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Hi, I was wondering how developers create loading screens. The only sensible way I can think of is if data is loaded in small, discrete chunks. However, the chunks would have to be able to be loaded in 1/60th of a second, so that the program can go through the render loop and present the loading screen at a good framerate. In other words, the first character's data would be loaded, the loading animation would render a frame, then the first character's data will be manipulated to suit the current level, then another frame is rendered, then the second character's data is loaded, etc. But this seems like a real pain, breaking up your data into such small pieces. Is there another way to do it? Thanks. -Gauvir_Mucca

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I'ld create another thread which were loading the data. Then when it's done loading it'ld set a bool variable to true and close itself, and when the first(rendering) thread sees that this flag has been set it'll go on to the menu screen(or whatever you need after the loading screen).

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Firstly have you ever seen a loading screen with smooth refresh? I haven't. Every load screen I know of (ok not every, but 95% of them anyway) jitter up in huge jumps and don't really give you any acurate feedback. BF1942 had a habbit of getting 95% complete in 5 seconds then sitting there for another 3 mins doing nothing in particular, and I think it was Tribes Vengance did the opposite - it would get to about 50% really slowly then suddenly finished. If you get something like fraps which tells you the framerate you'll see that most of them are only redrawn once or twice a second if that.

All you really need todo is update the loading screen after each mesh/texture rather than cutting them up. And even then you can probably get away with loading batches at a time. The loading screen is really just a nice way to say "Hey this program hasn't crashed - just wait a sec", so the accuracy isn't that important. Personally I just have a big list of models I want to load, then after each load I update the loading screen then move onto the next one. Its not smooth because each model takes a different amount of time to load (health pickup is almost instant where as the level mesh is a few seconds) but nobody has every complained or even noticed :D

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The loading system I saw for PS2 (where you can't do threads) basically did what the AP suggested. Each loadable item was split into small chunks. Then an OO-version would be:
class LoadableObject{StartLoad();ContinueLoad();EndLoad();}
So the first method did the initilisation, then ContinueLoad would load a small amount of data and finally EndLoad did any tidying.

I actually use a loading thread. By using Sleep(time) I can set the max fps for the load-screen rendering to optimise time sepnt actually loading the data. It works very well.

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Quote:
 Original post by d000hgThe loading system I saw for PS2 (where you can't do threads) basically did what the AP suggested. Each loadable item was split into small chunks.

The actual data itself wasn't split into seperate files though (which I think was what the AP wasn't happy with). You just load your one mesh file till either its finished or you've reached your time limit then just pause while loading screen updates then continue loading. So your while loop might looke something like:

while ((pos < fsize) && (time < MAX_LOAD_TIME)){   ... loading code ...}// if we've finished loading say soif (pos >= fsize)   return 1;else if (time >= MAX_LOAD_TIME)   return 0; // still need more loadingelse   return -1; // error or something

so you just keep calling this while rendering the loading screen every time it returns. When you get 1 you move onto your next model till your done. Personally I havent' bothered, but if your loading very large files (which I'm not) then its definatly a good option and much simpler than a second thread.

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To have it load smoothly, you're going to have to split the process into two different threads. One thread would be responsible for loading the data and outputing the status of what it's loading while the other would be responsible for displaying the status of the background process. If you have access to the .NET framework, you'll be using the System.Threading namespace.

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As already said - you don't need threads for this. But it's prabably easier to use them then rewrite how every different file type is loaded. Because the jobs of the two threads don't need to be related at all, you can escape all the nasty side of multithreaded programming.

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From a UI feedback point of view, I think I would argue that a jerky progress meter is actually BETTER than a very smooth one, particularly with long loads (e.g. 20+ seconds). No progress meters are ever continuous, and by now users have been conditioned to expect the jumping. Furthermore, they can hear the hard drive clicking away, and when the progress bar is tied to that clicking it provides a level of reinforcement (yes, this is really doing something, please be patient) that a calmly scrolling bar in the face of intermittent HD activitiy does not have.

It is a problem when games decide that they will assign each file a percentage of the progress bar (20 files, so each one is 5% right?), then load 95% of them in 4% of the time and sit there for 30 seconds without changing. I think it's important to identify which files take longer to load (file size alone may be enough) and assign them larger percentages. It's also important to make sure that the progress bar is never still for more than 5-10 seconds, to reassure people that nothing has frozen.

I've seen some programs (compression and installation programs, for example) that actually use two progress bars - one for the "total progress," and one for the processing of the current file after it's been loaded into memory (like when the program is creating a BSP or unencrypting or whatever). I always thought that that was a particularly effictive bit of UI.

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Quote:
 Original post by d000hgBecause the jobs of the two threads don't need to be related at all

Actually they do definatly need to be linked. How do you know what percentage to display if they aren't related at all? Thats also a problem in general that sometimes different things will load at different speeds on different machines. The first time you load a level the IO might be slowest, but then next time its already cached so the IO is pretty quick and its the BSP creator thats slower. So if you've said 50% is IO and 50% is bsp creation then on two different runs of your program the meter will move differently (first time 1st half will be slower, 2nd time 2nd half will be slower). You'll never be able to acuratly say how much of the loading is complete - a good guess is all you have which is why I've never bothered to split up loading into chunks or create another thread. Even if I do all that I still have to just 'best guess' at how much time each section takes so I don't thing you gain anything.

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I use threads, and while it can be tricky, I think it is worth it, because if you ever decide you want to stream data in without a loading screen, you can. This is useful to have large continuous worlds without breaking it up every few steps to load the next area. Also, nobody says you really need a percent done indicator on your loading screen. A lot of games don't have them, and really, all you need is some animation to show the program hasn't crashed.

tj963

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Well can you use a singleton object that handles the menu loading and refreshing?

So inside your loading loop you update your loading bar by calling the singleton object. This method works best if you have a single progress bar and a total progress bar.

I have never done threads (haha I need to start learning lol) but I would think using two threads is the best way to go.

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Quote:
 Original post by Vern777Well can you use a singleton object that handles the menu loading and refreshing?

Why a singleton? A normal object works just as well (IMHO: alot better), there is absolutely nothing that the singleton will solve in this matter. The problem is to know how much you've loaded and how much is left, as well as render every once in a while.

I think that threads are the way to go if you need max fps update on the loading bar, just poll the loading thread on how far it's gotten every time you render. A more "jumping" loading bar animation could be done while loading. Just render after each loaded object. Heck, you can even use that to get the filename before loading so you can have "Loading: SingletonsOhTehNoes.omg..." etc.

What I can't really figure out is, how do you know how much is left? Do you insert some extra info in the beginning of the main load file or do you go through the list of all things that need to be loaded, or how do you guys do it?

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I posted without much thinking.

But my idea is that you may have different loading code so you can call the singleton object from anywhere. But usually all your loading code belong in the same place...

So you have a loop, you know how many times to loop perhaps, or you know how many vertices you need to load, that would figure into you percentage equation...

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Rant! hehe I made it small so you can ignore it:
The singleton is the most misunderstood and missused pattern of all, it has become a "argh I'm to tired to design, let's just make this work!" pattern suddenly. It holds no advantage over being passed around like any other object you need to use. Unless you're force fitting it into something who's architecture / interface you can't / don't want to affect to much. Thus sneaking in a black-box communication system that there is no indication of that's it being used. To give an example of what I mean:

Which one uses the singleton (i.e. which one doesn't hint about using a loading bar?) and which one shows not just to the caller but the implementor of scene.load() that there's a loading bar and it should be used?

I've been thinking about "how do you know how big a step is?". Clearly there are differences in loading times, even for the same object made twice in a row or on two different computers (as pointed out by other posters). So there isn't even a point trying to make one object "worth more" than another. But if we (say 3D game) count triangles or models, when saving the game map this data is saved in a header part (i.e. in the beginning). Then you'd know how much to step after each modell. Other resources like sound and textures has to be taken into account too, of course. Easiest might just be having a "Total resources: 432" in the beginning and a small texture counts as much as a complex model. As mentioned earlier, we've almost come to expect jerky loading bars :)

I'm just speculating here. Has anyone tried this approach?

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Here is one approach you can try to make a smooth progress bar.

You need to know a few things about the machine that your app is running on:
1. The amount of data that you are going to be loading, in bytes for example
2. The spead at which your machine is loading that data (eg. bytes per millisecond)
3. A timer source (eg. in milliseconds)

The gist of this is to do an animation of the progress bar by calculating, on the fly, the spead at which the data is loading.

You can load the first chunk of data and figure out how long it took, and how many bytes were loaded. From this you can calculate the spead (bytes per millisecond, or what ever units you like).

Now that you know the spead at which your data is loading, you can do two things. If you assume that the rest of the data will load with a similar speed, then you can use the calculated spead to animate the progress bar. If not, then you can redo this calculation after each chunk of data is loaded.

To animate the progress bar, you smoothly interpolate it based on how much calculated time is left to load the remainder of the data.

If the platform that you are running your app on blocks during disk IO then you can either take the approach of using theads, or you can try to load data a byte or a few bytes at a time, do your progress bar calculations, and then continue on in a loop. If you write some library routines for doing your disk IO, then try incorporating this sort of progress bar functionality, and perhaps make it configurable for the times when you don't want the overhead of this kind of functionality.

Terry

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Quote:
 Original post by SeriemaRant! hehe I made it small so you can ignore it:The singleton is the most misunderstood and missused pattern of all, it has become a "argh I'm to tired to design, let's just make this work!" pattern suddenly. It holds no advantage over being passed around like any other object you need to use. Unless you're force fitting it into something who's architecture / interface you can't / don't want to affect to much. Thus sneaking in a black-box communication system that there is no indication of that's it being used. To give an example of what I mean:scene.load( filename ); VS scene.load( filename, loadingBar );Which one uses the singleton (i.e. which one doesn't hint about using a loading bar?) and which one shows not just to the caller but the implementor of scene.load() that there's a loading bar and it should be used?I've been thinking about "how do you know how big a step is?". Clearly there are differences in loading times, even for the same object made twice in a row or on two different computers (as pointed out by other posters). So there isn't even a point trying to make one object "worth more" than another. But if we (say 3D game) count triangles or models, when saving the game map this data is saved in a header part (i.e. in the beginning). Then you'd know how much to step after each modell. Other resources like sound and textures has to be taken into account too, of course. Easiest might just be having a "Total resources: 432" in the beginning and a small texture counts as much as a complex model. As mentioned earlier, we've almost come to expect jerky loading bars :)I'm just speculating here. Has anyone tried this approach?

