Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation
imc2000-day2-solutions(IMC国际数学竞赛题目2000第2天)

y = a0x4 + a1x3 + a2x2 + a3x + a4 x2 + a5x,
where the equation y = a5x determines the straight line A1A3. The abscissas of the points A1 and A3 are −a and a, a > 0, respectively. Since −a and a are points of tangency, the numbers −a and a must be double roots of the polynomial a0x4 + a1x3 + a2x2 + a3x + a4. It follows that the polynomial is of the form
µ(p, c) − µ(p c) = 1 or µ(p − 1, c) − µ(p c) = 1,
(3)
respectively. Putting (1), (2) and (3) together we obtain
S0 + S1 =
µ(p, c) − µ(p , c) +
µ(p − 1, c) − µ(p , c) =
Now observe that any dissection of a cube into n smaller cubes may be refined to give a dissection into n + (ad − 1) cubes, for any a ≥ 1. This refinement is achieved by picking an arbitrary cube in the dissection, and cutting it into ad smaller cubes. To prove the required result, then, it suffices to exhibit two relatively prime integers of form ad − 1. In the 2-dimensional case, a1 = 2 and a2 = 3 give the coprime numbers 22 − 1 = 3 and 32 − 1 = 8. In the general case, two such integers are 2d − 1 and (2d − 1)d − 1, as is easy to check.
polynomials 词根 -回复

polynomials 词根-回复Polynomials: Unraveling the Mysteries of Polynomial FunctionsIntroduction:Polynomials, derived from the Latin word "polynōma," which means "many terms," are a fundamental concept in mathematics with a rich history dating back thousands of years. As the name suggests, polynomials consist of several terms, each comprising a variable raised to a non-negative integer power, multiplied by a coefficient. These versatile mathematical expressions find applications in various fields, including algebra, calculus, physics, and computer science, making them an essential topic of study for students and researchers alike. In this article, we will delve into the intricate world of polynomials, unraveling their properties, operations, and practical applications.Definition and Structure:A polynomial function, often shortened to just a polynomial, is an algebraic expression consisting of variables, coefficients, and exponents. It follows a specific structure, with each term separatedby an addition or subtraction operator. The general form of a polynomial function is:f(x) = aₙxⁿ+ aₙ₋₁xⁿ⁻¹+ ... + a₁x + a₀Here, 'f(x)' represents the polynomial function, 'x' is the variable, 'a ₙ' (where 'n' is a non-negative integer) are the coefficients, and 'xⁿ' are the exponents. The highest power of the variable, 'n', is known as the degree of the polynomial. The coefficients can be any real or complex numbers and are essential for determining the shape and behavior of the polynomial function.Properties and Types of Polynomials:Polynomials possess several key properties that help in their classification and analysis. These properties include:1. Degree: As mentioned earlier, the degree of a polynomial represents the highest power of the variable. For example, a polynomial with the highest power of 'x' being 'x³' has a degree of 3. The degree aids in understanding the behavior and complexity of polynomial functions.2. Leading Coefficient: The coefficient that accompanies the term with the highest power of the variable is called the leading coefficient. It influences the overall shape and direction of the polynomial graph, providing valuable information about its behavior and end behavior.3. Roots or Zeros: The roots or zeros of a polynomial function represent the values of 'x' for which the function equals zero. These points provide insights into the intercepts and solutions of equations involving polynomials.Polynomials can be further classified based on their degree:1. Constant Polynomials: A polynomial with a degree of zero is known as a constant polynomial. It contains a single term, such as 'f(x) = 3,' and represents a horizontal line parallel to the x-axis.2. Linear Polynomials: A polynomial of degree one contains only one term raised to the power of one. It follows the form 'f(x) = mx + b,' where 'm' is the slope and 'b' is the y-intercept. Linear polynomials represent straight lines and have various applicationsin numerous fields.3. Quadratic Polynomials: Polynomials of degree two are called quadratic polynomials. They have the general form 'f(x) = ax²+ bx + c,' where 'a', 'b', and 'c' are coefficients. Quadratic polynomials represent parabolas, which find applications in physics, engineering, and optimization problems.4. Cubic, Quartic, and Quintic Polynomials: These polynomials have degrees three, four, and five, respectively. They are known as cubic, quartic, and quintic polynomials and exhibit unique shapes and features in their graphs. These polynomial types help model more complex functions found in real-world scenarios.Operations on Polynomials:Polynomials support various operations, enabling mathematicians to manipulate and combine them to solve equations, simplify expressions, and analyze functions. The primary operations on polynomials include:1. Addition and Subtraction: To add or subtract polynomials,combine like terms involving the same variable and degree. For example, adding '2x²+ 3x' and '4x²- 2x' results in '6x²+ x.'2. Multiplication: When multiplying polynomials, distribute each term in one polynomial to every term in the other polynomial, combining like terms afterward. For example, multiplying 'x + 2' and 'x - 3' results in 'x²- x - 6.'3. Division: Polynomial division involves dividing one polynomial by another, similar to long division. This process helps in finding factors, solving equations, and simplifying expressions.Applications of Polynomials:Polynomials have widespread applications across various domains, including:1. Engineering: Polynomials help model and solve engineering problems related to mechanics, circuit design, signal processing, and more.2. Physics: In physics, polynomial functions describe the motion ofobjects, electric and magnetic fields, waveforms, and other physical phenomena.3. Computer Science: Polynomials play a vital role in computer graphics, cryptography, error correction codes, and algorithms.4. Economics: Mathematically modeling economic phenomena often involves the use of polynomial functions to analyze trends, predict market behavior, and optimize decision-making processes.Conclusion:From their historical significance to their diverse applications in modern science and technology, polynomials have always played a crucial role in mathematical theory and practical problem-solving. Understanding the structure, properties, and operations of polynomials opens up avenues for exploring complex mathematical concepts and real-world phenomena. Whether it be graphing functions, optimizing processes, or unraveling the mysteries of the universe, polynomials continue to shape ourunderstanding of the world through their elegance and versatility.。
and

and M. Uhlmann†
Potsdam Institute for Climate Impact Research, D-14412 Potsdam
October 31, 2002
Abstract We construct an orthogonal wavelet basis for the interval using a linear combination of Legendre polynomials. The coefficients are taken as appropriate roots of Chebyshev polynomials of the second kind. The one-dimensional transform is applied to analytical data and appropriate definitions of a scalogram as well as local and global spectra are presented. The transform is then extended to the multi-dimensional case. Analyses of one- and twodimensional data from a direct numerical simulation of turbulent channel flow demonstrate the potential of the method.
Orthonormal polynomial wavelets on the interval and applications to the analysis of turbulent flow fields
ACT备考攻略:ACT数学主要考点

ACT备考攻略:ACT数学主要考点为了帮助大家更好的熟悉ACT,早日攻克ACT考试,三立教育为大家带来ACT备考攻略:数学主要考点一文,希望对大家的ACT备考学习有所帮助。
下面一起来学习一下吧!ACT数学主要考点1. 算术(Pre-Algebra):23%考核的内容为高中之前学习的知识。
例如,分数(fraction)、小数(decimal)、整数(integer)、平方根(square root)、比率(ratio)、百分比(percent)、整数的倍数(multiple)和因数(factor)、绝对值(absolute value)、一次方程式(linear equations with one variable)、概率(probability)等。
2. 初级代数(Elementary Algebra):17%考察变量表达式(use variables to express relationships)、代数表达式的代入法(substitute the value of a variable in an expression)、二次方程式的因式分解(solve simple quadratic equations by factoring)、解含有一个变量的线性不等式(solve linear inequalities with one variable)、应用指数和平方根(apply properties of integer exponents and square roots)等。
3. 中级代数(Intermediate Algebra):15%考察二次方程式(quadratic formula)公式的理解运用、根和有理数的表达式(radical and rational expressions)、不等式和绝对值等式(inequalities and absolute value equations)、序列(sequence)、二次不等式(quadratic inequality)、函数(function)、矩阵(matrix)、多项式的根(roots of polynomials)等。
Polynomial and Rational Functions - LeTourneau University 多项式与有理函数-拉托纽大学

•
•
•
End behavior
• Even degree
or
• Odd degree
or
Properties of Polynomial Functions
Even degree
• Leading coefficient positive
• Leading coefficient negative
bm xm
the quotient for extreme x
Example
Given
r(x)
3x2 8x 5x2 2x1
Graph on calculator
• Set window for -100 < x < 100, -5 < y < 5
Example
Note the value for a large x
3x2 5x2
How does this relate to the leading terms?
Try This One
Consider
r(x)
5x 2x2
6
Which terms dominate as x gets large
5x
What happens to 2 x 2
as x gets large?
• Horizontal asymptote at 0
When n – m = 1
• Diagonal asymptote
y ax
b
Vertical Asymptotes
A vertical asymptote happens when the
function R(x) is not defined
黎曼猜想英语

黎曼猜想英语The Riemann Hypothesis, named after the 19th-century mathematician Bernhard Riemann, is one of the most profound and consequential conjectures in mathematics. It is concerned with the distribution of the zeros of the Riemann zeta function, a complex function denoted as $$\zeta(s)$$, where $$s$$ is a complex number. The hypothesis posits that all non-trivial zeros of this analytical function have their real parts equal to $$\frac{1}{2}$$.To understand the significance of this conjecture, one must delve into the realm of number theory and the distribution of prime numbers. Prime numbers are the building blocks of arithmetic, as every natural number greater than 1 is either a prime or can be factored into primes. The distribution of these primes, however, has puzzled mathematicians for centuries. The Riemann zeta function encodes information about the distribution of primes through its zeros, and thus, the Riemann Hypothesis is directly linked to understanding this distribution.The zeta function is defined for all complex numbers except for $$s = 1$$, where it has a simple pole. For values of $$s$$ with a real part greater than 1, it converges to a sum over the positive integers, as shown in the following equation:$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$。
Algebra1中英版对照目录

分步和混合运算解一元一次方程 P92—P99 2—4:Solving Equations with Variables on Both Sides
变量在两边的一元一次方程的解法 P100—P106 2—5:Solving for a Variable 求变量的值(一元一次)P107—P113
指数幂的乘法运算(同底数幂相乘和积的乘方运算)P460—P466 7—4:Division Properties of Exponents 指数幂的除法 P467—P474 Quiz for Lessons 7-1 through 7-4 第七章 1—4 课小测试 P474—P475 7—5:Polynomials 多项式 P476—P483 7—6:Adding and Subtracting Polynomials 多项式的加减运算 P484—P491 7—7:Multiplying Polynomials 多项式的乘法(单项式与多项式相乘、多项
Algebra 1 与中文数学课本内容在顺序上的对照
Algebra 1
中教数学
CHAPTER 0---- To The Student 0—1:Geometry Formulas
几何公式(s quare,rec tangle,triangle,c irc le)Z3— Z4 0—2:Tree Diagrams 树状图 Z4—Z6 0—3:The Coordinate Plane 直角坐标系 Z7—Z8 0—4:Rounding and Estimating 四舍五入 Z9—Z11 0—5:Adding and Subtracting Decimals 小数的加减运算 Z12—Z13 0—6:Multiplying and Dividing Decimals 小数的乘除运算 Z14—Z16 0—7:Prime and Composite Numbers 素数与合数 Z17—Z18 0—8:Factoring 因数 Z19—Z20 0—9:GCF and LCM 最大公约数和最小公倍数 Z21—Z22
斯普林格数学研究生教材丛书

