The most fundamental results of this theory are Shannon's source coding theorem, which establishes that, on average, the number of bits needed to represent the result of an uncertain event is given by its entropy ; and Shannon's noisy-channel coding theorem, which states that reliable communication is possible over noisy channels provided that the rate of communication is below a certain threshold, called the channel capacity.

Shannon's entropy represents an absolute limit on the best possible lossless compression of any communication, under certain constraints: treating messages to be encoded as a sequence of independent and identically distributed random variables, Shannon's source coding theorem shows that, in the limit, the average length of the shortest possible representation to encode the messages in a given alphabet is their entropy divided by the logarithm of the number of symbols in the target alphabet.

Roughly speaking, Shannon's source coding theorem says that a lossless compression scheme cannot compress messages, on average, to have more than one bit of information per bit of message.

Shannon's version of the theorem states:

More recent statements of the theorem are sometimes careful to exclude the equality condition ; that is, the condition is if x ( t ) contains no frequencies higher than or equal to B ; this condition is equivalent to Shannon's except when the function includes a steady sinusoidal component at exactly frequency B.

According to Shannon's source coding theorem, the optimal code length for a symbol is − log < sub > b </ sub > P, where b is the number of symbols used to make output codes and P is the probability of the input symbol.

Shannon – Fano coding should not be confused with Shannon coding, the coding method used to prove Shannon's noiseless coding theorem, or with Shannon – Fano – Elias coding ( also known as Elias coding ), the precursor to arithmetic coding.

The theorem establishes Shannon's channel capacity for such a communication link, a bound on the maximum amount of error-free digital data ( that is, information ) that can be transmitted with a specified bandwidth in the presence of the noise interference, assuming that the signal power is bounded, and that the Gaussian noise process is characterized by a known power or power spectral density.

Building on Hartley's foundation, Shannon's noisy channel coding theorem ( 1948 ) describes the maximum possible efficiency of error-correcting methods versus levels of noise interference and data corruption.

Shannon's theorem shows how to compute a channel capacity from a statistical description of a channel, and establishes that given a noisy channel with capacity C and information transmitted at a line rate R, then if

Shannon's noisy coding theorem is general for all kinds of channels.

We also know from Shannon's channel coding theorem that if the source entropy is H bits / symbol, and the channel capacity is C ( where C < H ), then H − C bits / symbol will be lost when transmitting this information over the given channel.

Shannon's theorem –

* Shannon's theorem

Due to the limited bandwidth, information can only be transmitted very slowly, on the order of a few characters per minute ( see Shannon's coding theorem ).

Since then, there has been a surge of interest in extending Shannon's sampling theorem

In information theory, Shannon's source coding theorem ( or noiseless coding theorem ) establishes the limits to possible data compression, and the operational meaning of the Shannon entropy.

# redirect Shannon's source coding theorem

* Amiel Feinstein, the mathematician that proved the Shannon's theorem

It is also commonly called Shannon's interpolation formula and Whittaker's interpolation formula.

" Let's Party " ( released originally in the U. S. as " March of the Mods ") used " March of the Mods " ( also known as the Finnjenka Dance ), interpolating Del Shannon's " Runaway " and The Wrens ( R & B band )' " Come Back My Love " among others.

There are also many subplots, such as Anika's manipulation of her classmates and friends, Craig's various well-intentioned but ill-fated social projects, Shannon's sexuality crisis, the divorce of Mark, Travis, and Kat's parents, and the rise and fall of Chris's popularity.

He also had major supporting roles in Final Analysis ( 1992 ), The Specialist ( 1994 ), and the film Shannon's Rainbow ( 2009 ).

Sixteen wayside markers recounting aspects of Shannon's career were also placed along the trail, which runs through the region in which Shannon is thought to have wandered during his 1804 separation from the expedition.

His considerable influence using falsetto extended to Del Shannon, who paid homage to Jones and also The Ink Spots for Shannon's falsetto style.

The album also featured the Music Instructor Megamix and covers of Newcleus ' " Jam On It ", Freestyle's " Don't Stop The Rock ", Jonzun Crew's " Pack Jam ( Look Out For The OVC )" and Shannon's " Let The Music Play ".

Shannon's death not only left Sayid heartbroken, but also caused a rift between the survivors and the Tailies.

He also produced Scott Shannon's program on Pirate Radio in Los Angeles before working with Danny Bonaduce at WEGX in Philadelphia.

The name was originally intended for Shannon's and John " Tumor " Fahnestock's side project MF Pitbulls which also included 3 / 5 of the future band Snot.

The song also appears to have been strongly influenced by songs from the late 50s-early 60s (" when Rock was young "), including Del Shannon's 1962 " Cry Myself to Sleep " and " Little Darlin '", most famously recorded in 1957 by The Diamonds ( originally recorded by The Gladiolas.

Theory behind DSL, like many other forms of communication, can be traced back to Claude Shannon's seminal 1948 paper: A Mathematical Theory of Communication.

Depending on the model chosen, this may enable real computers to solve problems that are inextricable on digital computers ( for example, Hava Siegelmann's neural nets can have noncomputable real weights, making them able to compute nonrecursive languages ), or vice versa ( Claude Shannon's idealized analog computer can only solve algebraic differential equations, while a digital computer can solve some transcendental equations as well.

Shannon's model can be adapted to cope with this problem ).

The Hill cipher has achieved Shannon's diffusion, and an n-dimensional Hill cipher can diffuse fully across n symbols at once.

While in Shannon's theory the entropy of a composite system can never be lower than the entropy of any of its parts, in quantum theory this is not the case, i. e., it is possible that

In mathematics, Shannon's expansion or the Shannon decomposition is a method by which a Boolean function can be represented by the sum of two sub-functions of the original.

In Shannon's expansion the term x is very significant but problems can arise in simple equations.

Of course, you can perform Shannon's Expansion about any variable you desire, so long as you can provide for that variable in the expression without changing the truth value of the expression.

The king is sometimes given an arbitrary high value such as 200 points ( Shannon's paper ) or 1, 000, 000, 000 points ( 1961 USSR program ) to ensure that a checkmate outweighs all other factors.

In keeping with Shannon's claim to fame, a scavenger hunt was held, with tourists urged to " Find Private Shannon " by visiting all sixteen markers.

His first album, released in 1990, Bad of the Heart had many underground dance hits like Bad of the Heart (# 25 on the Billboard Hot 100 in 1990 ), Look Into My Eyes (# 4 Hot Dance Club Play, 1990 ) and Without You (# 4 Hot Dance, 1989 ), all co-produced by New York producer Chris Barbosa who had produced Shannon's string of hits a few years earlier.

When Shannon's boyfriend takes off with all of the money, Shannon goes to Boone's hotel room in Sydney, drunk, and Boone allows her to seduce him.