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...The image to the right illustrates the various possible dimensions that a string could vibrate in. (Currently, physicists accept the fact that there are at ...se strings actually exist. They are pretty much the main topic of [[string theory]]. ...

...This is a finite [[Sequence]] of letters from the alphabet. For example, a string of length 5 '''over''' ''{0,1}'' is ''01101''. ...tters (it is often written as <math>\lambda</math>). The empty string is a string over any alphabet. ...

...a string from <math>\boldsymbol{L}_{1}\,</math> and <math>w\,</math> is a string from <math>\boldsymbol{L}_{2}\,</math>. ...,</math> consists of all strings <math>v\,</math> for which there exists a string <math>w\,</math> in <math>\boldsymbol{L}_{2}\,</math> such that <math>vw\,< ...

In formal [[number theory]] a '''Gödel numbering''' is a [[function (mathematics)|function]] which as ...]]s can then represent some form or function. A [[numbering (computability theory)|numbering]] of the set of [[computable function]]s can then be represented ...

The fingerprint function we choose is as follows: by treating the input string <math>x\in\{0,1\}^n</math> as the binary representation of a number, let <m *Input: a string <math>x\in\{0,1\}^n</math> and a "pattern" <math>y\in\{0,1\}^m</math>. ...

...he problem and an output is called a '''solution''' to that instance. The theory of complexity deals almost exclusively with [http://en.wikipedia.org/wiki/D ...blems that the "yes" instances can be ''verified'' in polynomial time. The string <math>y</math> in the definition is called a '''certificate''' or a '''witn ...

.../math>. The time complexity of fastest matrix multiplication algorithm (in theory) is <math>O(n^{2.376})</math>, and so is the time complexity of this method The fingerprint function we choose is as follows: by treating the input string <math>x\in\{0,1\}^n</math> as the binary representation of a number, let <m ...

The new fingerprint function we design is as follows: by treating the input string <math>x\in\{0,1\}^n</math> as the binary representation of a number from <m *Input: a string <math>x\in\{0,1\}^n</math> and a "pattern" <math>y\in\{0,1\}^m</math>. ...

.../math>. The time complexity of fastest matrix multiplication algorithm (in theory) is <math>O(n^{2.376})</math>, and so is the time complexity of this method The fingerprint function we choose is as follows: by treating the input string <math>x\in\{0,1\}^n</math> as the binary representation of a number, let <m ...

== Principles in probability theory == * '''Basics of probability theory''': probability space, events, the union bound, independence, conditional p ...

== Principles in probability theory == * '''Basics of probability theory''': probability space, events, the union bound, independence, conditional p ...

A trivial way to solve EQ is to let Bob send his entire input string <math>b</math> to Alice and let Alice check whether <math>a=b</math>. This The new fingerprint function we design is as follows: by treating the input string <math>x\in\{0,1\}^n</math> as the binary representation of a number, let <m ...

A trivial way to solve EQ is to let Bob send his entire input string <math>b</math> to Alice and let Alice check whether <math>a=b</math>. This As before, we can define the fingerprint function as: for any bit-string <math>x\in\{0,1\}^n</math>, its random fingerprint is <math>\mathrm{FING}( ...

...f occurrences of the pattern <math>\pi</math> as a substring of the random string <math>X</math>. This problem has various important applications in both theory and practice. In many tasks, the data points are drawn from a high dimensio ...

...he problem and an output is called a '''solution''' to that instance. The theory of complexity deals almost exclusively with [http://en.wikipedia.org/wiki/D ...blems that the "yes" instances can be ''verified'' in polynomial time. The string <math>y</math> in the definition is called a '''certificate''' or a '''witn ...

We introduce a general theory of counting permutations with restricted positions. In the derangement prob We can equivalently represent each <math>S\subseteq U</math> by a boolean string <math>S\in\{0,1\}^U</math>, where <math>S(x)=1</math> if and only if <math> ...

We introduce a general theory of counting permutations with restricted positions. In the derangement prob We can equivalently represent each <math>S\subseteq U</math> by a boolean string <math>S\in\{0,1\}^U</math>, where <math>S(x)=1</math> if and only if <math> ...

We introduce a general theory of counting permutations with restricted positions. In the derangement prob We can equivalently represent each <math>S\subseteq U</math> by a boolean string <math>S\in\{0,1\}^U</math>, where <math>S(x)=1</math> if and only if <math> ...

We introduce a general theory of counting permutations with restricted positions. In the derangement prob We can equivalently represent each <math>S\subseteq U</math> by a boolean string <math>S\in\{0,1\}^U</math>, where <math>S(x)=1</math> if and only if <math> ...

We introduce a general theory of counting permutations with restricted positions. In the derangement prob We can equivalently represent each <math>S\subseteq U</math> by a boolean string <math>S\in\{0,1\}^U</math>, where <math>S(x)=1</math> if and only if <math> ...

We introduce a general theory of counting permutations with restricted positions. In the derangement prob ...th> function or '''the Euler totient function''', is fundamental in number theory. ...

...f occurrences of the pattern <math>\pi</math> as a substring of the random string <math>X</math>. This problem has various important applications in both theory and practice. In many tasks, the data points are drawn from a high dimensio ...

:In probability theory, an event <math>A</math> is said to be independent of events <math>B_1,B_2, The idea of the proof is to '''reconstruct''' a random string. ...

...e know this? Since it takes at most <math>d</math> steps to fix any binary string of length <math>d</math> bit-by-bit to any other.) This directly gives us t This problem has various important applications in both theory and practice. In many tasks, the data points are drawn from a high dimensio ...

...are fairly self-explanatory. In the 1906 4th edition of ''Probability and Theory of Errors'' <ref> ...formatting function, which allows one to convert a fractional number to a string, rounded to a user-specified number of decimal places (the ''precision''). ...

...hed 23 years later. It is a fundamental result in the area of extremal set theory, which studies the maximum (or minimum) possible cardinality of a set syste The idea of the proof is to '''reconstruct''' a random string. ...