Compatibility between superplasticizer admixtures and cements with mineral additions

混凝土外加剂的探讨摘要：在混凝土、砂浆或净浆的制备过程中，掺入不超过水泥用量5%(特殊情况除外)，能对混凝土、砂浆或净浆的正常性能要求而改性的一种产品，称为混凝土外加剂。

外加剂按其所对于应的功能不同分为减水剂、引气剂、僧水剂、促凝剂、早强剂、缓凝剂、发气剂、气泡剂、灌浆剂、着色列、超塑化剂、保水剂、粘结列、阻锈剂，喷射混纂土外加剂等。

下面就混凝土外加剂使用中应注意的几个问题进行了探讨。

关键字：混凝土；外加剂；相容性abstract: in the fabrication process of concrete, mortar or net plasma, a kind of modification product that can ensure the normal performance of concrete, mortar or net plasma and whose mixing is no more than 5% of cement content (special circumstances except),is called concrete admixtures. according to different functions, the admixture can be divided into water reducing agent, air-entraining agent, water increasing agent, coagulant agent, early strength agent, retarder, foaming agent, bubble agent, grouting agent, shading agent, superplasticizer, absorbent agent, the cementation agent, corrosion inhibitors, spraying mixing soil admixtures, etc. here will discusses some problems in the application of concrete admixtures.keywords: concrete; admixtures; compatibility中图分类号：f407.9文献标识码a 文章编号1 选择混凝土外加剂和检验外加剂产品的质量用户单位可可以根据工程设计、施工要求和技术指标进行比较，选择适合的生产厂家混凝土外加剂产品。

[官⽅]BeyondCompare ⾥⾯⼆进制⽐较的含义.Content ComparisonsActions > Compare Contents In the Actions menu, the Compare Contents command performs content comparisons on the selected pairs of files to determine if they match.Content comparison methodsCRC comparison compares files using their CRC values.Binary comparison compares files byte-by-byte.Rules-based comparison compares files based on their associations. It allows you to define unimportant differences, such as changes in whitespace or source code comments. A rules-based comparison can also ignore differences in file encoding or line endings.Results of content comparisonsCRC and Binary comparisons return one of these results:Binary same The files are exactly the same.Binary differences At least one byte is different between the files.Rules-based comparisons have a built-in binary comparison and return one of these results:Binary sameThe files are exactly the same.Rules-based same The files have binary differences, such as character encoding, that can be ignored.Unimportantdifferences A rules-based comparison found only unimportant differences.Importantdifferences A rules-based comparison found important differences.When content comparisons are performedContent comparisons are performed:·when a folder session is loaded, and its session settings call for automatic content comparisons·explicitly, when the Compare Contents command is used·when a pair of files is opened in a file session。

八年级英语议论文论证方法单选题40题1. In the essay, the author mentions a story about a famous scientist to support his idea. This is an example of _____.A.analogyB.exampleparisonD.metaphor答案：B。

本题主要考查论证方法的辨析。

选项A“analogy”是类比；选项B“example”是举例；选项C“comparison”是比较；选项D“metaphor”是隐喻。

文中提到一个关于著名科学家的故事来支持观点，这是举例论证。

2. The writer uses the experience of his own life to prove his point. This kind of method is called _____.A.personal storyB.example givingC.case studyD.reference答案：B。

选项A“personal story”个人故事范围较窄；选项B“example giving”举例；选项C“case study”案例分析；选项D“reference”参考。

作者用自己的生活经历来证明观点，这是举例论证。

3. The author cites several historical events to strengthen his argument. What is this method?A.citing factsB.giving examplesC.making comparisonsing analogies答案：B。

选项A“citing facts”引用事实，历史事件可以作为例子，所以是举例论证；选项B“giving examples”举例；选项C“making comparisons”比较；选项D“using analogies”使用类比。