《斯普林格数学研究生教材丛书》(Graduate Texts in Mathematics)GTM001《Introduction to Axiomatic Set Theory》Gaisi Takeuti, Wilson M.Zaring GTM002《Measure and Category》John C.Oxtoby(测度和范畴)(2ed.)GTM003《Topological Vector Spaces》H.H.Schaefer, M.P.Wolff(2ed.)GTM004《A Course in Homological Algebra》P.J.Hilton, U.Stammbach(2ed.)(同调代数教程)GTM005《Categories for the Working Mathematician》Saunders Mac Lane(2ed.)GTM006《Projective Planes》Daniel R.Hughes, Fred C.Piper(投射平面)GTM007《A Course in Arithmetic》Jean-Pierre Serre(数论教程)GTM008《Axiomatic set theory》Gaisi Takeuti, Wilson M.Zaring(2ed.)GTM009《Introduction to Lie Algebras and Representation Theory》James E.Humphreys(李代数和表示论导论)GTM010《A Course in Simple-Homotopy Theory》M.M CohenGTM011《Functions of One Complex VariableⅠ》John B.ConwayGTM012《Advanced Mathematical Analysis》Richard BealsGTM013《Rings and Categories of Modules》Frank W.Anderson, Kent R.Fuller(环和模的范畴)(2ed.)GTM014《Stable Mappings and Their Singularities》Martin Golubitsky, Victor Guillemin (稳定映射及其奇点)GTM015《Lectures in Functional Analysis and Operator Theory》Sterling K.Berberian GTM016《The Structure of Fields》David J.Winter(域结构)GTM017《Random Processes》Murray RosenblattGTM018《Measure Theory》Paul R.Halmos(测度论)GTM019《A Hilbert Space Problem Book》Paul R.Halmos(希尔伯特问题集)GTM020《Fibre Bundles》Dale Husemoller(纤维丛)GTM021《Linear Algebraic Groups》James E.Humphreys(线性代数群)GTM022《An Algebraic Introduction to Mathematical Logic》Donald W.Barnes, John M.MackGTM023《Linear Algebra》Werner H.Greub(线性代数)GTM024《Geometric Functional Analysis and Its Applications》Paul R.HolmesGTM025《Real and Abstract Analysis》Edwin Hewitt, Karl StrombergGTM026《Algebraic Theories》Ernest G.ManesGTM027《General Topology》John L.Kelley(一般拓扑学)GTM028《Commutative Algebra》VolumeⅠOscar Zariski, Pierre Samuel(交换代数)GTM029《Commutative Algebra》VolumeⅡOscar Zariski, Pierre Samuel(交换代数)GTM030《Lectures in Abstract AlgebraⅠ.Basic Concepts》Nathan Jacobson(抽象代数讲义Ⅰ基本概念分册)GTM031《Lectures in Abstract AlgebraⅡ.Linear Algabra》Nathan.Jacobson(抽象代数讲义Ⅱ线性代数分册)GTM032《Lectures in Abstract AlgebraⅢ.Theory of Fields and Galois Theory》Nathan.Jacobson(抽象代数讲义Ⅲ域和伽罗瓦理论)GTM033《Differential Topology》Morris W.Hirsch(微分拓扑)GTM034《Principles of Random Walk》Frank Spitzer(2ed.)(随机游动原理)GTM035《Several Complex Variables and Banach Algebras》Herbert Alexander, John Wermer(多复变和Banach代数)GTM036《Linear Topological Spaces》John L.Kelley, Isaac Namioka(线性拓扑空间)GTM037《Mathematical Logic》J.Donald Monk(数理逻辑)GTM038《Several Complex Variables》H.Grauert, K.FritzsheGTM039《An Invitation to C*-Algebras》William Arveson(C*-代数引论)GTM040《Denumerable Markov Chains》John G.Kemeny, urie Snell, Anthony W.KnappGTM041《Modular Functions and Dirichlet Series in Number Theory》Tom M.Apostol (数论中的模函数和Dirichlet序列)GTM042《Linear Representations of Finite Groups》Jean-Pierre Serre(有限群的线性表示)GTM043《Rings of Continuous Functions》Leonard Gillman, Meyer JerisonGTM044《Elementary Algebraic Geometry》Keith KendigGTM045《Probability TheoryⅠ》M.Loève(概率论Ⅰ)(4ed.)GTM046《Probability TheoryⅡ》M.Loève(概率论Ⅱ)(4ed.)GTM047《Geometric Topology in Dimensions 2 and 3》Edwin E.MoiseGTM048《General Relativity for Mathematicians》Rainer.K.Sachs, H.Wu伍鸿熙(为数学家写的广义相对论)GTM049《Linear Geometry》K.W.Gruenberg, A.J.Weir(2ed.)GTM050《Fermat's Last Theorem》Harold M.EdwardsGTM051《A Course in Differential Geometry》Wilhelm Klingenberg(微分几何教程)GTM052《Algebraic Geometry》Robin Hartshorne(代数几何)GTM053《A Course in Mathematical Logic for Mathematicians》Yu.I.Manin(2ed.)GTM054《Combinatorics with Emphasis on the Theory of Graphs》Jack E.Graver, Mark E.WatkinsGTM055《Introduction to Operator TheoryⅠ》Arlen Brown, Carl PearcyGTM056《Algebraic Topology:An Introduction》W.S.MasseyGTM057《Introduction to Knot Theory》Richard.H.Crowell, Ralph.H.FoxGTM058《p-adic Numbers, p-adic Analysis, and Zeta-Functions》Neal Koblitz(p-adic 数、p-adic分析和Z函数)GTM059《Cyclotomic Fields》Serge LangGTM060《Mathematical Methods of Classical Mechanics》V.I.Arnold(经典力学的数学方法)(2ed.)GTM061《Elements of Homotopy Theory》George W.Whitehead(同论论基础)GTM062《Fundamentals of the Theory of Groups》M.I.Kargapolov, Ju.I.Merzljakov GTM063《Modern Graph Theory》Béla BollobásGTM064《Fourier Series:A Modern Introduction》VolumeⅠ(2ed.)R.E.Edwards(傅里叶级数)GTM065《Differential Analysis on Complex Manifolds》Raymond O.Wells, Jr.(3ed.)GTM066《Introduction to Affine Group Schemes》William C.Waterhouse(仿射群概型引论)GTM067《Local Fields》Jean-Pierre Serre(局部域)GTM069《Cyclotomic FieldsⅠandⅡ》Serge LangGTM070《Singular Homology Theory》William S.MasseyGTM071《Riemann Surfaces》Herschel M.Farkas, Irwin Kra(黎曼曲面)GTM072《Classical Topology and Combinatorial Group Theory》John Stillwell(经典拓扑和组合群论)GTM073《Algebra》Thomas W.Hungerford(代数)GTM074《Multiplicative Number Theory》Harold Davenport(乘法数论)(3ed.)GTM075《Basic Theory of Algebraic Groups and Lie Algebras》G.P.HochschildGTM076《Algebraic Geometry:An Introduction to Birational Geometry of Algebraic Varieties》Shigeru IitakaGTM077《Lectures on the Theory of Algebraic Numbers》Erich HeckeGTM078《A Course in Universal Algebra》Stanley Burris, H.P.Sankappanavar(泛代数教程)GTM079《An Introduction to Ergodic Theory》Peter Walters(遍历性理论引论)GTM080《A Course in_the Theory of Groups》Derek J.S.RobinsonGTM081《Lectures on Riemann Surfaces》Otto ForsterGTM082《Differential Forms in Algebraic Topology》Raoul Bott, Loring W.Tu(代数拓扑中的微分形式)GTM083《Introduction to Cyclotomic Fields》Lawrence C.Washington(割圆域引论)GTM084《A Classical Introduction to Modern Number Theory》Kenneth Ireland, Michael Rosen(现代数论经典引论)GTM085《Fourier Series A Modern Introduction》Volume 1(2ed.)R.E.Edwards GTM086《Introduction to Coding Theory》J.H.van Lint(3ed .)GTM087《Cohomology of Groups》Kenneth S.Brown(上同调群)GTM088《Associative Algebras》Richard S.PierceGTM089《Introduction to Algebraic and Abelian Functions》Serge Lang(代数和交换函数引论)GTM090《An Introduction to Convex Polytopes》Ame BrondstedGTM091《The Geometry of Discrete Groups》Alan F.BeardonGTM092《Sequences and Series in BanachSpaces》Joseph DiestelGTM093《Modern Geometry-Methods and Applications》(PartⅠ.The of geometry Surfaces Transformation Groups and Fields)B.A.Dubrovin, A.T.Fomenko, S.P.Novikov (现代几何学方法和应用)GTM094《Foundations of Differentiable Manifolds and Lie Groups》Frank W.Warner(可微流形和李群基础)GTM095《Probability》A.N.Shiryaev(2ed.)GTM096《A Course in Functional Analysis》John B.Conway(泛函分析教程)GTM097《Introduction to Elliptic Curves and Modular Forms》Neal Koblitz(椭圆曲线和模形式引论)GTM098《Representations of Compact Lie Groups》Theodor Breöcker, Tammo tom DieckGTM099《Finite Reflection Groups》L.C.Grove, C.T.Benson(2ed.)GTM100《Harmonic Analysis on Semigroups》Christensen Berg, Jens Peter Reus Christensen, Paul ResselGTM101《Galois Theory》Harold M.Edwards(伽罗瓦理论)GTM102《Lie Groups, Lie Algebras, and Their Representation》V.S.Varadarajan(李群、李代数及其表示)GTM103《Complex Analysis》Serge LangGTM104《Modern Geometry-Methods and Applications》(PartⅡ.Geometry and Topology of Manifolds)B.A.Dubrovin, A.T.Fomenko, S.P.Novikov(现代几何学方法和应用)GTM105《SL₂ (R)》Serge Lang(SL₂ (R)群)GTM106《The Arithmetic of Elliptic Curves》Joseph H.Silverman(椭圆曲线的算术理论)GTM107《Applications of Lie Groups to Differential Equations》Peter J.Olver(李群在微分方程中的应用)GTM108《Holomorphic Functions and Integral Representations in Several Complex Variables》R.Michael RangeGTM109《Univalent Functions and Teichmueller Spaces》Lehto OlliGTM110《Algebraic Number Theory》Serge Lang(代数数论)GTM111《Elliptic Curves》Dale Husemoeller(椭圆曲线)GTM112《Elliptic Functions》Serge Lang(椭圆函数)GTM113《Brownian Motion and Stochastic Calculus》Ioannis Karatzas, Steven E.Shreve (布朗运动和随机计算)GTM114《A Course in Number Theory and Cryptography》Neal Koblitz(数论和密码学教程)GTM115《Differential Geometry:Manifolds, Curves, and Surfaces》M.Berger, B.Gostiaux GTM116《Measure and Integral》Volume1 John L.Kelley, T.P.SrinivasanGTM117《Algebraic Groups and Class Fields》Jean-Pierre Serre(代数群和类域)GTM118《Analysis Now》Gert K.Pedersen(现代分析)GTM119《An introduction to Algebraic Topology》Jossph J.Rotman(代数拓扑导论)GTM120《Weakly Differentiable Functions》William P.Ziemer(弱可微函数)GTM121《Cyclotomic Fields》Serge LangGTM122《Theory of Complex Functions》Reinhold RemmertGTM123《Numbers》H.-D.Ebbinghaus, H.Hermes, F.Hirzebruch, M.Koecher, K.Mainzer, J.Neukirch, A.Prestel, R.Remmert(2ed.)GTM124《Modern Geometry-Methods and Applications》(PartⅢ.Introduction to Homology Theory)B.A.Dubrovin, A.T.Fomenko, S.P.Novikov(现代几何学方法和应用)GTM125《Complex Variables:An introduction》Garlos A.Berenstein, Roger Gay GTM126《Linear Algebraic Groups》Armand Borel(线性代数群)GTM127《A Basic Course in Algebraic Topology》William S.Massey(代数拓扑基础教程)GTM128《Partial Differential Equations》Jeffrey RauchGTM129《Representation Theory:A First Course》William Fulton, Joe HarrisGTM130《Tensor Geometry》C.T.J.Dodson, T.Poston(张量几何)GTM131《A First Course in Noncommutative Rings》m(非交换环初级教程)GTM132《Iteration of Rational Functions:Complex Analytic Dynamical Systems》AlanF.Beardon(有理函数的迭代:复解析动力系统)GTM133《Algebraic Geometry:A First Course》Joe Harris(代数几何)GTM134《Coding and Information Theory》Steven RomanGTM135《Advanced Linear Algebra》Steven RomanGTM136《Algebra:An Approach via Module Theory》William A.Adkins, Steven H.WeintraubGTM137《Harmonic Function Theory》Sheldon Axler, Paul Bourdon, Wade Ramey(调和函数理论)GTM138《A Course in Computational Algebraic Number Theory》Henri Cohen(计算代数数论教程)GTM139《Topology and Geometry》Glen E.BredonGTM140《Optima and Equilibria:An Introduction to Nonlinear Analysis》Jean-Pierre AubinGTM141《A Computational Approach to Commutative Algebra》Gröbner Bases, Thomas Becker, Volker Weispfenning, Heinz KredelGTM142《Real and Functional Analysis》Serge Lang(3ed.)GTM143《Measure Theory》J.L.DoobGTM144《Noncommutative Algebra》Benson Farb, R.Keith DennisGTM145《Homology Theory:An Introduction to Algebraic Topology》James W.Vick(同调论:代数拓扑简介)GTM146《Computability:A Mathematical Sketchbook》Douglas S.BridgesGTM147《Algebraic K-Theory and Its Applications》Jonathan Rosenberg(代数K理论及其应用)GTM148《An Introduction to the Theory of Groups》Joseph J.Rotman(群论入门)GTM149《Foundations of Hyperbolic Manifolds》John G.Ratcliffe(双曲流形基础)GTM150《Commutative Algebra with a view toward Algebraic Geometry》David EisenbudGTM151《Advanced Topics in the Arithmetic of Elliptic Curves》Joseph H.Silverman(椭圆曲线的算术高级选题)GTM152《Lectures on Polytopes》Günter M.ZieglerGTM153《Algebraic Topology:A First Course》William Fulton(代数拓扑)GTM154《An introduction to Analysis》Arlen Brown, Carl PearcyGTM155《Quantum Groups》Christian Kassel(量子群)GTM156《Classical Descriptive Set Theory》Alexander S.KechrisGTM157《Integration and Probability》Paul MalliavinGTM158《Field theory》Steven Roman(2ed.)GTM159《Functions of One Complex Variable VolⅡ》John B.ConwayGTM160《Differential and Riemannian Manifolds》Serge Lang(微分流形和黎曼流形)GTM161《Polynomials and Polynomial Inequalities》Peter Borwein, Tamás Erdélyi(多项式和多项式不等式)GTM162《Groups and Representations》J.L.Alperin, Rowen B.Bell(群及其表示)GTM163《Permutation Groups》John D.Dixon, Brian Mortime rGTM164《Additive Number Theory:The Classical Bases》Melvyn B.NathansonGTM165《Additive Number Theory:Inverse Problems and the Geometry of Sumsets》Melvyn B.NathansonGTM166《Differential Geometry:Cartan's Generalization of Klein's Erlangen Program》R.W.SharpeGTM167《Field and Galois Theory》Patrick MorandiGTM168《Combinatorial Convexity and Algebraic Geometry》Günter Ewald(组合凸面体和代数几何)GTM169《Matrix Analysis》Rajendra BhatiaGTM170《Sheaf Theory》Glen E.Bredon(2ed.)GTM171《Riemannian Geometry》Peter Petersen(黎曼几何)GTM172《Classical Topics in Complex Function Theory》Reinhold RemmertGTM173《Graph Theory》Reinhard Diestel(图论)(3ed.)GTM174《Foundations of Real and Abstract Analysis》Douglas S.Bridges(实分析和抽象分析基础)GTM175《An Introduction to Knot Theory》W.B.Raymond LickorishGTM176《Riemannian Manifolds:An Introduction to Curvature》John M.LeeGTM177《Analytic Number Theory》Donald J.Newman(解析数论)GTM178《Nonsmooth Analysis and Control Theory》F.H.clarke, Yu.S.Ledyaev, R.J.Stern, P.R.Wolenski(非光滑分析和控制论)GTM179《Banach Algebra Techniques in Operator Theory》Ronald G.Douglas(2ed.)GTM180《A Course on Borel Sets》S.M.Srivastava(Borel 集教程)GTM181《Numerical Analysis》Rainer 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Complex Manifolds》Klaus Fritzsche, Hans Grauert(从正则函数到复流形)GTM214《Partial Differential Equations》Jüergen Jost(偏微分方程)GTM215《Algebraic Functions and Projective Curves》David M.Goldschmidt(代数函数和投影曲线)GTM216《Matrices:Theory and Applications》Denis Serre(矩阵:理论及应用)GTM217《Model Theory An Introduction》David Marker(模型论引论)GTM218《Introduction to Smooth Manifolds》John M.Lee(光滑流形引论)GTM219《The Arithmetic of Hyperbolic 3-Manifolds》Colin Maclachlan, Alan W.Reid GTM220《Smooth Manifolds and Observables》Jet Nestruev(光滑流形和直观)GTM221《Convex Polytopes》Branko GrüenbaumGTM222《Lie Groups, Lie Algebras, and Representations》Brian C.Hall(李群、李代数和表示)GTM223《Fourier Analysis and its Applications》Anders Vretblad(傅立叶分析及其应用)GTM224《Metric Structures in Differential Geometry》Gerard Walschap(微分几何中的度量结构)GTM225《Lie Groups》Daniel Bump(李群)GTM226《Spaces of Holomorphic Functions in the Unit Ball》Kehe Zhu(单位球内的全纯函数空间)GTM227《Combinatorial Commutative Algebra》Ezra Miller, Bernd Sturmfels(组合交换代数)GTM228《A First Course in Modular 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Grillet(抽象代数)GTM243《Topological Methods in Group Theory》Ross GeogheganGTM244《Graph Theory》J.A.Bondy, U.S.R.MurtyGTM245《Complex Analysis:In the Spirit of Lipman Bers》Jane P.Gilman, Irwin Kra, Rubi E.RodriguezGTM246《A Course in Commutative Banach Algebras》Eberhard KaniuthGTM247《Braid Groups》Christian Kassel, Vladimir TuraevGTM248《Buildings Theory and Applications》Peter Abramenko, Kenneth S.Brown GTM249《Classical Fourier Analysis》Loukas Grafakos(经典傅里叶分析)GTM250《Modern Fourier Analysis》Loukas Grafakos(现代傅里叶分析)GTM251《The Finite Simple Groups》Robert A.WilsonGTM252《Distributions and Operators》Gerd GrubbGTM253《Elementary Functional Analysis》Barbara D.MacCluerGTM254《Algebraic Function Fields and Codes》Henning StichtenothGTM255《Symmetry Representations and Invariants》Roe Goodman, Nolan R.Wallach GTM256《A Course in Commutative Algebra》Kemper GregorGTM257《Deformation Theory》Robin HartshorneGTM258《Foundation of Optimization》Osman GülerGTM259《Ergodic Theory:with a view towards Number Theory》Manfred Einsiedler, Thomas WardGTM260《Monomial Ideals》Jurgen Herzog, Takayuki HibiGTM261《Probability and Stochastics》Erhan CinlarGTM262《Essentials of Integration Theory for Analysis》Daniel W.StroockGTM263《Analysis on Fock Spaces》Kehe ZhuGTM264《Functional Analysis, Calculus of Variations and Optimal Control》Francis ClarkeGTM265《Unbounded Self-adjoint Operatorson Hilbert Space》Konrad Schmüdgen GTM266《Calculus Without Derivatives》Jean-Paul PenotGTM267《Quantum Theory for Mathematicians》Brian C.HallGTM268《Geometric Analysis of the Bergman Kernel and Metric》Steven G.Krantz GTM269《Locally Convex Spaces》M.Scott Osborne。
Vitali’s theorem and WWKL

