Combinatorial interpretations of the q-Faulhaber and q-Salié coefficients, preprint, arXiv

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Neural Networks and Evolutionary Computation. Part II Hybrid Approaches in the Neuroscience

Neural Networks and Evolutionary Computation. Part II:Hybrid Approaches in the NeurosciencesGerhard WeißAbstract—This paper series focusses on the intersection of neural networks and evolutionary computation.It is ad-dressed to researchers from artiﬁcial intelligence as well as the neurosciences.Part II provides an overview of hybrid work done in the neurosciences,and surveys neuroscientiﬁc theories that are bridging the gap between neural and evolutionary computa-tion.According to these theories evolutionary mechanisms like mutation and selection act in real brains in somatic time and are fundamental to learning and developmental processes in biological neural networks.Keywords—Theory of evolutionary learning circuits,theory of selective stabilization of synapses,theory of selective sta-bilization of pre–representations,theory of neuronal group selection.I.IntroductionIn the neurosciences biological neural networks are in-vestigated at diﬀerent organizational levels,including the molecular level,the level of individual synapses and neu-rons,and the level of whole groups of neurons(e.g.,[11, 36,44]).Several neuroscientiﬁc theories have been pro-posed which combine theﬁelds of neural and evolution-ary computation at these diﬀerent levels.These are the theory of evolutionary learning circuits,the theories of se-lective stabilization of synapses and pre–representations, and the theory of neuronal group selection.According to these theories,neural processes of learning and develop-ment strongly base on evolutionary mechanisms like mu-tation and selection.In other words,according to these theories evolutionary mechanisms play in real brains and nervous systems the same role in somatic time as they do in ecosystems in phylogenetic time.(Other neuroscientiﬁc work which is closely related to these theories is described in[28,46,47].)This paper overviews the hybrid work done in the neu-rosciences.Sections II to V survey the four evolutionary theories mentioned above.This includes a description of the major characteristics of these theories as well as a guide to relevant and related literature.Section VI concludes the paper with some general remarks on these theories and their relation to the hybrid approaches proposed in artiﬁ-cial intelligence.The author is with the Institut f¨u r Informatik(H2),Technische Uni-versit¨a t M¨u nchen,D-80290M¨u nchen,Germany.II.The Theory of EvolutionaryLearning CircuitsAccording to the theory of evolutionary learning circuits(TELC for short)neural learning is viewed as the grad-ual modiﬁcation of the information–processing capabilities of enzymatic neurons through a process of variation andselection in somatic time[12,13].In order to put this more precisely,ﬁrst a closer look is taken at enzymatic neurons,and then the fundamental claims of the TELCare described.The TELC starts from the point of view that the brainis organized into various types of local networks which contain enzymatic neurons,that is,neurons whoseﬁringbehavior is controlled by enzymes called excitases.(For details of this control and its underlying biochemical pro-cesses see e.g.[14,15].)These neurons incorporate the principle of double dynamics[15]by operating at two levels of dynamics:at the level of readin or tactilization dynam-ics,the neural input patterns are transduced into chemical–concentration patterns inside the neuron;and at the level of readout dynamics,these chemical patterns are recognized by the excitases.Consequently,the enzymatic neurons themselves are endowed with powerful pattern–recognition capabilities where the excitases are the recognition primi-tives.Both levels of dynamics are gradually deformable as a consequence of the structure–function gradualism(“slight changes in the structure cause slight changes in the func-tion”)in the excitases.As Conrad pointed out,this struc-ture–function gradualism is the key to evolution and evo-lutionary learning in general,and is a important condition for evolutionary adaptability in particular.(Evolutionary adaptability is deﬁned as the extent to which mechanisms of variation and selection can be utilized in order to survive in uncertain and unknown environments[16].)There are three fundamental claims made by the TESC: redundancy of brain tissue,speciﬁty of neurons,and ex-istence of brain–internal selection circuits.According to the claim for redundany,there are many replicas of each type of local network.This means that the brain consists of local networks which are interchangeable in the sense that they are highly similar with respect to their connec-tivity and the properties of their neurons.The claim for speciﬁty says that the excitases are capable of recognizing speciﬁc chemical patterns and,with that,cause the enzy-matic neurons toﬁre in response to speciﬁc input patterns. According to the third claim,the brain contains selectioncircuits which direct theﬁtness–oriented,gradual modiﬁ-cation of the local networks’excitase conﬁgurations.These selection circuits include three systems:a testing system which allows to check the consequences(e.g.,pleasure or pain)of the outputs of one or several local networks for the organism;an evaluation system which assignsﬁtness values to the local networks on the basis of these consequences; and a growth–control system which stimulates or inhibits the production of the nucleic acids which code for the lo-cal networks’excitases on the basis of theirﬁtness values. The nucleic acids,whose variability is ensured by random somatic recombination and mutation processes,diﬀuse to neighbouring networks of the same type(where they per-form the same function because of the interchangeability property mentioned above).These claims imply that neu-ral learning proceeds by means of the gradual modiﬁcation of the excitase conﬁgurations in the brain’s local networks through the repeated execution of the following evolution-ary learning cycle:1.Test and evaluation of the enzymatic neuron–basedlocal networks.As a result,aﬁtness value is assigned to each network.2.Selection of local networks.This involves theﬁtness–oriented regulation of the production of the excitase–coding nucleic acids,as well as their spreading to ad-jacent interchangeable networks.3.Application of somatic recombination and selection tothese nucleic acids.This maintains the range of the excitase conﬁgurations.The execution stops when a local network is found which has a suﬃciently highﬁtness.Conrad emphasized that this evolutionary learning cycle is much more eﬃcient than nat-ural evolution because the selection circuits enable an in-tensive selection even if there is hardly a diﬀerence between theﬁtness values of the interchangeable networks. Finally,some references to related work.The TESC is part of extensive work focussing on the diﬀerences be-tween the information processing capabilities of biological (molecular)systems and conventional computers;see e.g. [15,16,17].A computational speciﬁcation of the ESCM which concentrates on the pattern–processing capabilities of the enzymatic neurons,together with its sucessful appli-cation to a robot–control task,is contained in[29,30,31]. Another computational speciﬁcation which concentrates on the intraneuronal dynamics of enzymatic neurons is de-scribed in[32].A combination of these two speciﬁcations is described in[33].Further related work being of particu-lar interest from a computational point of view is presented in[1,18].III.The Theory of Selective Stabilizationof SynapsesThe theory of selective stabilization of synapses(TSSS for short)is presented in[7,8].This theory accounts for neural processes of learning and development by postulat-ing that a somatic,evolutionary selection mechanism acts at the level of synapses and contributes to the wiring pat-tern in the adult brain.Subsequently the neurobiological basis and the major claims of the TSSS are depicted. The neurobiological basis of the TSSS comprises aspects of both neurogenesis and neurogenetics.In vertebrates one can distinguish several processes of brain development. These are the cellular processes of cell division,movement, adhesion,diﬀerentiation,and death,and the synaptic pro-cesses of connection formation and elimination.(For de-tails see e.g.[19,20,38].)The TSSS focusses on the“synap-tic aspect”of neurogenesis;it deals with the outgrowth and the stabilization of synapses,and takes the developmental stage where maximal synaptic wiring exists as its initial state.The neurogenetic attidue of the TSSS constitutes a compromise between the preformist(“speciﬁed–by–genes”) and the empirist(“speciﬁed–by–activity”)view of brain development.It is assumed that the genes involved in brain development,the so–called genetic envelope,only specify the invariant characters of the brain.This includes,in particular,the connections between the main categories of neurons(i.e.,between groups of neurons which are of the same morphological and biochemical type)and the rules of synaptic growth and stabilization.These rules allow for an activity–dependent,epigenetic synapse formation within the neuronal categories.