Algebra Coding Theory Lecture6

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代数学的主要内容

代数学的主要内容Algebra is a fundamental branch of mathematics that deals with symbols and the rules for manipulating these symbols. 代数是数学的基础学科之一，涉及符号及其操作规则。

It is a broad field that encompasses various topics such as equations, polynomials, functions, and more. 代数是一个广泛的领域，涉及方程、多项式、函数等各种主题。

Algebra is essential in many areas of mathematics and science, providing a foundation for more advanced studies. 代数在许多数学和科学领域至关重要，为更深入的研究奠定了基础。

One of the main goals of algebra is to solve equations and analyze the properties of algebraic structures. 代数的一个主要目标是解方程和分析代数结构的性质。

By studying algebra, mathematicians gain insight into patterns and relationships that exist within mathematical systems. 通过研究代数，数学家可以深入了解存在于数学系统内的模式和关系。

Algebraic manipulation allows for the simplification of complex expressions and the solution of problems that may seem daunting at first. 代数运算允许简化复杂表达式并解决一开始看起来令人生畏的问题。

k阶指数哥伦布编码算法

k阶指数哥伦布编码算法k阶指数哥伦布编码算法（k-order exponential Golomb coding algorithm）是一种无损压缩算法，主要用于对非负整数序列进行编码。

本文将对该算法进行详细介绍和分析，包括算法原理、编码过程和编码效果评估。

一、算法原理k阶指数哥伦布编码算法的核心思想是通过对非负整数进行分组，然后对每个分组进行编码，并使用相应的标志位区分不同的分组。

该算法以k为阶数进行分组，将每个整数表示为两部分：前缀和后缀。

具体而言，设k为阶数，将整数n分解为一个前缀x和一个后缀y，其中x的位数为k-1，y的值范围为0到2^(k-1)。

前缀x表示该整数所属的分组，后缀y则表示该整数在分组中的位置。

对于非负整数n，其编码过程可以分为以下几个步骤：Step 1：计算前缀x和后缀y，其中x = n / 2^(k-1)，y = n mod 2^(k-1)。

Step 2：根据前缀x的位数和y的值，确定分组的编码长度，使用二进制表示的结果为g。

Step 3：将g的二进制表示加上前缀x的二进制表示作为最终的编码结果。

例如，当k=3时，对整数15进行编码：Step 1：计算x和y，其中x = 15 / 2^2 = 3，y = 15 mod 2^2 = 3。

Step 2：由于x = 3的二进制表示为"011"，y = 3的二进制表示为"011"，所以分组的编码长度为"011011"。

Step 3：将分组的编码长度"011011"与前缀x的二进制表示"011"拼接得到最终的编码结果"011011011"。

二、编码过程k阶指数哥伦布编码算法的编码过程可以总结为以下几个步骤：Step 1：确定阶数k和非负整数序列。

Step 2：对每个非负整数n进行编码。

Step 3：将每个编码的结果拼接得到最终的压缩结果。

群的阶与群中元素的阶的关系讲解

石家庄铁道学院毕业论文群的阶与其元素的阶的关系摘要近世代数虽是一门较新的，较抽象的学科，但如今它已渗透到科学的各个领域，解决了许多著名的数学难题:像尺规作图不能问题，用根式解代数方程问题，编码问题等等．而群是近世代数里面最重要的内容之一，也是学好近世代数的关键．本论文旨在从各个角度和方面来探讨群的阶与其元的阶之间的关系．具体地来说，本文先引入了群的概念，介绍了群及有关群的定义，然后着重讨论了有限群、无限群中关于元的阶的情况．并举了一些典型实例进行分析，之后又重点介绍了有限群中关于群的阶与其元的阶之间的关系的定理——拉格朗日定理，得出了一些比较好的结论．在群论的众多分支中，有限群论无论从理论本身还是从实际应用来说，都占据着更为突出的地位．同时，它也是近年来研究最多、最活跃的一个数学分支．因此，在本文最后，我们介绍了著名的有限交换群的结构定理，并给出了实例分析．关键词：群论有限群元的阶石家庄铁道学院毕业论文AbstractThe Modern Algebra is a relatively new and abstract subject, but now it has penetrated into all fields of science and solved a number of well-known mathematical problems, such as, the impossibility for Ruler Mapping problem, the solutions for algebraic equations with radical expressions, coding problems and so on. The group is one of the most important portions in the Modern Algebra, and also the key of learning it well.This paper aims at discussing the relations between the order of a group and the orders of its elements from all the angles and aspects. Specifically, this thesis firstly introduces the concept of a group and some relatives with it; secondly focuses on the orders of the elements in the finite group and the infinite group respectively, some typical examples are listed for analyses; thirdly stresses on the theorem - Lagrange's theorem on the relations between the order of a group and the orders of its elements in the finite group, accordingly obtaining some relatively good conclusion.In the many branches of group theory, the finite group theory, whether from the theory itself or from the practical applications, occupies a more prominent position. At the same time, it is also one of the largest researches and the most active branches of mathematics in the recent years. Therefore, in this paper finally, we introduce the famous theorem of the structures on the finite exchanging groups, and give several examples for analyses.Key words：group theory finite groups the orders of elements石家庄铁道学院毕业论文目录１绪论 (1)1.1 群论的概括 (1)1.2 群论的来源 (1)1.3 群论的思想 (2)2 预备知识 (2)2.1 群和子群 (2)2.1.1 群的定义 (2)2.1.2 群的阶的定义 (3)2.1.3 元的阶的定义 (4)2.1.4 子群、子群的陪集 (5)2.1.5 同构的定义 (6)2.2 不变子群与商群 (6)2.2.1 不变子群与商群 (6)2.2.2 Cayley(凯莱)定理 (7)2.2.3 内直和和外直积的定义 (8)3 群中元的阶的各种情况及其实例分析 (8)3.1 有限群中关于元的阶 (9)3.1.1 有限群中元的阶的有限性 (9)3.1.2 有限群中关于元的阶及其个数的关系 (9)3.2 无限群中关于元的阶 (10)3.2.1 无限群G中，除去单位元外，每个元素的阶均无限 (10)3.2.2 无限群G中，每个元素的阶都有限 (10)3.2.3 G为无限群，G中除单位元外，既有无限阶的元，又有有限阶的元 (11)4 群的阶与其元的阶之间的关系 (11)4.1 拉格朗日(Lagrange)定理 (11)4.1.1 拉格朗日定理 (11)4.1.2 相关结论 (12)4.2 有限交换群的结构定理 (13)4.2.1 有限交换群的结构定理 (13)石家庄铁道学院毕业论文4.2.2 相关例子 (14)参考文献 (15)致谢 ······································································错误！未定义书签。