VITALI’S THEOREM AND WWKLDOUGLAS K.BROWNMARIAGNESE GIUSTOSTEPHEN G.SIMPSONAbstract.Continuing the investigations of X.Yu and others,westudy the role of set existence axioms in classical Lebesgue mea-sure theory.We show that pairwise disjoint countable additivityfor open sets of reals is provable in RCA0.We show that sev-eral well-known measure-theoretic propositions including the VitaliCovering Theorem are equivalent to WWKL over RCA0.1.IntroductionThe purpose of Reverse Mathematics is to study the role of set ex-istence axioms,with an eye to determining which axioms are needed in order to prove specific mathematical theorems.In many cases,it is shown that a specific mathematical theorem is equivalent to the set existence axiom which is needed to prove it.Such equivalences are often proved in the weak base theory RCA0.RCA0may be viewed as a kind of formalized constructive or recursive mathematics,with full clas-sical logic but severely restricted comprehension and induction.The program of Reverse Mathematics has been developed in many publica-tions;see for instance[5,10,11,12,20].In this paper we carry out a Reverse Mathematics study of some aspects of classical Lebesgue measure theory.Historically,the subject of measure theory developed hand in hand with the nonconstructive, set-theoretic approach to mathematics.Errett Bishop has remarked that the foundations of measure theory present a special challenge to the constructive mathematician.Although our program of Reverse Mathematics is quite different from Bishop-style constructivism,we feel that Bishop’s remark implicitly raises an interesting question:Which nonconstructive set existence axioms are needed for measure theory?VITALI’S THEOREM AND WWKL 2This paper,together with earlier papers of Yu and others [21,22,23,24,25,26],constitute an answer to that question.The results of this paper build upon and clarify some early results of Yu and Simpson.The reader of this paper will find that familiarity with Yu–Simpson [26]is desirable but not essential.We begin in section 2by exploring the extent to which measure theory can be developed in RCA 0.We show that pairwise disjoint countable additivity for open sets of reals is provable in RCA 0.This is in contrast to a result of Yu–Simpson [26]:countable additivity for open sets of reals is equivalent over RCA 0to a nonconstructive set existence axiom known as Weak Weak K¨o nig’s Lemma (WWKL).We show in sections 3and 4that several other basic propositions of measure theory are also equivalent to WWKL over RCA 0.Finally in section 5we show that the Vitali Covering Theorem is likewise equivalent to WWKL over RCA 0.2.Measure Theory in RCA 0Recall that RCA 0is the subsystem of second order arithmetic with∆01comprehension and Σ01induction.The purpose of this section is toshow that some measure-theoretic results can be proved in RCA 0.Within RCA 0,let X be a compact separable metric space.We define C (X )= A,the completion of A ,where A is the vector space of rational “polynomials”over X under the sup-norm, f =sup x ∈X |f (x )|.For the precise definitions within RCA 0,see [26]and section III.E of Brown’s thesis [4].The construction of C (X )within RCA 0is inspired by the constructive Stone–Weierstrass theorem in section 4.5of Bishop and Bridges [2].It is provable in RCA 0that there is a natural one-to-one correspondence between points of C (X )and continuous functions f :X →R which are equipped with a modulus of uniform continuity ,that is to say,a function h :N →N such that for all n ∈N and x ,y ∈Xd (x,y )<12n .Within RCA 0we define a measure (more accurately,a nonnegative Borel probability measure)on X to be a nonnegative bounded linear functional µ:C (X )→R such that µ(1)=1.(Here µ(1)denotes µ(f ),f ∈C (X ),f (x )=1for all x ∈X .)For example,if X =[0,1],the unit interval,then there is an obvious measure µL :C ([0,1])→R given by µL (f )= 10f (x )dx ,the Riemann integral of f from 0to 1.We refer to µL as Lebesgue measure on [0,1].There is also the obvious generalization to Lebesgue measure µL on X =[0,1]n ,the n -cube.VITALI’S THEOREM AND WWKL 3Definition 2.1(measure of an open set).This definition is made in RCA 0.Let X be any compact separable metric space,and let µbe any measure on X .If U is an open set in X ,we defineµ(U )=sup {µ(f )|f ∈C (X ),0≤f ≤1,f =0on X \U }.Within RCA 0this supremum need not exist as a real number.(Indeed,the existence of µ(U )for all open sets U is equivalent to ACA 0over RCA 0.)Therefore,when working within RCA 0,we interpret assertions about µ(U )in a “virtual”or comparative sense.For example,µ(U )≤µ(V )is taken to mean that for all >0and all f ∈C (X )with 0≤f ≤1and f =0on X \U ,there exists g ∈C (X )with 0≤g ≤1and g =0on X \V such that µ(f )≤µ(g )+ .See also [26].Some basic properties of Lebesgue measure are easily proved in RCA 0.For instance,it is straightforward to show that the Lebesgue measure of the union of a finite set of pairwise disjoint open intervals is equal to the sum of the lengths of the intervals.We define L 1(X,µ)to be the completion of C (X )under the L 1-norm given by f 1=µ(|f |).(For the precise definitions,see [5]and[26].)In RCA 0we see that L 1(X,µ)is a separable Banach space,but to assert within RCA 0that points of the Banach space L 1(X,µ)represent measurable functions f :X →R is problematic.We shall comment further on this question in section 4below.Lemma 2.2.The following is provable in RCA 0.If U n ,n ∈N ,is a sequence of open sets,then µ∞ n =0U n ≥lim k →∞µ k n =0U n .Proof.Trivial.Lemma 2.3.The following is provable in RCA 0.If U 0,U 1,...,U k is a finite,pairwise disjoint sequence of open sets,then µ k n =0U n ≥k n =0µ(U n ).Proof.Trivial.An open set is said to be connected if it is not the union of two disjoint nonempty open sets.Let us say that a compact separable metric space X is nice if for all sufficiently small δ>0and all x ∈X ,the open ballB (x,δ)={y ∈X |d (x,y )<δ}VITALI’S THEOREM AND WWKL4 is connected.Such aδis called a modulus of niceness for X.For example,the unit interval[0,1]and the n-cube[0,1]n are nice, but the Cantor space2N is not nice.Theorem2.4(disjoint countable additivity).The following is prov-able in RCA0.Assume that X is nice.If U n,n∈N,is a pairwise disjoint sequence of open sets in X,thenµ∞n=0U n=∞n=0µ(U n).Proof.Put U= ∞n=0U n.Note that U is an open set.By Lemmas2.2and2.3,we have in RCA0thatµ(U)≥ ∞n=0µ(U n).It remainsto prove in RCA0thatµ(U)≤ ∞n=0µ(U n).Let f∈C(X)be suchthat0≤f≤1and f=0on X\U.It suffices to prove thatµ(f)≤∞n=0µ(U n).Claim1:There is a sequence of continuous functions f n:X→R, n∈N,defined by f n(x)=f(x)for all x∈U n,f n(x)=0for all x∈X\U n.To prove this in RCA0,recall from[6]or[20]that a code for a continuous function g from X to Y is a collection G of quadruples (a,r,b,s)with certain properties,the idea being that d(a,x)<r im-plies d(b,g(x))≤s.Also,a code for an open set U is a collection of pairs(a,r)with certain properties,the idea being that d(a,x)<r im-plies x∈U.In this case we write(a,r)<U to mean that d(a,b)+r<s for some(b,s)belonging to the code of U.Now let F be a code for f:X→R.Define a sequence of codes F n,n∈N,by putting(a,r,b,s) into F n if and only if1.(a,r,b,s)belongs to F and(a,r)<U n,or2.(a,r,b,s)belongs to F and b−s≤0≤b+s,or3.b−s≤0≤b+s and(a,r)<U m for some m=n.It is straightforward to verify that F n is a code for f n as required by claim1.Claim2:The sequence f n,n∈N,is a sequence of elements of C(X). To prove this in RCA0,we must show that the sequence of f n’s has a sequence of moduli of uniform continuity.Let h:N→N be a modulus of uniform continuity for f,and let k be so large that1/2k is a modulus of niceness for X.We shall show that h :N→N defined by h (m)=max(h(m),k)is a modulus of uniform continuity for all of the f n’s.Let x,y∈X and m∈N be such that d(x,y)<1/2h (m). To show that|f n(x)−f n(y)|<1/2m,we consider three cases.Case1:VITALI’S THEOREM AND WWKL5 x,y∈U n.In this case we have|f n(x)−f n(y)|=|f(x)−f(y)|<1VITALI’S THEOREM AND WWKL 6From (1)we see that for each >0there exists k such that µ(f )− ≤ kn =0µ(f n ).Thus we haveµ(f )− ≤kn =0µ(f n )≤k n =0µ(U n )≤∞ n =0µ(U n ).Since this holds for all >0,it follows that µ(f )≤ ∞n =0µ(U n ).Thus µ(U )≤ ∞n =0µ(U n )and the proof of Theorem 2.4is complete.Corollary 2.5.The following is provable in RCA 0.If (a n ,b n ),n ∈N is a sequence of pairwise disjoint open intervals,then µL ∞ n =0(a n ,b n ) =∞ n =0|a n −b n |.Proof.This is a special case of Theorem 2.4.Remark 2.6.Theorem 2.4fails if we drop the assumption that X is nice.Indeed,let µC be the familiar “fair coin”measure on the Cantor space X =2N ,given by µC ({x |x (n )=i })=1/2for all n ∈N and i ∈{0,1}.It can be shown that disjoint finite additivity for µC is equivalent to WWKL over RCA 0.(WWKL is defined and discussed in the next section.)In particular,disjoint finite additivity for µC is not provable in RCA 0.3.Measure Theory in WWKL 0Yu and Simpson [26]introduced a subsystem of second order arith-metic known as WWKL 0,consisting of RCA 0plus the following axiom:if T is a subtree of 2<N with no infinite path,thenlim n →∞|{σ∈T |length(σ)=n }|VITALI’S THEOREM AND WWKL 7see also Sieg [18].In this sense,every mathematical theorem provable in WKL 0or WWKL 0is finitistically reducible in the sense of Hilbert’s Program;see [19,6,20].Remark 3.2.The study of ω-models of WWKL 0is closely related to the theory of 1-random sequences,as initiated by Martin-L¨o f [16]and continued by Kuˇc era [7,13,14,15].At the time of writing of [26],Yu and Simpson were unaware of this work of Martin-L¨o f and Kuˇc era.The purpose of this section and the next is to review and extend the results of [26]and [21]concerning measure theory in WWKL 0.A measure µ:C (X )→R on a compact separable metric space X is said to be countably additive if µ∞ n =0U n =lim k →∞µ k n =0U n for any sequence of open sets U n ,n ∈N ,in X .The following theorem is implicit in [26]and [21].Theorem 3.3.The following assertions are pairwise equivalent over RCA 0.1.WWKL.2.(countable additivity)For any compact separable metric space Xand any measure µon X ,µis countably additive.3.For any covering of the closed unit interval [0,1]by a sequence of open intervals (a n ,b n ),n ∈N ,we have ∞n =0|a n −b n |≥1.Proof.That WWKL implies statement 2is proved in Theorem 1of [26].The implication 2→3is trivial.It remains to prove that statement 3implies WWKL.Reasoning in RCA 0,let T be a subtree of 2<N with no infinite path.PutT ={σ i |σ∈T,σ i /∈T,i <2}.For σ∈2<N put lh(σ)=length of σanda σ=lh(σ)−1n =0σ(n )2lh(σ).Note that |a σ−b σ|=1/2lh(σ).Note also that σ,τ∈2<N are incompa-rable if and only if (a σ,b σ)∩(a τ,b τ)=∅.In particular,the intervals (a τ,b τ),τ∈ T,are pairwise disjoint and cover [0,1)except for some of the points a σ,σ∈2<N .Fix >0and put c σ=a σ− /4lh(σ),d σ=a σ+ /4lh(σ).Then the open intervals (a τ,b τ),τ∈ T,(c σ,d σ),VITALI’S THEOREM AND WWKL 8σ∈2<N and (1− ,1+ )form a covering of [0,1].Applying statement 3,we see that the sum of the lengths of these intervals is ≥1,i.e. τ∈ T12lh(τ)=1.From this,equation (2)follows easily.Thus we have proved that state-ment 3implies WWKL.This completes the proof of the theorem.It is possible to take a somewhat different approach to measure the-ory in RCA 0.Note that the definition of µ(U )that we have given (Definition 2.1)is extensional in RCA 0.This means that if U and V contain the same points then µ(U )=µ(V ),provably in RCA 0.An alternative approach is the intensional one,embodied in Definition 3.4below.Recall that an open set U is given in RCA 0as a sequence of basic open sets.In the case of the real line,basic open sets are just intervals with rational endpoints.Definition 3.4(intensional Lebesgue measure).We make this defini-tion in RCA 0.Let U = (a n ,b n ) n ∈N be an open set in the real line.The intensional Lebesgue measure of U is defined by µI (U )=lim k →∞µL k n =0(a n ,b n ) .Theorem 3.5.It is provable in RCA 0that intensional Lebesgue mea-sure µI is countably additive on open sets.In other words,if U n ,n ∈N ,is a sequence of open sets,then µI∞ n =0U n =lim k →∞µI k n =0U n .Proof.This is immediate from the definitions,since ∞n =0U n is defined as the union of the sequences of basic open intervals in U n ,n ∈N .Returning now to WWKL 0,we can prove that intensional Lebesgue measure concides with extensional Lebesgue measure.In fact,we have the following easy result.Theorem 3.6.The following assertions are pairwise equivalent over RCA 0.VITALI’S THEOREM AND WWKL91.WWKL.2.