That's not what I meant.

What I wanted to do is to have a loading screen singleton... This has nothing to do with the actual loading.

If you have to refresh your screen while loading you need someway to refresh from inside you loading code. That's why it's best to use threads.

So at the beginning of loading some stuff I would do:

for(int i = 0; i < totalNumOfItems; i++)
{

}

{

//figure out how much each loop is worth in the total progress
//of this item pertange while. So if you have 100 vertes, 1 vert would
//be 1 % on the progress bar
for(int i = 0; i < numOfVerts; i++)
{

}

}

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just doing loadResource is trival..

What if I want to traverse a tree and update the progress of my tree traversal I would so something similar maybe.

I'm in no way an expert...

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My prefered method is just to total up all the media im loading, then work out what each one is in terms of percetage. Pretty easy to do this with code to give it some dynamic quality. Then just load the files one by one, and update the Loader.Percent attribute of the loader class each time a file is loaded... then when the screen refreshes i.e.: Loader.Render() is called from your main loop, then it draws a nice up-to-date loading bar... so what if its jerky... if your loading more than 100 files, you'll prolly get a fairly even spread and then no one is gonna notice anyways ;-)

Hope this helps.

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The other thing is that you don't have to do a percentage loaded kind of thing :- for example I have a suspicion that Quake doesn't do that. All the user is really interested in is that his computer hasn't crashed and that it is doing something normal (eg think of the busy cursor in most OS's [usually a sand timer]). For instance you can have an animation of some sort (say a spining hard-disk), and whilst you do that you can add messages about what you're doing ("Loading map...", "Loading textures...", "Loading models...", "Loading sounds..." etc.) [this incidentally gives you a bit of feedback when the user says 'it crashes in the load screen, blah blah blah']

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Im kinda of partial to those loading messages, definately a good idea :-) Also as posted above is alot of help with debugging.

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The singleton pattern does not fit this problem-space well. In order to use a singleton the object:

a) Needs to enforce a single instance.
b) Needs to have global access from the full program.

I would never use a singleton for this problem because it doesn't satisfy the two requirements to use the pattern... it actually doesn't satisfy either of them.

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It won’t be a linear progress bar but it will be accurate which is more important.
Another bonus is as multi core processors is coming the loading and decoding of data can be very efficient. Since I’m also using .Net the threading support should be sufficient for this layout I hope. But I may also be over my head here.

My to cents.

Jarle

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Okay, but how do I update the progress bar from abitrary loading functions??? I want to embed loading progress bar from anywhere...

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Although the idea of a background loading thread is quite cool, and probably a good way, properly optimized, to eliminate loading screens as much as possible, if you want to avoid using multiple threads, I'd recommend using coroutines.

You can either make a load loop that displays a loading screen, consistently updated, or you could probably even work it into your main loop and do "background" loading.

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Quote:
Original post by kaysik
Quote:
 Original post by d000hgBecause the jobs of the two threads don't need to be related at all

Actually they do definatly need to be linked. How do you know what percentage to display if they aren't related at all? Thats also a problem in general that sometimes different things will load at different speeds on different machines. The first time you load a level the IO might be slowest, but then next time its already cached so the IO is pretty quick and its the BSP creator thats slower. So if you've said 50% is IO and 50% is bsp creation then on two different runs of your program the meter will move differently (first time 1st half will be slower, 2nd time 2nd half will be slower). You'll never be able to acuratly say how much of the loading is complete - a good guess is all you have which is why I've never bothered to split up loading into chunks or create another thread. Even if I do all that I still have to just 'best guess' at how much time each section takes so I don't thing you gain anything.
I don't show a % complete, I just show a repeating animation (a bit like on the KnightRider car) to show something is happening. But even if I did, it's not much data overlap and probably not the kind which can cause problems. Either way it's simple as multithreaded applications go!


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# does the command evm --debug --code OPCODES still work?

I've been trying to run this command evm --debug --code 600580600b6000396000f36005600401, and some associated permutations of it, such as:

evm --debug --code 366020036101000a600035045b6001900380600c57 --input 0101

However, all I ever see as a result is the help menu, i.e.

Has this command been deprecated, or am I doing something wrong?

• Did you forget the run command? – Ismael Sep 17 '17 at 20:55
• what do you mean? I've not forgotten it- because I never knew it. can you maybe- show me an example? UPDATE wait- I did try that, for this evm run --debug --code 600580600b6000396000f36005600401 I get this error Incorrect Usage: flag provided but not defined: -debug flag provided but not defined: -debug – smatthewenglish Sep 17 '17 at 21:00
• I've tried evm --debug --code 366020036101000a600035045b6001900380600c57 --input 0101 run with evm version 1.6.7-stable-ab5646c5 and it works. – Ismael Sep 17 '17 at 21:21
• yeah- I tried with run in the front, you need to put run in the back! anyway- thanks for your help! :D – smatthewenglish Sep 17 '17 at 21:22

You've gotta put run in the BACK!
evm --debug --code 6000355b6001900380600357 --input 0000000000000000000000000000000000000000000000000000000000000005 run

format: plain


text:
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définition - Cross correlation

voir la définition de Wikipedia

Wikipedia

Cross-correlation

In signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long-signal for a shorter, known feature. It also has applications in pattern recognition, single particle analysis, electron tomographic averaging, cryptanalysis, and neurophysiology.

For continuous functions, f and g, the cross-correlation is defined as':

$(f \star g)(t)\ \stackrel{\mathrm{def}}{=} \int_{-\infty}^{\infty} f^*(\tau)\ g(t+\tau)\,d\tau,$

where f * denotes the complex conjugate of f.

Similarly, for discrete functions, the cross-correlation is defined as:

$(f \star g)[n]\ \stackrel{\mathrm{def}}{=} \sum_{m=-\infty}^{\infty} f^*[m]\ g[n+m].$

The cross-correlation is similar in nature to the convolution of two functions.

In an autocorrelation, which is the cross-correlation of a signal with itself, there will always be a peak at a lag of zero unless the signal is a trivial zero signal.

In probability theory and statistics, correlation is always used to include a standardising factor in such a way that correlations have values between −1 and +1, and the term cross-correlation is used for referring to the correlation corr(XY) between two random variables X and Y, while the "correlation" of a random vector X is considered to be the correlation matrix (matrix of correlations) between the scalar elements of X.

If $X$ and $Y$ are two independent random variables with probability density functions f and g, respectively, then the probability density of the difference $Y - X$ is formally given by the cross-correlation (in the signal-processing sense) $f \star g$; however this terminology is not used in probability and statistics. In contrast, the convolution $f * g$ (equivalent to the cross-correlation of f(t) and g(−t) ) gives the probability density function of the sum $X + Y$.

Explanation

As an example, consider two real valued functions $f$ and $g$ differing only by an unknown shift along the x-axis. One can use the cross-correlation to find how much $g$ must be shifted along the x-axis to make it identical to $f$. The formula essentially slides the $g$ function along the x-axis, calculating the integral of their product at each position. When the functions match, the value of $(f\star g)$ is maximized. This is because when peaks (positive areas) are aligned, they make a large contribution to the integral. Similarly, when troughs (negative areas) align, they also make a positive contribution to the integral because the product of two negative numbers is positive.

With complex-valued functions $f$ and $g$, taking the conjugate of $f$ ensures that aligned peaks (or aligned troughs) with imaginary components will contribute positively to the integral.

In econometrics, lagged cross-correlation is sometimes referred to as cross-autocorrelation[1]

Properties

• The correlation of functions f(t) and g(t) is equivalent to the convolution of f *(−t) and g(t).  I.e.:
$f\star g = f^*(-t)*g.$
• If f is Hermitian, then $f\star g = f*g.$
• $(f\star g)\star(f\star g)=(f\star f)\star (g\star g)$
$\mathcal{F}\{f\star g\}=(\mathcal{F}\{f\})^* \cdot \mathcal{F}\{g\},$

where $\mathcal{F}$ denotes the Fourier transform, and an asterisk again indicates the complex conjugate. Coupled with fast Fourier transform algorithms, this property is often exploited for the efficient numerical computation of cross-correlations. (see circular cross-correlation)

• The cross correlation of a convolution of f and h with a function g is the convolution of the correlation of f and g with the kernel h:
$(f * h) \star g = h(-)*(f \star g)$

Normalized cross-correlation

For image-processing applications in which the brightness of the image and template can vary due to lighting and exposure conditions, the images can be first normalized. This is typically done at every step by subtracting the mean and dividing by the standard deviation. That is, the cross-correlation of a template, $t(x,y)$ with a subimage $f(x,y)$ is

$\frac{1}{n-1} \sum_{x,y}\frac{(f(x,y) - \overline{f})(t(x,y) - \overline{t})}{\sigma_f \sigma_t}$.

where $n$ is the number of pixels in $t(x,y)$ and $f(x,y)$, $\overline{f}$ is the average of f and $\sigma_f$ is standard deviation of f. In functional analysis terms, this can be thought of as the dot product of two normalized vectors. That is, if

$F(x,y) = f(x,y) - \overline{f}$

and

$T(x,y) = t(x,y) - \overline{t}$

then the above sum is equal to

$\left\langle\frac{F}{\|F\|},\frac{T}{\|T\|}\right\rangle$

where $\langle\cdot,\cdot\rangle$ is the inner product and $\|\cdot\|$ is the L² norm. Thus, if f and t are real matrices, their normalized cross-correlation equals the cosine of the angle between the unit vectors F and T, being thus 1 if and only if F equals T multiplied by a positive scalar.

Normalized correlation is one of the methods used for template matching, a process used for finding incidences of a pattern or object within an image. It is also the 2-dimensional version of Pearson product-moment correlation coefficient.

