The Geometry of SU(3)

a r X i v :0803.2889v 2 [h e p -p h ] 14 J u l 2008Mapping Out SU (5)GUTs with Non-Abelian Discrete Flavor SymmetriesFlorian Plentinger ∗and Gerhart Seidl †Institut f¨u r Physik und Astrophysik,Universit¨a t W¨u rzburg,Am Hubland,D 97074W¨u rzburg,Germany(Dated:December 25,2013)We construct a class of supersymmetric SU (5)GUT models that produce nearly tribimaximal lepton mixing,the observed quark mixing matrix,and the quark and lepton masses,from discrete non-Abelian flavor symmetries.The SU (5)GUTs are formulated on five-dimensional throats in the flat limit and the neutrino masses become small due to the type-I seesaw mechanism.The discrete non-Abelian flavor symmetries are given by semi-direct products of cyclic groups that are broken at the infrared branes at the tip of the throats.As a result,we obtain SU (5)GUTs that provide a combined description of non-Abelian flavor symmetries and quark-lepton complementarity.PACS numbers:12.15.Ff,11.30.Hv,12.10.Dm,One possibility to explore the physics of grand unified theories (GUTs)[1,2]at low energies is to analyze the neutrino sector.This is due to the explanation of small neutrino masses via the seesaw mechanism [3,4],which is naturally incorporated in GUTs.In fact,from the perspective of quark-lepton unification,it is interesting to study in GUTs the drastic differences between the masses and mixings of quarks and leptons as revealed by current neutrino oscillation data.In recent years,there have been many attempts to re-produce a tribimaximal mixing form [5]for the leptonic Pontecorvo-Maki-Nakagawa-Sakata (PMNS)[6]mixing matrix U PMNS using non-Abelian discrete flavor symme-tries such as the tetrahedral [7]and double (or binary)tetrahedral [8]groupA 4≃Z 3⋉(Z 2×Z 2)and T ′≃Z 2⋉Q,(1)where Q is the quaternion group of order eight,or [9]∆(27)≃Z 3⋉(Z 3×Z 3),(2)which is a subgroup of SU (3)(for reviews see, e.g.,Ref.[10]).Existing models,however,have generally dif-ficulties to predict also the observed fermion mass hierar-chies as well as the Cabibbo-Kobayashi-Maskawa (CKM)quark mixing matrix V CKM [11],which applies especially to GUTs (for very recent examples,see Ref.[12]).An-other approach,on the other hand,is offered by the idea of quark-lepton complementarity (QLC),where the so-lar neutrino angle is a combination of maximal mixing and the Cabibbo angle θC [13].Subsequently,this has,in an interpretation of QLC [14,15],led to a machine-aided survey of several thousand lepton flavor models for nearly tribimaximal lepton mixing [16].Here,we investigate the embedding of the models found in Ref.[16]into five-dimensional (5D)supersym-metric (SUSY)SU (5)GUTs.The hierarchical pattern of quark and lepton masses,V CKM ,and nearly tribi-maximal lepton mixing,arise from the local breaking of non-Abelian discrete flavor symmetries in the extra-dimensional geometry.This has the advantage that theFIG.1:SUSY SU (5)GUT on two 5D intervals or throats.The zero modes of the matter fields 10i ,5H,24H ,and the gauge supermul-tiplet,propagate freely in the two throats.scalar sector of these models is extremely simple without the need for a vacuum alignment mechanism,while of-fering an intuitive geometrical interpretation of the non-Abelian flavor symmetries.As a consequence,we obtain,for the first time,a realization of non-Abelian flavor sym-metries and QLC in SU (5)GUTs.We will describe our models by considering a specific minimal realization as an example.The main features of this example model,however,should be viewed as generic and representative for a large class of possible realiza-tions.Our model is given by a SUSY SU (5)GUT in 5D flat space,which is defined on two 5D intervals that have been glued together at a common endpoint.The geom-etry and the location of the 5D hypermultiplets in the model is depicted in FIG.1.The two intervals consti-tute a simple example for a two-throat setup in the flat limit (see,e.g.,Refs.