Family of Boundary Poisson Brackets

描写亲前之间的英语作文Unwavering Bonds: The Enduring Power of Kinship.Within the tapestry of human existence, kinship weaves a vibrant thread, connecting individuals across time and space. It encompasses the sacred ties of family and the enduring bonds of friendship, forging a sanctuary of love, loyalty, and unwavering support.The Familial Covenant.The family, a primordial foundation of human society,is a crucible where kinship is forged in its purest form. Parents, siblings, children, and extended kin share an unbreakable bond forged through shared experiences, genetic heritage, and a deep sense of belonging.From the moment a child enters the world, the familial covenant is established. Parents become guardians, protectors, and nurturers, devoting themselves to theiroffspring's well-being. The love they impart is unconditional, transcending triumphs and tribulations alike. Siblings grow up together, forging an unbreakable bond through shared memories, laughter, and occasional sibling rivalry. They learn to rely on each other, offering support and encouragement through life's challenges.As families expand, extended kin form a wider circle of kinship. Aunts, uncles, cousins, and grandparentscontribute to the intricate web of relationships, enriching the lives of all involved. They provide sage advice, offera sense of continuity, and celebrate milestones together.The Bonds of Friendship.Friendship, another pillar of kinship, plays an equally vital role in human flourishing. While not bound by bloodor familial obligations, the ties of friendship are no less profound. True friends become chosen family, offering unwavering support, empathy, and a shared sense of purpose.Friendships are often characterized by common interests,values, or experiences. They provide a safe haven where individuals can share their innermost thoughts, fears, and aspirations without judgment. Friends offer a listening ear, a shoulder to cry on, and a helping hand when times are tough.Through the twists and turns of life, true friends remain steadfast. They celebrate successes together, commiserate during setbacks, and offer unwavering support through thick and thin. Their bond transcends distance and time, as they know that they can always count on each other.The Benefits of Kinship.The profound benefits of kinship cannot be overstated. Strong familial bonds have been associated with improved physical and mental health, greater happiness, and a longer life expectancy. Kinship provides a sense of stability, security, and purpose, which has a positive impact onoverall well-being.Friendships also contribute significantly to happinessand well-being. They provide emotional support, reduce stress, and encourage healthy behaviors. Having close friends has been linked to lower levels of depression, anxiety, and loneliness.Furthermore, kinship promotes social cohesion and cooperation. By fostering a sense of belonging, it encourages individuals to work together for the common good and support their community.Nurturing Kinship.While kinship is a precious gift, it requires nurturing to flourish. Here are some ways to cultivate and strengthen these bonds:Spend quality time together: Prioritize spending time with loved ones, engaging in meaningful conversations and shared activities.Communicate openly: Foster open and honest communication to strengthen understanding and resolveconflicts amicably.Show appreciation: Express gratitude for the people in your life and acknowledge their contributions.Be supportive: Offer support and encouragement during challenging times and celebrate achievements together.Set boundaries: Establish healthy boundaries toprotect and maintain the integrity of relationships.Conclusion.Kinship, in all its forms, is a fundamental pillar of human society. It provides a sanctuary of love, loyalty,and unwavering support that enriches our lives and empowers us to navigate life's challenges. By nurturing and valuing these bonds, we create a more compassionate, interconnected, and fulfilling world.。

家庭诗歌英文作文全文共2篇示例，仅供读者参考家庭诗歌英文作文1:Title: The Essence of Home: Exploring Family PoetryIn the vast expanse of literature, poetry stands as a timeless testament to the human experience. It captures emotions, thoughts, and moments with an unparalleled depth and brevity. Among the myriad themes that poetry delves into, one that resonates universally is the concept of family. Family, with its complexities, joys, and bonds, has inspired poets for generations. In this exploration of family poetry, we embark on a journey through verses that celebrate, reflect upon, and immortalize the essence of home.At the heart of family poetry lies the celebration of love and connection. Poets often weave intricate tapestries of emotions, painting portraits of familial love that transcend time and space. From Elizabeth Barrett Browning's tender verses in "How Do I Love Thee?" to Langston Hughes' poignant portrayal of a mother's unwavering support in"Mother to Son," the depth of familial affection knows no bounds. It is in these verses that we find solace, reassurance, and a sense of belonging.Yet, family poetry is not confined to idyllic portrayals of domestic bliss. It also delves into the complexities of familial relationships, acknowledging the struggles and conflicts that are an inherent part of human interaction. Robert Frost's "The Death of the Hired Man" and Sylvia Plath's "Daddy" are poignant examples of poems that grapple with the intricacies of family dynamics, exploring themes of loss, resentment, and longing. Through these verses, poets confront the darker aspects of familial bonds, inviting readers to confront their own emotions and experiences.Beyond individual relationships, family poetry also reflects upon the broader cultural and societal contexts in which families exist. Poets often use their verses to critique social norms, challenge traditions, and advocate for change. In Maya Angelou's "Still I Rise," the poet celebrates the resilience and strength of generations of women, overcoming adversity and oppression. Similarly, in Langston Hughes' "I, Too," the poetasserts the dignity and humanity of African Americans in the face of systemic racism. These poems serve as powerful reminders of the role that family plays in shaping identity and resilience in the face of adversity.In addition to reflecting upon the past and present, family poetry also looks towards the future, offering glimpses of hope, dreams, and aspirations. Poets envision a world where familial bonds transcend boundaries of race, class, and nationality, where love and understanding prevail. In Gwendolyn Brooks' "We Real Cool," the poet imagines a future where young people find acceptance and belonging, despite societal prejudices. Similarly, in Langston Hughes' "Dream Variations," the poet conjures images of a world where freedom and equality reign supreme.In conclusion, family poetry occupies a special place within the realm of literature, offering insights into the human experience unlike any other form of expression. Through its celebration of love, exploration of conflicts, and envisioning of a better future, family poetry serves as a timeless testament to the enduring power of familial bonds. As we navigate thecomplexities of our own lives, we can find solace, inspiration, and understanding in the verses that celebrate the essence of home.