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John–Nirenberg Inequality
In harmonic analysis in mathematics, a function of bounded mean oscillation, also known as a BMO function, is a real-valued function whose mean oscillation is bounded (finite). The space of functions of bounded mean oscillation (BMO), is a function space that, in some precise sense, plays the same role in the theory of Hardy spaces ''Hp'' that the space ''L''∞ of essentially bounded functions plays in the theory of ''Lp''-spaces: it is also called John–Nirenberg space, after Fritz John and Louis Nirenberg who introduced and studied it for the first time. Historical note According to , the space of functions of bounded mean oscillation was introduced by in connection with his studies of mappings from a bounded set \Omega belonging to \mathbb^n into \mathbb^n and the corresponding problems arising from elasticity theory, precisely from the concept of elastic strain: the basic notation was introduced in a closely following paper by , where several properties of this function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Analysis
Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded domains such as the full real line or by Fourier series for functions on bounded domains, especially periodic functions on finite intervals. Generalizing these transforms to other domains is generally called Fourier analysis, although the term is sometimes used interchangeably with harmonic analysis. Harmonic analysis has become a vast subject with applications in areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis, spectral analysis, and neuroscience. The term "harmonics" originated from the Ancient Greek word ''harmonikos'', meaning "skilled in music". In physical eigenvalue problems, it began to mean waves whose frequencies are integer multiples of one another, as are the freq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Akihito Uchiyama
Akihito (born 23 December 1933) is a member of the Imperial House of Japan who reigned as the 125th emperor of Japan from 1989 until 2019 Japanese imperial transition, his abdication in 2019. The era of his rule was named the Heisei era, Heisei being an expression of achieving peace worldwide. Born in 1933, Akihito is the fifth child and first son of Hirohito, Emperor Shōwa and Empress Nagako, Empress Kōjun. During the Second World War, he moved out of Tokyo with his classmates and remained in Nikkō until 1945. In 1952, his Coming-of-Age ceremony and investiture as crown prince were held, and he began to undertake official duties in his capacity as crown prince. The next year, he made his first journey overseas and represented Japan at the coronation of Elizabeth II in London. He completed his university education in 1956. In April 1959, he married Empress Michiko, Michiko Shōda, a commoner; it was the first imperial wedding to be televised in Japan, drawing about 15 mill ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Domain (mathematical Analysis)
In mathematical analysis, a domain or region is a non-empty, connected, and open set in a topological space. In particular, it is any non-empty connected open subset of the real coordinate space or the complex coordinate space . A connected open subset of coordinate space is frequently used for the domain of a function. The basic idea of a connected subset of a space dates from the 19th century, but precise definitions vary slightly from generation to generation, author to author, and edition to edition, as concepts developed and terms were translated between German, French, and English works. In English, some authors use the term ''domain'', some use the term ''region'', some use both terms interchangeably, and some define the two terms slightly differently; some avoid ambiguity by sticking with a phrase such as ''non-empty connected open subset''. Conventions One common convention is to define a ''domain'' as a connected open set but a ''region'' as the union of a domain w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integration (mathematics)
In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. the other being differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter. A definite integral computes the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an ''antiderivative'', a function whose deri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norm (mathematics)
In mathematics, a norm is a function (mathematics), function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the Origin (mathematics), origin: it Equivariant map, commutes with scaling, obeys a form of the triangle inequality, and zero is only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the #Euclidean norm, Euclidean norm, the #p-norm, 2-norm, or, sometimes, the magnitude or length of the vector. This norm can be defined as the square root of the inner product of a vector with itself. A seminorm satisfies the first two properties of a norm but may be zero for vectors other than the origin. A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a ''seminormed vector space''. The term pseudonorm has been used for several related meaning ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Mean Oscillation
In harmonic analysis in mathematics, a function of bounded mean oscillation, also known as a BMO function, is a real-valued function whose mean oscillation is bounded (finite). The space of functions of bounded mean oscillation (BMO), is a function space that, in some precise sense, plays the same role in the theory of Hardy spaces ''Hp'' that the space ''L''∞ of essentially bounded functions plays in the theory of ''Lp''-spaces: it is also called John–Nirenberg space, after Fritz John and Louis Nirenberg who introduced and studied it for the first time. Historical note According to , the space of functions of bounded mean oscillation was introduced by in connection with his studies of mappings from a bounded set \Omega belonging to \mathbb^n into \mathbb^n and the corresponding problems arising from elasticity theory, precisely from the concept of elastic strain: the basic notation was introduced in a closely following paper by , where several properties of this function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supremum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, and if ''b'' is a lower bound of S, then ''b'' is less than or equal to the infimum of S. Consequently, the term ''greatest lower bound'' (abbreviated as ) is also commonly used. The supremum (abbreviated sup; : suprema) of a subset S of a partially ordered set P is the least element in P that is greater than or equal to each element of S, if such an element exists. If the supremum of S exists, it is unique, and if ''b'' is an upper bound of S, then the supremum of S is less than or equal to ''b''. Consequently, the supremum is also referred to as the ''least upper bound'' (or ). The infimum is, in a precise sense, dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it coincides with the standard measure of length, area, or volume. In general, it is also called '-dimensional volume, '-volume, hypervolume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by \lambda(A). Henri Lebesgue described this measure in the year 1901 which, a year after, was followed up by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902. Definition For any interval I = ,b/math>, or I = (a, b), in the set \mathbb of real numbers, let \ell(I)= b - a denote its length. For any subset E\subseteq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volume (mathematics)
Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The definition of length and height (cubed) is interrelated with volume. The volume of a container is generally understood to be the capacity of the container; i.e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. By metonymy, the term "volume" sometimes is used to refer to the corresponding region (e.g., bounding volume). In ancient times, volume was measured using similar-shaped natural containers. Later on, standardized containers were used. Some simple three-dimensional shapes can have their volume easily calculated using arithmetic formulas. Volumes of more complicated shapes can be calculated with integral calculus if a formula exists for the shape's bou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. the other being Derivative, differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter. A definite integral computes the signed area of the region in the plane that is bounded by the Graph of a function, graph of a given Function (mathematics), function between two points in the real line. Conventionally, areas above the horizontal Coordinate axis, axis of the plane are positive while areas below are n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval (mathematics)
In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real interval can contain neither endpoint, either endpoint, or both endpoints, excluding any endpoint which is infinite. For example, the set of real numbers consisting of , , and all numbers in between is an interval, denoted and called the unit interval; the set of all positive real numbers is an interval, denoted ; the set of all real numbers is an interval, denoted ; and any single real number is an interval, denoted . Intervals are ubiquitous in mathematical analysis. For example, they occur implicitly in the epsilon-delta definition of continuity; the intermediate value theorem asserts that the image of an interval by a continuous function is an interval; integrals of real functions are defined over an interval; etc. Interval ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square (geometry)
In geometry, a square is a regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal sides. As with all rectangles, a square's angles are right angles (90 degrees, or /2 radians), making adjacent sides perpendicular. The area of a square is the side length multiplied by itself, and so in algebra, multiplying a number by itself is called squaring. Equal squares can tile the plane edge-to-edge in the square tiling. Square tilings are ubiquitous in tiled floors and walls, graph paper, image pixels, and game boards. Square shapes are also often seen in building floor plans, origami paper, food servings, in graphic design and heraldry, and in instant photos and fine art. The formula for the area of a square forms the basis of the calculation of area and motivates the search for methods for squaring the circle by compass and straightedge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
