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No rx tramadol bars, I came up with this surprising result thinking up probability examples for my class. It would be far too cruel to use in class, so I present it to you, dear reader.

Suppose we play a game called "I give you money." you repeatedly flip a fair coin until the first time it comes up tails. For every time you flipped heads, you win 25 cents.

Question 1: if Alice plays this game, what are her expected (average) winnings.

Answer: 25 cents, no rx tramadol bars.

Question 2: if Alice played this game and the tax-man took a fourth of her winnings, what's her expected net profit.

Answer: 80 cents.

So if we know the tax-man took some of her money, we expect she ends up with more. Tramadol Euphoria, Why. No rx tramadol bars, Believe it or not, the principle behind this question is actually useful in my line of work, which is reverse-engineering secret algorithms. Answer after the flip....

You might expect that this is some weird conditional probability scenario, and it is. The basic idea is that the probabilities change when you are given information. Here, the new information is that Alice's winnings must have been a multiple of four, or else the tax-man couldn't take exactly a fourth of her money.

You also know she hasn't won 0.00 dollars, which makes a big difference. You might dispute this based on the exact wording of the problem, no rx tramadol bars. We can work it out either way.

So prior to any extra information, Alice flips n heads with probability (0.5)^(n+1). The expected number of heads is 1, for 25 cents.

If I tell you that the number of heads was a multiple of 4 and greater than 0, Tramadol Hci Tablets, the probability of "most" numbers drops to zero, while the remaining probabilities are increased by a factor of 30 (so that all probabilities still add to 1.) Working out the sum gives you an expected value of 4*(16/15) heads, or 16/15 dollars won, and 12/15 dollars after the tax-man. No rx tramadol bars, 80 cents.

If instead you thought that Alice might have won 0 cents (and the tax-man took 0 cents, which is technically a fourth of her earnings,) then you end up dropping out a factor of 16, meaning her expected winnings are now 5 cents. Also counterintuitive, but not as dramatic.

Why is this interesting?

This is a phenomenon I call quantization leakage, meaning that you are given extra information by the fact that something in the real world tends to happen in whole numbers, or round numbers. What Is Ultram Used For, Sometimes this information is pretty significant. In reverse-engineering, we can often guess that the designer of a secret algorithm will pick round numbers for the sizes of certain things, no rx tramadol bars.

Here is a real-world example: we recently reverse-engineered a secret watermarking algorithm, as part of a contest that we won earlier this year (netting ourselves a digital camera and some great new theoretical results.) The watermark detector, it turns out, extracted roughly 50,000 statistical features from a computer image, and scanned them for a watermark. The watermark embedder tweaked these features in a detectable way---a pretty common approach to image watermarking. The images were standard monochrome test images of 512x512 pixels.

If you know enough about common techniques used in watermarking, "about 50,000" is a lot of information. No rx tramadol bars, First, 50,000 is not a round number in the computer world. That's round to human beings but not to machines. It is close to a common round number in hexadecimal, 49152 (C000 hex.) In some scenarios if you are told a number in a computer algorithm is "about 50, Overnight Tramadol Online,000," 49152 is a pretty good guess for the exact number.

The images used by the watermarking algorithm were 512 by 512, which have 4096 8x8 blocks. 49152 is 12x4096. Coincidence, no rx tramadol bars. 8x8 block transforms are quite common in image processing, particularly 8x8 DCTs. So you might guess that the watermark features are 8x8 block DCT coefficients, 12 per block, probably low-frequency AC coefficients. This was, in fact, the algorithm used. Tramadol Sent Cod, All this can be guessed from a rough estimate of the number of features.

I wish I could say that this was how we deduced the algorithm, and then you could all wow at our mad deductive abilities. In fact, we used different techniques to deduce the algorithm, which gave us a few papers worth of new ideas.

