Quaternion Algebras and Invariants of Virtual Knots and Links II The Hyperbolic Case

Models of Software Systems Fall 2004ObjectivesScientific foundations for software engineering depend on the use of precise, abstract models and logics for characterizing and reasoning about properties of software systems. There are a number of basic models and logics that over time have proven to be particularly important and pervasive in the study of software systems. This course is concerned with that body of knowledge. It considers many of the standard models for representing sequential and concurrent systems, such as state machines, algebras and traces. It shows how different logics can be used to specify properties of software systems, such as functional correctness, deadlock freedom, and internal consistency. Concepts such as composition mechanisms, abstraction relations, invariants, non-determinism, and inductive and denotational descriptions are recurrent themes throughout the course.By the end of the course you should be able to understand the strengths and weaknesses of certain models and logics, including state machines, algebraic and trace models, and temporal logics. You should be able to apply this understanding to select and describeabstract formal models for certain classes of systems. Further, you should be able to reason formally about the elementary properties of modeled systems.OrganizationLectures. Classes meet Monday & Wednesday, 5:30-6:50 am, in Newell-Simon 1305.Communication. We will be using the CMU Blackboard System this year for distributing most course materials, providing a general course bulletin board, and keeping track of student email addresses. In addition you can useOffice Hours: The instructor and the TAs have weekly office hours, listed above. We are also available other times by appointment.1.Email: We welcome email about the course at any time.2.Readings. Most lectures will have a reading assignment that we expect you to complete before you come to class. There is one required textbook for the course: Concurrency: State Models and Java Programs, by Magee and Kramer [MK99]. In addition, there is an optional companion text Using Z: Specification, Refinement, and Proof, by Woodcock and Davies [WD96]. This text is available on-line at /.An optional reference book may also be useful: The Z Notation: A Reference Manual, Second Edition, by J. M. Spivey (available on the web through /~mike/zrm). Some readings are in the form of handouts to supplement lectures; other additional readings are technical papers. These will be made available as needed throughout the course. Finally, for supplementary detail, there are a number of books noted in the References section at the end of this document.Homework Assignments. The course is organized around (roughly) weekly homework assignments and a set of three projects. The purpose of the assignments and projects is to give you practice in using the models, logics, and tools of the course. We encourage you to discuss your homework with other students, but the final write-up must be your own work.To give you the most opportunities to learn from the homework assignments, we will allow you to redo problems that didn't receive a passing grade. A redone homework must be turned in at the class following the one on which it is handed back. Problems done correctly the first time will be given more weight in the final grade.Projects. We will be assigning three group projects that are designed to give you a chance to apply the ideas of the course tosemi-realistic case studies. Each project will be completed by a team. Team members are expected to participate equally in the projects.We will distribute a team peer evaluation at the end of the semester.On-line materials. Most of the course materials will be available electronically via the CMU Blackboard System(/blackboard/). You will find copies of the lecture slides, handouts, homework, and readings. It will be your responsibility to make copies of these to bring to class or to use for homework.Some of the course materials have web sites. These are:Using Z: /Concurrency: State Models and Java Programs: /concurrency/The Z Notation: /~mike/zrm/PhD Option. Students taking the course for PhD credit students will be required to complete a course project. This project is described in separate handout.Exams. There will be a (take-home) mid-term (handed out Wednesday, October 20, due back Friday, October 22 by 5:00 p.m.) and a formal (in-class) final examination. Both exams will be open-book.Grading. The course grade will be determined as a combination of five factors: homework assignments (30%), projects (30%), midterm exam (15%), and final exam (25%). Final grades may be adjusted based on instructors judgment.Bold refers to Using Z [WD96].*Marks classes that follow a holiday.。