退位减法数学专业词汇英语词典

退位减法数学专业词汇英语词典Mathematics is a universal language that transcends cultural and linguistic boundaries. It is a language of logic, precision, and abstract reasoning that has shaped the course of human civilization. One of the fundamental concepts in mathematics is the operation of subtraction, which is often referred to as "taking away" or "removal." In the realm of advanced mathematics, subtraction takes on a more specialized term known as "retrograde operation" or "retrograde subtraction." This concept is particularly relevant in the study of number theory, where it forms the basis of various computational techniques and algorithms.Retrograde subtraction, also commonly referred to as "reverse subtraction" or "descending subtraction," is a method of subtracting a larger number from a smaller number. Unlike traditional subtraction, where the smaller number is subtracted from the larger number, retrograde subtraction involves subtracting the larger number from the smaller number. This operation is particularly useful in modular arithmetic, where numbers are considered within a specific range or modulus. By employing retrograde subtraction, mathematicians can work with negative numbers and effectively "wrap around" the modulus, ensuring that the result remains withinthe desired range.One of the key applications of retrograde subtraction is in the computation of modular inverses, which are essential in cryptography and other areas of mathematics. The modular inverse of a number is a value that, when multiplied by the original number and then taken modulo a specific value, results in a value of 1. Retrograde subtraction plays a crucial role in the extended Euclidean algorithm, which is commonly used to calculate modular inverses efficiently.Furthermore, retrograde subtraction finds applications in the study of congruences and number theory problems involving residues. It allows mathematicians to work with numbers in a more flexible manner, enabling them to navigate the intricate relationships between numbers within a specific modular system. This concept also has implications in the field of coding theory, where retrograde subtraction is used in the construction and analysis of error-correcting codes.Beyond its practical applications, retrograde subtraction also holds theoretical significance in the realm of abstract algebra and group theory. It is closely related to the concept of group inverses, where the inverse of an element in a group is defined as the element that, when combined with the original element through the groupoperation, results in the identity element. This connection highlights the deep algebraic underpinnings of retrograde subtraction and its role in the broader study of mathematical structures.In conclusion, retrograde subtraction, or "descending subtraction," is a specialized mathematical term that refers to the operation of subtracting a larger number from a smaller number. It is a fundamental concept in number theory, cryptography, coding theory, and abstract algebra, enabling mathematicians to work with numbers in a more flexible and efficient manner. The study of retrograde subtraction not only enhances our understanding of computational techniques but also sheds light on the intricate relationships between numbers and the underlying algebraic structures that govern them.。

