Texture Zeros and CP-violating Phases in the Neutrino Mass Matrix
第1章相空间重构
迟重构都可以用来进行相空间重构，但就实际应用而言，由于我们通常不知道混沌时间
序列的任何先验信息，而且从数值计算的角度看，数值微分是一个对误差很敏感的计算
问题，因此混沌时间序列的相空间重构普遍采用坐标延迟的相空间重构方法[2]。坐标延
迟法的本质是通过一维时间序列{x(n)}的不同时间延迟来构造 m 维相空间矢量：
以取任意值，但实际应用最后等时间序列都是含有噪声的有限长序列，嵌入维数和时间
延迟是不能任意取值，否则会严重影响重构的相空间质量。
有关时间延迟与嵌入维的选取方法，目前主要有两种观点。一种观点认为两者是互
-2-
不相关的，先求出时间延迟后再求出选择合适的嵌入维。求时间延迟τ 比较常用的方法 有自相关法[5]、平均位移法[5]、复自相关法[6]和互信息法[7, 8]等，目的是使原时间序列经 过时间延迟后可以作为独立坐标使用。一个好的重构相空间是使重构后的吸引子和系统 真正的吸引子尽可能做到拓扑等价，目前寻找最小嵌入维的方法主要是几何不变量法[9]、 虚假最临近点法[10](FNN)和它的改进形式 Cao 方法[11]。另一种观点认为时间延迟和嵌入 维是相关的，1996 年 Kugiumtzis 提出的时间窗长度是综合考虑两者的重要参数[12]。1999 年，Kim 等人基于嵌入窗法的思想提出了 C-C 方法，该方法使用关联积分同时估计出时 延与嵌入窗[13]。C-C 方法也是实际时间序列中比较常用的方法，针对该方法的缺陷，国 内学者作了相应的改进[14, 15]。
对于连续变量 x(t) ，其自相关函数(Autocorrelation function)定义为
T
∫ C(τ ) = lim T →∞
2 −T
x(t ) x(t
+τ
PACSnumbers1215Ff,1130Hv,1210Dm,1125Mj…
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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D.Carone and R.F.Lebed,Phys.Rev.D62,016009(2000);P.D.Carr and P.H.Frampton,arXiv:hep-ph/0701034;A.Aranda, Phys.Rev.D76,111301(2007).[9]I.de Medeiros Varzielas,S.F.King and G.G.Ross,Phys.Lett.B648,201(2007);C.Luhn,S.Nasri and P.Ramond,J.Math.Phys.48,073501(2007);Phys.Lett.B652,27(2007).[10]E.Ma,arXiv:0705.0327[hep-ph];G.Altarelli,arXiv:0705.0860[hep-ph].[11]N.Cabibbo,Phys.Rev.Lett.10,531(1963);M.Kobayashi and T.Maskawa,Prog.Theor.Phys.49, 652(1973).[12]M.-C.Chen and K.T.Mahanthappa,Phys.Lett.B652,34(2007);W.Grimus and H.Kuhbock,Phys.Rev.D77, 055008(2008);F.Bazzocchi et al.,arXiv:0802.1693[hep-ph];G.Altarelli,F.Feruglio and C.Hagedorn,J.High Energy Phys.0803,052(2008).[13]A.Y.Smirnov,arXiv:hep-ph/0402264;M.Raidal,Phys.Rev.Lett.93,161801(2004);H.Minakata andA.Y.Smirnov,Phys.Rev.D70,073009(2004).[14]F.Plentinger,G.Seidl and W.Winter,Nucl.Phys.B791,60(2008).[15]F.Plentinger,G.Seidl and W.Winter,Phys.Rev.D76,113003(2007).[16]F.Plentinger,G.Seidl and W.Winter,J.High EnergyPhys.0804,077(2008).[17]G.Cacciapaglia,C.Csaki,C.Grojean and J.Terning,Phys.Rev.D74,045019(2006).[18]K.Agashe,A.Falkowski,I.Low and G.Servant,J.HighEnergy Phys.0804,027(2008);C.D.Carone,J.Erlich and M.Sher,arXiv:0802.3702[hep-ph].[19]Y.Kawamura,Prog.Theor.Phys.105,999(2001);G.Altarelli and F.Feruglio,Phys.Lett.B511,257(2001);A.B.Kobakhidze,Phys.Lett.B514,131(2001);A.Hebecker and J.March-Russell,Nucl.Phys.B613,3(2001);L.J.Hall and Y.Nomura,Phys.Rev.D66, 075004(2002).[20]D.E.Kaplan and T.M.P.Tait,J.High Energy Phys.0111,051(2001).[21]C.D.Froggatt and H.B.Nielsen,Nucl.Phys.B147,277(1979).[22]Y.Nomura,Phys.Rev.D65,085036(2002).[23]H.Georgi and C.Jarlskog,Phys.Lett.B86,297(1979).[24]H.Arason et al.,Phys.Rev.Lett.67,2933(1991);H.Arason et al.,Phys.Rev.D47,232(1993).[25]D.S.Ayres et al.[NOνA Collaboration],arXiv:hep-ex/0503053;Y.Hayato et al.,Letter of Intent.[26]S.Antusch,S.F.King and R.N.Mohapatra,Phys.Lett.B618,150(2005).[27]W.Winter,Phys.Lett.B659,275(2008).[28]K.S.Babu and S.M.Barr,Phys.Rev.D48,5354(1993);K.Kurosawa,N.Maru and T.Yanagida,Phys.Lett.B 512,203(2001).[29]L.M.Krauss and F.Wilczek,Phys.Rev.Lett.62,1221(1989).[30]F.Plentinger and G.Seidl,in preparation.。