Compressive samplingEmamnuel J.Candès∗Abstract.Conventional wisdom and common practice in acquisition and reconstruction of images from frequency data follow the basic principle of the Nyquist density sampling theory. This principle states that to reconstruct an image,the number of Fourier samples we need to acquire must match the desired resolution of the image,i.e.the number of pixels in the image. This paper surveys an emerging theory which goes by the name of“compressive sampling”or “compressed sensing,”and which says that this conventional wisdom is inaccurate.Perhaps surprisingly,it is possible to reconstruct images or signals of scientific interest accurately and sometimes even exactly from a number of samples which is far smaller than the desired resolution of the image/signal,e.g.the number of pixels in the image.It is believed that compressive sampling has far reaching implications.For example,it suggests the possibility of new data acquisition protocols that translate analog information into digital form with fewer sensors than what was considered necessary.This new sampling theory may come to underlie procedures for sampling and compressing data simultaneously.In this short survey,we provide some of the key mathematical insights underlying this new theory,and explain some of the interactions between compressive sampling and otherfields such as statistics,information theory,coding theory,and theoretical computer science. Mathematics Subject Classification(2000).Primary00A69,41-02,68P30;Secondary62C65.pressive sampling,sparsity,uniform uncertainty principle,underdertermined systems of linear equations, 1-minimization,linear programming,signal recovery,error cor-rection.1.IntroductionOne of the central tenets of signal processing is the Nyquist/Shannon sampling theory: the number of samples needed to reconstruct a signal without error is dictated by its bandwidth–the length of the shortest interval which contains the support of the spectrum of the signal under study.In the last two years or so,an alternative theory of“compressive sampling”has emerged which shows that super-resolved signals and images can be reconstructed from far fewer data/measurements than what is usually considered necessary.The purpose of this paper is to survey and provide some of the key mathematical insights underlying this new theory.An enchanting aspect of compressive sampling it that it has significant interactions and bearings on somefields in the applied sciences and engineering such as statistics,information theory,coding ∗The author is partially supported by an NSF grant CCF–515362.Proceedings of the International Congressof Mathematicians,Madrid,Spain,2006©2006European Mathematical Society2Emmanuel J.Candès theory,theoretical computer science,and others as well.We will try to explain these connections via a few selected examples.From a general viewpoint,sparsity and,more generally,compressibility has played and continues to play a fundamental role in manyfields of science.Sparsity leads to efficient estimations;for example,the quality of estimation by thresholding or shrink-age algorithms depends on the sparsity of the signal we wish to estimate.Sparsity leads to efficient compression;for example,the precision of a transform coder depends on the sparsity of the signal we wish to encode[24].Sparsity leads to dimensionality reduction and efficient modeling.The novelty here is that sparsity has bearings on the data acquisition process itself,and leads to efficient data acquisition protocols.In fact,compressive sampling suggests ways to economically translate analog data into already compressed digital form[20],[7].The key word here is“economically.”Everybody knows that because typical signals have some structure,they can be com-pressed efficiently without much perceptual loss.For instance,modern transform coders such as JPEG2000exploit the fact that many signals have a sparse represen-tation in afixed basis,meaning that one can store or transmit only a small number of adaptively chosen transform coefficients rather than all the signal samples.The way this typically works is that one acquires the full signal,computes the complete set of transform coefficients,encode the largest coefficients and discard all the others.This process of massive data acquisition followed by compression is extremely wasteful (one can think about a digital camera which has millions of imaging sensors,the pixels,but eventually encodes the picture on a few hundred kilobytes).This raises a fundamental question:because most signals are compressible,why spend so much ef-fort acquiring all the data when we know that most of it will be discarded?Wouldn’t it be possible to acquire the data in already compressed form so that one does not need to throw away anything?“Compressive sampling”also known as“compressed sensing”[20]shows that this is indeed possible.This paper is by no means an exhaustive survey of the literature on compressive sampling.Rather this is merely an account of the author’s own work and thinking in this area which also includes a fairly large number of references to other people’s work and occasionally discusses connections with these works.We have done our best to organize the ideas into a logical progression starting with the early papers which launched this subject.Before we begin,we would like to invite the interested reader to also check the article[17]by Ronald DeV ore–also in these proceedings–for a complementary survey of thefield(Section5).2.Undersampled measurementsConsider the general problem of reconstructing a vector x∈R N from linear mea-surements y about x of the formy k= x,ϕk ,k=1,...,K,or y= x.(2.1)Compressive sampling3 That is,we acquire information about the unknown signal by sensing x against K vectorsϕk∈R N.We are interested in the“underdetermined”case K N,where we have many fewer measurements than unknown signal values.Problems of this type arise in a countless number of applications.In radiology and biomedical imaging for instance,one is typically able to collect far fewer measurements about an image of interest than the number of unknown pixels.In wideband radio frequency signal analysis,one may only be able to acquire a signal at a rate which is far lower than the Nyquist rate because of current limitations inAnalog-to-Digital Converter technology. Finally,gene expression studies also provide examples of this kind.Here,one would like to infer the gene expression level of thousands of genes–that is,the dimension N of the vector x is in the thousands–from a low number of observations,typically in the tens.Atfirst glance,solving the underdertermined system of equations appears hopeless, as it is easy to make up examples for which it clearly cannot be done.But suppose now that the signal x is compressible,meaning that it essentially depends on a number of degrees of freedom which is smaller than N.For instance,suppose our signal is sparse in the sense that it can be written either exactly or accurately as a superposition of a small number of vectors in somefixed basis.Then this premise radically changes the problem,making the search for solutions feasible.In fact,accurate and sometimes exact recovery is possible by solving a simple convex optimization problem.2.1.A nonlinear sampling theorem.It might be best to consider a concrete example first.Suppose here that one collects an incomplete set of frequency samples of a discrete signal x of length N.(To ease the exposition,we consider a model problem in one dimension.The theory extends easily to higher dimensions.For instance,we could be equally interested in the reconstruction of2-or3-dimensional objects from undersampled Fourier data.)The goal is to reconstruct the full signal f given only K samples in the Fourier domainy k=1√NN−1t=0x t e−i2πωk t/N,(2.2)where the‘visible’frequenciesωk are a subset (of size K)of the set of all frequencies {0,...,N−1}.Sensing an object by measuring selected frequency coefficients is the principle underlying Magnetic Resonance Imaging,and is common in manyfields of science,including Astrophysics.In the language of the general problem(2.1),the sensing matrix is obtained by sampling K rows of the N by N discrete Fourier transform matrix.We will say that a vector x is S-sparse if its support{i:x i=0}is of cardinality less or equal to S.Then Candès,Romberg and Tao[6]showed that one could almost alwaysrecover the signal x exactly by solving the convex program1( ˜x1:=Ni=1|˜x i|)(P1)min˜x∈R N ˜x1subject to ˜x=y.(2.3)1(P1)can even be recast as a linear program[3],[15].4Emmanuel J.Candès Theorem2.1([6]).Assume that x is S-sparse and that we are given K Fouriercoefficients with frequencies selected uniformly at random.Suppose that the number of observations obeysK≥C·S·log N.