Math. Logic Quart. June.
数学类SCI分区查询
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SIAM REV SIAM REVIEWJ AM MATH SOC JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETYANN MATH ANNALS OF MATHEMATICSB AM MATH SOC BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETYJ R STAT SOC B JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATIS J AM STAT ASSOC JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION STRUCT EQU MODELING STRUCTURAL EQUATION MODELING-A MULTIDISCIPLINARY JOURNAL MULTIVAR BEHAV RES MULTIVARIATE BEHAVIORAL RESEARCHINT J INFECT DIS DYNAMICS OF CONTINUOUS DISCRETE AND IMPULSIVE SYSTEMS-SE COMMUN PUR APPL MATH COMMUNICATIONS ON PURE AND APPLIED MATHEMATICSRISK ANAL RISK ANALYSISANN STAT ANNALS OF STATISTICSSIAM J SCI COMPUT SIAM JOURNAL ON SCIENTIFIC COMPUTINGMATH MOD METH APPL S MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCESSIAM J MATRIX ANAL A SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS MULTISCALE MODEL SIM MULTISCALE MODELING & SIMULATIONINVENT MATH INVENTIONES MATHEMATICAEJ R STAT SOC A STAT JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES A-STATIS STAT SCI 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0129-167X数学0.5310.50.3230.440333412405347388 0379-4024数学0.5270.30.490.446667507455474478.666667 1079-8986数学0.5250.40.2780.404333169129118138.666667 0895-4801数学0.5180.90.6360.679667875852676801 0039-3223数学0.5150.50.5270.5266671310110712051207.33333 0002-9939数学0.5130.40.5080.4833335758505851785331.33333 0949-5932数学0.5070.30.280.36866713692811030033-569X数学0.5060.30.8520.5611376129813091327.66667 0021-9002数学0.5040.60.6350.5733331684159917491677.33333 0021-9045数学0.50.50.360.4436671028949736904.333333 0219-8916数学0.50.30.274333331114.6666667 0233-1934数学0.50.30.330.385377349301342.333333 1026-0226数学0.50.10.4810.37233367675964.3333333 0378-3758数学0.4970.50.4460.4746671521129212181343.66667 0219-5259数学0.4910.60.368667166192119.333333 0030-6126数学0.4850.40.2140.351667523428456469 0378-620X数学0.4810.50.5110.494681589590620 0022-250X数学0.480.60.1670.418667296247254265.666667 0266-4763数学0.480.30.6650.483667642486467531.666667 0022-314X数学0.4790.40.3880.407984803857881.333333 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数理逻辑 第二章 算法、整数和矩阵 整数和除法
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三、素数
如果整数不能被小于或等于其平方根的 素数整除,它就是素数。
例5:证明101是素数。
解:不超过101的平方根的素数有2,3,5, 7。因为101不能被这些数整除,所以101是 素数。
三、素数
由于每个整数都有素因子分解,如何求 解整数n的素因子分解?
从最小的素数2开始,从小到大用一个个素 数去除n;
最常用的产生伪随机数的过程称为线性同 余法
xn+1=(axn+c) mod m P120
九、密码学
最重要的同余应用之一涉及研究信息保 密的密码学
解为 a p1a1 p2a2 pnan
b p1b1 p2b2 pnbn
每个指数都是非负整数,出现在a和b分解中的所有素数都包 含在两个分解之中,必要时以0为指数出现
gcd(a,b)
p p min(a1,b1) min(a2,b2)
1
2
p min( an ,bn ) n
五、最大公约数
证明:P116 例14:已知120和500的素因子分解分别
定理7:令m为正整数,若a≡b(mod m), c≡d(mod m),那么a+c≡b+d(mod m)以及 ac≡bd(mod m)。
证明:P118
例18:由于7≡2(mod 5)和11≡1(mod 5), 从定理7知: 18≡3(mod 5) 77≡2(mod 5)
八、同余应用
可以用同余为计算机分配内存地址 例19:散列(哈希)函数 散列就是无需查找,直接用元素的查找
数理逻辑
Mathematical Logic
第二章 算法、整数和矩阵
Chapter 2 Algorithm、Integer and Matrix
Discrete Math1.2-Logic and Math proof
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Example:
in Euclidean geometry we have undefined terms such as
Point
Line
Copyright 2011 @ by Xu Dezhi 7
Definitions
A definition is a proposition constructed from undefined terms and previously accepted concepts in order to create a new concept. Example: In Euclidean geometry the following are definitions:
Copyright 2011 @ by Xu Dezhi
13
Direct
proof: p q
A
direct method of attack that assumes the truth of proposition p, axioms and proven theorems so that the truth of proposition q is obtained.
Counterexample
ent x P(x) is false if x D such that P(x) is false. value x that makes P(x) false is called a counterexample to the statement x P(x).
(~q)(~p) is true.
Since
(~q) (~p) is logically equivalent to p q, then the
艺术与数学的关联英语作文
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艺术与数学的关联英语作文Art and mathematics, seemingly disparate disciplines, are in fact deeply intertwined. While art is often associated with creativity and expression, mathematics is hailed for its logic and structure. However, these two fields share fundamental principles that have shaped human understanding and innovation throughout history.To begin with, both art and mathematics involve patterns and structures. In art, patterns manifest in various forms—geometric designs, repetitions in motifs, or rhythmic arrangements of elements. These patterns evoke a sense of harmony and aesthetic pleasure, resonating with the mathematical concept of symmetry. Mathematics, on the other hand, studies patterns through numerical sequences, geometric shapes, and algebraic equations, aiming to uncover underlying rules and relationships.Furthermore, both disciplines require a meticulous attention to detail. Artists meticulously blend colors, refine textures, and manipulate light and shadow to convey their intended messages or emotions. Similarly, mathematicians delve into intricate proofs, scrutinize calculations, and analyze data with precision to derive meaningful insights and conclusions.Moreover, art and mathematics share a common quest for abstraction and representation. Artists often strive to depict abstract concepts, emotions, or philosophical ideas through symbolic imagery or unconventional forms. This parallels the mathematical pursuit of abstract concepts such as infinity, prime numbers, or complex geometries that transcend physical reality yet hold