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Portal:Mathematics

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Mathematics is the study of numbers, quantity, space, structure, and change. Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.

There are approximately 31,444 mathematics articles in Wikipedia.

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A Hilbert space is a real or complex vector space with a positive-definite Hermitian form, that is complete under its norm. Thus it is an inner product space, which means that it has notions of distance and of angle (especially the notion of orthogonality or perpendicularity). The completeness requirement ensures that for infinite dimensional Hilbert spaces the limits exist when expected, which facilitates various definitions from calculus. A typical example of a Hilbert space is the space of square summable sequences.

Hilbert spaces allow simple geometric concepts, like projection and change of basis to be applied to infinite dimensional spaces, such as function spaces. They provide a context with which to formalize and generalize the concepts of the Fourier series in terms of arbitrary orthogonal polynomials and of the Fourier transform, which are central concepts from functional analysis. Hilbert spaces are of crucial importance in the mathematical formulation of quantum mechanics.

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animation of dots of varying heights being sorted by height using the quicksort algorithm
Credit: RolandH

Quicksort (also known as the partition-exchange sort) is an efficient sorting algorithm that works for items of any type for which a total order (i.e., "≤") relation is defined. This animation shows how the algorithm partitions the input array (here a random permutation of the numbers 1 through 33) into two smaller arrays based on a selected pivot element (bar marked in red, here always chosen to be the last element in the array under consideration), by swapping elements between the two sub-arrays so that those in the first (on the left) end up all smaller than the pivot element's value (horizontal blue line) and those in the second (on the right) all larger. The pivot element is then moved to a position between the two sub-arrays; at this point, the pivot element is in its final position and will never be moved again. The algorithm then proceeds to recursively apply the same procedure to each of the smaller arrays, partitioning and rearranging the elements until there are no sub-arrays longer than one element left to process. (As can be seen in the animation, the algorithm actually sorts all left-hand sub-arrays first, and then starts to process the right-hand sub-arrays.) First developed by Tony Hoare in 1959, quicksort is still a commonly used algorithm for sorting in computer applications. On the average, it requires O(n log n) comparisons to sort n items, which compares favorably to other popular sorting methods, including merge sort and heapsort. Unfortunately, on rare occasions (including cases where the input is already sorted or contains items that are all equal) quicksort requires a worst-case O(n2) comparisons, while the other two methods remain O(n log n) in their worst cases. Still, when implemented well, quicksort can be about two or three times faster than its main competitors. Unlike merge sort, the standard implementation of quicksort does not preserve the order of equal input items (it is not stable), although stable versions of the algorithm do exist at the expense of requiring O(n) additional storage space. Other variations are based on different ways of choosing the pivot element (for example, choosing a random element instead of always using the last one), using more than one pivot, switching to an insertion sort when the sub-arrays have shrunk to a sufficiently small length, and using a three-way partitioning scheme (grouping items into those smaller, larger, and equal to the pivot—a modification that can turn the worst-case scenario of all-equal input values into the best case). Because of the algorithm's "divide and conquer" approach, parts of it can be done in parallel (in particular, the processing of the left and right sub-arrays can be done simultaneously). However, other sorting algorithms (including merge sort) experience much greater speed increases when performed in parallel.

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