Time series analysis

In time series analysis, as applied in statistics, the cross correlation between two time series describes the normalized cross covariance function.

Let $(X_t,Y_t)$ represent a pair of stochastic processes that are jointly wide sense stationary. Then the cross covariance is given by [2]

$\gamma_{xy}(\tau) = \operatorname{E}[(X_t - \mu_x)(Y_{t+\tau} - \mu_y)],$

where $\mu_x$ and $\mu_y$ are the means of $X_t$ and $Y_t$ respectively.

The cross correlation function $\rho_{xy}$ is the normalized cross-covariance function.

$\rho_{xy}(\tau) = \frac{\gamma_{xy}(\tau)}{\sigma_x \sigma_y}$
where $\sigma_x$ and $\sigma_y$ are the standard deviations of processes $X_t$ and $Y_t$ respectively.

Note that if $X_t=Y_t$ for all t, then the cross correlation function is simply the autocorrelation function.

Scaled correlation

In the analysis of time series scaled correlation can be applied to reveal cross-correlation exclusively between fast components of the signals, the contributions of slow components being removed.[3]

References

1. ^ Campbell, Lo, and MacKinlay 1996: The Econometrics of Financial Markets, NJ: Princeton University Press.
2. ^ von Storch, H.; F. W Zwiers (2001). Statistical analysis in climate research. Cambridge Univ Pr. ISBN 0-521-01230-9.
3. ^ Nikolić D, Muresan RC, Feng W, Singer W (2012) Scaled correlation analysis: a better way to compute a cross-correlogram. European Journal of Neuroscience, pp. 1–21, doi:10.1111/j.1460-9568.2011.07987.x http://www.danko-nikolic.com/wp-content/uploads/2012/03/Scaled-correlation-analysis.pdf

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Longitudinally Polarized Photoproduction of Heavy Flavors at Next-to-Leading Order of QCD

# Longitudinally Polarized Photoproduction of Heavy Flavors at Next-to-Leading Order of QCD

## Abstract

We present a phenomenological study of charm quark photoproduction in longitudinally polarized lepton-hadron collisions at next-to-leading order accuracy of QCD. Our results are based on a recently developed, flexible parton-level Monte Carlo program for spin-dependent heavy flavor hadroproduction, which we extend to deal also with both direct and resolved photon contributions. The subsequent hadronization into charmed mesons is modeled in our calculations, which allows us to compare with data on double-spin asymmetries for meson production taken by the COMPASS collaboration. In general, next-to-leading order QCD corrections are found to be very significant and do not cancel in spin asymmetries. We elucidate the role of the individual hard scattering subprocesses and determine the range of parton momentum fractions predominantly probed for charm production at COMPASS. Theoretical uncertainties are estimated by varying renormalization and factorization scales and parameters controlling the hadronization of the charm quarks.

###### pacs:
13.88.+e and 12.38.Bx and 13.85.Ni

## 1 Motivation and Introduction

The quest to understand the partonic structure of the nucleon spin remains to be one of the key research areas in Hadronic Physics even after more than 25 years of strenuous experimental and theoretical efforts. In particular, the polarized gluon density, defined as

 Δg(x,μ)≡g+(x,μ)−g−(x,μ), (1)

is still one of the most elusive quantities associated with the non-perturbative partonic structure of hadrons. Here, () denotes the probability of finding a gluon at a scale with light-cone momentum fraction and helicity () in a proton with helicity . The total, integrated gluon polarization,

 Δg(μ)≡∫10Δg(x,μ)dx, (2)

and a similar contribution from the sum of all quarks and antiquarks, enters the helicity sum rule of the nucleon along with the orbital angular momenta of quarks and gluons (1). The challenge is to precisely map in a wide range of in order to minimize extrapolation uncertainties in the first moment (2).

Currently, the best constraints on are derived from global QCD analyses (2); (3) which treat all available experimental probes simultaneously and consistently at a given order in the strong coupling in perturbative QCD. The availability of next-to-leading order (NLO) QCD corrections is essential for any meaningful, quantitative analysis of parton densities. In particular, single-inclusive pion (4) and jet production (5), measured in spin-dependent proton-proton collisions at BNL-RHIC (6), have started to put significant limits on the amount of gluon polarization in the nucleon (2). New, preliminary single and di-jet data from the STAR collaboration (7) show for the first time tantalizing hints for a non-zero (8); (9). Due to the given kinematics, the current probes mainly constrain in the medium-to-large region, , which is not sufficient to reliably determine its integral (2). A very significant contribution of up to one unit of , i.e., twice the proton spin, can still come from the unexplored small region (2). Narrowing down the uncertainties on and, at the same time, extending the range in continues to be the main objective of experimental efforts in the years to come. Eventually, only a future high-energy polarized electron-proton collider, such as the EIC project (10), will finally be able to quantitatively address all the remaining open questions related to the helicity structure in the small region (11).

World-data on polarized inclusive and semi-inclusive deep-inelastic scattering are pivotal in constraining the helicity quark and antiquark densities (2); (3); (8), but due to lack of sufficient kinematic coverage of present fixed-target experiments, information on the gluon density is notoriously difficult to obtain from QCD scaling violations. Viable probes of at fixed-target energies comprise of one- and two-hadron and open charm production and have been exploited by several experimental collaborations (12); (13); (14); (15). The proper theoretical description of these processes depends on the virtuality of the probing photon and is in general more involved than corresponding calculations for hadron-hadron scattering. In case of photoproduction, where a quasi-real photon is exchanged, one has to include also “resolved” contributions, where the photon fluctuates into a vector meson of the same quantum numbers before the hard scattering with partons in the proton takes place. If the virtuality of the photon is of or higher, resolved processes are sufficiently suppressed but the additional momentum scale greatly complicates the calculations of phase-space and loop integrals.

Not surprisingly, only very few calculations at NLO of QCD are available in case of polarized beams and targets (16); (17); (18), none of which for large virtualities . As a consequence, the available data sets on hadron and open charm production (12); (13); (14); (15) have not been included in global QCD analyses of helicity parton densities so far. Experiments have analyzed their data only in terms of the gluon polarization, , under certain simplifying assumptions and based on leading order (LO) matrix elements. Nonetheless, the results of these exercises, illustrated in Fig. 1, are in fairly good agreement with a NLO extraction of including the RHIC data (2). While determinations of at fixed-target energies are more involved and provide less of a constraint than collider data, they are crucial for further testing and establishing the assumed universality of helicity-dependent parton densities and hence for our understanding of the spin structure of the nucleon and QCD in general. On the one hand, lepton-nucleon scattering experiments are sensitive to different partonic hard scattering processes than jet or hadron production at RHIC and, on the other hand, the relevant momentum fractions fall within the range already probed by data.

In this paper, we present a comprehensive phenomenological analysis of longitudinally polarized photoproduction of heavy flavors at NLO of QCD. To this end, we extend our recently developed flexible parton-level Monte Carlo program for spin-dependent heavy flavor hadroproduction (19) by including all relevant subprocesses for direct (16); (17) and resolved (19); (20) photon contributions. To facilitate the comparison to data, the hadronization of the produced charm quarks into charmed mesons and, as an additional option, their semi-leptonic decays can be modeled in our calculations based on the phenomenological functions used in Ref. (21). The presented results will allow one to consistently include available and upcoming data on spin-dependent open charm photoproduction into future global QCD analyses of helicity parton densities at NLO accuracy.

As a first application, we examine in detail all aspects of open charm production in the kinematic regime accessible with the COMPASS experiment (15). We will demonstrate that NLO corrections are very significant and do not cancel in double-spin asymmetries used to extract information on the polarized gluon density . We elucidate the relevance of the individual direct and resolved partonic subprocesses and estimate theoretical uncertainties by varying renormalization and factorization scales and parameters controlling the hadronization of the produced charm quarks into experimentally observed mesons. Finally, we shall illustrate the sensitivity of the existing data on the double-spin asymmetry for charm quark photoproduction (15); (22); (23); (24) to and compare our estimates of the relevant range of momentum fractions with those obtained by the COMPASS collaboration.

We note that our flexible Monte Carlo code is capable of computing any infrared safe heavy flavor photoproduction cross section at in longitudinally polarized lepton-nucleon collisions, including correlations of the produced heavy quark pair. Our results complement and significantly extend previously existing spin-dependent NLO calculations of single-inclusive heavy quark yields based on largely analytical methods (16); (17), where the resolved contribution was neglected, any information on the partonic recoil system was lost, and most experimental cuts could not be implemented. The results presented here will be also useful for future studies of spin-dependent charm photoproduction at an EIC (10).

The outline of the paper is as follows: in Sec. II we briefly review some of the technical aspects of setting up a parton-level Monte Carlo program for heavy flavor photoproduction in polarized lepton-nucleon collisions at NLO accuracy. In Sec. III we present a detailed phenomenological study of open charm quark production at COMPASS, including the relevance of NLO corrections, the role of the different hard scattering subprocesses, an estimate of the relevant momentum fractions , and an assessment of theoretical uncertainties. We summarize our results in Sec. IV.

## 2 Technical Framework

We consider heavy quark photoproduction in longitudinally polarized lepton-nucleon collisions at NLO accuracy of QCD by consistently including for the first time both direct and resolved photon contributions. In order to compute arbitrary infrared-safe observables within flexible experimental acceptance cuts and to account for the hadronization of the produced heavy quark pair, all phase-space integrations are performed numerically with appropriate Monte Carlo techniques. To this end, we follow closely the subtraction method devised and used in Refs. (25); (26). In the following, we will only briefly outline the technical details relevant for the case of polarized photoproduction.

