Question: In modular arithmetic, two integers are added and then the sum is divided by a positive integer called the modulus. The result of modular addition is the remainder of that division. For any modulus, n, the set of integers from 0 to n 1 forms a group under modular addition: the inverse of any element a is n a, and 0 is the identity element. This is familiar from the addition of hours on the face of a clock: if the hour hand is on 9 and is advanced 4 hours, it ends up on 1, as shown at the right. This is expressed by saying that 9 + 4 equals 1 "modulo 12" or, in symbols,
Try to answer this question if possible: What type of device can be use to demonstrate modular addition?
Answer: a clock
Question: On August 1, 2012, Windows 8 (build 9200) was released to manufacturing with the build number 6.2.9200.16384 . Microsoft planned to hold a launch event on October 25, 2012 and release Windows 8 for general availability on the next day. However, only a day after its release to manufacturing, a copy of the final version of Windows 8 Enterprise N (a version for European markets lacking bundled media players to comply with a court ruling) leaked online, followed by leaks of the final versions of Windows 8 Pro and Enterprise a few days later. On August 15, 2012, Windows 8 was made available to download for MSDN and TechNet subscribers. Windows 8 was made available to Software Assurance customers on August 16, 2012. Windows 8 was made available for students with a DreamSpark Premium subscription on August 22, 2012, earlier than advertised.
Try to answer this question if possible: What was the final build Microsoft decided to manufacture?
Answer: 9200
Question: According to the United States Census Bureau, the city has a total area of 656.3 square miles (1,700 km2); this comprises 634.0 square miles (1,642 km2) of land and 22.3 square miles (58 km2) of water. The Piney Woods is north of Houston. Most of Houston is located on the gulf coastal plain, and its vegetation is classified as temperate grassland and forest. Much of the city was built on forested land, marshes, swamp, or prairie which resembles the Deep South, and are all still visible in surrounding areas. Flatness of the local terrain, when combined with urban sprawl, has made flooding a recurring problem for the city. Downtown stands about 50 feet (15 m) above sea level, and the highest point in far northwest Houston is about 125 feet (38 m) in elevation. The city once relied on groundwater for its needs, but land subsidence forced the city to turn to ground-level water sources such as Lake Houston, Lake Conroe and Lake Livingston. The city owns surface water rights for 1.20 billion gallons of water a day in addition to 150 million gallons a day worth of groundwater.
Try to answer this question if possible: What kind of weather event is a recurring problem for Houston?
Answer: flooding
Question: Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions, that is, g h, and g1 must not vary wildly if g and h vary only little. Such groups are called topological groups, and they are the group objects in the category of topological spaces. The most basic examples are the reals R under addition, (R {0}, ), and similarly with any other topological field such as the complex numbers or p-adic numbers. All of these groups are locally compact, so they have Haar measures and can be studied via harmonic analysis. The former offer an abstract formalism of invariant integrals. Invariance means, in the case of real numbers for example:
Try to answer this question if possible: What group operations must occur for group law and topology to integrate well?
Answer:
continuous functions