|
Rising Sun Lemma
In mathematical analysis, the rising sun lemma is a lemma due to Frigyes Riesz Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> mathema ..., used in the proof of the Hardy–Littlewood maximal theorem. The lemma was a precursor in one dimension of the Calderón–Zygmund lemma. The lemma is stated as follows:See: * * * * :Suppose ''g'' is a real-valued continuous function on the interval 'a'',''b''and ''S'' is the set of ''x'' in 'a'',''b''such that there exists a ''y''∈(''x'',''b''] with ''g''(''y'') > ''g''(''x''). (Note that ''b'' cannot be in ''S'', though ''a'' may be.) Define ''E'' = ''S'' ∩ (''a'',''b''). :Then ''E'' is an open set, and it may be written as a countable union of disjoint intervals ::E=\bigcup_k (a_k,b_k) :such that ''g''(''a''''k'') = ''g''(''b''''k''), unle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rising Sun Lemma
In mathematical analysis, the rising sun lemma is a lemma due to Frigyes Riesz Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> mathema ..., used in the proof of the Hardy–Littlewood maximal theorem. The lemma was a precursor in one dimension of the Calderón–Zygmund lemma. The lemma is stated as follows:See: * * * * :Suppose ''g'' is a real-valued continuous function on the interval 'a'',''b''and ''S'' is the set of ''x'' in 'a'',''b''such that there exists a ''y''∈(''x'',''b''] with ''g''(''y'') > ''g''(''x''). (Note that ''b'' cannot be in ''S'', though ''a'' may be.) Define ''E'' = ''S'' ∩ (''a'',''b''). :Then ''E'' is an open set, and it may be written as a countable union of disjoint intervals ::E=\bigcup_k (a_k,b_k) :such that ''g''(''a''''k'') = ''g''(''b''''k''), unle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemma (mathematics)
In mathematics, informal logic and argument mapping, a lemma (plural lemmas or lemmata) is a generally minor, proven proposition which is used as a stepping stone to a larger result. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem". In many cases, a lemma derives its importance from the theorem it aims to prove; however, a lemma can also turn out to be more important than originally thought. The word "lemma" derives from the Ancient Greek ("anything which is received", such as a gift, profit, or a bribe). Comparison with theorem There is no formal distinction between a lemma and a theorem, only one of intention (see Theorem terminology). However, a lemma can be considered a minor result whose sole purpose is to help prove a more substantial theorem – a step in the direction of proof. Well-known lemmas A good stepping stone can lead to many others. Some powerful results in mathematics are known as lemmas, first named for their originally min ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frigyes Riesz
Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> mathematician who made fundamental contributions to functional analysis, as did his younger brother Marcel Riesz. Life and career He was born into a Jewish family in Győr, Austria-Hungary and died in Budapest, Hungary. Between 1911 and 1919 he was a professor at the Franz Joseph University in Kolozsvár, Austria-Hungary. The post-WW1 Treaty of Trianon transferred former Austro-Hungarian territory including Kolozsvár to the Kingdom of Romania, whereupon Kolozsvár's name changed to Cluj and the University of Kolozsvár moved to Szeged, Hungary, becoming the University of Szeged. Then, Riesz was the rector and a professor at the University of Szeged, as well as a member of the Hungarian Academy of Sciences. and the Polish Academy of Learning. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calderón–Zygmund Lemma
In mathematics, the Calderón–Zygmund lemma is a fundamental result in Fourier analysis, harmonic analysis, and singular integrals. It is named for the mathematicians Alberto Calderón and Antoni Zygmund. Given an integrable function , where denotes Euclidean space and denotes the complex numbers, the lemma gives a precise way of partitioning into two sets: one where is essentially small; the other a countable collection of cubes where is essentially large, but where some control of the function is retained. This leads to the associated Calderón–Zygmund decomposition of , wherein is written as the sum of "good" and "bad" functions, using the above sets. Covering lemma Let be integrable and be a positive constant. Then there exists an open set such that: :(1) is a disjoint union of open cubes, , such that for each , ::\alpha\le \frac \int_ , f(x), \, dx \leq 2^d \alpha. :(2) almost everywhere in the complement of . Calderón–Zygmund decomposition Given as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The American Mathematical Society
''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. As a requirement, all articles must be at most 15 printed pages. According to the ''Journal Citation Reports'', the journal has a 2018 impact factor of 0.813. Scope ''Proceedings of the American Mathematical Society'' publishes articles from all areas of pure and applied mathematics, including topology, geometry, analysis, algebra, number theory, combinatorics, logic, probability and statistics. Abstracting and indexing This journal is indexed in the following databases: 2011. American Mathematical Society. * |
Graduate Studies In Mathematics
Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published ihardcoverane-bookformats. List of books *1 ''The General Topology of Dynamical Systems'', Ethan Akin (1993, ) *2 ''Combinatorial Rigidity'', Jack Graver, Brigitte Servatius, Herman Servatius (1993, ) *3 ''An Introduction to Gröbner Bases'', William W. Adams, Philippe Loustaunau (1994, ) *4 ''The Integrals of Lebesgue, Denjoy, Perron, and Henstock'', Russell A. Gordon (1994, ) *5 ''Algebraic Curves and Riemann Surfaces'', Rick Miranda (1995, ) *6 ''Lectures on Quantum Groups'', Jens Carsten Jantzen (1996, ) *7 ''Algebraic Number Fields'', Gerald J. Janusz (1996, 2nd ed., ) *8 ''Discovering Modern Set Theory. I: The Basics'', Winfried Just, Martin Weese (1996, ) *9 ''An Invitation to Arithmetic Geometry'', Dino Lorenzini (1996, ) *10 ''Representations of Finite and Compact Groups'', Barry Simon (199 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability. Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions. Scope Construction of the real numbers The theorems of real analysis rely on the properties of the real number system, which must be established. The real number system consists of an uncountable set (\mathbb), together with two binary operations denoted and , and an order denoted . The operations make the real numbers a field, and, along with the order, an ordered field. The real number system is the unique ''complete ordered field'', in the sense that any other complete ordered field is isomorphic to it. Intuitively, completeness means ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
