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In 1968 Shirley Chisholm became the first Black woman elected to the United States Congress. In 1972 she launched a campaign for the Democratic presidential nomination, breaking another political barrier.
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 coding theory, triangular network coding (TNC) is a non-linear network coding based packet coding scheme introduced by Qureshi, Foh & Cai (2012).[1] Previously, packet coding for network coding was done using linear network coding (LNC). The drawback of LNC over large finite field is that it resulted in high encoding and decoding computational complexity. While linear encoding and decoding over GF(2) alleviates the concern of high computational complexity, coding over GF(2) comes at the tradeoff cost of degrading throughput performance.
The main contribution of triangular network coding is to reduce the worst-case decoding computational complexity of to (where n is the total number of data packets being encoded in a coded packet) without degrading the throughput performance, with code rate comparable to that of optimal coding schemes.
Triangular code has also been proposed as Fountain code[2] to achieve near-optimal performance with encoding and decoding computational complexity of . It has been further shown that triangular based fountain code can even outperform optimized Luby transform code.[2]
In TNC, coding is performed in two stages. First redundant "0" bits are added at the head and tail of each packet such that all packets are of uniform bit length. Then the packets are XOR coded, bit-by-bit. The "0" bits are added in such a way that these redundant "0" bits added to each packet generate a triangular pattern.
In essence, the TNC decoding process, like the LNC decoding process involves Gaussian elimination. However, since the packets in TNC have been coded in such a manner that the resulting coded packets are in triangular pattern, the computational process of triangularization,[3] with complexity of , where is the number of packets, can be bypassed. The receiver now only needs to perform back-substitution,[3] with worst-case complexity given as for each bit location.
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In coding theory, triangular network coding (TNC) is a non-linear network coding based packet coding scheme introduced by Qureshi, Foh & Cai (2012). Previously, packet coding for network coding was done using linear network coding (LNC). The drawback of LNC over large finite field is that it resulted in high encoding and decoding computational complexity. While linear encoding and decoding over GF(2) alleviates the concern of high computational complexity, coding over GF(2) comes at the tradeoff cost of degrading throughput performance. The main contribution of triangular network coding is to reduce the worst-case decoding computational complexity of O ( n 3 ) {\displaystyle O(n^{3})} to O ( n 2 ) {\displaystyle O(n^{2})} (where n is the total number of data packets being encoded in a coded packet) without degrading the throughput performance, with code rate comparable to that of optimal coding schemes. Triangular code has also been proposed as Fountain code to achieve near-optimal performance with encoding and decoding computational complexity of O ( n log n ) {\displaystyle O(n\log n)} . It has been further shown that triangular based fountain code can even outperform optimized Luby transform code.
In coding theory, fountain codes (also known as rateless erasure codes) are a class of erasure codes with the property that a potentially limitless sequence of encoding symbols can be generated from a given set of source symbols such that the original source symbols can ideally be recovered from any subset of the encoding symbols of size equal to or only slightly larger than the number of source symbols. The term fountain or rateless refers to the fact that these codes do not exhibit a fixed code rate. A fountain code is optimal if the original k source symbols can be recovered from any k successfully received encoding symbols (i.e., excluding those that were erased). Fountain codes are known that have efficient encoding and decoding algorithms and that allow the recovery of the original k source symbols from any k’ of the encoding symbols with high probability, where k’ is just slightly larger than k. LT codes were the first practical realization of fountain codes. Raptor codes and online codes were subsequently introduced, and achieve linear time encoding and decoding complexity through a pre-coding stage of the input symbols. Triangular network coding achieve linear encoding and decoding using non-linear encoding, and decoding using the back-substitution method.
In computer networking, linear network coding is a program in which intermediate nodes transmit data from source nodes to sink nodes by means of linear combinations. Linear network coding may be used to improve a network's throughput, efficiency, and scalability, as well as reducing attacks and eavesdropping. The nodes of a network take several packets and combine for transmission. This process may be used to attain the maximum possible information flow in a network. It has been proven that, theoretically, linear coding is enough to achieve the upper bound in multicast problems with one source. However linear coding is not sufficient in general; even for more general versions of linearity such as convolutional coding and filter-bank coding. Finding optimal coding solutions for general network problems with arbitrary demands is a hard problem, which can be NP-hard and even undecidable.
The triangular numbers or triangle numbers are the sequence of positive integers that can be represented as a lattice of points arranged in an equilateral triangle. The triangular lattice representing the n {\displaystyle n} th triangular number contains n {\displaystyle n} rows: the first row contains one point, the second row contains two, and this pattern continues up to the n {\displaystyle n} th row, which contains n {\displaystyle n} . Therefore, the triangular numbers may also be represented by the formula. T n = 1 + 2 + 3 + ⋯ + ( n − 1 ) + n = ∑ k = 1 n k . {\displaystyle T_{n}=1+2+3+\cdots +(n-1)+n=\sum _{k=1}^{n}k.} Triangular numbers are the simplest kind of figurate number – figurate numbers generalize their concept to other two-dimensional polygons, such as the pentagonal numbers, as well as higher-dimensional polyhedra, such as the tetrahedral numbers. Taking T 0 = 0 {\displaystyle T_{0}=0} (see empty sum), the first few terms are (sequence A000217 in the OEIS)
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.
