Manifold

The surface of the Earth requires (at least) two charts to include every point. Here the globe is decomposed into charts around the North and South Poles. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows more complicated structures to be described and understood in terms of the relatively well-understood properties of Euclidean space. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. Manifolds may have additional features. One important class of manifolds is the class of differentiable manifolds. Motivational examples[edit] Circle[edit] Figure 1: The four charts each map part of the circle to an open interval, and together cover the whole circle. The top and right charts overlap: their intersection lies in the quarter of the circle where both the x- and the y-coordinates are positive. Figure 2: A circle manifold chart based on slope, covering all but one point of the circle. and Related: .caisson test

Curse of dimensionality The curse of dimensionality refers to various phenomena that arise when analyzing and organizing data in high-dimensional spaces (often with hundreds or thousands of dimensions) that do not occur in low-dimensional settings such as the three-dimensional physical space of everyday experience. The term curse of dimensionality was coined by Richard E. Bellman when considering problems in dynamic optimization.[1][2] The "curse of dimensionality" depends on the algorithm[edit] The "curse of dimensionality" is not a problem of high-dimensional data, but a joint problem of the data and the algorithm being applied. When facing the curse of dimensionality, a good solution can often be found by changing the algorithm, or by pre-processing the data into a lower-dimensional form. Curse of dimensionality in different domains[edit] Combinatorics[edit] , exponential in the dimensionality. Sampling[edit] Optimization[edit] Machine learning[edit] Bayesian statistics[edit] Distance functions[edit] and dimension .

Vector space Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w. An example of a vector space is that of Euclidean vectors, which may be used to represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dimensional space also form vector spaces. Vectors in vector spaces do not necessarily have to be arrow-like objects as they appear in the mentioned examples: vectors are best thought of as abstract mathematical objects with particular properties, which in some cases can be visualized as arrows. Introduction and definition[edit] The concept of vector space will first be explained by describing two particular examples: and

Dimensions in Philosophy Fourth dimension in literature The idea of a fourth dimension has been a factor in the evolution of modern art, but use of concepts relating to higher dimensions has been little discussed by academics in the literary world.[1] From the late 1800s onwards, many writers began to make use of possibilities opened up by the exploration of such concepts as hypercubes and non-Euclidian geometry. While many writers took the fourth dimension to be one of time (as it is commonly considered today), others preferred to think of it in spatial terms, and some associated the new mathematics with wider changes in modern culture. Early influence[edit] Theoretical physicist James Clerk Maxwell is best known for his work in formulating the equations of electromagnetism. He was also a prize-winning poet,[2] and in his last poem Paradoxical Ode; Maxwell muses on connections between science, religion and nature, touching upon higher-dimensions along the way:[3] "..shall we stay our upward course? H.G. Other works[edit] References[edit]

Networks and dimension Dimension A diagram showing the first four spatial dimensions. 1-D: Two points A and B can be connected to a line, giving a new line segment AB. 2-D: Two parallel line segments AB and CD can be connected to become a square, with the corners marked as ABCD. 3-D: Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH. 4-D: Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube, with the corners marked as ABCDEFGHIJKLMNOP. In physical terms, dimension refers to the constituent structure of all space (cf. volume) and its position in time (perceived as a scalar dimension along the t-axis), as well as the spatial constitution of objects within—structures that correlate with both particle and field conceptions, interact according to relative properties of mass—and are fundamentally mathematical in description. The concept of dimension is not restricted to physical objects. A tesseract is an example of a four-dimensional object.

Function space Examples[edit] Function spaces appear in various areas of mathematics: Functional analysis[edit] Functional analysis is organized around adequate techniques to bring function spaces as topological vector spaces within reach of the ideas that would apply to normed spaces of finite dimension. Bibliography[edit] Kolmogorov, A. See also[edit] Hilbert space The state of a vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. Definition and illustration[edit] Motivating example: Euclidean space[edit] One of the most familiar examples of a Hilbert space is the Euclidean space consisting of three-dimensional vectors, denoted by R3, and equipped with the dot product. The dot product satisfies the properties: Completeness means that if a particle moves along the broken path (in blue) travelling a finite total distance, then the particle has a well-defined net displacement (in orange). Definition[edit]

Level II: Universes with different physical constants Level IV: Ultimate ensemble The quilted universe