# Matthew van Eerde's web log

• #### Why is 1 Pascal equal to 94 dB Sound Pressure Level? (1 Pa = 94 dB SPL)

Last time we talked about why a full-scale digital sine wave has a power measurement of -3.01 dB FS (Spoiler: because it's not a square wave.)

This time we'll discuss why an atmospheric sound which generates a root-mean-square pressure of 1 Pascal has a power measurement 94 dB SPL.

As before, dB is defined as 10 log10(PA2 / PB2) where PB is a reference level.

Before, we had a digital measurement with an obvious ceiling: sample values of -1 and 1. So the reference point 0 dB FS was defined in terms of the signal with the greatest possible energy.

In the analog domain, there isn't an obvious ceiling. We instead consider the floor - the quietest possible signal that is still audible by human ears.

This is a rather wishy-washy definition, but the convention is to take PB = 20 μPa = 0.00002 Pa exactly.

So our 0 dB SPL reference point is when PA = PB: 0 dB SPL = 10 log10(0.000022 / 0.000022) = 10 log10(1) = 10 (0) = 0.

What if the pressure level is 1 Pascal? This is a quite loud sound, somewhere between heavy traffic and a jackhammer.

1 Pa in dB SPL =

10 log10(12 / PB2) =

20 log10(1 / PB) =

-20 log10(PB) =

-20 log10(2(10-5)) =

-20 (log10 2 + log10 10-5) =

-20 ((log10 2) - 5) =

100 - 20 log10 2 ≈ 93.9794 dB SPL

So 1 Pa is actually a tiny bit less than 94 dB SPL; it's closer to 93.98 = (100 - 6.02) dB SPL.

• #### Playing audio to the earpiece from a Windows Phone 8.1 universal app

Some time ago I blogged about the Windows Phone AudioRoutingManager API which allows you to put a "Speakerphone" and "Bluetooth" button in your Windows Phone app.

A common question that I get now is "I'm trying to play audio to the earpiece from my app, but AudioRoutingManager::SetAudioEndpoint(AudioRoutingEndpoint_Earpiece) is failing."

It's an interesting question, because Windows Phone will automatically route your audio to the earpiece if you have everything set up right - and if you don't have everything set up just right, you can't route it to the earpiece at all!

So how do you "set things up right?"

There are two things you have to do.

1. Tag the audio in question as "communications"
2. Tag your app as either a "voice over IP" app or a "voicemail" app

If you do both of these things, audio will flow to the earpiece automatically - no call to SetAudioEndpoint needed.

(At this point, if you want to get fancy, you can put a "Speakerphone" in your app and wire it up to a call to SetAudioEndpoint(AudioRoutingEndpoint_Speakerphone), but that's up to you.)

Let's look at the two things in a little more detail.

Tag the audio in question as "communications"

How to do this depends on what API you're using. It could be as simple as <audio src="..." msAudioCategory="communications" msAudioDeviceType="communications" />. Or you might have to call IAudioClient2::SetClientProperties with an AudioClientProperties structure whose AudioClientProperties.eCategory = AudioCategory_Communications.

Tag your app as "voice over IP" or "voicemail"

You'll need to set either the ID_CAP_VOIP or ID_CAP_VOICEMAIL Phone capability on your app. (The docs reference an ID_CAP_AUDIOROUTING capability, but that doesn't exist.)

If you're writing a Silverlight app, you can do that directly in the manifest.

If you're writing a Windows Phone 8.1 (non-Silverlight) or Universal app, you have to create a WindowsPhoneReservedAppInfo.xml file and add it to your application package. It should look like this.

<?xml version="1.0" encoding="utf-8"?>
<WindowsPhoneReservedAppInfo xmlns="http://schemas.microsoft.com/phone/2013/windowsphonereservedappinfo">
<SoftwareCapabilities>
<SoftwareCapability Id="ID_CAP_VOIP" />
</SoftwareCapabilities>
</WindowsPhoneReservedAppInfo>

• #### Using the Speech API to convert speech to text

Today I had to debug an issue with SAPI reading from a .wav file, so I updated it to accept a listen.exe --file foo.wav argument; this consumes the audio in the .wav file instead of listening to the microphone.

Pseudocode for the difference:

CoCreate(ISpRecognizer);
CoCreate(ISpStream);
pSpStream->BindToFile(file);
pSpRecognizer->SetInput(pSpStream);

Also, we have to tell the ISpRecoContext that we're interested in SPEI_END_SR_STREAM events as well as SPEI_RECOGNITION events.

Full source and binaries attached.

A gotcha: the .wav file has to have a WAVEFORMATEX.wFormatTag = WAVE_FORMAT_PCM. If it's anything else, ISpRecoGrammar::SetDictationState fails with SPERR_UNSUPPORTED_FORMAT. Neither WAVE_FORMAT_IEEE_FLOAT nor (WAVE_FORMAT_EXTENSIBLE with SubFormat = KSDATAFORMAT_SUBTYPE_PCM) work.

• #### More on audio buffer alignment requirements

I chatted in the past about how audio device alignment requirements impact the buffer size and the WASAPI alignment dance.

There are three alignment requirements on audio buffers:

1. The buffer size must be a multiple of WAVEFORMATEX.nBlockAlign. This allows individual audio frames to be copied around without worrying about them being cut in half and then having to glue them together at the end.
2. KSPROPERTY_RTAUDIO_BUFFER must be a multiple of a page - that is, 4096 bytes. This allows multiply mapping the buffer into consecutive pages, which in turn simplifies memory copies where the buffer is the source or the destination. KSPROPERTY_RTAUDIO_BUFFER is for timer-driven streaming; there is an event-driven analog, KSPROPERTY_RTAUDIO_BUFFER_WITH_NOTIFICATION, which has no corresponding alignment requirement.
3. HD Audio buffer allocations must be a multiple of 256 bytes. For timer-driven buffers, this applies to the whole buffer. For event-driven buffers, this applies to the sum of the "ping" and "pong" buffers, so the individual "ping" or "pong" buffer must be a multiple of 128 bytes.

Consider a 5.1 16-bit 48 kHz stream playing to HD Audio hardware via KSPROPERTY_RTAUDIO_BUFFER. Where multiple alignment requirements apply, the effective alignment requirement is the least common multiple of all the applicable requirements.

From the nBlockAlign requirement, the buffer must be a multiple of (6 * 16) / 8 bytes = 12 bytes.

From the KSPROPERTY_RTAUDIO_BUFFER requirement, the buffer must be a multiple of PAGE_SIZE = 4096 bytes.

From the HD Audio requirement, the buffer must be a multiple of 256 bytes (this is timer-driven, so we do not divide by 2.)