家庭诗歌英文作文2:Family Poetry: Celebrating the Bonds of LovePoetry has always been a powerful medium to express emotions and feelings. It is an art form that transcends language barriers and cultural differences. In this article, we will explore the beauty of family poetry and how it can celebrate the bonds of love that hold families together.Family is the cornerstone of our lives. It is where we learn to love, share, and grow. It is where we find comfort and support during challenging times. Family poetry captures the essence of these relationships and showcases the depth of emotions that we feel towards our loved ones.One of the most famous family poems is "Mother to Son" by Langston Hughes. The poem describes the struggles and hardships that a mother has faced in her life and how she has persevered through them. She tells her son that life is not easy,but he must keep climbing and never give up. The poem is a testament to the love and strength that a mother has for her child.Another popular family poem is "The Road Not Taken" by Robert Frost. Although the poem is not specifically about family, it touches on the theme of making choices and the impact they have on our lives. Family is often at the center of these choices, and the poem reminds us that the choices we make can shape our future.Family poetry can also be funny and lighthearted. "My Father's Hats" by Mark Irwin is a humorous poem that describes the different hats that his father wore throughout his life. Each hat represents a different phase of his father's life, from a cowboy hat to a fedora. The poem is a tribute to the unique personality of his father and the memories they shared together.Writing family poetry can be a way to express gratitude and appreciation for our loved ones. It can also be a way to process difficult emotions and heal from past traumas. Writingpoetry can be a cathartic experience that allows us to connect with our feelings and share them with others.In conclusion, family poetry is a beautiful way to celebrate the bonds of love that hold families together. It can capture the essence of our relationships and showcase the depth of emotions that we feel towards our loved ones. Whether it's a serious or lighthearted poem, family poetry can bring us closer together and remind us of the importance of cherishing our family bonds.。

a rXiv:h ep-th/005232v125M a y2Quantum Spin Dynamics (QSD):VII.Symplectic Structures and Continuum Lattice Formulations of Gauge Field Theories T.Thiemann ∗MPI f.Gravitationsphysik,Albert-Einstein-Institut,Am M¨u hlenberg 1,14476Golm near Potsdam,Germany Preprint AEI-2000-026Abstract Interesting non-linear functions on the phase spaces of classical field theories can never be quantized immediately because the basic fields of the theory become operator valued distributions.Therefore,one is usually forced to find a classical substitute for such a function depending on a regulator which is expressed in terms of smeared quantities and which can be quantized in a well-defined ly,the smeared functions define a new symplectic manifold of their own which is easy to quantize.Finally one must remove the regulator and establish that the final operator,if it exists,has the correct classical limit.In this paper we investigate these steps for diffeomorphism invariant quantum field theories of connections.We introduce a generalized projective family of symplectic mani-folds,coordinatized by the smeared fields,which is labelled by a pair consisting of a graph and another graph dual to it.We show that there exists a generalized projective sequence of symplectic manifolds whose limit agrees with the symplectic manifold that one started from.This family of symplectic manifolds is easy to quantize and we illustrate the programmeoutlined above by applying it to the Gauss constraint.The framework developed here is the classical cornerstone on which the semi-classical analysis developed in a new series of papers called “Gauge Theory Coherent States”is based.This article also complements,as a side result,earlier work by Ashtekar,Corichi and Zapata who observed that certain operators are non-commuting on certain states although the Poisson brackets between the classical functions that these authors based the quantization on,vanish.We show that there are other functions on the classical phase space which give rise to the same operators but whose Poisson algebra precisely mirrors the quantum commutator algebra.1IntroductionQuantum General Relativity (QGR)has matured over the past decade to a mathematically well-defined theory of quantum gravity.In contrast to string theory,by definition,GQR is a man-ifestly background independent,diffeomorphism invariant and non-perturbative theory.The obvious advantage is that one will never have to postulate the existence of a non-perturbativeextension of the theory,which in string theory has been called the still unknown M(ystery)-Theory.The disadvantage of a non-perturbative and background independent formulation is,of course,that one is faced with new and interesting mathematical problems so that one cannot just go ahead and“start calculating scattering amplitudes”:As there is no background around which one could perturb,rather the full metric isfluctuating,one is not doing quantumfield theory on a spacetime but only on a differential manifold.Once there is no(Minkowski) metric at our disposal,one loses familiar notions such as causality structure,locality,Poincar´e group and so forth,in other words,the theory is not a theory to which the Wightman axioms apply.Therefore,one must build an entirely new mathematical apparatus to treat the resulting quantumfield theory which is drastically different from the Fock space picture to which particle physicists are used to.As a consequence,the mathematical formulation of the theory was the main focus of research in thefield over the past decade.The main achievements to date are the following(more or less in chronological order):i)Kinematical FrameworkThe starting point was the introduction of newfield variables[1]for the gravitational field which are better suited to a background independent formulation of the quantum theory than the ones employed until that time.In its original version these variables were complex valued,however,currently their real valued version,consideredfirst in[2]for classical Euclidean gravity and later in[3]for classical Lorentzian gravity,is preferred because to date it seems that it is only with these variables that one can rigorously define the dynamics of Euclidean or Lorentzian quantum gravity[4].These variables are coordinates for the infinite dimensional phase space of an SU(2) gauge theory subject to further constraints besides the Gauss law,that is,a connection and a canonically conjugate electricfield.As such,it is very natural to introduce smeared functions of these variables,specifically Wilson loop and electricflux functions.(Notice that one does not need a metric to define these functions,that is,they are background independent).This had been done for ordinary gaugefields already before in[5]and was then reconsidered for gravity(see e.g.[6]).The next step was the choice of a representation of the canonical commutation relations between the electric and magnetic degrees of freedom.This involves the choice of a suitable space of distributional connections[7]and a faithful measure thereon[8]which, as one can show[9],isσ-additive.The proof that the resulting Hilbert space indeed solves the adjointness relations induced by the reality structure of the classical theory as well as the canonical commutation relations induced by the symplectic structure of the classical theory can be found in[10].Independently,a second representation,called the loop representation,of the canonical commutation relations had been advocated(see e.g.[11]and especially[12]and references therein)but both representations were shown tobe unitarily equivalent in[13](see also[14]for a different method of proof).This is then thefirst major achievement:The theory is based on a rigorously defined kinematical framework.ii)Geometrical OperatorsThe second major achievement concerns the spectra of positive semi-definite,self-adjoint geometrical operators measuring lengths[15],areas[16,17]and volumes[16,18,19,20,11] of curves,surfaces and regions in spacetime.These spectra are pure point(discete)and imply a discrete Planck scale structure.It should be pointed out that the discreteness is,in contrast to other approaches to quantum gravity,not put in by hand but it is a prediction!iii)Regularization-and Renormalization TechniquesThe third major achievement is that there is a new regularization and renormalization technique[21,22]for diffeomorphism covariant,density-one-valued operators at our dis-posal which was successfully tested in model theories[23].This technique can be applied, in particular,to the standard model coupled to gravity[24,25]and to the Poincar´e gener-ators at spatial infinity[26].In particular,it works for Lorentzian gravity while all earlier proposals could at best work in the Euclidean context only(see,e.g.