Hua's Lemma

	Hua's Lemma In mathematics, Hua's lemma, named for Hua Loo-keng, is an estimate for exponential sum In mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually expressed by means of the function :e(x) = \exp(2\pi ix).\, Therefore, a typi ...s. It states that if ''P'' is an integral-valued polynomial of degree ''k'', \varepsilon is a positive real number, and ''f'' a real function defined by :f(\alpha)=\sum_^N\exp(2\pi iP(x)\alpha), then :\int_0^1, f(\alpha), ^\lambda d\alpha\ll_ N^, where (\lambda,\mu(\lambda)) lies on a polygonal line with vertices :(2^\nu,2^\nu-\nu+\varepsilon),\quad\nu=1,\ldots,k. References Lemmas Analytic number theory Hua Luogeng {{mathanalysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Hua Loo-keng Hua Luogeng or Hua Loo-Keng (; 12 November 1910 – 12 June 1985) was a Chinese mathematician and politician famous for his important contributions to number theory and for his role as the leader of mathematics research and education in the People's Republic of China. He was largely responsible for identifying and nurturing the renowned mathematician Chen Jingrun who proved Chen's theorem, the best known result on the Goldbach conjecture. In addition, Hua's later work on mathematical optimization and operations research made an enormous impact on China's economy. He was elected a foreign associate of the US National Academy of Sciences in 1982. He was elected a member of the Standing Committee of the 1st through 6th National People's Congresses, Vice-Chairman of the 6th National Committee of the Chinese People's Political Consultative Conference (April 1985) and vice-chairman of the China Democratic League (1979). He joined the Chinese Communist Party in 1979. Hua did not recei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Exponential Sum In mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually expressed by means of the function :e(x) = \exp(2\pi ix).\, Therefore, a typical exponential sum may take the form :\sum_n e(x_n), summed over a finite sequence of real numbers ''x''''n''. Formulation If we allow some real coefficients ''a''''n'', to get the form :\sum_n a_n e(x_n) it is the same as allowing exponents that are complex numbers. Both forms are certainly useful in applications. A large part of twentieth century analytic number theory was devoted to finding good estimates for these sums, a trend started by basic work of Hermann Weyl in diophantine approximation. Estimates The main thrust of the subject is that a sum :S=\sum_n e(x_n) is ''trivially'' estimated by the number ''N'' of terms. That is, the absolute value :, S, \le N\, by the triangle inequality, since each summand has absolute v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Integral-valued Polynomial In mathematics, an integer-valued polynomial (also known as a numerical polynomial) P(t) is a polynomial whose value P(n) is an integer for every integer ''n''. Every polynomial with integer coefficients is integer-valued, but the converse is not true. For example, the polynomial : P(t) = \frac t^2 + \frac t=\fract(t+1) takes on integer values whenever ''t'' is an integer. That is because one of ''t'' and t + 1 must be an even number. (The values this polynomial takes are the triangular numbers.) Integer-valued polynomials are objects of study in their own right in algebra, and frequently appear in algebraic topology.. See in particular pp. 213–214. Classification The class of integer-valued polynomials was described fully by . Inside the polynomial ring \Q /math> of polynomials with rational number coefficients, the subring of integer-valued polynomials is a free abelian group. It has as basis the polynomials :P_k(t) = t(t-1)\cdots (t-k+1)/k! for k = 0,1,2, \dots, i.e., ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Lemmas Lemma (from Ancient Greek ''premise'', ''assumption'', from Greek ''I take'', ''I get'') may refer to: Language and linguistics * Lemma (morphology), the canonical, dictionary or citation form of a word * Lemma (psycholinguistics), a mental abstraction of a word about to be uttered Science and mathematics * Lemma (botany), a part of a grass plant * Lemma (mathematics), a proven proposition used as a step in a larger proof Other uses * ''Lemma'' (album), by John Zorn (2013) See also Analemma, a diagram showing the variation of the position of the Sun in the sky Dilemma Lema (other) Lemmatisation Neurolemma Neurilemma (also known as neurolemma, sheath of Schwann, or Schwann's sheath) is the outermost cell nucleus, nucleated cytoplasmic layer of Schwann cells (also called neurilemmocytes) that surrounds the axon of the neuron. It forms the outermost la ... , part of a neuron {{Disambiguation ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]
	Analytic Number Theory In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet ''L''-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem). Branches of analytic number theory Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique. * Multiplicative number theory deals with the distribution of the prime numbers, such as estimating the number of primes in an interval, and includes the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. Additive numb ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]