In all, then, the buffer must be a multiple of LCM(12, 4096, 256) = 12288 bytes.

Since WAVEFORMATEX.nAvgBytesPerSec = ((6 * 16) / 8) * 48000 = 576000 byte/sec, this corresponds to 12288 / 576000 * 1000 = 21.333 milliseconds.

• #### Using StrCmpLogicalW to sort strings the way the shell does

I have various scripts which use the "dir" command to enumerate files and then act on the most recent file that meets certain criteria. I noticed that "dir" and the Windows shell sometimes have different orders.

For example, if I have a folder containing files { track1.mp3, track2.mp3, ..., track9.mp3, track10.mp3, track11.mp3 } then the Windows shell will enumerate them in the natural order, but "dir" will enumerate them in the rather strange order { track1.mp3, track10.mp3, track11.mp3, track2.mp3, ...}.

Windows provides the StrCmpLogicalW API which is smart enough to realize that track9.mp3 < track10.mp3, so I whipped up a quick shellsort.exe utility that takes its standard input, breaks it up into lines, sorts those lines according to StrCmpLogicalW, and prints the resulting lines on its standard output. For bonus points I gave it a -reverse command-line option to print the output in reverse order (although it would have been more in the "toolbox" spirit to create a separate reverse.exe.)

>dir /b
track1.mp3
track10.mp3
track11.mp3
track2.mp3
track3.mp3
track4.mp3
track5.mp3
track6.mp3
track7.mp3
track8.mp3
track9.mp3

>dir /b | shellsort -reverse
track11.mp3
track10.mp3
track9.mp3
track8.mp3
track7.mp3
track6.mp3
track5.mp3
track4.mp3
track3.mp3
track2.mp3
track1.mp3

As with my previous exercise to find the longest duplicated substring in a given text, most of the code is taken up in reading the contents of the standard input in an efficient way.

Source and binaries attached.

• #### Linearity of Windows volume APIs - render session and stream volumes

We have talked about some of the volume APIs Windows exposes. We have also talked about what it means for a volume control to be linear in magnitude, linear in power, or linear in dB. We have also talked about how to read IAudioMeterInformation and how the limiter can attenuate full-scale signals.

The last post had a volume-linearity.exe which, when called with --signal, showed that IAudioMeterInformation is linear in amplitude.

Today we'll look at the --stream, --channel, and --session arguments, which explore the linearity of IAudioStreamVolume, IChannelAudioVolume, and ISimpleAudioVolume respectively. Each of these modes plays a half-scale square wave, then set the volume API to various levels, and reads the resulting IAudioMeterInformation. We use a half-scale square wave to avoid running afoul of the limiter; we expect a meter reading of 0.5 when the volume is set to 1.  The graphs below have their meter readings doubled to account for the fact that we're using a half-scale square wave rather than a full-scale.

Here's what we get for IAudioStreamVolume, graph-inated for your convenience:

And IChannelAudioVolume:

And ISimpleAudioVolume:

We already know that IAudioMeterInformation is linear in amplitude. We now know that IAudioStreamVolume, IChannelAudioVolume, and ISimpleAudioVolume have a linear effect (with slope 1 and intercept 0) on IAudioMeterInformation. We infer that IAudioStreamVolume, IChannelAudioVolume, and ISimpleAudioVolume are linear in amplitude.

• #### Nitpicking Sam Loyd - a wheel within a wheel

In August 1878 Sam Loyd published this mate in two and dedicated it to a friend of his named Wheeler:

Mate in two; Black to move and mate in two; Selfmate in two; Black to move and selfmate in two

While the mates appear to stand up, the problem position is not legal. White has three a-pawns; this implies at least three Black pieces were captured by a White pawn. But Black has fifteen pieces on the board; only one is missing!

Looking at Black pawn captures - the b2-, c-, and d- pawns together account for three pawn captures. This seems OK at first glance since White has three pieces missing. But all the missing White pieces are pawns, and they are from the right half of the board... so they must have promoted. This implies more pawn captures to either get the Black pawns out of the way or to get the White pawns around them. (The promoted pieces could have been captured by the Black pawns, or the original pieces could have been captured in which case the promoted pieces are on the board now.)

Finally, the h-pawns on h5 and h6 could not have got into their present position without at least one pawn capture by White, or at least two pawn captures by Black.

• #### Mark your variadic logging function with __format_string to have PREfast catch format specifier errors

There are a handful of Problems (with a capital P) which occur over and over again in programming. One of them is Logging.

It is incredibly convenient to use the variadic printf function to log strings with values of common types embedded in them:

// spot the bug
LOG(L"Measurement shows %lg% deviation", 100.0 * abs(expected - actual) / expected);

However, printf is very error prone. It is very easy to use the wrong format specifier like %d instead of %Id, or to forget to escape a special character like % or \.
In particular, the above line contains a bug.

Static code analysis tools like PREfast are quite good at catching these kinds of errors. If my LOG macro was something like this, PREfast would catch the bug:

#define LOG(fmt, ...) wprintf(fmt L"\n", __VA_ARGS__)

This works because PREfast knows that the first argument to wprintf is a format string, and can match up the format specifiers with the trailing arguments and verify that they match.

If you implement your own variadic logger function, though, PREfast doesn't necessarily know that the last explicit argument is a format specifier - you have to tell it. For example, PREfast will NOT catch format specifier issues if the LOG macro is defined like this:

// PREfast doesn't know Format is a format string
interface IMyLogger { virtual void Log(LPCWSTR Format, ...) = 0; };
extern IMyLogger *g_Logger;
#define LOG(fmt, ...) g_Logger->Log(fmt, __VA_ARGS__)

How do you tell it? Well, let's look at the declaration of wprintf. It's in (SDK)\inc\crt\stdio.h:

_CRTIMP __checkReturn_opt int __cdecl wprintf(__in_z __format_string const wchar_t * _Format, ...);

The relevant part here is __format_string. So the fixed IMyLogger declaration looks like this:

// Now PREfast can catch format specifier issues
interface IMyLogger { virtual void Log(__format_string LPCWSTR Format, ...) = 0; };
extern IMyLogger *g_Logger;
#define LOG(fmt, ...) g_Logger->Log(fmt, __VA_ARGS__)

• #### Beep sample

A question came in today about the Beep(...) API1 not being able to set the frequency of the beep that was generated. In order to confirm that it worked I whipped up a quick sample which would take the frequency (and duration) on the command line. Source and binaries attached.