[12]and references therein).The algebra of important operators of the resulting quantumfield theories was shown to be consistent[27].Most surprisingly,these operators are UV and IRfinite!No-tice that this result,at least as far as these operators are concerned,is stronger than the believed but unprovedfiniteness of scattering amplitudes order by order in perturbation theory of thefive critical string theories,figuratively speaking,we claim that our pertur-bation series converges.The absence of the divergences that usually plague interacting quantumfields propagating on a Minkowski background can be understood intuitively from the diffeomorphism invariance of the theory:“short and long distances are gauge equivalent”.We will elaborate more on this point in future publications.The classical limit of the above mentioned operators will be studied in our companion paper[28]. iv)Spin Foam ModelsAfter the construction of the densely defined Hamiltonian constraint operator of[21,22],a formal,Euclidean functional integral was constructed in[29]and gave rise to the so-called spin foam models(a spin foam is a history of a graph with faces as the history of edges)[30].Spin foam models are in close connection with causal spin-network evolutions[31],state sum models[32]and topological quantumfield theory,in particular BF theory[33].To date most results are at a formal level and for the Euclidean version of thetheory only but the programme is exciting since it may restore manifest four-dimensional diffeomorphism invariance which in the Hamiltonian formulation is somewhat hidden. v)Finally,thefifth major achievement is the existence of a rigorous and satisfactory frame-work[34,35,36,37,38,39,40]for the quantum statistical description of black holes which reproduces the Bekenstein-Hawking Entropy-Area relation and applies,in particular,to physical Schwarzschild black holes while stringy black holes so far are under control only for extremal charged black holes.Summarizing,the work of the past decade has now culminated in a promising starting point for a quantum theory of the gravitationalfield plus matter and the stage is set to address physical questions.In particular,one would like to make contact with the language that particle physicists are more familiar with,that is,perturbation theory.In other words,one should be able to define something like gravitons and photons propagating on afluctuating quantum spacetime.By this we mean the following:Suppose we want to study the semi-classical limit of our quantum gravity theory,that is,a limit in which the gravitationalfield behaves almost classical.This does not mean that we want to treat gravity as a backgroundfield[41],rather we take all the quantumfluctuations into account but try tofind a state with respect to which thosefluctuations(around the Minkowski metric) are minimal.With respect to such a“background state”one can study relative excitations of the gravitationalfield(gravitons)or of matterfields(such as photons).In order to do this we mustfirst develop an appropriate semi-classical framework which we will do in[42,43,44].But even before doing this we must examine the following issue which seems not to have been sufficiently appreciated throughout the literature so far:Namely,the quantum theory is based on certain configuration and conjugate momentum vari-ables respectively,specifically holonomy–and electricflux variables.We stress that we use here non-standardflux variables not previously considered in the literature.These non-localfunctions on the classical phase space are,in particular,used to regularize more complicated composite operators such as the geometrical operators mentioned above.It is already quite remarkable that one can remove the regulator without encountering any UV divergencies! However,in order to be convinced that this regulator-free operator really has the correct clas-sical limit one has to check,for instance,that it has the correct expectation values with respect to semi-classical states.The question arises how such semi-classical states should look like. Now,since thefinal regulator-free operator is actually only densely defined(since it is usually unbounded)one has to employ states which are semi-classical and simultaneously belong to a dense subspace of the Hilbert space.The states which belong to the domain of definition of the operator are labelled by graphsγ.Given such a graphγone can define unambiguously holonomies along its edges as the basic configuration operators labelled byγ,however,the associated(conjugate)momentum operators are largely ambiguous in the sense that any choice of surfaces which are mutually disjoint and are intersected by precisely one edge ofγgives rise to completely identical Poisson brackets between the canonical variables.This then leads to the following problem:Suppose we define a semi-classical state by requiring that the expectation value of the basic holonomy and electricflux operators associated withγtake certain values and satisfy a minimal uncertainty property.Given a point in the classical phase space those values should be the values of the holonomy andflux functions evaluated at that point.However,this makes sense only when we specify the surfaces with respect to which we calculate theflux.We are thus led to invent a new kind of generalized projective family labelled not only by graphs but also by so-called“dual”faces.In particular,we wish to do this already at the classical level by introducing a new kind of generalized projective family of symplectic manifolds.The idea behind all of this is that these symplectic manifolds enable us to discuss in a clean way the quantization procedure and its inverse,the process of taking the classical limit:1.Classical RegularizationSuppose we are given a function on the classical phase space(M,Ω),usually a function F(m)of the connection and the electricfield,m=(A,E).Here M denotes the set of connections and electricfields respectively(a differentiable manifold modelled on a Banach space,see below)andΩis a strong symplectic structure on M.As we cannot defineˆA,ˆE on our Hilbert space directly as operators,we mustfirstfind a substitute Fγ(m)for F which can be written entirely in terms of holonomy andflux variables associated withγ.These variables coordinatize a symplectic manifold(Mγ,Ωγ).We will say that Fγ(m)isa substitute for F(m)provided that1)Fγconverges to F pointwise on M asγ→∞(thegraph becomes infinitelyfine,we will specify this limit below)and2)that the Hamiltonian vectorfield of Fγwith respect toΩγconverges pointwise on M to that of F with respect toΩ.2.Regularized OperatorsThe classical phase spaces(Mγ,Ωγ)turn out to be(in)finite direct products of(copies of)cotangent spaces over the gauge group G equipped with a non-standard symplectic structure and allow for a bonafide quantization by usual geometrical quantization tech-niques.By substituting classical variables for operators defined on a subspace Hγof the Hilbert space and Poisson brackets with respect toΩγby commutators we obtain an operatorˆFγunambiguously defined on Hγup to factor ordering ambiguities.Thus,the phase spaces Mγare much better suited for the quantization of interesting functions F on M as they are automaticallyfinite and we have always control that the quantization has the correct classical limit on Mγ.In other words,quantization and regularization can be neatly separated as individual processes.3.Unregularized OperatorIt turns out that for a large class of functions F including the ones of physical interest the family of operatorsˆFγso obtained provides an operatorˆF consistently defined on a dense subspace of the whole Hilbert space in the sense that its restriction to Hγcoincides withˆFγ.This will be our candidate for a well-defined continuum