Assuming, as usual, factorization, the inclusive cross section for producing a heavy quark in spin-dependent lepton-proton collisions at a center-of-mass system (c.m.s.) energy can be written as a convolution,

 dΔσQ ≡ 12[dσQ++−dσQ+−] (3) = ∑a,b∫dxadxbΔfla(xa,μf)Δfpb(xb,μf)S × dΔ^σab(xa,xb,S,mQ,k1,k2,μf,μr),

where the subscripts in (3) label the helicity states of the lepton and proton. In analogy to Eq. (1), the denote the spin-dependent parton distribution functions (PDFs) of flavor in a lepton or proton . The sum in (3) is over all contributing partonic processes , including those with a direct photon, i.e., , with the associated polarized hard scattering cross sections. The required spin-dependent matrix elements squared at NLO accuracy in dimensional regularization for the direct and resolved photon processes can be taken from (16) and (20), respectively. Corresponding unpolarized results can be found in Refs. (25); (26); (27). At , parton can be either a gluon or a light (anti-)quark producing an additional jet or hadron which is usually not observed in experiment. denote the four-momenta of the heavy quark and antiquark with mass , i.e., . and are the renormalization and factorization scale, respectively, which are typically chosen as a combination of the hard scales characterizing the process.

The “measurement function” in Eq. (3) defines the observable, for instance, through a set of step functions implementing the experimental cuts imposed on the final-state particles and selecting a certain bin in a kinematical distribution of interest. In case of the COMPASS experiment, single-inclusive spectra of mesons, differential in transverse momentum , are measured (15); (22); (23); (24). Thus, the cross section (3) at the heavy quark-level is not yet sufficient for comparing theory with experimental results and needs to be convoluted with an additional phenomenological function modeling the hadronization of the charm quark:

 dΔσHQ=dΔσQ⊗DQ→HQ. (4)

We will specify our choice for , along with all other non-perturbative inputs, in Sec. 3. Even though not relevant for the phenomenological applications considered in this paper, our flexible Monte Carlo program can also account for the semi-leptonic decays of mesons if needed.

For the integration of the fully exclusive partonic cross sections in Eq. (3), we generalize the framework used for unpolarized heavy quark production in (25); (26) to deal with singular regions of phase-space. The gist of the method is to add and subtract appropriate spin-dependent counter terms to which can be integrated analytically with respect to momenta of unresolved partons. For all resolved photon processes, , we can adopt the expressions obtained in our recent calculation of polarized heavy quark hadroproduction at (19), since these processes share the same hard scattering matrix elements. Hence, we only need to consider processes with a direct photon, , in the following.

In general, for a numerically efficient implementation of the subtraction method it is convenient (25); (26); (19) to express the three-body phase-space in terms of variables where soft and collinear singularities are particularly transparent. This is achieved by choosing , the invariant mass of the pair scaled by the available partonic c.m.s. energy squared, i.e., . In addition, one uses , the cosine of the angle between the -direction, aligned with the spatial direction of parton , and , the momentum of parton , in the c.m.s. of the incoming partons. Soft and collinear regions of phase-space are then simply associated with and , respectively, and the hard scattering matrix element for the process can be written as

 Δ|Mγb|2=Δfγb(s,mQ,x,y,θ1,θ2)s2(1−x)2(1−y2), (5)

where is regular for and . The angles are used to parametrize the spatial orientation of with respect to the plane span by the other three momenta in the c.m.s. of the pair; for further details, see (25); (26); (19). We note that the genuine NLO subprocess can have only collinear singularities at .

The partonic subprocesses contributing to the direct photon cross section at , and , can be decomposed as (26)

 dΔ^σγb = dΔ^σ(b)γb+dΔ^σ(c+)γb+dΔ^σ(c−)γb (6) + dΔ^σ(s)γb+dΔ^σ(v)γb+dΔ^σ(f)γb.

Here, and denote the Born contribution and the one-loop corrections to the process, respectively. Analytic expressions for the virtual contributions in dimensions, with ultraviolet divergences being subtracted at the renormalization scale , can be found in Ref. (16).

In Eq. (6), denotes the soft gluon emission part of the scattering cross section, which can be obtained from the full -dimensional matrix elements squared in the limit where the phase-space integrations can be performed analytically (26). We obtain the same result for the squared three-body amplitude in the soft limit as given in Eq. (A.14) of Ref. (26) but with the Born contribution for being replaced by its spin-dependent, color-averaged counterpart, which reads in dimensions:

 Δ|Mγg|2=12s(4παs)e2e2QΔBγg (7)

where

 ΔBγg=(t1u1+u1t1)(2m2Qst1u1−1). (8)

Here, and are the usual tree-level Mandelstam variables. Contrary to the unpolarized case, receives no contributions (16). is the electromagnetic charge of the heavy quark in units of the coupling , i.e., , and .

All processes exhibit singularities related to collinear splittings off the incoming photon () and parton () which need to be factorized into the bare photon and proton PDFs at a scale , respectively. As for the soft contributions discussed above, the kinematics collapses to the much simpler case of scattering, such that compact analytical expressions, summarized by in Eq. (6), can be obtained:

 dΔ^σ(c±)γb = −(4π)ε−2Γ[1+ε](2ω)εs−1−ε4εdPS2 (9) × Missing or unrecognized delimiter for \left × Δf(c±)γb(s,mQ,x,θ1),

where

 Δf(c+)γg(x,θ1) = 0, (10) Δf(c−)γg(x,θ1) = 32παss(1−x) (11) × Δ|Mγg|2∣∣p2→xp2ΔPgg(x), Δf(c+)γq(x,θ1) = 32παe2Qs(1−x) (12) × Δ|Mq¯q|2∣∣p1→xp1ΔPqγ(x), Δf(c−)γq(x,θ1) = 32παss(1−x) (13) × Δ|Mγg|2∣∣p2→xp2ΔPgq(x).

Here, and denote the momenta of the photon and parton , respectively. Both, the standard two-body phase-space in (9) and the Born matrix elements squared in Eqs. (10)-(13) are to be evaluated with appropriate collinear kinematics as indicated by the shift of momenta , etc; further details can be found in Refs. (25); (26); (19).

The resulting divergence in (9) assumes the form dictated by the factorization theorem, i.e., a convolution of -dimensional helicity-dependent LO splitting functions and Born matrix elements . By adding appropriate counter cross sections to (9), which to schematically read

 dΔ^σ~cγb(μf) = −αs2π∑i∫dxx[ΔPiγ(x,μf)dΔ^σ(b)ib(xs) (14) + ΔPib(x,μf)dΔ^σ(b)γi(xs)],

where

 Pij(x,μf)=ΔPij(x)[−1ε+γE−ln4π+lnμ2fμ2]+Δgij(x), (15)

all collinear singularities can be consistently factorized into the scale evolution of the bare photon and proton PDFs, depending on whether they originate from collinear configurations involving the initial-state photon or parton . The required Born cross sections and LO are listed in the Appendix of Ref. (19), except for

 ΔPqγ=CA[2x−1−2ε(1−x)]. (16)

in (16) and, below, are the usual QCD color factors

The factorization scheme is fully specified by the choice of in (15) for which we take , to guarantee helicity conservation within the HVBM prescription for in dimensions (28), and otherwise. The Euler constant and , both, like the scale , artifacts of dimensional regularization, are subtracted along with the singularity. This defines the scheme in the polarized case, see, e.g., Ref. (29).

Finally, the last term in Eq. (6), , contains all the remaining, finite contributions, and the phase-space integration can be performed numerically in four dimensions. Monte Carlo integrations of (3) for different measurement functions can be done in parallel by randomly generating a sufficiently large sample of final-state configurations characterized by and to account for the possibly large cancellations among the various terms. Remnants of the regularization of soft and collinear regions of phase are the mathematical distributions appearing in the different contributions to Eq. (6). Their proper definitions through a test function, properties, and numerical treatment are discussed at length in Refs. (25); (26); (19) and need not be repeated here.

## 3 Phenomenological Studies

### 3.1 Preliminaries

Based on our Monte Carlo code described above, we present a comprehensive study of charm quark photoproduction in longitudinally polarized muon-deuterium collisions at a c.m.s. energy of relevant for the COMPASS experiment at CERN. Unless specified otherwise, we will show results for single-inclusive cross sections, differential in the transverse momentum of the observed meson.

The hadronization of the heavy quarks produced in the hard scattering is modeled using Eq. (4) with (30)

 DQ→HQ(z)=NQzαQ(1−z). (17)

The fragmentation function is normalized such that its first moment is unity, i.e., . For the remaining free parameter in (17) we take as the default value in all our calculations; see Table 4 in Ref. (31). To estimate the uncertainties associated with the choice of , we will vary it in the range (31). Like in the experimental analyses (15); (22); (23); (24), we adopt a cut throughout our phenomenological studies, where is the fraction of the energy of the photon taken by the meson in the laboratory frame.

In the computation of the LO and NLO unpolarized cross sections we use the LO and NLO CTEQ6 parton densities (32) and values for the strong coupling , respectively. In the polarized case, we adopt the best fit from the comprehensive global analysis performed by the DSSV group (2), which is the only set of helicity PDFs including constraints on from data. The resulting PDFs are characterized by a small in the -range predominantly probed by RHIC experiments, , with a node at (2). To study the sensitivity of charm photoproduction data to different , we also use the GRSV “standard” set (33), which has a positive that is larger in size than the one of DSSV. We note that the most recent RHIC data (7); (9) tend to prefer a polarized gluon density somewhere in between the ones obtained in the DSSV and GRSV fits (8); (9).

Equation (3) consistently includes both direct and resolved photon contributions to the spin-dependent photoproduction cross section by defining as the convolution of the polarized lepton-to-photon splitting function and the structure functions of a circularly polarized photon, i.e.,

 Δfla(xa,μf)=∫1xadyyΔPγl(y)Δfγa(xγ=xay,μf). (18)

Nothing is known experimentally about the , which account for the hadronic structure of a photon in the resolved case where in (3). However, as will be demonstrated below, uncertainties associated with turn out to be negligible for charmed meson photoproduction at COMPASS kinematics. Contrary to inclusive hadron production (18), the resolved contribution turns out to be numerically small even when estimated with the “maximum” model for introduced in Ref. (34) and based on saturating the positivity limit at some low scale using the unpolarized set of GRV photon distributions (35) as reference. The direct part of the cross section (3) where is obtained by setting

 Δfγa(x,μf)=δ(1−x). (19)

in Eq. (18).

