Mathematical diagrams, such ascharts andgraphs, are mainly designed to conveymathematical relationships—for example, comparisons over time.[1]
Acomplex number can be visually represented as a pair of numbers forming a vector on a diagram called anArgand diagram.Thecomplex plane is sometimes called theArgand plane because it is used inArgand diagrams. These are named afterJean-Robert Argand (1768–1822), although they were first described by Norwegian-Danish land surveyor and mathematicianCaspar Wessel (1745–1818).[2] Argand diagrams are frequently used to plot the positions of thepoles andzeroes of afunction in the complex plane.
The concept of the complex plane allows ageometric interpretation of complex numbers. Underaddition, they add likevectors. Themultiplication of two complex numbers can be expressed most easily inpolar coordinates — the magnitude ormodulus of the product is the product of the twoabsolute values, or moduli, and the angle orargument of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation.
In the context offast Fourier transform algorithms, abutterfly is a portion of the computation that combines the results of smallerdiscrete Fourier transforms (DFTs) into a larger DFT, or vice versa (breaking a larger DFT up into subtransforms). The name "butterfly" comes from the shape of the data-flow diagram in the radix-2 case, as described below. The same structure can also be found in theViterbi algorithm, used for finding the most likely sequence of hidden states.
Thebutterfly diagram show a data-flow diagram connecting the inputsx (left) to the outputsy that depend on them (right) for a "butterfly" step of a radix-2Cooley–Tukey FFT algorithm. This diagram resembles abutterfly as in theMorpho butterfly shown for comparison, hence the name.
In mathematics, and especially incategory theory, a commutative diagram is a diagram ofobjects, also known as vertices, andmorphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition.
Commutative diagrams play the role in category theory that equations play in algebra.
AHasse diagram is a simple picture of a finitepartially ordered set, forming adrawing of the partial order'stransitive reduction. Concretely, one represents each element of the set as a vertex on the page and draws a line segment or curve that goes upward fromx toy precisely whenx <y and there is noz such thatx <z <y. In this case, we say ycovers x, or y is an immediate successor of x. In a Hasse diagram, it is required that the curves be drawn so that each meets exactly two vertices: its two endpoints. Any such diagram (given that the vertices are labeled) uniquely determines a partial order, and any partial order has a unique transitive reduction, but there are many possible placements of elements in the plane, resulting in different Hasse diagrams for a given order that may have widely varying appearances.
InKnot theory a useful way to visualise and manipulate knots is to project the knot onto a plane — to imagine the knot casting a shadow on the wall. A small perturbation in the choice of projection will ensure that it isone-to-one except at the double points, calledcrossings, where the "shadow" of the knot crosses itself once transversely.[3]
At each crossing we must indicate which section is "over" and which is "under", so as to be able to recreate the original knot. This is often done by creating a break in the strand going underneath. If by following the diagram the knot alternately crosses itself "over" and "under", then the diagram represents a particularly well-studied class of knot,alternating knots.
AVenn diagram is a representation of mathematical sets: a mathematical diagram representing sets as circles, with their relationships to each other expressed through their overlapping positions, so that all possible relationships between the sets are shown.[4]
The Venn diagram is constructed with a collection of simple closed curves drawn in the plane. The principle of these diagrams is that classes be represented by regions in such relation to one another that all the possible logical relations of these classes can be indicated in the same diagram. That is, the diagram initially leaves room for any possible relation of the classes, and the actual or given relation, can then be specified by indicating that some particular region is null or is not null.[5]
AVoronoi diagram is a special kind of decomposition of ametric space determined by distances to a specified discrete set of objects in the space, e.g., by adiscrete set of points. This diagram is named afterGeorgy Voronoi, also called a Voronoitessellation, a Voronoi decomposition, or a Dirichlet tessellation afterPeter Gustav Lejeune Dirichlet.
In the simplest case, we are given a set of points S in the plane, which are the Voronoi sites. Each site s has a Voronoi cell V(s) consisting of all points closer to s than to any other site. The segments of the Voronoi diagram are all the points in the plane that are equidistant to two sites. The Voronoi nodes are the points equidistant to three (or more) sites.
Awallpaper group orplane symmetry group orplane crystallographic group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art. There are 17 possible distinctgroups.
Wallpaper groups are two-dimensionalsymmetry groups, intermediate in complexity between the simplerfrieze groups and the three-dimensionalcrystallographic groups, also calledspace groups. Wallpaper groups categorize patterns by their symmetries. Subtle differences may place similar patterns in different groups, while patterns which are very different in style, color, scale or orientation may belong to the same group.
AYoung diagram orYoung tableau, also calledFerrers diagram, is a finite collection of boxes, or cells, arranged in left-justified rows, with the row sizes weakly decreasing (each row has the same or shorter length than its predecessor).
Listing the number of boxes in each row gives apartition (lambda) of a positive integern, the total number of boxes of the diagram. The Young diagram is said to be of shape, and it carries the same information as that partition. Listing the number of boxes in each column gives another partition, theconjugate ortranspose partition of; one obtains a Young diagram of that shape by reflecting the original diagram along its main diagonal.
Young tableaux were introduced byAlfred Young, amathematician atCambridge University, in 1900. They were then applied to the study of symmetric group byGeorg Frobenius in 1903. Their theory was further developed by many mathematicians.
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