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c aPNAProbability, Networks and AlgorithmsProbability, Networks and AlgorithmsAn image retrieval system based on adaptive waveletliftingP.J. Oonincx, P.M. de ZeeuwR EPORT PNA-R0208 M ARCH 31, 2002CWI is the National Research Institute for Mathematics and Computer Science. It is sponsored by the Netherlands Organization for Scientific Research (NWO).CWI is a founding member of ERCIM, the European Research Consortium for Informatics and Mathematics. CWI's research has a theme-oriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms.Probability, Networks and Algorithms (PNA)Software Engineering (SEN)Modelling, Analysis and Simulation (MAS)Information Systems (INS)Copyright © 2001, Stichting Centrum voor Wiskunde en InformaticaP.O. Box 94079, 1090 GB Amsterdam (NL)Kruislaan 413, 1098 SJ Amsterdam (NL)Telephone +31 20 592 9333Telefax +31 20 592 4199ISSN 1386-3711CWIP.O.Box94079,1090GB Amsterdam,The NetherlandsPatrick.Oonincx,Paul.de.Zeeuw@cwi.nl1.I NTRODUCTIONContent-based image retrieval(CBIR)is a widely used term to indicate the process of retrieving desired images from a large collection on the basis of features.The extraction process should be automatic(i.e. no human interference)and the features used for retrieval can be either primitive(color,shape,texture) or semantic(involving identity and meaning).In this paper we confine ourselves to grayscale images of objects against a background of texture.This class of images occurs for example in various databases created for the combat of crime:stolen objects[21],tyre tracks and shoe sole impressions[1].In this report we restrict ourselves to the following problem.Given an image of an object(a so-called query)we want to identify all images in a database which contain the same object irrespective of translation,rotation or re-sizing of the object,lighting conditions and the background texture.One of the most classical approaches to the problem of recognition of similar images is by the use of moment invariants[11].This method is based on calculating moments in both the-and-direction of the image density function up to a certain order.Hu[11]has shown that certain homogeneous polynomials of these moments can be used as statistical quantities that attain the same values for images that are of the same class,i.e.,that can be obtained by transforming one single original image(affine transforms and scaling).However,this method uses the fact that such images consists of a crisp object against a neutral background.If the background contains‘information’(noise,environment in a picture)the background should be the same for all images in one class and should also be obtained from one background using the same transformations.In general this will not be the case.The kind of databases we consider in this paper consists of classes of different objects pasted on different background textures.To deal with the problem of different backgrounds one may use somefiltering process as a preprocessing step.In Do et al.[7]the wavelet transform modulus maxima is used as such preprocessing step.To measure the(dis)similarity between images,moments of the set of maxima points are determined(per scale)and subsequently Hu’s invariants are computed.Thus,each image is indexed by a vector in the wavelet maxima moment space.By its construction,this vector predominantly represents shapes.In this report we propose to bring in adaptivity by using different waveletfilters for smooth and unsmooth parts of the image.Thefilters are used in the context of the(redundant)lifting scheme[18].The degree2of”smoothness”is determined by measuring the relative local variance(RLV),which indicates whether locally an image behaves smoothly or not.Near edges low order predictionfilters are activated which lead to large lifting detail coefficients along thin curves.At backgrounds of texture high order predictionfilters are activated which lead to negligible detail coefficients.Moments and subsequently moment invariants are computed with respect to these wavelet detail coefficients.With the computation of the detail coefficients a certain preprocessing is required to make the method robust for shifts over a non-integer number of gridpoints.Further we introduce the homogeneity condition which means that we demand a homogeneous change in the elements of a feature vector if the image seen as a density distribution is multiplied by a scalar.The report is organized as follows.In Sections2and3we discuss the lifting scheme and its adaptive version.Section4is devoted to the topic of affine invariances of the coefficients obtained from the lifting scheme.In Section5the method of moment invariants is recapitulated.The homogeneity condition is in-troduced which leads to a normalization.Furthermore,the mathematical consequences for the computation of moments of functions represented byfields of wavelet(detail)coefficients are investigated.Section6 discusses various aspects of thefinal retrieval algorithm,including possible metrics.Numerical results of the algorithm for a synthetic database are presented in Section7.Finally,some conclusions are drawn in Section8.2.T HE L IFTING S CHEMEThe lifting scheme as introduced by Sweldens in1997,see[18],is a method for constructing wavelet transforms that are not necessarily based on dilates and translates of one function.In fact the construction does not rely on the Fourier transform which makes it also suitable for functions on irregular grids.The transform also allows a fully in-place calculation,which means that no auxiliary memory is needed for the computations.The idea of lifting is based on splitting a given set of data into two subsets.In the one-dimensional case this can mean that starting with a signal the even and odd samples are collected into two new signals,i.e.,,where and,for all.The next step of the lifting scheme is to predict the value of given the sequence.This prediction uses a prediction operator acting on.The predicted value is subtracted from yielding a ‘detail’signal.An update of the odd samples is needed to avoid aliassing problems.This update is performed by adding to the sequence,with the update operator.The lifting procedure can also be seen as a2-bandfilter bank.This idea has been depicted in Figure1.The inverse lifting scheme can/.