[17,18]),where the two 5D inter-vals,or throats,have the lengths πR 1and πR 2,and the coordinates y 1∈[0,πR 1]and y 2∈[0,πR 2].The point at y 1=y 2=0is called ultraviolet (UV)brane,whereas the two endpoints at y 1=πR 1and y 2=πR 2will be referred to as infrared (IR)branes.The throats are supposed to be GUT-scale sized,i.e.1/R 1,2 M GUT ≃1016GeV,and the SU (5)gauge supermultiplet and the Higgs hy-permultiplets 5H and2neously broken to G SM by a 24H bulk Higgs hypermulti-plet propagating in the two throats that acquires a vac-uum expectation value pointing in the hypercharge direc-tion 24H ∝diag(−12,13,15i ,where i =1,2,3is the generation index.Toobtainsmall neutrino masses via the type-I seesaw mechanism [3],we introduce three right-handed SU (5)singlet neutrino superfields 1i .The 5D Lagrangian for the Yukawa couplings of the zero mode fermions then readsL 5D =d 2θ δ(y 1−πR 1) ˜Y uij,R 110i 10j 5H +˜Y d ij,R 110i 5H +˜Y νij,R 15j5i 1j 5H +M R ˜Y R ij,R 21i 1j+h.c. ,(3)where ˜Y x ij,R 1and ˜Y x ij,R 2(x =u,d,ν,R )are Yukawa cou-pling matrices (with mass dimension −1/2)and M R ≃1014GeV is the B −L breaking scale.In the four-dimensional (4D)low energy effective theory,L 5D gives rise to the 4D Yukawa couplingsL 4D =d 2θ Y u ij 10i 10j 5H +Y dij10i 5H +Y νij5i ∼(q i 1,q i 2,...,q i m ),(5)1i ∼(r i 1,r i 2,...,r im ),where the j th entry in each row vector denotes the Z n jcharge of the representation.In the 5D theory,we sup-pose that the group G A is spontaneously broken by singly charged flavon fields located at the IR branes.The Yukawa coupling matrices of quarks and leptons are then generated by the Froggatt-Nielsen mechanism [21].Applying a straightforward generalization of the flavor group space scan in Ref.[16]to the SU (5)×G A represen-tations in Eq.(5),we find a large number of about 4×102flavor models that produce the hierarchies of quark and lepton masses and yield the CKM and PMNS mixing angles in perfect agreement with current data.A distri-bution of these models as a function of the group G A for increasing group order is shown in FIG.2.The selection criteria for the flavor models are as follows:First,all models have to be consistent with the quark and charged3 lepton mass ratiosm u:m c:m t=ǫ6:ǫ4:1,m d:m s:m b=ǫ4:ǫ2:1,(6)m e:mµ:mτ=ǫ4:ǫ2:1,and a normal hierarchical neutrino mass spectrumm1:m2:m3=ǫ2:ǫ:1,(7)whereǫ≃θC≃0.2is of the order of the Cabibbo angle.Second,each model has to reproduce the CKM anglesV us∼ǫ,V cb∼ǫ2,V ub∼ǫ3,(8)as well as nearly tribimaximal lepton mixing at3σCLwith an extremely small reactor angle 1◦.In perform-ing the group space scan,we have restricted ourselves togroups G A with orders roughly up to 102and FIG.2shows only groups admitting more than three valid mod-els.In FIG.2,we can observe the general trend thatwith increasing group order the number of valid modelsper group generally increases too.This rough observa-tion,however,is modified by a large“periodic”fluctu-ation of the number of models,which possibly singlesout certain groups G A as particularly interesting.Thehighly populated groups would deserve further system-atic investigation,which is,however,beyond the scopeof this paper.From this large set of models,let us choose the groupG A=Z3×Z8×Z9and,in the notation of Eq.