The collinear emission of a quasi-real photon with low virtuality and momentum fraction off a muon with mass is given by the Weizsäcker-Williams equivalent photon spectrum which reads in the polarized case (36)

 ΔPγl(y) = α2π[1−(1−y)2ylnQ2max(1−y)m2μy2] (20) + 2m2μy2(1Q2max−1−ym2μy2)].

The upper limit is determined by experimental conditions. For COMPASS we take and, in addition, restrict the fraction of the muon’s momentum taken by the quasi-real photon to the range .

We use as the value of the charm quark mass for all our results. For the factorization and renormalization scales in Eq. (3) we take with as the central value and where denotes the transverse momentum of the charm quark. As is commonly done, we vary them simultaneously in the range to estimate the residual scale dependence at NLO which represents the dominant source of theoretical uncertainty and can be taken as a rough measure of yet unknown higher order corrections.

### 3.2 Numerical Results

We begin our detailed numerical studies with a discussion of the scale dependence of the unpolarized and polarized photoproduction cross sections for mesons in muon-deuterium collisions at COMPASS. Figure 2 shows the scale ambiguity from varying simultaneously in the range for the dominant direct photon contribution () only. The solid lines denote our default choice of scales, . Since the published COMPASS data (15) are divided into three bins in the energy of the produced meson, , , and , we impose the cut in Fig. 2. Results for the other two bins in are very similar and hence not shown.

As can be seen, the theoretical uncertainties due to the choice of scale are quite sizable, both in the unpolarized and in the polarized case and can be even further inflated by varying and independently as was done, for instance, in our study of heavy quark hadroproduction (19). Also, possible variations of , which we do not pursue here, would add to the theoretical error. By comparing the solid and dashed lines in Fig. 2 one can infer the relevance of the resolved photon cross section (). As was argued in Sec. 3.1, its contribution is rather small compared to the direct photon cross section and potential uncertainties due to the unknown do not matter.

Next, we study the role of the individual partonic subprocesses and their contribution to the photoproduction cross section in Eq. (3). Figure 3 displays the decomposition of the unpolarized direct photon cross section for , as shown in Fig. 2, in terms of the and processes. Again, we choose , and results for the other two bins in are very similar. As expected, the photon-gluon fusion mechanism, which is already present at LO, gives the main contribution to the cross section. The genuine NLO photon-quark channel yields a negative but small correction. The NLO result for the meson production cross section is roughly a factor of two larger than the corresponding estimate at LO as can be gathered from comparing the dashed and solid lines in Fig. 3.

A similar exercise in the polarized case is shown in Fig. 4 for two different choices of helicity PDFs. First, one notices that the results obtained with the DSSV and GRSV sets differ in sign and magnitude of the cross section, which is readily explained by the very different gluon densities in both sets. The positive definite of GRSV leads to a similar decomposition into and subprocesses as was observed in the unpolarized case in Fig. 3. Again, the cross section is strongly dominated by photon-gluon fusion, and the channel only yields a small correction. On the contrary, the oscillating of the DSSV set of helicity PDFs leads to a negative . Since the DSSV gluon is much smaller in size than the one of GRSV, the genuine NLO photon-quark contribution, which is numerically very similar in both PDF sets, is more important and yields more than a quarter of the cross section at small . Another important observation concerns the relevance of NLO corrections which appears to be very different for the DSSV and GRSV helicity PDFs. This implies that higher order QCD effects do not cancel in the experimentally relevant double-spin asymmetry,

 ALL≡dΔσDdσD, (21)

which we shall discuss in more detail below. Clearly, for a reliable quantitative analysis of charm photoproduction in terms of polarized PDFs, preferably as part of a global QCD fit, it is indispensable to properly include NLO corrections.

Figure 6 illustrates the uncertainties due to our choice of in the fragmentation of the charm quark into the observed meson. As in Fig. 2, we show the direct photon contribution for , but here we do not impose any cuts on the meson to allow for a better comparability with the cross sections on the charm quark level which are presented as well. Compared to the factorization scale ambiguities, the dependence of our results on is fairly moderate for both the unpolarized and the polarized cross sections. A similar observation was made in the case of hadroproduction (19). Since charm quarks lose only very little of their momentum during hadronization, i.e., is peaked at fairly large values of , the convolution (4) introduces only a rather small shift in the transverse momentum spectrum of the charm quarks. This can be inferred from the curves labeled in Fig. 6.

Another interesting question concerns the range of momentum fractions predominantly probed in the PDFs by the COMPASS data. Due to the dominance of the photon-gluon fusion process, charm photoproduction will mainly lead to a constraint on the gluon helicity distribution, which is the prime motivation for such measurements. The distribution in LO and NLO, for both the unpolarized and polarized direct photon contribution to the cross section (3) is explored in Fig. 6. No cuts are imposed in this calculation and the meson spectrum is integrated over the entire phase space. As has to be expected from the fact that the PDFs enter the cross section (3) through a convolution, a broad range of values is sampled. It turns out, however, that the mean value of , where the distribution is peaked, is fairly independent not only of the order in perturbation theory, LO or NLO, but to a large extent also of the chosen set of polarized or unpolarized PDFs. We roughly estimate the average momentum fraction to be with an error of about .

Our results differ from preliminary estimates of by the COMPASS collaboration (22); (23); (24), where NLO results have been obtained based on some parton shower Monte Carlo to approximate the phase space for the NLO matrix elements of Ref. (16). Significant differences between estimated in LO and NLO are found in this way. While their LO result for agrees with our estimate of about 0.08, their preliminary NLO result is . Since the details of the method are not yet published, it is not yet clear how these results can be compared to our full NLO calculation. We also note that once data on photoproduction processes are implemented in global QCD analyses of helicity PDFs, information on , though useful, is no longer required or relevant as the fits automatically impose the constraints from data for any given functional form assumed for the .

Next, we turn to the experimentally measured double-spin asymmetry, defined in Eq. (21), which was analyzed at LO accuracy and under certain simplifying assumptions in terms of the mean gluon polarization by the COMPASS collaboration (15); see the discussion below. Preliminary NLO estimates are also available at NLO with the hybrid method outlined above (22); (23); (24). In Fig. 7 we show for the two sets of helicity PDFs used throughout this paper, by computing the ratio of the cross sections shown in Figs. 2 and 3 for . To resolve the differences in obtained with the DSSV and GRSV PDFs, which mainly stem from , an experimental precision of at least needs to be achieved.

We also compare NLO and LO estimates of for both sets of helicity PDFs in the lower panel of Fig. 7 and find rather different patterns depending on . At small , the NLO is about a factor of two larger than the LO estimate for the DSSV set whereas a reduction by roughly the same amount is found with the GRSV PDFs. This illustrates that any approximations for the spin asymmetry, either to assume a cancellation of NLO corrections or a constant pattern independent of the choice of PDFs, are not justified and should not be used for analyzing data. Again, only a global analysis will lead to consistent results. Qualitatively very similar results have been obtained for other cuts on the energy of the observed meson, and .

Finally, we compare our calculations at NLO accuracy with the available results from the COMPASS experiment. Figure 8 shows the data in three bins of the energy of the detected meson as a function of its transverse momentum , imposing the cut . Note that instead of using the published data (15), we show new, preliminary results presented recently in Ref. (22); (23); (24). A weighted average is performed to combine the results for the three decay channels , , and listed in (22); (23); (24). We adopt the photon polarization dilution factors tabulated in Ref. (22); (23); (24) to convert the data for given in the photon-nucleon system to the double-spin asymmetries for muon-nucleon scattering computed with our Monte Carlo code.

Clearly, present experimental uncertainties are too large to discriminate between different spin-dependent gluon densities. Apart from the DSSV and GRSV “standard” sets, we also use an extreme GRSV set, characterized by a very large and negative gluon density based on setting at some low bound-state like input scale (33) for the evolution. Despite leading to a distinctively different than our two default sets, all results are compatible with data within the experimental precision.

To overcome the statistical limitations, fewer bins have to be used or all data need to be combined. COMPASS has performed such an analysis (15) yielding the result for the gluon polarization quoted in Fig. 1, which is compatible with similar extractions of from hadron production data. These kind of analyses are LO estimates, assuming, in addition, that the convolutions of the PDFs with the partonic hard cross sections in Eq. (3) can be approximated as

 ALL=Δg⊗dΔ^σγgg⊗dΔ^σγg≈Δgg⟨dΔ^σγgd^σγg⟩. (22)

Here, the “analyzing power” is evaluated at some average kinematics. As mentioned before, in a proper global analysis of helicity PDFs one does not need to resort to any of these approximations whose validity is difficult to estimate or justify. With our new Monte Carlo program it is now possible to perform such an analysis at NLO accuracy in the future. As we have demonstrated in some detail, NLO corrections are indispensable for a quantitative analysis. Most importantly, they do not cancel in the spin asymmetry as one might naively expect.

Apart from the phenomenological applications presented here, our code can be also used to estimate spin asymmetries for charm and bottom photoproduction and their impact on our knowledge of the spin structure of the nucleon at higher c.m.s. energies intended for a first polarized lepton-ion collider. Details on the EIC project can be found in (10). Such a machine will be indispensable to finally unravel the quark and gluon contributions to the nucleon spin (11).

## 4 Summary and Outlook

We have presented a flexible parton-level Monte Carlo program to compute heavy flavor distributions at NLO accuracy in longitudinally polarized lepton-nucleon collisions in the photoproduction regime. For the first time, we consistently include both direct and resolved photon contributions and found the latter to be negligibly small for charm production at COMPASS kinematics. Experimental acceptance cuts, the hadronization of the produced heavy quark pair, and, if needed, their subsequent semi-leptonic decays can be included in phenomenological applications.

Heavy flavor photoproduction receives its importance for the field of spin physics from its expected strong sensitivity to the polarized gluon density which we confirm. In general, higher order corrections are found to be sizable and strongly dependent on the chosen set of helicity PDFs. This is also true for the experimentally relevant spin asymmetry despite naive expectations that QCD corrections cancel in the ratio. Theoretical uncertainties due to the choice of the factorization scale are sizable even at next-to-leading order accuracy while ambiguities from the exact form of the charm quark fragmentation function are less important.