µI(U)=µL(U)for all open sets U⊆[0,1].3.µI is extensional on open sets.In other words,for all open setsU,V⊆[0,1],if∀x(x∈U↔x∈V)thenµI(U)=µI(V).4.For all open sets U⊇[0,1],we haveµI(U)≥1.Proof.This is immediate from Theorems3.3and3.5.4.More Measure Theory in WWKL0In this section we show that a good theory of measurable functions and measurable sets can be developed within WWKL0.Wefirst consider pointwise values of measurable functions.Our ap-proach is due to Yu[21,24].Let X be a compact separable metric space and letµ:C(X)→R be a positive Borel probability measure on X.Recall that L1(X,µ)is defined within RCA0as the completion of C(X)under the L1-norm.In what sense or to what extent can we prove that a point of the Banach space L1(X,µ)gives rise to a function f:X→R?In order to answer this question,recall that f∈L1(X,µ)is given by a sequence f n∈C(X),n∈N,which converges to f in the L1-norm; more preciselyf n−f n+1 1≤12nfor all n,and|f m(x)−f m (x)|≤12k.VITALI’S THEOREM AND WWKL10 Then for x∈C fnand m ≥m≥n+2k+2we have|f m(x)−f m (x)|≤m −1i=m|f i(x)−f i+1(x)|≤∞i=n+2k+2|f i(x)−f i+1(x)|≤12k.We need a lemma:Lemma4.2.The following is provable in RCA0.For f∈C(X)and >0,we haveµ({x|f(x)> })≤ f 1/ .Proof.Put U={x|f(x)> }.Note that U is an open set.If g∈C(X),0≤g≤1,g=0on X\U,then we have g≤|f|, hence µ(g)=µ( g)≤µ(|f|)= f 1,henceµ(g)≤ f 1/ .Thus µ(U)≤ f 1/ and the lemma is proved.Using this lemma we haveµ(X\C fnk )=µx∞i=n+2k+2|f i(x)−f i+1(x)|>12i=1VITALI’S THEOREM AND WWKL 11hence by countable additivityµ(X \C f n )≤∞ k =0µ(X \C f nk )≤∞k =012n .This completes the proof of Proposition 4.1.Remark 4.3(Yu [21]).In accordance with Proposition 4.1,forf = f n n ∈N ∈L 1(X,µ)and x ∈ ∞n =0C f n ,we define f (x )=lim n →∞f n (x ).Thus we see thatf (x )is defined on an F σset of measure 1.Moreover,if f =g in L 1(X,µ),i.e.if f −g 1=0,then f (x )=g (x )for all x in an F σset of measure 1.These facts are provable in WWKL 0.We now turn to a discussion of measurable sets within WWKL 0.We sketch two approaches to this topic.Our first approach is to identify measurable sets with their characteristic functions in L 1(X,µ),accord-ing to the following definition.Definition 4.4.This definition is made within WWKL 0.We say that f ∈L 1(X,µ)is a measurable characteristic function if there exists a sequence of closed setsC 0⊆C 1⊆···⊆C n ⊆...,n ∈N ,such that µ(X \C n )≤1/2n for all n ,and f (x )∈{0,1}for all x ∈ ∞n =0C n .Here f (x )is as defined in Remark 4.3.Our second approach is more direct,but in its present form it applies only to certain specific situations.For concreteness we consider only Lebesgue measure µL on the unit interval [0,1].Our discussion can easily be extended to Lebesgue measure on the n -cube [0,1]n ,the “fair coin”measure on the Cantor space 2N ,etc .Definition 4.5.The following definition is made within RCA 0.Let S be the Boolean algebra of finite unions of intervals in [0,1]with rational endpoints.For E 1,E 2∈S we define the distanced (E 1,E 2)=µL ((E 1\E 2)∪(E 2\E 1)),the Lebesgue measure of the symmetric difference of E 1and E 2.Thus d is a pseudometric on S ,and we define S to be the compact separable metric space which is the completion of S under d .A point E ∈ S is called a Lebesgue measurable set in [0,1].VITALI’S THEOREM AND WWKL 12We shall show that these two approaches to measurable sets (Defi-nitions 4.4and 4.5)are equivalent in WWKL 0.Begin by defining an isometry χ:S →L 1([0,1],µL )as follows.For 0≤a <b ≤1defineχ([a,b ])= f n n ∈N ∈L 1([0,1],µL )where f n (0)=f n (a )=f n (b )=f n (1)=0and f n a +b −a 2n +1=1and f n ∈C ([0,1])is piecewise linear otherwise.Thus χ([a,b ])is a measurable characteristic function corresponding to the interval [a,b ].For 0≤a 1<b 1<···<a k <b k ≤1defineχ([a 1,b 1]∪···∪[a k ,b k ])=χ([a 1,b 1])+···+χ([a k ,b k ]).It is straightforward to prove in RCA 0that χextends to an isometryχ: S→L 1([0,1],µL ).Proposition 4.6.The following is provable in WWKL 0.If E ∈ Sis a Lebesgue measurable set,then χ(E )is a measurable characteristic function in L 1([0,1],µL ).Conversely,given a measurable characteristic function f ∈L 1([0,1],µL ),we can find E ∈ Ssuch that χ(E )=f in L 1([0,1],µL ).Proof.It is straightforward to prove in RCA 0that for all E ∈ S , χ(E )is a measurable characteristic function.For the converse,let f be a measurable characteristic function.By Definition 4.4we have that f (x )∈{0,1}for all x ∈ ∞n =0C n .ByProposition 4.1we have |f (x )−f 3n +3(x )|<1/2n for all x ∈C f n .Put U n ={x ||f 3n +3(x )−1|<1/2n }and V n ={x ||f 3n +3(x )|<1/2n }.Then for n ≥1,U n and V n are disjoint open sets.Moreover C n ∩C f n ⊆U n ∪V n ,hence µL (U n ∪V n )≥1−1/2n −1.By countable additivity(Theorem 3.3)we can effectively find E n ,F n ∈S such that E n ⊆U n and F n ⊆V n and µL (E n ∪F n )≥1−1/2n −2.Put E = E n +5 n ∈N .It is straightforward to show that E belongs to S and that χ(E )=f in L 1([0,1],µL ).This completes the proof.Remark 4.7.We have presented two notions of Lebesgue measurable set and shown that they are equivalent in WWKL 0.Our first notion (Definition 4.4)has the advantage of generality in that it applies to any measure on a compact separable metric space.Our second no-tion (Definition 4.5)is advantageous in other ways,namely it is more straightforward and works well in RCA 0.It would be desirable to find a single definition which combines all of these advantages.VITALI’S THEOREM AND WWKL 135.Vitali’s TheoremLet S be a collection of sets.A point x is said to be Vitali covered by S if for all >0there exists S ∈S such that x ∈S and the diameter of S is less than .The Vitali Covering Theorem in its simplest form says the following:if I is a sequence of intervals which Vitali covers an interval E in the real line,then I contains a countable,pairwise disjoint set of intervals I n ,n ∈N ,such that ∞n =0I n covers E except for a set of Lebesgue measure 0.The purpose of this section is to show that various forms of the Vitali Covering Theorem are provable in WWKL 0and in fact equivalent to WWKL over RCA 0.Throughout this section,we use µto denote Lebesgue measure.Lemma 5.1(Baby Vitali Lemma).The following is provable in RCA 0.Let I 0,...,I n be a finite sequence of intervals.Then we can find a pair-wise disjoint subsequence I k 0,...,I k m such thatµ(I k 0∪···∪I k m )≥1VITALI’S THEOREM AND WWKL 14I =[2a −b,2b −a ].)Thusµ(I 0∪···∪I n )≤µ(I k 0∪···∪I k m )≤µ(I k 0)+···+µ(I k m )=3µ(I k 0)+···+3µ(I k m )=3µ(I k 0∪···∪I k m )and the lemma is proved.Lemma 5.2.The following is provable in WWKL 0.Let E be an in-terval,and let I n ,n ∈N ,be a sequence of intervals.If E ⊆ ∞n =0I n ,then µ(E )≤lim k →∞µ k n =0I n .Proof.If the intervals I n are open,then the desired conclusion follows immediately from countable additivity (Theorem 3.3).Otherwise,fix >0and let I n be an open interval with the same midpoint as I n andµ(I n )=µ(I n )+µ(E \A ).(3)VITALI’S THEOREM AND WWKL 15To prove the claim,use Lemma 5.2and the Vitali property to find a finite set of intervals J 1,...,J l ∈I such that J 1,...,J l ⊆E \A andµ(E \(A ∪J 1∪···∪J l ))<13µ(J 1∪···∪J l ).We then have µ(E \(A ∪I 1∪···∪I k ))<212µ(E \A )≤212µ(E \A )=34nµ(E ).Then by countable additivity we have µ E \∞ n =1A n =0and the lemma is proved.Remark 5.4.It is straightforward to generalize the previous lemma to the case of a Vitali covering of the n -cube [0,1]n by closed balls or n -dimensional cubes.In the case of closed balls,the constant 3in the Baby Vitali Lemma 5.1is replaced by 3n .Theorem 5.5.The Vitali theorem for the interval [0,1](as stated in Lemma 5.3)is equivalent to WWKL over RCA 0.Proof.Lemma 5.3shows that,in RCA 0,WWKL implies the Vitali theorem for intervals.It remains to prove within RCA 0that the Vitali theorem for [0,1]implies WWKL.Instead of proving WWKL,we shall prove the equivalent statement 3.3.3.Reasoning in RCA 0,suppose thatVITALI’S THEOREM AND WWKL 16(a n ,b n ),n ∈N ,is a sequence of open intervals which covers [0,1].Let I be the countable set of intervals (a nki ,b nki )= a n +i k(b n −a n ) where i,k,n ∈N and 0≤i <k .Then I is a Vitali covering of [0,1].By the Vitali theorem for intervals,I contains a sequence of pairwise disjoint intervals I m ,m ∈N ,such that µ ∞ m =0I m ≥1.By disjoint countable additivity (Corollary 2.5),we have∞m =0µ(I m )≥1.From this it follows easily that∞n =0|a n −b n |≥1.Thus we have 3.3.3and our theorem is proved.We now turn to Vitali’s theorem for measurable sets.Recall our discussion of measurable sets in section 4.A sequence of intervals I is said to almost Vitali cover a Lebesgue measurable set E ⊆[0,1]if for all >0we have µL (E \O )=0,where O = {I |I ∈I ,diam(I )< }.Theorem 5.6.The following is provable in WWKL 0.Let E ⊆[0,1]be a Lebesgue measurable set with µ(E )>0.Let I be a sequence of intervals which almost Vitali covers E .Then I contains a pairwise disjoint sequence of intervals I n ,n ∈N ,such that µ E \∞ n =0I n =0.Proof.The proof of this theorem is similar to that of Lemma 5.3.The only modification needed is in the proof of the claim.Recall from Definition 4.5that E =lim n →∞E n where each E n is a finite union of intervals in [0,1].Fix m so large thatµ((E \E m )∪(E m \E ))<1VITALI’S THEOREM AND WWKL 17andµ(E m \(A ∪J 1∪···∪J l ))<136µ(E \A )<236µ(E \A )≤236µ(E \A )<236µ(E \A )=3,The Baire category theorem in weak subsystems of second order arith-metic ,Journal of Symbolic Logic 58(1993),557–578.7.O.Demuth and A.Kuˇc era,Remarks on constructive mathematical analysis ,[3],1979,pp.81–129.8.H.-D.Ebbinghaus,G.H.M¨u ller,and G.E.Sacks (eds.),Recursion Theory Week ,Lecture Notes in Mathematics,no.1141,Springer-Verlag,1985,IX +418pages.VITALI’S THEOREM AND WWKL189.Harvey Friedman,unpublished communication to Leo Harrington,1977.10.Harvey Friedman,Stephen G.Simpson,and Rick L.Smith,Countable algebraand set existence axioms,Annals of Pure and Applied Logic25(1983),141–181.11.,Randomness and generalizations offixed point free functions,[1],1990, pp.245–254.15.,Subsystems of Second Order Arithmetic,Perspectives in Mathematical Logic,Springer-Verlag,1998,XIV+445pages.21.Xiaokang Yu,Measure Theory in Weak Subsystems of Second Order Arithmetic,Ph.D.thesis,Pennsylvania State University,1987,vii+73pages.22.,Riesz representation theorem,Borel measures,and subsystems of sec-ond order arithmetic,Annals of Pure and Applied Logic59(1993),65–78. 24.,A study of singular points and supports of measures in reverse mathe-matics,Annals of Pure and Applied Logic79(1996),211–219.26.Xiaokang Yu and Stephen G.Simpson,Measure theory and weak K¨o nig’slemma,Archive for Mathematical Logic30(1990),171–180.E-mail address:dkb5@,giusto@dm.unito.it,simpson@ The Pennsylvania State University。
韦达定理 整系数多项式