-,()*+/.-,()*+/.-,()*+/.-,()*+/.-,()*+Figure1:The lifting scheme:splitting,predicting,updating.immediately be found by undoing the prediction and update operators.In practice,this comes down in Figure1to simply changing each into a and vice versa.Compared to the traditional wavelet transform the sequence can be regarded as detail coefficients of the signal.The updated sequence can be regarded as the approximation of at a coarse ing again as input for the lifting scheme yields detail and approximation signals at lower resolution levels.We observe that every discrete wavelet transform can also be decomposed into a finite sequence of lifting steps[6].To understand the notion of vanishing moments in terms of the prediction and update operators,we com-pare the lifting scheme with a two-channelfilter bank with analysisfilters(lowpass)and(highpass) and synthesisfilters and.Such afilter bank has been depicted in Figure2.Traditionally we say that a389:;?>=<89:;?>=</.-,()*+89:;?>=<89:;?>=<Figure2:Classical2-band analysis/synthesisfilter bank.filter bank has primal and vanishing moments ifandwhere denotes the space of all polynomial sequences of order.Given thefilter operators and, the correspondingfilters and can be computed by(2.1)(2.2)where and denote thefilter sequences of the operators and respectively.In[12]Kovacevic and Sweldens showed that we can always use lifting schemes with primal vanishing moments and dual vanishing moments by taking for a Nevillefilter of order with a shift and for half the adjoint of a Nevillefilter of order and shift,see[17].Example2.1We take and.With these operators we getThefilter bank has only one vanishing moment.The lifting transform corresponds in this example to the Haar wavelet transform.Example2.2For more vanishing moments,i.e.,smoother approximation signals,we takeThese Nevillefilters give rise to a2-channelfilter bank with2primal and4dual vanishing moments. The lifting scheme can also be used for higher dimensional signals.For these signals the lifting scheme consists of channels,where denotes the absolute value of the determinant of the dilation matrix,that is used in the corresponding discrete wavelet transform.In each channel the signal is translated along one of the coset representatives from the unit cell of the corresponding lattice,see[12]. The signal in thefirst channel is then used for predicting the data in all other channels by using possible different prediction operators.Thereafter thefirst channel is updated using update operators on the other channels.Let us consider an image as a two-dimensional signal.An important example of the lifting scheme applied to such a signal is one that involves channels().We subdivide the lattice on which the signal has been defined into two sets on quincunx grids,see Figure3.This division is also called ”checkerboard”or”red-black”division.The pixels on the red spots()are used to predict the samples on the black spots(),while updating of the red spots is performed by using the detailed data on the black spots.An example of a lifting transform with second order prediction and updatefilters is given by4Figure3:A rectangular grid composed of two quincunx grids.order200000040000060008Table1:Quincunx Nevillefilter coefficientsThe algorithm using the quincunx lattice is also known as the red-black wavelet transform by Uytterhoeven and Bultheel,see[20].In general can be written as(2.3) with a subset of mod and,a set of coefficients in. In this case a general formula for reads(2.4)with depending on the number of required primal vanishing moments.For several elements in the coefficients attain the same values.Therefore we take these elements together in subsets of, i.e.,(2.5)Table1indicates the values of all,for different values of(2through8)when using quincunx Nevillefilters,see[12],which are thefilters we use in our approach.We observe that and so a44tapsfilter is used as prediction/update if the requiredfilter order is8.For an illustration of the Nevillefilter of order see Figure4.Here the numbers,correspond to the values of thefilter coefficients as given in and respectively at that position.The left-handfilter can be used to transform a signal defined on a quincunx grid into a signal defined on a rectangular grid,the right-hand filter is the degrees rotated version of the left-handfilter and can be used to transform a signal from a rectangular grid towards a quincunx grid.We observe that the quincunx lattice yields a non separable2D-wavelet transform,which is also sym-metric in both horizontal and vertical direction.Furthermore,we only need one prediction and one update operator for this2D-lifting scheme,which reduces the number of computations.The prediction and update operators for the quincunx lattice do also appear in schemes for other lattices, like the standard2D-separable lattice and the hexagonal lattice[12].The algorithm for the quincunx lattice can be extended in a rather straightforward