(5),thecharge assignment101∼(1,1,6),102∼(0,3,1),103∼(0,0,0),52∼(0,7,0),52↔4FIG.3:Effect of the non-Abelian flavor symmetry on θ23for a 10%variation of all Yukawa couplings.Shown is θ23as a function of ǫfor the flavor group G A (left)and G A ⋉G B (right).The right plot illustrates the exact prediction of the zeroth order term π/4in the expansion θ23=π/4+ǫ/√2and the relation θ13≃ǫ2.The important point is that in the expression for θ23,the leading order term π/4is exactly predicted by thenon-Abelian flavor symmetry G F =G A ⋉G B (see FIG.3),while θ13≃θ2C is extremely small due to a suppression by the square of the Cabibbo angle.We thus predict a devi-ation ∼ǫ/√2,which is the well-known QLC relation for the solar angle.There have been attempts in the literature to reproduce QLC in quark-lepton unified models [26],however,the model presented here is the first realization of QLC in an SU (5)GUT.Although our analysis has been carried out for the CP conserving case,a simple numerical study shows that CP violating phases (cf.Ref.[27])relevant for neutri-noless double beta decay and leptogenesis can be easily included as well.Concerning proton decay,note that since SU (5)is bro-ken by a bulk Higgs field,the broken gauge boson masses are ≃M GUT .Therefore,all fermion zero modes can be localized at the IR branes of the throats without intro-ducing rapid proton decay through d =6operators.To achieve doublet-triplet splitting and suppress d =5pro-ton decay,we may then,e.g.,resort to suitable extensions of the Higgs sector [28].Moreover,although the flavor symmetry G F is global,quantum gravity effects might require G F to be gauged [29].Anomalies can then be canceled by Chern-Simons terms in the 5D bulk.We emphasize that the above discussion is focussed on a specific minimal example realization of the model.Many SU (5)GUTs with non-Abelian flavor symmetries,however,can be constructed along the same lines by varying the flavor charge assignment,choosing different groups G F ,or by modifying the throat geometry.A de-tailed analysis of these models and variations thereof will be presented in a future publication [30].To summarize,we have discussed the construction of 5D SUSY SU (5)GUTs that yield nearly tribimaximal lepton mixing,as well as the observed CKM mixing matrix,together with the hierarchy of quark and lepton masses.Small neutrino masses are generated only by the type-I seesaw mechanism.The fermion masses and mixings arise from the local breaking of non-Abelian flavor symmetries at the IR branes of a flat multi-throat geometry.For an example realization,we have shown that the non-Abelian flavor symmetries can exactly predict the leading order term π/4in the sum rule for the atmospheric mixing angle,while strongly suppress-ing the reactor angle.This makes this class of models testable in future neutrino oscillation experiments.In addition,we arrive,for the first time,at a combined description of QLC and non-Abelian flavor symmetries in SU (5)GUTs.One main advantage of our setup with throats is that the necessary symmetry breaking can be realized with a very simple Higgs sector and that it can be applied to and generalized for a large class of unified models.We would like to thank T.Ohl for useful comments.The research of F.P.is supported by Research Train-ing Group 1147“Theoretical Astrophysics and Particle Physics ”of Deutsche Forschungsgemeinschaft.G.S.is supported by the Federal Ministry of Education and Re-search (BMBF)under contract number 05HT6WWA.∗********************************.de †**************************.de[1]H.Georgi and S.L.Glashow,Phys.Rev.Lett.32,438(1974);H.Georgi,in Proceedings of Coral Gables 1975,Theories and Experiments in High Energy Physics ,New York,1975.[2]J.C.Pati and A.Salam,Phys.Rev.D 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sketchup实用快捷键su逐字母快捷键解析四、快捷键横向解析1、字母A字母A单独使用时表示弧线工具，与Shift、Ctrl配合使用时A=All，表示与所有物体有关的命令操作。