profound significance in theoretical frameworks.Beyond abstraction, both disciplines are integral to technological advancements and scientific breakthroughs.Mathematics provides the theoretical foundation for physics, engineering, and computer science, enabling the development of sophisticated algorithms, simulations, and models. In parallel, art inspires innovation in design, architecture, and visual communication, pushing boundaries of creativity and aesthetics in the digital age.Moreover, interdisciplinary collaborations between artists and mathematicians have led to groundbreaking discoveries and innovations. From the Renaissance period, when artists like Leonardo da Vinci explored anatomy and perspective using mathematical principles, to contemporary digital art and fractal geometry, these collaborations have enriched both fields by fostering new perspectives and methodologies.In conclusion, the relationship between art and mathematics transcends mere parallels; it represents a symbiotic fusion of creativity and logic, imagination, andanalysis. By understanding and appreciating their interconnectedness, we can cultivate a holistic approach to education, innovation, and human expression, bridging the perceived gap between the arts and sciences for a more enriched and interconnected future.。
Handbook of Mathematical Functions The Handbook of Mathematical Functions with Formulas, Gr
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Handbook of Mathematical Functions The Handbook of Mathematical Functions withFormulas,Graphs,and Mathematical T ables[1]was theculmination of a quarter century of NBS work on coremathematical tools.Evaluating commonly occurringmathematical functions has been a fundamental need aslong as mathematics has been applied to the solution ofpractical problems.In1938,NBS initiated its Mathe-matical T ables Project to satisfy the increasing demandfor extensive and accurate tables of functions[2].Located in New Y ork and administered by the W orksProjects Administration,the project employed not onlymathematicians,but also a large number of additionalstaff who carried out hand computations necessary toproduce tables.From1938until1946,37volumes of theNBS Math T ables Series were issued,containing tablesof trigonometric functions,the exponential function,natural logarithms,probability functions,and relatedinterpolation formulae.In1947,the Math T ables Projectwas moved to W ashington to form the ComputationLaboratory of the new National Applied MathematicsLaboratories of NBS.Many more tables subsequentlywere published in the NBS Applied Mathematics Series;the first of these,containing tables of Bessel functions[3],appeared in1948.On May15,1952,the NBS Applied MathematicsDivision convened a Conference on T tonAbramowitz of NBS,who had been a member of thetechnical planning staff for the Math T ables Project, described preliminary plans for a compendium of mathematical tables and related material.Abramowitz indicated that the Bureau was in need of both technical advice and financial support to carry out the project. With the support of the National Science Foundation (NSF),a two-day Conference on T ables was held at the Massachusetts Institute of T echnology on September 15-16,1954,to discuss the prospects for such an under-taking.Twenty-eight persons attended,including both table producers and users from the science and engineer-ing community.The report of the conference concluded that“an outstanding need is for a Handbook ofT ables for the Occasional Computer,withtables of usually encountered functions and aset of formulas and tables for interpolationand other techniques useful to the occasionalcomputer.”(Note that here the term computer refers to a person performing a calculation by hand.)The report recom-mended that NBS manage the production of the Hand book and that NSF provide financial assistance.The conference elected the following committee to carry out its recommendations:P.M.Morse(Chair), M.Abramowitz,J.H.Curtiss,R.W.Hamming,D.H. Lehmer,C.B.Tompkins,and J.W.Tukey.The commit-tee was successful in persuading both NBS and NSF to support the project,and it began officially in December of1956.The Mathematics Division of the National Research Council also had an interest in mathematical tables. Since1943,they had been publishing a quarterly journal entitled Mathematical Tables and Other Aids to Computation(today known as Mathematics of Compu-tation).To provide technical assistance to NBS,as well as independent oversight for NSF,the NRC established a Committee on Revision of Mathematical Tables.Its members were P.M.Morse(Chair),A.Erde´lyi,M.C.Fig.1.Portrait of Milton Abramowitz.Gray,N.C.Metropolis,J.B.Rosser,H.C.Thacher,Jr., John T odd,C.B.T ompkins,and J.W.Tukey.This group of luminaries in the fields of applied mathematics and physics provided guidance to NBS throughout the project to produce the Handbook.Milton Abramowitz,who was then Chief of the Computation Laboratory of the NBS Applied Mathe-matics Division,led the project.Abramowitz was born in Brooklyn,NY,in1915.He received a B.A.from Brooklyn College in1937and an M. A.in1940. He joined the NBS Math T ables Project in1938and in 1948received a Ph.D.in Mathematics from New Y ork University.Abramowitz’dedication,enthusiasm,and boundless energy led to substantial progress in the project during its first year.The proposed outline for the Handbook called for