nerf-texture解读 -回复
nerf-texture解读-回复什么是nerftexture（韧纤维纹理）？Nerftexture（韧纤维纹理）是一个用于优化计算机图形渲染的技术。
它使用了一种称为纹理映射的过程，通过在三维物体表面应用一种纹理图像，将具有不同纹理和颜色的模型转换为更真实、更具细节的表现形式。
nerftexture的目标是增强图形的真实感和感知深度，为用户提供更沉浸式的虚拟体验。
那么，nerftexture是如何工作的？nerftexture的核心原理是通过将二维纹理映射到三维物体的表面来实现。
映射的过程中，图像的像素信息与三维物体表面的顶点进行相互匹配，从而实现纹理的贴图。
这种贴图是基于计算机生成的纹理坐标，该坐标定义了二维纹理图像在三维物体表面上的位置。
在渲染过程中，nerftexture使用了逐像素纹理映射方法，通过对每个像素应用相应的纹理信息，从而最大程度地提升渲染的精细度和真实感。
nerftexture在图形渲染中的应用领域是广泛的。
对于虚拟现实(VR)和增强现实(AR)技术来说，nerftexture的效果尤为显著。
在VR和AR应用中，用户希望能够沉浸在一个逼真的虚拟世界中，这就要求图形能够准确地模拟真实世界的纹理和细节。
nerftexture能够实现这一目标，使得虚拟现实和增强现实应用更加逼真、生动。
另外，nerftexture还常常应用于电影和游戏制作中。
在电影制作过程中，通过nerftexture技术可以为特效场景增加更多的细节，使得电影更具观赏性。
对于游戏制作来说，nerftexture能够提高游戏画面的质量，提供更真实、更细腻的游戏体验。
纹理的细致程度能够直接影响游戏场景和角色的真实感，nerftexture通过提供更高质量的纹理映射，提升了游戏的视觉呈现。
总之，nerftexture是一项优化计算机图形渲染的技术，它使用纹理映射的过程，将二维纹理图像应用到三维物体表面，从而实现增强真实感和细节感的目标。
shader is corrupt or in an unrecognized format
shader is corrupt or in an unrecognized formatShader corruption or unrecognized format errors can occur when working with shaders in various graphic applications or game engines. These errors can be frustrating, but there are several possible causes and solutions that can help resolve these issues. Let's explore them in more detail.1. Incorrect Shader File Format:One possible cause of shader corruption or unrecognized format errors is the use of an incorrect shader file format. Different software or game engines may have specific requirements for the file format of shaders. Ensure that you are using the correct file format, such as .shader, .cg, .hlsl, or .glsl, depending on the software or engine you are using.2. File Corruption:Another possible cause is file corruption. It is essential to ensure that the shader file is not corrupt. Try to open the shader file in a text editor or shader development tool to see if it appears as expected. If the file does not open correctly or shows gibberish characters, it is likely corrupted. In such cases, try restoring the file from a backup or redownload it if available.3. Shader Compiler Compatibility:Different shader compilers may have varying levels of compatibility. If you are facing shader corruption or unrecognized format errors, check if the shader compiler is compatible with the version of your software or game engine. You may need to update the shader compiler or use alternative compilers that are compatible with your environment.4. Shader Syntax Errors:Shader syntax errors can also lead to corruption or unrecognized format