(As Changeux formulated:“The genetic envelope oﬀers a hazily outlined network,the ac-tivity deﬁnes its angles.”[3,p.193])The TSSS makes three major claims.First,at the crit-ical stage of maximal connectivity there is a signiﬁcant but limited redundany within the neuronal categories as regards the speciﬁty of the synapses.Second,at this time of so–called“structural redundany”any synapse may exist under(at least)three states of plasticity:labile,stable,and degenerate.Only the labile and stable synapses transmit nerve impulses,and the acceptable state transitions are those from labile to either stable or degenerate and from stable to labile.Especially,the state transition of a synapse is epigenetically regulated by all signals received by the postsynaptic soma during a given time interval.(The max-imal synaptic connectivity,the mechanisms of its develop-ment,and the regulative and integrative properties of the soma are determinate expressions of the genetic envelope.) Third,the total activity of the developing network leads to the selective stabilization of some synapses,and to the re-gression of their functional equivalents.As a consequence, structural redundancy decreases and neuronal singularity (i.e.,individual connectivity)increases.This provides a plausible explanation of the naturally occuring connection elimination occuring during neural development.For further readings in the TSSS see e.g.[4,5,6,10].IV.The Theory of Selective Stabilizationof Pre–RepresentationsThe theory of selective stabilization of pre–representa-tions(TSSP for short)can be viewed as an extension of the TSSS.This theory provides a selectionist view of neurallearning and development in the adult brain by postulating that somatic selection takes place at the level of neural networks[5,10,27].Similar to the theory of neuronal group selection(section V),the TSSP may be viewed as an attempt to show how neurobiology and psychology are related to each other.There are two major claims made by the TSSP.Theﬁrst claim is that there exist mental objects or“neural repre-sentations”in the brain.A mental object is deﬁned as a physical state achieved by the correlated and transitory (both electrical and chemical)activity of a cell assembly consisting of a large number of neurons having diﬀerent singularities.According to the TSSP,three classes of men-tal objects are distinguished.First,primary percepts;these are labile mental objects whose activation depends on the direct interaction with the outside world and is caused by sensory stimulations.Second,stored representations;these are memory objects whose evocation does not demand en-vironmental interaction and whose all–or–none activity be-havior results from a stable,cooperative coupling between the neurons.And third,pre–representations;these are mental objects which are generated before and concomitant with any environmental interaction.Pre–representations are labile and of great variety and variability;they result from the spontaneous but correlatedﬁring of neurons or groups of neurons.The second claim made by the TSSP is that learning in the adult brain corresponds to the se-lective stabilization of pre–representations,that means,the transition from selected pre–representations to stored rep-resentations.This requires,in the simplest case,the inter-action with the environment,the criterion of selection is the resonance(i.e.,spatial overlapping orﬁring in phase) between a primary percept and a pre–representation.Further literature on the TSSP.In[9]the two selective–stabilization theories,TSSS and TSSP,are embedded in more general considerations on the neural basis of cogni-tion.A formal model of neural learning and development on the basis of the TSSP is described in[22,43].V.The Theory of Neuronal Group SelectionThe theory of neuronal group selection(TNGS for short) or“neural Darwinism”[23,25]is the most rigorous and elaborate hybrid approach in the neurosciences.This the-ory,which has attracted much attention especially in the last few years,bridges the gap between biology and psy-chology by postulating that somatic selection is the key mechanism which establishes the connection between the structure and the function of the brain.As done in the preceding sections,below the major ideas of the TNGS are described.There are three basic claims.First,during prenatal and early postnatal development,primary repertoires of degen-erate neuronal groups were formed epigenetically by selec-tion.According to the TNGS a neuronal group is consid-ered as a local anatomical entity which consists of hundreds to thousands of strongly connected neurons,and degener-ate neuronal groups are groups that have diﬀerent struc-tures but carry out the same function more or less well (they are nonisomorphic but isofunctional).The concept of degeneracy is fundamental to the TNGS;it implies both structural diversity and functional redundancy and,hence, ensures both a wide range of recognition and the reliabil-ity against the loss of neural tissue.Degeneracy naturally origins from the processes of brain development which are assumed to occur in an epigenetic manner and to elabo-rate several selective events at the cellular level.According to the regulator hypothesis,these complex developmental processes,as well as the selective events accompaning these processes,are guided by a relatively small number of cell adhesion molecules.Second,in the(postnatal)phase of be-havioral experience,a secondary repertoire of functioning neuronal groups is formed by selection among the preexist-ing groups of each primary repertoire.This group selection is accomplished by epigenetic modiﬁcations of the synap-tic strenghts without change of the connectivity pattern. According to the dual rules model,these modiﬁcations are realized by two synaptic rules that operate upon popula-tions of synapses in a parallel and independent fashion: a presynaptic rule which applies to long–term changes in the whole target neuron and which aﬀects a large num-ber of synapses;and a postsynaptic rule which applies to short–term changes at individual synapses.The function-ing groups are more likely to respond to identical or simi-lar stimuli than the non–selected groups and,hence,con-tribute to the future behavior of the organism.A funda-mental operation of the functional groups is to compete for neurons that belong to other groups;this competition aﬀects the groups’functional properties and is assumed to play a central role in the formation and organization of cerebral cortical maps.Third,reentry–phasic sig-naling over re–entrant(reciprocal and cyclic)connections between diﬀerent repertoires,in particular between topo-graphic maps–allows for the spatiotemporal correlation of the responses of the repertoires at all levels in the brain. This kind of phasic signaling is viewed as an important mechanism supporting group selection and as being essen-tial both to categorization and the development of con-sciousness.Reentry implies two fundamental neural struc-tures:ﬁrst,classiﬁcation couples,that is,re–entrant reper-toires that can perform classiﬁcations more complex than a single involved repertoire could do;and second,global map-pings,that is,re-entrant repertoires that correlate sensory input and motor activity.Some brief notes on how the TNGS accounts for psy-chological functions.Following Edelman’s argumentation, categories do not exist apriori in the world(the world is “unlabeled”),and categorization is the fundamental prob-lem facing the nervous system.This problem is solved by means of group selection and reentry.Consequently,cat-egorization largely depends on the organism’s interaction with its environment and turns out to be the central neural operation required for all other operations.Based on this view of categorization,Edelman suggests that memory is “the enhanced ability to categorize or generalize associa-tively,not the storage of features or attributes of objects as a list”[25,p.241]and that learning,in the minimal case,is the“categorization of complexes of adaptive value under conditions of expectancy”[25,p.293].There is a large body of literature on the TNGS.The most detailed depiction of the theory is provided in Edel-man’s book[25].In order to be able to test the TNGS, several computer models have been constructed which em-body the theory’s major ideas.These models are Darwin I[24],Darwin II[26,39,25],and Darwin III[40,41].Re-views of the TNGS can be found in e.g.