way for these two other well-known lattices.5111122222222111122222222Figure 4:Neville filter of order :rectangular (left)and quincunx (right)Figure 5illustrates the possibility of the use of more than channels in the lifting scheme.Herechannels are employed,using a four-colour division of the 2D-lattice.It involves (interchange-able)prediction steps.Each of the subsets with colours ,and respectively,is predicted by application of a prediction filter on the subset with colour .Figure 5:Separable grid (four-colour division).3.A DAPTIVE L IFTINGWhen using the lifting scheme or a classical wavelet approach,the prediction/update filters or wavelet/scaling functions are chosen in a fixed fashion.Generally they can be chosen in such way that a signal is approximated with very high accuracy using only a limited number of coefficients.Discontinuities mostly give rise to large detail coefficients which is undesirable for applications like compression.For our purpose large detail coefficients near edges in an images are desirable,since they can be identified with the shape of objects we want to detect.However,they are undesirable if such large coefficients are related to the background of the image.This situation occurs if a small filter is used on a background of texture that contains irregularities locally.In this case a large smoothing filter gives rise to small coefficients for the background.These considerations lead to the idea of using different prediction filters for different parts of the signal.The signal itself should indicate (for example by means of local behavior information)whether a high or low order prediction filter should be used.Such an approach is commonly referred to as an adaptive approach.Many of these adaptive approaches have been described already thoroughly in the literature,e.g.[3,4,8,13,19].In this paper we follow the approach proposed by Baraniuk et al.in [2],called the space-adaptive approach.This approach follows the scheme as shown in Figure 6.After splitting all pixels of a given image into two complementary groups and (red/black),thepixels inare used to predict the values in .This is done by means of a prediction filter acting on ,i.e.,.In the adaptive lifting case this prediction filter depends on local information of the image pixels .Choices for may vary from high to low order filters,depending on the regularity of the image locally.For the update operator,we choose the update filter that corresponds to the prediction filter with lowest order from all possible to be chosen .The order of the update filter should be lower or equal to the order of the prediction filter as a condition to provide a perfect reconstruction filter bank.As with the classical wavelet filter bank approach,the order of the prediction filter equals the number of dual vanishing6/.-,()*+/.-,()*+Figure6:Generating coefficients via adaptive liftingmoments while the order of the updatefilter equals the number of primal vanishing moments,see[12].The above leads us to use a second order Nevillefilter for the update step and an th order Nevillefilter for the prediction step,where.In our application the reconstruction part of the lifting scheme is not needed.In[2],Baraniuk et al.choose to start the lifting scheme with an update operator followed by an adaptively chosen prediction operator.The reason for interchanging the prediction and update operator is that this solves stability and synchronization problems in lossy coding applications.We will not discuss this topic in further detail,but only mention that they took for thefilters of and the branch of the Cohen-Daubechies-Feauveau(CDF)filter family[5].The order of the predictionfilter was chosen to be,,or,depending on the local behavior of the signal.Thefilter orders of the CDFfilters in their paper correspond to thefilter orders of the Nevillefilters we are using in our approach.Relative local variance We propose a measure on which the decision operator in the2D adaptive lifting scheme can be based on,namely the relative local variance of an image.This relative local variance(RLV) of an image is given byrlv var(3.1) with(3.2) For the window size we take,since with this choice all that are used for the prediction of contribute to the RLV for,even for the8th order Nevillefilter.When the RLV is used at higher resolution levels wefirst have to down sample the image appropriately.Thefirst time the predictionfilter is applied(to the upper left pixel)we use the8th order Nevillefil-ter on the quincunx lattice as given in Table1.For all other subsequent pixels to be predicted,we first compute rlv.Then quantizing the values of the RLV yields a decisionmap indicating which predictionfilter should be used at which positions.Values above the highest quantizing level induce a 2nd order Nevillefilter,while values below the lowest quantizing levels induce an8th order Nevillefil-ter.For the quantizing levels we take multiples of the mean of the RLV.Test results have shown that rlv rlv rlv are quantizing levels that yield a good performance in our application.In Figure7we have depicted an image(left)and its decision map based on the RLV(right).4.A FFINE I NVARIANT L IFTINGAlthough both traditional wavelet analysis and the lifting scheme yield detail and approximation coeffi-cients that are localised in scale and space,they are both not translation invariant.This means that if a signal or image is translated along the grid,its lifting coefficients may not be just be given by a translation of the original coefficients.Moreover,in general the coefficients will attain values in the same range of the original values(after translation),but