For fun I added the ability to pass in the frequency using Scientific pitch notation. Note that A4 is about 431 Hz using this scale, rather than the more standard 440 Hz2.

for (int i = 1; i + 1 < argc; i += 2) {

ULONG frequency;
HRESULT hr = HertzFromScientificPitchNotation(argv[i], &frequency);
if (FAILED(hr)) { return -__LINE__; }

ULONG duration;
hr = UlongFromString(argv[i + 1], &duration);
if (FAILED(hr)) { return -__LINE__; }

if (!Beep(frequency, duration)) {
LOG(L"Beep(%u, %u) failed: GetLastError() = %u", frequency, duration, GetLastError());
return -__LINE__;
}
}

So, for example, you can play a certain well-known tune via Beep() using this command:

>beep.exe C3 2000 G3 2000 C4 4000 E4 500 Eb4 3500 C3 500 G2 500 C3 500 G2 500 C3 500 G2 500 C3 2000

1 More on the Beep(...) API:

The official Beep(...) documentation

A couple of blog posts from Larry Osterman:
Beep Beep
What’s up with the Beep driver in Windows 7?

2 If you want the more standard pitch, change this line:

double freq = 256.0;

To this:

double freq = 440.0 * pow(semitoneRatio, -9.0); // C4 is 9 semitones below A4

• #### Command-line app to set the desktop wallpaper

Working on Windows, I find myself installing Windows a lot.

I find that I like to change a lot of the settings that Windows offers to non-default values.  (That is, I'm picky.)

I have a script which automates some of these things, which I add to now and again.  Some of the bits of the script are straightforward, but once in a while the tweak itself is of interest.

One of the things I love about my work setup is the many large monitors.  So, one of the things I like to change is the desktop wallpaper image.

Changing the desktop wallpaper required some code, which makes it "of interest."  Here's the code.

// main.cpp

#include <windows.h>
#include <winuser.h>
#include <stdio.h>

int _cdecl wmain(int argc, LPCWSTR argv[]) {
if (1 != argc - 1) {
wprintf(L"expected a single argument, not %d\n", argc);
return -__LINE__;
}

if (!SystemParametersInfo(
SPI_SETDESKWALLPAPER,
0,
const_cast<LPWSTR>(argv[1]),
SPIF_SENDCHANGE
)) {
DWORD dwErr = GetLastError();
wprintf(L"SystemParametersInfo(...) failed with error %d\n", dwErr);
return -__LINE__;
}

wprintf(L"Setting the desktop wallpaper to %s succeeded.\n", argv[1]);
return 0;
}

Binaries attached.

Warning: if you pass a relative path to this tool, it won't qualify it for you, and the SystemParametersInfo call won't either - so the wallpaper you want won't be set, though all the calls will succeed.  Make sure to specify a fully-qualified path.

• #### How to create a shortcut from the command line

Working on Windows, I install Windows a lot.  This means a lot of my customizations have to be re-applied every time I install.  To save myself some time I created a script which applies some of them. Last time I showed how to set the desktop wallpaper from a command-line app.

This time, a script to create a shortcut.  The example usage creates a shortcut to Notepad and puts that in the "SendTo" folder.  I find this very useful because I often need to edit text files that have non-".txt" assocations.  (There are also other shortcuts I create with it.)

Here's the script:

>create-shortcut.vbs

If WScript.Arguments.Count < 2 Or WScript.Arguments.Count > 3 Then
WScript.Echo "Expected two or three arguments; got " & WScript.Arguments.Count
WScript.Echo "First argument is the file to create"
WScript.Echo "Second is the command to link to"
WScript.Echo "Third, if present, is the arguments to pass"
WScript.Quit
End If

Set shell = WScript.CreateObject("WScript.Shell")

If WScript.Arguments.Count = 3 Then
End If

• #### Generating primes using the Sieve of Eratosthenes plus a few optimizations

When solving Project Euler problems I frequently need to iterate over prime numbers less than a given n. A Sieve of Eratosthenes method quickly and easily finds the small prime numbers; there are more complicated methods that find larger prime numbers, but with a couple of tweaks the Sieve of Eratosthenes can get quite high.

A naive implementation for finding the set of primes below n will:

1. Allocate an array of n booleans, initialized to false.
2. Allocate an empty list
3. For each i in the range 2 to n:
1. If the boolean value at this index in the array is true, i is composite. Skip to the next value and check that.
2. If the boolean value at this index in the array is false, i is prime!
3. Add i to the list of primes
4. For each multiple of i in the range 2i to n, set the boolean value at that index in the array to true

There are a handful of simple optimizations that can be made to this naive implementation:

1. Step 3d) will have no effect until the multiple of i reaches i2, so the range can be changed to "i2 to n"
2. As a direct consequence of this, step 3d) can be skipped entirely once i2 passes n.
3. Instead of allocating an array of n booleans, an array of nbits will suffice.
4. All the even-indexed bits are set to true on the first pass. Manually recognize that 2 is prime, and only allocate bits for odd-numbered values. Change the outer loop in 3) to "in the range 3 to n", incrementing by two each time. Change the loop 3d) to increment by 2i each time.
5. Storing the list of primes takes a lot of memory - more than the sieve. Don't bother creating a list of primes, just write an enumerator that travels the sieve directly.

With these optimizations I can enumerate primes from 2 up to 5 billion (5 * 109) in about seven minutes.  Source and binaries attached.

>primes 5000000000
Will enumerate primes <= 5000000000 = 5e+009
Memory for sieve: 298.023 MB
Initialization complete: 983 milliseconds since start
Sieving to 70711
Sieving complete: 4.70292 minutes since start
Picking up the rest to 5000000000
Pickup complete: 6.12252 minutes since start
Primes: 234954223
1: 2
23495423: 442876981
46990845: 920233121
70486267: 1410555607
93981689: 1909272503
117477111: 2414236949
140972533: 2924158169
164467955: 3438252577
187963377: 3955819157
211458799: 4476550979
234954221: 4999999883
Enumerating complete: 7.43683 minutes since start
Freeing CPrimes object

There are more complicated sieves like the Sieve of Atkin which perform better but at the cost of being much more complex. So far I haven't had to resort to any of those.

• #### Programmatically grabbing a screenshot of the primary display

It's sometimes difficult to explain to people what my job actually is. "I test Windows sound." "Cool.  How does that work?"

A product like Windows has a lot of components that interact with each other.  If everything works, the user doesn't know that most of these components even exist; everything is invisible and seamless.

Most testing involves the connection ("interface") between two components.  "I test APIs."  To the uninitiated, this is just a word.  It sounds like "I test wakalixes."  "You test what, now?"