[21,34,35,42,37].VI.Concluding RemarksThis paper overviewed neuroscientiﬁc theories which view real brains as evolutionary systems or“Darwin machines”[2].This point of view is radically opposed to traditional in-structive theories which postulate that brain development is directed epigenetically during an organism’s interaction with its environment by rules for a more or less precise brain wiring.Nowadays most researchers agree that the instructive theories are very likely to be wrong und unre-alistic,and that the evolutionary theories oﬀer interesting and plausible alternatives.In particular,there is an in-creasing number of neurobiological facts and observations described in the literature which indicate that evolutionary mechanisms(and in particular the mechanism of selection) as postulated by the evolutionary theories are indeed fun-damental to the neural processes in our brains.Someﬁnal notes on the relation between the hybrid work done in the neurosciences and the hybrid work done in artiﬁcial intelligence(see part I of this paper series[45]). Whereas the neuroscientiﬁc approaches aim at a better un-derstanding of the developmental and learning processes in real brains,the artiﬁcial intelligence approaches typically aim at the design of artiﬁcial neural networks that are ap-propriate for solving speciﬁc real-world tasks.Despite this fundamental diﬀerence and its implications,however,there are several aspects and questions which are elementary and signiﬁcant to both the neuroscientiﬁc and the artiﬁcial in-telligence approaches:•Symbolic–subsymbolic intersection(e.g.,“What are the neural foundations of high-level,cognitive abili-ties like concept formation?”and“How are symbolic entities encoded in the neural tissue?”),•Brain wiring(e.g.,“What are the principles of neural development?”and“How are the structure and the function of neural networks related to each other?”),•Genetic encoding(e.g.,“How and to what extend are neural networks genetically encoded?”),and •Evolutionary modiﬁcation(e.g.,“At what network level and at what time scale do evolutionary mechanisms operate?”and“In how far do the evolutionary mech-anisms inﬂuence the network structure?”).Because of this correspondence of interests and research topics it would be useful and stimulating for the neurosci-entiﬁc and the artiﬁcial intelligence community to be aware of each others hybrid work.This requires an increased in-terdisciplinary transparency.To oﬀer such a transparency is a major intention of this paper series.References[1]Akingbehin,K.&Conrad,M.(1989).A hybrid architecture forprogrammable computing and evolutionary learning.Parallel Distributed Computing,6,245–263.[2]Calvin,W.H.(1987).The brain as a Darwin machine.Nature,330,pp.33–43.[3]Changeux,J.–P.(1980).Genetic determinism and epigenesisof the neuronal network:Is there a biological compromise be-tween Chomsky and Piaget?In M.Piatelli–Palmarini(Ed.), Language and learning–The debate between Jean Piaget and Noam Chomsky(pp.184–202).Routledge&Kegan Paul. [4]Changeux,J.–P.(1983).Concluding remarks:On the“singu-larity”of nerve cells and its ontogenesis.In J.–P.Changeux, J.Glowinski,M.Imbert,&F.E.Bloom(Eds.),Molecular and cellular interactions underlying higher brain function(pp.465–478).Elsevier Science Publ.[5]Changeux,J.–P.(1983).L’Homme neuronal.Fayard.[6]Changeux,J.–P.(1985).Remarks on the complexity of thenervous system and its ontogenesis.In J.Mehler&R.Fox (Eds.),Neonate cognition.Beyond the blooming buzzing con-fusion(pp.263–284).Lawrence Erlbaum.[7]Changeux,J.–P.,Courrege,P.,&Danchin,A.(1973).A theoryof the epigenesis of neuronal networks by selective stabilization of synapses.In Proceedings of the National Academy of Sciences USA,70(10),2974–2978.[8]Changeux,J.–P.,&Danchin,A.(1976).Selective stabilizationof developing synapses as a mechanism for the speciﬁcation of neuronal networks.Nature,264,705–712.[9]Changeux,J.–P.,&Dehaene,S.(1989).Neuronal models ofcognitive functions.Cognition,33,63–109.[10]Changeux,J.–P.,Heidmann,T.,&Patte,P.(1984).Learningby selection.In P.Marler&H.S.Terrace(Eds.),The biology of learning(pp.115–133).Springer.[11]Changeux,J.–P.,&Konoshi,M.(Eds.).(1986).The neural andmolecular basis of learning.Springer.[12]Conrad,M.(1974).Evolutionary learning cicuits.Journal ofTheoretical Biology,46,167–188.[13]Conrad M.(1976).Complementary models of learning andmemory.BioSystems,8,119–138.[14]Conrad,M.(1984).Microscopic–macroscopic interface in bio-logical information processing.BioSystems,16,345–363. [15]Conrad,M.(1985).On design principles for a molecular com-munications of the ACM,28(5),464–480.[16]Conrad,M.(1988).The price of programmability.In R.Herken(Ed.),The universal Turing machine–A half–century survey (pp.285–307).Kammerer&Unverzagt.[17]Conrad,M.(1989).The brain–machine disanalogy.BioSystems,22,197–213.[18]Conrad,M.,Kampfner,R.R.,Kirby,K.G.,Rizki, E.N.,Schleis,G.,Smalz,R.,&Trenary,R.(1989).Towards an ar-tiﬁcial brain.BioSystems,23,175–218.[19]Cowan,W.M.(1978).Aspects of neural development.Interna-tional Reviews of Physiology,17,150–91.[20]Cowan,W.M.,Fawcett,J.W.,O’Leary,D.D.M.,&Stanﬁeld,B.B.(1984).Regressive events in neurogenesis.Science,225,1258–1265.[21]Crick,F.(1989).Neural Edelmanism.Trends in Neurosciences,12,240–248.(Reply from G.N.Reeke,R.Michod and F.Crick: Trends in Neurosciences,13,11–14.)[22]Dehaene,S.,Changeux,J.–P.,&Nadal,J.–P.(1987).Neuralnetworks that learn temporal sequences by selection.In Proceed-ings of the National Academy of Sciences USA,84,pp.2727–2731.[23]Edelman,G.M.(1978).Group selection and phasic reentrantsignaling:A theory of higher brain function.In G.M.Edelman &V.B.Mountcastle(Eds.),The mindful brain.Cortical organi-zation and the group–selective theory of higher brain functions (pp.51–100).The MIT Press.[24]Edelman,G.M.(1981).Group selection as the basis for higherbrain function.In F.O.Schmitt,F.G.Worden,G.Adelman&S.G.Dennis(Eds.),The organization of the cerebral cortex (pp.535–563).The MIT Press.[25]Edelman,G.M.(1987).Neural Darwinism.The theory of neu-ronal group selection.Basic Books.[26]Edelman,G.M.,&Reeke,G.N.(1982).Selective networkscapable of representative transformations,limited generaliza-tions,and associative memory.In Proceedings of the National Academy of Sciences USA,79,2091–2095.[27]Heidmann, A.,Heidmann,T.M.,&Changeux,J.–P.(1984).Stabilisation selective de representations neuronales par reso-nance entre“presepresentations”spontanes du reseau cerebral et“percepts”.In C.R.Acad.Sci.Ser.III,299,839–844. [28]Jerne,N.K.(1967).Antibodies and learning:selection vs.in-struction.In G.C.Quarton,T.Melnechuk&F.O.Schmitt (Eds.),The neurosciences:a study program(pp.200–205).Rockefeller University Press.[29]Kampfner,R.R.(1988).Generalization in evolutionary learn-ing with enzymatic neuron–based systems.In M.Kochen&H.M.Hastings(Eds.),Advances in cognitive science.Steps to-ward convergence(pp.190–209).Westview Press,Inc.[30]Kampfner,R.R.,&Conrad,M.(1983).Computational model-ing of evolutionary learning processes in the brain.Bulletin of Mathematical Biology,45(6),931–968.[31]Kampfner,R.R.,&Conrad,M.(1983).Sequential behaviorand stability properties of enzymatic neuron networks.Bulletin of Mathematical Biology,45(6),969–980.[32]Kirby,K.G.,&Conrad,M.(1984).The enzymatic neuron asa reaction–diﬀusion network of cyclic nucleotides.Bulletin ofMathematical Biology,46,765–782.[33]Kirby,K.G.,&Conrad,M.(1986).Intraneuronal dynamics asa substrate for evolutionary learning.In Physica22D,205–215.[34]Michod,R.E.(1989).Darwinian selection in the brain.Evolu-tion,(3),694–696.[35]Nelson,R.J.(1989).Philosophical issues in Edelman’s neuralDarwinism.Journal of Experimental and Theoretical Artiﬁcial Intelligence,1,195–208.[36]Neuroscience Research(1986).Special issue3:Synaptic plastic-ity,memory and learning.[37]Patton,P.,&Parisi,T.(1989).Brains,computation,and selec-tion:An essay review of Gerald Edelman’s Neural Darwinism.Psychobiology,17(3),326–333.[38]Purves,D.,&Lichtman,J.W.(1985).Principles of neural de-velopment.Sinauer Associates Inc.[39]Reeke,G.N.jr.,&Edelman,G.M.(1984).Selective networksand recognition automata.In Annals of the New York Academy of Sciences,426(Special issue on computer culture),181–201.[40]Reeke,G.N.jr.,&Edelman,G.M.(1988).Real brains andartiﬁcial intelligence.Deadalus,117(1),143–173.[41]Reeke,G.N.jr.,Sporns,O.,&Edelman,G.M.(1988).Syn-thetic neural modeling:a Darwinian approach to brain theory.In R.Pfeifer,Z.Schreter,F.Fogelman–Soulie&L.Steels(Eds.), Connectionism in perspective.Elsevier.[42]Smoliar,S.(1989).Book review of[25].Artiﬁcial Intelligence,39,121–139.[43]Toulouse,G.,Dehaene,S.,&Changeux,J.–P.(1986).Spin glassmodels of learning by selection.In Proceedings of the National Academy of Sciences USA,83,1695–1698.[44]Trends in Neurosciences(1988).Vol.11(4),Special issue:Learn-ing,memory.[45]Weiß,G.(1993,submitted to IEEE World Congress on Compu-tational Intelligence).Neural networks and evolutionary com-putation.Part I:Hybrid approaches in artiﬁcial intelligence. [46]Young,J.Z.(1973).Memory as a selective process.In Aus-tralian Academy of Science Report:Symposium on Biological Memory;Australian Academy of Science(pp.25–45).[47]Young,J.Z.(1975).Sources of discovery in neuroscience.InF.G.Worden,J.P.Swazey&G.Edelman(Eds.),The neu-rosciences:Paths of discovery(pp.15–46).Oxford University Press.。