they will be totally different.7a) original image b) decision map (RLV)Figure7:An object on a wooden background and its rel.local variance(decision map):white=8th order, black=2nd order.For studying lifting coefficients of images a desirable property would also be invariance under reflections and rotations.However,for these two transformations we have to assumefirst that the values of the image on the grid points is not affected a rotation or reflection.In practice,this means that we only consider reflections in the horizontal,the vertical and the diagonal axis and rotations over multiples of.4.1Redundant LiftingFor the classical wavelet transform a solution for translation invariance is given by the redundant wavelet transform[15],which is a non-decimated wavelet(at all scales)transform.This means that one gets rid of the decimation step.As a consequence the data in all subbands have the same size as the size as the input data of the transform.Furthermore,at each scaling level,we have to use zero padding to thefilters in order to keep the multiresolution analysis consistent.Not only more memory is used by the redundant transform,also the computing complexity of the fast transform increases.For the non-decimated transform computing complexity is instead of for the fast wavelet transform.Whether the described redundant transform is also invariant under reflections and rotations depends strongly on thefilters(wavelets)themselves.Symmetry of thefilters is necessary to guarantee certain rotation and reflection invariances.This is a condition that is not satisfied by many well-known wavelet filters.The redundant wavelet transform can also be translated into a redundant lifting scheme.In one dimension this works out as follows.Instead of partitioning a signal into and we copy to both and.The next step of the lifting scheme is to predict by(4.1) The predictionfilter is the samefilter as used for the non-redundant case,however now it depends on the resolution level,since at each level zero padding is applied to.This holds also for the updatefilters .So,the update step reads(4.2)For higher dimensional signals we copy the data in all channels of the usedfilter bank. Next the-channel lifting scheme is applied on the data,using zero padding for thefilters at each resolu-8................ ........Figure8:Tree structure of the-channel lifting scheme.tion level.Remark,that for each lifting step in the redundant-channel lifting scheme we have to store at each scaling level times as much data as in the non-redundant scheme,see Figure8.We observe that in our approach Nevillefilters on a quincunx lattice are used.Due to their symmetry properties,see Table1,the redundant scheme does not only guarantee translation invariance,but also invariance under rotations over multiples of and reflections in the horizontal,vertical and diagonal axis is assured.Invariance under other rotations and reflections can not be guaranteed by any prediction and updatefilter pair,since the quincunx lattice is not invariant under these transformations.4.2An Attempt to Avoid Redundancy:Fixed Point LiftingAs we have seen the redundant scheme provides a way offinding detail and approximation coefficients that are invariant under translations,reflections and rotations,under which the lattice is also invariant.Due to its redundancy this scheme is stable in the sense that it treats all samples of a given signal in the same way.However redundancy also means additional computational costs and perhaps even worse additional memory to store the coefficients.Therefore we started searching for alternative schemes that are also invariant under the described class of affine transformation.Although we did not yet manage to come up with an efficient stable scheme,we would like to stretch the principal idea behind the building blocks of such approach.In the sequel we will only use the redundant lifting scheme as described in the preceding section.Before we start looking for possible alternative schemes we examine why the lifting scheme is not translation invariant.Assume we have a signal that is analysed with an-band lifting scheme. Then after one lifting step we have approximation data and detail data.Whether one sample,is determined to become either a sample of or a sample ofdepends only on its position on the lattice and the way we partition the lattice into groups.Of course, this partitioning is rather arbitrary.The more channels we use the higher the probability is that for afixed partitioning one sample that was determined to be used for predicting other samples,will become a sample of after translating.Following Figure8it is clear that any sample,can end up after lifting steps in ways,either in approximation data at level or in detail data at some level.The idea of the alternative scheme we propose here is to partition a signal not upon its position on the lattice but upon its structure.This means that for each individual signal we indicate afixed point for which we demand that it will end up in the approximation data after lifting steps.If this point can be chosen independent of its coordinates on the lattice,the lifting scheme based on this partitioning will then translation invariant.For higher dimensional signals we can also achieve invariance under the other discussed affine transformations,however then we have tofix more points,depending on the number ofchannels.In our approach the quincunx lattice is used and thereforefixing one approximation sample on scales will immediatelyfix the partitioning of all other samples on the quincunx lattice at scale. As a result thefixed point lifting scheme is