a series of some20chapters,each with a separate author.Authors were drawn from NBS staff and guest researchers,as well as external researchers working under contract.Most chapters would focus on a particular class of functions,providing formulas,graphs,and tables.Listed formulas would include differential equations,definite and indefinite integrals,inequalities,recurrence relations,power series,asymptotic expansions,and polynomial and rational approximations.Material would be carefully selected in order to provide information most important in applications,especially in physics.Consequently, the higher mathematical functions,such as Bessel functions,hypergeometric functions,and elliptic functions,would form the core of the work.Additional chapters would provide background on interpolation in tables and related numerical methods for differentiation and quadrature.Philip J.Davis of NBS first prepared Chapter6,on the gamma and related functions,to serve as a model for other authors.This chapter portrayed the telegraphic style that is a hallmark of the Handbook,i.e.,the material is displayed with a minimum of textual descrip-tion.In the course of developing his chapter,Davis became interested in the history of the topic.This led to a historical profile published in1959[4],which won the prestigious Chauvenet Prize for distinguished mathematical exposition from the Mathematical Association of America.The Handbook project occurred during the period when general-purpose electronic computing machinery was first coming into use in government research laboratories.(Early computer development of SEAC at NBS is described elsewhere in this volume.)Never-theless,most of the tables in the Handbook were gener-ated by hand on desk calculators.However,even at that time it was clear to the developers of the Handbook that the need for tables themselves would eventually be superseded by computer programs which could evaluate functions for specified arguments on demand.By the summer of1958,substantial work had been completed on the project.Twelve chapters had been completed,and the remaining ones were well underway. The project experienced a shocking setback one week-end in July1958when Abramowitz suffered a heart attack and died.Irene Stegun,who was Assistant Chief of the Computation Laboratory,took over management of the project.Stegun,who was born in Y onkers,NY in 1919,had received an M.A.from Columbia University in1941,and joined NBS in1943.The exacting work of assembling the many chapters,checking tables and formulas,and preparing the work for printing took much longer than anticipated.Nevertheless,the Handbook of Mathematical Functions,with Formulas, Graphs,and Mathematical T ables was finally issued as Applied Mathematics Series Number55in June1964 [1].The volume,which is still in print at the U.S. Government Printing Office and stocked by many bookstores and online booksellers,is1046pages in length.The chapters and authors are as follows.Fig.2.Portrait of Irene Stegun.1.Mathematical Constants,D.S.Liepman.2.Physical Constants and Conversion Factors,A.G.McNish.3.Elementary Analytical Methods,M.Abramowitz.4.Elementary Transcendental Functions,R.Zucker.5.Exponential Integral and Related Functions,W.Gautschi(American University)and William F.Cahill.6.Gamma Function and Related Functions,P.J.Davis.7.Error Function and Fresnel Integrals,W.Gautschi(American University).8.Legendre Functions,I.A.Stegun.9.Bessel Functions of Integer Order,F.W.J.Olver.10.Bessel Functions of Fractional Order,H. A.Antosiewicz.11.Integrals of Bessel Functions,Y.L.Luke.12.Struve Functions and Related Functions,M.Abramowitz.13.Confluent Hypergeometric Functions,L.J.Slater(Cambridge University).14.Coulomb W ave Functions,M.Abramowitz.15.Hypergeometric Functions,F.Oberhettinger.16.Jacobian Elliptic Functions and Theta Functions,L.ne-Thomson(University of Arizona). 17.Elliptic Integrals,ne-Thomson(Univer-sity of Arizona).18.W eierstrass Elliptic and Related Functions,T.H.Southard.19.P arabolic Cylinder Functions,ler(Cam-bridge University).20.Mathieu Functions,G.Blanch(Wright-PattersonAir Force Base).21.Spheroidal W ave Functions,A.N.Lowan(Y eshivaUniversity).22.Orthogonal P olynomials,U.W.Hochstrasser(American University).23.Bernoulli and Euler P olynomials—Riemann ZetaFunction,E.V.Haynsworth and K.Goldberg.binatorial Analysis,K.Goldberg,M.Newman,and E.Haynsworth.25.Numerical Interpolation,Differentiation,and Inte-gration,P.J.Davis and I.Polonsky.26.Probability Functions,M.Zelen and N.C.Severo27.Miscellaneous Functions,I.A.Stegun.28.Scales of Notation,S.Peavy and A.Schopf(American University).place Transforms.The public reaction to the publication of the Hand-book was overwhelmingly positive.In a preface to the ninth printing in November1970,NBS Director Lewis Branscomb wrote“The enthusiastic reception accorded the‘Handbook of Mathematical Functions’islittle short of unprecedented in the longhistory of mathematical tables that beganwhen John Napier published his tables oflogarithms in1614.Only four and one-halfyears after the first copy came from the pressin1964,Myron Tribus,the Assistant Secre-tary for Commerce for Science and T echnol-ogy,presented the100,000th copy of theHandbook to Lee A.DuBridge,then ScienceAdvisor to the President.”The Handbook has had enormous impact on science and engineering.Likely the most widely distributed NBS/NIST technical publication of all