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1paring two images, or an image and a model, is the fundamental operation for many image processing and computer vision systems. In most systems of interest, a simple pixelby-pixel comparison won’t do: the diﬀerence measurement that we determine must bear some correlation with the perceptual diﬀerence between the two images, or with the diﬀerence between two adequate interpretations of the two images. In order to compute meaningful diﬀerences between images, the ﬁrst step is usually the determination of a suitable set of features which encode the characteristics that we intend to measure. Measuring meaningful image similarity is a dichotomy that rests on two elements: ﬁnding the right set of features and endowing the feature space with the right metric. Since the same feature space can be endowed with an inﬁnity of metrics, the two problems are by no means equivalent, nor does the ﬁrst subsume the second. In this paper we consider the problem of measuring distances in feature spaces. In a number of cases, after having selected the right set of features extracted, and having characterized an

Binomial coefficient

Binomial coefficientFrom Wikipedia, the free encyclopediaJump to: navigation, searchThe binomial coefficients form the entries of Pascal's triangle.In mathematics, the binomial coefficient is the coefficient of the x k term in the polynomial expansion of the binomial power (1 + x)n.In combinatorics, is interpreted as the number of k-element subsets (the k-combinations) of an n-element set, that is the number of ways thatk things can be "chosen" from a set of n things. Hence, is often read as "n choose k" and is called the choose function of n and k.The notation was introduced by Andreas von Ettingshausen in 1826,[1] although the numbers were already known centuries before that (seePascal's triangle). Alternative notations include C(n, k), n C k, n C k,,[2] in all of which the C stands for combinations or choices.Contents[hide]∙ 1 Definitiono 1.1 Recursive formulao 1.2 Multiplicative formulao 1.3 Factorial formulao 1.4 Generalization and connection to the binomial series ∙ 2 Pascal's triangle∙ 3 Combinatorics and statistics∙ 4 Binomial coefficients as polynomialso 4.1 Binomial coefficients as a basis for the space of polynomialso 4.2 Integer-valued polynomialso 4.3 Example∙ 5 Identities involving binomial coefficientso 5.1 Powers of -1o 5.2 Series involving binomial coefficientso 5.3 Identities with combinatorial proofso 5.4 Continuous identities∙ 6 Generating functionso 6.1 Ordinary generating functions∙7 Divisibility properties∙8 Bounds and asymptotic formulas∙9 Generalizationso9.1 Generalization to multinomialso9.2 Generalization to negative integerso9.3 Taylor serieso9.4 Binomial coefficient with n=1/2o9.5 Identity for the product of binomial coefficientso9.6 Partial Fraction Decompositiono9.7 Newton's binomial serieso9.8 Two real or complex valued argumentso9.9 Generalization to q-serieso9.10 Generalization to infinite cardinals∙10 Binomial coefficient in programming languages∙11 See also∙12 Notes∙13 References[edit] DefinitionFor natural numbers(taken to include 0) n and k, the binomial coefficientcan be defined as the coefficient of the monomial X k in the expansion of (1 + X)n. The same coefficient also occurs (if k≤ n) in the binomial formula(valid for any elements x,y of a commutative ring), which explains the name "binomial coefficient".Another occurrence of this number is in combinatorics, where it gives the number of ways, disregarding order, that a k objects can be chosen from among n objects; more formally, the number of k-element subsets (ork-combinations) of an n-element set. This number can be seen to be equal to the one of the first definition, independently of any of the formulas below to compute it: if in each of the n factors of the power (1 + X)n one temporarily labels the term X with an index i(running from 1 to n), then each subset of k indices gives after expansion a contribution X k, and the coefficient of that monomial in the result will be the number ofsuch subsets. This shows in particular that is a natural number for any natural numbers n and k. There are many other combinatorial interpretations of binomial coefficients (counting problems for which the answer is given by a binomial coefficient expression), for instance the number of words formed of n bits(digits 0 or 1) whose sum is k, but most of these are easily seen to be equivalent to counting k-combinations.Several methods exist to compute the value of without actually expanding a binomial power or counting k-combinations.[edit] Recursive formulaOne has a recursive formula for binomial coefficientswith as initial valuesThe formula follows either from tracing the contributions to X k in (1 + X)n−1(1 + X), or by counting k-combinations of {1, 2, ..., n} that containn and that do not contain n separately. It follows easily thatwhen k > n, and for all n, so the recursion can stop when reaching such cases. This recursive formula then allows the construction of Pascal's triangle.[edit] Multiplicative formulaA more efficient method to compute individual binomial coefficients is given by the formulaThis formula is easiest to understand for the combinatorial interpretation of binomial coefficients. The numerator gives the number of ways to select a sequence of k distinct objects, retaining the order of selection, from a set of n objects. The denominator counts the number of distinct sequences that define the same k-combination when order is disregarded.[edit] Factorial formulaFinally there is a formula using factorials that is easy to remember:where n! denotes the factorial of n. This formula follows from the multiplicative formula above by multiplying numerator and denominator by (n−k)!; as a consequence it involves many factors common to numerator and denominator. It is less practical for explicit computation unless common factors are first canceled (in particular since factorial values grow very rapidly). The formula does exhibit a symmetry that is less evident from the multiplicative formula (though it is from the definitions)[edit] Generalization and connection to the binomial seriesThe multiplicative formula allows the definition of binomial coefficients to be extended[note 1] by replacing n by an arbitrary number α (negative, real, complex) or even an element of any commutative ring in which all positive integers are invertible:With this definition one has a generalization of the binomial formula (with one of the variables set to 1), which justifies still calling thebinomial coefficients:This formula is valid for all complex numbers α and X with |X| < 1. It can also be interpreted as an identity of formal power series in X, where it actually can serve as definition of arbitrary powers of series with constant coefficient equal to 1; the point is that with this definition all identities hold that one expects for exponentiation, notablyIf αis a nonnegative integer n, then all terms with k > n are zero, and the infinite series becomes a finite sum, thereby recovering the binomial formula. However for other values of α, including negative integers and rational numbers, the series is really infinite.[edit] Pascal's triangleMain article: Pascal's ruleMain article: Pascal's trianglePascal's rule is the important recurrence relationwhich can be used to prove by mathematical induction that is a natural number for all n and k, (equivalent to the statement that k! divides the product of k consecutive integers), a fact that is not immediately obvious from formula (1).Pascal's rule also gives rise to Pascal's triangle:0: 11: 1 12: 1 2 13: 1 3 3 14: 1 4 6 4 15: 1 5 10 10 5 16: 1 6 15 20 15 6 17: 1 7 21 35 35 21 7 18: 1 8 28 56 70 56 28 8 1Row number n contains the numbers for k= 0,…,n. It is constructed by starting with ones at the outside and then always adding two adjacent numbers and writing the sum directly underneath. This method allows the quick calculation of binomial coefficients without the need for fractions or multiplications. For instance, by looking at row number 5 of the triangle, one can quickly read off that(x + y)5 = 1x5 + 5x4y + 10x3y2 + 10x2y3 + 5x y4 + 1y5.The differences between elements on other diagonals are the elements in the previous diagonal, as a consequence of the recurrence relation (3) above.[edit] Combinatorics and statisticsBinomial coefficients are of importance in combinatorics, because they provide ready formulas for certain frequent counting problems:There are ways to choose k elements from a set of n elements.See Combination.∙There are ways to choose k elements from a set of n if repetitions are allowed. See Multiset.