AlgebraicGraphTheory

Algebraic Graph TheoryChris Godsil(University of Waterloo),Mike Newman(University of Ottawa)April25–291Overview of the FieldAlgebraic graph theory comprises both the study of algebraic objects arising in connection with graphs, for example,automorphism groups of graphs along with the use of algebraic tools to establish interesting properties of combinatorial objects.One of the oldest themes in the area is the investigation of the relation between properties of a graph and the spectrum of its adjacency matrix.A central topic and important source of tools is the theory of association schemes.An association scheme is,roughly speaking,a collection of graphs on a common vertex set whichﬁt together in a highly regular fashion.These arise regularly in connection with extremal structures:such structures often have an unex-pected degree of regularity and,because of this,often give rise to an association scheme.This in turn leads to a semisimple commutative algebra and the representation theory of this algebra provides useful restrictions on the underlying combinatorial object.Thus in coding theory we look for codes that are as large as possible, since such codes are most effective in transmitting information over noisy channels.The theory of association schemes provides the most effective means for determining just how large is actually possible;this theory rests on Delsarte’s thesis[4],which showed how to use schemes to translate the problem into a question that be solved by linear programming.2Recent Developments and Open ProblemsBrouwer,Haemers and Cioabˇa have recently shown how information on the spectrum of a graph can be used to proved that certain classes of graphs must contain perfect matchings.Brouwer and others have also investigated the connectivity of strongly-regular and distance-regular graphs.This is an old question,but much remains to be done.Recently Brouwer and Koolen[2]proved that the vertex connectivity of a distance-regular graph is equal to its valency.Haemers and Van Dam have worked on extensively on the question of which graphs are characterized by the spectrum of their adjacency matrix.They consider both general graphs and special classes,such as distance-regular graphs.One very signiﬁcant and unexpected outcome of this work was the construction,by Koolen and Van Dam[10],of a new family of distance-regular graphs with the same parameters as the Grassmann graphs.(The vertices of these graphs are the k-dimensional subspaces of a vector space of dimension v over theﬁniteﬁeld GF(q);two vertices are adjacent if their intersection has dimension k1.The graphs are q-analog of the Johnson graphs,which play a role in design theory.)These graphs showed that the widely held belief that we knew all distance-regular graphs of“large diameter”was false,and they indicate that the classiﬁcation of distance-regular graphs will be more complex(and more interesting?)than we expected.1Association schemes have long been applied to problems in extremal set theory and coding theory.In his(very)recent thesis,Vanhove[14]has demonstrated that they can also provide many interesting results in ﬁnite geometry.Recent work by Schrijver and others[13]showed how schemes could used in combination with semidef-inite programming to provide signiﬁcant improvements to the best known bounds.However these methods are difﬁcult to use,we do not yet have a feel for we might most usefully apply them and their underlying theory is imperfectly understood.Work in Quantum Information theory is leading to a wide range of questions which can be successfully studied using ideas and tools from Algebraic Graph Theory.Methods fromﬁnite geometry provide the most effective means of constructing mutually unbiased bases,which play a role in quantum information theory and in certain cryptographic protocols.One important question is to determine the maximum size of a set of mutually unbiased bases in d-dimensional complex space.If d is a prime power the geometric methods just mentioned provide sets of size d+1,which is the largest possible.But if d is twice an odd integer then in most cases no set larger than three has been found.Whether larger sets exist is an important open problem. 3Presentation HighlightsThe talks mostlyﬁtted into one of four areas,which we discuss separately.3.1SpectraWillem Haemers spoke on universal adjacency matrices with only two distinct eigenvalues.Such matrices are linear combinations of I,J,D and A(where D is the diagonal matrix of vertex degrees and A the usual adjacency matrix).Any matrix usually considered in spectral graph theory has this form,but Willem is considering these matrices in general.His talk focussed on the graphs for which some universal adjacency matrix has only two eigenvalues.With Omidi he has proved that such a graph must either be strong(its Seidel matrix has only two eigenvalues)or it has exactly two different vertex degrees and the subgraph induced by the vertices of a given degree must be regular.Brouwer formulated a conjecture on the minimum size of a subset S of the vertices of a strongly-regular graph X such that no component of X\S was a single vertex.Cioabˇa spoke on his recent work with Jack Koolen on this conjecture.They proved that it is false,and there are four inﬁnite families of counterexamples.3.2PhysicsAs noted above,algebraic graph theory has many applications and potential applications to problems in quantum computing,although the connection has become apparent only very recently.A number of talks were related to this connection.One important problem in quantum computing is whether there is a quantum algorithm for the graph isomorphism problem that would be faster than the classical approaches.Currently the situation is quite open.Martin Roetteler’s talk described recent work[1]on this problem.For our workshop’s viewpoint,one surprising feature is that the work made use of the Bose-Mesner algebra of a related association scheme; this connection had not been made before.Severini discussed quantum applications of what is known as the Lov´a sz theta-function of a graph.This function can be viewed as an eigenvalue bound and is closely related to both the LP bound of Delsarte and the Delsarte-Hoffman bound on the size of an independent set in a regular graph.Severini’s work shows that Lov´a sz’s theta-function provides a bound on the capacity of a certain channel arising in quantum communication theoryWork in quantum information theory has lead to interest in complex Hadamard matrices—these are d×d complex matrices H such that all entries of H have the same absolute value and HH∗=dI.Both Chan and Sz¨o ll˝o si dealt with these in their talks.Aidan Roy spoke on complex spherical designs.Real spherical designs were much studied by Seidel and his coworkers,because of their many applications in combinatorics and other areas.The complex case languished because there were no apparent applications,but now we have learnt that these manifest them-selves in quantum information theory under acronyms such as MUBs and SIC-POVMs.Roy’s talk focussedon a recent 45page paper with Suda [12],where (among other things)they showed that extremal complex designs gave rise to association schemes.One feature of this work is that the matrices in their schemes are not symmetric,which is surprising because we have very few interesting examples of non-symmetric schemes that do not arise as conjugacy class schemes of ﬁnite groups.3.3Extremal Set TheoryCoherent conﬁgurations are a non-commutative extension of association schemes.They have played a sig-niﬁcant role in work on the graph isomorphism problem but,in comparison with association schemes,they have provided much less information about interesting extremal structures.The work presented by Hobart and Williford may improve matters,since they have been able to extend and use some of the standard bounds from the theory of schemes.Delsarte [4]showed how association schemes could be used to derive linear programs,whose values provided strong upper bounds on the size of codes.Association schemes have both a combinatorial structure and an algebraic structure and these two structures are in some sense dual to one another.In Delsarte’s work,both the combinatorial and the algebraic structure had a natural linear ordering (the schemes are both metric and cometric)and this played an important role in his work.Martin explained how this linearity constraint could be relaxed.This work is important since it could lead to new bounds,and also provide a better understanding of duality.One of Rick Wilson’s many important contributions to combinatorics was his use of association schemes to prove a sharp form of the Erd˝o s-Ko-Rado theorem [15].The Erd˝o s-Ko-Rado theorem itself ([5])can certainly be called a seminal result,and by now there are many analogs and extensions of it which have been derived by a range of methods.More recently it has been realized that most of these extensions can be derived in a very natural way using the theory of association schemes.Karen Meagher presented recent joint work (with Godsil,and with Spiga,[8,11])on the case where the subsets in the Erd˝o s-Ko-Rado theorem are replaced by permutations.It has long been known that there is an interesting association scheme on permutations,but this scheme is much less manageable than the schemes used by Delsarte and,prior to the work presented by Meagher,no useful combinatorial information had been obtained from it.Chowdhury presented her recent work on a conjecture of Frankl and F¨u redi.This concerns families F of m -subsets of a set X such that any two distinct elements of have exactly λelements in common.Frankl and F¨u redi conjectured that the m -sets in any such family contain at least m 2 pairs of elements of X .Chowdhury veriﬁed this conjecture in a number of cases;she used classical combinatorial techniques and it remains to see whether algebraic methods can yield any leverage in problems of this type.3.4Finite GeometryEric Moorhouse spoke on questions concerning automorphism groups of projective planes,focussing on connections between the ﬁnite and inﬁnite case.Thus for a group acting on a ﬁnite plane,the number of orbits on points must be equal to the number of orbits on lines.It is not known if this must be true for planes of inﬁnite order.Is there an inﬁnite plane such that for each positive integer k ,the automorphism group has only ﬁnitely many orbits on k -tuples?This question is open even for k =4.Simeon Ball considered the structure of subsets S of a k -dimensional vector space over a ﬁeld of order q such that each d -subset of S is a basis.The canonical examples arise by adding a point at inﬁnity to the point set of a rational normal curve.These sets arise in coding theory as maximum distance separable codes and in matroid theory,in the study of the representability of uniform matroids (to mention just two applications).It is conjectured that,if k ≤q −1then |S |≤q +1unless q is even and k =3or k =q −1,in which case |S |≤q +2.Simeon presented a proof of this theorem when q is a prime and commented on the general case.He developed a connection to Segre’s classical characterization of conics in planes of odd order,as sets of q +1points such that no three are collinear.There are many analogs between ﬁnite geometry and extremal set theory;questions about the geometry of subspaces can often be viewed as q -analogs of questions in extremal set theory.So the EKR-problem,which concerns characterizations of intersecting families of k -subsets of a ﬁxed set,leads naturally to a study of intersecting families of k -subspaces of a ﬁnite vector space.In terms of association schemes this means we move from the Johnson scheme to the Grassmann scheme.This is fairly well understood,with thebasic results obtained by Frankl and Wilson[6].But inﬁnite geometry,polar spaces form an important topic. Roughly speaking the object here is to study the families of subspaces that are isotropic relative to some form, for example the subspaces that lie on a smooth quadric.In group theoretic terms we are now dealing with symplectic,orthogonal and unitary groups.There are related association schemes on the isotropic subspaces of maximum dimension.Vanhove spoke on important work from his Ph.D.thesis,where he investigated the appropriate versions of the EKR problem in these schemes.4Outcome of the MeetingIt is too early to offer much in the way of concrete evidence of impact.Matt DeV os observed that a conjecture of Brouwer on the vertex connectivity of graphs in an association scheme was wrong,in a quite simple way. This indicates that the question is more complex than expected,and quite possibly more interesting.That this observation was made testiﬁes to the scope of the meeting.On a broader level,one of the successes of the meeting was the wide variety of seemingly disparate topics that were able to come together;the ideas of algebraic graph theory touch a number of things that would at ﬁrst glance seem neither algebraic nor graph theoretical.There was a lively interaction between researchers from different domains.The proportion of post-docs and graduate students was relatively high.This had a positive impact on the level of excitement and interaction at the meeting.The combination of expert and beginning researchers created a lively atmosphere for mathematical discussion.References[1]A.Ambainis,L.Magnin,M.Roetteler,J.Roland.Symmetry-assisted adversaries for quantum state gen-eration,arXiv1012.2112,35pp.[2]A.E.Brouwer,J.H.Koolen.The vertex connectivity of a distance-regular graph.European bina-torics30(2009),668–673.[3]A.E.Brouwer,D.M.Mesner.The connectivity of strongly regular graphs.European binatorics,6(1985),215–216.[4]P.Delsarte.An algebraic approach to the association schemes of coding theory.Philips Res.Rep.Suppl.,(10):vi+97,1973.[5]P.Erd˝o s,C.Ko,R.Rado.Intersection theorems for systems ofﬁnite sets.Quart.J.Math.Oxford Ser.(2),12(1961),313–320.[6]P.Frankl,R.M.Wilson.The Erd˝o s-Ko-Rado theorem for vector binatorial Theory,SeriesA,43(1986),228–236.[7]D.Gijswijt,A.Schrijver,H.Tanaka.New upper bounds for nonbinary codes based on the Terwilligeralgebra and semideﬁnite binatorial Theory,Series A,113(2006),1719–1731. [8]C.D.Godsil,K.Meagher.A new proof of the Erd˝o s-Ko-Rado theorem for intersecting families of per-mutations.arXiv0710.2109,18pp.[9]C.D.Godsil,G.F.Royle.Algebraic Graph Theory,Springer-Verlag,(New York),2001.[10]J.H.Koolen,E.R.van Dam.A new family of distance-regular graphs with unbounded diameter.Inven-tiones Mathematicae,162(2005),189-193.[11]K.Meagher,P.Spiga.An Erdos-Ko-Rado theorem for the derangement graph of PGL(2,q)acting onthe projective line.arXiv0910.3193,17pp.[12]A.P.Roy,plex spherical Codes and designs,(2011),arXiv1104.4692,45pp.[13]A.Schrijver.New code upper bounds from the Terwilliger algebra and semideﬁnite programming.IEEETransactions on Information Theory51(2005),2859–2866.[14]F.Vanhove.Incidence geometry from an algebraic graph theory point of view.Ph.D.Thesis,Gent2011.[15]R.M.Wilson.The exact bound in the Erds-Ko-Rado binatorica,4(1984),247–257.。

