Mathematics Essay Example | Topics and Well Written Essays - 1500 words

Mathematics - Essay Example If we let "s" stand for the number of kth powers, then g(k) is the least such "s" powers. Some examples of g(k) are: g(1) = 1; g(2) = 4, since from Lagranges 4-square theorem, every natural number is the sum of atleast 4 squares. In addition it was found that 7 requires 4 squares and 23 requires 9 cubes. Progress was made on Warings Problem by establishing bounds, or the maximum number of powers. For instance, Liouville found that g(4) is at most 53. The work of Hardy and Littlewood also led to other bounds; in particular, they found the upper bound for g(k) to be O(k2k+1). The work of Hardy and Littlewood also led to the realization that the number G(k) is more fundamental than g(k). Here, G(k) is the least positive integer s such that every sufficiently large integer (greater than some constant) is a sum of at most s kth powers of positive integers. A formula for the exact value of G(k) for all k has not been found, but there have been many bounds established. 1, 4, 9, 19, 37, 73, 143, 279, 548, 1079, 2132, 4223, 8384, 16673, 33203, 66190, 132055, 263619, 526502, 1051899, 2102137, 4201783, 8399828, 16794048, 33579681, 67146738, 134274541, 268520676, 536998744, 1073933573, 2147771272 Hilberts proof of Warings Problem for all positive k can be seen as proving an equivalent theorem: There are positive integers A and M and positive rationals 1, ..., M, depending only on k, such that each integer N A can be written in the form Many generalizations of Waring’s Problem have been made. For instance, there is the prime Waring’s problem, and generalizations of the problem to algebraic number fields and arbitrary fields. The problem known as the â€Å"easier† Waring’s Problem takes the integer n to be a sequence of numbers x, each to the kth power. All of these variations have led to a Mathematics Subject Classification 11P05 entitled â€Å"Waring’s Problem and variants.† In 1742, Goldbach suggested that every

Women's History Essay Example | Topics and Well Written Essays - 750 words

Women's History - Essay Example Even in cabinet or parliaments, women are underrepresented. It should be noted that population wise, the number of women in this world is more than the number of men. Yet they were deprived of equal opportunities in political circles with men. The conditions of the women were not so good in the past also. It should be noted that slavery and other social evils were prevailing not only in America but also in other parts of the world during past few centuries. Slavery was hyperactive during the period 1600-1820. Women community struggled more than male community because of slavery related problems as well as other social evils. Participation in politics was only a distant dream for women during the past. This paper analyses the political roles of women during the period 1600-1820. Carol Berkin in her article, â€Å"The Effect of Slavery on African American Women Lives† clearly explains the social and political life of women in America during 1600-1820. ‘Denied by any legal abilities to control the conditions of their life, women labored according to their master’s demands. Women were vulnerable to brutal punishments and to the separation of their families. They had no choice but to accept the marginal food and clothing† (â€Å"Berkin, p.59). These facts clearly illustrate the typical lives of black women during the period of 1600-1820 in America. Many people have the illusion that the conditions of white women was so good in America during this period. In fact white women also faced lot of discriminations during this period even though they never faced harsh experiences as faced by the black women explained above. When talking about women in Africa and other countries of the global south (formerly known as the Third World), feminists often point out that these women must manage multiple forms of disadvantage or oppression. Not only do they suffer the universal subordination shared by women across the world, but also they must contend wit h living in poorer or less-developed countries(Women, Politics, and Power, p.27). Even today, African women are leading pathetic life compared to women in other parts of the world. They were forced to look after their family single handedly. African males treat their counterparts as inferior citizens. They never allow women to work in the mainstreams of social or political life. The case is not much different in Asia or Europe. In all these regions, the participation of women in politics was negligible during 1600-1820. Women all over the world started to get some kind of recognition in politics from the latter parts of 19 th century onwards. Women’s modern-day participation in politics begins with the acquisition of voting rights (suffrage). The first country to fully enfranchise women, and the only country to give women’s suffrage in the 19th century, was New Zealand in 1893. In 1902, Australia was the second country to give women’s suffrage and was followed b y a variety of Western and Eastern European states. By 1945, 46% of the world’s countries allowed women to vote (Women, Politics, and Power, p.16). These statistics clearly suggest that women started to get voting rights only during the latter periods of 19 th century. Before that, it was unimaginable for women to think in terms of participating in politics. Discrimination against women was evident even in some of the most civilized societies during the 1600-1820 period. Women were used only for the reproductive work during this period. In majority of the countries, women were treated as commodities rather than human beings in this period. It should not be forgotten that the conditions of women is not so good even at present in some of the Middle Eastern countries. Even