∙There are strings containing k ones and n zeros.∙There are strings consisting of k ones and n zeros such that no two ones are adjacent.∙The Catalan numbers are∙The binomial distribution in statistics is∙The formula for a Bézier curve.[edit] Binomial coefficients as polynomialsFor any nonnegative integer k, the expression can be simplified and defined as a polynomial divided by k!:This presents a polynomial in t with rational coefficients.As such, it can be evaluated at any real or complex number t to define binomial coefficients with such first arguments. These "generalized binomial coefficients" appear in Newton's generalized binomial theorem.For each k, the polynomial can be characterized as the unique degree k polynomial p(t) satisfying p(0) = p(1) = ... = p(k− 1) = 0 and p(k) = 1.Its coefficients are expressible in terms of Stirling numbers of the first kind, by definition of the latter:The derivative of can be calculated by logarithmic differentiation:[edit] Binomial coefficients as a basis for the space of polynomialsOver any field containing Q, each polynomial p(t) of degree at most d isuniquely expressible as a linear combination . The coefficient a k is the k th difference of the sequence p(0), p(1), …, p(k). Explicitly,[note 2][edit] Integer-valued polynomialsEach polynomial is integer-valued: it takes integer values at integer inputs. (One way to prove this is by induction on k, using Pascal's identity.) Therefore any integer linear combination of binomial coefficient polynomials is integer-valued too. Conversely, (3.5) shows that any integer-valued polynomial is an integer linear combination of these binomial coefficient polynomials. More generally, for any subring R of a characteristic 0 field K, a polynomial in K[t] takes values in R at all integers if and only if it is an R-linear combination of binomial coefficient polynomials.[edit] ExampleThe integer-valued polynomial 3t(3t + 1)/2 can be rewritten as[edit] Identities involving binomial coefficientsFor any nonnegative integers n and k,This follows from (2) by using (1 + x)n = x n·(1 + x−1)n. It is reflected in the symmetry of Pascal's triangle. A combinatorial interpretation of this formula is as follows: when forming a subset of k elements (from a set of size n), it is equivalent to consider the number of ways you can pick k elements and the number of ways you can exclude n−k elements. The factorial definition lets one relate nearby binomial coefficients. For instance, if k is a positive integer and n is arbitrary, thenand, with a little more work,[edit] Powers of -1A special binomial coefficient is ; it equals powers of -1:[edit] Series involving binomial coefficientsThe formulais obtained from (2) using x = 1. This is equivalent to saying that the elements in one row of Pascal's triangle always add up to two raised to an integer power. A combinatorial interpretation of this fact involving double counting is given by counting subsets of size 0, size 1, size 2, and so on up to size n of a set S of n elements. Since we count the number of subsets of size i for 0 ≤ i≤ n, this sum must be equal to the number of subsets of S, which is known to be 2n. That is, Equation 5 is a statement that the power set for a finite set with n elements has size 2n.The formulasandfollow from (2), after differentiating with respect to x (twice in the latter) and then substituting x = 1.Vandermonde's identityis found by expanding (1 + x)m (1 + x)n−m = (1 + x)n with (2). As is zero if k > n, the sum is finite for integer n and m. Equation (7a) generalizes equation (3). It holds for arbitrary, complex-valued m and n, the Chu-Vandermonde identity.A related formula isWhile equation (7a) is true for all values of m, equation (7b) is true for all values of j between 0 and k inclusive.From expansion (7a) using n=2m, k = m, and (4), one findsLet F(n) denote the n th Fibonacci number. We obtain a formula about the diagonals of Pascal's triangleThis can be proved by induction using (3) or by Zeckendorf's representation (Just note that the lhs gives the number of subsets of {F(2),...,F(n)} without consecutive members, which also form all the numbers below F(n+1)).Also using (3) and induction, one can show thatAgain by (3) and induction, one can show that for k = 0, ... , n−1as well aswhich is itself a special case of the result from the theory of finite differences that for any polynomial P(x) of degree less than n,[citation needed]Differentiating (2) k times and setting x = −1 yields this for, when 0 ≤ k< n, and the general case follows by taking linear combinations of these.When P(x) is of degree less than or equal to n,where a n is the coefficient of degree n in P(x).More generally for 13b,where m and d are complex numbers. This follows immediately applying (13b) to the polynomial Q(x):=P(m + dx)instead of P(x), and observing that Q(x) has still degree less than or equal to n, and that its coefficient of degree n is d n a n.The infinite seriesis convergent for k≥ 2. This formula is used in the analysis of the German tank problem. It is equivalent to the formula for the finite sumwhich is proved for M>m by induction on M.Using (8) one can deriveand.[edit] Identities with combinatorial proofsMany identities involving binomial coefficients can be proved by combinatorial means. For example, the following identity for nonnegativeintegers (which reduces to (6) when q = 1):can be given a double counting proof as follows. The left side counts the number of ways of selecting a subset of [n] of at least q elements, and marking q elements among those selected. The right side counts the same parameter, because there are ways of choosing a set of q marks and they occur in all subsets that additionally contain some subset of the remaining elements, of which there are 2n−q.The recursion formulawhere both sides count the number of k-element subsets of {1, 2, . . . , n} with the right hand side ﬁrst grouping them into those which contain element n and those which don’t.The identity (8) also has a combinatorial proof. The identity readsSuppose you have 2n empty squares arranged in a row and you want to mark(select) n of them. There are ways to do this. On the other hand, you may select your n squares by selecting k squares from among the first n and n−k squares from the remaining n squares. This givesNow apply (4) to get the result.[edit] Continuous identitiesCertain trigonometric integrals have values expressible in terms of binomial coefficients:For andThese can be proved by using Euler's formula to convert trigonometric functions to complex exponentials, expanding using the binomial theorem, and integrating term by term.[edit] Generating functions[edit] Ordinary generating functionsFor a fixed n, the ordinary generating function of the sequenceis:For a fixed k, the ordinary generating function of the sequenceis:The bivariate generating function of the binomial coefficients is:[edit] Divisibility propertiesIn 1852, Kummer proved that if m and n are nonnegative integers and p isa prime number, then the largest power of p dividing equals p c, where c is the number of carries when m and n are added in base p. Equivalently,the exponent of a prime p in equals the number of positive integers j such that the fractional part of k/p j is greater than the fractionalpart of n/p j. It can be deduced from this that is divisible byn/gcd(n,k).A somewhat surprising result by David Singmaster (1974) is that any integer divides almost all binomial coefficients. More precisely, fix aninteger d and let f(N) denote the number of binomial coefficientswith n < N such that d divides . ThenSince the number of binomial coefficients with n < N is N(N+1) / 2, this implies that the density of binomial coefficients divisible by d goes to 1.Another fact: An integer n≥ 2 is prime if and only if all the intermediate binomial coefficientsare divisible by n.Proof: When p is prime, p dividesfor all 0 < k < pbecause it is a natural number and the numerator has a prime factor p but the denominator does not have a prime factor p.When n is composite, let p be the smallest prime factor of n and let k = n/p. Then 0 < p < n andotherwise the numerator k(n−1)(n−2)×...×(n−p+1) has to be divisible by n = k×p, this can only be the case when (n−1)(n−2)×...×(n−p+1) is divisible by p. But n is divisible by p, so p does not divide n−1, n−2, ..., n−p+1 and because p is prime, we know that p does not divide(n−1)(n−2)×...