字母A单独使用时表示弧线工具，与Shift、Ctrl配合使用时A=All，表示与所有物体有关的命令操作。

字母A单独使用时表示弧线工具，与Shift、Ctrl配合使用时A=All，表示与所有物体有关的命令操作。

（1）Draw>Arc - A二、圆弧工具。

*S 定义弧的段数，*R 定义弧的半径，其中*为需要输入的数字。

（2）Edit>Select All - Ctrl+A全选，Windows系统命令。

（3）Edit>UnHide>All - Shift+A显示所有物体。

2、字母B字母B单独使用时表示矩形工具，B=Box，盒子的意思。

（1）Draw>Rectangle - B矩形工具。

矩形是特殊的多边形，因此将多边形设置为Alt+B。

（2）Draw>Polygon - Alt+B多边形工具。

与画圆的命令类似，参数相同。

*S定义正多边形的边数，其中*为需要输入的数字。

3、字母C字母C单独使用时表示画圆工具。

与Alt键一起使用，表示设定相机位置。

与Ctrl 键一起使用，是Windows的系统命令。

（1）Draw>Circle – C 画圆工具。

*S 定义圆的段数，其中*为需要输入的数字。

（2）Camera>Position Camera - Alt+C 相机位置工具，用来设定相机位置。

这里的C=Camera，相机的意思。

（3）Edit>Copy - Ctrl+C 复制，是Windows的系统命令。

4、字母D字母D单独使用时表示路径跟随工具，与Alt结合，表示删除关键帧页面。

（1）T ools>Follow Me – D 路径跟随工具，即沿路径放样。

D象形为剖面的断面。

（2）View>T ourguide>Delete Page - Alt+D 删除页面。

Geometric ModelingGeometric modeling is a branch of mathematics that deals with the representation of objects in space. It is a fundamental tool in computer graphics, computer-aided design (CAD), and other applications that require the creation of 3D models. Geometric modeling involves the use of mathematical equations and algorithms to create and manipulate objects in space. In this essay, we will explore the different aspects of geometric modeling, including its history, applications, and challenges.The history of geometric modeling can be traced back to the early 19th century when mathematicians began to study the properties of curves and surfaces. In the early 20th century, the development of calculus and differential geometry led to the creation of new methods for representing complex objects in space. The introduction of computers in the mid-20th century revolutionized the field of geometric modeling, making it possible to create and manipulate 3D models with greater precision and ease.Today, geometric modeling is used in a wide range of applications, including computer graphics, animation, video games, virtual reality, and CAD. In computer graphics and animation, geometric modeling is used to create realistic 3D models of objects, characters, and environments. In video games, geometric modeling is used to create the game world and characters. In virtual reality, geometric modeling is used to create immersive environments that simulate real-world experiences. In CAD, geometric modeling is used to create precise 3D models of mechanical parts and assemblies.One of the biggest challenges in geometric modeling is the representation of complex shapes and surfaces. Many real-world objects, such as cars, airplanes, and human bodies, have complex shapes that are difficult to represent using simple geometric primitives such as spheres, cylinders, and cones. To overcome this challenge, researchers have developed advanced techniques such as NURBS (non-uniform rational B-splines), which allow for the creation of complex curves and surfaces by combining simple geometric primitives.Another challenge in geometric modeling is the optimization of models for efficient rendering and simulation. As the complexity of models increases, so doesthe computational cost of rendering and simulating them. To address this challenge, researchers have developed techniques such as level-of-detail (LOD) modeling,which involves creating multiple versions of a model at different levels of detail to optimize rendering and simulation performance.In conclusion, geometric modeling is a fundamental tool in computer graphics, animation, video games, virtual reality, and CAD. Its history can be traced backto the early 19th century, and its development has been closely tied to the advancement of mathematics and computer technology. Despite its many applications and successes, geometric modeling still faces challenges in the representation of complex shapes and surfaces, as well as the optimization of models for efficient rendering and simulation. As technology continues to advance, it is likely that new techniques and approaches will emerge to overcome these challenges and pushthe field of geometric modeling forward.。

