Portal:Mathematics
The Mathematics Portal
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.
Selected article  Selected picture  Did you know...  Topics in mathematics
Categories  WikiProjects  Things you can do  Index  Related portals
There are approximately 31,444 mathematics articles in Wikipedia.
Selected article
A Hilbert space is a real or complex vector space with a positivedefinite 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.
View all selected articles  Read More... 
Selected picture
Quicksort (also known as the partitionexchange 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 subarrays 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 subarrays; 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 subarrays longer than one element left to process. (As can be seen in the animation, the algorithm actually sorts all lefthand subarrays first, and then starts to process the righthand subarrays.) 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 worstcase O(n^{2}) 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 subarrays have shrunk to a sufficiently small length, and using a threeway partitioning scheme (grouping items into those smaller, larger, and equal to the pivot—a modification that can turn the worstcase scenario of allequal 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 subarrays can be done simultaneously). However, other sorting algorithms (including merge sort) experience much greater speed increases when performed in parallel.
Did you know...
 ...that in graph theory, a pseudoforest can contain trees and pseudotrees, but cannot contain any butterflies, diamonds, handcuffs, or bicycles?
 ...that it is not possible to configure two mutually inscribed quadrilaterals in the Euclidean plane, but the Möbius–Kantor graph describes a solution in the complex projective plane?
 ...that the six permutations of the vector (1,2,3) form a hexagon in 3D space, the 24 permutations of (1,2,3,4) form a truncated octahedron in four dimensions, and both are examples of permutohedra?
 ...that the Rule 184 cellular automaton can simultaneously model the behavior of cars moving in traffic, the accumulation of particles on a surface, and particleantiparticle annihilation reactions?
 ...that a cyclic cellular automaton is a system of simple mathematical rules that can generate complex patterns mixing random chaos, blocks of color, and spirals?
 ...that a nonconvex polygon with three convex vertices is called a pseudotriangle?
 ...that the axiom of choice is logically independent of the other axioms of Zermelo–Fraenkel set theory?
WikiProjects
The Mathematics WikiProject is the center for mathematicsrelated editing on Wikipedia. Join the discussion on the project's talk page.
Project pages
Essays
Subprojects
Related projects
Things you can do
Categories
Algebra  Arithmetic  Analysis  Complex analysis  Applied mathematics  Calculus  Category theory  Chaos theory  Combinatorics  Dynamic systems  Fractals  Game theory  Geometry  Algebraic geometry  Graph theory  Group theory  Linear algebra  Mathematical logic  Model theory  Multidimensional geometry  Number theory  Numerical analysis  Optimization  Order theory  Probability and statistics  Set theory  Statistics  Topology  Algebraic topology  Trigonometry  Linear programming
Mathematics (books)  History of mathematics  Mathematicians  Awards  Education  Literature  Notation  Organizations  Theorems  Proofs  Unsolved problems
Topics in mathematics
General  Foundations  Number theory  Discrete mathematics 



Algebra  Analysis  Geometry and topology  Applied mathematics 
Index of mathematics articles
ARTICLE INDEX:  A B C D E F G H I J K L M N O P Q R S T U V W X Y Z (0–9) 
MATHEMATICIANS:  A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 
Related portals
Algebra  Analysis  Category theory 
Computer science 
Cryptography  Discrete mathematics 
Geometry 
Logic  Mathematics  Number theory 
Physics  Science  Set theory  Statistics  Topology 
In other Wikimedia projects
Return to Fuhz Home  This article covering Portal:Mathematics is enhanced for the visually impaired.
The text of this Fuhz article is released under the GNU Free Documentation License
Privacy Policy  Latest Page William Tragni SCAM