韦达定理整系数多项式The Vieta's theorem, also known as Vieta's formulas, provides a powerful method for relating the roots of a polynomial to its coefficients. This theorem is named after the French mathematician François Viète, who made significant contributions to algebra in the 16th century. By understanding and applying Vieta's theorem, mathematicians and students can solve complex problems involving polynomial equations with ease.韦达定理,也称作韦达公式,为将多项式的根与其系数进行关联提供了强大的方法。
这个定理以十六世纪的法国数学家弗朗索瓦·韦达的名字命名,他对代数学做出了重大的贡献。
通过理解和应用韦达定理,数学家和学生可以轻松解决涉及多项式方程的复杂问题。
Vieta's theorem states that the sum and product of the roots of a polynomial equation with real or complex coefficients can be expressed in terms of the coefficients of the polynomial. By examining the relationship between the roots and coefficients, mathematicians can derive useful formulas to simplify the manipulation of polynomial equations. This theorem is fundamentalin the study of algebra and has applications in various branches of mathematics, such as number theory, calculus, and geometry.韦达定理陈述了具有实数或复数系数的多项式方程的根的和与积可以用多项式的系数来表达。
2011年丘成桐大学生数学竞赛——代数,数论与组合卷

S.-T.Yau College Student Mathematics Contests 2011Algebra,Number Theory andCombinatoricsIndividual2:30–5:00pm,July 10,2011(Please select 5problems to solve)For the following problems,every example and statement must be backed up by proof.Examples and statements without proof will re-ceive no-credit.1.Let K =Q (√−3),an imaginary quadratic field.(a)Does there exists a finite Galois extension L/Q which containsK such that Gal(L/Q )∼=S 3?(Here S 3is the symmetric group in 3letters.)(b)Does there exists a finite Galois extension L/Q which containsK such that Gal(L/Q )∼=Z /4Z ?(c)Does there exists a finite Galois extension L/Q which containsK such that Gal(L/Q )∼=Q ?Here Q is the quaternion group with 8elements {±1,±i,±j,±k },a finite subgroup of the group of units H ×of the ring H of all Hamiltonian quaternions.2.Let f be a two-dimensional (complex)representation of a finite group G such that 1is an eigenvalue of f (σ)for every σ∈G .Prove that f is a direct sum of two one-dimensional representations of G3.Let F ⊂R be the subset of all real numbers that are roots of monic polynomials f (X )∈Q [X ].(1)Show that F is a field.(2)Show that the only field automorphisms of F are the identityautomorphism α(x )=x for all x ∈F .4.Let V be a finite-dimensional vector space over R and T :V →V be a linear transformation such that(1)the minimal polynomial of T is irreducible;(2)there exists a vector v ∈V such that {T i v |i ≥0}spans V .Show that V contains no non-trivial proper T -invariant subspace.5.Given a commutative diagramA →B →C →D →E↓↓↓↓↓A →B →C →D →E1Algebra,Number Theory and Combinatorics,2011-Individual2 of Abelian groups,such that(i)both rows are exact sequences and(ii) every vertical map,except the middle one,is an isomorphism.Show that the middle map C→C is also an isomorphism.6.Prove that a group of order150is not simple.。
国家重点基础研究发展计划(973)项目