There are two interfaces which are easier to explain.  There's the software-to-hardware interface, where the driver talks to the hardware.  "I test the HD Audio, USB Audio, and Bluetooth audio class drivers."  "Huh?" "They make the speakers and the microphone work."  "Oh, cool.  So you sit around and use Skype all day?"

But the easiest of all to explain is the user interface.  "I make sure that the Sound Recorder app, the volume slider, and the Sound control panel work." "Oh, that!  I had this annoying problem once where..."

What does the test result for an invisible interface look like? A lot of logging.  "I expected this call to succeed; it returned this HRESULT."  "I poked the hardware like this and got a bluescreen."  "There seems to be an infinite loop here."  Lots of text.

Boring.

UI testing has logging too.  But with UI testing you can also... TAKE PICTURES!  A UI bug is a lot easier to understand (triage, and fix) if there's a screenshot attached (preferably with a big red rectangle highlighting the problem.)

It is therefore valuable to have an automatable utility that can take a screenshot and dump it to a file.  Here's one I cribbed together from the "Capturing an Image" sample code on http://msdn.microsoft.com/en-us/library/dd183402(v=VS.85).aspx.  Source and binaries attached.

This version only captures the main display, and not secondary monitors (if any.)

Pseudocode:

screen_dc = GetDC(nullptr);

memory_dc = CreateCompatibleDC(screen);

rect = GetClientRect(GetDesktopWindow());

hbmp = CreateCompatibleBitmap(screen_dc, rect);

SelectObject(memory_dc, hbmp);

BitBlt(memory_dc, rect, screen_dc);

bmp = GetObject(hbmp);

bytes = allocate enough memory

bytes = GetDIBits(screen_dc, bmp, hbmp)

file = CreateFile();

WriteFile(bytes);

• #### Excruciating rhymes

I was watching How the Grinch Stole Christmas and I was struck by this rhyme (from the song  "You're a Mean One, Mr. Grinch:")

You're a nauseous super naus!
You're a dirty crooked jockey, and you drive a crooked hoss

-- Dr. Seuss, How the Grinch Stole Christmas

I tried to see what other particularly excruciating rhymes I could remember.  I came up with two:

You know, that little guy, he's got me feeling all contempt-ey:
He takes his boat out loaded up and brings it back in empty.

-- Phil Vischer, Lyle the Kindly Viking

And then of course:

In short, when I've a smattering of elemental strategy,
You'll say a better Major-General had never sat a-gee!

-- W. S. Gilbert, The Pirates of Penzance

• #### Teaching someone to fish and the AKS primality test

This morning my wife (whom I love and adore) woke me up at 3:00 AM with an urgent question.

"Hey!" she said, shaking me awake.

"Is 19 prime?"

...

Like a fool, I answered her.  "Yes.  Yes it is."  Off she went.

This is a true response, but not a correct response.  I realized shortly afterwards that a correct response would look more like:

I'm glad you asked me that. dear.  Eratosthenes, the Greek mathematician, discovered a very efficient way to list primes in about 200 BC that is still in use today.  You start by writing out all the numbers from 1 to 19: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19.  1 is a very special number (it's the multiplicative identity, or what algebraists would call a unit) so we put a square around it.  The first number we didn't consider was 2, so we circle it - that means it's prime - and then cross out every multiple of 2 after that.  Going back, the first number we didn't consider was 3... and so on until we get 1 2 3 X 5 X 7 X X X 11 X 13 X X X 17 X 19.  A common optimization is to realize that after circling a prime p, the first number you cross out (that wasn't crossed out before) is always p2, which means that after circling a number you can immediately jump to its square, and also means you can stop crossing out altogether once you hit p> N...

This would allow her to fish rather than waking me up when she wanted a fish.

An even better response would have been:

It's funny you've asked me that.  Number theorists and cryptanalysts have considered this question for thousands of years.  Eratosthenes' method (see above) is a very simple way to find all the primes below a given number, but an efficient way to determine whether a given number is prime was found only very recently.

In practice, the test that is usually used is the randomized version of the Miller-Rabin test.  Although this is nondeterministic, it is very fast indeed, and will tell you to a very high degree of certainty whether the given number is prime.  This usually suffices.

There is a deterministic version of the Miller-Rabin test too, which is guaranteed to tell you with perfect certainty whether the given number is prime.  But it only works if you believe in the generalized Riemann hypothesis.  Most mathematicians nowadays believe the hypothesis, but no-one has (yet) been able to prove it.

Amazingly, in 2002 three mathematicians named Manindra Agrawal, Neeraj Kayal, and Nitin Saxena came up with a deterministic, proven, polynomial-time (specifically, polynomial in the number of digits in the input) method for telling whether a given number is prime.   This is known as the AKS primality test.  The most striking thing about this test is its simplicity - if something this straightforward can be found after thousands of years of looking, what else out there remains to be found?

Such a response would probably prevent her from waking me again for any mathematical problem at all.  Boo-ya.

My own Perl implementation follows:

use strict;
# use bignum;

print <<USAGE and exit 0 unless @ARGV;
$0 [-v] n Use the AKS primality test to check whether n is prime -v adds verbose log spew USAGE sub is_power($$); sub ceil_log2(); sub first_r($$); sub check_gcds($$); sub check_polynomials($$$);
sub gcd($$); sub totient(); sub polypow($$\@);
sub polymult($$\@\@); sub polyeq(\@\@); my verbose = ARGV[0] eq "-v"; shift @ARGV if verbose; die "Expected only one argument" unless 1 == @ARGV; my n = shift; # step 0: restrict to integers >= 2 print "n is not an integer and so is NEITHER PRIME NOR COMPOSITE\n" and exit 0 unless int(n) == n; print "n < 2 and so is NEITHER PRIME OR COMPOSITE\n" and exit 0 unless n >= 2; # step 1: check if the number is a power of some lower number. # this can be done quickly by iterating over the exponent (2, 3, ...) # and doing a binary search on the base. # we start at the top and work down for performance reasons; # several subroutines need to know ceil(log2(n)) so we calculate it once and pass it around. my log2_n = ceil_log2(n); is_power(n, log2_n) and exit 0; print "Not a power.\n"; # step 2: find the smallest r such that o_r(n) > (log2 n)^2 # where o_r(n) is the multiplicative order of n mod r # that is, the smallest k such that n^k == 1 mod r my r = first_r(n, log2_n); print "r = r\n"; # step 3: for all a between 2 and r inclusive, check whether gcd(a, n) > 1 check_gcds(n, r) or exit 0; # step 4: if r >= n, we're done if (r >= n) { print "r >= n so n is PRIME\n"; exit 0; } # step 5: for all a between 1 and floor( sqrt(phi(r)) log2(n) ) # check whether (x + a)^n = x^n + a mod x^r - 1, n check_polynomials(n, r, log2_n) or exit 0; # step 6: if we got this far, n is prime print "n is PRIME\n"; sub is_power($$) {
my $n = shift; my$log2_n = shift; # actually ceil(log2(n))

print "Checking for power-ness...\n";

