = 0 are c1,c2,c3 all 0" is UNTRUE

i have exhibited a counter-example.