operator.4.Classical LimitIn order to study the classical limit ofˆF we introduce a generalized projective family of semi-classical statesψtγ,m∈Hγlabelled by the graphγ,a point in m∈M and a classicality parameter t∝¯h.We say thatˆFγis a quantization of Fγ(m)provided that lim t→0<ψtγ,m,ˆFγψtγ,m>=Fγ(m)for each m∈M and thatˆF is a quantization of F provided that lim t→0[limγ→∞<ψtγ,m,ˆFγψtγ,m>]=F(m)for each m∈M.Theses four steps provide then a closed path of how to go from a classical phase space function to an operator and back.As we see,this procedure requires as a classical cornerstone the analysis of the phase spaces(Mγ,Ωγ)which is the subject of the present paper.In particular, one must show that these symplectic manifolds contain a generalized projective sequence that can be identified with(M,Ω).The outline of the paper is as follows:In section two we recall a working collection of material from the kinematical framework of the theory.In section three we derive from the symplectic manifold(M,Ω)for gauge theories with compact gauge groups(in any dimension and on any(globally hyperbolic)manifold)a gen-eralized projective family of(in)finite dimensional symplectic manifolds(Mγ,Ωγ)labelled by graphsγembedded in that manifold.We show that the generalized projective limit symplec-tic manifold of a certain generalized projective sequence agrees with the standard symplectic manifolds(M,Ω)for gauge theories(weighted Sobolev spaces).The purpose of doing this is that the generalized family of symplectic manifolds is much better suited to quantization than the standard gauge theory phase space as outlined above.In section four we propose a substitute Gγfor an important function G on the phase space of any gauge theory,namely the Gauss constraint and show that Gγconverges to G pointwise on M in the generalized projective limit.In sectionfive we derive the quantization of(Mγ,Ωγ)and Gγ.We show thatˆGγis a consistently defined system of cylindrical projections of an operatorˆG whose constraint algebra closes without anomalies.Finally we sketch the last step of the above programme applied to ˆG concerning the classical limit.The proof that this step can be completed will be found in [42,43,44].In section six we complement earlier results obtained by Ashtekar,Corichi and Zapata[45] :These authors considered certain classical functions F on M and quantized them usingΩas a starting point.They obtained operatorsˆF this way which do not commute on certain states fγ∈Hγalthough the classical functions F have vanishing Poisson brackets(with respect toΩ) among each other.This seeming quantum“anomaly”was explained by pointing out that the connection and electricfield of the theory are smeared with distributional rather than smooth test functions.If one uses a smearing with smooth functions then the“anomaly”vanishes, allowing the interpretation that the Poisson brackets of the unsmearedfields is non-vanishing, proportional to a distribution with support contained in a measurable subset of Lebesgue measure zero which is therefore detectable only when smearing with distributional smearing functions.This interpretation therefore removes the apparent contradiction.However,then onenotices that this extended Poisson bracket does not close in an obvious way(the Jacobi identity is not obeyed in an obvious way).This was shown not to be an obstacle to quantization by recalling that it is not necessary to base the quantization on Poisson brackets but that one can instead base it on the Lie algebra of vectorfields on M which always obey the Jacobi identity and is always closed.We show that the non-commutativity of these operators has a natural explanation from the point of view of the symplectic manifolds(Mγ,Ωγ):1)We observe that we canfind,for each of the above choices ofγ,functions Fγ=F which can be considered as functions on Mγas well.Furthermore,the functions Fγdo have non-vanishing Poisson brackets among each other,both with respect toΩand with respect toΩγ(actually, their brackets with respect toΩγfollow from those with resepct toΩ).2)The quantization of these new functions is such thatˆFγandˆF agree on Hγ.3)The commutator algebra of theˆF on Hγis precisely the one to be expected from the Poisson bracket structure of the Fγ.In conclusion,the unexpected non-commutativity observed in[45]can be related to a quan-tization ambiguity.If we insist on a Poisson bracket–comutator correspondence principle, however,then one cannot accept the F as classical limit ofˆF but must instead consider the Fγ.Finally,in an appendix we write the symplectic structureΩγfor G=U(1),SU(2)in the language of differential forms which could be useful for future research.2PreliminariesIn this section we will recall the main ingredients of the mathematical formulation of diffeo-morphism invariant quantumfield theories of connections with local degrees of freedom in any dimension and for any compact gauge group.See[10]and references therein for more details.Let G be a compact gauge group,Σa D−dimensional manifold which admits a principal G−bundle with connection overΣ.Let us denote the pull-back toΣof the connection by local sections by A i a where a,b,c,..=1,..,D denote tensorial indices and i,j,k,..=1,..,dim(G) denote indices for the Lie algebra of G.We will denote the set of all smooth connections by A and endow it with a globally defined metric topology of the Sobolev kinddρ[A,A′]:= N Σd D xpγ(A)={h e(A)}e∈E(γ).The set of such functions is denoted byΦγ.Holonomies are invariant under reparameterizations of the edge and in this article we take edges always to be analytic diffeomorphisms between[0,1]and a one-dimensional submanifold ofΣ.Gauge transformations are functions g:Σ→G;x→g(x)and they act on holonomies as h e→g(e(0))h e g(e(1))−1.A particularly useful set of cylindrical functions are the so-called spin-netwok functions [48,49,13].A spin-network function is labelled by a graphγ,a set of irreducible representations π={πe}e∈E(γ)(choose from each equivalence class of equivalent representations once and for all afixed representant),one for each edge ofγ,and a set c={c v}v∈V(γ)of contraction matrices, one for each vertex ofγ,which contract the indices of the tensor product⊗e∈E(γ)πe(h e)in such a way that the resulting function is gauge invariant.We denote spin-network functions as T I where I={γ, π, c}is a compound label.One can show that these functions are linearly independent.The set offinite linear combinations of spin-network functions forms an Abelian∗algebra B of functions on A.By completing it with respect to the sup-norm topology it becomes an Abelian C∗algebra(here the compactness of G is crucial).The spectrumC is called the quantum configuration space. This space is equipped with the Gel’fand topology,that is,the space of continuous functions C0(A is given by the Gel’fand transforms of elements of B.Recall that the Gel’fand transform is given by˜f(¯A):=¯A(f)∀¯A∈A with this topology is a compact Hausdorffspace.Obviously,the elements of A are contained inA are,however,distributional.The idea is now to construct a Hilbert space consisting of square integrable functions onA,dµ0)obtained by completing their finite linear spanΦ.An equivalent definition ofA is in one to one correspondence,via the surjective map H defined below,with the setA∋¯A→H¯A∈Hom(X,G)where X∋e→H¯A(e):=¯A(h e)=˜h e(¯A)∈G and˜h e is the Gel’fand transform of the function A∋A→h e(A)∈G.Consider now the restriction of X to Xγ,the groupoid of composable edges of the graphγ.One can then show that the projective limit of the corresponding cylindrical sets A′.Moreover,we have{{H(e)}e∈E(γ);H∈A}=G|E(γ)|.Let now f∈B be a function cylindrical overγthen(˜f)=χµA which has measure zero with respect toµ0.This concludes the definition of the kinematical Hilbert space H,of the quantum configu-ration space3Symplectic Manifolds Labelled by Graphs3.1Standard Continuum Symplectic StructuresLet usfirst recall the usual infinite dimensional symplectic geometry that underlies gauge the-ories.Let F a i be a Lie algebra valued vector density testfield of weight one and let f i a be a Lie algebra