issues. Carefully review the shader code for any syntax errors, missing semicolons, or incorrect variable declarations. Even a minor mistake can cause the shader to fail. Use shader development tools or integrated development environments (IDEs) that provide syntax highlighting and error checking to aid in identifying and resolving syntax errors.5. Shader Compilation Issues:Shader compilation errors can indicate problems with the shader code or compilation settings. Check the debugger or log files for any error messages related to shader compilation. This information can help pinpoint the cause of the problem. Fix any compilation errors by resolving issues such as undefined variables, incompatible data types, or incorrect macro usage.6. Graphics Driver Compatibility:In some cases, shader corruption or unrecognized format errors can be related to outdated or incompatible graphics drivers. Ensure that you have the latest drivers installed for your graphics card. Visit the manufacturer's website to download and install the appropriate driver version. Updating the drivers can resolve many graphics-related issues.7. Hardware Compatibility:Certain shaders may require specific hardware capabilities, and if your hardware does not support them, corruption or unrecognized format errors might occur. Check the shader documentation orrequirements to ensure your hardware meets the specifications. If your hardware is not compatible, you may need to find alternative shaders or adjust your shader settings.In conclusion, shader corruption or unrecognized format errors can be caused by a variety of factors, including incorrect file formats, file corruption, shader compiler compatibility, syntax errors, compilation issues, graphics driver compatibility, and hardware compatibility. By carefully examining and addressing these potential causes, you can troubleshoot and resolve these errors, allowing you to work with shaders effectively within your software or game engine.。
shader的ao渲染原理
shader的ao渲染原理
Shader的AO渲染原理
AO（Ambient Occlusion）是一种用于增强场景真实感的技术，它可以模拟光线在物体表面的反射和散射，从而产生阴影效果。
在游戏开发中，AO技术被广泛应用于场景渲染、角色渲染等方面，可以大大提高游戏的视觉效果。
Shader是一种用于实现图形渲染的编程语言，它可以在GPU上运行，实现高效的图形渲染。
在AO渲染中，Shader起到了至关重要的作用，它可以通过计算光线在物体表面的反射和散射，来模拟出阴影效果。
AO渲染的原理是基于光线追踪的，它通过计算光线在物体表面的反射和散射，来确定物体表面的阴影强度。
在Shader中，可以通过计算物体表面的法线向量和光线向量的夹角，来确定光线在物体表面的反射和散射情况。
当夹角越小时，反射和散射越强，阴影效果也越明显。
为了实现AO渲染，Shader需要对场景中的每个物体进行计算，确定它们的阴影强度。
在计算过程中，Shader会考虑物体之间的相互作用，以及光线在物体表面的反射和散射情况。
通过这些计算，Shader可以模拟出真实的阴影效果，使场景更加逼真。
Shader的AO渲染原理是基于光线追踪的，通过计算光线在物体表
面的反射和散射情况，来模拟出阴影效果。
在游戏开发中，AO技术被广泛应用于场景渲染、角色渲染等方面，可以大大提高游戏的视觉效果。
UnityShader曲面细分简介
UnityShader曲⾯细分简介⽬录Unity Shader 曲⾯细分简介概念介绍曲⾯细分⼀种对输⼊的图元（三⾓形、四边形、线段）进⾏细化，产⽣出更多的顶点，使其变得更精细的技术。
这⼀功能在渲染管线中完成，通常会由显卡硬件⽀持。
从阶段上讲它位于顶点着⾊器之后，像素着⾊器之前。
它的内部流程分为三部分：可编程的Hull Shader（控制细分参数），不可编程的Generator（细分顶点），可编程的Domain Shader（顶点位置计算）。