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 KressGTM182《Ordinary Differential Equations》Wolfgang WalterGTM183《An introduction to Banach Spaces》Robert E.MegginsonGTM184《Modern Graph Theory》Béla Bollobás（现代图论）GTM185《Using Algebraic Geomety》David A.Cox, John Little, Donal O’Shea（应用代数几何）GTM186《Fourier Analysis on Number Fields》Dinakar Ramakrishnan, Robert J.Valenza GTM187《Moduli of Curves》Joe Harris, Ian Morrison（曲线模）GTM188《Lectures on the Hyperreals：An Introduction to Nonstandard Analysis》Robert GoldblattGTM189《Lectures on Modules and Rings》m（模和环讲义）GTM190《Problems in Algebraic Number Theory》M.Ram Murty, Jody Esmonde（代数数论中的问题）GTM191《Fundamentals of Differential Geometry》Serge Lang（微分几何基础）GTM192《Elements of Functional Analysis》Francis Hirsch, Gilles LacombeGTM193《Advanced Topics in Computational Number Theory》Henri CohenGTM194《One-Parameter Semigroups for Linear Evolution Equations》Klaus-Jochen Engel, Rainer Nagel（线性发展方程的单参数半群）GTM195《Elementary Methods in Number Theory》Melvyn B.Nathanson（数论中的基本方法）GTM196《Basic Homological Algebra》M.Scott OsborneGTM197《The Geometry of Schemes》David Eisenbud, Joe HarrisGTM198《A Course in p-adic Analysis》Alain M.RobertGTM199《Theory of Bergman Spaces》Hakan Hedenmalm, Boris Korenblum, Kehe Zhu（Bergman空间理论）GTM200《An Introduction to Riemann-Finsler Geometry》D.Bao, S.-S.Chern, Z.Shen GTM201《Diophantine Geometry An Introduction》Marc Hindry, Joseph H.Silverman GTM202《Introduction to Topological Manifolds》John M.LeeGTM203《The Symmetric Group》Bruce E.SaganGTM204《Galois Theory》Jean-Pierre EscofierGTM205《Rational Homotopy Theory》Yves Félix, Stephen Halperin, Jean-Claude Thomas（有理同伦论）GTM206《Problems in Analytic Number Theory》M.Ram MurtyGTM207《Algebraic Graph Theory》Chris Godsil, Gordon Royle（代数图论）GTM208《Analysis for Applied Mathematics》Ward CheneyGTM209《A Short Course on Spectral Theory》William Arveson（谱理论简明教程）GTM210《Number Theory in Function Fields》Michael RosenGTM211《Algebra》Serge Lang（代数）GTM212《Lectures on Discrete Geometry》Jiri Matousek（离散几何讲义）GTM213《From Holomorphic Functions to 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 Forms》Fred Diamond, Jerry Shurman（模形式初级教程）GTM229《The Geometry of Syzygies》David Eisenbud（合冲几何）GTM230《An Introduction to Markov Processes》Daniel W.Stroock（马尔可夫过程引论）GTM231《Combinatorics of Coxeter Groups》Anders Bjröner, Francesco Brenti（Coxeter 群的组合学）GTM232《An Introduction to Number Theory》Graham Everest, Thomas Ward（数论入门）GTM233《Topics in Banach Space Theory》Fenando Albiac, Nigel J.Kalton（Banach空间理论选题）GTM234《Analysis and Probability：Wavelets, Signals, Fractals》Palle E.T.Jorgensen（分析与概率）GTM235《Compact Lie Groups》Mark R.Sepanski（紧致李群）GTM236《Bounded Analytic Functions》John B.Garnett（有界解析函数）GTM237《An Introduction to Operators on the Hardy-Hilbert Space》Rubén A.Martínez-Avendano, Peter Rosenthal（哈代-希尔伯特空间算子引论）GTM238《A Course in Enumeration》Martin Aigner（枚举教程）GTM239《Number Theory：VolumeⅠTools and Diophantine Equations》Henri Cohen GTM240《Number Theory：VolumeⅡAnalytic and Modern Tools》Henri Cohen GTM241《The Arithmetic of Dynamical Systems》Joseph H.SilvermanGTM242《Abstract Algebra》Pierre Antoine 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。

Algebra1中英版对照目录

GCF：greatest common factor LCM：least common multiple 0—10：Equivalent Fractions 等值分数 Z23—Z24 0—11：Decimals ,Fractions ,and Percents 小数，分数，百分数 Z25—Z27 0—12：Finding a Common Denominator 公分母求值 Z28—Z29 0—13：Adding and Subtracting Fractions 分数的加减运算 Z30—Z31 0—14：Multiplying and Dividing Fractions 分数的乘除运算 Z32—Z33
解两侧有变量的一元一次不等式 P194—P201 3—6：Solving Compound Inequalities
解混合一元一次不等式（一元一次不等式组）P202—P209 Quiz for Chapter 3 第三章小测试 P210—P225 CHAPTER 4—Functions 函数 4—1：Graphing Relationships 关系图（函数关系图）P230—P235 4—2：Relations and Functions 函数的对应关系 P236—P244 4—3：Writing Functions 函数的表达式 P245—251 4—4：Graphing Functions 函数的图像（直线、二次函数）P252—P260 CHAPTER 4—Functions 函数 4—5：Scatter Plots and Trend Lines 点的散布图与趋势线 P262—P271 4—6：Arithmetic Sequences 等差数列（等差数列的项和通项）P272—P278 Quiz for Chapter 4 第四章小测试 P279—P291 CHAPTER 5— Linear Functions 一次函数 5—1：Identifying Linear Functions 识别一次函数 P296—P302 5—2：Using Intercepts 截距式（一次函数与坐标轴的交点）P303—P309 CHAPTER 5— Linear Functions 一次函数 5—2：Using Intercepts 截距式（一次函数与坐标轴的交点）P303—P309 5—3：Rate of Change and Slope 变化率和斜率 P310—P319 5—4：The Slope Formula 斜率的公式 P320—P325 5—5：Direct Variation 正比例函数 P326—P332 5—6：Slope-Intercept Form 斜截式方程 P334—P340 5—6：Point-Slope Form 点斜式方程 P341—P348 5—8：Slope of Parallel and Perpendicular Lines 相互平行与垂直直线的斜率

大学数学专业英语教材

大学数学专业英语教材IntroductionMathematics plays a crucial role in various fields and industries, and studying mathematics at the university level requires a solid foundation in both the subject itself and the English language. A well-designed mathematics textbook for university students in the field of mathematics can effectively integrate mathematical concepts with English language learning. In this article, we will explore the essential features and requirements of a comprehensive English textbook for mathematics students at the university level.Chapter 1: Fundamental ConceptsThe first chapter of the textbook should cover the fundamental concepts of mathematics, introducing students to the basic principles that underpin the subject. It should provide concise explanations and definitions, supplemented with examples and illustrations to aid comprehension. Additionally, this chapter should include exercises to reinforce learning and promote critical thinking.Chapter 2: AlgebraAlgebra is a cornerstone of mathematics, and this chapter should delve into its key theories and principles. It should cover topics such as equations, inequalities, functions, and matrices. The textbook should present clear explanations of concepts, accompanied by real-life applications to demonstrate the practical relevance of algebra.Chapter 3: CalculusCalculus is essential for advanced mathematics and the study of other disciplines such as physics and engineering. The textbook should guide students through both differential and integral calculus, ensuring a thorough understanding of concepts like limits, derivatives, and integrals. Practical examples and exercises should be incorporated to enhance students' problem-solving skills.Chapter 4: Probability and StatisticsIn this chapter, the textbook should introduce students to probability theory and statistical analysis. The content should cover topics such as probability distributions, hypothesis testing, and regression analysis. The inclusion of real-world data sets and case studies can foster students' ability to apply statistical methods effectively.Chapter 5: Discrete MathematicsDiscrete mathematics is vital in areas like computer science and cryptography. This chapter should explore concepts such as set theory, logic, graph theory, and combinatorics. The textbook should present clear explanations of these topics, accompanied by relevant examples and exercises to consolidate understanding.Chapter 6: Linear AlgebraLinear algebra is widely applicable in various fields, including computer science and physics. This chapter should cover vector spaces, linear transformations, and eigenvalues. Emphasis should be placed on theconnections between linear algebra and other mathematical disciplines, demonstrating its practical significance.Chapter 7: Number TheoryNumber theory explores the properties and relationships of numbers, and it forms the basis for cryptographic algorithms and computer security systems. This chapter should introduce students to prime numbers, modular arithmetic, and cryptographic algorithms. Examples and exercises should be given to develop students' problem-solving skills in the realm of number theory.Chapter 8: Numerical AnalysisNumerical analysis involves using algorithms to solve mathematical problems on computers. This chapter should cover topics such as interpolation, numerical integration, and numerical solutions of equations. The textbook should provide step-by-step guidance on implementing numerical algorithms, allowing students to develop practical coding skills.ConclusionA comprehensive English textbook for university-level mathematics students should provide a solid foundation in mathematical concepts while simultaneously enhancing students' English language proficiency. By incorporating clear explanations, practical examples, and engaging exercises, this textbook can foster a deep understanding of mathematics within an English language learning context. Such a resource will empower students to pursue further studies in mathematics and apply their skills in various professional domains.。