The results obtained in this paper allow one to include data on charm photoproduction consistently into future global analyses of helicity PDFs. We have shown that data from COMPASS can lead in principle to a constraint on the polarized gluon density at a momentum fraction of about 0.1. Currently available data will have, however, very little impact on existing fits due to the size of the experimental uncertainties which are too large to discriminate between different gluon densities.

In addition to the phenomenological studies performed in this paper, our code will be useful in assessing the physics impact of heavy quark photoproduction at a possible future polarized lepton-ion collider like the EIC project.

## Note Added

While completing our analysis, the preliminary COMPASS results (22); (23); (24) discussed in this paper have been published in (37). Our conclusions remain unchanged.

## Acknowledgements

J.R. was supported by a grant of the “Cusanuswerk”, Bonn, Germany. M.S. acknowledges support by the U.S. Department of Energy (DOE) under contract No. DE-AC02-98CH10886 and, in part, by a BNL “Laboratory Directed Research and Development Program” (LDRD 12-034). This work was supported in part by the “Bundesministerium für Bildung und Forschung”, Germany.

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9. E.C. Aschenauer et al., report on “The RHIC Spin Program: Achievements and Future Opportunities”, BNL, 2012, http://www.bnl.gov/npp/docs/ RHIC-Spin-WriteUp-121105.pdf
10. D. Boer et al., “INT report on EIC Science”, arXiv:1108.1713.
11. E. C. Aschenauer, R. Sassot, and M. Stratmann, Phys. Rev. D 86 (2012) 054020.
12. A. Airapetian et al. [HERMES Collaboration], Phys. Rev. Lett. 84, 2584 (2000); JHEP 1008, 130 (2010).
13. B. Adeva et al. [Spin Muon Collaboration (SMC)], Phys. Rev. D 70, 012002 (2004).
14. E. S. Ageev et al. [COMPASS Collaboration], Phys. Lett. B 633, 25 (2006); C. Adolph et al. [COMPASS Collaboration], arXiv:1202.4064.
15. M. Alekseev et al. [COMPASS Collaboration], arXiv:0802.3023; Phys. Lett. B 676, 31 (2009).
16. I. Bojak and M. Stratmann, Phys. Lett. B 433, 411 (1998); Nucl. Phys. B 540, 345 (1999) [Erratum-ibid. B 569, 694(E) (2000)].
17. Z. Merebashvili, A. P. Contogouris, and G. Grispos, Phys. Rev. D 62, 114509 (2000) [Erratum-ibid. D 69, 019901 (2004)]; Phys. Lett. B 482, 93 (2000).
18. B. Jager, M. Stratmann, and W. Vogelsang, Phys. Rev. D 68, 114018 (2003); Eur. Phys. J. C 44, 533 (2005); C. Hendlmeier, A. Schafer, and M. Stratmann, Eur. Phys. J. C 55, 597 (2008).
19. J. Riedl, A. Schafer, and M. Stratmann, Phys. Rev. D 80, 114020 (2009).
20. I. Bojak and M. Stratmann, Phys. Rev. D 67, 034010 (2003).
21. M. Cacciari, P. Nason and R. Vogt, Phys. Rev. Lett. 95, 122001 (2005).
22. K. Kurek, Habilitation Thesis, National Centre of Nuclear Research, Swierk, November 2011.
23. C. Franco, Ph.D. Thesis, Universidade Tecnica de Lisboa, December 2011.
24. C. Franco [for the COMPASS Collaboration], arXiv:1208.6567.
25. M. L. Mangano, P. Nason, and G. Ridolfi, Nucl. Phys. B 373, 295 (1992).
26. S. Frixione, M. L. Mangano, P. Nason, and G. Ridolfi, Nucl. Phys. B 412, 225 (1994).
27. R. K. Ellis and P. Nason, Nucl. Phys. B 312, 551 (1989); J. Smith and W. L. van Neerven, Nucl. Phys. B 374, 36 (1992); W. Beenakker, H. Kuijf, W. L. van Neerven, and J. Smith, Phys. Rev. D 40, 54 (1989); W. Beenakker, W. L. van Neerven, R. Meng, G. A. Schuler, and J. Smith, Nucl. Phys. B 351, 507 (1991); P. Nason, S. Dawson, and R. K. Ellis, Nucl. Phys. B 327, 49 (1989) [Erratum-ibid. B335, 260 (1990)].
28. G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B 44, 189 (1972); P. Breitenlohner and D. Maison, Commun. Math. Phys. 52, 11 (1977).
29. R. Mertig and W. L. van Neerven, Z. Phys. C 70, 637 (1996); W. Vogelsang, Phys. Rev. D 54, 2023 (1996); Nucl. Phys. B 475, 47 (1996).
30. V. G. Kartvelishvili, A. K. Likhoded, and V. A. Petrov, Phys. Lett. B 78, 615 (1978).
31. For a review, see, e.g., J. Baines et al., summary report of the “Heavy Quarks Working Group” for the “HERA-LHC Workshop” proceedings, arXiv:hep-ph/0601164 and references therein.
32. J. Pumplin, D. R. Stump, J. Huston, H. L. Lai, P. M. Nadolsky, and W. K. Tung, JHEP 0207, 012 (2002).
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34. M. Glück and W. Vogelsang, Z. Phys. C 55, 353 (1992); C 57, 309 (1993); M. Glück, M. Stratmann, and W. Vogelsang, Phys. Lett. B 337, 373 (1994); M. Stratmann and W. Vogelsang, Phys. Lett. B 386, 370 (1996).
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36. D. de Florian and S. Frixione, Phys. Lett. B 457, 236 (1999).
37. C. Adolph et al. [COMPASS Collaboration], arXiv:1211.6849.
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• Gurobi Staff

m.addConstr(x[t] == max_(y[t], z[t]))

m.addGenConstrMax(x[t], [y[t], z[t]])

Thank you very much. The first syntax does not work for me but the second one does.

• Gurobi Staff

Glad to hear it works. There was a ")" missing in the first one. I updated it accordingly. It should work too.

I get the same error with the first syntax, i.e. "name 'max_ ' not defined'

Could it be a limitation of the editor I am using?

• Gurobi Staff

The $$\texttt{max_}$$ function is part of the $$\texttt{gurobipy}$$ module. Thus, in your case, you have to access it via

gurobipy.max_(y[t], z[t])


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### Tom Gilat (Bar-Ilan University)

Thursday, January 23, 2020, 11:10 – 12:00, -101

Abstract:

The main result in this talk is a decomposition theorem for a measure on the one-dimensional torus. Given a sufficiently large subset S of the positive integers, an arbitrary measure on the torus is decomposed as the sum of two measures. The first one $\mu_1$ has the property that the random walk with initial distribution $\mu_1$ evolved by the action of S equidistributes very fast. The second measure $\mu_2$ in the decomposition is concentrated on very small neighborhoods of a small number of points.


text:
# Tips And Tricks And Shortcuts For Number Series

## Number series Tips and Tricks and Shortcuts

Here are Specific quick and Easy Tips and Tricks for Number Series to solve different types of number series:

In Number Series When we identify pattern then It is Easy to Find any particular term.This Page helps you to Solve Questions in less time.

So basically there are five different types of number series. There are mentioned below with one suitable example:

## Number series Tips and Tricks and Shortcuts

Here are Specific quick and Easy Tips and Tricks for Number Series to solve different types of number series:

In Number Series When we identify pattern then It is Easy to Find any particular term.This Page helps you to Solve Questions in less time.

So basically there are five different types of number series. There are mentioned below with one suitable example:

### Tips and Tricks – Perfect Square Series

Consists of the perfect square of some numbers arranged in a specific order, with one number missing. Now we’ve to find the pattern the series is following and fill up the blank accordingly by finding that number.

Question 1.

100,121,144,__, 196

Solution:

This series consists of a perfect square of consecutive numbers 10, 11, 12, 13

Hence 169 will come in the blank.

### Tips and Tricks – Perfect Cube Series

It consists of a cube of numbers sequenced in a particular order.

Question 1.

9,64, 125, __, 343

Solution:

This series consists of a series of numbers with perfect cubes the is (3 x 3 x 3), (4 x 4 x 4), (5 x 5 x 5), (6 x 6 x 6), (7 x 7 x 7)

Hence 216 will be coming in the blank, as the series is following a trend of cubes of numbers in sequential order.

### Tips and Tricks and Shortcuts- Ration Series

This type of series consists of numbers arranged in sequential order (following a particular trend, i.e., Either increasing or decreasing). Now, all we have to do here is to trace out this trend,(which could be *, /, +, -) of each number of the series with a fixed number) by finding the proportional difference between the numbers of the series.

Question 1.

3, 6, 9, 12, __, 18, 21

Solution:

Here the series is following an increasing trend in which three is added to each number of the series.

3

6 (3+3)

9 (6+3)

12 (9+3)

15 (12+3)

18 (15+3)

21 (18+3)

### Tips and Tricks and Shortcuts – Arithmetic series

In this kind of sequences, each number is found by ( + , -) each term by a constant number.

The formula of  A S = {a, a+d, a +2d.….}

Where a= first term of the series

d = common difference

Question 1.

3, 6, 9 , 12

Solution:

Here a = 3(first term of the series)

d = 3

Hence we get:

3 + 3 = 6

6 + 3 = 9

9 + 3 = 12

12 + 3 = 15

### Tips and Tricks and Shortcuts – Geometric series

In this kind of sequences, each number is found by (*, /) each term by a constant number.

The formula of G S= {a, ar, ar2, ar3,….}

Where a= first term of the series

R= factor or difference between the term, also known as the common ratio.

Question 1.

1, 2, 4, 8, 16, 32

Solution:

Here a = 1 (first term of the series)

r = 2 (a standard number that is multiplied with the consecutive number of the series)

Hence we get:

1

1 x 2

1 x 22

1 x 23, ….)