invariant under translations,rotations and reflections that leave the quincunx lattice invariant.In the sequel of this chapter we will only discuss the lifting scheme for for the quincunx lattice.Although the proposedfixed point lifting scheme may seem to be a powerful tool for affine invariant lifting,it will be hard to deal with in practice.The problem we will have to face is how to choose afixed point in every image.In other words we have tofind a suitable decision operator that adds to every a unique,itsfixed point,i.e.,If we demand to depend only on and not on the lattice(coordinate free)it will be hard tofind such that is well defined.This independence of the coordinates is necessary for rotation invariances.However, this is not the only difficulty we have to face.Stability of the scheme is an other problem.If for some reason afixed point has been wrongly indicated,for example due to truncation errors,the whole scheme might collapse down.Although we cannot easily solve the problem of determining incorrectfixed points we can increase the stability of the scheme by not imposing that at each scale should be an index number of the coarse scale data after zero padding.Instead of this procedure we rather determine afixed point for both the original signal()and for the coarse scale data(at each scale.Then we impose that should be used for prediction in the th lifting step,for and with.Furthermore,stability may be increased by using decision operators that generate a set offixed points.However,since no stable method (uniform decision operator)is available yet,we will use the redundant lifting scheme in our approach and do not work out the idea offixed point lifting here at this moment.5.M OMENT I NVARIANTSAt the outset of this section we give a brief introduction into the theory of statistical invariants for imag-ing purposes,based on centralized moments.Traditionally,these features have been widely used in pat-tern recognition applications to recognize the geometrical shapes of different objects[11].Here,we will compute invariants with respect to the detail coefficients as produced by the wavelet lifting schemes of Sections2–4.We use invariants based on moments of the coefficients up to third order.We show how to construct a feature vector from the obtained wavelet coefficients at several scales It is followed by proposals for normalization of the moments to keep them in comparable range.5.1Introduction and recapitulationWe regard an image as a density distribution function,the Schwartz class.In order to obtain translation invariant statistics of such we use central moments of for our features.The order central moment of is given by(5.1) with the center of massand(5.2)Computing the centers of mass and of yieldsand bining this with(5.1)showsi.e.,the central moments are translation invariant.We also require that the features should be invariant under orthogonal transformations(rotations).For deriving these features we follow[11]using a method with homogeneous polynomials of order.These are given by(5.3) Now assume that the variables are obtained from other variables under some linear transformation,i.e.,then is an algebraic invariant of weight if(5.4) with the new coefficients obtained after transforming by.For orthogonal transformations we have and therefore is invariant under rotations ifParticularly we have from[11],that if is an algebraic invariant,then also the moments of order have the same invariant,i.e.,(5.5) From this equation2functions of second order can be derived that are invariant under rotations,see[11]. For we have the invariantsandIt was also shown that these two functions are also invariant under reflections,which can be a useful property for identifying reflected images.Since the way of deriving these invariants may seem a bit technical and artificial,we illustrate with straightforward calculus that and are indeed invariant under rotations.The invariance under reflections is left to the reader,since showing this follows the same calculations.We consider the rotated distribution functionand the corresponding invariants and,which are and but now based on moments calculated from.So what we have to show is that and.It follows from(5.1)and(5.2)that if and only ifwith and.Obviously this holds true,considering the trigonometric rule .To do the same for we also have to introduce and.Because we have to take products of integrals that define,we cannot use and in both integrals.As for we can now derive from(5.1)and(5.2)that if and only ifWe simplify the right-hand side term by term.Thefirst term,that is related to becomes The second term(related to)becomesAdding these two terms gives uswhich demonstrates that indeed also is invariant under rotations.Similar calculus shows that invariance under reflections also holds.From Equation(5.5)four functions of third order and one function of both second and third order can be derived that are invariant under both rotations and reflecting,namelyandwithThe last polynomial that is invariant under both rotations and reflections consists of both second and third order moments and is given bywith and as above.To these six invariants we can add a seventh one,which is only invariant under rotations and changes sign under reflections.It is given bySince we want to include reflections as well in our set of invariant transformations we will use instead of in our approach.From now on,we will identify with.We observe that all possible linear combinations of these invariants are invariant under proper orthogonal transformations and translations.Therefore we can call these seven invariants also invariant generators.。