×(n−p+1) and so the numerator cannot be divisible by n. [edit] Bounds and asymptotic formulasThe following bounds for hold:Stirling's approximation yields the bounds:and, in general,for m≥ 2 and n≥ 1,and the approximationasThe infinite product formula (cf. Gamma function, alternative definition)yields the asymptotic formulasas .This asymptotic behaviour is contained in the approximationas well. (Here H k is the k th harmonic number and γis the Euler–Mascheroni constant).The sum of binomial coefficients can be bounded by a term exponential in n and the binary entropy of the largest n/ k that occurs. More precisely,for and , it holdswhere is the binary entropy of ε.[3]A simple and rough upper bound for the sum of binomial coefficients is given by the formula below (not difficult to prove)[edit] Generalizations[edit] Generalization to multinomialsBinomial coefficients can be generalized to multinomial coefficients. They are defined to be the number:whereWhile the binomial coefficients represent the coefficients of (x+y)n, the multinomial coefficients represent the coefficients of the polynomialSee multinomial theorem. The case r = 2 gives binomial coefficients:The combinatorial interpretation of multinomial coefficients is distribution of n distinguishable elements over r (distinguishable) containers, each containing exactly k i elements, where i is the index of the container.Multinomial coefficients have many properties similar to these of binomial coefficients, for example the recurrence relation:and symmetry:where (σi) is a permutation of (1,2,...,r).[edit] Generalization to negative integersIf , thenextends to all n.[edit] Taylor seriesUsing Stirling numbers of the first kind the series expansion around any arbitrarily chosen point z0 is[edit] Binomial coefficient with n=1/2The definition of the binomial coefficients can be extended to the case where n is real and k is integer.In particular, the following identity holds for any non-negative integer k :This shows up when expanding into a power series using the Newtonbinomial series :[edit] Identity for the product of binomial coefficientsOne can express the product of binomial coefficients as a linear combination of binomial coefficients:where the connection coefficients are multinomial coefficients. In terms of labelled combinatorial objects, the connection coefficients represent the number of ways to assign m+n-k labels to a pair of labelled combinatorial objects of weight m and n respectively, that have had their first k labels identified, or glued together, in order to get a new labelled combinatorial object of weight m+n-k. (That is, to separate the labels into 3 portions to be applied to the glued part, the unglued part of the first object, and the unglued part of the second object.) In this regard, binomial coefficients are to exponential generating series what falling factorials are to ordinary generating series.[edit] Partial Fraction DecompositionThe partial fraction decomposition of the inverse is given byand[edit] Newton's binomial seriesNewton's binomial series, named after Sir Isaac Newton, is one of the simplest Newton series:The identity can be obtained by showing that both sides satisfy the differential equation (1+z) f'(z) = αf(z).The radius of convergence of this series is 1. An alternative expression iswhere the identityis applied.[edit] Two real or complex valued argumentsThe binomial coefficient is generalized to two real or complex valued arguments using the gamma function or beta function viaThis definition inherits these following additional properties from Γ:moreover,The resulting function has been little-studied, apparently first being graphed in (Fowler 1996). Notably, many binomial identities fail:but for n positive (so −n negative). The behavior is quite complex, and markedly different in various octants (that is, with respect to the x and y axes and the line y= x), with the behavior for negative x having singularities at negative integer values and a checkerboard of positive and negative regions:∙in the octant it is a smoothly interpolated form of the usual binomial, with a ridge ("Pascal's ridge").∙in the octant and in the quadrant the function is close to zero.∙in the quadrant the function is alternatingly very large positive and negative on the parallelograms with vertices ( −n,m + 1),( −n,m),( −n− 1,m− 1),( −n− 1,m) ∙in the octant 0 > x > y the behavior is again alternatingly very large positive and negative, but on a square grid.∙in the octant − 1 > y > x + 1 it is close to zero, except for near the singularities.[edit] Generalization to q-seriesThe binomial coefficient has a q-analog generalization known as the Gaussian binomial coefficient.[edit] Generalization to infinite cardinalsThe definition of the binomial coefficient can be generalized to infinite cardinals by defining:where A is some set with cardinalityα. One can show that the generalized binomial coefficient is well-defined, in the sense that no matter whatset we choose to represent the cardinal number α, will remain the same. For finite cardinals, this definition coincides with the standard definition of the binomial coefficient.Assuming the Axiom of Choice, one can show that for any infinite cardinal α.[edit] Binomial coefficient in programming languagesThe notation is convenient in handwriting but inconvenient for typewriters and computer terminals. Many programming languages do not offer a standard subroutine for computing the binomial coefficient, but for example the J programming language uses the exclamation mark: k ! n .Naive implementations of the factorial formula, such as the following snippet in C:int choose(int n, int k) {return factorial(n) / (factorial(k) * factorial(n - k));}are prone to overflow errors, severely restricting the range of input values. A direct implementation of the multiplicative formula works well:unsigned long long choose(unsigned n, unsigned k) {if (k > n)return 0;if (k > n/2)k = n-k; // Take advantage of symmetrylong double accum = 1;unsigned i;for (i = 1; i <= k; i++)accum = accum * (n-k+i) / i;return accum + 0.5; // avoid rounding error}Another way to compute the binomial coefficient when using large numbers is to recognize thatlnΓ(n) is a special function that is easily computed and is standard in some programming languages such as using log_gamma in Maxima, LogGamma in Mathematica, or gammaln in MATLAB. Roundoff error may cause the returned value not to be an integer.[edit] See also∙Central binomial coefficient∙Binomial transform∙Star of David theorem∙Table of Newtonian series∙List of factorial and binomial topics∙Multiplicities of entries in Pascal's triangle∙Sun's curious identity[edit] Notes1.^See (Graham, Knuth & Patashnik 1994), which also defines fork< 0. Alternative generalizations, such as to two real or complex valued arguments using the Gamma function assign nonzero values tofor k< 0, but this causes most binomial coefficient identities to fail, and thus is not widely used majority of definitions. Onesuch choice of nonzero values leads to the aesthetically pleasing "Pascal windmill" in Hilton, Holton and Pedersen, Mathematicalreflections: in a room with many mirrors, Springer, 1997, but causes even Pascal's identity to fail (at the origin).2.^This can be seen as a discrete analog of Taylor's theorem. It isclosely related to Newton's polynomial. Alternating sums of this form may be expressed as the Nörlund–Rice integral.[edit] References1.^Nicholas J. Higham. Handbook of writing for the mathematicalsciences. SIAM. p. 25. ISBN0898714206.2.^ G. E. Shilov (1977). Linear algebra. Dover Publications.ISBN9780486635187.3.^ see e.g. Flum & Grohe (2006, p. 427)∙Fowler, David(January 1996). "The Binomial Coefficient Function".The American Mathematical Monthly(Mathematical Association ofAmerica) 103 (1): 1–17. doi:10.2307/2975209./stable/2975209∙Knuth, Donald E.(1997). The Art of Computer Programming, Volume 1: Fundamental Algorithms (Third ed.). Addison-Wesley.pp. 52–74. ISBN0-201-89683-4.∙Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994).Concrete Mathematics (Second ed.). Addison-Wesley.pp. 153–256. ISBN0-201-55802-5.∙Singmaster, David (1974). "Notes on binomial coefficients. III.Any integer divides almost all binomial coefficients". J. LondonMath. Soc. (2)8: 555–560. doi:10.1112/jlms/s2-8.3.555.∙Bryant, Victor (1993). Aspects of combinatorics. Cambridge University Press.∙Arthur T. Benjamin; Jennifer Quinn, Proofs that Really Count: The Art of Combinatorial Proof, Mathematical Association of America,2003.∙Flum, Jörg; Grohe, Martin (2006). Parameterized Complexity Theory.Springer. ISBN978-3-540-29952-3./east/home/generic/search/results?SGWID=5-40109-22-141358322-0.This article incorporates material from the following PlanetMath articles, which are licensed under the Creative Commons Attribution/Share-Alike License: Binomial Coefficient, Bounds for binomial coefficients, Proof that C(n,k) is an integer, Generalized binomial coefficients.。