因此，它基于顶点着⾊器阶段后的顶点数据，并可以控制⽣成更多的顶点数据交由像素着⾊器使⽤。
作⽤曲⾯细分技术降低了模型精细度要求，在模型制作、存储、加载上都可带来节省。
通过曲⾯细分⽣成更精细的顶点后，可以应⽤“置换贴图”技术来渲染出更加精细、逼真的模型。
置换贴图不同于法线贴图修改顶点的法线⽅向来造成光影错觉，它会真正修改顶点的位置来产⽣凹凸感，因此当视线与⽹格切⾯⽅向接近时法线贴图基本上失效⽽置换贴图仍可产⽣逼真的效果。
例如以下分别是【标准】材质、【未细分仅置换】材质、【固定数⽬细分置换】材质的效果：关于Unity⽀持unity⽂档⾥只提了Built-in Render Pipeline的Surface Shader⽀持曲⾯细分，并且只⽀持三⾓形。
同时它需要Shader Model 4.6target，因此不⽀持更⽼的设备。
官⽅案例在使⽤Surface Shader的前提下使⽤曲⾯细分⾮常简单，以下的每种细分⽅法都提供了内置的函数，直接调⽤就完事。
总的来说是在shader中增加两个函数：tessellate:FunctionNamevertex:FunctionNametessellate声明的函数返回细分控制参数，这是⼀个float4类型，前三个值分别表⽰输⼊三⾓形的三个边被分成⼏段，第四个值是Inside因⼦，算法使⽤，不太清楚。
vertex声明的函数可以拿到细分后的所有顶点，可以在这⾥修改其位置。
ue4 refraction算法
ue4 refraction算法UE4 Refraction算法是虚幻引擎4中用于实现折射效果的一种算法。
折射是指光线在穿过介质界面时改变方向的现象，而UE4的Refraction算法则可以模拟出这种效果，使得游戏中的物体看起来更加逼真和真实。
在游戏开发中，折射效果是非常重要的，它可以让玩家更加身临其境地体验游戏世界。
例如，当光线穿过水面时，会因为水的折射效应而发生弯曲，这样就可以通过Refraction算法来模拟出这种效果。
在UE4中，要实现折射效果，需要对物体的材质进行设置。
在UE4中创建一个材质球，并将其应用到需要实现折射效果的物体上。
然后，在材质球的属性面板中，可以找到一个叫做Refraction 的选项。
通过调整Refraction的参数，可以控制折射效果的强度和变形程度。
例如，可以通过增加Refraction的值来增强折射效果，使得物体看起来更加扭曲和逼真。
除了调整Refraction的参数外，还可以通过调整其他材质属性来进一步增强折射效果。
例如，可以通过调整Opacity属性来控制物体的透明度，从而使得折射效果更加明显。
此外，还可以通过调整Roughness属性来改变物体的光滑程度，从而影响折射效果的表现。
在UE4中，Refraction算法是基于物理的，并且可以与其他效果进行组合使用。
例如，可以将折射效果与反射效果结合起来，以增强物体的真实感。
同时，还可以通过调整光照和环境光等参数，使得折射效果与场景中的其他元素相匹配，从而达到更好的视觉效果。
UE4的Refraction算法是一种用于实现折射效果的强大工具。
通过调整材质的属性和参数，可以在游戏中模拟出逼真的折射效果，使得物体看起来更加真实和生动。
无论是在水面、玻璃窗或者其他透明介质中，Refraction算法都可以帮助开发者实现出更加引人入胜的游戏世界。
Esri+CityEngine中文教程V1.0
1 基础内容 ..............................................................................................................1
1.1 基本操作.....................................................................................................1 1.2 教程与示例数据下载.................................................................................2 1.3 基本概念.....................................................................................................3
2.10 导出模型...................................................................................................17
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February 2, 2008
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has six independent entries. If n of them is/are taken to be vanishing, we will arrive at
ZHI-ZHONG XING Institute of High Energy Physics, Chinese Academy of Sciences P.O. Box 918 (4), Beijing 100039, China E-mail: xingzz@
We stress that specific texture zeros of lepton mass matrices, which might dynamically arise from a new kind of flavor symmetry, can help us to establish simple and testable relations between the lepton flavor mixing parameters and lepton mass ratios. We present a brief review of one-zero, two-zero and three-zero textures of the neutrino mass matrix. Their phenomenological consequences on neutrino mixing and CP violation are also discussed.