数学中的对称性研究数学是一门关注于数字、结构、变化和空间等概念的学科，而对称性则是数学中一个重要的概念。

在数学中，对称性是指某个物体或者某个系统在某种变换下保持不变的特性。

对称性的研究不仅在数学本身具有广泛的应用，也在解决实际问题中起到关键的作用。

本文将探讨数学中对称性的研究以及其在不同领域中的应用。

1. 对称性的概念及分类对称性是指某个物体或系统经过某种变换后能够保持不变。

在数学中，对称性可以分为几个类别。

首先是平移对称性，即物体在平移后仍然保持不变；其次是旋转对称性，即物体在旋转一定角度后仍然保持不变；还有镜像对称性，即将物体沿着某条线对折后两边完全相等。

2. 对称性与数学定义对称性在数学中有着严格的定义。

对称性可以通过变换矩阵来描述，例如在平面几何中，对称性可以通过矩阵乘法来实现。

此外，对称性还与数学中的群论、张量等概念联系紧密。

3. 对称性在几何中的应用对称性在几何中有着广泛的应用。

在二维空间中，对称性能帮助我们研究各种图形的性质，比如正方形、圆等具有对称性的图形。

在三维空间中，对称性对于解决空间几何问题非常重要，例如研究立方体、球体等具有对称性的立体。

4. 对称性在代数中的应用对称性在代数中同样具有重要的应用。

代数中的对称群是研究对称性的一个重要工具，它对应于物体在各种变换下可以保持不变的所有变换的集合。

对称群在研究各种代数结构、群、环等方面都具有重要作用。

5. 对称性在物理学中的应用对称性在物理学中也起着关键的作用。

物理学中的许多基本定律和原理都与对称性有关，例如守恒定律，即能量、动量、角动量等在空间变换下具有不变性。

对称性在物理学中的应用也涉及到相对论、量子力学等领域。

6. 对称性在其他领域中的应用除了上述几个领域，对称性在其他领域中也有广泛的应用。

在密码学中，对称密钥加密算法可以保证数据的安全性；在图像处理中，对称性能够帮助我们实现图像的压缩和加密；在信息科学中，对称性能够帮助我们提高数据传输的效率。

理论物理电子书理论物理-电子书0000理论物理基础彭桓武Simons B. Concepts in theoretical physics (Cambridge lecture notes, 2002)(T)(273s)Principles of Modern Physics-N E I L A S H B Y-S T A N L E Y C . M I L L E R-University of ColoradoFUNDAMENTALS OF physics-J. Richard Christman0-mathematical physics李代数李超代数及在物理学中的应用孙洪洲群论.及其在粒子物理学中的应用,.高崇寿.1992群论及其在固体物理中的应用【徐婉棠,喀兴林】群论及其在物理中的应用(马中骐)群论习题精解+(马中骐)群论与量子力学物理系群论讲义物理学中的群论(上册).陶瑞宝物理学中的群论基础 A W 约什Geometry_Topology_and Physics-NakaharaGeometry+and+Physics+(Jürgen Jost)Lee J.M. Differential and physical geometry (draft)(721s)数学物理中的微分几何与拓扑学_汪容.浙大版.1998Differential Geometry, Analysis and Physics 。

