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Dyadic Cubes
In mathematics, the dyadic cubes are a collection of cubes in R''n'' of different sizes or scales such that the set of cubes of each scale partition R''n'' and each cube in one scale may be written as a union of cubes of a smaller scale. These are frequently used in mathematics (particularly harmonic analysis) as a way of discretizing objects in order to make computations or analysis easier. For example, to study an arbitrary subset of ''A'' of Euclidean space, one may instead replace it by a union of dyadic cubes of a particular size that cover the set. One can consider this set as a pixelized version of the original set, and as smaller cubes are used one gets a clearer image of the set ''A''. Most notable appearances of dyadic cubes include the Whitney extension theorem and the Calderón–Zygmund lemma. Dyadic cubes in Euclidean space In Euclidean space, dyadic cubes may be constructed as follows: for each integer ''k'' let Δ''k'' be the set of cubes in R''n'' of sidelength 2− ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy-Littlewood Maximal Inequality
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime'' is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger. However, it is unknown whether there are infinitely many twin primes (the so-called twin prime conjecture) or if there is a largest pair. The breakthrough work of Yitang Zhang in 2013, as well as work by James Maynard, Terence Tao and others, has made substantial progress towards proving that there are infinitely many twin primes, but at present this remains unsolved. Properties Usually the pair (2, 3) is not considered to be a pair of twin primes. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wavelet Transform
In mathematics, a wavelet series is a representation of a square-integrable (real number, real- or complex number, complex-valued) function (mathematics), function by a certain orthonormal series (mathematics), series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform. Definition A function \psi \,\in\, L^2(\mathbb) is called an orthonormal wavelet if it can be used to define a Hilbert space#Orthonormal bases, Hilbert basis, that is a orthonormal basis, complete orthonormal system, for the Hilbert space L^2\left(\mathbb\right) of Square-integrable function, square integrable functions. The Hilbert basis is constructed as the family of functions \ by means of Dyadic transformation, dyadic translation (geometry), translations and dilation (operator theory), dilations of \psi\,, :\psi_(x) = 2^\frac \psi\left(2^jx - k\right)\, for integers j,\, k \,\in\, \mathbb. If under the standard ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadtree
A quadtree is a tree data structure in which each internal node has exactly four children. Quadtrees are the two-dimensional analog of octrees and are most often used to partition a two-dimensional space by recursively subdividing it into four quadrants or regions. The data associated with a leaf cell varies by application, but the leaf cell represents a "unit of interesting spatial information". The subdivided regions may be square or rectangular, or may have arbitrary shapes. This data structure was named a quadtree by Raphael Finkel and J.L. Bentley in 1974. A similar partitioning is also known as a ''Q-tree''. All forms of quadtrees share some common features: * They decompose space into adaptable cells * Each cell (or bucket) has a maximum capacity. When maximum capacity is reached, the bucket splits * The tree directory follows the spatial decomposition of the quadtree. A tree-pyramid (T-pyramid) is a "complete" tree; every node of the T-pyramid has four child nodes excep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doubling Measures
In mathematics, a metric space with metric is said to be doubling if there is some doubling constant such that for any and , it is possible to cover the ball with the union of at most balls of radius . The base-2 logarithm of is called the doubling dimension of . Euclidean spaces \mathbb^d equipped with the usual Euclidean metric are examples of doubling spaces where the doubling constant depends on the dimension . For example, in one dimension, ; and in two dimensions, . In general, Euclidean space \mathbb^d has doubling dimension \Theta(d). Assouad's embedding theorem An important question in metric space geometry is to characterize those metric spaces that can be embedded in some Euclidean space by a bi-Lipschitz function. This means that one can essentially think of the metric space as a subset of Euclidean space. Not all metric spaces may be embedded in Euclidean space. Doubling metric spaces, on the other hand, would seem like they have more of a chance, since ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Spaces
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vitali Covering Lemma
In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem. The covering theorem is credited to the Italian mathematician Giuseppe Vitali.. The theorem states that it is possible to cover, up to a Lebesgue-negligible set, a given subset ''E'' of R''d'' by a disjoint family extracted from a ''Vitali covering'' of ''E''. Vitali covering lemma There are two basic version of the lemma, a finite version and an infinite version. Both lemmas can be proved in the general setting of a metric space, typically these results are applied to the special case of the Euclidean space \mathbb^d. In both theorems we will use the following notation: if B = B(x,r) is a ball and c \in \mathbb, we will write cB for the ball B(x,cr). Finite version Theorem (Finite Covering Lemma). Let B_, \dots, B_ be any finite coll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integrable
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, such that its behaviour has far fewer degrees of freedom than the dimensionality of its phase space; that is, its evolution is restricted to a submanifold within its phase space. Three features are often referred to as characterizing integrable systems: * the existence of a ''maximal'' set of conserved quantities (the usual defining property of complete integrability) * the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability) * the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability) Integrable systems may be seen as very different in qualitative character from more ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Integrable Function
In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar to spaces, but its members are not required to satisfy any growth restriction on their behavior at the boundary of their domain (at infinity if the domain is unbounded): in other words, locally integrable functions can grow arbitrarily fast at the domain boundary, but are still manageable in a way similar to ordinary integrable functions. Definition Standard definition .See for example and . Let be an open set in the Euclidean space \mathbb^n and be a Lebesgue measurable function. If on is such that : \int_K , f , \, \mathrmx <+\infty, i.e. its |
Ball (mathematics)
In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ''ball'' in dimensions is called a hyperball or -ball and is bounded by a ''hypersphere'' or ()-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment. In other contexts, such as in Euclidean geometry and informal use, ''sphere'' is sometimes used to mean ''ball''. In the field of topology the closed n-dimensional ball is often denoted as B^n or D^n while the open n-dimensional ball is \operatorname B^n or \ope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron a 3-zonohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron. It has cubical or octahedral symmetry. The cube is the only convex polyhedron whose faces are all squares. Orthogonal projections The ''cube'' has four special orthogonal projections, centered, on a vertex, edges, face and normal to its vertex figure. The first and third correspond to the A2 and B2 Coxeter planes. Spherical tiling The cube can also be represented as a spherical tiling, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