the greeks assumed that the structure of language

The Greeks Assumed That the Structure of LanguageIntroductionLanguage is a fundamental aspect of human communication and plays a significant role in shaping our thoughts and ideas. The Greeks, renowned for their contributions to philosophy and literature, also pondered over the nature and structure of language. This article aims to delve intothe Greek assumptions regarding the structure of language, exploringtheir theories and implications.Origins of Greek Linguistic ThoughtThe Greek fascination with language can be traced back to prominent philosophers such as Plato and Aristotle. Plato believed that language was not a mere tool for communication but a reflection of the ultimate reality. According to him, words and their meanings were not arbitrarybut had a deeper connection to the essence of objects or concepts. Aristotle, on the other hand, studied language from a more empirical perspective, focusing on its function and structure.Greek Assumptions about Language StructureThe Greeks made several assumptions about the structure of language,which had a profound impact on subsequent linguistic thought. These assumptions include:1. Words Reflect RealityThe Greeks assumed that words had an inherent connection to the objectsor concepts they represented. They believed that through language, individuals could access and understand the true nature of reality. This assumption laid the foundation for the philosophical concept of “logos,” which refers to the relationship between words and reality.2. Language Is Composed of Basic ElementsThe Greeks recognized that language could be broken down into smaller units with distinctive meanings. They postulated that these basic elements, known as morphemes, combined to form words. This assumption paved the way for the development of morphological analysis in linguistics, which studies the internal structure of words.3. Syntax and Grammar Govern LanguageAncient Greek philosophers acknowledged the importance of syntax and grammar in organizing and conveying meaning. They recognized that language followed specific rules and structures that determined the relationships between words in a sentence. This assumption laid the groundwork for syntactical analysis, which explores the arrangement of words and phrases in a sentence.4. Language Is InnateThe Greeks assumed that the ability to acquire and understand language was innate to humans. They believed that language proficiency stemmed from natural predispositions rather than external influences. This assumption aligns with modern theories of language acquisition, such as Noam Chomsk y’s concept of a Universal Grammar.Implications of Greek Linguistic ThoughtThe Greek assumptions about language structure had far-reaching implications for various disciplines, including linguistics, philosophy, and literature. Some of these implications are:1. Language as a Mirror of RealityThe concept of language reflecting reality influenced subsequent philosophical and metaphysical thought. It prompted thinkers to explore the relationship between language, perception, and knowledge. This exploration ultimately shaped diverse philosophical schools, such as phenomenology and hermeneutics.2. Development of Linguistic AnalysisThe Greek assumptions regarding the composition of language elements and the importance of syntax and grammar laid the groundwork for linguistic analysis. These assumptions influenced the development of structural linguistics, generative grammar, and other linguistic theories that investigate the form and function of language.3. Influence on Literary StylesGreek linguistic thought permeated literary works, influencing writing styles and literary devices. Writers began incorporating rhetorical techniques, such as metaphors and analogies, to convey deeper meanings and evoke emotional responses. These techniques shaped the foundations of poetry, prose, and dramatic literature.4. Evolution of Language EducationThe Greek assumptions about language being innate and governed by rules contributed to the development of language education methodologies. They inspired instructional approaches that emphasize the systematic teaching of grammar, syntax, and vocabulary. These approaches continue to influence language teaching methodologies worldwide.ConclusionThe Greeks’ assumptions about the structure of language have left an indelible mark on human understanding and exploration of linguistic phenomena. Their belief that language reflects reality, the recognition of basic language elements, the importance of syntax and grammar, and the innate nature of language have shaped various disciplines. From philosophy to linguistics, and literature to education, the Greek assumptions continue to shape our understanding and appreciation of language.。

Definition and Interpretation(定义及解释规则翻译)

