Civil Rights
Movements, leaders, victories and the continuing fight for equality.
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Explore the people, places, events, achievements, struggles and stories that shaped our journey.
Movements, leaders, victories and the continuing fight for equality.
Innovation, patents, science, technology and world-changing contributions.
Pioneers, champions, Negro Leagues, records, activism and excellence.
Meet the people whose lives, choices and achievements shaped the journey.
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Mansa Musa was the ruler of the Mali Empire in West Africa. Details recorded here should be sourced; unknown information is left blank.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
In information theory, Shannon's source coding theorem (or noiseless coding theorem) establishes the statistical limits to possible data compression for data whose source is an independent identically-distributed random variable, and the operational meaning of the Shannon entropy. Named after Claude Shannon, the source coding theorem shows that, in the limit, as the length of a stream of independent and identically-distributed random variable (i.i.d.) data tends to infinity, it is impossible to compress such data such that the code rate (average number of bits per symbol) is less than the Shannon entropy of the source, without it being virtually certain that information will be lost. However it is possible to get the code rate arbitrarily close to the Shannon entropy, with negligible probability of loss. The source coding theorem for symbol codes places an upper and a lower bound on the minimal possible expected length of codewords as a function of the entropy of the input word (which is viewed as a random variable) and of the size of the target alphabet. Note that, for data that exhibits more dependencies (whose source is not an i.i.d. random variable), the Kolmogorov complexity, which quantifies the minimal description length of an object, is more suitable to describe the limits of data compression. Shannon entropy takes into account only frequency regularities while Kolmogorov complexity takes into account all algorithmic regularities, so in general the latter is smaller. On the other hand, if an object is generated by a random process in such a way that it has only frequency regularities, entropy is close to complexity with high probability (Shen et al. 2017).
In information theory, an entropy coding (or entropy encoding) is any lossless data compression method that attempts to approach the lower bound declared by Shannon's source coding theorem, which states that any lossless data compression method must have an expected code length greater than or equal to the entropy of the source. More precisely, the source coding theorem states that for any source distribution, the expected code length satisfies E x ∼ P [ ℓ ( d ( x ) ) ] ≥ E x ∼ P [ − log b ( P ( x ) ) ] {\displaystyle \operatorname {E} _{x\sim P}[\ell (d(x))]\geq \operatorname {E} _{x\sim P}[-\log _{b}(P(x))]} , where ℓ {\displaystyle \ell } is the function specifying the number of symbols in a code word, d {\displaystyle d} is the coding function, b {\displaystyle b} is the number of symbols used to make output codes and P {\displaystyle P} is the probability of the source symbol. An entropy coding attempts to approach this lower bound. Two of the most common entropy coding techniques are Huffman coding and arithmetic coding. If the approximate entropy characteristics of a data stream are known in advance (especially for signal compression), a simpler static code may be useful. These static codes include universal codes (such as Elias gamma coding or Fibonacci coding) and Golomb codes (such as unary coding or Rice coding). Since 2014, data compressors have started using the asymmetric numeral systems (ANS) family of entropy coding techniques, which allows combination of the compression ratio of arithmetic coding with a processing cost similar to Huffman coding. ANS has been adopted by compressors developed by Facebook (Zstandard), Apple (LZFSE), and Google (Draco), among others.
The Sloot Digital Coding System (SDCS) is an alleged technique for data encoding claimed to have been invented in 1995 by Romke Jan Bernhard Sloot (1944–1999), an electronics engineer in the Netherlands. Sloot claimed his system could represent an entire feature film with only one kilobyte of data, a level of compression which is mathematically impossible according to Shannon's source coding theorem. Sloot demonstrated the technology by recording and playing back video, apparently using a small smart card which stored a unique identifier by which any possible video could be referenced. He convinced entrepreneurs Roel Pieper, Marcel Boekhoorn, and Tom Perkins to invest in the technology. However, several days before the conclusion of a contract to sell his invention, Sloot died of a sudden heart attack, and his source code was lost forever. His claims have never been reproduced or verified. Contrary to his claims, the playback device he used was discovered to contain a hard drive. Software engineer Adam Gordon Bell speculated that Sloot had come upon a variation of shared dictionary compression, a known technique. He speculates that Sloot believed in his idea because he failed to understand its mathematical limitations, and faked the demonstrations in order to buy time while refining the details.
In information theory, the entropy of a random variable quantifies the average level of uncertainty or information associated with the variable's potential states or possible outcomes. This measures the expected amount of information needed to describe the state of the variable, considering the distribution of probabilities across all potential states. Given a discrete random variable X {\displaystyle X} , which may be any member x {\displaystyle x} within the set X {\displaystyle {\mathcal {X}}} and is distributed according to p : X → [ 0 , 1 ] {\displaystyle p\colon {\mathcal {X}}\to [0,1]} , the entropy is H ( X ) := − ∑ x ∈ X p ( x ) log p ( x ) , {\displaystyle \mathrm {H} (X):=-\sum _{x\in {\mathcal {X}}}p(x)\log p(x),} where Σ {\displaystyle \Sigma } denotes the sum over the variable's possible values. The choice of base for log {\displaystyle \log } , the logarithm, varies for different applications. Base 2 gives the unit of bits (or "shannons"), while base e gives "natural units" nat, and base 10 gives units of "dits", "bans", or "hartleys". An equivalent definition of entropy is the expected value of the self-information of a variable. The concept of information entropy was introduced by Claude Shannon in his 1948 paper "A Mathematical Theory of Communication", and is also referred to as Shannon entropy. Shannon's theory defines a data communication system composed of five elements: an information source, a transmitter, a communication channel, a receiver, and a destination. The "fundamental problem of communication" – as expressed by Shannon – is for the receiver to be able to identify what data was generated by the source, based on the signal it receives through the channel. Shannon considered various ways to encode, compress, and transmit messages from a data source, and proved in his source coding theorem that the entropy represents an absolute mathematical limit on how well data from the source can be losslessly compressed onto a perfectly noiseless channel. Shannon strengthened this result considerably for noisy channels in his noisy-channel coding theorem. Entropy in information theory is directly analogous to the entropy in statistical thermodynamics. The analogy results when the values of the random variable designate energies of microstates, so Gibbs's formula for the entropy is formally identical to Shannon's formula. Entropy has relevance to other areas of mathematics such as combinatorics and machine learning. The definition can be derived from a set of axioms establishing that entropy should be a measure of how informative the average outcome of a variable is. For a continuous random variable, differential entropy is analogous to entropy. The definition E [ − log p ( X ) ] {\displaystyle \mathbb {E} [-\log p(X)]} generalizes the above.
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →Shirley Chisholm, elected in 1968.