time,the govern-ment edition has never gone out of print,and it has appeared as a Dover reprint since1965.It has been reprinted(in all or part)by other publishers,such as Moscow Nauka,V erlag Harri Deutsch,and Wiley ernment sales exceed150,000copies,with commercial sales estimated at three to six times this number.The Handbook’s citation record is also remark-able.More than23,000citations have been logged by Science Citation Index(SCI)since1973.Remarkably, the number of citations to the Handbook continues to grow,not only in absolute numbers,but also as aFig.3.Photograph of Handbook.fraction of the total number of citations made in the sciences and engineering.During the mid-1990s,for example,about once every1.5hours of each working day some author,somewhere,made sufficient use of the Handbook to list it as a reference.The success of the Handbook was due to several factors.It collected in one place,and in a well-organized way,the most important information needed to make use of mathematical func-tions in practical applications.It served to standardize notations and normalizations for the special functions of applied mathematics,thus easing the communication of scientific results.In1965,Irene Stegun was awarded a Gold Medal from the Department of Commerce for her efforts in completing the project.A number of difficult mathematical problems that emerged in the course of developing the Handbook engaged researchers in the NBS Applied Mathematics Division for a number of years after its publication. Two of these are especially noteworthy,the first having to do with stability of computations and the second with precision.Mathematical functions often satisfy recurrence relations(difference equations)that have great potential for use in computations.However,if used improperly, recurrence relations can quickly lead to ruinous errors. This phenomenon,known as instability,has tripped up many a computation that appeared,superficially,to be straightforward.The errors are the result of subtle interactions in the set of all possible solutions of the difference equation.Frank Olver,who wrote the Handbook’s chapter on Bessel functions of integer order,studied this problem in great detail.In a paper published in1967[5],Olver provided the first(and only) stable algorithm for computing all types of solutions of a difference equation with three different kinds of behavior:strongly growing,strongly decaying,and showing moderate growth or decay.Part of the impact of this work is reflected today in the existence of robust software for higher mathematical functions.Olver worked on such topics in the Mathematical Analysis Division of NBS,and this work provided the foundation for his very influential later book on asymptotic analysisFig.4.Screen shot of the NIST Digital Library of Mathematical Functions.and special functions[6].This book has been cited more than800times,according to SCI.Another important problem in mathematical compu-tation is the catastrophic loss of significance caused by the fixed length requirement for numbers stored in computer memory.Morris Newman,who co-authored the Handbook’s chapter on combinatorial analysis, sought to remedy this situation.He proposed storing numbers in a computer as integers and performing oper-ations on them exactly.This contrasts with the standard approach in which rounding errors accumulate with each arithmetic operation.Newman’s approach had its roots in classical number theory:First perform the computations modulo a selected set of small prime numbers,where the number of primes required is deter-mined by the problem.These computations furnish a number of local solutions,done using computer numbers represented in the normal way.At the end,only one multilength computation is required to construct the global solution(the exact answer)by means of the Chinese Remainder Theorem.This technique was first described in a paper by Newman in1967[7];it was employed with great success in computing and checking the tables in Chapter24of the Handbook.T oday,this technique remains a standard method by which exact computations are performed.Newman’s research on this and other topics,performed at NBS,formed the basis for his1972book[8],which quickly became a standard reference in the applications of number theory to computation.Research into the functions of applied mathematics has continued actively in the36years since the Hand-book appeared.New functions have emerged in impor-tance,and new properties of well-known functions have been discovered.In spite of the fact that sophisticated numerical methods have been embodied in well-designed commercial software for many functions,there continues to be a need for a compendium of information on the properties of mathematical functions.T o address this need,NIST is currently developing a successor to the Handbook to be known as the Digital Library of Mathematical Functions(DLMF)[9].Based upon a completely new survey of the literature,the DLMF will provide reference data in the style of the Handbook in a freely available online format,with sophisticated mathematical search facilities and interactive three-dimensional graphics.Prepared by Ronald F.Boisvert and Daniel W.Lozier. Bibliography[1]Milton Abramowitz and Irene A.Stegun,eds.,Handbook of Math-ematical Functions With Formulas,Graphs,and Mathematical T ables,NBS Applied Mathematics Series55,National Bureau of Standards,W ashington,DC(1964).