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systems10:50-11:15李邦和酶动力学中的拟稳态假设10:50-11:15张树功多元有理插值的Groebner基方法11:15-11:40王怀富A criterion for the similarity of length-two elements in a PID11:15-11:40LEI YANG, 李树荣Optimization of injectionstrategies for polymerflooding based on areal-coded geneticalgorithm11:15-11:40Erich Kaltofen, 李斌, 杨争锋, 支丽红Exact Certification ofGlobal Optimality ofApproximateFactorizations ViaRationalizingSums-Of-Squares withFloating Point Scalars11:40-12:05郑大彬, 吴敏Testing algebraicdependence of hyperexponential elements11:40-12:05侯春望因子优化法在控制系统根轨迹绘制中的应用11:40-12:05王明生Prime factorization ofmultivariate polynomialmatrices12:00-2:00 午餐10月25日2:00-3:40 分组报告:**会议室**会议室**会议室分组4:微分代数(主席:李志斌)分组5:编码与密码(主席:邢朝平)分组6:应用与算法(主席:齐东旭)2:00-2:25朝鲁DifferentialCharacteristic Set Algorithm for the Complete Symmetry Classificationof PDEs2:00-2:25林东岱, 邓炎炎密码学理论中的挑战2:00-2:25黄雷, 李洪波基于共形几何和复数法的几何计算新方法2:25-2:50刘姜, 李洪波, 曹源昊涉及坐标变换的微分多项式在求和约定下的化简和标准型2:25-2:50吴文玲Improved ImpossibleDifferentialCryptanalysis ofReduced-Round Camellia2:25-2:50廖啟征四元数的复数形式及其在机构求解中的应用2:50-3:15李子明, Martin Ondera, 王怀富Simplifying skewfractions modulodifferential anddifference relations2:50-3:15刘峰, 武传坤, 林喜军Color Visual CryptographySchemes2:50-3:15李忠, 王爱玲一种基于D-S证据推理的信息融合改进算法3:15-3:40袁春明差分素理想的一个判定准则3:15-3:40邓映蒲攻破Cai-Cusick基于格的公钥密码系统3:15-3:403:40-4:00 休息10月25日4:00-5:40 分组报告:**会议室**会议室**会议室分组7:组合与图论(主席:王明生)分组8:有限域(主席:刘卓军)分组9:计算机视觉与模式识别(主席:查红彬)4:00-4:25陈永川, 唐凌, 王星炜, 杨立波Schur positivity and q-log-convexity4:00-4:25高小山, 黄震宇有限域上求解多项式方程的特征列方法4:00-4:25阮秋琦基于偏微分方程的最具可分性人脸特征融合的预处理算法4:25-4:50Burcin Erocal, 侯庆虎,Peter PauleAn implement of MacMahon's partition analysis4:25-4:50赵尚威有限域上二次方程组求解的近似算法4:25-4:50罗定生汉语词汇的一体化联合分析方法研究4:50-5:15冯荣权Enumerating typicalabelian prime-fold coverings of a circulantgraph4:50-5:15孙瑶, 王定康有限域F2上Groebner基的计算4:50-5:15张超Multivariate LaplaceFilter: a Heavy-TailedModel for Target Tracking5:15-5:40谢应泰A polynomial time algorithm for judgingH-graph5:15-5:40张艳硕基于身份的短代理签名方案及其扩展5:15-5:40许超多媒体检索中的转移学习10月26日主会场2(主席:李洪波)08:30-09:15 邀请报告: 邢朝平, 南洋理工大学Space-time codes--introduction and constructions09:15-10:00 邀请报告: 张健, 中科院软件所有限模型和反例的搜索10:00-10:20 休息10月26日10:20-12:00 分组报告:**会议室**会议室**会议室分组10:实代数方法(主席:冯勇)分组11:计算机图形学与辅助设计(主席:陈发来)分组12:优化算法(主席:支丽红)10:20-10:45张景中直观几何代数基础问题10:20-10:45陈冲, 徐国良几何设计中的水平集方法10:20-10:45黄文奇, 叶涛等圆Packing问题完全拟物算法的进一步研究10:45-11:10 邵俊伟, 侯晓荣基于区间分析的不等式自动证明系统10:45-11:10汪国昭混合B样条的统一表示10:45-11:10谢福鼎时序波动周期关联规则挖掘的一个算法11:10-11:35曾振柄基于区域剖分的不等式证明11:10-11:35李华基于几何不变量的三维形状分析和检索11:10-11:35纪哲基于层次分析法的购房策略模型11:35-12:00 张志海, 马蕾, 夏壁灿判定一类线性程序终止性的加速算法11:35-12:00宋瑞霞数字图象自适应非均匀分割及其应用11:35-12:00刘新平, 刘颖基于最大最小距离的改进遗传算法12:00-2:00 午餐10月26日2:00-3:40 分组报告:**会议室**会议室**会议室分组13:逻辑与网络(主席:张健)分组14:模式识别(主席:李华)分组15:微分方程(主席:李子明)2:00-2:25吴尽昭基于代数符号计算的形式化验证方法及其若干关键问题研究2:00-2:25杨国为, 王守觉判定一点是否属于高维复杂形体的算法2:00-2:25张鸿庆一类非线性偏微分方程组的解析解2:25-2:50Guang Zheng, 李廉,吴尽昭, Wenbo Chen Weaker bisimulation: how to make a+b and tau.a+bequivalent?2:25-2:50査红彬, 裴玉茹The CraniofacialReconstruction from theLocal Structural Diversityof Skulls2:25-2:50李志斌Darboux变换与多孤子解算法研究2:50-3:15杜玉越逻辑工作流网及其应用2:50-3:15林通流形学习理论与应用2:50-3:15陆征一Computer aided anal- ysisfor differentialpolynomial systems3:15-3:40 刘家保, 潘向峰Estrada Index of Hypercubes Networks3:15-3:40邓九英, 王钦若,毛宗源, 杜启亮基于粗糙集的支持向量回归机混合算法3:15-3:40闫振亚The MKdV eqs with variablecoefficients: Exactuni/bi-variable travelingwave-like solutions3:40-4:00 休息10月26日4:00-5:40 分组报告:**会议室**会议室**会议室分组16:实代数方法(主席:曾振柄)分组17:计算机辅助设计与数控(主席:徐国良)分组18:控制方法(主席:李树荣)4:00-4:25符红光Dixon结式的三类多余因子4:00-4:25杨周旺点云曲线/曲面的微分信息计算4:00-4:25王峰, 杨永青多目标随机规划在区域水资源优化调度中的应用4:25-4:50冯勇, 张景中Obtaining Exact Inter- polation Multivariate Polynomial byApproximations4:25-4:50韩丽基于复杂截面点云的三角网格模型重建和特征检测方法研究4:25-4:50张玉斌基于MPI的迭代动态规划并行化4:50-5:15 Zhen-Yi Ji, 李永彬Some Improvements upon Unmixed Decomposition of An Algebraic Variety4:50-5:15张梅, 曹源昊数控系统中的数据压缩4:50-5:15田华阁, 车荣杰, 王平, 田学民基于FP-EFCM的聚丙烯熔融指数软测量5:15-5:40王云诚, 方伟武,吴天骄A New Bisection-Newton Method for Finding Real Roots of UnivariatePolynomials5:15-5:40李家代数曲线与曲面拓扑的确定与逼近5:15-5:40张晓东聚合物驱最优控制问题的必要条件及数值求解第二届全国计算机数学大会报告摘要10月25日主会场1(主席:高小山)09:00-09:45 邀请报告: 徐宗本, 西安交通大学题目:基于视觉认知的数据建模摘要:数据建模是信息技术的共有基础,是当今信息化社会数学应用的主要形式之一,其目的是揭示数据中所隐含的信息(结构、模式与规律等)。
Eigenvalues of a real supersymmetric tensor

Abstract In this paper, we define the symmetric hyperdeterminant, eigenvalues and E-eigenvalues of a real supersymmetric tensor. We show that eigenvalues are roots of a one-dimensional polynomial, and when the order of the tensor is even, E-eigenvalues are roots of another one-dimensional polynomial. These two one-dimensional polynomials are associated with the symmetric hyperdeterminant. We call them the characteristic polynomial and the E-characteristic polynomial of that supersymmetric tensor. Real eigenvalues (E-eigenvalues) with real eigenvectors (E-eigenvectors) are called H-eigenvalues (Z-eigenvalues). When the order of the supersymmetric tensor is even, H-eigenvalues (Z-eigenvalues) exist and the supersymmetric tensor is positive definite if and only if all of its H-eigenvalues (Z-eigenvalues) are positive. An m th-order n -dimensional supersymmetric tensor where m is even has exactly n (m − 1)n −1 eigenvalues, and the number of its E-eigenvalues is strictly less than n (m − 1)n −1 when m ≥ 4. We show that the product of all the eigenvalues is equal to the value of the symmetric hyperdeterminant, while the sum of all the eigenvalues is equal to the sum of the diagonal elements of that supersymmetric tensor, multiplied by (m − 1)n −1 . The n (m − 1)n −1 eigenvalues are distributed in n disks in C. The centers and radii of these n disks are the diagonal elements, and the sums of the absolute values of the corresponding off-diagonal elements, of that supersymmetric tensor. On the other hand, E-eigenvalues are invariant under orthogonal transformations. © 2005 Elsevier Ltd. All rights reserved.
Polynomialfunctions

Polynomial functionsMany common functions are polynomial functions.In this unit we describe polynomial functions and look at some of their properties.In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.After reading this text,and/or viewing the video tutorial on this topic,you should be able to:•recognise when a rule describes a polynomial function,and write down the degree of the polynomial,•recognize the typical shapes of the graphs of polynomials,of degree up to4,•understand what is meant by the multiplicity of a root of a polynomial,•sketch the graph of a polynomial,given its expression as a product of linear factors.Contents1.Introduction22.What is a polynomial?23.Graphs of polynomial functions34.Turning points of polynomial functions65.Roots of polynomial functions71.IntroductionA polynomial function is a function such as a quadratic,a cubic,a quartic,and so on,involving only non-negative integer powers of x.We can give a general defintion of a polynomial,and define its degree.2.What is a polynomial?A polynomial of degree n is a function of the formf(x)=a n x n+a n−1x n−1+...+a2x2+a1x+a0where the a’s are real numbers(sometimes called the coefficients of the polynomial).Although this general formula might look quite complicated,particular examples are much simpler.For example,f(x)=4x3−3x2+2is a polynomial of degree3,as3is the highest power of x in the formula.This is called a cubic polynomial,or just a cubic.Andf(x)=x7−4x5+1is a polynomial of degree7,as7is the highest power of x.Notice here that we don’t need every power of x up to7:we need to know only the highest power of x tofind out the degree.An example of a kind you may be familiar with isf(x)=4x2−2x−4which is a polynomial of degree2,as2is the highest power of x.This is called a quadratic. Functions containing other operations,such as square roots,are not polynomials.For example,√f(x)=4x3+3.Graphs of polynomial functionsWe have met some of the basic polynomials already.For example,f(x)=2is a constant function and f(x)=2x+1is a linear function.It is important to notice that the graphs of constant functions and linear functions are always straight lines.We have already said that a quadratic function is a polynomial of degree2.Here are some examples of quadratic functions:f(x)=x2,f(x)=2x2,f(x)=5x2.What is the impact of changing the coefficient of x2as we have done in these examples?One way tofind out is to sketch the graphs of the functions.You can see from the graph that,as the coefficient of x2is increased,the graph is stretched vertically(that is,in the y direction).What will happen if the coefficient is negative?This will mean that all of the positive f(x)values will now become negative.So what will the graphs of the functions look like?The functions are nowf(x)=−x2,f(x)=−2x2,f(x)=−5x2.−x22Notice here that all of these graphs have actually been reflected in the x-axis.This will always happen for functions of any degree if they are multiplied by−1.Now let us look at some other quadratic functions to see what happens when we vary the coefficient of x,rather than the coefficient of x2.We shall use a table of values in order to plot the graphs,but we shallfill in only those values near the turning points of the functions.x−5−3−112060−40−8−8You can see the symmetry in each row of the table,demonstrating that we have concentrated on the region around the turning point of each function.We can now use these values to plot the graphs.As you can see,increasing the positive coefficient of x in this polynomial moves the graph down and to the left.What happens if the coefficient of x is negative?x−1135602x2−4x−3−3−8−8−2022063177511−2−42Our table of values is particularly easy to complete since we can use our answers from the x2+x column tofind everything else.We can use these tables of values to plot the graphs of the functions.x 2 + x + 5x 2 + x − 4x 2 + x + 1x 2 + x As we can see straight away,varying the constant term translates the x 2+x curve vertically.Furthermore,the value of the constant is the point at whichthe graph crosses the f (x )axis.4.T urning points of polynomial functionsA turning point of a function is apoint where the graph of the function changes from sloping downwards to sloping upwards,or vice versa.So the gradient changes from negative to positive,or from positive to negative.Generally speaking,curves ofdegree n can have up to (n −1)turning points.For instance,a quadratic has only one turning point.A cubic could have up to two turning points,and so would look something like this.However,some cubics have fewer turning points:for example f (x )=x 3.But no cubic has more than two turning points.In the same way,a quartic could have up tothree turning turning points,and so wouldlook something like this.Again,some quartics have fewer turning points,but none has more.Key PointA polynomial of degree n can have up to(n−1)turning points.5.Roots of polynomial functionsYou may recall that when(x−a)(x−b)=0,we know that a and b are roots of the function f(x)=(x−a)(x−b).Now we can use the converse of this,and say that if a and b are roots, then the polynomial function with these roots must be f(x)=(x−a)(x−b),or a multiple of this.For example,if a quadratic has roots x=3and x=−2,then the function must be f(x)= (x−3)(x+2),or a constant multiple of this.This can be extended to polynomials of any degree. For example,if the roots of a polynomial are x=1,x=2,x=3,x=4,then the function must bef(x)=(x−1)(x−2)(x−3)(x−4),or a constant multiple of this.Let us also think about the function f(x)=(x−2)2.We can see straight away that x−2=0, so that x=2.For this function we have only one root.This is what we call a repeated root, and a root can be repeated any number of times.For example,f(x)=(x−2)3(x+4)4has a repeated root x=2,and another repeated root x=−4.We say that the root x=2has multiplicity3,and that the root x=−4has multiplicity4.The useful thing about knowing the multiplicity of a root is that it helps us with sketching the graph of the function.If the multiplicity of a root is odd then the graph cuts through the x-axis at the point(x,0).But if the multiplicity is even then the graph just touches the x-axis at the point(x,0).For example,take the functionf(x)=(x−3)2(x+1)5(x−2)3(x+2)4.•The root x=3has multiplicity2,so the graph touches the x-axis at(3,0).•The root x=−1has multiplicity5,so the graph crosses the x-axis at(−1,0).•The root x=2has multiplicity3,so the graph crosses the x-axis at(2,0).•The root x=−2has multiplicity4,so the graph touches the x-axis at(−2,0).To take another example,suppose we have the function f(x)=(x−2)2(x+1).We can see that the largest power of x is3,and so the function is a cubic.We know the possible general shapes of a cubic,and as the coefficient of x3is positive the curve must generally increase to the right and decrease to the left.We can also see that the roots of the function are x=2and x=−1.The root x=2has even multiplicity and so the curve just touches the x-axis here, whilst x=−1has odd multiplicity and so here the curve crosses the x-axis.This means we can sketch the graph as follows.Key PointThe number a is a root of the polynomial function f(x)if f(a)=0,and this occurs when (x−a)is a factor of f(x).If a is a root of f(x),and if(x−a)m is a factor of f(x)but(x−a)m+1is not a factor,then we say that the root has multiplicity m.At a root of odd multiplicity the graph of the function crosses the x-axis,whereas at a root of even multiplicity the graph touches the x-axis.Exercises1.What is a polynomial function?2.Which of the following functions are polynomial functions?(a)f(x)=4x2+2(b)f(x)=3x3−2x+√3.Write down one example of each of the following types of polynomial function:(a)cubic(b)linear(c)quartic(d)quadratic4.Sketch the graphs of the following functions on the same axes:(a)f(x)=x2(b)f(x)=4x2(c)f(x)=−x2(d)f(x)=−4x25.Consider a function of the form f(x)=x2+ax,where a represents a real number.The graph of this function is represented by a parabola.(a)When a>0,what happens to the parabola as a increases?(b)When a<0,what happens to the parabola as a decreases?6.Write down the maximum number of turning points on the graph of a polynomial function of degree:(a)2(b)3(c)12(d)n7.Write down a polynomial function with roots:(a)1,2,3,4(b)2,−4(c)12,−1,−68.Write down the roots and identify their multiplicity for each of the following functions:(a)f(x)=(x−2)3(x+4)4(b)f(x)=(x−1)(x+2)2(x−4)39.Sketch the following functions:(a)f(x)=(x−2)2(x+1)(b)f(x)=(x−1)2(x+3)Answers1.A polynomial function is a function that can be written in the formf(x)=a n x n+a n−1x n−1+a n−2x n−2+...+a2x2+a1x+a0,where each a0,a1,etc.represents a real number,and where n is a natural number(including 0).2.(a)f(x)=4x2+2is a polynomial(b)f(x)=3x3−2x+√x(c)f(x)=12−4x5+3x2is a polynomial(d)f(x)=sin x+1is not a polynomial,because of sin x(e)f(x)=3x12−2/x is not a polynomial,because of2/x(f)f(x)=3x11−2x12is a polynomial3.(a)The highest power of x must be3,so examples might be f(x)=x3−2x+1or f(x)=x3−2.(b)The highest power of x must be1,so examples might be f(x)=x or f(x)=6x−5.(c)The highest power of x must be4,so examples might be f(x)=x4−3x3+2x orf(x)=x4−x−5.(d)The highest power of x must be2,so examples might be f(x)=x2or f(x)=x2−5. 4.5.(a)When a>0,the parabola moves down and to the left as a increases.(b)When a<0,the parabola moves down and to the right as a decreases. 6.(a)1turning point(b)2turning points(c)11turning points (d)(n−1)turning points.7.(a)f(x)=(x−1)(x−2)(x−3)(x−4)or a multiple(b)f(x)=(x−2)(x+4)or a multiple(c)f(x)=(x−12)(x+1)(x+6)or a multiple.8.(a)x=2odd multiplicityx=−4even multiplicity(b)x=1odd multiplicityx=−2even multiplicityx=4odd multiplicity9)− 1)2(x + 3)x− 2)2(x + 1)11c math centre July18,2005。
抽象代数

