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Chaos and Fractals, Part 2: Pascal Triangle

The notation was introduced by Albert von Ettinghausen in 1826, although these numbers were already known centuries before that (see Pascal's triangle). The function is often called the choose function, and is often read as "n choose k". The binomial coefficients are the coefficients of the series expansion of a power of a binomial. If the exponent nonnegative integer then this infinite series is actually a finite sum as all terms with k > n are 0 (zero), but if the exponent n is negative or a non-integer, then it is an infinite series. (REF: textbook 'Chaos and Fractals: the New Frontiers of Science') Applying the formula to (1 + xⁿ we immediately obtain the k^th coefficient b_k (k runs from 0 to n) of row number n of Pascal's triangle. For example, the coefficient for k = 3 in row n = 7 is: The calculation of the binomial coefficient is conveniently arranged like this: ((((5/1)·6)/2)·7)/3, alternately dividing and multiplying with increasing integers. Each division produces an integer result which is itself a binomial coefficient. The recipe to compute the coefficients of a row is thus very simple. The first and last lines are copied from the line above. These will always be equal to 1. The other coefficients are just the sum of the two coefficients in the row above. In this scheme it is most convenient to write Pascal's triangle in the form with the top vertex centered on a line above it as shown. Another identity is easy to derive: the sum of all coefficients in row number n of Pascal's triangle is equal to 2_n, which is seen by setting x = y = 1 in the binomial formula. (NOTE: while Christian Kramp (Algrebra textbook, 1808) and Euler wrote [n]; Gauss (German mathematician and physicist) used the notation π(n). (REF: Wikipedia Binomial coefficient) Pascal's rule is the important recurrence relation which follows directly from the definition: The recurrence relation just proved can be used to prove by mathematical induction that C (n, k) is a natural number (and C = combinations or choices) for all n and k. Pascal's rule also gives rise to Pascal's triangle: Row number n contains the numbers C(n,k) for k = 0,…,n. It is constructed by starting with ones at the outside and then always adding two adjacent numbers and writing the sum directly underneath. This method allows the quick calculation of binomial coefficients without the need for fractions or multiplications. For instance, by looking at row number 5 of the triangle, one can quickly read off that: (x + y)⁵ = 1 x⁵ + 5 x⁴y + 10 x³y² + 10 x²y³ + 5 x y⁴ + 1 y⁵.

The differences between elements on other diagonals are the elements in the previous diagonal, as a consequence of the recurrence relation above.

In the 1303 AD treatise Precious Mirror of the Four Elements, Zhu Shijie mentioned the triangle as an ancient method for evaluating binomial coefficients indicating that the method was known to Chinese mathematicians five centuries before Pascal.

Chaos and Fractals, Part 1

The Pascal Triangle Blaise Pascal (1623-1662) was a great French mathemetician and scientist. When he was twenty years old, he built some 10 mathematical machines for the addition of integers, a precursor to modern computers. What is known as the arithmetic triangle (or Pascal's Triangle) was not originally his discovery. The first printed form of the arithmetic triange in Europe dates back to 1527. A Chinese version had already been published in 1303 (see image.) But Pascal used the arithmetic triangle to solve some problems related to chances in gambling, which he had discussed with Pierre de Fermat in 1654. This research later became the foundations of probability theory. The arithmetic triangle is a triangular array of numbers composed of the coefficients of the expansion of the polynomial (1 + x)ⁿ. Here n denotes the row starting from n = 0. Row n has n + 1 entries. For example, for n = 3 the polynomial is (1 + x)³ = 1 + 3x + 3x² + x³. Thus, row number 3 reads: 1,3,3,1. There are two ways to compute the coefficients. The first one inductively computes one row based on the entries of the previous row. Assume that the coefficients a0,...,an in row n are given:(1 + x)n = a0 + a1x + ...+ anxn, and the coefficients b0,...,bn + 1 of the following row are required: (1 + x)^{n + 1} = b₀ + b_1x + ...+_{n + 1}x^{n + 1} Comparing coefficients we obtain the result b₀ = a₀, b_k = a_{k - 1}a_k for k = 1,...,n, b_{n + 1} = a_n. Wikipedia Binomial coefficient In Mathematics, the binomial coefficient is the coefficient of the x^k term in the polynomial expansion of the binomial power (1 + x)ⁿ. is often called the choose function of n. k is the number of k-element subsets (the k-combinations) of an n-element set; that is, the number of ways that k things can be 'chosen' from a set of n things.

Given a non-negative integer n and an integer k, the binomial coefficient is defined to be the natural number

and where n denotes the factorial (factorial: the product of all positive integers less than or equal to n of n.

Alternatively, a recursive defition can be written as where .