(2.4) Then minimizing 1reconstructs x exactly with overwhelming probability.In details, if the constant C is of the form22(δ+1)in(2.4),then the probability of success exceeds1−O(N−δ).Thefirst conclusion is that one suffers no information loss by measuring just aboutany set of K frequency coefficients.The second is that the signal x can be exactlyrecovered by minimizing a convex functional which does not assume any knowledgeabout the number of nonzero coordinates of x,their locations,and their amplitudeswhich we assume are all completely unknown a priori.While this seems to be a great feat,one could still ask whether this is optimal,or whether one could do with even fewer samples.The answer is that in general,we cannot reconstruct S-sparse signals with fewer samples.There are examplesfor which the minimum number of samples needed for exact reconstruction by anymethod,no matter how intractable,must be about S log N.Hence,the theorem istight and 1-minimization succeeds nearly as soon as there is any hope to succeed byany algorithm.The reader is certainly familiar with the Nyquist/Shannon sampling theory and onecan reformulate our result to establish simple connections.By reversing the roles oftime and frequency in the above example,we can recast Theorem1as a new nonlinearsampling theorem.Suppose that a signal x has support in the frequency domainwith B=| |.If is a connected set,we can think of B as the bandwidth of x.Ifin addition the set is known,then the classical Nyquist/Shannon sampling theoremstates that x can be reconstructed perfectly from B equally spaced samples in the timedomain2.The reconstruction is simply a linear interpolation with a“sinc”kernel.Now suppose that the set ,still of size B,is unknown and not necessarily con-nected.In this situation,the Nyquist/Shannon theory is unhelpful–we can onlyassume that the connected frequency support is the entire domain suggesting thatall N time-domain samples are needed for exact reconstruction.However,Theo-rem2.1asserts that far fewer samples are necessary.Solving(P1)will recover x perfectly from about B log N time samples.What is more,these samples do not haveto be carefully chosen;almost any sample set of this size will work.Thus we have anonlinear analog(described as such since the reconstruction procedure(P1)is non-linear)to Nyquist/Shannon:we can reconstruct a signal with arbitrary and unknown frequency support of size B from about B log N arbitrarily chosen samples in the time domain.Finally,we would like to emphasize that our Fourier sampling theorem is onlya special instance of much more general statements.As a matter of fact,the results2For the sake of convenience,we make the assumption that the bandwidth B divides the signal length N evenly.Compressive sampling5 extend to a variety of other setups and higher dimensions.For instance,[6]shows how one can reconstruct a piecewise constant(one or two-dimensional)object from incomplete frequency samples provided that the number of jumps(discontinuities) obeys the condition above by minimizing other convex functionals such as the total variation.2.2.Background.Now for some background.In the mid-eighties,Santosa and Symes[44]had suggested the minimization of 1-norms to recover sparse spike trains, see also[25],[22]for early results.In the last four years or so,a series of papers[26], [27],[28],[29],[33],[30]explained why 1could recover sparse signals in some special setups.We note though that the results in this body of work are very different than the sampling theorem we just introduced.Finally,we would like to point out important connections with the literature of theoretical computer science.Inspired by[37],Gilbert and her colleagues have shown that one could recover an S-sparse signal with probability exceeding1−δfrom S·poly(log N,logδ)frequency samples placed on special equispaced grids[32].The algorithms they use are not based on optimization but rather on ideas from the theory of computer science such as isolation, and group testing.Other points of connection include situations in which the set of spikes are spread out in a somewhat even manner in the time domain[22],[51].2.3.Undersampling structured signals.The previous example showed that the structural content of the signal allows a drastic“undersampling”of the Fourier trans-form while still retaining enough information for exact recovery.In other words,if one wanted to sense a sparse object by taking as few measurements as possible,then one would be well-advised to measure randomly selected frequency coefficients.In truth, this observation triggered a massive literature.To what extent can we recover a com-pressible signal from just a few measurements.What are good sensing mechanisms? Does all this extend to object that are perhaps not sparse but well-approximated by sparse signals?In the remainder of this paper,we will provide some answers to these fundamental questions.3.The Mathematics of compressive sampling3.1.Sparsity and incoherence.In all what follows,we will adopt an abstract and general point of view when discussing the recovery of a vector x∈R N.In practical instances,the vector x may be the coefficients of a signal f∈R N in an orthonormalbasisf(t)=Ni=1x iψi(t),t=1,...,N.(3.1)For example,we might choose to expand the signal as a superposition of spikes(the canonical basis of R N),sinusoids,B-splines,wavelets[36],and so on.As a side6Emmanuel J.Candès note,it is not important to restrict attention to orthogonal expansions as the theory and practice of compressive sampling accommodates other types of expansions.For example,x might be the coefficients of a digital image in a tight-frame of curvelets[5]. To keep on using convenient matrix notations,one can write the decomposition(3.1)as x= f where is the N by N matrix with the waveformsψi as rows or equivalently,f= ∗x.We will say that a signal f is sparse in the -domain if the coefficient sequence is supported on a small set and compressible if the sequence is concentrated near a small set.Suppose we have available undersampled data about f of the same form as beforey= f.Expressed in a different way,we collect partial information about x via y= x where = ∗.In this setup,one would recover f byfinding–among all coefficient sequences consistent with the data–the decomposition with minimum 1-normmin ˜x1such that ˜x=y.Of course,this is the same problem as(2.3),which justifies our abstract and general treatment.With this in mind,the key concept underlying the theory of compressive sampling is a kind of uncertainty relation,which we explain next.3.2.Recovery of sparse signals.In[7],Candès and Tao introduced the notion of uniform uncertainty principle(UUP)which they refined in[8].The UUP essentially states that the K×N sensing matrix obeys a“restricted isometry hypothesis.”Let T,T⊂{1,...,N}be the K×|T|submatrix obtained by extracting the columns of corresponding to the indices in T;then[8]defines the S-restricted isometry constantδS of which is the smallest quantity such that(1−δS) c 22≤ T c 22≤(1+δS) c 22(3.2)for all subsets T with|T|≤S and coefficient sequences(c j)j∈T.This property es-sentially requires that every set of columns with cardinality less than S approximately behaves like an orthonormal system.An important result is that if the columns of the sensing matrix are approximately orthogonal,then the exact recovery phenomenon occurs.Theorem3.1([8]).Assume that x is S-sparse and suppose thatδ2S+δ3S<1or, better,δ2S+θS,2S<1.Then the solution x to(2.3)is exact,i.e.,x =x.In short,if the UUP holds at about the level S,the minimum 1-norm reconstruction is provably exact.Thefirst thing one should notice when comparing this result with the Fourier sampling theorem is that it is deterministic in the sense that it does not involve any probabilities.It is also universal in that all sufficiently sparse vectorsCompressive sampling7 are exactly reconstructed from x.In Section3.4,we shall give concrete examples of sensing matrices obeying the exact reconstruction property for large values of the sparsity level,e.g.for S=O(K/log(N/K)).Before we do so,however,we would like to comment on the slightly better version δ2S+θS,2S<1,which is established in[10].The numberθS,S for S+S ≤N is called the S,S -restricted orthogonality constants and is the smallest quantity such that| T c, T c |≤θS,S · c2 c2(3.3)holds for all disjoint sets T,T ⊆{1,...,N}of cardinality|T|≤S and|T |≤S . ThusθS,S is the cosine of the smallest angle between the two subspaces spanned by the columns in T and T .Small values of restricted orthogonality constants that disjoint subsets of covariates span nearly orthogonal subspaces.Theδ2S+θS,2S<1is better thanδ2S+δ3S<1since it is not hard to see thatS+S−δS ≤θS,S ≤δS+S for S ≥S[8,Lemma1.1].Finally,now that we have introduced all the quantities needed to state our recovery theorem,we would like to elaborate on the conditionδ2S+θS,2S<1.Suppose thatδ2S=1which may indicate that there is a matrix T1∪T2with2S columns (|T1|=S,|T2|=S)that is rank-deficient.If this is the case,then there is a pair (x1,x2)of nonvanishing vectors with x1supported on T1and x2supported on T2 obeying(x1−x2)=0⇐⇒ x1= x2.In other words,we have two very distinct S-sparse vectors which are indistinguishable. This is why any method whatsoever needsδ2S<1.For,otherwise,the model is not identifiable to use a terminology borrowed from the statistics literature.With this in mind,one can see that the conditionδ2S+θS,2S<1is only slightly stronger than this identifiability condition.3.3.Recovery of compressible signals.In general,signals of practical interest may not be supported in space or in a transform domain on a set of relatively small size. Instead,they may only be concentrated near a sparse set.For example,a commonly discussed model in mathematical image or signal processing assumes that the coef-ficients of elements taken from a signal class decay rapidly,typically like a power law.Smooth signals,piecewise signals,images with bounded variations or bounded Besov norms are all of this type[24].A natural question is how well one can recover a signal that is just nearly sparse. For an arbitrary vector x in R N,denote by x S its best S-sparse approximation;that is,x S is the approximation obtained by keeping the S largest entries of x and setting the others to zero.It turns out that if the sensing matrix obeys the uniform uncertainty principle at level S,then the recovery error is not much worse than x−x S 2.8Emmanuel J.Candès Theorem3.2([9]).Assume that x is S-sparse and suppose thatδ3S+δ4S<2.Then the solution x to(2.3)obeysx∗−x2≤C·x−x S1√S.