一、课程目的与教学基本要求本课程是在学生已学习大学一年级“几何与代数”必修课的基础上,进一步学习群、环、域三个基本的抽象的代数结构。
要求学生牢固掌握关于这三种抽象的代数结构的基本事实、结果、例子。
对这三种代数结构在别的相关学科,如数论、物理学等的应用有一般了解。
二、课程内容第1章准备知识(Things Familiar and Less Familiar)10课时复习集合论、集合间映射及数学归纳法知识,通过学习集合间映射为继续学习群论打基础。
1、几个注记(A Few Preliminary Remarks)2、集论(Set Theory)3、映射(Mappings)4、A(S)(The Set of 1-1 Mappings of S onto Itself)5、整数(The Integers)6、数学归纳法(Mathematical Induction)7、复数(Complex Numbers)第2章群(Groups) 22课时建立关于群、子群、商群及直积的基本概念及基本性质;通过实例帮助建立抽象概念,掌握群同态定理及其应用;了解有限阿贝尔群的结构。
1、群的定义和例子(Definitions and Examples of Groups)2、一些简单注记(Some Simple Remarks)3、子群(Subgroups)4、拉格朗日定理(Lagrange’s Theorem)5、同态与正规子群(Homomorphisms and Normal Subgroups)6、商群(Factor Groups)7、同态定理(The Homomorphism Theorems)8、柯西定理(Cauchy’s Theorem)9、直积(Direct Products)10、有限阿贝尔群(Finite Abelian Groups) (选讲)11、共轭与西罗定理(Conjugacy and Sylow’s Theorem)(选讲)第3章对称群(The Symmetric Group) 8课时掌握对称群的结构定理,了解单群的概念及例子。
Figure Captions

V = fz = x + iy : ?0:501 x ?0:497; 0:537 y 0:541g :
Numerical computation (evaluating polynomials of degrees 9 with 0; 1 coe cients at a 41 41 uniform grid, and bounding derivatives) shows that if w 2 V \ W , then w can only be a zero of a power series of the form
27
14] P. X. Gallagher, The large sieve and probabilistic Galois theory, pp. 91{101 in Analytic Number Theory, H. G. Diamond, ed., Proc. Symp. Pure Math., vol. 24, Amer. Math. Soc., 1973. 15] J. von Scheidt and A. T. Bharucha-Reid, On the averaging problem for the roots of random algebraic polynomials, Wiss. Beitr. Ingenieurhochschule Zwickau 9 (1983), 1{43. 16] B. Solomyak, Conjugates of beta-numbers and the zero-free domain for a class of analytic functions, to be published. 17] E. C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford Univ. Press, 1939.
2011年丘成桐大学生数学竞赛——代数,数论与组合卷

S.-T.Yau College Student Mathematics Contests 2011Algebra,Number Theory andCombinatoricsIndividual2:30–5:00pm,July 10,2011(Please select 5problems to solve)For the following problems,every example and statement must be backed up by proof.Examples and statements without proof will re-ceive no-credit.1.Let K =Q (√−3),an imaginary quadratic field.(a)Does there exists a finite Galois extension L/Q which containsK such that Gal(L/Q )∼=S 3?(Here S 3is the symmetric group in 3letters.)(b)Does there exists a finite Galois extension L/Q which containsK such that Gal(L/Q )∼=Z /4Z ?(c)Does there exists a finite Galois extension L/Q which containsK such that Gal(L/Q )∼=Q ?Here Q is the quaternion group with 8elements {±1,±i,±j,±k },a finite subgroup of the group of units H ×of the ring H of all Hamiltonian quaternions.2.Let f be a two-dimensional (complex)representation of a finite group G such that 1is an eigenvalue of f (σ)for every σ∈G .Prove that f is a direct sum of two one-dimensional representations of G3.Let F ⊂R be the subset of all real numbers that are roots of monic polynomials f (X )∈Q [X ].(1)Show that F is a field.(2)Show that the only field automorphisms of F are the identityautomorphism α(x )=x for all x ∈F .4.Let V be a finite-dimensional vector space over R and T :V →V be a linear transformation such that(1)the minimal polynomial of T is irreducible;(2)there exists a vector v ∈V such that {T i v |i ≥0}spans V .Show that V contains no non-trivial proper T -invariant subspace.5.Given a commutative diagramA →B →C →D →E↓↓↓↓↓A →B →C →D →E1Algebra,Number Theory and Combinatorics,2011-Individual2 of Abelian groups,such that(i)both rows are exact sequences and(ii) every vertical map,except the middle one,is an isomorphism.Show that the middle map C→C is also an isomorphism.6.Prove that a group of order150is not simple.。
RootsandRadicalExpressions7根和根的表达式课件

Notice that the index is an odd number here . . .
ex :
3 x3 y6 3 x3( y2)3 x y2
Absolute value symbols must not be used here. If x is negative, then the radicand is negative and the root must also be negative.
Here's another way tolookat it : IF x 5, x2 52 25 5 x BUT if x 5, x2 (5)2 25 5 x so we want tomakesure that xis alwaysa positive with evenindexesso thatit is truebackwards and forwards.Weuse theabsolutevalue tomakesure it stays positive.
radicals • Complex numbers • Conjugates
What is a monomial?
An expression that is a number, that may or may not include a variable.
MONOMIALS
10xy
2x2
4x
NOT MONOMIALS
index
Radical sign
na
radicand
How to find the root (other than a square root), using a graphing calculator
晒晒我们班的数学牛人英语作文600字英语作文