二项式定理公式概念

二项式定理公式概念The binomial theorem is a fundamental concept in algebra and mathematics. It provides a formula for expanding the power of a binomial expression, which is an algebraic expression with two terms. The theorem is often used in various fields of mathematics, including calculus, combinatorics, and probability theory. Understanding the binomial theorem is essential for solving problems in these areas and is a crucial part of a student's mathematical education.The binomial theorem states that for any positive integer n, the expansion of (a + b)^n can be calculated using the formula (a + b)^n = C(n,0)a^n + C(n,1)a^(n-1)b + C(n,2)a^(n-2)b^2 + ... + C(n,n-1)ab^(n-1) + C(n,n)b^n, where C(n,k) represents the binomial coefficient, which is the number of ways to choose k items from a set of n distinct items. This formula allows for the efficient calculation of the coefficients in the expansion of a binomial expression to any power, making it a powerful toolin algebraic manipulations.One of the key applications of the binomial theorem isin the field of probability theory. In probability, the binomial distribution describes the number of successes ina fixed number of independent Bernoulli trials, each with the same probability of success. The binomial theorem provides a way to calculate the probabilities of different outcomes in such trials, making it an essential tool for analyzing and predicting the likelihood of events invarious real-world scenarios.In addition to its applications in probability theory, the binomial theorem also plays a crucial role in combinatorics, the branch of mathematics concerned with counting, arranging, and organizing objects. The theorem provides a systematic way to expand binomial expressionsand calculate the number of combinations of items, which is essential for solving problems in combinatorial mathematics. This makes the binomial theorem an indispensable tool for solving problems in areas such as graph theory, coding theory, and cryptography.Furthermore, the binomial theorem has practical applications in fields such as engineering, physics, and computer science. In engineering and physics, the theorem is used to expand and simplify expressions in mathematical models and calculations, allowing for the efficient analysis and design of systems and structures. In computer science, the binomial theorem is applied in algorithms and data structures, where efficient calculations and manipulations of binomial expressions are required for various computational tasks.Overall, the binomial theorem is a fundamental concept with wide-ranging applications in mathematics and its related fields. Understanding and mastering the theorem is essential for students and professionals in various disciplines, as it provides a powerful tool for solving problems and analyzing real-world phenomena. Whether it's calculating probabilities in a game of chance, solving combinatorial puzzles, or designing complex systems, the binomial theorem is an indispensable tool that underpins many aspects of modern mathematics and its applications.。

信息论与编码理论中的英文单词和短语

信息论与编码理论Theories of Information
and Coding
第一章介绍
Chapter 1 Introduction
第二章信息理论Chapter 2 Information Theory
信息论与编码理论中的英文单词和短语
第三章离散无记忆信
道和容量成本方程
Chapter 3 Discrete Memory less Channels and their Capacity -Cost Equations
第四章离散无记忆信
源和扭曲率方程
Chapter 4 Discrete Memoryless Sources and their Rate -Distortion Equations
第五章高斯信道和信
源
Chapter 5 Gaussian Channel and Source
第六章信源-信道编码
理论
Chapter 6 Source -Channel Coding Theory
第七章第一部分访问
先进标题
Chapter 7 Survey of Advanced Topics for Part One
第八章线性码Chapter 8 Linear codes
第九章循环码Chapter 9 Cyclic Codes
第十章香农码和相关
的码
Chapter 10 Shannon Codes and Related Codes
第十一章卷积码
Chapter 11 Convolution Codes
第十二章变量长度源编
码
Chapter 12 Variable-length Source Coding。

Coding Theory(part two-1)

F2 represents
{0,1} .
1
input data（binary）
u mK
umK−1
…
um 2
u m1
um
( K − dim . binary )
+
vm1
binary Symbol
xm1
×
ϕ1 (t )
vm 2
+
… …
to Channel
xm 2
…
×
ϕ 2 (t )
Σ
xm (t )
Symbol mapping xmN
Linear Block Codes
Encoding process is an one to one mapping from the set
{U m = [u m1 , u m 2 ,..., u mK ]; m = 1, 2,..., M } , where
the source data alphabet. In to the set Q=2, u mk ∈ {0,1} .
8
the “parity” of the data symbol “1”s added to form vml
Obviously: The Hamming distance between
Vm & Vk
EX:
V 1 = ( 01101 ) W (V 1 ) = 3
is just W (V m ⊕ V k )

n:v m 'n ≠ v mn
∑
ln
P0 ( − z n ) >0 P0 ( z n )
12
Pem =
z∈Λ m n =1
∑ ∏P ( z )