(3.4)For reasonable values ofδ4S,the constant in(3.4)is well behaved;e.g.C≤8.77for δ4S=1/5.Suppose further thatδS+2θS,S+θ2S,S<1,we also havex∗−x1≤C x−x S1,(3.5)for some positive constant C.Again,the constant in(3.5)is well behaved.Roughly speaking,the theorem says that minimizing 1recovers the S-largest entries of an N-dimensional unknown vector x from K measurements only.As a side remark,the 2-stability result(3.4)appears explicitly in[9]while the‘ 1instance optimality’(3.5)is implicit in[7]although it is not stated explicitely.For example,it follows from Lemma2.1–whose hypothesis holds because of Lemma2.2.in[8]–in that paper.Indeed,let T be the set where x takes on its S-largest values.Then Lemma2.1in[7]gives x∗·1T c 1≤4 x−x S 1and,therefore, (x∗−x)·1T c 1≤5 x−x S 1.We conclude by observing that on T we have(x∗−x)·1T1≤√S (x∗−x)·1T 2≤C x−x S 1,where the last inequality follows from(3.4).For information,a more direct argument yields better constants.To appreciate the content of Theorem3.2,suppose that x belongs to a weak- p ball of radius R.This says that if we rearrange the entries of x in decreasing order of magnitude|x|(1)≥|x|(2)≥···≥|x|(N),the i th largest entry obeys|x|(i)≤R·i−1/p,1≤i≤N.(3.6) More prosaically,the coefficient sequence decays like a power-law and the parame-ter p controls the speed of the decay:the smaller p,the faster the decay.Classical calculations then show that the best S-term approximation of an object x∈w p(R)obeysx−x S2≤C2·R·S1/2−1/p(3.7) in the 2norm(for some positive constant C2),andx−x S1≤C1·R·S1−1/pin the 1-norm.For generic elements obeying(3.6),there are no fundamentally better estimates available.Hence,Theorem3.2shows that with K measurements only,we can achieve an approximation error which is as good as that one would obtain by knowing everything about the signal and selecting its S-largest entries.Compressive sampling 93.4.Random matrices.Presumably all of this would be interesting if one could design a sensing matrix which would allow us to recover as many entries of x as possible with as few as K measurements.In the language of Theorem 3.1,we would like the condition δ2S +θS,2S <1to hold for large values of S ,ideally of the order of K .This poses a design problem.How should one design a matrix –that is to say,a collection of N vectors in K dimensions –so that any subset of columns of size about S be about orthogonal?And for what values of S is this possible?While it might be difficult to exhibit a matrix which provably obeys the UUP for very large values of S ,we know that trivial randomized constructions will do so with overwhelming probability.We give an example.Sample N vectors on the unit sphere of R K independently and uniformly at random.Then the condition of Theorems 3.1and 3.2hold for S =O(K/log (N/K))with probability 1−πN where πN =O(e −γN )for some γ>0.The reason why this holds may be explained by some sort of “blessing of high-dimensionality.”Because the high-dimensional sphere is mostly empty,it is possible to pack many vectors while maintaining approximate orthogonality.•Gaussian measurements.Here we assume that the entries of the K by N sensing matrix are independently sampled from the normal distribution with mean zero and variance 1/K .Then ifS ≤C ·K/log (N/K),(3.8)S obeys the condition of Theorems 3.1and 3.2with probability 1−O(e −γN )for some γ>0.The proof uses known concentration results about the singular values of Gaussian matrices [16],[45].•Binary measurements.Suppose that the entries of the K by N sensing matrix are independently sampled from the symmetric Bernoulli distribution P ( ki =±1/√K)=1/2.Then it is conjectured that the conditions of Theorems 3.1and 3.2are satisfied with probability 1−O(e −γN )for some γ>0provided that S obeys (3.8).The proof of this fact would probably follow from new concentration results about the smallest singular value of a subgaussian matrix[38].Note that the exact reconstruction property for S -sparse signals and (3.7)with S obeying (3.8)are known to hold for binary measurements [7].•Fourier measurements.Suppose now that is a partial Fourier matrix obtained by selecting K rows uniformly at random as before,and renormalizing the columns so that they are unit-normed.Then Candès and Tao [7]showed that Theorem 3.1holds with overwhelming probability if S ≤C ·K/(log N)6.Recently,Rudelson and Vershynin [43]improved this result and established S ≤C ·K/(log N)4.This result is nontrivial and use sophisticated techniques from geometric functional analysis and probability in Banach spaces.It is conjectured that S ≤C ·K/log N holds.10Emmanuel J.Candès •Incoherent measurements.Suppose now that is obtained by selecting K rows uniformly at random from an N by N orthonormal matrix U and renormalizing the columns so that they are unit-normed.As before,we could think of U as the matrix ∗which maps the object from the to the -domain.Then the arguments used in[7],[43]to prove that the UUP holds for incomplete Fourier matrices extend to this more general situation.In particular,Theorem3.1holds with overwhelming probability provided thatS≤C·1μ2·K(log N)4,(3.9)whereμ:=√N max i,j|U i,j|(observe that for the Fourier matrix,μ=1which gives the result in the special case of the Fourier ensemble above).WithU= ∗,μ:=√maxi,j| ϕi,ψj |(3.10)which is referred to as the mutual coherence between the measurement basis and the sparsity basis [27],[28].The greater the incoherence of the measurement/sparsity pair( , ),the smaller the number of measurements needed.In short,one can establish the UUP for a few interesting random ensembles and we expect that in the future,many more results of this type will become available.3.5.Optimality.Before concluding this section,it is interesting to specialize our recovery theorems to selected measurement ensembles now that we have established the UUP for concrete values of S.Consider the Gaussian measurement ensemble in which the entries of are i.i.d.N(0,1/K).Our results say that one can recover any S-sparse vector from a random projection of dimension about O(S·log(N/S)),see also[18].Next,suppose that x is taken from a weak- p ball of radius R for some 0<p<1,or from the 1-ball of radius R for p=1.Then we have shown that for all x∈w p(R)x −x2≤C·R·(K/log(N/K))−r,r=1/p−1/2,(3.11) which has also been proven in[20].An important question is whether this is op-timal.In other words,can wefind a possibly adaptive set of measurements and a reconstruction algorithm that would yield a better bound than(3.11)?By adaptive, we mean that one could use a sequential measurement procedure where at each stage, one would have the option to decide which linear functional to use next based on the data collected up to that stage.It proves to be the case that one cannot improve on(3.11),and we have thus identified the optimal performance.Fix a class of object F and let E K(F)be the best reconstruction error from K linear measurementsE K(F)=inf supf∈F f−D(y)2,y= f,(3.12)where the infimum is taken over all set of K linear functionals and all reconstruction algorithms D.Then it turns out E K(F)nearly equals the Gelfand numbers of a class F defined asd K(F)=infV {supf∈FP V f :codim(V)<K},(3.13)where P V is the orthonormal projection on the subspace V.Gelfand numbers play animportant role in approximation theory,see[40]for more information.If F=−Fand F=F+F≤c F F,then d K(F)≤E K(F)≤c F d K(F).Note that c F=21/p in the case where F is a weak- p ball.The thing is that we know the approximatevalues of the Gelfand numbers for many classes of interest.Suppose for examplethat F is the 1-ball of radius R.A seminal result of Kashin[35]and improved byGarnaev and Gluskin[31]shows that for this ball,the Gelfand numbers obeyC1·R·log(N/K)+1K≤d k(F)≤C2·R·log(N/K)+1K,(3.14)where C1,C2are universal constants.Gelfand numbers are also approximately knownfor weak- p balls as well;the only difference is that((log(N/K)+1)/K)r substitutes((log(N/K)+1)/K)1/2.Hence,Kashin,Garnaev and Gluskin assert that with K measurements,the minimal reconstruction error(3.12)one can hope for is boundedbelow by a constant times(K/log(N/K))−r.Kashin’s arguments[35]also usedprobabilistic functionals which establish the existence of recovery procedures forwhich the reconstruction error is bounded above by the right-hand side of(3.14).Similar types of recovery have also been known to be possible in the literature oftheoretical computer science,at least in principle,for certain types of random mea-surements[1].In this sense,our results–specialized to Gaussian measurements–are optimalfor weak- p norms.The novelty is that the information about the object can beretrieved from random coefficients by minimizing a simple linear program(2.3),andthat the decoding algorithm adapts automatically to the weak- p signal class,withoutknowledge thereof.Minimizing the 1-norm is adaptive and nearly gives the bestpossible reconstruction error simultaneously over a wide range of sparse classes ofsignals;no information about p and the radius R are required.4.Robust compressive samplingIn any realistic application,we cannot expect to measure x without any error,and wenow turn our attention to the robustness of compressive sampling vis a vis measure-ment errors.This is a very important issue because any real-world sensor is subjectto at least a small amount of noise.And one thus immediately understands that tobe widely applicable,the methodology needs to be stable.Small perturbations in the。