# we consider numbers of the form b^i
# we iterate over the exponent i
# starting at i = ceil(log2(n)) and working down to i = 2
#
# for each exponent we do a binary search on the base
# the lowest the base can be is 2
# and the highest the base can be (initially) is 2
#
# we set up bounds on the base that are guaranteed to
# surround the actual base
my $b_low = 1; # 1 ^ ceil(log2(n)) = 1 < n my$b_high = 3; # 3 ^ ceil(log2(n)) > 2 ^ log2(n) = n

for (my $i =$log2_n; $i >= 2;$i--) {
print "\tb^$i\n" if$verbose;

# let's check that the bounds are really correct
die "$b_low ^$i is not < $n" unless$b_low ** $i <$n;
die "$b_high ^$i is not > $n" unless$b_high ** $i >$n;

# do a binary search to find b such that b ^ i = n
while ($b_high -$b_low > 1) {
print "\t\tb^$i: b is between$b_low and $b_high\n" if$verbose;
my $b = int(($b_low + $b_high)/2); my$t = $b **$i;
if ($t ==$n) {
print "$n =$b^$i;$n is COMPOSITE\n";
return 1;
}

($t >$n ? $b_high :$b_low) = $b; } # as we pass from the exponent (say, 5) # to the exponent below (say, 4) # we need to reconsider our bounds # # b_low can remain the same because b ^ (i - 1) is even less than b ^ i # OPEN ISSUE: can we even raise b_low? # # but we need to raise b_high since b ^ i > n does NOT imply b ^ (i - 1) > n # # we'll square b_high; b ^ i > n => (b ^ 2) ^ (i - 1) = b ^ (2 i - 2) > n # since i >= 2 # # OPEN ISSUE: is there a better way to raise this higher bound? Does this help much?$b_high *= $b_high; } # nope, not a power return 0; } sub ceil_log2($) {
my $n = shift; my$i = 0;
my $t = 1; until ($t >= $n) {$i++;
$t *= 2; } return$i;
}

sub first_r($$) { my n = shift; my log2_n = shift; # actually ceil(log2(n)) my s = log2_n ** 2; print "Looking for the first r where o_r(n) > s...\n"; # for each r we want to find the smallest k such that # n^k == 1 mod r my r; for (r = 2; ; r++) { # print "\tTrying r...\n"; # find the multiplicative order of n mod r my k = 1; my t = n % r; until (1 == t or k > s) { t = (t * n) % r; k++; } if (k > s) { # print "\to_r(n) is at least k\n"; last; } else { # print "\to_r(n) = k\n"; } } return r; } sub check_gcds($$) {
my ($n,$r) = @_;

print "Checking GCD($n, a) for a = 2 to$r...\n";

for (my $a = 2;$a <= $r;$a++) {
my $g = gcd($n, $a); next if ($g == $n); # this is OK if (1 !=$g) {
print "gcd($n,$a) = $g;$n is COMPOSITE\n";
return 0;
}
}

print "All GCDs are 1 or $n\n"; return 1; } sub gcd($$) { my (x, y) = @_; (x, y) = (y, x) unless x > y; while (y) { (x, y) = (y, x % y); } return x; } sub check_polynomials($$$) {
my $n = shift; my$r = shift;
my $log2_n = shift; # actually ceil(log2(n)) # iterate over a from 1 to floor( sqrt(phi(r)) log2(n) ) # for each a, check whether the polynomial equality holds: # (x + a)^n = x^n + a mod (x^r - 1, n) # if it fails to hold, the number is composite # # first we need to evaluate phi(r) so we can determine the upper bound # OPEN ISSUE: this seems to be a potential weakness in the algorithm # because the usual way to evaluate phi(r) is to find the prime factorization of r # and then form the product r*PI(1 - 1/p) where the product ranges over all primes # which divide r my$phi = totient($r); # a < sqrt(phi(r)) * log2(n) => a^2 < phi(r) * (log2(n))^2 my$a2_max = $phi *$log2_n * $log2_n; print "Checking polynomials up to roughly ", int sqrt($a2_max), "...\n";

for (my $a = 1;$a * $a <=$a2_max; $a++) { print "\ta =$a...\n" if $verbose; # polynomials are of the form (c0, c1, c2, ..., ci, ...) # which corresponds to c0 + c1 x + c2 x^2 + ... + ci x^i + ...) my @x = (0, 1); my @x_plus_a = ($a % $n, 1); my @lhs = polypow($n, $r, @x_plus_a); # POTENTIAL OPTIMIZATION: # x^n + a mod (x^r - 1) is just x^(n % r) + a # and we know n % r != 0 my @rhs = polypow($n, $r, @x); # x^n$rhs[0] = ($rhs[0] +$a) % $n; # + a next if polyeq(@lhs, @rhs); print "(x +$a)^$n is not equal to x^$n + $a mod(x^$r - 1, $n)\n"; print "So$n is COMPOSITE\n";
return 0;
}

return 1;
}

sub totient($) { my$r = shift;

print "Finding the Euler totient of $r\n"; # we'll do a trial division to find the totient # there are faster ways that use a sieve # but we don't know how big r is my$t = $r; # by construction p will always be prime when it is used # OPEN ISSUE: this might be slow for (my$p = 2; $r > 1;$p++) {
next if $r %$p;

print "\t$p is a factor\n" if$verbose;
# decrease the totient
$t /=$p;
$t *=$p - 1;

# decrease r
$r /=$p; # we know there's at least one factor of p
$r /=$p until $r %$p; # there might be more
}

print "Totient is $t\n"; return$t;
}

sub polypow($$\@) { my n = shift; # this is both the mod and the exponent my r = shift; my @base = @{ +shift }; my exp = n; my @result = (1); # 1 # print "\t(", join(" ", @base), ")^exp mod (x^r - 1, n)\n" if verbose; # basic modpow routine, but with polynomials while (exp) { if (exp % 2) { @result = polymult(n, r, @result, @base); } exp = int (exp / 2); @base = polymult(n, r, @base, @base); } # print "\t= (", join(" ", @result), ")\n" if verbose; return @result; } sub polymult($$\@\@) {
my $n = shift; my$r = shift;
my @first = @{ +shift };
my @second = @{ +shift };