(TO BE CONTINUED)

Nigeria, Africa Blog

(names from Wikipedia)

Orílẹ̀-èdè Olómìnira Àpapọ̀ Naìjírìà Republik Nijeriya جمهورية نيجيريا Republic ndi Naigeria Republik Federaal bu Niiseriya Federal Republic of Nigeria

History from: Lonely Planet

Pre-20th-Century History The earliest Nigerians were the Nok people, skilled artisans from around the Jos area. By the beginning of the second millennium the Nok had virtually disappeared and the state of Kanem-Borno, to the north east of Lake Chad, was flourishing. Much of Kanem-Borno was Islamic, as were the kingdoms around Kano and Katsina, and its wealth came from control of the trans-Saharan trade route from West Africa to the Mediterranean. These northern Islamic states remained untouched by Europeans until well into the 19th century. By contrast the southern states were dominated in the 14th and 15th centuries by a number of Yoruba empires with traditional Obas (kings) who cultivated European contact through the Portuguese spice trade. At the end of the 18th century Fulani religious zealots in the north, sick of being dominated by the Islamic Hausa states, took over and created the single Islamic state of the Sokoto Caliphate. This original division between the Islamic government in the north and the Yoruba tribes in the south has never healed, and over the years intertribal fighting and civil wars have rubbed salt into the wounds. Even today Nigerian politics is riddled with tribal rivalries and ancient axes to grind. After the bottom fell out of the spice trade, the Portuguese, and then the British, began a miserable trade in slaves, but by 1807 slavery had been banned and the British began to look for other ways to turn a buck - palm oil along the coast, and tin mining around Jos in the centre. The British also appointed chiefs in the southern Ibo communities to run the area but this was like hammering square pegs into round holes. These 'invented chiefs' had little in common with the people and simmering hostility and resentment was the usual result.

Modern History In 1960 Nigeria declared independence. Unfortunately the British system of colonialism had done nothing to unify Nigeria or prepare it for independence. The historical conflicts between north and south, and other inter-regional fighting, made the idea of a unified republic unworkable. By 1966 the dream of a flourishing democracy was floundering amidst a series of massacres, inter-regional hostilities and, finally, a military coup that installed the first of a series of military governments. The Ibo responded by seceding from the federation and declaring the independent republic of Biafra, kick-starting an all-out civil war that lasted for nearly three years before federal Nigeria won and reintegrated Biafra. The war left behind nearly 1,000,000 dead and 'Biafra' as a byword in mass destruction and famine.

Given Nigeria's seesawing fortunes it was almost predictable that it would follow one of the world's worst famines with a champagne period of excessive prosperity. Rocketing oil prices provided the Nigerian government with a chance to go on a spending spree of reckless proportions and the country quickly became a hotbed of foreigners rushing to Nigeria with their dash (bribe) money. Corruption became de rigueur, crime rampant, and chaos spread like cancer. By the early 1980s the world recession sent oil prices plummeting again and plunged Nigeria into a cycle of massive debt, soaring inflation, large-scale unemployment and widespread corruption. In 1993 the country came under the iron-fisted rule of General Abacha. Far from delivering on the promise of a US-style democracy, Abacha earned the wrath of human rights group and the censure of the Commonwealth nations for executing well-known playwright Ken Saro-Wiwa and eight others for seditious political activity. This and other despotic actions sparked rioting and civil unrest across Nigeria. In June 1998 Abacha died and was immediately replaced by Major General Abubakar. Abubakar promised a return to civilian rule. He kept his promise and in 1999 Olusegun Obasanjo, a former military leader and - until 1998 - a political prisoner, was elected president.

Recent History Upon the election of Olusegun Obasanjo, Nigerians were euphoric, as it seemed they were finally free from military rule. It was not long before things deteriorated as several rival groups (religious and tribal), no longer threatened by army intervention, settled down to protracted conflict. In one night of carnage during the Sharia'a riots (over full implementation of Islamic law) in February 2000, over 300 people were killed in hand-to-hand rioting between Igbo Christians and Hausa Muslims in Kaduna. The emergency was exacerbated by fuel shortages and extended power blackouts that left the country in darkness for weeks. Little improved under the new democracy. Obasanjo consolidated Nigeria's position as West Africa's political heavyweight and a key player in the Commonwealth, but the country was still beset by ethnic and religious violence, especially in Lagos, the Central Plateau and the southern oilfields. In presidential elections in April 2003 Obasanjo claimed an overwhelming victory, although independent observers expressed reservations over irregularities and intimidation. The constitution prevents him standing for a third term, and many of the old political players are currently jockeying for a run at the top prize in the 2007 elections.

MORE about Nigeria (from Wikipedia) The capital city is Abuja. The people of Nigeria have an extensive history, and archaeological evidence shows that human habitation of the area dates back to at least 9000 BC.[2] The Benue-Cross River area is thought to be the original homeland of the Bantu migrants who spread across most of central and southern Africa in waves between the 1st millennium BC and the 2nd millennium AD. Nigeria is the most populous country in Africaand the eighth most populous country in the world with a population of over 140 million. It is a regional power and it is also listed among the "Next Eleven" economies. The economyof Nigeria is one of the fastest growing in the world with the International Monetary Fundprojecting a growth of 9% in 2008 and 8.3% in 2009. OTHER LINKS: http://www.africa.upenn.edu/Country_Specific/Nigeria.html