Jeffrey M. Lee微分几何学及其在物理学中的应用物理学家用微分几何-侯伯宇-侯伯元物理中的张量孙志铭Arnold vol1，2A Guided Tour Of Mathematical Physics (By Roel Snieder, Department Of Geophysics, Utrecht UniversAbramovitz M., Stegun I.A. (eds.) Handbook of mathematical functions (10ed., NBS, 1972)(T)(1037s)Academic Press, Methods of Modern Mathematical Physics -- Vol. 1, Functional AnCourant, Hilbert - Methods of Mathematical Physics Vol. 1 ENG (578p)Introduction+to+Applied+Mathematics-GilbertStrangIntroduction+to+Mathematical+Physics+(Laurie+Cosse y)Math_method_for_Phy_Ken Riley, Michael Hobson and Stephen Bence Cambridge, 1997Szekeres, Peter - A Course in Modern Mathematical Physics - Groups, Hilbert Spaces and Differenti数学物理方法梁昆淼数学物理方法（R.+柯朗、D.+希尔伯特）数学物理方法吴崇试数学物理学中的微分形式数学物理中的几何方法(B·F·舒茨)特殊函数概论王竹溪物理学中的非线性方程刘式适物理学中的数学方法（李政道）1-Classical Mechanics and Fluid MechanicsClassical Mechanics - Goldstein古典力学(戈德斯坦)Hand, Finch Analytical Mechanics (Cup, 1998)(T)(590S)Structure and Interpretation of Classical Mechanics-Gerald Jay Sussman and Jack Wisdom with Meinhard E. Mayer -MIT Press经典力学张启仁2-Statistical And Thermal Physics理论物理学基础教程丛书统计物理学(苏汝铿)量子统计力学 by 张先蔚量子统计物理学(北京大学物理系)统计物理现代教程（上、下册）(雷克)统计物理中的蒙特卡罗模拟方法（含有热力学，难度适中）Reif. Fundamentals of Statistical And Thermal PhysicsBratteli O , Robinson D W Vol 1 Operator Algebras And Quantum Statistical Mechanics (2Ed , SpringHuang K. Statistical mechanics (2ed., Wiley, 1987)(T)(506s)Reichl L.E. A modern course in statistical physics (2ed, Wiley, 1998)(T)(840s)3-Electrodynamics赵凯华-电磁学上宇宙电动力学_阿尔芬引力论和宇宙论：广义相对论的原理和应用-温伯格相对论物理宇宙学讲义俞允强天体物理学【李宗伟、肖兴华】+时空的大尺度结构（原版）- 霍金简明天文学手册-刘步林广义相对论引论广义相对论dirac广义相对论（刘辽）大众天文学【法】弗拉马利翁Jackson J.D. Classical electrodynamics (3ed., Wiley,1999)(ISBN 047130932X)(600dpi)(K)(T)(833s).d（研究生程度的必读教材）JACKSON经典电动力学（上册）（经典之作）J.A.Wheeler E.F.Taylor Spacetime_PhysicsHerbert Neff - Introductory ElectromagneticsElectromagnetics (Rothwell & Cloud, 2001 CRC Press)Electricity+and+Magnetism-MITcourseCohen-Tannoudji Introduction to quantum electrodynamicsBuch_John Wiley. Sons_An Introduction to Modern Cosmology4-Optics（光学经典，全面、很厚，很难）光学原理上册、下册(m.玻恩 e.沃耳夫)Bass M , Et Al (Eds) Osa Handbook Of Optics, Vol 1 (Mgh, 1995)(1606s)Goodman - Geometrical Optics--p1628 - cambridgeWiley,.Modern.Nonlinear.Optics.Part.I.Advances.in. Chemical.Physics.Volume.119.(2001),.2Ed5-Quantum MechanicsClassical and Quantum ChaosCohen-Tannoudji Quantum Mechanics, Vol 1Galindo A., Pascual P. Quantum mechanics I (Springer,1990)(ISBN 0387514066)(T) (431s)量子系统中的几何相位-A.Bohm等Jack_Simons_-_Quantum MechanicsJohn_Norbury_-_Quantum_Mechanics_for_Undergraduate sMathematics+of+Quantum+Computation-Goong.ChenModern Quantum Mechanics And Solutions For The Exercices (J J Sakurai)Nuclear And Particle Physics-NielsWaletPhillips.-.Introduction.to.quantum.mechanics.(2003 )(T)(284s)Quantum Mechanics - Concepts and Applications-Tarun.BiswasShankar-Principles Of Quantum Mechanics 2nd EditionThe Basic Tools Of Quantum MechanicsThe+Physics+of+Phase+Transitions-P. Papon J. Leblond P.H.E. MeijerLecture Notes in Physics-Time+in+Quantum++Mechanics+1J.G. Muga.R. Sala Mayato?I.L. Egusquiza (Eds.)Zaarur E. Schaum's Outline of Quantum Mechanics.. Including Hundreds of Solved Problems (Schaum,1喀兴林-高等量子力学席夫量子力学-繁体中文版量子力学(Messiah)Vol1量子力学（卷I）.曾谨言量子力学“天龙八部”-张永德量子力学+(苏汝铿)量子力学Fermi量子力学讲义（张永德）量子力学原理(狄拉克)量子论的物理原理量子论与原子结构-吴大遒量子物理学导论(MIT)物理学引论Vol4-A.P.French By Tsungp Lee量子物理-赵凯华高等量子力学-张永德6-Field theory量子场论-温伯格1,2,3An Introduction to Quantum FieldTheory(Peskin,Schroeder)(full and revised)Banks,Modern+Quantum+Field+Theory--A+Concise+Intro ductionField.theory,.Roman.S..(2ed.,.Springer,.2005)Giachetta,Advanced+Classical+Field+Theory经典场论Kleinert H. Quantum field theory and particle physicsItep-PARTICLE-PHYSICS-and-field-theory场论I-M.A.ShifmanQuantum Field Theory R ClarksonQuantum+Field+Theory+(M.Srednicki) Quantum+Field+Theory-David McMahon Sundaresan. Handbook of particle physics (CRC, 2001)(T)(439 Tong-Quantum Field Theory Zinn-Justin. Quantum field theory and critical phenomena (1ed., 1989)(K)(150dpi)(T)(924s) 北大2005量子场论讲义（赵光达）量子场论-清华王青讲义规范场论（胡瑶光）粒子和场【卢里着,董明德等译】量子场论(上)【依捷克森,祖柏尔着,杜东生等译】量子场论A.Zee量子场论F.Mandl-G.Shaw量子场论LEWIS-H.RYDER实时统计场论-徐宏华统计物理学中的量子场论方法-Abrikosov微分几何-统一场论超弦理论导论Elias-Kiritsis张秋光《场论》上册朱洪元+量子场论On Wittens 3-manifold Invariants-Kevin WalkerLectures on Topological Quantum Field Theory-J. M. F. Labastidaa-Carlos LozanobGEOMETRY OF 2D TOPOLOGICAL FIELD THEORIES-Boris DUBROVIN-SISSA, TriesteDunne(1999)-Aspects of Chern-Simons Theorylabastida(1998)-Chern-Simons Gauge Theory-- Ten Years After7-Solid state physics（非常好的书）固体物理学（黄昆）固体物理导论C.KittelMechanics Of Solids-Bela I. Sandor-University of Wisconsin-MadisonKleinert H. Gauge fields in condensed matter physics part1(T)(252s)Ashcroft, Neil W, Mermin, David N - Solid State PhysicsAltland & Simons - Concepts Of Theoretical Solid State Physics。