# print "\t\t(", join(" ", @first), ") * (", join(" ", @second), ") mod (x^$r - 1,$n)\n" if $verbose; my @result = (); # first do a straight multiplication first * second my$s = @second - 1;
for (my $i = @first - 1;$i >= 0; $i--) { for (my$j = $s;$j >= 0; $j--) { my$k = $i +$j;
$result[$k] += $first[$i] * $second[$j];
$result[$k] %= $n; } } # then do a straight mod x^r - 1 # consider a polynomial # c0 + ... + ck x^k # with k >= r # we can subtract ck (x^r - 1) # without changing the mod value # the net effect is to eliminate the x^k term # and add ck to the x^(k - r) term for (my$i = @result - 1; $i >=$r; $i--) { my$j = $i -$r;
$result[$j] += $result[$i];
$result[$j] %= $n; pop @result; } # eliminate any leading zero terms for (my$i = @result - 1; 0 == $result[$i]; $i--) { pop @result; } # print "\t\t= (", join(" ", @result), ")\n" if$verbose;
return @result;
}

sub polyeq(\@\@) {
my @lhs = @{ +shift };
my @rhs = @{ +shift };

# print "(", join(" ", @lhs), ") = (", join(" ", @rhs), ")?\n" if $verbose; return 0 unless @lhs == @rhs; for (my$i = @lhs - 1; $i >= 0;$i--) {
return 0 unless $lhs[$i] == $rhs[$i];
}

return 1;
}

Here's the output when I run it on 19:

>perl -w aks.pl 19
Checking for power-ness...
Not a power.
Looking for the first r where o_r(19) > 25...
r = 19
Checking GCD(19, a) for a = 2 to 19...
All GCDs are 1 or 19
19 >= 19 so 19 is PRIME

And here's the output with a bigger input:

>perl -w aks.pl 99
Checking for power-ness...
Not a power.
Looking for the first r where o_r(997) > 100...
r = 103
Checking GCD(997, a) for a = 2 to 103...
All GCDs are 1 or 997
Finding the Euler totient of 103
Totient is 102
Checking polynomials up to roughly 100...
997 is PRIME
• #### Enumerating MIDI devices

In addition to audio playback and recording, Windows Multimedia (WinMM) provides a Musical Instrument Digital Interface (MIDI) API.

Here's how to make a list of all the MIDI devices on the system, their capabilities, and the hardware device interface associated with each of them.

Source and binaries attached.

Pseudocode:

midiInGetNumDevs or midiOutGetNumDevs
for each device
midiInGetDevCaps or midiOutGetDevCaps
log device capabilities
midiInMessage or midiOutMessage
with DRV_QUERYDEVICEINTERFACESIZE
and DRV_QUERYDEVICEINTERFACE
log the device interface

Output:

>midienum.exe
midiIn devices: 1
-- 0: USB2.0 MIDI Device --
Device ID: 0
Manufacturer identifier: 65535
Product identifier: 65535
Driver version: 1.6
Product name: USB2.0 MIDI Device
Support: 0x0
Device interface: "\\?\usb#vid_xxxx&pid_yyyy&..."
midiOut devices: 2
-- 0: Microsoft GS Wavetable Synth --
Device ID: 0
Manufacturer identifier: 1
Product identifier: 27
Driver version: 1.0
Product name: Microsoft GS Wavetable Synth
Technology: 7 (MOD_SWSYNTH)
Voices: 32
Notes: 32
Support: 0x1
MIDICAPS_VOLUME
Device interface: ""
-- 1: USB2.0 MIDI Device --
Device ID: 1
Manufacturer identifier: 65535
Product identifier: 65535
Driver version: 1.6
Product name: USB2.0 MIDI Device
Technology: 1 (MOD_MIDIPORT)
Voices: 0
Notes: 0
Support: 0x0
Device interface: "\\?\usb#vid_xxxx&pid_yyyy&..."

(Actual device interface string suppressed.)

Note the Microsoft GS Wavetable Synth device, which is always present.

Why would you want to know the device interface? In our case, because we want to test all the audio-related interfaces of a particular device on the system.

• #### Enumerating mixer devices, mixer lines, and mixer controls

The WinMM multimedia APIs include an API for enumerating and controlling all the paths through the audio device; things like bass boost, treble control, pass-through audio from your CD player to your headphones, etc.  This is called the "mixer" API and is the forerunner of the IDeviceTopology API.

I wrote a quick app to enumerate all the mixer devices on the system; for each mixer device, enumerate each mixer line (that is, each source and destination); for each mixer line, enumerate all the controls (volume, mute, equalization, etc.); and for each control, query the associated text (if any) and the current value.

Source and binaries attached.

Pseudocode:

mixerGetNumDevs()
for each mixer device
mixerGetDevCaps(dev)
for each destination (line) on the device
mixerGetLineInfo(dest)
mixerGetLineControls(dest)
for each control on the line
if the control supports per-item description
mixerGetControlDetails(control, MIXER_GETCONTROLDETAILSF_LISTTEXT)
log the per-item description
mixerGetControlDetails(control, MIXER_GETCONTROLDETAILSF_VALUE)
log the value(s)

Usage:

>mixerenum.exe
Mixer devices: 5
Device ID: 0
Manufacturer identifier: 1
Product identifier: 104
Driver version: 6.2
Support: 0x0
Destinations: 1
-- Destination 0: Master Volume --
Destination: 0
Source: -1
Line ID: 0xffff0000
Status: MIXERLINE_LINEF_ACTIVE (1)
User: 0x00000000
Channels: 1
Connections: 2
Controls: 2
Short name: Volume
Long name: Master Volume
-- Target:  --
Type: MIXERLINE_TARGETTYPE_UNDEFINED (0)
Device ID: 0
Manufacturer identifier: 65535
Product identifier: 65535
Driver version: 0.0
Product name:
-- Control 1: Mute --
Type: MIXERCONTROL_CONTROLTYPE_MUTE (0x20010002)
Status: MIXERCONTROL_CONTROLF_UNIFORM (0x1)
Item count: 0
Short name: Mute
Long name: Mute
-- Values --
FALSE
-- Control 2: Volume --
Type: MIXERCONTROL_CONTROLTYPE_VOLUME (0x50030001)
Status: MIXERCONTROL_CONTROLF_UNIFORM (0x1)
Item count: 0
Short name: Volume
Long name: Volume
-- Values --
0xffff on a scale of 0x0 to 0xffff
-- 1: HDMI Audio (Contoso --
Device ID: 1
Manufacturer identifier: 1
Product identifier: 104
Driver version: 6.2
Product name: HDMI Audio (Contoso
Support: 0x0
Destinations: 1
-- Destination 0: Master Volume --
Destination: 0
Source: -1
Line ID: 0xffff0000
Status: MIXERLINE_LINEF_ACTIVE (1)
User: 0x00000000
Component Type: MIXERLINE_COMPONENTTYPE_DST_DIGITAL (1)
Channels: 1
Connections: 2
Controls: 2
Short name: Volume
Long name: Master Volume
-- Target:  --
Type: MIXERLINE_TARGETTYPE_UNDEFINED (0)
Device ID: 0
Manufacturer identifier: 65535
Product identifier: 65535
Driver version: 0.0
Product name:
-- Control 1: Mute --
Type: MIXERCONTROL_CONTROLTYPE_MUTE (0x20010002)
Status: MIXERCONTROL_CONTROLF_UNIFORM (0x1)
Item count: 0
Short name: Mute
Long name: Mute
-- Values --
FALSE
-- Control 2: Volume --
Type: MIXERCONTROL_CONTROLTYPE_VOLUME (0x50030001)
Status: MIXERCONTROL_CONTROLF_UNIFORM (0x1)
Item count: 0
Short name: Volume
Long name: Volume
-- Values --
0xffff on a scale of 0x0 to 0xffff
-- 2: Speakers (Contoso --
Device ID: 2
Manufacturer identifier: 1
Product identifier: 104
Driver version: 6.2
Product name: Speakers (Contoso
Support: 0x0
Destinations: 1
-- Destination 0: Master Volume --
Destination: 0
Source: -1
Line ID: 0xffff0000
Status: MIXERLINE_LINEF_ACTIVE (1)
User: 0x00000000
Component Type: MIXERLINE_COMPONENTTYPE_DST_SPEAKERS (4)
Channels: 1
Connections: 2
Controls: 2
Short name: Volume
Long name: Master Volume
-- Target:  --
Type: MIXERLINE_TARGETTYPE_UNDEFINED (0)
Device ID: 0
Manufacturer identifier: 65535
Product identifier: 65535
Driver version: 0.0
Product name:
-- Control 1: Mute --
Type: MIXERCONTROL_CONTROLTYPE_MUTE (0x20010002)
Status: MIXERCONTROL_CONTROLF_UNIFORM (0x1)
Item count: 0
Short name: Mute
Long name: Mute
-- Values --
FALSE
-- Control 2: Volume --
Type: MIXERCONTROL_CONTROLTYPE_VOLUME (0x50030001)
Status: MIXERCONTROL_CONTROLF_UNIFORM (0x1)
Item count: 0
Short name: Volume
Long name: Volume
-- Values --
0xffff on a scale of 0x0 to 0xffff
Device ID: 3
Manufacturer identifier: 1
Product identifier: 104
Driver version: 6.2
Support: 0x0
Destinations: 1
-- Destination 0: Master Volume --
Destination: 0
Source: -1
Line ID: 0xffff0000
Status: MIXERLINE_LINEF_ACTIVE (1)
User: 0x00000000
Component Type: MIXERLINE_COMPONENTTYPE_DST_WAVEIN (7)
Channels: 1
Connections: 1
Controls: 2
Short name: Volume
Long name: Master Volume
Type: MIXERLINE_TARGETTYPE_WAVEIN (2)
Device ID: 0
Manufacturer identifier: 1
Product identifier: 101
Driver version: 6.2
-- Control 1: Mute --
Type: MIXERCONTROL_CONTROLTYPE_MUTE (0x20010002)
Status: MIXERCONTROL_CONTROLF_UNIFORM (0x1)
Item count: 0
Short name: Mute
Long name: Mute
-- Values --
FALSE
-- Control 2: Volume --
Type: MIXERCONTROL_CONTROLTYPE_VOLUME (0x50030001)
Status: MIXERCONTROL_CONTROLF_UNIFORM (0x1)
Item count: 0
Short name: Volume
Long name: Volume
-- Values --
0xf332 on a scale of 0x0 to 0xffff
-- 4: Microphone (Contoso --
Device ID: 4
Manufacturer identifier: 1
Product identifier: 104
Driver version: 6.2
Product name: Microphone (Contoso
Support: 0x0
Destinations: 1
-- Destination 0: Master Volume --
Destination: 0
Source: -1
Line ID: 0xffff0000
Status: MIXERLINE_LINEF_ACTIVE (1)
User: 0x00000000
Component Type: MIXERLINE_COMPONENTTYPE_DST_WAVEIN (7)
Channels: 1
Connections: 1
Controls: 2
Short name: Volume
Long name: Master Volume
-- Target: Microphone (Contoso --
Type: MIXERLINE_TARGETTYPE_WAVEIN (2)
Device ID: 1
Manufacturer identifier: 1
Product identifier: 101
Driver version: 6.2
Product name: Microphone (Contoso
-- Control 1: Mute --
Type: MIXERCONTROL_CONTROLTYPE_MUTE (0x20010002)
Status: MIXERCONTROL_CONTROLF_UNIFORM (0x1)
Item count: 0
Short name: Mute
Long name: Mute
-- Values --
FALSE
-- Control 2: Volume --
Type: MIXERCONTROL_CONTROLTYPE_VOLUME (0x50030001)
Status: MIXERCONTROL_CONTROLF_UNIFORM (0x1)
Item count: 0
Short name: Volume
Long name: Volume
-- Values --
0xf332 on a scale of 0x0 to 0xffff

• #### Muting all audio outputs with IAudioEndpointVolume

I have a selfhost tool that I use to mute all audio outputs programmatically.

Pseudocode:

IMMDeviceEnumerator::EnumAudioEndpoints
for each device:
IMMDevice::Activate(IAudioEndpointVolume)
IAudioEndpointVolume::SetMute(TRUE)

Source and binaries attached.

• #### Implementing a "say" command using ISpVoice from the Microsoft Speech API

I've known for a while that Microsoft Windows comes with text-to-speech and speech-to-text APIs, which power the Narrator and Speech Recognition features respectively.

This forum post prompted me to mess around with them a little.

I came up with this implementation of a say.exe command which takes a single argument as text, and then uses the ISpVoice text-to-speech API to have the computer speak it aloud.

Source and binaries attached.

Pseudocode:

CoInitialize(nullptr);
CoCreateInstance(ISpVoice)
pSpVoice->Speak(text);

Usage:

• #### unattend.xml: turning on Remote Desktop automatically

Here are the portions of my unattend.xml file which are needed to turn on Remote Desktop automatically.

This piece flips the "no remote desktop" kill switch to "allow."