algebra的概念 -回复

algebra的概念-回复什么是代数？代数是数学中的一个重要分支，研究数、符号和运算关系的结构以及它们之间的变化规律和性质。

它是几何学、数论、数学分析和其他数学分支的基础之一，并且广泛应用于工程学、物理学、计算机科学等领域。

代数的基本概念在介绍代数的基本概念之前，我们首先需要明确一些符号和术语:1. 数: 数是代数中最基本的概念，代表了一个确定的量或数目。

例如，1、2、3等都是数。

2. 字母: 字母是代数中常用的符号，用来表示未知数或变量。

例如，x、y、z等都是字母。

3. 表达式: 表达式是由数、字母和运算符组成的符号序列。

例如，2x + 3是一个表达式，其中2和3是数，x是字母，+是运算符。

4. 方程: 方程是等号连接的两个表达式，表示相等的关系。

例如，2x + 3 = 7是一个方程。

5. 约束: 约束是指对变量的限制条件，用来确定变量的取值范围。

例如，x > 0是一个约束，表示x必须大于0。

6. 函数: 函数是一种特殊的关系，将一个变量的值映射到另一个变量的值。

例如，y = f(x)表示x和y之间的函数关系。

代数的基本运算代数中有四种基本运算，它们分别是加法、减法、乘法和除法。

这些运算符在代数中有特定的符号表示:1. 加法: 加法的基本符号是"+”，它表示将两个数或表达式相加。

例如，3 + 4 = 7。

2. 减法: 减法的基本符号是"-"，它表示将一个数或表达式减去另一个数或表达式。

例如，5 - 2 = 3。

3. 乘法: 乘法的基本符号是"*" 或"×"，它表示将两个数或表达式相乘。

例如，2 * 3 = 6。

4. 除法: 除法的基本符号是"÷" 或"/"，它表示将一个数或表达式除以另一个数或表达式。

例如，6 ÷2 = 3。

除了这四种基本运算，代数中还有指数运算和根号运算等其他运算，它们用于处理更复杂的数学问题。

信息论与编码理论中的英文单词和短语

《科技服务业人才能力要求》英文

《科技服务业人才能力要求》英文全文共6篇示例，供读者参考篇1The Skill Requirements for the Technology Services IndustryHey there, kids! Today, we're going to explore an exciting industry that's shaping our world – the technology services industry. This industry is all about using computers, software, and other cool gadgets to help businesses and people solve problems and make their lives easier. But to be part of this industry, you need to have some special skills. Let me tell you all about them!First and foremost, you need to be a whiz at using computers and technology. This means you should know how to use different types of software, like word processors, spreadsheets, and presentation tools. You should also be able to navigate the internet like a pro and understand how to use search engines, email, and other online tools. Oh, and don't forget about coding! Learning how to write computer programs and code is like learning a brand new language, and it's super important in the tech world.But that's not all! In the technology services industry, you'll also need to have excellent problem-solving skills. You see, businesses and people often come to you with all sorts oftech-related problems, and it's your job to figure out how to solve them. This might involve troubleshooting computer issues, fixing software bugs, or finding creative ways to use technology to make things better. It's like being a detective, but instead of solving mysteries, you're solving tech puzzles!Another important skill is being able to communicate clearly and effectively. You'll need to explain complex technical concepts to people who might not be as tech-savvy as you are. This means using simple language, providing clear instructions, and being patient when someone doesn't understand something right away. It's like being a teacher, but your classroom is the whole world!Speaking of the whole world, in the technology services industry, you'll also need to be comfortable working with people from different cultures and backgrounds. Technology has made our world smaller and more connected than ever before, so you'll often find yourself collaborating with people from all over the globe. Being respectful, open-minded, and able to appreciate different perspectives is crucial.But wait, there's more! You'll also need to be creative and think outside the box. Technology is constantly evolving, and new challenges and opportunities arise all the time. Being able to come up with innovative ideas, solutions, and ways of using technology is what will make you stand out in this industry.And let's not forget about being able to learn quickly and adapt to change. The tech world moves at lightning speed, with new software, hardware, and trends popping up all the time. If you want to stay ahead of the game, you'll need to be a lifelong learner, always curious and eager to learn new things.Phew, that's a lot of skills, isn't it? But don't worry, with hard work, dedication, and a love for technology, you can develop all of these abilities and more. And who knows, maybe one day you'll be the one creating the next big thing in the technology services industry!So, what do you say? Are you ready to embark on this exciting journey and become a tech superhero? Remember, with great skills comes great responsibility – the responsibility to use technology in ways that make the world a better place. Keep learning, keep exploring, and never stop dreaming big!篇2Technology is Cool!Hi there, friends! Have you ever wondered how those amazing smartphones, computers, and video games work? It's all thanks to the awesome people working in the technology services industry. They create and maintain the tech gadgets and software that make our lives so much fun and easier!But being part of this exciting industry isn't as simple as playing video games all day (though that would be pretty sweet!). There are certain skills and abilities that these tech wizards need to have. Let me tell you all about them!A Brilliant MindFirst and foremost, people in the tech industry need to be really, really smart. They have to understand complex ideas and solve tricky problems every day. It's like doing a huge puzzle, but with numbers, codes, and computer languages instead of pieces.Luckily, you don't have to be a genius from birth. With hard work, dedication, and a love for learning, anyone can develop a brilliant mind for technology. It's all about practicing your problem-solving skills and never stopping to learn new things.Creativity Rocks!While being smart is important, it's not the only thing that matters. Tech whizzes also need to be super creative. They have to come up with innovative ideas and find new ways to make things better, faster, and more fun.Imagine if someone hadn't been creative enough to think of smartphones or video games! We'd still be using old-fashioned phones and playing boring board games all the time. No, thank you!Being creative means thinking outside the box and seeing things from different angles. It's like being an artist, but instead of painting or sculpting, you create amazing gadgets and software.Team Players Win the GameEven though technology might seem like a solo activity (you know, just you and your computer), it actually involves a lot of teamwork. Tech professionals often have to collaborate with others to bring their ideas to life.Just like in sports, being a team player is crucial. You need to communicate clearly, listen to others, and work together towards a common goal. After all, even the coolest gadget won't work if everyone involved can't get along.So, if you're the type who loves playing with friends and working as a team, you're already on the right track!Stay Curious, Keep LearningThe tech world is constantly changing and evolving. New inventions, updates, and advancements happen all the time. That's why people in this industry need to be curious and always eager to learn new things.Imagine if someone stopped learning after inventing the first computer. We'd still be using those huge, clunky machines instead of the sleek laptops and tablets we have today. Boring!Staying curious means never being satisfied with what you already know. It's about constantly asking questions, exploring new ideas, and embracing change. If you're the type of kid who loves discovering new things and learning about the world,you've got what it takes!Problem-Solving SuperpowersLast but not least, working in the tech industry requires some serious problem-solving skills. Every day, these professionals face new challenges and obstacles that need to be overcome.It's like playing a video game on the hardest level – you need to think quickly, stay calm, and come up with creative solutionsto move forward. And just like in games, sometimes you'll face setbacks and have to start over. But that's all part of the fun!If you're the type of kid who loves puzzles, brain teasers, and figuring things out, you've got the perfect mindset for tackling tech problems.So, there you have it, friends! Those are the main skills and abilities needed to be a rockstar in the technology services industry. With a brilliant mind, creativity, teamwork, curiosity, and problem-solving prowess, you'll be well on your way to creating the next big thing in tech.Who knows, maybe one day you'll invent a holographic video game or a flying smartphone! The possibilities are endless when you've got the right skills and a passion for technology. Keep learning, keep exploring, and keep having fun!篇3Technology Rules! Requirements for Talents in the Tech Services WorldHey kids! Do you love playing video games, using apps, or surfing the internet? Then you're going to want to learn all about the amazing technology services industry! This is where thecoolest gadgets, games, and websites are created by some super talented people.But what kind of skills do you need to work in this awesome field? Let me break it down for you into a few key areas:Coding NinjasIf you want to build the next hit game or app, you've got to master coding! Coding is like giving instructions to computers using special programming languages. The best coders are like martial arts masters who can write code flawlessly to make computers do exactly what they want.You'll need to learn languages like Python, Java, C++, and more. It's like learning different languages to speak to computers! The more coding languages you know, the more powerful you become. Coding takes patience, logic, and crazy problem-solving skills.Creative WizardsTechnology is all about combining code with amazing creativity to make something new and exciting. So along with those coding skills, you need a huge dose of creativity and innovation!The tech geniuses come up with unique ideas for apps, games, websites and then bring them to life through design and coding. They have to think outside the box and dream up experiences that no one has ever seen before. A brilliant creative mind is a must!Communication RockstarsWorking in technology means you can't just hide behind a computer screen all day. You have to collaborate with teammates, understand what customers want, and sell your ideas to others.The top tech talents are Communication Rockstars who can clearly explain complex topics, give presentations like pros, and really listen to feedback. If you dream of being a tech leader, you've got to master these people skills along with the technical abilities.Math MastersA lot of the coding and technology development relies on math - geometry, algebra, statistics, and more! If you can crunch numbers quickly and apply mathematical concepts to real problems, you'll have a huge advantage.The ability to think logically and look for patterns using math is essential. Tech geniuses use math skills every day to analyze data, optimize systems, and quantify pretty much everything! Time to make math your friend.Learning LegendsOne of the most important traits for tech talents is an insatiable curiosity and desire to constantly learn new things. Technology moves at lightning speed, so you can never stop expanding your knowledge.The top professionals are always studying up on the latest coding languages, tools, frameworks, trends and more. They read books, take courses, attend events - they are true Learning Legends who never stop feeding their brains!If you can combine coding talents with creativity, communication skills, math abilities and a passion for constant learning - then you'll be unstoppable in the tech services world!Maybe you'll design the next Fortnite, build an incredible new AI assistant, or launch some revolutionary tech product that changes the world. The opportunities are endless if you have what it takes.So start practicing those coding, creative, and math skills now. Read tons, communicate effectively with others, and never stop learning. The future of technology needs more talented kids like you!Who's ready to join the tech revolution? Let's goooo!