Hua's Lemma
In mathematics, Hua's lemma, named for Hua Loo-keng, is an estimate for exponential sums. It states that if ''P'' is an 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 ... 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 {{mathanalysis-stub ...
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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 ...
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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 National People's Congress, standing Committee of the first to sixth National people's Congress, Vice-Chairman of the sixth National Committee of the Chinese People's Political Consultative Conference (April 1985) and Vice-Chairman of the China Democratic League (1979). He joined the ...
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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 va ...
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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 : \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., the bi ...
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Lemmas
Lemma 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 type of proposition Other uses * ''Lemma'' (album), by John Zorn (2013) * Lemma (logic), an informal contention 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 nucleated cytoplasmic layer of Schwann cells (also called neurilemmocytes) that surrounds the axon of the neuron. It forms the outermost layer of the ne ...