### Tips and Tricks and Shortcuts – Mixed series

In such series, while calculating the difference, there can be two steps involved in getting the next consecutive number of the series. So we’ve to follow the same pattern to get the perpetual number of the series.

Question 1.

$1, \frac{5}{2}, \frac{10}{3}, \frac{17}{4}, —$

Options:

A.  $\frac{26}{5}$

B.  $\frac{24}{5}$

C.  $\frac{21}{5}$

D.  5

Correct Option: A

Explanation:

Here we’ve to observe the trend that this series is following, as each term is divided by a specific number, i.e., 1 then 2 then3 then 4 and so on. Hence we can make out that the denominator must be 5.

Now comes the numerator, where the 1’st number is 1.

Then comes 5, now how can we get 5 from the term number 2. It can come by $\frac{2^{2} + 1 }{2} = \frac{5}{2}$ Next comes $\frac{3^{2} + 1 }{3} = \frac{10}{3}$,$\frac{4^{2} + 1 }{4} = \frac{17}{4}$, $\frac{5^{2} + 1 }{5} = \frac{26}{5}$

Hence option A is the correct one.

Read AlsoHow to solve Number series questions

Formulas for How to Solve Number Series

Questions and Answers of Number Series

### 7 comments on “Tips And Tricks And Shortcuts For Number Series”

• pratip109

In Perfect Cube Series- Q1.) 9,64, 125, __, 343
9 is not a perfect cube. Its a square of 3. The pattern should be 27, 64, 125, ___, 343

• Rupa

hi sir/mam…..
this tips are very useful to every one and me also, iam following to learn and improve my aptitude sir/mam
we want so many tips of apptitude to develope our speed to solve
thank you so much sir/mam

• Savitri

solving tricks is very super Sir

• HelpPrepInsta

Thank you for your appreciation and also we want you to know that we are more than happy to help you and serve you better!!!

• Shweta

Tysm for the tips😍

• Vanitha B Mulimani

the explanation for the mixed series is not appropriate
when we solve for (4^2+1)/2 for 4 condition its will be 17/4 not 15/4

• Nishu Sagar

@Vanitha B Mulimani
thank you for your feedback. We have resolve the issue.


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2. STERN* started out in 1998 as a dj under the name Julastic. He was part of the Parisian nightlife in 2001 in such clubs as Café Chéri(e) where…

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- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnline Submission ㆍMy Manuscript - For Reviewers - For Editors
 Numerical Solution of Equilibrium Equations Commun. Korean Math. Soc. 2000 Vol. 15, No. 1, 133-142 Ho-Jong Jang Hanyang University Abstract : We consider some numerical solution methods for equilibrium equations $Af+E^T\lambda=r$, $Ef=s$. Algebraic problems of this form evolve from many applications such as structural optimization, fluid flow, and circuits. An important approach, called the force method, to the solution to such problems involves dimension reduction nullspace computation for $E$. The purpose of this paper is to investigate the substructuring method for the solution step of the force method in the context of the incompressible fluid flow. We also suggests some iterative methods based upon substructuring scheme. Keywords : equilibrium equation, substructuring method, force method, fluid flow, parallel computation MSC numbers : 65F10, 65F50, 76D05 Downloads: Full-text PDF

 Copyright © Korean Mathematical Society. The Korea Science Technology Center (Rm. 411), 22, Teheran-ro 7-gil, Gangnam-gu, Seoul 06130, Korea Tel: 82-2-565-0361  | Fax: 82-2-565-0364  | E-mail: paper@kms.or.kr   | Powered by INFOrang Co., Ltd


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# DetStrategy -- DetStrategy is a strategy for allowing the user to choose how determinants (or rank), is computed

## Description

Passing the option DetStrategy => Symbol controls how certain functions compute determinants (or in some cases, rank). For all methods, Bareiss, Cofactor and Recursive are valid options. The first two come included with Macaulay2, the third is just a call to recursiveMinors.

Additionally, for the methods getSubmatrixOfRank and isRankAtLeast, one can also pass DetStrategy the option Rank (the default). Not using Rank tells the functions to check rank by computing the determinant, instead of using the internal rank function. This is not usually recommended, but sometimes it can be effective.


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2 added 73 characters in body

This is an example when proving "locally free" instead of merely "Noetherian" flat" is easier and more straightforward, and no Noetherian assumption on the base is needed. The point is that if some coefficient $a$ of a polynomial $f\in R[x_1,\dotsc x_n]$ is nonzero at $p\in Spec R$ (i.e. nonzero in $R/p$) then it is invertible in an open neighborhood $D(a)\ni p$.

So let $f\in R[x_1,\dotsc,x_n]$ be a polynomial , of degree $d$, $p$ be a point of $Spec R$ (i.e. a prime ideal in $R$) and $k$ be the quotient field of $R/p$. Let $\bar f \in k[x_1,\dotsc,x_n]$ be the reduction of $f$ modulo $p$.

Using a change of coordinates in $k[x_1,\dotsc,x_n]$, put $\bar f$ in a Weierstrass form w.r.t. to the variable $x_n$. This means that

$$\bar f= \bar a ( x_n^d + p_{d-1}x_n^{d-1} + \dots + p_0 )$$

for some polynomials $p_j$ in the remaining variables $x_1,\dotsc,x_{n-1}$, and $\bar a\in k$, $\bar a\ne 0$.

If $k$ is infinite, this can be done by a linear change of coordinates. If $k$ is finite, there is a little trick.

If $r_i/s_i\in k$ are the coefficients involved in the change of coordinates ($r_i,s_i\in R$) then this change of coordinates can be done already in the ring $R'=R[1/a \prod s_i]$, i.e. over the open set $Spec R'= D(a\prod s_i)$ in $Spec R$ containing $[p]$. Further, $a$ is invertible over this set.

Now, over $R'$ the quotient $R'[x_1,\dotsc,x_n]/(f)$ is a free $R'[x_1,\dotsc,x_{n-1}]$-module with a basis $1,x_n,\dotsc, x_n^{d-1}$. Hence, it is a free $R'$-module. QED

This proves the statement for a family of nonzero hypersurfaces in $\mathbb A^n$. For a family of nonzero hypersurfaces in $\mathbb P^n$, cover $\mathbb P^n$ by $\mathbb A^n$ appropriately.

1

This is an example when proving "locally free" instead of merely "Noetherian" is easier and more straightforward. The point is that if some coefficient $a$ of a polynomial $f\in R[x_1,\dotsc x_n]$ is nonzero at $p\in Spec R$ (i.e. nonzero in $R/p$) then it is invertible in an open neighborhood $D(a)\ni p$.

So let $f\in R[x_1,\dotsc,x_n]$ be a polynomial, $p$ be a point of $Spec R$ (i.e. a prime ideal in $R$) and $k$ be the quotient field of $R/p$. Let $\bar f \in k[x_1,\dotsc,x_n]$ be the reduction of $f$ modulo $p$.

Using a change of coordinates in $k[x_1,\dotsc,x_n]$, put $\bar f$ in a Weierstrass form w.r.t. to the variable $x_n$. This means that

$$\bar f= a( x_n^d + p_{d-1}x_n^{d-1} + \dots + p_0)$$

for some polynomials $p_j$ in the remaining variables $x_1,\dotsc,x_{n-1}$, and $a\in k$, $a\ne 0$.

If $k$ is infinite, this can be done by a linear change of coordinates. If $k$ is finite, there is a little trick.

If $r_i/s_i\in k$ are the coefficients involved in the change of coordinates ($r_i,s_i\in R$) then this change of coordinates can be done already in the ring $R'=R[1/a \prod s_i]$, i.e. over the open set $Spec R'= D(a\prod s_i)$ in $Spec R$ containing $[p]$. Further, $a$ is invertible over this set.

Now, over $R'$ the quotient $R'[x_1,\dotsc,x_n]/(f)$ is a free $R'[x_1,\dotsc,x_{n-1}]$-module with a basis $1,x_n,\dotsc, x_n^{d-1}$. Hence, it is a free $R'$-module. QED

This proves the statement for a family of nonzero hypersurfaces in $\mathbb A^n$. For a family of nonzero hypersurfaces in $\mathbb P^n$, cover $\mathbb P^n$ by $\mathbb A^n$ appropriately.


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# Obscurantism

For those without comedic tastes, the self-proclaimed experts at Wikipedia have an article about Obscurantism.

Obscurantism, also known as the politico-economic effect of trihydrostatic integration on ursine urine and technical terminology, is quasi-linearly analyzable trochaically, ichtyologically, phenomenologically, comparatively by Boltzmann's rule of marine linguistics, obsessive-compulsively, paleobotanically, and phallologically (Spears et al. 1124-1635) a fundamental component of triglycerol deprotonation. The end result of this arduous process is the extraction of various cycloalkenes and pseudopsychological innaccuracies in the trigonometrically disproportional dingbat insertion in the "Flux vs. Time" postulate, detailing the creation of heliocentrism from circa 442 B.C. to 956 A.D., a controversial glutaminal contamination at the least. By utilizing the tertiary qualities of Fourier transforms on Platonic and semi-Platonic internal character metaelectrohydrolysis, we further synthesized the several epistemological constraints of dynamic ornithological hexameter.

## editIchtyological Multivariable Analysis

The per capita gross domestic product of ursine urine in 2006 was 7,304.4534 Coriolis cephalopods (Quayle et al. 258-789). Following alienation en masse of prokaryotic entities with high Hitler index, secondary pectoral fins underwent in vitro fertilization with numerous non-governmental organizations ad nauseum.

Whereas ethnographic investigation reveals higher rates of decapitation among low tidal nested loops, data transfer mechanisms between Belgium and illiterate owls varied Germanically with rate of mastication. Thus, the subdivisions within the quasilumniferous bodies of radiation will hyperpermutate with proportion to an iota function (n=1). Thus, we have demonstrated the reflexive axiom of ursine urine, that is, urine is urine.