[2]Arnold N.Lowan,The Computation Laboratory of the NationalBureau of Standards,Scripta Math.15,33-63(1949).[3]T ables of the Bessel Functions Y0(x),Y1(x),K0(x),K1(x)0<=x<=1,NBS Applied Mathematics Series1,National Bureau of Stan-dards,W ashington,DC(1948).[4]Philip J.Davis,Leonhard Euler’s Integral:A Historical Profile ofthe Gamma Function,Am.Math.Monthly66,849-869(1959).[5]F.W.J.Olver,Numerical Solution of Second-Order Linear Differ-ence Equations,J.Res.Natl.Bur.Stand.71B,111-129(1967).[6]F.W.J.Olver,Asymptotics and Special Functions,AcademicPress,New Y ork(1974).Reprinted by A.K.Peters,W ellesley,MA (1997).[7]Morris Newman,Solving Equations Exactly,J.Res.Natl.Bur.Stand.71B,171-179(1967).[8]Morris Newman,Integral Matrices,Academic Press,New Y ork(1972).[9]D.Lozier,F.W.J.Olver,C.Clark,and R.Boisvert(eds.),Digital Library of Mathematical Functions,() National Institute of Standards and T echnology,.。
数理逻辑 第一章 逻辑、集合和函数 函数
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集合B到集合C的函数,函数f和g的组 合用f ◦g表示,定义为:
(f ◦g)(a)=f ( g (a) )
三、反函数和函数组合
如果g的值域不是f 的定义域的子集, 就无法定义f ◦g。
P62 - 例17-18。 对函数组合而言交换律不成立。
函数f是一对一的,当且仅当只要x≠y, 就有f(x)≠f(y)
P59 - 例6 - 例8
二、一对一函数和映上函数
定义域和伴域都是实数集合子集的函 数f称为严格递增的,如果对f定义域 中的x和y,只要x<y就有f(x)<f(y)。
f称为严格递减的,如果对f定义域中 的x和y,只要x<y就有f(x)>f(y)。
一、引言
在许多情况下,我们都会为一个集合的每 个元素指派另一个集合的一个特定元素。
例如:假定为学习数理逻辑课的每个学生 从{A,B,C,D}中选择一个字母作为他的得分。 再假定张三的得分为A,李四的得分为C, 王五的得分为A,赵六的得分为D。
这种打分就是一个函数。
一、引言
令A和B为集合。从A到B的函数f是对 元素的一种指派,对A的每个元素恰 好指派B的一个元素。如果f指派给A 中元素a的唯一的B的元素是b,就写 成f(a)=b。如果f是从A到B的函数, 就写成 f: A→B。
二、一对一函数和映上函数
例:A={1,2,3,4},B={a,b,c},如果f : A→B为 f(1)=a,f(2)=c,f(3)=b,f(4)=c,
则f是满射。
例:A={1,2,3},B={a,b,c,d},如果f : A→B为 f(1)=a,f(2)=c,f(3)=b,
则f是单射。
TI—83 plus图形计算器常用功能简介
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TI—83 plus图形计算器常用功能简介一、基本代数功能(一)MATH运算访问MATH(数学)菜单,可按 。
菜单中有四个子菜单,它们分别是:MATH(数学)、NUM(数值)、CPX(复数)以及PRB(概率)。
(二)角度和关系运算访问ANGLE(角度)菜单,可按ψ[ANGLE]。
ANGLE菜单显示与角度有关的指令。
但要注意的是:Radian/Degree方式设置会影响计算器对指令的解释。
访问TEST(关系)菜单,可按ψ [TEST]。
TEST菜单有两个子菜单:TEST(关系)和LOGIC(布尔)。
1.ANGLE菜单介绍1:。
度表示。
5:R④Pr(给定x和y,返回r。
2:’DMS分表示。
6:R④Pθ(给定x和y,返回θ。
3:r 弧度表示。
7:P④Rx(给定R和θ,返回x。
4:④DMS 以度/分/秒显示。
8:P④Ry(给定R和θ,返回y。
2.TEST菜单1:= 相等。
3:> 大于。
5:< 小于。
2:≠不等。
4:≥大于等于。
6:≤小于等于。
关系运算符比较两边的值,当判断为真时返回1,判断为假时返回0。
二、函数作图功能【例】用边长为60cm的正方形铁皮在四角各剪去一个小正方形做成一个无盖水箱,问水箱底面边长取多少时,才能使所得的水箱容量最大,并求出最大容量。
1.设置函数Func 按ζ(图1)图1 图2 图3ζ菜单介绍分类按屏幕顺序数字记数法:Normal(通常)Sci(科学)Eng(工程)小数位数:Float(浮点)0~9(小数位数)角度单位:Radian(弧度)Degree(角度)作图类型:Func(函数)Par(参数)Pol(极坐标)Seq(数列)Connected(用线连接)Dot(点)是否连接图象点:是否同时绘图:Sequential(一个接一个)Simul(同时)显示结果:Real(实数)a+bi(直角坐标复数)re^θi(极坐标复数)屏幕方式:Full(整屏)Horiz(水平)G—T(图形—表)2.建立体积y与x的关系:在ο编缉器中输入函数解析式:y=(60-2x)(60-2x)x(图2)3.设置窗口变量:考虑到x的变化范围为0<x<30,估计y的范围是0<y<19000。
QuartusII基本设计流程
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第2章 Quartus Ⅱ应用向导Quartus II 是Altera公司的综合性PLD开发软件,支持原理图、VHDL、VerilogHDL以及AHDLAltera Hardware Description Language等多种设计输入形式,内嵌自有的综合器以及仿真器,可以完成从设计输入到硬件配置的完整PLD设计流程.Quartus II可以在XP、Linux以及Unix上使用,除了可以使用Tcl脚本完成设计流程外,提供了完善的用户图形界面设计方式.具有运行速度快,界面统一,功能集中,易学易用等特点.Quartus II支持Altera的IP核,包含了LPM/MegaFunction宏功能模块库,使用户可以充分利用成熟的模块,简化了设计的复杂性、加快了设计速度.对第三方EDA工具的良好支持也使用户可以在设计流程的各个阶段使用熟悉的第三放EDA工具.此外,Quartus II 通过和DSP Builder工具与Matlab/Simulink相结合,可以方便地实现各种DSP应用系统;支持Altera的片上可编程系统SOPC开发,集系统级设计、嵌入式软件开发、可编程逻辑设计于一体,是一种综合性的开发平台.Maxplus II 作为Altera的上一代PLD设计软件,由于其出色的易用性而得到了广泛的应用.目前Altera已经停止了对Maxplus II 的更新支持,Quartus II 与之相比不仅仅是支持器件类型的丰富和图形界面的改变.Altera在Quartus II 中包含了许多诸如SignalTap II、Chip Editor和RTL Viewer的设计辅助工具,集成了SOPC和HardCopy设计流程,并且继承了Maxplus II 友好的图形界面及简便的使用方法.Altera Quartus II 作为一种可编程逻辑的设计环境, 由于其强大的设计能力和直观易用的接口,越来越受到数字系统设计者的欢迎.Altera的Quartus II可编程逻辑软件属于第四代PLD开发平台.该平台支持一个工作组环境下的设计要求,其中包括支持基于Internet的协作设计.Quartus平台与Cadence、ExemplarLogic、 MentorGraphics、Synopsys 和Synplicity等EDA供应商的开发工具相兼容.改进了软件的LogicLo ck模块设计功能,增添了FastFit编译选项,推进了网络编辑性能,而且提升了调试能力.支持MAX7000/MAX3000等乘积项器件基本设计流程本节以十进制计数器为例,通过实现流程,详细介绍Quartus II的重要功能和使用方法建立工作库文件和编辑设计文件任何一项设计都是一项工程Project,都必须首先为此工程建立一个放置与此工程相关的所有设计文件的文件夹.此文件夹将被EDA软件默认为工作库Work Library.一般,不同的设计项目最好放在不同的文件夹中,而同一工程的所有文件都必须放在同一文件夹中.在建立了文件夹后就可以将设计文件通过QuartusII的文本编辑器编辑并存盘.注意不要将文件夹设在计算机已有的安装目录中,更不要将工程文件直接放在安装目录中1新建一个文件夹.这里假设本项设计的文件夹取名为jsq,在E盘中,路径为E:\jsq .注意,文件夹名不能用中文,也最好不要用数字.2输入源程序.打开计算机桌面上图表,选择菜单File→New,出现如图所示见面,在New窗口Device Design Files中选择编译文件的语言类型,这里选择VHDL File,选好后用鼠标左键单击OK按钮,出现源程序输入窗口如图所示以十进制为例.图选择编译文件的语言类型图源程序输入窗口十进制计数器源程序如下:LIBRARY IEEE;USE CNT10 ISPORT CLK,RST,EN : IN STD_LOGIC;CQ : OUT STD_LOGIC_VECTOR3 DOWNTO 0;COUT : OUT STD_LOGIC ;END CNT10;ARCHITECTURE behav OF CNT10 ISBEGINPROCESSCLK, RST, ENVARIABLE CQI : STD_LOGIC_VECTOR3 DOWNTO 0;BEGINIF RST = '1' THEN CQI := OTHERS =>'0' ; --计数器异步复位 ELSIF CLK'EVENT AND CLK='1' THEN --检测时钟上升沿IF EN = '1' THEN --检测是否允许计数同步使能IF CQI < 9 THEN CQI := CQI + 1; --允许计数, 检测是否小于9ELSE CQI := OTHERS =>'0'; --大于9,计数值清零END IF;END IF;END IF;IF CQI = 9 THEN COUT <= '1'; --计数大于9,输出进位信号ELSE COUT <= '0';END IF;CQ <= CQI; --将计数值向端口输出END PROCESS;END behav;3文件存盘选择File→Save As命令,找到已建立的文件夹E:\ jsq,存盘文件名应与实体的名字一致,即CNT10,其界面窗口如图所示.图文件存盘单击“否N”按钮,则按以下方法进入创建工程流程.创建工程使用New Project Wizard可以为工程指定工作目录、分配工程名称以及指定最高层设计实体的名称,还可以指定要在工程中使用的设计文件、其他源文件、用户库和EDA工具,以及目标器件系列和具体器件等.1打开建立新工程管理窗选择File→New Preject Wizard工具选项创建设计工程命令,即弹出“工程设置”对话框如图所示,单击对话框最上第一栏右侧的“…”按钮,找到文件夹E:\jsq,选种已存盘的文件CNT10,再单击打开按钮,既出现如图所示的设置情况.对话框中第一行表示工程所在的工作库文件夹,第二行表示此项工程的工程名,第三行表示顶层文件的实体名.图利用New Preject Wizard创建工程CNT102将设计文件加入工程中单击图中下方的Next按钮,出现如图所示的对话框,在弹出的对话框中单击File name栏的按钮,将与工程相关的所有VHDL文件加入此工程,加入完成后单击Next按钮.