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Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
Each homology class is an equivalence class over cycles and two cycles in the same homology class are said to be homologous. [6] A chain complex is said to be exact if the image of the (n+1)th map is always equal to the kernel of the nth map. The homology groups of X therefore measure "how far" the chain complex associated to X is from being ...
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the ...
Gottfried Wilhelm Leibniz (or Leibnitz; [a] 1 July 1646 [O.S. 21 June] – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to many other branches of mathematics, such as binary arithmetic and statistics.
It is a declared nuclear-weapons state, and is ranked amongst the emerging and growth-leading economies, [18] with a large and rapidly growing middle class. [19] [20] Pakistan's political history since independence has been characterized by periods of significant economic and military growth as well as those of political and economic instability.
In the United States in the wake of 9/11, the Terrorism Risk Insurance Act 2002 (TRIA) set up a federal program providing a transparent system of shared public and private compensation for insured losses resulting from acts of terrorism. The program was extended until the end of 2014 by the Terrorism Risk Insurance Program Reauthorization Act ...
Of the 50 U.S. states, Hawaii is the fourth-smallest in land area and the 11th-least populous; but with 1.4 million residents, it ranks 13th in population density. Two-thirds of Hawaii residents live on O'ahu, home to the state's capital and largest city, Honolulu .
A comparatively destitute country in 1951, [62] India has become a fast-growing major economy and a hub for information technology services; it has an expanding middle class. [63] Indian movies and music increasingly influence global culture. [64] India has reduced its poverty rate, though at the cost of increasing economic inequality. [65]