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The Underhanded C Contest Discount tramadol without prescription, is a challenge to write innocent-looking C source code implementing nonobvious malicious behavior. From the site:

Every year, Tramadol And Acetaminophen, Tramadol Withdraw, we will propose a challenge to coders to solve a simple data processing problem, but with covert malicious behavior, 50mg Dose Hcl Tramadol. Drugs Similar To Ultram, Examples include miscounting votes, shaving money from financial transactions, Order Tramadol Online, or leaking information to an eavesdropper. The main goal, however, is to write source code that easily passes visual inspection by other programmers.

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No prescription tramadol bars, Back in grad school I did a little research on the key distibution of online tunebooks. This was lost for a while, but I recently found the article and graphs buried in my computer.

So to test image tags and floating bodies, here follows a statistical snapshot of Henrik Norbeck's online tunebook, from about 4 years ago.

If you don't know ABC notation, I recommend you learn it. It's a way of notating folk melodies using ordinary text, in a simple and straightforward manner. ABC is human readable/writable, and can be converted to sheet music or MIDI files. Now, Cheap Overnight Tramadol, I think it's best to learn tunes from people rather than from paper whenever humanly possible, but as a computer scientist I think ABC notation is just a work of genius. It's everything a file format needs to be and not a drop more.Anyway, since ABC is text, I can grep it, no prescription tramadol bars. In particular, I can type

$ cat *.abc | grep "^K:" | sort | uniq -c

...which strips out all the key fields, sorts and counts them, giving me a handy breakdown of the number of tunes in each key. [A slightly more elaborate string of commands is really needed to handle extra annoyances such as capital/lowercase letters, etc.]So, I headed over to Henrik Norbeck's tune index, a large and very well-formatted collection of traditional tunes in ABC format. Mr. Norbeck provides all the tunes in one big zip file, Tramadol Cor 127, to boot. So I download and grep.

No prescription tramadol bars, Enough chit-chat, here are the results:To your left are the tunes sorted by key signature. E.g., A Mixolydian == 2 sharps, corresponding to D major, so all the AMix tunes and Dmaj tunes (and Edor tunes and Baol tunes) are all binned together under "D".

When the key signatures are arranged on the circle of 5ths, you see a pretty clear bell curve. And look. G and D tunes are pert-near neck-and-neck. G is actually ahead of D by a tiny margin, not what I expected.

To the right is the same data, viewed by tonic (e.g, D major, D minor, D dorian, D neptunian ... Free tramadol online without prescription, are all binned under "D".) Now, D reigns supreme, which is what you'd expect given the standard modes for Irish music.The typical modes of Irish music are Ionian ("major",) Mixolydian, Aeolian ("minor") and Dorian (also often called "minor".) These four modes fall in a row on the circle of 5ths: C major == G mix == D dorian == A aeolian, in the sense that they all have the same key signature. It thus stands to reason that the right half of the graph is going to be bigger, and the left half smaller, when we look at tonics rather than key signatures.

One of the reasons I used Henrik Norbeck's tune index is that careful attention is paid to both the formatting and the key information, no prescription tramadol bars. Nowadays, many people just bin mixolydian tunes as "major," and don't distinguish between different kinds of "minor" tunes. The distinction matters in Irish music, though, as "minor" very often means dorian, Ultram And Surgery, not aeolian.

Here's both together:

And if you're interested in modes, check this out:

No surprise that Ionian (the standard "major" key) is the most common, but check out the minor keys. The big minor key is actually the dorian mode, while the "natural" minor is less common even than mixolydian.

Given the type of instruments used in Irish music, it's no surprise that the dorian mode is the common one. Consider a whistle, which plays a D scale with room for a few accidentals and a usable range of a couple octaves. This isn't the only instrument that is geared to a few keys and a few octaves: the melodeon and pipes are similar. With these types of instruments, Generic tramadol online legally, the dorian minor is much more comfortable because it's right in the same place, just a note up the scale. If you can flatten a note you can make it an aolian minor, but nevertheless the dorian is the natural choice.

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