篇4Title: What Does It Take to Become a Tech Superstar?Hey there, awesome kids! Do you love playing video games, using cool apps, or surfing the internet? Well, guess what? All those amazing things you enjoy are created by people who work in the technology services industry. These tech wizards are the ones who make our digital world so much fun and fascinating!But have you ever wondered what it takes to become one of these tech superstars? Well, buckle up, because I'm about to spill the beans on the awesome skills and abilities you need to have if you want to join this super cool club!First and foremost, you've got to be a math whiz! Yep, that's right – math is the language of technology. All those fancy gadgets and programs run on complex mathematical algorithms and equations. So, if you want to be a tech genius, you betterstart practicing your addition, subtraction, multiplication, and division like a boss!Next up, you need to be a coding master. Coding is like speaking a secret language that tells computers what to do. It's like giving them a set of instructions, and they follow them to the letter. Pretty cool, right? Learning to code is like unlocking a whole new world of possibilities. You can create your own games, apps, and websites – how awesome is that?But wait, there's more! You also need to have rock-solid problem-solving skills. Tech superstars are like detectives, constantly hunting down and fixing bugs (that's what we call errors in computer programs). They have to think outside the box and come up with creative solutions to all sorts of puzzles and challenges. It's like being a real-life Sherlock Holmes, but with way cooler gadgets!Communication skills are also super important. Tech wizards don't work alone – they're part of awesome teams that collaborate and share ideas. You need to be able to explain complex concepts in simple terms, so everyone can understand what you're talking about. It's like being a translator between the human world and the digital realm!Curiosity and a love for learning are essential too. Technology is always changing and evolving, so you need to be a lifelong learner, constantly exploring new ideas and stayingup-to-date with the latest trends and developments. It's like going on an endless adventure, where you never know what exciting new discovery is waiting around the corner!Last but not least, you need to be passionate about technology. This isn't just a job – it's a way of life. Tech superstars eat, sleep, and breathe all things digital. They're the ones who can't wait to get their hands on the latest gadgets and spend hours tinkering and experimenting with new tech toys.So, there you have it, my friends! If you want to become a tech superstar, you need to be a math genius, coding wizard, problem-solving detective, communication pro, curious explorer, and technology fanatic all rolled into one! It's a tall order, but with hard work, dedication, and a whole lot of passion, you can make your dreams of joining the tech revolution a reality!篇5Technology is Super Cool!Hi friends! Today I want to tell you all about the really neat world of technology services. It's so awesome and full ofamazing gadgets, computers, and all sorts of crazy cool things. But to work with technology, you need to have some special skills and abilities. Let me share what I've learned about the talents you need in this field.First off, you have to be really good at using computers and technology stuff. I'm talking being a total wiz at typing, navigating programs, fixing glitches, that kind of thing. You can't be afraid to dive in and fiddle with the settings to make things work perfectly. My dad is a tech expert and he can make our computer and devices do all sorts of wild tricks. I'm still learning, but I practice a ton to get really skilled.Another major talent is knowing math and science like the back of your hand. Technology is pretty much applied math and science, so you have to understand all those number games and science rules. Physics, calculus, coding languages – it's like a second language you must master. My brain hurts just thinking about it, but lots of practice makes you a math/science superstar.Speaking of languages, you also need mad communication skills. Technology involves explaining super complicated things to non-techy people in simple ways they'll understand. You have to be a clear writer, speaker and listener. I have to explain how touse the TV remote to my grandma like every week. It's trickier than you think to make tech easy!Then there's creative problem-solving. Technology is all about finding new solutions to problems and making our lives better. You have to think outside the box and tinker with things in new ways. Like my friend's dad created this crazy home security system by combining a bunch of devices. Or those Silicon Valley geniuses designing the next big app or robot. Wild imagination unlocks new possibilities!Don't forget teamwork. Technology is rarely a solo gig nowadays. You need to collaborate with other experts, compromise on ideas, and work together efficiently. Those Amazon workers, Apple developers, and Google squads all hustle as tight teams to create their killer products and services. You won't go far alone in tech.Finally, lifelong learning is mandatory. Technology keeps evolving at light speed, so you can never stop expanding your knowledge and skills. You have to read up on all the latest trends, software updates, coding techniques, you name it. My parents are always taking classes and watching tutorials to stay ahead of the curve. Learning is never finished in this fast-paced field.Phew, that's a lot to take in, huh? Tech may seem complicated, but it's honestly super fun if you have the right mindset and talents. You get to push boundaries, solve problems creatively, and make a career out of your passions for math, science and gadgets. Who knew following your curiosity could lead to such an awesome profession?篇6Title: What It Takes to Work with Cool Tech StuffHi there! Do you like playing video games, watching movies with awesome special effects, or using super fun apps on your tablet? All that amazing technology is created by people working in the technology service industry. It's a field where you get to dream up and build the latest gadgets, games, and digital services that make life more entertaining and convenient.But what does it really take to land one of those awesome tech jobs when you grow up? You might think you just need to be a computer wiz who's good at coding and math. While those skills are definitely important, there's actually a whole bunch of other abilities you need to develop too. Let me break it down for you:Curiosity and CreativityThe tech world is all about coming up with new and innovative ideas. The best developers, engineers, and designers are those who have an insatiable curiosity about how things work and a wild imagination for thinking up fresh concepts. You have to be willing to tinker, take things apart, ask "What if?" and dream up crazy new ways to solve problems.Critical ThinkingTechnology is super complex, with different systems having to integrate seamlessly. To build robust solutions, you need powerful critical thinking abilities. This means carefully analyzing issues, anticipating potential problems, weighing options, and making well-reasoned decisions. You have to be a master at breaking down complicated topics in a logical way.Communication SkillsYou might think tech nerds just stare at computer screens all day without talking to anyone. But actually, collaboration and communication are vital. Tech projects involve lots of people, from graphic designers to customer service reps. You need excellent speaking, writing, and presentation abilities to clearly explain your ideas and work well with your teammates.Love of LearningTechnology keeps evolving at lightning speed, with new programming languages, tools, and technical standards constantly emerging. People in this field can never stop acquiring new knowledge. You have to be a self-motivated learner who thrives on continuously expanding your skills through classes, books, and hands-on experimentation.Problem-Solving ProwessPretty much every day, you'll encounter coding bugs, system crashes, and other issues that need troubleshooting. You have to be able to calmly identify the root cause of complex problems and create effective solutions, drawing upon both your technical knowledge and creative thinking.Attention to DetailMissing a single line of code or fudging a minor configuration setting could lead to massive headaches and system failures. You need to be hyper-vigilant about precision and accuracy when developing and testing technology. Excellent organizational abilities also help you juggle multiple tasks and deliverables.Passion for TechLet's face it - building the latest app, designing a hot new computer game, or implementing a company's whole network infrastructure can be challenging work at times. You'll likely need to put in long hours, endure plenty of frustration when things go wrong, and have endless patience. A genuine enthusiasm for technology that borders on obsession will motivate you to persevere.So as you can see, there's a lot more to success in the tech service industry than just coding chops. You need a diverse skill set that covers both technical abilities and more general professional competencies like communication andproblem-solving.The awesome part is that if you have a natural inclination toward math, science, and all things digital, then you're already on the right path. By developing all those other supporting skills too, you'll be equipped to turn your passion for technology into an amazing career creating the mind-blowing products, services, and experiences of the future.Pretty cool, right? So start flexing your creativity, critical thinking, and curiosity muscles now. With dedication, you could end up being one of the tech wizards who changes the world with incredible new innovations. Just don't forget all of usregular folks who will be eagerly awaiting your next must-have app or game!。