a rXiv:h ep-th/061122v11O ct26Superalgebras of (split-)division algebras and the split octonionic M -theory in (6,5)-signature Zhanna Kuznetsova ∗and Francesco Toppan †∗ICE-UFJF,cep 36036-330,Juiz de Fora (MG),Brazil †CBPF,Rua Dr.Xavier Sigaud 150,cep 22290-180,Rio de Janeiro (RJ),Brazil February 2,2008Abstract The connection of (split-)division algebras with Clifford algebras and super-symmetry is investigated.At first we introduce the class of superalgebras con-structed from any given (split-)division algebra.We further specify which real Clifford algebras and real fundamental spinors can be reexpressed in terms of split-quaternions.Finally,we construct generalized supersymmetries admitting bosonic tensorial central charges in terms of (split-)division algebras.In particu-lar we prove that split-octonions allow to introduce a split-octonionic M -algebra which extends to the (6,5)signature the properties of the 11-dimensional octo-nionic M -algebras (which only exist in the (10,1)Minkowskian and (2,9)signa-tures).CBPF-NF-014/061IntroductionThe connection between division algebras and supersymmetry is well established since the[1]paper.Along the years,this connection has been further clarified,see e.g.[2].It is also well-known that division algebras are at the core of the classification of Clifford algebras and spinors,[3]and[4].The inter-relation of these mathematical structures has played a major role in a vast set of physical applications,ranging from supergravity (as well as superstrings and M-theory)to the construction of supersymmetric integrable paratively less is known,however,when we consider the case of the split version of division algebras(see[5]for an introduction to split-division algebras). Split-division algebras have been used with profit both in mathematical applications like,e.g.,the generalization of the Tits-Freudenthal magic square construction to split division algebras[6],as well as in more physically motivated applications,like the recent interesting reformulation of electrodynamics,see[7]and[8],in terms of split-octonions.The purpose of this paper is to clarify the interrelation between(split)-division algebras and the graded algebras that can be obtained from them,as well as the role played by the split-division algebras in the construction of Clifford algebras and spinors. Finally,we will construct the generalized supersymmetries associated to them.As an extra bonus,we will be able to solve a puzzle concerning the octonionic version of the M-theory,see[9]and[10],proving the existence of a split-octonionic M-algebra existing in the exotic(6,5)signature(more on that later).This paper is so conceived,atfirst we introduce the whole set of(split)-division algebras through the unified framework provided by the(generalized)Cayley-Dickson doubling construction.We further point out which graded algebras can be obtained as (anti-)commutators algebras from a given split-division algebra composition law.Ta-bles will further be produced,extending to split-quaternions the results of[3]concerning the division-algebra character of Clifford algebras and spinors.In given space-time sig-natures spinors which are valued either in the split-quaternions or in the split-octonions are produced.This allows to extend to split-quaternions or to split-octonions the con-struction of(constrained)generalized supersymmetries presented in[11],[12]and[13]. The split-octonionic M-algebra,existing in the(6,5)signature,is a particular example of this construction.In the Conclusions we make further comments about the implica-tions and the physical relevance of the results here obtained.For the moment we point out that higher-dimensional(generalized)supersymmetries formulated in space-time dimensions D≥8admit the peculiar feature that they come in several related ver-sions in given signatures.The associated supersymmetric theories are all dually related (“the space-time dualities”of ref.[14]).The10-dimensional superstrings,e.g.,only exist in three((9,1),(1,9)and(5,5))signatures,the latter presentingfive time di-rections.The11-dimensional supergravities are encountered,besides the Minkowskian (10,1)signature,also in the exotic(2,9)and(6,5)signatures.It was proven in[15] that such dually-related versions are in consequence of the triality of the D=8dimen-sions(the dually related theories are indeed triality related and close the S3group). We recall that it is eight-dimensional the transverse space of both the light-cone for-mulation of the10-dimensional superstrings and of the supermembranes evolving in aflat11-dimensional target spacetime.In D=8,the triality allowed signatures are(8,0),(0,8)and the exotic(4,4).Both the original Cartan’s triality,see[16],and the space-time triality of ref.[15]are a consequence of the octonions.On the other hand, it was quite puzzling that,while the standard M-algebra(based on real spinors)existsfor the whole set of above signatures,the octonionic M-algebra of ref.[9]and[10]only exists in(10,1)and(2,9)(the(6,5)signature is missing).The reason being essentiallydue to the fact that the seven imaginary octonions have to be accommodated either in the time-like or in the space-like directions.Obviously7cannot enter either6or5.By relaxing the condition of dealing with division-algebras,we can here solve the puzzleby expressing the counterpart of the octonionic M-algebra in the exotic signature(6,5) in terms of the split-octonions.It is worth mentioning that the(6,5)space-time alsocarries a supersymmetry based on split-quaternionic spinors.The paper is structured as follows:in Section2we revisit the split-division alge-bras.In Section3we construct the(graded)-algebras associated to the(split)-divisionalgebras.The tables relating Clifford algebras and spinors to split-quaternions are furnished in Section4.The generalized supersymmetries based on split-division alge-bras and the split-octonionic M-algebra are presented in Section5.We produce herealso the free actions for split-quaternionic and split-octonionic spinors.In the Conclu-sions,we provide comments on the results here obtained.In order to make the paperself-consistent we present in appendix the generalized Cayley-Dickson doubling.2Split-division algebras revisitedThe construction of split-division algebras in terms of the Cayley-Dickson doubling pro-cedure is reviewed in the Appendix.For later purposes it is useful to explicitly present here the(split-)division algebras structure constants,conjugations and quadratic forms in the case of quaternions(H),split-quaternions( H),octonions(O)and split-octonions ( O).Complex(C)and split-complex( C)numbers are immediately recovered as sub-algebra of,let’s say,the split-quaternions.Let us introduce atfirst the quaternions.The three imaginary quaternions e i∈H (i=1,2,3)satisfy the relationse i·e j=−δij1+ǫijk e k(2.1) (ǫijk is the totally antisymmetric tensor,normalized s.t.ǫ123=1).The conjugation and the quadratic form(norm)are respectively given bye i∗=−e i,N(e i)=1.(2.2) For what concerns the octonions,we can introduce them as as O=H−(see Ap-pendix).Therefore,the seven imaginary octonions E i are recovered through the posi-tionsE i=(e i,0)E3+i=(0,e i)E7=−(0,1).(2.3)They satisfy the relationsE i·E j=−δij1+C ijk E k,(2.4) while their conjugation and quadratic form are respectively given byE i∗=−E i,N(E i)=1.(2.5) In the above(2.4)formula the C ijk’s are the totally antisymmetric octonionic structure constants,non-vanishing only for the triples∗C123=C147=C165=C246=C257=C354=C367=1.(2.6) With a similar procedure the split octonions can be expressed through the identificaion O=H+.The seven imaginary split-octonions E i are given,as before,byE i=(e i,0)E3+i=(0,e i)E7=−(0,1).(2.7) They satisfy the relationsE i· E j=−ηij1+C ijkηkr E r,(2.8) together withE∗i=− E i,N( E i)=ηii.(2.9) In the above formulasηij denotes the diagonal matrix(+++−−−−)with three positive and four negative eigenvalues(normalized to±1).The quaternionic subalgebra H of the split octonions is obtained by restricting the imaginary split-octonions E i to the values i=1,2,3.On the other hand,the split-quaternionic subalgebra H is recovered by taking any subset of three elements lying in the six other lines of the Fano’s projective plane (namely,the triples(147),(165),(246),(257),(354)and(367)).The split-quaternions subalgebra can be explicitly presented as follows,in terms of the three generators e i(i=1,2,3),e i· e j=−ηij1+ǫijkηkr e r,(2.10) with conjugation and quadratic form given bye∗i=− e i,N( e i)=ηii.(2.11)ηij is now the diagonal matrix(−−+).The split quaternions admit a faithful representation in terms of2×2real matrices given byτ1= 0110 τ2= 100−1τA= 01−10 12= 1001 (2.12) The conjugate element of a generic split-quaternion X∈ H is represented byX∗=−τA X TτA.