<settings pass="specialize">
...
<component name="Microsoft-Windows-TerminalServices-LocalSessionManager" ...>
<fDenyTSConnections>false</fDenyTSConnections>

That's not enough; it is also necessary to poke a hole in the firewall to allow inbound connections.  I use an indirect string for the Group name, to allow for installing localized builds.  This points to the "Remote Desktop" feature group.

<settings pass="specialize">
...
<component name="Networking-MPSSVC-Svc" ...>
<FirewallGroups>
<Active>true</Active>
<Profile>all</Profile>
<Group>@FirewallAPI.dll,-28752</Group>

If your user account is a member of the "Administrators" group, you're done:

<settings pass="oobeSystem">
<component name="Microsoft-Windows-Shell-Setup" ...>
<UserAccounts>
<LocalAccounts>
<PlainText>true</PlainText>

But if you're like me and you don't want to live in the Administrators group, you need to join the Remote Desktop Users group to be able to log in remotely:

<settings pass="oobeSystem">
<component name="Microsoft-Windows-Shell-Setup" ...>
<UserAccounts>
<DomainAccounts>
<Domain>redmond.corp.microsoft.com</Domain>
<Group>RemoteDesktopUsers</Group>
<Name>MatEer</Name>

• #### Weighing the Sun and the Moon

In an earlier post I mentioned how the Cavendish experiment allowed us to weigh the Earth - to determine the mass of the Earth mE.  Newton knew the acceleration due to gravity on the surface of the Earth and was able to use that to find the product G mE; Cavendish determined G directly, and was thus able to solve for mE.  He would also have been able to find the mass of the sun as follows:

mE aE = G mE mS / rE2

G, rE, and aE = vE2 / rE are known, so we can solve for mS.

But calculating the mass of the moon is trickier.

Once we were able to put a satellite around the moon we could measure its orbital radius and speed, deduce the acceleration, and use that plus the known G to calculate the mass of the moon.  But prior to that we were limited to techniques like:

The moon does not exactly orbit the Earth, but instead orbits the center of mass of the moon/Earth system.  By careful observation we can determine where this center of mass is.  We can then measure the distance between the center of mass and the Earth's center.  This plus the known mass of the Earth and the distance of the Earth from the Moon allows us to determine the mass of the Moon.

If we're lucky enough to see a foreign object come close to the moon, we can determine how much it is accelerated by the Moon.  This will allow us to determine the mass of the Moon using the technique above.  (We won't be able to determine the mass of the foreign object, but we don't need it.)

When the USSR launched Sputnik, American scientists really wanted to know what its mass was.  But because none of the techniques above were useful, they were unable to determine it.

• #### Programmatically rearranging displays

Most of my test machines and my laptop have a single display; but I have two dev machines which are each connected to two displays.

When I clean install Windows, I sometimes need to rearrange the displays:

Since I clean install Windows frequently, I wrote myself a little C++ app which does this programmatically using EnumDisplayDevices / EnumDisplaySettings / ChangeDisplaySettingsEx.

Source and binaries attached.

Pseudocode:

for (each device returned by EnumDisplayDevices) {

grab the position and the height/width using EnumDisplaySettings

}

calculate the desired position of the secondary monitor

Set it using ChangeDisplaySettingsEx with DM_POSITION

Call as:

>swapmonitors
Moved secondary monitor to (1920, 0)

EDIT: Oops: 0 should be CDS_GLOBAL | CDS_UPDATEREGISTRY, to make the settings apply to all users, and to persist across display resets / reboots

LONG status = ChangeDisplaySettingsEx(
nameSecondary,
&mode,
nullptr, // reserved
CDS_GLOBAL | CDS_UPDATEREGISTRY, // was 0
nullptr // no video parameter
);

• #### Windows Sound test team rowing morale event

Last Friday the Windows Sound test team went kayaking.  We went to the Agua Verde paddle club and kayaked around Union Bay for a while.

Here's the route we took:

More detail:

http://connect.garmin.com/activity/179545084

• #### Why square waves have ears: Gibbs' phenomenon (Wilbraham's phenomenon)

In a recent post I sung the praises of square waves as a way to get a heckuva lot of power (3 dB more power than a sine wave) into a sample-range-limited signal.  It's time to take them down a notch now.

A problem with square waves is they're impossible to generate in the analog domain.  In fact, you can't even get close.

Signals generated in the analog domain are subject to physical laws regarding continuity (no teleportation) and whatnot. A common way to model these is to express (periodic) analog signals using a basis of sine waves with integral periods.  Suppose I want to generate the square wave:

 f(x) = 1, -π < x < 0 -1, 0 < x < π

Graphed below are some approximations of this function using sine waves as a basis.  Note that only odd values of n are used in the sums of sin(nx).

The sums converge to +/-1 quite well, but there are definite "ears" at x near 0 where there's overshoot.  This doesn't appear to die down.  A closeup of one of the "ears":

If anything, rather than dying down, the "ears" converge to max(fn) → about 1.18, or about 9% of the "jump" from -1 to 1.  (Dym and McKean, in their 1972 book Fourier Series and Integrals, get the 9% right but incorrectly assert that the convergence is to 1.09.)

This mathematical phenomenon - Gibbs' phenomenon - is a good illustration of the difference between convergence of a series of functions and uniform convergence.

In this case, the series of partial sums pointwise converge to the square wave... for any given point x > 0 and ε > 0, the ear will eventually move to the left, and I can choose an N such that fn(x) is within of ε of 1 for all n > N...

... but the series does not uniformly converge to the square wave.  The following assertion is false: "for any given ε > 0, I can pick an N such that fn(x) is within of ε of 1 for all n > N and all x > 0."  This can be demonstrated by picking ε = 0.17, say.  For any n, even, say, n = 10100, there is an x close to 0 where f1e100(x) > 1.17.

• #### Minimal unsatisfiable regular expression, XPath query

Regular expressions are a tool for matching generic text.  XPath queries are a tool for matching chunks of XML.  Both are search technologies.

When using search technologies it is occasionally quite useful to have a query that will never match anything - for SQL, this would be something like "SELECT 1 WHERE 1 = 0".

My candidates for minimal unsatisfiable regular expression:

/a\bc/
\b is a zero-width assertion that matches a boundary at the beginning or end of a word - specifically, it is true between a word-ish character (\w) and a non-wordish character (\W).  Since literal "a" and "c" are both wordish characters, this will never match.

Or, if you allow Perl extensions:

/(?!)/
This is a negative lookahead for an empty string.  Since the empty string always matches everywhere, this will never match.

My candidate for minimal unsatisfiable XPath query:

/parent::*
This matches everything at or under the parent of the root element.  Since, by definition, the root element has no parent, this will never match.

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