Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size deciding. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, and combinatorics also has many applications in optimization, computer science, ergodic theory and statistical physics. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century however powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is graph theory, which also has numerous natural connections to other areas. Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms. Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. In elementary number theory, study integers without use of techniques from other mathematical fields. Questions of divisibility, use of the Euclidean algorithm to compute greatest common divisors, integer factorizations into prime numbers, investigation of perfect numbers and congruences belong here. Several important discoveries of this field are Fermat's little theorem, Euler's theorem, the Chinese remainder theorem and the law of quadratic reciprocity. The properties of multiplicative functions such as the Möbius function and Euler's φ function, integer sequences, factorials, and Fibonacci numbers all also fall into this area.

Gasoline Engines

Gasoline engines have the advantage over diesel in being lighter and able to work at higher rotational speeds and they are the usual choice for fitting in high-performance sports cars. Continuous development of gasoline engines for over a hundred years has produced improvements in efficiency and reduced pollution. The carburetor was used on nearly all road car engines until the 1980s but it was long realized better control of the fuel/air mixture could be achieved with fuel injection. Indirect fuel injection was first used in aircraft engines from 1909, in racing car engines from the 1930s, and road cars from the late 1950s.

Gasoline Direct Injection (GDI) is now starting to appear in production vehicles such as the 2007 BMW MINI. Exhaust gases are also cleaned up by fitting a catalytic converter into the exhaust system. Clean air legislation in many of the car industries most important markets has made both catalysts and fuel injection virtually universal fittings. Most modern gasoline engines are also capable of running with up to 15% ethanol mixed into the gasoline - older vehicles may have seals and hoses that can be harmed by ethanol. With a small amount of redesign, gasoline-powered vehicles can run on ethanol concentrations as high as 85%. 100% ethanol is used in some parts of the world (such as Brazil), but vehicles must be started on pure gasoline and switched over to ethanol once the engine is running. Most gasoline engines cars can also run on LPG with the addition of an LPG tank for fuel storage and carburetion modifications to add an LPG mixer. LPG produces fewer toxic emissions and is a popular fuel for fork lift trucks that have to operate inside buildings.