(2.13) 3Graded algebras from(split-)division algebrasThe multiplication“·”of a composition algebra A induces on A the structure of a (graded)algebra A×A→A of(anti)ly,for a,b∈A,we can introduce the algebra of graded brackets defined through[a,b}=ab+(−1)ǫaǫb ba,(3.14) whereǫa,b≡0,1mod2corresponds to the Z2grading of the generators a,b respectively. The(anti)commutator algebra is a(graded)Lie algebra if the multiplication is asso-ciative.If the multiplication is alternative(see the Appendix),the(anti)commutator algebra is a(graded)Malcev algebra(see[17]for its definition).The Z2grading implies for A the decomposition A=A0⊕A1such that,for non-vanishing[a,b},deg([a,b})=deg(a)+deg(b)≡ǫa+ǫb(mod2)(3.15) We can easily list the set of admissible Z2gradings for each one of the four division algebras(the R case is trivial).As a corollary,this gives us the list of the admissible superalgebras based on each division algebra.From the previous section results we know that the split-division algebras structure constants are recovered,up to a nor-malization factor,from the structure constants of their corresponding division algebra. For this reason the list of the admissible Z2gradings(and associated superalgebras) of division algebras can also be regarded as the list of admissible Z2gradings(and associated superalgebras)of the split-division algebras.The identity is necessarily an even(bosonic)element of the(super)algebra and corresponds to a central term.The (split)imaginary numbers close graded subalgebras of dimension1(for C and C),3 (for H and H)and7(for O and O).It is worth noticing that we can regard the(anti)commutators algebras induced by the composition law as abstract(super)algebras.In particular this implies that the Z2 superalgebra grading does not necessarily coincide with a Z2grading of the composition law(which requires satisfying deg(ab)=deg(a)+deg(b)mod2).This point can bebetter illustrated with an explicit example.Let’s take the three imaginary quaternions e i’s.If we assign odd-grading(fermionic character)to e1and e2,then e3,appearing on the r.h.s.of the multiplication e1·e2,is necessarily even-graded(bosonic).On the other hand,the anticommutator{e1,e2}is vanishing.As far as the anticommutators alone are concerned,we can consistently assign odd-grading to e3as well.In the following we will denote as“compatible”the restricted class of(super)algebras whose Z2grading is an acceptable Z2grading for the composition law.The admissible Z2gradings are expressed by the following table(the last column refers to the compatible superalgebras).We have(super)algebraC, C yes0B+1F yes3B+0F yesno1B+2F yesyesO, O yes6B+1F−no4B+3F−no3B+4F(b)yesno1B+6F noyeswithηij a diagonal matrix of(s,t)signature(i.e.s positive,+1,and t negative,−1, diagonal entries,with s and t denoting,respectively,the number of space-like and time-like dimensions).The most general irreducible real matrix representation of a Clifford algebra is classified according to the property of the most general S matrix commuting with all theΓ’s([S,Γi]=0for all i).If the most general S is a multiple of the identity,we get the normal(R)case.Otherwise,S can be the sum of two matrices,the second one multiple of the square root of−1(this is the almost complex,C case)or the linear combination of4matrices closing the quaternionic algebra(this is the H case).We obtain,for s,t≤8,the following table,see[3]s\t13570C2H C(4)2R(8) 1R(2)H(2)H(4)R(16)22R(2)C(4)2H(4)C(16)3R(4)R(8)H(8)H(16)4C(4)2R(8)C(16)2H(16) 5H(4)R(16)R(32)H(32)62H(4)C(16)2R(32)C(64)7H(8)H(16)R(64)R(128) 8C(16)2H(16)C(64)2R(128)The spinors carry a representation of the Spin(s,t)spin group(see[3]),whose Lie algebra generators are given by the gamma matrices commutators.As a result,the division algebra structure of Gamma matrices extends to spinors.There is,however,for certain signatures of the space-time,a mismatch between division-algebra properties of the fundamental spinors and their associated Clifford algebras,see[9]and[18]. The mismatch is due to the existence of a Weyl-projection.We recall that,following [11],the fundamental spinors belong to the representation of the spin group admitting maximal division algebra structure.A table,presenting the division-algebra properties of spinors for s,t≤8,is here produceds\t13570R W H H(2)W R(8) 1R W C(2)W H(2)W C(8)W 2R(2)R(4)W H(4)H(8)W 3C(2)W R(4)W C(8)W H(8)W 4H(2)W R(8)R(16)W H(16) 5H(2)W C(8)W R(16)W C(32)W 6H(4)H(8)W R(32)R(64)W 7C(8)W H(8)W C(32)W R(64)W 8R(16)W H(16)H(32)W R(128)At the end we obtain the table of split-quaternionic Clifford algebras given bys\t2468H0002 H(2)00 H(16)H(2) H(4)004 H(4) H(8)000 H(8) H(16)060 H(16) H(32)00 H(32) H(64)800 H(64) H(128) Similarly,the split-quaternionic table for fundamental spinors is given by s\t246800002 H W0000 H(2)W0040 H(4)W0000 H(8)W0600 H(16)W0000 H(32)W8000 H(64)W Both tables above can be extended for s,t>8due to the mod8property of Clifford algebras.5Split-division algebras and generalized supersym-metriesIn this Section we discuss a physical application of both split-quaternions and split-octonions.Essentially,we will prove that the constructions concerning quaternionic and octonionic spinors,carried out in[11],can be extended to the split-quaternionic and the split-octonionic cases.The main results in[11]include i)the construction of free invariant actions for quaternionic and octonionic spinors and ii)the construction of quaternionic and octonionic generalized supersymmetries.We explicitly discuss here which modifications have to be introduced in the split cases w.r.t.their non split counterparts.The notion of octonionic and split-octonionic spinors requires the introduction of the (split-)octonionic realizations of the(4.17)relation,in terms of matrices with(split-)octonionic entries.The meaning of the octonionic realizations of(4.17)has been fully described in[11].The same considerations apply to the split-octonionic cases as well.The results furnished in this paper will concern the so-called maximal Cliffordalgebras leading to the oxidized supersymmetries,see[13]for a definition.Basically, the maximal Clifford algebras correspond to the space-times of maximal dimension supporting spinors of a given size.The results for the non-maximal space-times are simply obtained via a dimensional reduction of the oxidized cases by applying the formulas produced in[13](see this reference for a full discussion).A feature distinguishing the split-quaternionic and octonionic cases w.r.t.the non-split ones is the fact that,in the split case,the same space-time of a given signa-ture can carry both split-quaternionic and split-octonionic spinors.In the non-split case,quaternionic and octonionic spinors are carried by space-times of different sig-natures.For instance,in D=11dimensions,the octonionic spinors can be intro-duced in the Minkowskian(10,1)signature,while quaternionic spinors are associated with the Euclidean(0,11)space-time.On the other hand,the(6,5)signature carries both split-quaternionic and split-octonionic spinors.The“oxidized”split-quaternionic space-times are given by formulas(4.20)and(4.21),while the oxidized split-octonionic space-times are restricted by the conditionss=4+k,t=3+8m+k,for m,k=0,1,2,...(5.22) ands=5+8m+k,t=4+k,for m,k=0,1,2,...(5.23) We further point out that the D generating gamma-matrices entering(4.17)for a D-dimensional space-time are given,in the split-quaternionic case,by D−3purely real matrices,while the remaining three matrices are given by the three split-imaginary split-quaternions multiplying the same purely real matrix(let’s call it“T”).In the split-octonionic case they are given by D−7purely real matrices plus the seven matrices obtained by the seven split-imaginary split-octonions multiplying a single,purely real matrix T.We need atfirst to introduce the two matrices,usually denoted as C and A in the literature(see[11]),which are related to the transposition and the hermitian con-jugation respectively.C is the charge-conjugation matrix.For the maximal Clifford algebras it is uniquely defined and is given,up to a sign factor,by the product of all symmetric(or all antisymmetric)generating gamma matrices.The matrix A in the split-quaternionic and split-octonionic cases can be defined as follows.If the purely real matrix T introduced above is symmetric,then A is given by the product of the subset of purely real gamma matrices of space-like type.Conversely,if T is antisymmetric,A is the product of the subset of purely real gamma matrices of time-like type.The importance of a non-trivial conjugation is reflected by the fact that the su-persymmetry algebra can be decomposed in the three relations below.We have,for split-quaternionic or split-octonionic supercharges Q a,the following relations{Q a,Q b}=W ab,{Q a∗,Q b∗}=W ab∗,{Q a,Q b∗}=Z ab.(5.24) The r.h.s.matrices W ab and Z ab contain the bosonic degrees of freedom of the su-peralgebra.W ab is a symmetric matrix,while Z ab is hermitian.The bosonic r.h.s.can be expanded in terms of the antisymmetrized products of the gamma matrices, namely W ab= k(CΓ[µ1...µk])ab W[µ1...µk]and Z ab= k(AΓ[µ1...µk])ab Z[µ1...µk],with the sum over k restricted to symmetric or hermitian matrices.For split-quaternions and split-octonions,just like their quaternionic and octonionic counterparts(and unlike the real and complex cases)the decomposition of the symmetric matrix W ab has tobe limited to,at most,k=0and k=1.The reasons discussed in[11]applytothe split-cases as well.No such a limitation exists for the hermitian Z ab matrices.The admissible integers k label the higher-rank tensors sectors of the bosonic r.h.s..These sectors will be compactly denoted as“M k”.Due to the non-associativity,in the split-octonionic case(similarly as for the oc-tonionic case),it must be consistently specified the order in taking the antisymmetric product of k>2(split-)octonionic valued gamma matrices.The correct prescription, given in[11],applies also to the split-octonions.It is given by the formula[Γ1·Γ2·...·Γk]≡12(.((Γ1Γ2)Γ3...)