valued covector testfield.Let,as before A i a be a the pull-back of a connection toΣand consider a vector bundle of electricfields,that is,of Lie algebra valued vector densities of weight one whose bundle projection toΣwe denote by E a i.We consider the smeared quantitiesF(A):= Σd D xF a i A i a and E(f):= Σd D xE a i f i a(3.1) While both are diffeomorphism covarinat it is only the latter which is gauge covariant,one reason to consider the singular smearing functions discussed below.The choice of the space of pairs of testfields(F,f)∈S depends on the boundary conditions on the space of connections and electricfields which in turn depends on the topology ofΣand will not be specified in what follows.Consider the set M of all pairs of smooth functions(A,E)onΣsuch that(3.1)iswelldefined for any(F,f)∈S.We wish to endow it with a manifold structure and a symplectic structure,that is,we wish to turn it into an infinite dimensional symplectic manifold.We define a topology on M through the metric:dρ,σ[(A,E),(A′,E′)]:= N Σd D x[dρσ[(A,E),(A(0),E(0))](3.4) The norm(4.4)is of course no longer gauge and diffeomorphism covariant since thefields A(0),E(0)do not transform,they are backgrondfields.We need it,however,only in order to encode the fall-offbehaviour of thefields which are independent of gauge–and diffeomorphism covariance.Notice that the metric induced by this norm coincides with(3.2).In the terminology of weighted Sobolev spaces the completion of E in the norm(3.4)is called the Sobolev spaceH20,ρ×H20,σ−1,see e.g.[51].We will call the completed space E again and its image under ϕ−1,M again(the dependence ofϕon(A(0),E(0))will be suppressed).Thus,E is a normed, complete vector space,that is,a Banach space,in fact it is even a Hilbert space.Moreover, we have modelled M on the Banach space E,that is,M acquires the structure of a(so far only topological)Banach manifold.However,since M can be covered by a single chart and the identity map on E is certainly C∞,M is actually a smooth manifold.The advantage of modelling M on a Banach manifold is that one can take over almost all the pleasant properties from thefinite dimensional case to the infinite dimensional one(in particular,the inverse function theorem).Next we study differential geometry on M with the standard techniques of calculus on infinite dimensional manifolds(see e.g.[52,53]).We will not repeat all the technicalities of the definitions involved,the interested reader is referred to the literature quoted.i)A function f:M→C isdifferentiable at u=ϕ(m),that is,there exist bounded linear operators Dg u,Rg u:E→||v||=0.(3.5) Df m:=Dg u is called the functional derivative of f at m(notice that we identify,as usual,the tangent space of M at m with E).The definition extends in an obvious way to the case wherev)A differential form of degree p on M or p−form is a cross section of thefibre bundle of completely skew continuous p−linear forms.Exterior product,pull-back,exterior differ-ential,interior product with vectorfields and Lie derivatives are defined as in thefinite dimensional case.Definition3.1Let M be a differentiable manifold modelled on a Banach space E.A weak respectively strong symplectic structureΩon M is a closed2-form such that for all m∈M the mapΩm:T m(M)→T′m(M);v m→Ω(v m,.)(3.7) is an injection respectively a bijection.Strong symplectic structures are more useful because weak symplectic structures do not allowus to define Hamiltonian vectorfields through the definition DL+iχL Ω=0for differentiableL on M and Poisson brackets through{f,g}:=Ω(χf,χg).Thus we definefinally a strong symplectic structure for our case byΩ((f,F),(f′,F′)):= Σd D x[F a i f i′a−F a′i f i a](x)(3.8)for any(f,F),(f′,F′)∈E.To see thatΩis a strong symplectic structure we observefirst that the integral kernel ofΩis constant so thatΩis clearly exact,so,in particular,closed.Next,let θ∈E′≡E.To show thatΩis a bijection it suffices to show that it is a surjection(injectivity follows trivially from linearity).We mustfind(f,F)∈E so thatθ(.)=Ω((f,F),.).Now by the Riesz lemma there exists(fθ,Fθ)∈E such thatθ(.)=<(fθ,Fθ),.>where<.,.>is the inner product induced by(3.4).Comparing(3.4)and(3.8)we see that we have achieved our goal provided that the functionsF a i:=ρabdet(ρ)and det(ρ)σcdρcaρdb/det(σ)andρaband DE a i(x)m·(f,F)=F a i(x)as follows from the definition.Coming back to the choice of S,it will in general be a subspace of E so that(3.1)still converges.We can now compute the Poisson brackets between the functions F(A),E(f)on M andfind{E(f),E(f′)}={F(A),F′(A)}=0,{E(f),A(F)}=F(f)(3.13) Remark:In physicists notation one often writes(DL m)i a(x):=δLR D−1.2)S has no boundary,∂S=∅,i.e.it is open.3)S is contained in the domain of a chart ofΣ.4)S is maximal,that is,there does not exist any S′⊂∂∆properly containing S which satisfies 1),2)and3).ii)As S is an open submanifold ofΣof codimension one and for some∆∈P,S⊂∂∆is contained in the domain of a chart(U,ϕ)of an atlas ofΣthere exists an open subset V⊂U containing S,divided by S into two halves and a diffeomorphismϕ′that maps V,S respectively to。

Modern Applied Statistics with S-PlusBenny YakirAbstractThese notes are based on the book Modern Applied Statistics withS-Plus by W.N.Venables and B.D.Ripley.No originality is claimed.***************************************************************************************** Contents1Starting R41.1Installing R (4)1.2Basic R Commands (4)1.3An Example of an R Session (4)2Objects in R62.1Vectors and matrices (6)2.2Lists (7)2.3Factors (8)2.4Data frames (8)2.5Homework (8)3Functions103.1Built-in functions (10)3.2Writing functions (11)3.3Plotting functions (13)3.4Home Work (13)4Linear models144.1A simple regression example (14)4.2Model formulae (14)4.3Diagnostics (16)4.4Model selection (17)4.5Homework (18)15Generalized Linear Models(GLM)195.1The basic model (19)5.2Analysis of deviance (20)5.3Fitting the model (21)5.4Generic functions and method functions (21)5.5A small Binomial example (21)5.6Fitting other families (23)5.7Frames (23)5.8A Poisson example (23)5.9Home Work (24)6Robust methods266.1Introduction (26)6.2The lqs function for resistant regression (26)6.3Some examples (28)6.4Home work (31)7Non-linear Regression337.1The basic model (33)7.2A small example (33)7.3Attributes (35)7.4Prediction (35)7.5Profiles (38)7.6Home work (39)8Non-parametric and semi-parametric Regression408.1Non-parametric smoothing (40)8.2The projection-pursuit model (41)8.3A simulated example of PP-regression (42)8.4Home work (44)9Tree-based models459.1An example of a regression tree (46)9.2An example of a classification tree (49)9.3Homework (52)10Multivariate analysis5310.1Multivariate data (53)10.2Graphical methods (54)10.3Principal components analysis (54)210.4cluster analysis (55)10.5Home work (57)10.6Classification (58)10.7Multivariate linear models (64)10.8Canonical correlations (65)10.9Project3 (66)31Starting R1.1Installing RBasic information on R can be found at the URL/r/r.htmlInformation on how to download and install R can be found in thefile:installing-R.doc1.2Basic R Commandsfix()edits the function in an emacs window.ls()list thefiles in.Data.rm(filename)delete thefilefilename.q()quit R.When the program is ready for your commands you will get the“>”prompt.When you push Enter,but the command is not completed yet, you will get the“+”prompt.You can change the working directory using File→Change dir.The current environment(image)using File→Save Image.You can then restore the image with File→Load Image.To get help either use Help→R language(standard),and then type the name of the comand/function you like to get more information on.Alter-natively,you can use Help→R language(html),which uses an internet-style browser,1.3An Example of an R Session>bpiq<-read.table("bpiq.dat",s=c("dep","iq","bp")) >hist(bpiq[,"iq"],xlab="IQ",main="IQ:all children")>iq.d<-bpiq[bpiq[,"dep"]=="d","iq"]>iq.nd<-bpiq[bpiq[,"dep"]=="nd","iq"]>hist(iq.d,xlab="IQ",main="IQ:d children")4>hist(iq.nd,xlab="IQ",main="IQ:nd children")>t.test(iq.d,iq.nd)Welch Two Sample t-testdata:iq.d and iq.nd t=-1.6388,df=15.491,p-value=0.1214 alternative hypothesis:true difference in means is not equal to0 95percent confidence interval:-26.922823 3.481156sample estimates:mean of x mean of y101.0667112.7875>t.test(iq.d,iq.nd,var.equal=T)Two Sample t-testdata:iq.d and iq.nd t=-2.4801,df=93,p-value=0.01494 alternative hypothesis:true difference in means is not equal to0 95percent confidence interval:-21.105819-2.335847sample estimates:mean of x mean of y101.0667112.7875>q()52Objects in R2.1Vectors and matricesObjects are identified in S-Plus by their attributes.