algebra number theory

algebra number theory"Algebra Number Theory" is a branch of mathematics that focuses on the study of integers and their properties. It explores various topics such as prime numbers, divisors, congruences, and Diophantine equations.One of the central concepts in Algebra Number Theory is prime numbers. Prime numbers are integers greater than 1 that have only two distinct divisors, namely 1 and themselves. The study of prime numbers includes their distribution, properties, and connections to other mathematical areas.Another important topic in Algebra Number Theory is divisors. A divisor of a number is any integer that divides the number evenly. The study of divisors includes investigating their properties, relationships, and applications in number theory.Congruences are also a significant part of Algebra Number Theory. Congruence refers to the equality of two numbers modulo a given integer. This concept is used to study properties of numbers and solve various problems, including Diophantine equations.Diophantine equations are polynomial equations with integer coefficients and solutions in integers. Solving Diophantine equations involves finding integer solutions to these equations, which often leads to interesting number-theoretic results.Algebra Number Theory has numerous applications in areas such as cryptography, coding theory, and computer science. It provides essential tools and techniques for studying and manipulating integers and their properties.Overall, Algebra Number Theory is a fascinating field that combines elements of algebra and number theory. It offers a deeper understanding of integers and their relationships, opening up new avenues of exploration in mathematics.。

algebraic number theory

algebraic number theoryAlgebraic number theory is a branch of mathematics that studies the properties of algebraic numbers. Algebraic numbers are complex numbers that are the roots of a non-zero polynomial with rational coefficients. Algebraic number theory explores the relationships between these numbers and the structures that they form.The roots of a polynomial with rational coefficients are called algebraic numbers because they are solutions of algebraic equations. For example, the roots of the quadratic equation ax^2 + bx + c = 0, where a, b, and c are rational numbers and a is not zero, are algebraic numbers. There are several key concepts in algebraic number theorythat are used to study these numbers.The first concept is the ring of algebraic integers. An algebraic integer is a complex number that is the root of a monic polynomial with integer coefficients. The ring of algebraic integers is the set of all algebraic integers together with their addition, subtraction, and multiplicationoperations. This ring is important in algebraic number theory because it has many properties that are similar to those of the integers, such as unique factorization and Euclidean division.The second concept is the ideal. An ideal is a subset of the ring of algebraic integers that is closed under addition, subtraction, and multiplication by elements of the ring. Ideals are useful in algebraic number theory because they provide a way to study the divisibility of algebraic integers. In particular, the prime ideals of the ring of algebraic integers play a role similar to that of prime numbers in the theory of integers.The third concept is the Galois group. The Galois group is a group of automorphisms of a field that fixes the base field. In the context of algebraic number theory, the Galois group of a number field is the group of automorphisms of that field that fix the rational numbers. It is an important tool in algebraic number theory becauseit provides a way to study the algebraic propertiesof a number field by studying the properties of its Galois group.One of the main goals of algebraic number theory is to understand the distribution of prime numbers among algebraic integers. In particular, it is of interest to determine which prime numbers can be written as the norm of an algebraic integer. The norm of an algebraic integer is the product of all its Galois conjugates, which are the roots of its minimal polynomial.The study of algebraic number theory has many applications outside of pure mathematics. For example, it is used in cryptography, coding theory, and theoretical physics. In cryptography, algebraic number theory is used to construct secure cryptographic systems based on the hardness of factorization and the discrete logarithm problem. In coding theory, algebraic number theory is used to construct error-correcting codes with good properties. In theoretical physics, algebraic number theory is used to study the symmetries of physical systems.In conclusion, algebraic number theory is a fascinating and important branch of mathematicsthat explores the properties of algebraic numbers and the structures that they form. It has many applications in diverse areas of science and engineering, and its insights have led to the development of sophisticated cryptographic systems, error-correcting codes, and theoretical models of physical systems. Its continued study is essential for the advancement of science and the improvement of human life.。

6603_Lecture6

• Capacity of continuous-time AWGN channel:
1 P P C 2W log 2 (1 ) W log 2 (1 ) 2 N 0W N 0W
Semester A, 2010
bit/s
Wireless Communication Technologies
n! n 2nHb ( p ) (np )!(n(1 p ))! np H b ( p ) p log 2 p (1 p) log 2 (1 p )
0 1
1-p p p 1-p
0 1
Choose a subset of all possible codewords, so that the possible error codewords for each element of this subset is NOT overlapping! • The maximum size of the subset:
1 L Pl P. L l 1
•
Suppose the receiver has CSI.
Semester A, 2010
Wireless Communication Technologies
11
Ergodic Capacity without CSIT
At each coherence time period, the reliable communication rate is
•
If we can increase the bandwidth without limit, can we get an infinitely large channel capacity?

有限域上的多项式理论

有限域上的多项式理论Polynomial Theory of Finite Fields摘要域的概念的提出为代数学中的讨论的方便提供了条件，而作为在域中占有重要地位的有限域而言，更是在组合设计、编码理论、密码学、计算机代数和通信系统等领域发挥着自己的作用。

多项式理论又是代数学中的基础，它的应用在其它领域也是常见的，本文的主要思想就是将高等代数中建立在数域中的多项式理论进行推广，将有关的性质、定理在有限域上进行验证，进而形成一套建立在有限域上的多项式理论。

当下，通信技术已经飞速发展，而保证信息在传输过程中的准确性是通信安全的一个重要前提。

本文在第三章给出了有限域上的多项式在该领域的一个具体应用——利用本原多项式来进行纠错码的操作。

正文部分的结构组成包括：有限域的基本知识、一元多项式、多项式的整除和带余除法、最大公因式、因式分解定理、重因式、多元多项式及本原多项式在纠错码中的应用。

本文通过大量理论证明，验证了关于多项式的定理，性质，将数域上的多项式理论建立在有限域上。

从结果中可以看出，对于建立在一般数域的多项式理论，大部分的结果在有限域上也是普遍成立的，但是不排除一些特殊的情况。

同时，在部分章节的最后也给出了一些只有在有限域中成立，在普通数域中不成立的结论。

关键词：有限域；多项式；带余除法；纠错码AbstractWith the concept of the field being raised, it has provided the conditions for the convenience of the discussion in Algebra. Meanwhile, the finite field also plays an important role in combination of design, coding theory, cryptography, commuter and communications systems. Polynomial theory is the basis of Algebra. The main idea is to put the polynomial theory to the finite field and check the related properties and theorems.Nowadays, the communicational technology has developed rapidly. Keeping accuracy is an important prerequisite for communication security. In the third chapter, this paper introduces the primitive polynomial’s applications: Error-correcting code.The text contains: The basis knowledge of finite field, polynomial, divisibility of polynomials, greatest common factor, factorization theorem, repeated divisors, multivariate polynomial and the primitive polynomial’s applications: Error-correcting code.In this paper, a number of properties and theorems are checked by theoretical proof. We will establish the polynomial theory of finite field. According to it, we can see that the most parts of the polynomial theory of number field are established in finite field except in some special situations. At the same time, some conclusions which only established in finite field are given in some chapters.Keywords: finite fields; polynomial; divisibility of polynomials; Error-correcting code目录摘要 (I)Abstract (II)第1章绪论 (1)1.1有限域的发展 (1)1.2 有限域的基础理论 (2)第2章有限域上的多项式 (5)2.1 一元多项式 (5)2.2 多项式的整除和带余除法 (9)2.4 最大公因式 (14)2.5 因式分解定理 (18)2.6 重因式 (21)2.7 多元多项式 (22)第3章有限域上的多项式的应用 (27)第4章结论 (33)参考文献 (34)致谢..................................................................................................... 错误!未定义书签。

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algebra number theory

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6603_Lecture6

有限域上的多项式理论

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