Γk)+1M0+M1D=55−D=91M0+M1D=1313(5.27)ii)Split-octonionic symmetric case,D=71+7M1D=11−M0M0D=51+5M1+M2(5.29)D=936+84M0+M3+M4D=131+13+715+1287iv)Split-octonionic hermitian case,D=71M0+M1≡M4D=1111+41=52M2+M3≡M6Let us conclude this section producing the formulas for the free action of the split-quaternionic and split-octonionic spinorsΨ.Following[11]we can express the la-grangian L as a sum L=K+M,where the kinetic term K and the massive term M are given byK=12tr[Ψ†(AΓµ∂µΨ)],M=tr(Ψ†AΨ)(5.31) In the above formulas the“trace”tr refers to the projection over the identity for elements of the(split-)division algebra(i.e.tr(x0+x j e j)=x0).The brackets are inserted to take care of the correct ordering when dealing with split-octonions,due to their non-associativity.The formula(5.31)clearly holds also for the split-quaternionic case.One can easily prove that no massive terms are allowed in the split-quaternionic space-times(2,1)or(3,2).Similarly,no massive terms are allowed for the(4,3)and (5,4)split-octonionic spacetimes.The most interesting spacetime for the connection with the M-theory and its triality properties,as discussed in the Introduction,is(6,5). It allows a non-vanishing mass-term for split-octonionic spinors.6ConclusionsIn this work we have analyzed the connection of(split-)division algebras with Clifford algebras,spinors and supersymmetry.More specifically,after having reviewed the generalized Cayley-Dickson double construction which allows to introduce,in a unified framework,all seven inequivalent(split-)division algebras,we analyzed the following points.Atfirst we derived the(graded)algebras of(anti)commutators which can be constructed from the(split-)division algebras composition law.Next,we produced the tables of Clifford algebras and fundamental spinors related to the split-quaternions.In Section5we made use of the conjugation properties of the split-division al-gebras in order to construct,in the case of space-time signatures supporting either split-quaternionic or split-octonionic valued spinors,the corresponding(constrained) generalized supersymmetries.We also produced the free lagrangians for both the split-quaternionic and the split-octonionic spinors.Concerning supersymmetries,we pointed out that in several examples the same space-time can carry both split-quaternionic and split-octonionic supersymmetries.The most interesting application concerned the con-struction of the split-octonionic M-algebra,available in the exotic(6,5)-signature.The construction of split-octonionic generalized supersymmetries parallels the construction, already available in the literature([9,10,11]),of the octonionic generalized supersym-metries.As recalled in the Introduction,higher-dimensional(for D≥8)supersymmetries hide a fundamental ambiguity,due to the existence of triality-related formulations in different signatures.This feature is exemplified,for instance,in the case of super-strings.The criticality condition allows to determine the overall dimension(=10)of the target spacetime,but not its signature,which could be either(9,1),(1,9)or theexotic(5,5).A mathematically consistent superstring theory can be formulated not only for a Minkowski target,but also in a(5,5)spacetime.In[14]the web of dualities of the corresponding theories(and their consequences for the dimensionally reduced theories)were explored.So far this construction only existed for theories formulated in terms of real-valued (Majorana or Majorana-Weyl)spinors,but not for their octonionic counterparts.The introduction of split-octonionic supersymmetries in exotic signatures allows to extend the web of dualities to the(split-)octonionic formulations.It immmediately implies,e.g.,the existence(for the exotic(5,5)signature)of a split-octonionic version of the[19]octonionic formulation of the superstrings.In the case of generalized supersym-metries with tensorial central charges,we obtain the split-octonionic version of the M-algebra in(6,5)signature,sharing the same features as its octonic counterpart(the most noticeable property being the dependence of the rank-5totally antisymmetric tensors in terms of a combination of the rank-1and rank-2antisymmetric tensors). The11-dimensional M-algebra admits an equivalent12-dimensional presentation,the F-algebra formulation[11]which,in case of the split-octonions,is realized in the(6,6) signature.It is worth reminding that it has been suggested,see e.g.[20]and references therein,that since octonions are at the very core of many mathematical exceptional structures,a possible exceptional formulation for a“Theory Of Everything”would re-quire an octonionic formulation.If this is indeed the case,it would be expected that the octonionic and split-octonionic versions of the M-algebra should play a major role. AppendixWe collect here for convenience,following[5]and[6],the main properties and definition of(split-)division algebras.An algebra A over the reals(R)is a composition algebra if it possesses a unit (denoted as1A)and a non-degenerate quadratic form N satisfyingN(1A)=1,N(xy)=N(x)N(y),∀x,y∈A.(A.1) A composition algebra is alternative if the following left and right alternative properties are satisfied[22](x2)y=x(xy),yx2=(yx)x.(A.2) A positive definite quadratic form(norm)is a mapping N:A→R+s.t.N(x)=0⇔x=0.(A.3) A composition algebra with positive quadratic form is a division algebra,satisfying the propertyxy=0⇒x=0∨y=0.(A.4)Due to the Hurwitz’s theorem,the only division algebras are R,C,H and O.A∗-algebra possesses a conjugation(i.e.an involutive automorphism A→A)s.t.,denoted as x∗the conjugate of x∈A,we have(x∗)∗=x,(xy)∗=y∗x∗.(A.5) The norm N(x)of an element of a division algebra is expressed asN(x)=xx∗.(A.6) Besides division algebras,we can introduce their split forms[5]as a new set of algebras.The split-division algebras are∗-algebras with unit.The quadratic form N is no longerpositive-definite and the property(A.4)is no longer valid.The algebras of split complex numbers,split quaternions and split octonions are respectively denoted as C, H and O. The total number of inequivalent(split)-division algebras over R is7(the4divisionalgebras and their3split forms above).(Split)-division algebrasfind a unified description through the Cayley-Dickson dou-bling construction.Given an algebra A over R,possessing a“·”multiplication,a“∗”conjugation and a quadratic form N,the Cayley-Dickson doubled algebra A2over R is defined in terms of the operations in A.The multiplication,the conjugation and the quadratic form in A2are respectively given byi)multiplication in A2:(x,y)·(z,w)=(xz+εw∗y,wx+yz∗),ii)conjugation in A2:(x,y)∗=(x∗,−y),iii)norm in A2:N(x,y)=N(x)−εN(y).The unit element1A2of A2is represented by1A2=(1A,0).In the above formulasεis just a sign(ε=±1).It is convenient to denote the Cayley-Dickson’s double of an algebra A by writing theεsign on the right of the original algebra.For division algebrasεis always negative (ε=−1).We can therefore writeC=R−,H=C−=R−−,O=H−=C−−=R−−−.The split division algebras are obtained by taking a positive(ε=+1)sign.We have C=R+,H=C+=R−+O=H+=C−+=R−−+.Other choices of the sign produce,at the end,isomorphic algebras.We can,e.g., also write H=R++,as well as O=R+++.All(split-)division algebras over R are obtained by iteratively applying the Cayley-Dickson’s construction starting from R.AcknowledgmentsThis work received support from CNPq(Edital Universal19/2004).Z.K.acknowl-edges FAPEMIG forfinancial support and CBPF for hospitality. 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刻意训练试题及答案英语一、选择题1. What does the term "deliberate practice" refer to?A. Casual practiceB. Planned and structured practiceC. Random practiceD. Unstructured practice答案：B2. According to the theory of deliberate practice, which of the following is NOT a characteristic?A. It is goal-oriented.B. It requires focused attention.C. It involves repetition of random tasks.D. It is beyond one's current level of performance.答案：C3. Who is often credited with developing the concept of deliberate practice?A. Albert EinsteinB. Malcolm GladwellC. Anders EricssonD. Stephen Hawking答案：C二、填空题4. Deliberate practice is a type of practice that is__________, __________, and __________.(目标导向的，需要集中注意力的，超出当前表现水平的)答案：目标导向的，需要集中注意力的，超出当前表现水平的5. The concept of deliberate practice emphasizes the importance of __________ and __________ in the pursuit of excellence.(持续改进，逐步提升)答案：持续改进，逐步提升三、简答题6. Explain the difference between deliberate practice and mindless practice.答案：Deliberate practice is characterized by its structure, goal orientation, and the requirement for focused attention, which is designed to push the individual beyond their current abilities. In contrast, mindless practice lacks structure and does not challenge the individual to improve, often involving repetitive tasks without the intent to enhance skills or performance.四、论述题7. Discuss the role of deliberate practice in achieving expertise in any field.答案：Deliberate practice plays a crucial role in achievingexpertise by providing a systematic approach to skill development. It involves setting specific goals, receiving immediate feedback, and engaging in activities that are challenging but achievable. This type of practice is essential for continuous improvement and for reaching the highest levels of performance in any field. It also helps in identifying and overcoming weaknesses, refining techniques, and maintaining a growth mindset, which are all critical for becoming an expert.。