>mydata<-c(2.9,3.4,3.4,3.7)>colour<-c("red","green","blue")>x1<-25:30>length(mydata)[1]4>mode(mydata)[1]"numeric">mode(x1)[1]"numeric">colour[2:3][1]"green""blue">x1[-1][1]2627282930>colour!="green"[1]TRUE FALSE TRUE>colour[colour!="green"][1]"red""blue">names(mydata)<-c(’a’,’b’,’c’,’d’)>mydataa b c d2.93.43.43.7>letters [1]"a""b""c""d""e""f""g""h""i""j""k""l""m""n""o" "p""q""r"[19]"s""t""u""v""w""x""y""z">mydata[letters[1:2]]a b2.93.4>matrix(x1,2,3)[,1][,2][,3][1,]252729[2,]262830>matrix(x1,2,3,byrow=T)[,1][,2][,3]6[1,]252627[2,]282930>x2<-matrix(x1,2,3)>x2[1,1][1]25>x1*3[1]757881848790>x1+x1[1]505254565860>x1+x2[,1][,2][,3][1,]505458[2,]525660>rbind(x2,x2)[,1][,2][,3][1,]252729[2,]262830[3,]252729[4,]2628302.2ListsCollections of other S-Plus objects.>alist<-list(c(0,1,2),1:10,"Name")>alist[[1]]:[1]012[[2]]:[1]12345678910[[3]]:[1]"Name">blist<-list(x=matrix(1:10,ncol=2),y=c("A","B"),z=alist) >blist$x[,1][,2][1,]167[2,]27[3,]38[4,]49[5,]510>blist$z[[3]][1]"Name"2.3FactorsSpecial type of vectors.Hold categorical variables.>height<-factor(c("H","L","M","L","H"),levels=c("L","M","H","VH")) >height[1]H L M L HLevels:L M H VH>height<-ordered(c("H","L","M","L","H"),levels=c("L","M","H","VH")) >height[1]H L M L HLevels:L<M<H<VH2.4Data framesUsed to store Data matrix.List of variables of the same length.>bpiq[1:5,c(2,3)]iq bp11034212412312494104359632.5Homework1.Turn thefile bpiq.dat into an R data frame with the function read.table.2.Read the helpfiles on the functions hist and t.test.3.Produce an histogram with a different number of bars.84.What would happen if we would choose the argument probability=Tin the function hist?5.Perform a one-side t.test,forα=0.1,0.05,0.01.6.Test(α=0.05)that the mean of iq.d is100.Repeat this analysis foriq.nd.93FunctionsThere are many built-in functions in S-Plus.One can write his/her own functions easily.The structure of a function is:function(arguments)expressions3.1Built-in functions>data.ex<-matrix(1:50,ncol=5)>sum(data.ex)[1]1275>mean(data.ex)[1]25.5>var(data.ex)[,1][,2][,3][,4][,5][1,]9.1666679.1666679.1666679.1666679.166667[2,]9.1666679.1666679.1666679.1666679.166667[3,]9.1666679.1666679.1666679.1666679.166667[4,]9.1666679.1666679.1666679.1666679.166667[5,]9.1666679.1666679.1666679.1666679.166667>cor(data.ex)[,1][,2][,3][,4][,5][1,]11111[2,]11111[3,]11111[4,]11111[5,]11111>apply(data.ex,2,mean)[1] 5.515.525.535.545.5>apply(data.ex,1,sum)[1]105110115120125130135140145150>max(data.ex)[1]50>min(data.ex)[1]1>range(data.ex)10[1]1503.2Writing functions>sumsq<-function(){}>fix(sumsq)(In the editor window write:)function(x){ssq<-sum(x*x)scb<-sum(x*x*x)result<-list(ssq=ssq,scb=scb)result}(Exit and save.)>sumsq(1:10)$ssq:[1]385$scb:[1]3025>grid.cal<-function(){}>fix(grid.cal)function(x,y){grid<-matrix(0,length(x),length(y))for(i in1:length(x)){for(j in1:length(y))grid[i,j]<-sqrt(x[i]^2+y[j]^2) }grid}>grid.cal(1:3,1:4)[,1][,2][,3][,4][1,]1.4142142.2360683.1622784.123106[2,]2.2360682.8284273.6055514.472136[3,]3.1622783.6055514.2426415.000000>outer(1:3,1:4,"+")11[,1][,2][,3][,4][1,]2345[2,]3456[3,]4567>h.dist<-function(x,y)sqrt(x^2+y^2)>outer(1:3,1:4,h.dist)[,1][,2][,3][,4][1,]1.4142142.2360683.1622784.123106[2,]2.2360682.8284273.6055514.472136[3,]3.1622783.6055514.2426415.000000>system.time(grid.cal(1:100,1:100))[1]NA NA2.46NA NA>system.time(grid.cal(1:500,1:500))[1]NA NA61.41NA NA>system.time(outer(1:100,1:100,h.dist))[1]NA NA0.04NA NA>system.time(outer(1:500,1:500,h.dist))Error:heap memory(6144Kb)exhausted[needed1953Kb more] See"help(Memory)"on how to increase the heap size. Timing stopped at:NA NA0.53NA NA>sumpow<-function(){}>fix(sumpow)function(x,pow=2:3){l<-length(pow)result<-vector("list",length=l)for(i in1:l)result[[i]]<-sum(x^pow[i])result}>sumpow(1:10)[[1]]:[1]385[[2]]:[1]302512>sumpow(1:3,c(3,5,6))[[1]]:[1]36[[2]]:[1]276[[3]]:[1]7943.3Plotting functionsThe R environment provides comprehensive graphical facilities:Many higher-level plotting functions which are controlled by a verity of parameters and lower-level plotting functions which can be used in order to add to existing plots.>x<-rnorm(10)>y<-rnorm(10)>l<-paste("(x",1:10,",y",1:10,")",sep="")>plot(x,y)>text(x,y,l)>title("Ten points")3.4Home Work1.Write a function that calculates central moments of a vector.2.Read the helpfile on the function tapply.e the function tapply to calculate thefirst4central moments of thend and d children iq and bp.4.Write a function that returns the indices of elements of a vector whichare more then2std from the vector mean.5.Identify the children which are more than2std from the mean(a)in the full list.(b)within each subgroup.6.Write a function that produces a bubble plot,i.e.given3vectors x,yand r it plots circles of radius r,centered at(x,y).(Use the function symbols.)Generate3random vectors of length20and try the function.134Linear models4.1A simple regression exampleLinear models are the core of classical statistics:Regression,ANOVA,Anal-ysis of covariance and much more.R’S basicfitting function is lm(and aovfor ANOVA).Here we consider a simple case of linear regression.>trees.dat<-read.table("trees.dat",s=c("Diameter", +"Height","Volume"))>attach(trees.dat)>trees.fit<-lm(Volume~Diameter+Height)>summary(trees.fit)Call:lm(formula=Volume~Diameter+Height)Residuals:Min1Q Median3Q Max-6.4065-2.6493-0.2876 2.20038.4847Coefficients:Estimate Std.Error t value Pr(>t)(Intercept)-57.98778.6382-6.7132.75e-07***Diameter4.70820.264317.816<2e-16***Height0.33930.1302 2.6070.0145*---Signif.codes:0‘***’0.001‘**’0.01‘*’0.05‘.’0.1‘’1Residual standard error:3.882on28degrees of freedom Multiple R-Squared:0.948,Adjusted R-squared:0.9442F-statistic:255 on2and28degrees of freedom,p-value:04.2Model formulaeThe basic formula for a linear model is:Y=β X+ ,14where Y is a random vector(or matrix),X is a design matrix of known constants,βis an unknown vector of parameters and is a random vector.For example,for multiple regression with three explanatory variables the modelY=β0+β1x1+β2x2+β3x3+is specified by the formulay˜x1+x2+x3Note that•The left hand side is a vector or a matrix.•The intercept is implicit.•In the right hand side vectors or matrices,separated by+.•To remove intercept use y˜−1+x1+x2+x3.Factors generate columns in X to allow separate parameters for each level.Hence,if a1and a2are two factors theny˜a1+x1is a model for parallel regression(analysis of covariance)andy˜a1+a2is a two-way ANOVA,with no interaction.To get the model with interaction usey˜a1+a2+a1:a2,ory˜a1∗a2.The nested modely=β0+αa1+βa1x1+is formulated by y˜a1+a1:x1,whereas the formula y˜a1∗x1corresponds toy=β0+αa1+β1x1+βa1x1+Terms like a1∗a2∗a3would generate the modela1+a2+a3+a1:a2+a1:a3+a2:a3+a1:a2:a3,15and terms like(a1+a2+a3)ˆ2correspond toa1+a2+a3+a1:a2+a1:a3+a2:a3.To allow operators to be used with their arithmetic meaning use the I(·) function.For exampleprofit^I(dollar.inc+1.55*pound.inc)Tofit polynomials use the function poly,hencey~poly(x1,x2,3)corresponds to orthogonal polynomials of degree3in x1and x2.4.3DiagnosticsThe basic tool for examining thefit is the residuals.They are not indepen-dent.var(e)=σ2[I−H],where H=X(X X)−1X is the hat matrix and e is the vector of residuals.If the leverage h ii of the observation is large(more than3or2times p/n,where p is the number of explanatory variables and n the number of observations) than the observation is influential.The standardized residualse =es√1−hor the jackknifed residualse∗=y−ˆy(·)var(y−ˆy(·))=e n−p−e 2n−p−11/2are used to examine normality and identify outlayers with the normal or half-normal probability plot.The residual plot is used to identify non-random patterns.