三维设计英语试题及答案一、选择题（每题2分，共20分）1. Which of the following is NOT a common 3D modeling software?A. AutoCADB. SketchUpC. PhotoshopD. Blender2. The process of creating a 3D model is known as:A. RenderingB. ModelingC. TexturingD. Lighting3. What does UV mapping refer to in 3D design?A. The process of applying colors to a 3D modelB. The process of mapping a 2D image onto a 3D modelC. The process of creating a wireframeD. The process of adding details to a 3D model4. Which of the following is NOT a type of 3D printing material?A. PLAB. ABSC. InkD. Resin5. In 3D animation, what does 'keyframe' mean?A. The starting point of an animationB. A point in time where an object's position is setC. The end point of an animationD. The speed at which an object moves6. What is the term for the process of making a 3D model appear more realistic by adding surface details?A. SmoothingB. SubdivisionC. DisplacementD. Extrusion7. Which of the following is a unit of measurement used in 3D design?A. PixelB. MeterC. KilogramD. Bit8. What does LOD stand for in 3D modeling?A. Level of DetailB. Line of DefenseC. Light of DayD. Long Overdue9. In 3D design, what is the purpose of a 'rig'?A. To create a skeleton for a characterB. To set the lighting of a sceneC. To define the camera's viewD. To apply textures to a model10. What is the term used to describe the process of converting a 3D model into a 2D image?A. ProjectionB. ExtrusionC. TexturingD. Rendering二、填空题（每空2分，共20分）11. The ________ is a tool in 3D modeling software that allows you to move objects around in the workspace.(答案: Move Tool)12. When creating a 3D model, the first step is usually to create a basic shape known as a ________.(答案: Primitive)13. The process of adding color and texture to a 3D model is called ________.(答案: Texturing)14. In animation, the ________ is the main character or object that the story revolves around.(答案: Protagonist)15. The ________ is the process of adjusting the camera angle and position to frame a scene.(答案: Camera Setup)16. To create a 3D model of a complex object, you may need to use a technique called ________.(答案: Boolean Operations)17. The ________ is the process of adding motion to a 3D model.(答案: Animation)18. In 3D printing, the ________ is the layer-by-layer process of building an object.(答案: Additive Manufacturing)19. The ________ is a tool in 3D modeling software that allows you to modify the shape of a model by dragging points. (答案: Sculpt Tool)20. When a 3D model is complete, it is often saved in a file format that ends with the extension ________.(答案: .obj)三、简答题（每题10分，共20分）21. Explain the difference between a 'polygon mesh' and a'NURBS' in 3D modeling.(答案: A polygon mesh is a collection of vertices, edges, and faces that form a 3D shape. It is commonly used in video games and animation. NURBS, on the other hand, stands for Non-Uniform Rational B-Splines and is a mathematical model used to create smooth, curved surfaces. It is often used in industrial design and automotive applications.)22. What are the advantages and disadvantages of using a'real-time rendering' engine in 3D animation?(答案: Advantages of real-time rendering include theability to see the final product as you work, which can save time and provide immediate feedback. It is also computationally less intensive than pre-rendering. Disadvantages include potential limitations in visual quality compared to pre-rendered scenes, and the fact that it may。