全文分为作者个人简介和正文两个部分:作者个人简介:Hello everyone, I am an author dedicated to creating and sharing high-quality document templates. In this era of information overload, accurate and efficient communication has become especially important. I firmly believe that good communication can build bridges between people, playing an indispensable role in academia, career, and daily life. Therefore, I decided to invest my knowledge and skills into creating valuable documents to help people find inspiration and direction when needed.正文:晒晒我们班的数学牛人英语作文600字英语作文全文共3篇示例,供读者参考篇1My Class Math ProdigyYou know how every class has that one kid who just seems to be operating on another level when it comes to math? The one who solves complicated equations in their head while therest of us are still trying to borrow from the tens column? Well, in my class, that freakishly gifted mathlete is Aiden Patel.I still remember the first time I realized Aiden was a legitimate math genius. It was during Mrs. Thompson's algebra class back in 9th grade. She had written out a hairy polynomial equation on the whiteboard and asked if anyone could simplify it. Most of us just stared at it blankly, completely overwhelmed. But Aiden's hand shot straight up. Mrs. Thompson called on him, probably expecting him to at least attempt an algebraic solution. Instead, Aiden just blurted out the correct simplified form immediately. He didn't show any work or explain his reasoning - he had somehow arrived at the final answer through pure mental calculation!Mrs. Thompson and the rest of the class were dumbfounded. After an awkward silence, she reluctantly confirmed that Aiden was indeed correct. From that day on, he solidified his reputation as the undisputed math whiz of our grade. Anytime a teacher posed a remotely challenging math problem, all eyes immediately turned to Aiden. Without fail, his hand would shoot up and he would effortlessly recite the solution, even for the most labyrinthine equations and proofs.What makes Aiden's talents even more remarkable is that he doesn't come off as a natural-born genius in other subjects. In English class, he struggles with essays and literary analysis just like the rest of us. He's an okay athlete but nothing special on the sports teams. In areas outside of mathematics, he seems...well, pretty normal and averagely intelligent. But when it comes to numbers, logic, patterns, probability - anything even remotely related to math - Aiden is in a completely different stratosphere. It's honestly mind-boggling.His skills extend far beyond simply being a human calculator, too. In Geometry last year, he didn't just memorize formulas and theorems - he seemed to derive an innate, intuitive FEEL for shapes, angles, spatial reasoning, and proofs. Sketching out an elegant geometric proof on the board was like artistic expression for Aiden. He would get this intense look of rapturous focus, like he was conceiving dimensions and axioms that the rest of us couldn't even perceive.Then in AP Calculus this year, he took his talents to even higher realms of abstract mathematics. Aiden treated integrals, derivatives, and limits not just as headache-inducing computations, but as frameworks for modeling and understanding the entire universe. To him, calculus was abeautiful language for describing motion, change, and the infinite. Our teacher Mr. Singh says Aiden grasps calculus concepts that most students don't fully understand until graduate-level real analysis.Despite his otherworldly skills, Aiden carries himself with an easygoing humility. He's not an eccentric math obsessive or arrogant teacher's pet - just a regular teenager who happens to possess savant-level aptitudes. Aiden is honestly a solid,down-to-earth guy who loves video games, alt rock, and pizza as much as the next kid. His freakish talents don't define his whole personality.That said, there's no doubt Aiden is destined for greatness in mathematics and anything else quantitative. Most of us will likely end up in fairly conventional career paths - Aiden is practically guaranteed to become an ingenious researcher, pioneering scientific pioneer, or era-defining technological innovator. Maybe he'll map ultra-dense computational networks, reformulate economic theory through a numerical lens, or even help unify physics' grand unified theory. Nothing seems beyond the lofty scope of Aiden's supreme mathematical mind.For now though, Aiden is just enjoying his teenage years alongside the rest of us average Joes. Every so often in class, ourMath Maestro still leaves us all slack-jawed at the depth and dexterity of his numerical abilities. His gift is both incredibly fascinating and utterly mystifying to the mere mortal students around him. While the rest of us get migraines from quadratics and polynomials, Aiden casually crests stratospheres of mathematical brilliance that we can scarcely fathom. We're all just lucky we get to witness his precocious genius bloom firsthand during these formative high school years. Who knows what cosmic truths and revelations that incredible mind will one day unlock for us?篇2The Mathletes of Class 3BThere's something special about Class 3B – we're a bunch of math whizzes! I'm not just saying that to brag, although a little bragging is definitely allowed. We've got a squad of number nerds who can calculate like calculators on steroids. Let me introduce you to the Mathletes of 3B.First up, we have Aisha, our resident human calculator. This girl can rattle off square roots of 10-digit numbers faster than I can say "mathematic-ally challenged." Rumor has it she does complex equations in her head for fun during recess. I'vewitnessed her resolving quarrels over pocket money by crunching numbers mid-argument. "You owe Jayden 3.75 for that candy bar trade last week," she'll announce, and just like that, the matter is settled. Aisha is, without a doubt, our undisputed Math Queen.Then there's Thomas, a soft-spoken boy who becomes a math beast when challenged. His genius lies in geometric constructions and 3D visualizations. You should see the masterpieces he can create using just a compass, ruler, and well, his brilliant mind. Last year's Math Fair had everyone gawking at his mind-bending geometric art installments. Who knew angles and lines could be sculpted into such hypnotic patterns? That's our Thomas – a modern-day Euclid with a knack for shapes.We can't forget Samantha, whose superpower islightning-fast mental math. This human calculator can crush arithmetic problems at warp speed, all while braiding her hair or doodling geometric doodles. Show her a string of numbers, and within seconds, she'll burst out with the precise answer…and maybe a silly math pun too. "Whatdo you call a line that really loves pizza? A slice of pi!" Samantha's brain is basically a parallel processing unit dedicated to crunching numbers at turbo speeds.Of course, every elite squad has its unsung heroes too. In our case, it's the tireless efforts of Ethan and Priya. These two form an unstoppable equation-solving duo, dividing and conquering every challenge thrown their way. Ethan is a master at algebraic manipulation, while Priya has a supernatural intuition for patterns and sequences. Together, they're an unmatchable math force, often staying back late to tackle bonus problems "just for fun." Their dedication, teamwork, and sheer love for the subject inspires all of us mathsters.As you can tell, Class 3B isn't just about books and blackboards. We're a squad of number nerds, mental math masters, and problem-solving prodigies. We revel in the beauty of mathematics, finding solutions where others only see garbles of numbers and symbols. Some may call us "uncool" for getting excited over mathematical puzzles, but hey, to each their own. We're the Mathletes, and we wear that badge with geeky pride!篇3Math Prodigies in Our ClassYou know that feeling when the math teacher announces a surprise test, and a collective groan echoes through the classroom? Well, not for the math geniuses in our class! Whilethe rest of us break into a cold sweat, these remarkable individuals barely bat an eyelash. They're the ones who make complex equations look like child's play, leaving us mere mortals in awe of their brilliance.At the top of the leaderboard, we have Emily, our resident math prodigy. From the moment she stepped into our classroom, it was clear that numbers held no secrets from her. She's like a human calculator, effortlessly crunching through complex problems that would leave the rest of us scratching our heads in bewilderment. Emily's ability to grasp advanced mathematical concepts with ease is nothing short of extraordinary.Then there's Michael, the master of mental math. This guy can solve multi-digit calculations in his head faster than most of us can punch numbers into a calculator. It's like he has a built-in supercomputer in his brain, processing intricate equations at lightning speed. We've all witnessed him breeze through challenging problems on the whiteboard, leaving the teacher in a state of utter amazement.Don't get me started on Samantha, the queen of logic and reasoning. Her ability to dissect word problems and unravel their hidden mathematical gems is truly remarkable. She has a knack for breaking down complex scenarios into a series of logicalsteps, making even the most convoluted problems seem straightforward. Samantha's analytical prowess is the envy of our entire class.And let's not forget about David, the geometry master. This guy can visualize and manipulate shapes in his mind like a seasoned architect. Whether it's calculating the area of irregular polygons or solving intricate proofs, David navigates the world of geometry with an ease that defies comprehension. His spatial reasoning skills are off the charts!But what truly sets these math whizzes apart is not just their exceptional abilities; it's their passion for the subject. They genuinely love the thrill of tackling complex problems and finding elegant solutions. You can see the excitement in their eyes when a new mathematical challenge presents itself, and they dive into it with an enthusiasm that's nothing short of infectious.Of course, being surrounded by such mathematical brilliance can be both inspiring and intimidating for the rest of us. But you know what? We've learned to embrace and appreciate their talents. After all, they're the ones who keep our class on its toes, pushing us to strive for excellence and never settle for mediocrity.So, here's to the math prodigies in our class – the ones who make numbers dance and equations sing. They may have left us in the dust academically, but their dedication and love for mathematics have undoubtedly inspired us to reach for greater heights. Who knows, maybe one day we'll join their ranks and become math whizzes ourselves!。
四次函数

四次函数形如y=ax^4+bx^3+cx^2+dx+e(a≠0,b,c,d,e为常数)的函数叫做四次函数。
四次函数的图像a*x^4+b*x^3+c*x^2+b*x+a=0的求解方法,对于一般的四次方程a*x^4+b*x^3+c*x^2 +d*x+e=0,先求解三次方程8y^3-4cy^2+(2bd-8e)y+e(4c-b^2)-d^2=0,得到的y的任一实根分别代入下面两个方程:x^2+(b+sqrt(8y+b^2-4c))x/2+(y+(by-d)/sqrt(8y+b^2-4c))=0及x^2+(b-sqrt(8y+b^2-4c))x/2+(y-(by-d)/sqrt(8y+b^2-4c))=0就可得到原方程的四个根。
在数学中, 四次方程是令一个四次函数等于零的结果.四次方程的一个例子如下<math>2x^4+4x^3-26x^2-28x+48=0;</math>它的通式是<math>a_0x^4+a_1x^3+a_2x^2+a_3x+a_4=0,\qquad\mboxa_0\ne0.</ math>代数基本定理告诉我们, 一个四次方程总有四个解(根). 它们可能是复数而且可能有等根.[编辑]解决四次方程自然,人们为了找到这些根做了许多努力. 就像其它多项式, 有时可能对一个四次方程分解出因式;但更多的时候这样的工作是极困难的,尤其是当根是无理数或复数时.因此找到一个通式解法或运算法则(就像二次方程那样, 能解所有的一元二次方程)是很有用的. After much effort, such a formula was indeed found for quarti cs —but since then it has been proven (by Evariste Galois) that such an approach dead-ends with quartics; they are the highest-degree polynomial eq uations whose roots can be expressed in a formula using a finite number of arithmetic operators and n-th roots. From quintics on up, one requires more powerful methods if a general algebraic solution is sought, as explained u nder quintic equations.Given the complexity of the quartic formulae (see below), they are not often used. If only the real rational roots are needed, they can be found (a s is true for polynomials of any degree) via trial and error, using Ruffini's r ule (so long as all the polynomial coefficients are rational). In the modern a ge of computers, furthermore, good numerical approximations for the roots a re rapidly obtainable via Newton's method. But if the quartic must be solved entirely and precisely, the procedures are outlined below.特殊情况名义上的四次方程如果a4 = 0,那么其中一个根为x = 0,其它根可以通过消去四次项,并解产生的三次方程,<math>a_0x^3+a_1x^2+a_2x+a_3=0.</math>双二次方程四次方程式中若a3 和a1 均为0 者有下列型态:<math>a_0x^4+a_2x^2+a_4=0\,\!</math>因此它是一个双二次方程式。
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arXiv:0803.4396v1 [cond-mat.stat-mech] 31 Mar 2008
I.
INTRODUCTION
Despite several decades of research, understanding the zero crossing properties of non-Markov stochastic processes remains a challenging issue. Among them, the persistence probability p0 (t) received a particular attention, especially in the context of many-body non-equilibrium statistical physics, both analytically [1] as well as experimentally [2, 3, 4, 5]. The persistence p0 (t) for a time dependent stochastic process with zero mean is defined as the probability that it has not changed sign up to time t. In various physical situations, p0 (t) has a power law tail p0 (t) ∼ t−θ where θ turns out to be a non-trivial exponent whenever the stochastic process under study has a non Markovian dynamics. One such example is the diffusion, or heat equation in space dimension d where a scalar field φ(x, t) evolves according to the deterministic equation ∂t φ(x, t) = ∇2 φ(x, t) , (1)
2 correlations [cosh(T /2)]−d/2 decays for large T as P0 (T ) ∼ exp [−θ(d)T ] where θ(d) is the same as the persistence exponent in diffusion equation. This exponent θ(d) was measured in numerical simulations [6, 7], yielding for instance θsim (1) = 0.12050(5), θsim (2) = 0.1875(1). The case of dimension d = 1 is particularly interesting because θ(1) was determined experimentally using NMR techniques to measure the magnetization of spin polarized Xe gas [5], yielding θexp (1) = 0.12 in good agreement with numerical simulations. √ In the limit of large dimension d, which will be of interest in the following, one can show that θ(d) = 2−3/2 θ∞ d where θ∞ is the decay constant associated with the no zero crossing probability of the GSP with Gaussian correlations exp (−T 2 /2), which was studied in the past by engineers, in particular in the context of fading of long-wave radio signals (see for instance Ref. [8]). A seemingly unrelated topic concerns the study of random algebraic equations which, since the first work by Bloch and P´ olya [9] in the 30’s, has now a long story [10, 11]. Recently it has attracted a renewed interest in the context of probability and number theory [12] as well as in the field of quantum chaos [13]. In a recent letter [14], we have established a close connection between zero crossing properties of the diffusion equation with random initial conditions (1) and the real roots of real random polynomials (i.e. polynomials with real random coefficients). In Ref. [14], we focused on a class of real random polynomials Kn (x) of degree n, the so called generalized Kac polynomials, indexed by an integer d
Real Roots of Random Polynomialsቤተ መጻሕፍቲ ባይዱand Zero Crossing Properties of Diffusion Equation.
Gr´ egory Schehr1 and Satya N. Majumdar2
Laboratoire de Physique Th´ eorique (UMR du CNRS 8627), Universit´ e de Paris-Sud, 91405 Orsay Cedex, France 2 Laboratoire de Physique Th´ eorique et et Mod` eles Statistiques, Universit´ e Paris-Sud, Bˆ at. 100, 91405 Orsay Cedex, France (Dated: March 31, 2008) We study various statistical properties of real roots of three different classes of random polynomials which recently attracted a vivid interest in the context of probability theory and quantum chaos. We first focus on gap probabilities on the real axis, i.e. the probability that these polynomials have no real root in a given interval. For generalized Kac polynomials, indexed by an integer d, of large degree n, one finds that the probability of no real root in the interval [0, 1] decays as a power law n−θ(d) where θ(d) > 0 is the persistence exponent of the diffusion equation with random initial conditions in spatial dimension d. For n ≫ 1 even, the probability that they have no real root on the full real axis decays like n−2(θ(2)+θ(d)) . For and Binomial polynomials, √Weyl polynomials √ this probability √ decays respectively like exp (−2θ∞ n) and exp (−πθ∞ n) where θ∞ is such that θ(d) = 2−3/2 θ∞ d in large dimension d. We also show that the probability that such polynomials have exactly k roots on a given interval [a, b] has a scaling form given by exp (−Nab ϕ ˜(k/Nab )) where Nab is the mean number of real roots in [a, b] and ϕ ˜(x) a universal scaling function. We develop a simple Mean Field (MF) theory reproducing qualitatively these scaling behaviors, and improve systematically this MF approach using the method of persistence with partial survival, which in some cases yields exact results. Finally, we show that the probability density function of the largest absolute value of the real roots has a universal algebraic tail with exponent −2. These analytical results are confirmed by detailed numerical computations. Some of these results were announced in a recent letter [G. Schehr and S. N. Majumdar, Phys. Rev. Lett. 99, 060603 (2007)].