>trees.res<-residuals(trees.fit)>trees.prd<-predict(trees.fit)>s<-summary(trees.fit)$sigma>h<-lm.influence(trees.fit)$hat>trees.res<-trees.res/(s*sqrt(1-h))16>par(mfrow=c(2,1))>plot(trees.dat[,"Diameter"],trees.res,xlab="Diameter",+ylab="Std.Residuals")>abline(h=0,lty=2)>title("Std.Residuals vs.Diameter")>plot(trees.dat[,"Height"],trees.res,xlab="Height",+ylab="Std.Residuals")>abline(h=0,lty=2)>title("Std.Residuals vs.Height")>par(mfrow=c(1,1))>plot(trees.prd,trees.res,xlab="Predicted Volume",+ylab="Std.Residuals")>abline(h=0,lty=2)>title("Std.Residuals vs.Fitted Values")>par(pty="s")>qqnorm(trees.res,ylab="Std.Residuals")>title("Normal plot of Std.Residuals")>plot(1:31,h,type="n",xlab="index",+ylab="Diagonal of hat matrix")>abline(h=mean(h))>segments(1:31,h,1:31,mean(h))>title("Leverage measure")4.4Model selectionStatistical hypothesis testing is all about comparing between models.>trees1.fit<-lm(Volume~Diameter+I(Diameter^2)+Height)>anova(trees1.fit,trees.fit)Analysis of Variance TableModel1:Volume~Diameter+I(Diameter^2)+Height Model2: Volume~Diameter+HeightRes.Df Res.Sum Sq Df Sum Sq F value Pr(>F)127186.01228421.92-1-235.9134.2433.13e-06***---Signif.codes:0‘***’0.001‘**’0.01‘*’0.05‘.’0.1‘’117A disadvantage of this approach that it does not take into account the complexity of the models.In principle,one prefers simpler models—i.e. models with less parameters.Hence,one may want to consider introducing a penalty which grows with the number of parameters which are used for thefit.This is known as Information Criteria.Examples include:ACI:2[maxθ∈K (θ)−maxθ∈H (θ)]+2[dim(K)−dim(H)].BCI:2[maxθ∈K (θ)−maxθ∈H (θ)]+log n[dim(K)−dim(H)].C p:ˆσ−2[Sum of Squares Diff.]+2[dim(K)−dim(H)].Given a nested sequence H⊂K1⊂K2⊂···⊂K J the model with the smallest information criteria should be preferred4.5Homework1.The data frame cars.dat gives the speed of cars and the distances takento stop.Note that the data were recorded in the1920s.It contains50 observations on2variable:speed:(numeric)Speed(mph).dist:(numeric)Stopping distance(ft).Analyze the relation between these two variables.2.The data frame cereales.dat gives the calories,fat andfiber content ofdifferent types of cereals.It contains77observations on3variable:cal:(numeric)Calories.fat:(numeric)Total fat.fiber:(factor)Fibers.Analyze the calories as a function of total fat and thefiber content.185Generalized Linear Models(GLM)5.1The basic modelGLM extends linear models to accommodate both non-normal response and transformations to linearity.Assumptions:•The distribution of Y i is of the form:(y i)=exp[A i{y iθi−γ(θi)}/ψ+τ(y i,ψ/A i)],fθiwhereψis a(nuisance)scale parameter,A i known weights.The distri-bution is controlled by theθi s•The mean of Y,µ,is a function of a linear combination of the predictors:µ=m(β x)=l−1(β x).l is called the link function.If l=(γ )−1thenθ=β x and l is called the canonical link function.Examples:Normal:log f(y)={yµ−µ2/2}/σ2−12{y2/σ2+log[2πσ2]}.θ=µγ(θ)=θ2/2(µ)=µlψ=σ2.Poisson:log f(y)=y logµ−µ−log(y!).θ=logµγ(θ)=eθµ=eθl(µ)=log(µ)ψ=1.19Binomial:Y=k/n.log f(y)=n y log p1−p+log(1−p) +log n ny .γ(θ)=log 1+eθµ=eθ/ 1+eθl(µ)=log(µ/(1−µ))ψ=1A=n.The parameters are estimated using MLE.The maximum is calculated using iterative regression.5.2Analysis of devianceThe parameters in the saturated model S are not constrained.The mean of the i th observation is estimated by the observation itself.The deviance of the model M,M⊂S,is:D M=2ni=1A i y iˆθS−γ(ˆθS) − y iˆθM−γ(ˆθM) .This is the unscaled generalized log-likelihood-ratio statistic.In the Gaussian familyD M/ψ∼χ2n−p,henceˆψ=D M/(n−p)is unbiased.In some other cases these relations maybe approximately true or not true at all.Whenψis known,testing thefit of the model M relative to the nullmodel M0,where M0⊂M,is performed with chi-square test of(D M0−D M)/ψ.Whenψis unknown,testing is performed with a F test of(D M0−D M)/(ˆψ(p−q)).An alternative approach is to apply the AIC criteria,whereAIC=D M+2pˆψ(ˆψshould be the same across all models.)205.3Fitting the modelThe basicfitting function is glm,for which the basic arguments areglm(formula,family,data,weights,control)formula:of the linear predictor.family:gives the family name with additional information.For example: function=binomial(link=probit)fits a binomial response with the probit link.control:of the iterative process.For example maxit determines the maxi-mal number of iterations.5.4Generic functions and method functionsR functions are designed to be as general as possible.The function summary, for example,can be used on many objects.This is achieved by a construction which includes a generic function which is associated with method functions.A method function performs the appropriate operation on an object of a specific class.For example,summary.glm produces a summary of objects of class glm.Each object has among its attributes a vector class,which is used by the generic function in order to call the appropriate method function.Generic functions with methods for glm include coef,resid,print,sum-mary,anova,predict,deviance.5.5A small Binomial example>options(contrast=c("contr.treatment","contr.poly"))>ldose<-rep(0:5,2)>numdead<-c(1,4,9,13,18,20,0,2,6,10,12,16)>sex<-factor(rep(c("M","F"),c(6,6)))>SF<-cbind(numdead,numalive=20-numdead)>budworm.fit<-glm(SF~sex*ldose,family=binomial)>summary(budworm.fit)21Call:glm(formula=SF~sex*ldose,family=binomial)Deviance Residuals:Min1Q Median3Q Max-1.39849-0.32094-0.075920.38220 1.10375Coefficients:Estimate Std.Error z value Pr(>z)(Intercept)-2.99350.5525-5.4186.02e-08***sex0.17500.77810.2250.822ldose0.90600.16715.4245.84e-08***sex.ldose0.35290.2699 1.3070.191---Signif.codes:0‘***’0.001‘**’0.01‘*’0.05‘.’0.1‘’1 (Dispersion parameter for binomial family taken to be1)Null deviance:124.8756on11degrees of freedomResidual deviance: 4.9937on8degrees of freedom AIC:43.104Number of Fisher Scoring iterations:3>plot(c(1,32),c(0,1),type="n",xlab="dose",ylab="prob",log="x")>text(2^ldose,numdead/20,as.character(sex))>ld<-seq(0,5,0.1)>lines(2^ld,predict(budworm.fit,data.frame(ldose=ld,+sex=factor(rep("M",length(ld)),levels=levels(sex))),+type="response"))>lines(2^ld,predict(budworm.fit,data.frame(ldose=ld,+sex=factor(rep("F",length(ld)),levels=levels(sex))),+type="response"))>anova(update(budworm.fit,.~sex+ldose+factor(ldose)),test="Chisq") Analysis of Deviance TableModel:binomial,link:logitResponse:SFTerms added sequentially(first to last)22Df Deviance Resid.Df Resid.Dev P(>Chi)NULL11124.876sex1 6.07710118.7990.014ldose1112.0429 6.7570.000factor(ldose)4 1.7445 5.0130.7835.6Fitting other familiesThe families of distribution available for glm are the gaussian,binomial,poisson,inverse.gaussian and gamma.Other families can be defined.The function make.family can be used with arguments:name:The name of the family.link:A list with the link function,its inverse,and its derivative.variance:A list with the variance and deviance functions.5.7FramesAn evaluation frame is a list that associates names with values.The frameof the session is frame0.Any expression evaluated in the interactive levelis evaluated in frame1.Frames are added as evaluations become morecomplex.A name is searched for in the local frame.If it is not there thesearch moves to frame1,then to frame0and then through the search path.5.8A Poisson example>quine.dat<-read.table("quine.dat",header=T)>attach(quine.dat)>glm(Days~.^4,family=poisson,data=quine.dat)Call:glm(formula=Days~.^4,family=poisson,data=quine.dat) Coefficients:(Intercept)Eth Sex AgeF13.0564-0.1386-0.4914-0.622723AgeF2AgeF3Lrn Eth.Sex-2.3632-0.3784-1.9577-0.7524Eth.AgeF1Eth.AgeF2Eth.AgeF3Eth.Lrn0.1029-0.55460.0633 2.2588Sex.AgeF1Sex.AgeF2Sex.AgeF3Sex.Lrn0.4092 3.1098 1.1145 1.5900AgeF1.Lrn AgeF2.Lrn AgeF3.Lrn Eth.Sex.AgeF12.6421 4.8585NA-0.3105Eth.Sex.AgeF2Eth.Sex.AgeF3Eth.Sex.Lrn Eth.AgeF1.Lrn0.34690.8329-0.1639-3.5493Eth.AgeF2.Lrn Eth.AgeF3.Lrn Sex.AgeF1.Lrn Sex.AgeF2.Lrn -3.3315NA-2.4285-4.1914 Sex.AgeF3.Lrn Eth.Sex.AgeF1.Lrn Eth.Sex.AgeF2.Lrn Eth.Sex.AgeF3.Lrn NA 2.1711 2.1029NA Degrees of Freedom:145Total(i.e.Null);118ResidualNull Deviance:2074Residual Deviance:1174AIC:1818>days.mean<-tapply(Days,list(Eth,Sex,Age,Lrn),mean)>days.var<-tapply(Days,list(Eth,Sex,Age,Lrn),var)>days.std<-sqrt(days.var)>plot(mays.mean,days.std)5.9Home Worke the rep and seq functions to produce the vectors:123412341234123444443333222211111223334444555552.Re-analyze the budworm data.Replace ldose by ldose-3and see that sexis significant.What is the predicted death rate at dose=7(i)if the sex isunknown?