空间解析几何英语Spatial Analytic Geometry.Spatial analytic geometry is a branch of mathematics that deals with the study of geometric objects in three-dimensional space. It extends the concepts and techniques of two-dimensional analytic geometry to the three-dimensional realm, allowing for a more comprehensive understanding of spatial relationships and structures. In this article, we will explore the fundamental principles and applications of spatial analytic geometry.1. Coordinate Systems in Three Dimensions.In spatial analytic geometry, the fundamental tool is the three-dimensional coordinate system. This system consists of three perpendicular axes, typically denoted as the x, y, and z axes. Any point in three-dimensional space can be uniquely identified by its coordinates (x, y, z) relative to these axes.2. Vectors in Three Dimensions.Vectors play a crucial role in spatial analytic geometry. A vector is a mathematical object that represents both magnitude and direction. In three dimensions, a vector can be represented as an ordered triplet of numbers (a, b, c), where each number corresponds to the component of the vector along one of the coordinate axes. Vectors can be used to represent displacements, forces, velocities, and other quantities that have both magnitude and direction.3. Geometric Objects in Three Dimensions.Spatial analytic geometry deals with a variety of geometric objects in three dimensions, including points, lines, planes, and more complex shapes such as spheres, cylinders, and cones. Each of these objects can be described and analyzed using the language and techniques of spatial analytic geometry.4. Equations of Geometric Objects.In spatial analytic geometry, equations are used to describe the geometric objects of interest. For example,the equation of a line in three dimensions can be expressed as a system of two linear equations in x, y, and z. Similarly, the equation of a plane can be expressed as a linear equation in x, y, and z. These equations provide a means to study the properties and relationships ofgeometric objects in a rigorous and systematic manner.5. Applications of Spatial Analytic Geometry.Spatial analytic geometry finds applications in various fields, including computer graphics, robotics, physics, and engineering. In computer graphics, for example, spatial analytic geometry is used to represent and manipulatethree-dimensional objects on a computer screen. In robotics, it is employed to model and control the movement of robotsin three-dimensional space. In physics and engineering, spatial analytic geometry is fundamental to the understanding and analysis of complex systems and structures.6. Conclusion.Spatial analytic geometry is a powerful tool for understanding and analyzing geometric objects in three dimensions. It extends the principles of two-dimensional analytic geometry to the three-dimensional realm, enabling the study of complex spatial relationships and structures. With its wide range of applications, spatial analytic geometry plays a crucial role in fields such as computer graphics, robotics, physics, and engineering. By mastering the concepts and techniques of spatial analytic geometry, one can gain a deeper understanding of the geometric world and apply this understanding to solve real-world problems.。