此工程加入的方法有两种:第一种是单击Add All按钮,将设定的工程目录中的所有VHDL文件加入到工程文件栏中;第二种方法是单击“Add …”按钮,从工程目录中选出相关的VHDL文件.图将所有的工程VHDL文件加入此工程图3选择仿真器、综合器和目标器件的类型单击图中Next按钮,即弹出如图所示的仿真器和综合器及目标器件对话框.其仿真器和综合器及目标器件设置如图所示.首先在 Famil 栏选芯片系列,在此选Cyclone系列,在有效器件列表中选择专用器件,分别选择封装形式为PQFP,引脚输出240,器件速度级别为8,选择此系列的具体芯片是EP1C6Q240C8, 这里EP1C6表示Cyclone系列及此器件的规模.设计完成后单击Finish按钮.图仿真器和综合器类型设置图4工具设置.单击图中的Next按钮后,弹出图所示工具设置窗口,此窗口有3项选择.EDA design entry/synthesis用于选择输入的HDL类型和综合工具.EDA simulation用于选择仿真工具.EDA timing analysis tool用于选择时序分析工具,这是除Cyclone自含的所有设计工具以外的外加的工具,因此,如果都不做选择,表示选择Cyclone自含的所有工具.在此例中这3项都不做选择,单击Next后即弹出图所示”工程设置统计”窗口.最后单击图中Finish,即已设定好此工程,并出现CNT10的工程管理窗口.Quartus Ⅱ将工程信息存储在工程配置文件中,它包含有关Quartus Ⅱ工程的所有信息,包括设计文件、波形文件、Signa1Tap Ⅱ文件、内存初始化文件等,以及构成工程的编译器、仿真器和软件构建设置.建立工程后,可以使用工具栏的Project→ADD/Remove Files Project页在工程中添加和删除、设计其它文件,在执行Quartus Ⅱ的Analysis &Synthesis期间,Quartus Ⅱ将按ADD/Remove Files Project页中显示的顺序处理文件.图工具设置窗口图工程设置统计”窗口编译前设置选择FPGA目标芯片.目标芯片的选择也可以这样来实现:选择Assignmemts菜单中的settings项,可以弹出图对话框.选择配置器件的工作方式.单击图中的Device & Pin Options按钮,进入选择窗,这将弹出Device & Pin Options窗口,其对话框如图所示.在Configuration选项页,选择配置器件为EPCS4,其配置模式可选择Active Serial.这种方式只对专用的Flash技术的配置器件专用于Cyclone系列FPGA的EPCS4和EPCS1等进行编程.注意,PC机对FPGA的直接配置方式都是JTAG方式,而对于FPGA进行所谓“掉电保护式”编程通常有两种:主动串行模式AS Mode和被动串行模式PS Mode.对EPCS1/EPCS4的编程必须用AS Mode.图选择FPGA目标芯片图选择配置器件工作方式图全程编译Quartus II编译器是由一系列处理模块构成的,这些模块负责对设计项目的检错,逻辑综合、结构综合、输出结果的编辑配置,以及时序分析.在这一过程中,将设计项目适配到FPGA/CPLD目标器中,同时产生多种用途的输出文件,如功能和时序信息文件、器件编程的目标文件等.编译器首先检查出工程设计文件中可能错误信息,供设计者排除.然后产生一个结构化的以网表文件表达的电路原理图文件.编译前首先选择Processing菜单的Start Compilation项,启动全程编译.这里所谓的全程编译Compilation包括以上提到的Quartus II对设计输入的多项处理操作,其中包括排错、数据网表文件提取、逻辑综合、适配、装配文件仿真文件与编程配置文件生成,以及基于目标器件的工程时序分析等.编译过程中要注意工程管理窗下方的“Processing”栏中的编译信息.如果工程中的文件有错误,启动编译后在下方的Processing处理栏中会显示出来,如图所示.对于Processing栏显示出的语句格式错误,可双击错误信息条文,即弹出对应的vhdl文件,在深色标记条处即为文件中的错误,再次进行编译直至排除所有错误.如果编译成功,可以见到如图所示的工程管理窗的左上角显示了工程cnt10的层次结构和其中结构模块耗用的逻辑宏单元数;在此栏下是编译处理流程,包括数据网表建立、逻辑综合、适配、配置文件装配和时序分析等.最下栏是编译处理信息;中栏Compilation Report栏是编译报告项目选择菜单,点击其中各项可以详细了解编译与分析结果.图全程编译后信息图时序仿真对工程编译通过后,必须对其功能和时序性质进行仿真测试,以了解设计结果是否满足原设计要求.以VWF文件方式的仿真流程的详细步骤如下:1打开波形编辑器.选择菜单File中的New项,在New窗口中选择Other Files 中的Vector Waveform File如图所示,单击OK按钮,即出现空白的波形编辑器如图所示,注意将窗口扩大,以利观察.图选择编辑矢量波形文件图波形编辑器2设置仿真时间区域,对于时序仿真来说,将仿真时间设置在一个合理的时间区域上十分重要.通常设置时间范围在数十微妙间.首先在Edit菜单中选择End Time 项,即弹出如图所示窗口.在此例中整个仿真时间设置为10μs,单击OK按钮结束设置.图设置仿真时间长度3将工程CNT10的端口信号节点选入波形编辑器中.方法是首先选择View 菜单中的Utility Windows项的Node Finder项.弹出的对话框如图所示,在Filter框中选Pins : all通常已默认选此项,然后单击List按钮,于是在下方的Nodes Found 窗口中出现设计中的CNT10工程的所有端口引脚名.图 CNT10的信号节点注意如果此对话框中的“List”不显示CNT10工程的端口引脚名,需要重译一次,即选择Processing→Start Compilation,然后再重复以上操作过程.最后,用鼠标将重要的端口节点CLK、EN、RST、COUT和输出总线信号CQ分别拖到波形编辑窗,结束后关闭Nodes Found窗口.单击波形窗左侧的“全屏显示”按钮,使全屏显示,并单击“放大缩小”按钮后注意:左键放大,右键缩小,再用鼠标在波形编辑区域右键单击,使仿真坐标处于适当位置,如图所示,这时仿真时间横坐标设定在数十微秒数量级.设定仿真时间宽度,选择Edit项及其End time选项,在End time选择窗中选择适当的仿真时间域,如可选10us,以便有足够长的观察时间.图拖入节点后波形编辑器4波形文件存盘.选择File中的Save as,将以默认名为的波形文件存入文件夹E:\jsq中,即出现如图所示的激励波形文件存盘窗口.图 vwf激励波形文件存盘5编辑输入波形输入激励信号.用鼠标左键单击图所示窗口的时钟信号名CLK,使之变成蓝色条,再单击左列的时钟设置键,即弹出如图时钟脉冲周期及占空比设置窗口,在图中的上部份是已经设置好了的仿真时间区域为10μs,这里不需要改变,下部分CLK的时钟周期设置为50ns;Clock窗口中的Duty cycle是占空比,默认为50,即50%占空比.然后再分别设置EN和RST的电平,RST为复位端,EN为使能端.最后设置好的激励信号波形如图所示.图时钟脉冲周期及占空比设置窗口图设置好的激励信号波形图6总线数据格式设置.单击如图所示的输出信号“CQ”左旁的“+”,则能展开此总线中的所有信号;如果双击此“+”号左旁的信号标记,将弹出对该信号数据格式设置的对话框如图所示.在该对话框的Radix栏有4种选择,这里可选择无符号十进制整数Unsigned Decimal表达方式.最后对波形文件再次存盘.图信号数据格式设置图7仿真方式的选择在QuarturⅡ软件中仿真方式有两种,功能仿真和时序仿真,此例选择功能仿真,方法是:在工具栏中选择processing→Simulater Tool即弹出如图仿真方式选择窗口,在窗口Simulater mode处是时序仿真和功能仿真选择窗口,此例选择功能仿真Functionl.选好后单击Generate Functional Simulater Nellist按钮,再单击确定按钮,最后再单击图中的start按钮,即完成仿真方式的确定.图仿真方式选择窗口8仿真器参数设置选择菜单Assignment→Settings,即弹出如图选择仿真参数设置窗口,此例中选择的参数如图所示.图选择仿真参数设置窗口9启动仿真器.现在所有设置进行完毕,在菜单Processing项下选择Start Simulation,也可以选择工具栏上的图表.8观察仿真结果.仿真波形文件“Simulation Report”通常会自动弹出如所示仿真结果.同时在图窗口中用鼠标右键单击选择Zoom→Fit in window即选择全时域显示.如果在启动仿真运行后,并没有出现仿真完成后的波形图,而是出现文字“Can’t open Simulation Report Window”,但报告仿真成功,则可自己打开波形报告,选择Processing→Simulation Report.图仿真波形输出结果应用RTL电路图观察器选择方法是Tools→Netlist Viewers在出现的下拉菜单中有四个选项,此例中选择第一项RTL Viewer,即HDL的RTL级图形观测器,选好后将自动弹出如图所示RTL电路.图 RTL电路对于较复杂的RTL电路,可利用功能过滤器Filter简化电路,即用右健单击该模块,在弹出的下拉菜单中选择Filter项的Sources或Destinations,由此产生相应的电路.。
英文期刊名称常用缩略词语对照表
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Rep. Res. Reson. Resour. Rev. Robot. Roy. Saf. Satell. Scand. Sci. Sect. Secur. Seismol. Sel.
Structure Studies Superconductivity Supplement Surface Survey Sustainable Symposium Systems Technical Techniques Technology Telecommunications Television Temperature
Abstr. Acad. Accel. Acoust. Act. Admin. Administ. Adv. Aeronaut. Aerosp. Affect. Afr. Aircr. Algebr. Amer.
Atomic, Atoms Australasian Australia Automatic Automation Automotive Autonomous Behavior(al) Belgian Biochemical Bioinformatics Biology, Biological Biomedical Biophysics British
Innovation Institute Instrument Instrumentation Insulation Integrated Intelligence Intelligent Interactions International Isotopes Israel Japan Journal Knowledge
Mathematical Mathematics
Digit. Discl. Discuss. Diss. Distrib. Dyn. Earthq. Econ.
左孝凌离散数学
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辩证逻辑是研究反映客观世界辩证发展过程的人类思 维的形态的。
第一部分 数理逻辑(Mathematical Logic)
❖ 形式逻辑是研究思维的形式结构和规律的科学,它撇 开具体的、个别的思维内容,从形式结构方面研究概 念、判断和推理及其正确联系的规律。
上例中的命题)。 复合命题:由简单命题通过联结词联结而成的命题。
联结词就是复合命题中的运算符。
第一章 命题逻辑(Propositional Logic)
1.1 命题及其表示方法
注意:
(1)一个符号(如P), 它表示的是命题常量还是命题变 元,一般由上下文来确定。
(2)命题变元可以表示任意命题,它不能确定真值, 故命题变元不是命题。这与“变数x不是数”是一样 的道理。
离散数学( ) Discrete Mathematics
第一部分 数理逻辑(Mathematical Logic)
❖ 逻辑:是研究推理的科学。公元前四世纪由希腊的哲 学家亚里斯多德首创。作为一门独立科学,十七世纪, 德国的莱布尼兹(Leibniz)给逻辑学引进了符号, 又称 为数理逻辑(或符号逻辑)。
哈哈, 这句话不是命
题哦!