Comparison Between Synchronous and Asynchronous Implementation of ParallelGenetic ProgrammingShisanu Tongchim Prabhas Chongstitvatana Department of Computer Engineering Department of Computer Engineering Chulalongkorn University Chulalongkorn UniversityBangkok10330,Thailand Bangkok10330,Thailandg41stc@cp.eng.chula.ac.th prabhas@chula.ac.thAbstractAn evolutionary method such as Genetic Program-ming(GP)can be used to solve a large number of com-plex problems in various application domains.However, one obvious shortcoming of GP is that it usually uses a substantial amount of processing time to arrive at a solution.In this paper,we present the parallel imple-mentations that can reduce the processing time by using a coarse-grained model for parallelization and an asyn-chronous migration.The problem chosen to examine the parallel GP is a mobile robot navigation problem.The experimental results show that superlinear speedup of GP can be achieved.1IntroductionGenetic Programming was successfully used to per-form automatic generation of mobile robot programs[1]. The use of the perturbation to improve robustness of the robot programs was proposed.Each robot program was evaluated under many environments that were different from the original one.As a result,the substantial pro-cessing time was required to evaluate thefitness of the robot programs.To reduce the computational time,this study pro-posed two parallel implementations.Asynchronous and synchronous parallelization approaches were examined. We also compared the quality of the solutions generated from the serial and parallel GP.The earlier work of parallel GP was implemented on a network of transputers by Koza and Andre[2].Their result showed that the parallel speedup was greater than linear.Dracopoulos and Kent[3]proposed the use of the Bulk Synchronous Parallel Programming(BSP)model to parallelize genetic programming.Two approaches of parallel GP were examined on a cluster of Sun worksta-tions.Thefirst was based on a master-slave model while the second was based on a coarse-grained model.The results showed that the achieved speedup was close to linear.A recent paper by Punch[4]presented the em-pirical study about some problem-specific factors which affect the effectiveness of parallel GP.Punch concluded that the achieved performance of parallel GP by using a coarse-grained model may vary according to the nature of problems.The remaining sections are organized as follows:The next section is a description of the mobile robot naviga-tion problem.Section3describes a problem represen-tation in the serial GP.Section4shows the parallel so-lutions.Section5presents the experimental results and discussion.Finally,section6provides the conclusions of this work.2Mobile Robot Navigation Problem Our previous work[1],GP was used to generate a robot control program for the mobile robot navigation problem.The task was to control a mobile robot from a starting point to a target point in a simulated envi-ronment.The environment wasfilled with the obstacles which had several geometrical shapes.The aim of the work was to generate robust control programs.In the evolution process,each individual was evaluated under many environments that were different from the original one.The result showed that the ro-bustness of the robot programs was improved by such an approach.3Serial AlgorithmThe terminal set is composed of three primitive move-ment controls{move,left,right}and one sensor in-formation{isnearer}.The function set is composed of three functions{if-and,if-or,if-not}.The GP pa-rameters are shown in Table1.Thefitness function is a sum of thefitness value in each environment which is based on the distance of the final position and the number of moves.In the evolution process,the percent of disturbance is20%and the number of training environments is8.Table1:GP parametersTotal population6000Crossover probability0.9Mutation probability0.1Reproduction5%of Total populationMaximum generation2004Parallel Genetic Programming In a coarse-grained model,the population is divided into subpopulations and are maintained by different pro-cessors.The model is also known as Island model and the subpopulation is called deme[2].Some works in parallelization of GA and GP using a coarse-grained model[2,5]show that the results can achieve superlinear speedup1.This is caused by two factors;the speedup from the populations distributed across different processors and the speedup obtained by increasing the probability infinding the correct solution, as the number of populations is increased.In applying the parallel approach to the previous work [1]by using a conventional coarse-grained model,the re-sult achieves only linear speedup[6]since the amount of work isfixed–the algorithm is terminated when it reaches the maximum generation.Hence,the parallel algorithm does not exploit the probabilistic advantage that the answer may be obtained before the maximum generation.We reduce redundant jobs by dividing the environments among the processing nodes.After a spe-cific number of generations,every subpopulations are mi-grated between processors using a fully connected topol-ogy.However,this scheme leads to the reduction of ro-bustness since each individual has a shorter period in each training environment.To mend this problem,we increase the number of environments in each node.How-ever,the number of environments per node should be less than the number of environments per node in the general coarse-grained model.We implement our parallel algorithm on a dedicated cluster of PC workstations with350MHz Pentium II pro-cessors,each with32Mb of RAM,and running Linux as an operating system.These machines are connected via 10Mbs ethernet cabling.We extend the program used in[1]to run under a clustered computer by using MPI as a message passing library.Several trials are examined tofind an appropriate value for the number of environments per node(see Ta-ble2).The migration is carried out as follows:each 1Superlinear speedup means that speedup is greater than the number of processors used.Table2:Experimental ParametersNum.of Processors124610 Pop.size∗6000300015001000600 Environments∗87432 Migration interval NA100503420∗per nodeprocedure Migrationbeginbarrier1wait all nodes readyfor i=1to nbeginif(my process id=i)broadcast sendelsebeginbroadcast receiveendbarrier2wait for the next broadcastendendFigure1:The migration processnode broadcasts its subpopulation to all other nodes by MPI Bcast function,this is repeated for every node.The top5%of individuals from each subpopulation are ex-changed during the migration.Pseudo-code for the mi-gration is shown infigure1.The detail will be discussed in the timing analysis section.The total population is hold constantly for the task and is divided equally among workstations.The number of selected individuals,crossover operation,mutation op-eration,reproduction are a percentage of the amount of the total population.The parallel efficiency is measured by varying a number of nodes and the results are aver-aged over20runs for each number of nodes.In thefirst implementation,the migration between subpopulations is synchronized.Each node is blocked by MPI Barrier function until all subpopulations evolve to the same number of generations.However,the syn-chronizing migration results in uneven work loads among the processors since the time required to complete the evaluation varies,with the least effective programs tak-ing the longest period and the best programs taking the shortest period.4050607080901001020304050607080901001 node2 nodes (Syn)2 nodes (Asyn)4 nodes (Syn)4 nodes (Asyn)6 nodes (Syn)6 nodes (Asyn)10 nodes (Syn)10 nodes (Asyn)Disturbance (%)R o b u s t n e s s (%)Figure 2:RobustnessIn the second implementation,we attempt to fur-ther improve the speedup of the parallel algorithm by the asynchronous migration.When the fastest node reaches predetermined generation numbers,the migra-tion request is sent to all subpopulations.The migration takes place at the end of the current generation.In this state,if any populations are still in the fitness evaluation phase,the other nodes must wait.The waiting time will be at most less than one generation.5Results and Discussion5.1SpeedupTo make an adequate comparison between the serial algorithm and parallel algorithm,Cant´u -Paz [7]suggests that the two must give the same quality of the solution.In this paper,we define the robustness of the gener-ated program from the serial algorithm as a baseline.In addition,if the generated program from the paral-lel algorithm gives the same robustness as the program from the serial algorithm,the equal amount of work to achieve the same quality of the answer is done.From the robustness graph in figure 2,the generated program from the parallel GP is better than the serial GP.Hence,the amount of work from the parallel algorithm is not less than the serial algorithm.The parallel speedup is defined as the ratio of the serial execution time to the parallel execution time.Speedup =Serial timeP arallel time(1)Figure 3illustrates the speedup observed on the two implementations as a function of the number of pro-cessors used.The performance is less than we ex-pect,although both implementations exhibit superlinear speedup.The speedup curves taper offfor 10processors and the performance of the asynchronous implementa-tion is slightly better than the performance of the syn-chronous implementation.In order to discern the cause14710131619222512345678910IdealSynchronous AsynchronousNumber of ProcessorsS p e e d u pFigure 3:SpeedupFigure 4shows the relative time spent in each section of the implementations.The communication overhead –the sum of the barrier time and broadcast time –goes up considerably as the number of processors increases.The asynchronous implementation does not help much on re-ducing the communication overhead at large numbers of processors.Thus,we investigate the further detailed analysis of the communication overhead.Figure 5shows the absolute time spent in major func-tions of the communication.The time spent in barriers indicates the time spent on waiting for all processes to reach the same point.From pseudo-code of the migra-tion in figure 1,the barrier time consists of the time spent to wait for all nodes to be ready for the migration and the next broadcast.In the synchronous implementation,the time spent in barriers reduces as the number of processors increases.This is because the barrier time depends on the variationis plementation increases as the number of processors in-creases.This is due to the fact that the time spent in the second barrier(waiting for the next broadcast)increases with the number of nodes.However,the asynchronous implementation eliminates thefirst barrier therefore it reduces the total time in the barriers compared to the synchronous implementation.The absolute time spent in a broadcast increases con-siderably–greater than linear.From the inspection in the trace information by using a visualization tool,we found that the transmission of the broadcast functions in the implementation of MPI that we use may be exe-cuted more than once,especially for a large number of processors.After obtaining some timing analyses,the results re-veal the cause of the problem.The performance degra-dation in10processors is caused by the excessive com-munication time due to the broadcast function.Al-though the asynchronous migration reduces the barrier time effectively compared to the synchronous migration, the increase in the communication time in10processors obliterates this advantage.In case of the small number of processors(2,4,6),the gain from the asynchronous mi-gration is considerable as the evolution proceeds at the speed of the fastest node.As the size of the work increases(i.e.,the number of training environments increases),the serial and par-allel computation time will be increased when the time spent in the communication is constant.If the ratio of the computation/communication can be kept large (large work load),then one can expect that the parallel performance will be improved.6ConclusionsThe result presented in this paper shows a success of speeding up the Genetic Programming process by means of parallel processing.The parallel implemen-tations of Genetic Programming successfully exploit the computing resource of a dedicated cluster of PC work-stations.Superlinear speedup of GP can be acquired by improving a coarse-grained model for parallelization as less computational work needs to be done.Furthermore, the timing analyses indicate the scalability of the paral-lel approaches,as the size of the problem increases,the speedup will be improved.References[1]Chongstitvatana P(1998),Improving robustness ofrobot programs generated by genetic programming for dynamic environments.Proc.of IEEE Asia Pa-cific Conference on Circuits and Systems,p.523–526 [2]Koza JR,Andre D(1995),Parallel genetic program-ming on a network of transputers.Proc.of the Work-shop on Genetic Programming:From Theory to Real-World Applications,University of Rochester, National Resource Laboratory for the Study of Brain and Behavior,Technical Report95-2,p.111-120 [3]Dracopoulos DC,Kent S(1996),Bulk synchronousparallelisation of genetic programming.Proc.of the Third International Workshop on Applied Parallel Computing in Industrial Problems and Optimization (PARA’96),Springer Verlag,Berlin[4]Punch B(1998),How effective are multiple poplu-lations in genetic programming.Proc.of the Third Annual Conference in Genetic Programming,pp.308-313[5]Lin S-C,Punch WF,Goodman ED(1994),Coarse-grain parallel genetic algorithms:Categorization and new approach.Proc.of the Sixth IEEE SPDP,pp.28-37[6]Tongchim S,Chongstitvatana P(1999),Speedup Im-provement on Automatic Robot Programming by Parallel Genetic Programming.Proc.of1999IEEE International Symposium on Intelligent Signal Pro-cessing and Communication Systems(ISPACS’99), Phuket,Thailand[7]Cant´u-Paz E(1999),Designing efficient and accurateparallel genetic algorithms.PhD thesis,University of Illinois at Urbana-Champaign。