Homework33.1: P177Ex4: set the answer to be . For i going from 1 through n-1, computer the value of the (i+1)st element in the list minus the i th element in the list. If this is larger than the answer, reset the answer to be this value.Ex27ternary(V, s, e)if s > ereturn -1elsem1 ←(e-s)/3 + sm2 ←2*(e-s)/3 + sif V = A[ m1 ]return m1else if V = A[ m2 ]return m2else if V < A[ m1 ]return ternary(V, s, m1-1)else if V < A[ m2 ]return ternary(V, m1+1, m2-1)elsereturn ternary(V, m2+1, e)3.2: P191Ex2 (a) Yes (c) Yes (e) NoEx20 The approach in these problems is to pick out the most rapidly growing term in each sum and discard the rest (including the multiplicative constants).a) O(n3log(n)) b) O(6n) c) O(n n n!)Ex28 (a) Choose C1=1 and C2=2, for all x>1, we have C1*3x2 <=3x2+x+1<=C2*3x2.Ex36 This does not follow. Let f(x)=2x and g(x)=x. then f(x) is O(g(x)). Now 2f(x)=4x, and 2g(x)=2x. The ratio 4x/2x=2x grows without bound as x grows—it is not bounded by a constant.3.3: P200Ex8a) Initially y:=3, for i=1 we set y to 7, for i=2 we set y to 15, and we are done.b) There is one multiplication and one addition for each of the n passes through the loop, so there are n multiplication and n additions in all.Ex10a) 1.224*10-6 seconds b) 1.05*10-3 seconds c) 1.13*106 seconds d) 1.27*1021 seconds3.4: P208Ex6 Under the hypotheses, we have c=as and d=bt for some s and t. Multiplying we obtain cd=ab(st), which means ab|cd, as desired.Ex24 Write n=2k+1 for some integer k. Then n 2=4k(k+1)+1. Since either k or k+1 is even, therefore n 2-1 is a multiple of 8, so n 2≡1 (mod 8).3.5: P217Ex10 These are 1,5,7,and 11.Ex32 From a ≡b (mod m) we know that b=a+sm for some integer s. Now if d is a common divisor of a and m, then it divides the right-hand side of this equation, so it dived b. We can rewrite the equation as a=b-sm, and then by similar reasoning, we see that every common divisor of b and m is also a divisor of a. This shows that the set of common divisors of a and m is equal to the set of common divisors of b and m, so certainly gcd(a,m)=gcd(b,m).3.6:P230Ex8 a) 1111 0111 becomes F7Ex28 d) 79=81-3+13.7: P244Ex4 Since 13*937-1=12180=2436*5, we have 13*937≡1(mod 2436).Ex12 We know from exercise 6 that 9 is an inverse of 2 modulo 17. Therefore if we multiply both sides of this equation by 9, we will get x ≡12 (mod 17)Ex18 x=2*20*2+1*15*3+3*12*3≡53 (mod 60).3.8: P255Ex16 The (i,j)th entry of (A t )t is the (j,i)th entry of (A t ), which is (i,j)th entry of A.Homework44.1: P279Ex6. The basis step is clear, since 1*1!=2!-1. Assuming the inductive hypothesis, we then have 1*1!+2*2!+….+k*k!+(k+1)*(k+1)!=(k+1)!-1++(k+1)*(k+1)!=(k+2)!-1.Ex38. The basis step is trivial, as usual: A 1⊆B 1. Assume the inductive hypothesis that if A j ⊆B j for j=1,2,…,k, then 11kkj j j j A B ==⊆U U . We want to show that if A j ⊆B j for j=1,2,…,k+1, then1111k kj j j j A B ++==⊆U U . To show that one set is a subset of another we show that an arbitraryelement of the first set must be an element of the second set. So letx ∈11k j j A +=U =11()kj j k A A =+U U . Either x ∈1kj j A =U or x ∈1k A +. In the first case we know bythe inductive hypothesis that x ∈1kj j B =U ; in the second case, we know from the given fact thatA k+1⊆B k+1 that x ∈1k B +. Therefore in either case x ∈11k j j B +=U .Ex49. In the inductive hypothesis, it assume x and y are positive integer. Therefore we can ’t conclude x-1=y-1 from max(x-1,y-1)=k since x-1 and y-1 can be not positive.4.2: P293Ex12. The basis step is to note that 1=20. Assume the inductive hypothesis, that every positive integer up to k can be written as a sum of distinct powers of 2. We must show that k+1 can be written as a sum of distinct powers of 2. if k+1 is odd, then k is even, so 20is not part of the sum for k. therefore the sum for k+1 is the same as the sum for k with the extra term 20added. If k+1 is even, then (k+1)/2 is a positive integer, so by the inductive hypothesis (k+1)/2 can be written as a sum of distinct powers of 2. Increasing each exponent by 1 doubles the value and give us the desired sum for k+1.4.3: P308Ex6. a) valid b) valid c) invalid d) invalid e) invalid.Ex28.a) basis step: (1,2)∈S, (2,1)∈S. recursive step: if (a,b) ∈S then (a+2,b)∈S, (a,b+2)∈S and (a+1,b+1)∈S.b) basis step: (1,1)∈S. recursive step: if (a,a) ∈S then (a+1,a+1)∈S; if (a,b) ∈S then (a,a+b)∈S.c) basis step: (1,2)∈S, (2,1)∈S. recursive step: if (a,b) ∈S then (a+3,b)∈S, (a,b+3)∈S, (a+1,b+2)∈S and (a+2,b+1)∈S.4.4: P321Ex12.Procedure power(x, n, m:positive intergers)If n=1 then power(x,n,m):=x mod mElse power(x,n,m):=(x*power(x,n-1,m)) mod m4.5: P327Ex2. There are two cases. If x>=0 initially, then nothing is executed, so x>=0 at the end. If x<0 initially, then x is set equal to 0, so x=0 at the end; hence again x>=0 at the end.。