第一章 命题逻辑(Propositional Logic)
1.1 命题及其表示方法
因而又可以称命题是具有唯一真值的陈述ቤተ መጻሕፍቲ ባይዱ。
判断命题的两个步骤:
1、是否为陈述句;
2、是否有确定的、唯一的真值。
例:判断下列句子是否为命题。 (1). 100是自然数。 T (2). 太阳从西方升起。 F
判断的陈述句构成了推理的基本单位。
秒选数学记得英语
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秒选数学记得英语。
Mathematics is an essential part of life. It is an essential part of technology and science, and can be found everywhere in everyday life. It is the language of the universe, enabling us to understand and express the world around us in ways no other form of communication can.Mathematics is the language of the physical world. It is the language of numbers, equations and geometric shapes. Mathematics is used to describe the behavior of physical objects and to explain the relationships between them. It is also used to evaluate the likelihood of certain outcomes and to calculate probability, as well as to measure distances and angles. Mathematics enables us to answer questions regarding the size, shape, and motion of objects, and it is used in many fields, such as engineering, medicine and finance.Mathematics is the language of logic and reasoning. Mathematics teaches us how to think and organize our thoughts. It can help us understand complicated theories, analyze data, andsolve problems. Mathematics also helps motivate problem-solving skills, critical thinking, and decision making.Mathematics is also the language of computers. Computers rely on mathematics and algorithms to create digital systems that can perform complex tasks quickly and efficiently. This allows us to store, analyze, and display large amounts of data quickly. It also helps us create more interactive and engaging experiences with digital media and gaming.Finally, mathematics is the language of the arts. Artists use mathematics to create designs in art, music, and architecture. They use formulas to create beautiful shapes, proportions, and textures. Filmmakers, photographers, and game developers use line and shape to create moods and evoke emotions.In short, mathematics is the language of our universe. It serves as a way of understanding, expressing, and manipulating the things around us. It is used in every field and discipline, including engineering and medicine, as well as the arts and humanities. Mathematics helps us gain a better understanding of our world and how things work, which in turn helps us make smarter decisions and solve problems.。
有关圆周率的英语作文
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有关圆周率的英语作文Pi, you know, that number we use for circles? It's like the ultimate circle superhero. It's irrational, meaning it goes on and on, never repeating. Just when you think you've got a handle on it, it throws you a curveball. But hey,that's math for you, always keeping us on our toes.Ever wonder how people first stumbled upon pi? Picture this: ancient civilizations, scratching their heads over circles and their mysterious ratios. They probably stared at the moon, the sun, and the wheels of their carts, trying to make sense of it all. And then, bam! Someone decides to divide the circumference by the diameter, and voila, pi is born. Talk about a lightbulb moment.Pi isn't just a number; it's a celebrity. People throw parties for it every March 14th, because, you know, 3/14, get it? They recite digits of pi like it's some kind of sacred chant. Some folks have even memorized thousands of digits, which is impressive and kind of insane at the sametime. But hey, to each their own.Let's talk about pi's role in the real world. It's not just hanging out in math textbooks; it's doing stuff. Engineers use it to build bridges that won't collapse, astronomers use it to explore the cosmos, and even your GPS relies on pi to pinpoint your location. So yeah, next time you're lost in the middle of nowhere, you can thank pi for getting you back on track.But here's the kicker: despite its fame and usefulness, we still can't pin pi down. It's like trying to catch a slippery fish with your bare hands. Sure, we've calculated trillions of digits, but we still haven't found a pattern. It's the ultimate enigma, teasing us with its infinite complexity.In the end, pi is more than just a number; it's a symbol of human curiosity and perseverance. We may never fully unravel its mysteries, but that's okay. Maybe the joy is in the chase, the endless quest to understand somethingas elusive as pi. So here's to pi, the ultimate circle superhero, forever keeping us guessing.。
无处不在的数学:音乐与节奏的数学之美
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无处不在的数学:音乐与节奏的数学之美The Ubiquitous Beauty of Mathematics in Music andRhythmMathematics, a language of precision and logic, often finds its way into unexpected corners of life. Amidst the melodies and harmonies that envelope our ears, there is an intricate dance of mathematical principles at play—a silent symphony where numbers and patterns govern the beats and scales. It's remarkable how such abstract concepts as fractions, ratios, and wave forms can translate into such emotional and visceral experiences as music.The foundation of much western music lies in the octave system, a harmonious scale divided precisely by halves. This simple yet profound ratio of 2:1 underlies the construction of most musical instruments and defines the structure of melodies we cherish. Just like the golden ratio found in nature’s most aesthetically plea sing shapes, these mathematical proportions create a sense of harmony and balance within music that resonates deeply with us. Rhythm, another integral aspect of music, also exhibits stunning mathematical properties. Beats per minute, tempos, and measures are all quantifiable elements that structure the flow of sound. These metrical systems allow composers to orchestrate precise patterns of silence and noise, creating dynamic tension and release that move listeners emotionally. Whether it's the subtle shifts in tempothat build anticipation or the repeating rhythmic motifs that establish a groove, mathematics underpins the artistry of rhythm in ways both obvious and sublime.Moreover, modern advancements in digital audio workstations have further blurred the lines between music and mathematics. Waveforms, frequencies, and sample rates can now be manipulated with surgical precision using algorithms and software—tools that draw heavily on complex mathematical principles. This marriage of art and science has birthed new genres and styles of music that explore previously unimaginable sonic landscapes.In conclusion, the beauty of mathematics in music and rhythm is not just about cold calculations but about the warmth of human expression. It's about understanding how abstract principles can manifest into something tangible and emotionally charged—something that moves us deeply and connects us across cultures and eras. As long as people continue to compose and appreciate music, the elegant dance between these two seemingly disparate fields will persist, reminding us once again of the ubiquity and wonder of mathematics in every facet of life.。
Mathematisches Seminar
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Abstract
1 Introduction
Divide and conquer algorithms are characterized by recursive calls of the same algorithm on a lower level. After a nite time the algorithm terminates. The termination time may depend on the input. If the input is unknown and modelled with a stochastic component then the termination time becomes a random variable. This kind of randomness is external random. In contrast to external randomness we are mainly concerned with internal randomness, that is the randomness within the algorithm itself. A stochastic (random) divide and conquer algorithm is a divide and conquer algorithm with a stochastic component within the algorithm itself. The termination time is a random variable and may additionally depend on the input or other external in uence. Often, and this is the default setting, the built-in stochastic component within the algorithm itself ensures a distribution of the performance time (and other features) not dependent on the actual input or other external in uence. In the following we consider such stochastic divide and conquer algorithms and analyse their performance time, on average and in distribution. The rst example in this direction, Quicksort (Rosler (1991) 17]), will guide the reader to the general setup. Let Xn; n 2 IN; denote the performance costs like money, time or resources to solve completely a task of level n: These costs are deterministic or random. They satisfy a recursive relation. The cost Xn for a problem of level n is the sum of the cost Cn for calling the algorithm once and the costs for the ( nitely) many problems of levels Zn;i into which the algorithm splitts the level n task, D X Xn = XZn;i;i + Cn: (1)