(ii)if sex=“M”?3.Write the density of negative-binomial family in the exponential-familyform.What areθ,γ(θ),ψ,A,and the canonical link function?4.The data set quine.dat contains the outcome of an observational study onnumber of days absent from school during one year in a town in Australia.The kids are classified by four factors:24Eth:Ethnic group.A=aboriginal,N=non-aboriginal.Sex:M,F.Age:F0=Primary,F1=first form,F3=second form,F4=third form. Lrn:AL=average learner,SL=slow learner.Investigate this data set using glm.Identify important factors and inter-actions.256Robust methods6.1IntroductionRobust methods are methods which will work under a wide spectrum of circumstances.Two of the main aspects of robustness are:1.Resistance to outliers.Methods with high breakdown point.The estimatesare not effected much even if the value of a large portion of the observationis given an arbitrary value.2.Robustness to distributional assumptions.The method is still efficienteven if the distributional assumptions are violated.The usual least squares methods are neither resistant nor robust.6.2The lqs function for resistant regressionUsagelqs(x,...)lqs.formula(formula,data=NULL,...,method=c("lts","lqs","lms","S","model.frame"),subset,na.action=na.fail,model=TRUE,x=FALSE,y=FALSE,contrasts=NULL)lqs.default(x,y,intercept,method=c("lts","lqs","lms","S"),quantile,control=lqs.control(...),k0=1.548,seed,...) lmsreg(...)ltsreg(...)print.lqs(x,digits,...)residuals.lqs(x)Argumentsformula a formula of the form y~x1+x2+...{}{}.data data frame from which variables specified in formula are preferentially to be taken.subset An index vector specifying the cases to be used in fitting.(NOTE:If given,this argument must be named exactly.)na.action A function to specify the action to be taken if NAs are found.The default action is for the procedure to fail.26An alternative is na.omit,which leads to omission of caseswith missing values on any required variable.(NOTE:Ifgiven,this argument must be named exactly.)x a matrix or data frame containing the explanatory variables.y the response:a vector of length the number of rows of x.intercept should the model include an intercept?method the method to be used.model.frame returns the model frame: for the others see the Details ing lmsreg or ltsregforces"lms"and"lts"respectively.quantile the quantile to be used:see Details.This is over-ridden if method="lms".control additional control items:see Details.seed the seed to be used for random sampling:see.Random.seed.The current value of.Random.seed will be preserved if it is set.. ...arguments to be passed to lqs.default or lqs.control.DescriptionFit a regression to the good points in the dataset,thereby achievinga regression estimator with a high breakdown point.lmsreg and ltsregare compatibility wrappers.DetailsSuppose there are n data points and p regressors,including any intercept. The first three methods minimize some function of the sorted squared residuals.For methods"lqs"and"lms"is the quantile squared residual, and for"lts"it is the sum of the quantile smallest squared residuals. "lqs"and"lms"differ in the defaults for quantile,which arefloor((n+p+1)/2)and floor((n+1)/2)respectively.For"lts"the defaultis‘floor(n/2)+floor((p+1)/2)’.The"S"estimation method solves for the scale s such that the average of a function chi of the residuals divided by s is equal to a given constant. The control argument is a list with components item{psamp}{the size of each sample.Defaults to p.}item{nsamp}{the number of samples or"best" or"exact"or"sample".If"sample"the number chosen is min(5*p,3000), taken from Rousseeuw and Hubert(1997).If"best"exhaustive enumeration is done up to5000samples:if"exact"exhaustive enumeration will be attempted however many samples are needed.}item{adjust}{should the intercept be optimized for each sample?}27。

（考试时间：90分钟，满分：100分）一、选择题（每题2分，共30分）1. Which word has the same sound as "weather"?A. WhetherB. WearC. WhereD. Wire2. Choose the correct form of the verb in brackets.He ______ (go) to the library every weekend.A. goB. goesC. goingD. gone15. What is the plural form of "child"?A. ChildsB. ChildrenC. ChildesD. Childs'二、判断题（每题1分，共20分）1. "Cook" and "book" have the same pronunciation. ( )2. "She don't like apples" is a correct sentence. ( )20. "The cat is playing with its toy" is an example of a passive sentence. ( )三、填空题（每空1分，共10分）1. I ______ (to be) a teacher when I grow up.2. They ______ (to watch) a movie last night.10. There ______ (to be) a lot of traffic in the city center.四、简答题（每题10分，共10分）1. What is the past participle of "go"?2. Write a sentence using the future perfect tense.五、综合题（1和2两题7分，3和4两题8分，共30分）1. Translate the following sentences into English:我昨天买了一本书。

家庭最不可缺作文题目：The Unbreakable Bond of FamilyHome, a sanctuary from the tempests of life, is much more than a physical structure; it is a living entity, pulsating with emotions, memories, and aspirations. At its core lies the most indispensable element of all: the unbreakable bond of family. This essay delves into the essence of family, examining its multifaceted role in shaping individual lives, fostering emotional well-being, and providing an anchor of stability in an ever-changing world.The foundation of the family bond is love, an unconditional and enduring affection that transcends the ebb and flow of daily life. It is the glue that binds parents, siblings, and extended relatives together, creating a safe haven where vulnerabilities can be exposed without fear of judgment, and strengths celebrated without reservation. Love in the family manifests in countless ways: through the warm embrace of a parent after a trying day, the laughter shared over a family dinner, or the quiet presence of a sibling during a difficult moment. These expressions of love imbue the home with a warmth and comfort that cannot be replicated elsewhere, making it an irreplaceable source of solace and rejuvenation.Beyond emotional sustenance, the family unit serves as a crucible for personal growth and character development. Within the familial sphere, children learn essential life skills, values, and moral principles that guide them throughout their lives. Parents and elders model resilience, responsibility, and empathy, instilling these traits in younger generations through their words and actions. Moreover, the inevitable conflicts and resolutions that arise within families teach vital lessons about compromise, communication, and forgiveness, preparing individuals for the complexities of interpersonal relationships outside the home.Family also provides a crucial sense of identity and belonging. It is within the family that we first encounter our cultural heritage, inherit traditions, and develop a sense of continuity with past generations. Our unique family stories, customs, and rituals contribute to our individuality, giving us a distinct place in the world. This connection to our roots fosters a deep sense of pride and self-worth, anchoring us amidst the often disorienting currents of societal change.In an increasingly fast-paced and disconnected society, the stability and predictability offered by family become even more precious. The family unit remains a steadfast source ofsupport, offering a refuge from the pressures of work, school, and social obligations. It is where we find solace in times of crisis, celebrate triumphs, and share life's mundane yet meaningful moments. The unwavering presence of loved ones, their unwavering belief in our capabilities, and their willingness to weather life's storms alongside us, fortify us against the vagaries of fortune and give us the courage to navigate life's challenges.题目：坚不可摧的家庭纽带家，是生活风暴中的避风港，远不只是一个物理结构；它是一个有生命力的实体，充满了情感、记忆与憧憬。