### editDeontological vs. Neomystical Phenomena

We are presented with an immensely irresolvable Kantian ideological and ichtyological spheroidal dilemma. In Finnegan's Wake, Joyce, the great hippopotomonstrosesquippedaliophobiac of our time, writes,

 “ Now (to forebare for ever solittle of Iris Trees and Lili O'Ran- 1 gans), concerning the genesis of Harold or Humphrey Chimp- 2 den's occupational agnomen (we are back in the presurnames 3 prodromarith period, of course just when enos chalked halltraps) 4 and discarding once for all those theories from older sources which 5 would link him back with such pivotal ancestors as the Glues, the 6 Gravys, the Northeasts, the Ankers and the Earwickers of Sidles- 7 ham in the Hundred of Manhood or proclaim him offsprout of 8 vikings who had founded wapentake and seddled hem in Herrick 9 or Eric, the best authenticated version, the Dumlat, read the 10 Reading of Hofed-ben-Edar, has it that it was this way. ”

This reflects the semi-Marlovian, or paleo-opthalomogical, tradition of canonical reductionist synthetic ontology. The alchemical symbolism employed by the author parallels the discovery of trinitoluene surfactants in the Silurian period (Scheisskopf et al. 792-821). Hence, we can analyze the Hitler index of a non-Newtonian macroscopic entity using the multidimensional reduction formula:

$\int^\infty_\pi \sum_{n=0}^{\infty} \sqrt{{{x n^{\theta-1} }\over {(n^2+1)!}} f(x-\theta^2)} d\theta$

where $\theta$ and x represent the psychoses of a calculus student suffering from various disorders of category CVN-501(c)(93)(3%630-z)(T). It is then evident that trihydrostatic integration of ursine urine can only be performed under a dynamic equilibrium exceeding the Lavoisier-Descartes-Feynman Limit.

## editThe Einstein-Beer Stein Paradox: Phallological/Paleobotanical Solution

Let us examine the nature of the premises.

$\partial_n: C_n(X) \to C_{n-1}(X): \, (\sigma: [v_0,\ldots,v_n] \to X) \mapsto (\partial_n \sigma = \sum_{i=0}^n (-1)^i \sigma|[v_0,\ldots, \hat v_i, \ldots, v_n]),$

We use X to denote the rate of quasi-rotational entropy. By $\infty$ we denote the degree of perspicuousness of this article. Thus, we apply a series of conspiratorial and hyperesoteric transcendental operations and obtain:

$\gamma = \lim_{n \rightarrow \infty } \left( \left( \sum_{k=1}^n \frac{1}{k} \right) - \log (n) \right)=\int_1^\infty\left({1\over\lfloor x\rfloor}-{1\over x}\right)\,dx$

If we substitute Z for X, then we obtain the solution to numerous theogastrological problems of antiquity. Hence the paradox reduces itself to a Miltonian singularity, reminiscent of the work of Pollock. Within the Alighieri Equations, there are several variables which can be solved using the methods of cephalalgic calculus. This leads to several corollaries concerning the interplay of lex talionis and Tyrannosaurus rex. 33.458% of subjects in the trial chose Tyrannosaurus rex when the corpus callosum was ionized (Scheisskopf et al. 320-452). When the fractious gravitons orbit the sub-noxious vapor dominions at the above prescribed angle (whilst exposed to gamma and ultraviolet radiation, of course), the Dinonychus, closely related to its jurassic counterpart, Velociraptor, and its miniature triassic variant, Coelophysis, the postmodernist approach to the modernists' increases asymptotically, leaving little room for doubt that there is (without the ethnomonologogically relevant substitutions necessary for prompt revival) no such thing as a scietific method.

## editConclusion

Although methodological and epistemological constraints limited the number of photosynthetic polar bears we were able to produce by the synthesis of 3,7-dihydro-3,7-dimethyl-1H-purine-2,6-dione, a pattern of allegorical pandemic eschatology was established. Furthermore, man's inhumanity of man is supremely symbolized by the ontological nature of the actions of the antagonist. Consequently, we are able to conclude with 75.23% certainty that ursine urine is not equivalent to canine urine.


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how do I clear a string?

This topic is 5208 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

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when my strings are printed with text out they have black rectangles in it, I tried char szPrime01 [12] = {" "}; which got rid of some of the rectangles in the middle but the first one and last one are still there, if I add more spaces I get an error about the size

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Do you mean the gdi function TextOut? What kind of rectangles? Do you mean a background rectangle, similar to how selected text looks, or do you mean rectangles that indicate a missing font character? The artifact could be the result of the way that many Windows GUI text out put functions render new line characters. Try using "\r\n" instead of just "\n". Maybe that will do the trick.

[edited by - lessbread on February 20, 2004 7:15:54 PM]

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szPrime01 [12] = {"\0"};

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I dont understand why you would get non-printing chars at the start though. Any chance you post all the relivant code?

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char szPrime01 [12] = {"\0"}; didn't work

strcpy ( szPrime04, szPrime05);

sprintf ( szPrime05, TEXT (" %u "), realnum);

TextOut (hdc, cxChar * 20, cyChar * 0, szPrime01, 12) ;

I think thats all the relivant stuff

oh yea and the rectangles look like fat I's black

[edited by - ppcthug on February 20, 2004 12:17:20 AM]

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Your code makes no sense. You print szPrime01 but you never show what you've set it to.

The "fat Is" mean you're trying to print invalid characters -- specifically, characters that aren't in the current font. When that happens, Windows defaults to an outlined square or filled-in rectangle. Example:

Please show us how szPrime01 is set.

~CGameProgrammer( );

[edited by - CGameProgrammer on February 20, 2004 12:24:14 AM]

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Use ZeroMemory() on the string

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quote:
Original post by CProgrammer
Use ZeroMemory() on the string

He''s not trying to make the string blank... if he was, doing "szPrime01[0] = 0" would clear it, there''s no need for clearing the entire array anyway.

~CGameProgrammer( );

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sorry, I'm a bit distracted

void PrimeGenerator (){if (divisor >= realnum)   {                            // std::cout << "Prime number: " << realnum << "\n";  doesn't work in win32   /* cannot copy strings or arrays!!!   szPrime01 = szPrime02;   szPrime02 = szPrime03;   szPrime03 = szPrime04;   szPrime04 = szPrime05;   */      strcpy ( szPrime01, szPrime02);   strcpy ( szPrime02, szPrime03);   strcpy ( szPrime03, szPrime04);   strcpy ( szPrime04, szPrime05);      sprintf ( szPrime05, TEXT (" %u  "), realnum);      // szPrime05 = realnum; can't convert from unsigned int to char string      InvalidateRect (hwnd, NULL, TRUE);                                                                                realnum += 2;   divisor = 2;   count++;      if (count >= 15)      {              OnOff = false;                                         // std::cout << "Do you want to quit?(y to quit)\n"; doesn't work in win32       // std::cin >> quit; same here              /*                    all work and no play yada yada yada       if ( quit == 'y' )                 break;          else                 count = 0;        */                  }                   }      if ((realnum % divisor) != 0)      {      divisor++;      }         else      {         realnum += 2;      divisor = 2;                  /*                        for some reasion this         if (fuse = 60000)         counter quit the program          {                         before it should have!?          break;          }        */      }            }

[edited by - ppcthug on February 21, 2004 2:43:12 AM]

[edited by - ppcthug on February 21, 2004 2:43:37 AM]

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Are any of those strings ever initialized? You really can''t just declare strings and start using them. Show use where you have char* szPrime01 = "string"; or something like that.

Tolerance is a drug. Sycophancy is a disease.

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Tuesday, September 27, 2011

More OPERA II

Some people need to see the pretty pictures, so let us draw the full pink line for tachyons.
Although I have not worried about the correct scaling for the curve, the only essential adjustment to relativity is the existence of a minimum $v > c$, given by $\Delta v/c = 2.5 \times 10^{-5}$. If photons always travel at the same local speed, why not particles of very small rest mass.

8 comments:

1. Sorry, just a quick clarification. What is this crazy theory exactly? I follow a bunch of different physics blogs (including this one), and it's unclear to me what the theory is. Sorry if I've just missed it...

2. Philosopher, if you are not a physicist and/or not reading the literature, I'm afraid you cannot expect a two line answer to your question.

3. FYI, Dennis Overbye linked to you from the New York Times.

4. What's unique about neutrinos is that they are the only particle whose speed we've accurately measured that **does not** participate in electromagnetic interactions.

The next non-electromagnetic particle whose speed we can expect to measure is the graviton (through the speed of gravity waves). I'm betting that they're going to go a lot faster than c.

5. Yes, Mitchell. Believe it or not, the New York Times link did result in a not insubstantial increase in blog traffic.

Carl, as I have said many times before, I do not believe in gravitational waves. Neutrinos are a lot more interesting, because they exist.

6. The simplest interpretation of the curve, though definitely wrong, might be worth stating.

It's the straightforward combination of two proposed explanations regarding OPERA's observations: neutrinos are tachyons, and photons have a small mass. The high-energy asymptote for the pink curve is the true relativistic speed-limit, and standard light-speed (corresponding to delta-v/c = 0, in your graph) is something like 1/1.025 =approx 0.975 of the true speed-limit.

But this is definitely wrong, because if the true speed-limit was *that* much greater than measured light-speed, then all measured relativistic effects would be palpably different, e.g. the variation in measured lifetime of unstable particles at different momenta, or the slowdown of atomic clocks in gravitational fields, or anything to do with gravitational red-shift, or anything to do with relativistic binding energies.

7. I can think of two more hypotheses, but they will require modifications of relativity that I don't know how to realize.

One option: measured light-speed is the true relativistic speed-limit, neutrinos are tachyons, but they are an odd sort of tachyon which asymptotes to the new minimal superluminal speed rather than to c.

The other option: there are two special speeds - call them c1 and c2 - and two types of Lorentz transformation, and the asymptote in the diagram is for c2 > c1. So photons move at c1, and tachyonic neutrinos asymptote to c2.

8. I thought of yet another possibility. :-) Measured light-speed is the relativistic speed-limit, neutrinos are tachyons that asymptote to light-speed at high energies, and the data point from OPERA around 43 GeV is due to some interaction. You might even be able to fit it by supposing neutrino oscillations between different imaginary masses.

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