数学规律在音乐节奏中的体现
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数学规律在音乐节奏中的体现The Manifestation of Mathematical Patterns in MusicalRhythmIn the wonderful symphony of music, mathematical laws are often silently yet profoundly present. They weave a complex tapestry of notes and beats, creating rhythmical patterns that enchant our ears.The foundation of musical rhythm lies in numbers. Beats per minute—the metric for tempo—is an inherent numerical concept. Whether it's a lively jig or a somber ballad, each piece adheres to a specific speed, determined by precise fractions of time. These fractions dictate how frequently notes are played or sung, creating the backbone of any melody.Furthermore, the harmony within melodies is deeply rooted in mathematical principles. Scales, chords, and intervals all abide by rules that can be expressed through ratios and frequencies. For instance, the consonant sounds we find pleasing are often those whose frequencies align with simple ratios like whole numbers or halves. This harmonious blend creates a sense of unity and balance within the composition.Moreover, mathematical patterns govern even the most intricate rhythmic variations. In complex compositions, musicians employ various techniques such as polyrhythms—where multiple independent rhythmsoverlap—to create dynamic and exciting performances. These polyrhythmic structures, though seemingly chaotic at first glance, actually adhere to strict mathematical formulas that ensure their coherence and harmony.It could be concluded then, that mathematics isn't just a language of logic but also one of beauty; its invisible hand shapes the audible world of music, giving rise to melodies that move us emotionally while satisfying our intellectual curiosity. As composers and performers continue to explore new sonic territories, they will undoubtedly discover more fascinating intersections between these two seemingly disparate fields: the science of mathematics and the art of music.。
圆周率日英语阅读
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圆周率日英语阅读圆周率日在英语中的阅读材料可能包含与圆周率(π)相关的历史、数学原理、应用和庆祝活动等内容。
以下是一个关于圆周率的英语阅读材料:Title: The Importance of Pi DayOn March 14th, people around the world celebrate Pi Day, a special occasion dedicated to the mathematical constant pi. Pi is an irrational num ber that represents the ratio of a circle's circumference to its diameter. It has been an essential element of mathematics since ancient times and play s a crucial role in various scientific and engineering fields.The origins of Pi Day can be traced back to the year 1738 when A merican mathematician and physicist Benjamin Franklin proposed the idea of celebrating the day. Since then, the celebration has grown in popularity, with numerous events and activities held each year to mark the occasion.One of the most popular traditions of Pi Day is the consumption of pie. People across the globe enjoy various types of pies, including fruit pi es, cream pies, and even pizza pie. This delicious tradition has become a symbol of the holiday, as it represents the mathematical concept of pi in a fun and tasty way.In addition to enjoying pies, many schools and universities organize math competitions, puzzles, and workshops to celebrate Pi Day. These eve nts aim to promote interest in mathematics and celebrate the importance o f pi in our lives.Pi Day is also an opportunity to recognize the contributions of promi nent mathematicians who have advanced our understanding of pi. Some of these mathematicians include Archimedes, who calculated pi to 20 decimal places using a method industrialization; Chinese mathematician Zu Chong zhi, who calculated pi to 7 decimal places; and Indian mathematician Srini vasa Ramanujan, who discovered numerous properties of pi.As we celebrate Pi Day, let us remember the significance of this rem arkable mathematical constant and the brilliant minds that have contributed to its study. Whether it's enjoying a slice of pie, participating in a math competition, or simply appreciating the beauty of pi, there are countless w ays to celebrate the day and honor the universal language of mathematics.在这个材料中,我们了解了圆周率日的由来、庆祝活动以及与圆周率相关的历史和数学原理。
中英音译字滙
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帶氧運動 (健身舞) 阿哥哥舞
球 芭蕾舞
樂隊
預訂
回目錄
保齡 Bowling
卜成 霹靂舞 笨豬跳
Boxing
Breakdance Bungeejumping
嘉年華 Carnival
喳喳 Cha-cha
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保齡球 拳擊 霹靂舞
磞極跳 狂歡節 喳喳舞
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刁士 安哥 否 分唇 高爾夫 呼拉圈
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Deuce Encore
BBQ 羅宋湯 暴飛 周打 唂爹 咖哩
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Barbecue Borsch Buffet Chowder Cocktail Curry
燒烤 俄羅斯菜湯
自助餐 雜燴 雞尾/雜混 咖哩
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吉列 免翁 免治 奄列
沙律
月展
西冷
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Cutlet Mignon Mince Omelet
Salad Shin Sirloin
馬拉松 Marathon 馬拉松長跑
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Olympic Outside
奧林匹克 出界
乒乓
Ping-pong 乒乓球
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拉力賽 樂與怒
倫巴 森巴
做騒 士碌架
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Rally
汽車競賽
Rock-and-roll 街頭舞
Rumba
倫巴舞
Samba
森巴舞
Show
做秀
Snooker
英式桌球
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追捧者
法西斯主義者 領班
男同性戀者 納粹德國 秘密警察 尼泊爾人 吉卜賽人
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黑客 印度人 嬉皮士 荷李活
可汗 凱撒 卡啦OK
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专为中国人写的单词密码
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第1天空杯心态经典的记忆训练通常是从一个实验开始的:桌子上放着三个相同大小的杯子,一个是空杯.一个杯子中有半杯水,还有一个杯子是装满了水的。
此时,用容器分别向王个杯子中加水。
结果是,第一个杯子被注入了整杯的水,第二个杯子只能加入半杯水,而第二个杯子一滴水也加不进去了。
实验很简单,几乎所有的人都能预见到结果。
因为加入的水量是由杯子剩余的空间所决定的,装满了水的杯子没留下任何空间,自然一滴也加不进去。
请暂时清空我们杯中的水.以积极开放的心态面对新事物。
”善人者不善人之师,不善者善人之资。
”谦虚好学开放——这就是空杯心态。
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Additions to Part VI:Bibliography
Change the entry Fossy/Morillon[1995]to:
Fossy,J./Morillon,M.
[1998]The Baire category property and some notions of compactness,J.London Math.Soc.57,1–19.
Change“Bacsich[1972]”to“Bacsich[1972a]”
Add the following:
Alas,O.T.
[1994]On countably compact product spaces,Proceedings of the Eleventh In-ternational Conference of Topology(Trieste1993),Rend.Mat.Univ.
Trieste25,1–4.
Bacsich,P.D.
[1972b]Effective equivalents of the Rasiowa-Sikorski lemma,J.London.Math.
Soc.(2)4,513–518.
Banaschewski,B.
[1961]On some theorems equivalent to the axiom of choice,Z.Math.Logik Grundlagen Math.,7,279–282.
[1990]On proving the Tychonoffproduct theorem,Kyungpook Math.J.30, 65–73.
Dodu,J./Morillon,M.
[1999]The Hahn-Banach property and the axiom of choice,Math.Logic Quart.
45,299–314.
smallskip Howard,P./Keremedis,K.,/Rubin,J./Stanley,A.
[1999a]Compactness in countable Tychonoffproducts and choice,Accepted Math.Logic Quart.January.
Halpern,J.D./Howard,P.
[1976]The law of infinite cardinal addition is weaker than the axiom of choice, Trans.Amer.Math.Soc.220,195–204.
Keremedis,K.
[1999d]The multiple choice axiom does not imply every vector space over Q has
a basis,Preprint.
Howard,P./Rubin,J./Stanley,A.
[1999]Von Rimsha’s transitivity conditions,Accepted Math.Logic Quart.August.
Add to Howard/Keremedis/Rubin/Stanley/Tachtsis[1999]:
Submitted Math.Logic Quart.August.
1
2
Change Howard/Keremedis/Rubin/Stanley[1997]to
Howard,P./Keremedis,K.,/Rubin,J./Stanley,A.
[1999b]Paracompactness of metric spaces and the axiom of choice,Accepted Math.Logic Quart.June.
Change Keremedis[1996b]to
Keremedis,K.
[1998a]Filters,antichains and towers in topological spaces and the axiom of choice,Math.Logic Quart.44359-366.
Change Keremedis[1996c]to[1996b].
Change Keremedis[1996d]to[1996c].
Change Keremedis[1998]to[1998b].。