1. Introduction Impressively robust evidence in favor of neutrino oscillations has been accumulated from the solar1, atmospheric2 , reactor3 and accelerator4 neutrino experiments in the past few years. We are now convinced that neutrinos are massive and lepton flavors are mixed. In spite of such exciting progress in neutrino physics, our quantitative knowledge about neutrino masses and lepton flavor mixing remain rather poor – for example, the absolute scale of three neutrino masses, the smallest lepton mixing angle and three CPviolating phases are still unknown. In the lack of a convincing flavor theory, five approaches have been tried towards a deeper understanding of fermion mass generation5: (a) radiative mechanisms6 ; (b) texture zeros7 ; (c) flavor symmetries8 ; (d) seesaw mechanisms9 ; and (e) extra dimensions10 . Some of them can be correlated with one another. For instance, lepton mass matrices may have a few texture zeros as a natural consequence of a new kind of flavor symmetry11 , and those texture zeros may guarantee some calculability and allow us to predict the neutrino mass spectrum and lepton flavor mixing parameters via the seesaw mechanism. Note that texture zeros of a fermion mass matrix dynamically mean that the corresponding matrix elements are sufficiently suppressed in comparison with their neighboring counterparts. A very good
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lesson from the quark sector is remarkable: reasonable zeros of quark mass matrices allow us to establish some simple and testable relations between the flavor mixing angles and quark mass ratios12 – if such relations are more or less unique and experimentally favored, they may have a good chance to be close to the truth – namely, the same or similar relations should be predicted by the underlying (true) theory with much fewer fundamental parameters. Hence a phenomenological study of possible texture zeros of fermion mass matrices does make some sense to get useful hints about flavor dynamics responsible for the generation of fermion masses and the origin of CP violation. The phenomenology of lepton masses and flavor mixing at low energies can be formulated in terms of the charged lepton mass matrix Ml and the (effective) neutrino mass matrix Mν . The lepton flavor mixing matrix V arises from the mismatch between diagonalizations of Ml and Mν . There are totally twelve physical parameters: three charged lepton masses (me , mµ , mτ ), three neutrino masses (m1 , m2 , m3 ), three flavor mixing angles (θ12 , θ23 , θ13 ), and three CP-violating phases (δ, ρ, σ ). Besides me , mµ 2 and mτ , preliminary values of ∆m2 21 , |∆m32 |, θ12 and θ23 have essentially been extracted from solar and atmospheric neutrino oscillations. How small θ13 is remains an open question13 . The sign of ∆m2 32 is unknown and the magnitudes of δ , ρ and σ are entirely unrestricted. It seems impossible to fully determine Ml and Mν from the feasible experiments at present or in the near future. In this situation, we hope that possible texture zeros of Ml and Mν may help us out. We remark that texture zeros of lepton mass matrices can lead to some simple and testable relations between unknown and known parameters of neutrino oscillations. Of course, such zeros may not be preserved to all orders or at any energy scales in the unspecified interactions from which lepton masses are generated. At the one-loop level and in the flavor basis where Ml is diagonal and positive, however, the renormalization-group evolution of Mν from the seesaw scale (i.e., the mass scale of the lightest right-handed Majorana neutrino) to the electroweak scale does allow its texture zeros to preserve12 . Once the approach of texture zeros is combined with the seesaw mechanism, it is possible to simultaneously account for the cosmological baryon number asymmetry via leptogenesis14 and current neutrino oscillation data. This talk is subject to a phenomenological analysis of texture zeros of lepton mass matrices at low energy scales. For simplicity, we restrict ourselves to symmetric Ml and Mν . A symmetric lepton mass matrix totally