借助于SketchUp 工具的ArcGIS三维建模(1)使用ArcGIS桌面，即ArcMap，加载矢量数据；(2)在ArcMap环境中，利用插件工具，将所需要建模的区域导入SketchUp中。

(3)在SketchUp创建模型。

(4)在SketchUp中将模型转成ArcGIS的Multipatch模型要素文件并保存于Personal GeoDatabase（后面统称为PGDB）中。

软件环境ArcGIS桌面产品和服务器产品；SketchUP 6专业版三维建模软件（建议安装版本6 pro）；SketchUp ESRI插件；以及图像处理软件Photoshop，用来制作材质文件。

软件安装及配置步骤(1)安装ArcGIS Desktop软件，如ArcInfo。

（过程略）(2)安装草图大师Goolge SketchUp 6 Pro软件。

（过程略）(3)安装SketchUp6 ESRI 插件1.双击“SketchUp6ESRI.exe”，开始安装，2.接受协议，点击“Next”3.第一个组件“GIS Plugin”，使用户能够在SketchUp中将模型以Multipatch要素的形式导入GDB。

第二个组件“3D Analyst SketchUp 3D Symbol Support”，用户可以在ArcMap中将GIS数据导入SketchUp中。

上述两个组件的安装位置尽量不要改变，可能会导致在SketchUp中导出3D模型失败。

4.执行组件安装(4)在ArcGIS环境中激活SketchUp6 ESRI插件1.启动ArcMap界面，在工具栏上右键，单击“Customize”2.点击“Add from file”，找到SketchUp ArcGIS Plugin安装目录下的Features To SKP.dll3．添加插件动态库后，在Toolbars项中可以找到SketchUp6的功能项。

4.选中“SketchUp 6 Tools”组件以后，在桌面上会弹出组件的功能按钮。

数学英语练习题### 数学英语练习题#### 数学部分一、选择题1. What is the value of \( 2^3 \) ?- A. 4- B. 6- C. 8- D. 102. If the perimeter of a square is 20 units, what is the length of one side?- A. 5 units- B. 4 units- C. 3 units- D. 2 units3. Solve for \( x \) in the equation \( 3x - 7 = 8 \) . - A. 3- B. 5- C. 7- D. 9二、填空题1. The sum of the first \( n \) natural numbers is given by the formula \( \frac{n(n+1)}{2} \). If \( n = 10 \), the sum is ______.2. The area of a rectangle is calculated by multiplying its length by its width. If the area is 48 square meters and the width is 8 meters, the length is ______ meters.3. If a right triangle has legs of 3 and 4 units, the hypotenuse is \( \sqrt{3^2 + 4^2} \), which equals ______ units.三、解答题1. A store has a sale where all items are 20% off. If a jacket originally costs $100, what is the sale price?2. A train travels at a constant speed of 60 miles per hour. How long will it take to travel 180 miles?3. A company has 120 employees. If the company decides to increase the workforce by 25%, how many new employees will be hired?#### 英语部分一、选择题1. Which word is a synonym for "diligent"?- A. Careless- B. Lazy- C. Hardworking- D. Idle2. The phrase "break the ice" is used to describe what?- A. Shattering a frozen lake- B. Starting a conversation- C. Cooling a room- D. Freezing a beverage3. The word "altruistic" is most closely related to which of the following?- A. Selfish- B. Generous- C. Greedy- D. Stingy二、填空题1. The word "____" can be used to describe someone who is always ready to help others.- (Hint: Begins with 'w')2. The past tense of "begin" is ______.3. The phrase "____" is used to express that something is very easy.- (Hint: Contains the word 'piece')三、作文题Write a short essay about the importance of learning a secondlanguage. Your essay should include:- The benefits of being bilingual.- How learning a new language can broaden one's perspective. - Your personal experience or opinion on language learning.Please attempt the above questions to test your understanding of mathematical concepts and English language skills. Remember, practice makes perfect!。