Definition and Interpretation（定义及解释规则翻译）翻译实践1. "Affiliate" means any person or company that directly or indirectly controls a Party or is directly or indirectly controlled by a Party, including a Party's parent or subsidiary, or is under direct or indirect common control with such Party. For the purpose of this Agreement, "control" shall mean either the ownership of fifty per cent (50%) or more of the ordinary share capital of the company carrying the right to vote at general meetings or the power to nominate a majority of the board of directors of the Company.2. "Proprietary Know-how" shall mean processes, methods and manufacturing techniques, experience and other information and materials including but not limited to the Technical Information and Technical Assistance supplied or rendered by the Licensor to the Licensee hereunder which have been developed by and are known to the Licensor on the date hereof and/or which may be further developed by the Licensor or become known to it during the continuance of this Agreement excepting, however, any secret know-how acquired by the Licensor from third parties which the Licensor is precluded from disclosing to the Licensee.3. "Proprietary Information" means the information, whether patentable or not, disclosed to the CJV by either Party or its Affiliates or disclosed by the CJV to either Party or its Affiliates during the term of this Contract, including technology, inventions, creations, know-how, formulations, recipes, specifications, designs, methods, processes, techniques, data, rights, devices, drawings, instructions, expertise, trade practices, trade secrets and such commercial, economic, financial or other information as is generally treated as confidential by the disclosing Party, its Affiliates, or the CJV, as the case may be; provided that when such information is in unwritten or intangible form, the disclosing Party, its Affiliates or the CJV shall, within one month of making the disclosure, provide the other Partyand/or the CJV with a written confirmation that such information constitutes its Proprietary Information.4. "Encumbrances" include any option, right to acquire, right of preemption, mortgage, charge, pledge, lien, hypothecation, title creation, right of set-off, counterclaim, trust arrangement or other security or any equity or restriction (including any relevant restriction imposed under the relevant law).5. In this Agreement, unless the context otherwise requires:a) headings are for convenience only and shall not affect the interpretation - 27 -of this Agreement;b) words importing the singular include the plural and vice versa; c) words importing a gender include any gender;d) an expression importing a natural person includes any company, partnership, joint venture, association, corporation or other body corporate and any governmental agency;e) a reference to any law, regulation or rule includes all laws, regulations, or rules amending, consolidating or replacing them, and a reference to a law includes all regulations and rules under that law;f) a reference to a document includes an amendment or supplement to, or replacement or novation of, that document;g) a reference to a party to any document includes that party's successors and permitted assigns;h) a reference to an agreement includes an undertaking, agreement or legally enforceable arrangement or understanding whether or not in writing;i) a warranty, representation, undertaking, indemnity, covenant or agreement on the part of two or more persons binds them jointly and severally; andj) the schedules, annexures and appendices to this Agreement shall form an integral part of this Agreement.参考译文:定义、解释1. “关联公司”指直接或间接控制一方(包括其母公司或子公司)或受一方直接或间接控制，或与该方共同受直接或间接控制的任何人或公司。

隐含作者、第二自我与自我的多重性--小议叙事理论中“隐含作者”概念的争议与潜力

隐含作者、第二自我与自我的多重性----小议叙事理论中“隐含作者”概念的争议与潜力吕琪摘要：“隐含作者”这一小说叙事理论中的概念自布斯提出后即引发了叙事理论界的持续探索”对于“隐含作者”的身份阐释，不同的理论家给出了不同的理解，并因此又提出了更多新颖而具有意义的概念。

然而，“隐含作者”这一理论中的“第二自我”这一层面并没有得到充分发掘，本文认为这是这一概念引起争议的一个重要原因”第二自我本身的多重性是“隐含作者”这一概念的活力与潜力所在”本文分析比较了布斯、查特曼和申丹对“隐含作者”这一概念的阐释，结合“自我”概念在心理学和社会学中的多重定义，指出“隐含作者”与“第二自我”间存在多重关系，正确理解这种“自我”的多重性，有助于疏通对“隐含作者”进行的各种阐释中自相矛盾或费解之处”关键词：隐含作者第二自我叙事自我的多重性The Implied Author,the Second-Self and the Multiple Identities of Self:On the Disputes and Potentials in the Concept of“the Implied Author"Lyu QiAbstract:Since it was proposed by Wayne Booth,the concept of"the implied author”has inspired many relevant researches in narratology.Aboutthe identity of the implied author ,theorists have tried to interpret t inmany ways ,which has led to the proposition of other original andmeaningful concepts.However,the idea of the second-self in theconcept of the implied author has not been thoroughly discussed yet,whichthise s aytends oconsiderasoneof hemosimporan reasonsforthedispuesaroundthisconcep.In some way,he dynamics andpoenials ofthe conceptofthe implied auhorlieinthe mulipleidentities embedded in the second-self.This essay compares theinterpretations of three theorists,Booth,Chatman and Shen,andpointsoutthatthecomplicatedrelationshipbetweentheimpliedauthorandthesecond-selfhasbeencommonlyneglectedandthushasarousedcontradictionsorconfusionintheirinterpretationsKeywords:the Implied Author;the Second-self;Narratology;Multiple identities ofself“隐含作者”这一叙事理论概念自韦恩•布斯(WayneBooth)在《小说修辞学》(1961)中提出后即引起了批评家和理论家极大的关注。

第九讲 interpretation Guiding Lesson

III Interpretation and the Interpreter
Professional classification: Consecutive interpretation Simultaneous interpretation Sight interpretation (Sight simultaneous interpretation) Court interpretation (庭审)
III Interpretation and the Interpreter
1)
According to interpreting modes Alternating interpretation Consecutive interpretation Simultaneous interpretation Whispering interpretation Sight interpretation Sign language interpretation
III Interpretation and the Interpreter
The definition of Interpretation
1) Interpretation is a vocal translation to the information delivered by vocal utterance and text. (Shuttleworth & Cowie, 1997:82) 2) In its purest form, consecutive interpretation is a mode in which the interpreter begins their interpretation of a complete message after the speaker has stopped producing the source utterance. At the time that the interpretation is rendered the interpreter is the only person in the communication environment who is producing a message. By Roberto Santiago

Enumerations of Permutations by Circular Descent Sets

Keyword: Circular Descent; Generating Tree; Permutation; Permutation Tableaux;
∗ †
Email address of the corresponding author: giannic@.tw Partially supported by NSC 96-2115-M-006-012
Institute of Mathematics, Academia Sinica, Taipei, Taiwan.
Abstract The circular descent of a permutation σ is a set {σ (i) | σ (i) > σ (i + 1)}. In this paper, we focus on the enumerations of permutations by the circular descent set. Let cdesn (S ) be the number of permutations of length n which have the circular descent set S . We derive the explicit formula for cdesn (S ). We describe a class of generating binary trees Tk with weights. We ﬁnd that the number of permutations in the set CDESn (S ) corresponds to the weights of Tk . As a application of the main results in this paper, we also give the enumeration of permutation tableaux according to their shape.

[整理]Binomialcoefficient.

Binomial coefficientFrom Wikipedia, the free encyclopediaJump to: navigation, searchThe binomial coefficients form the entries of Pascal's triangle.In mathematics, the binomial coefficient is the coefficient of the x k term in the polynomial expansion of the binomial power (1 + x)n.In combinatorics, is interpreted as the number of k-element subsets (the k-combinations) of an n-element set, that is the number of ways thatk things can be "chosen" from a set of n things. often read as "n choose k" and is called the choose function of n and k.was introduced by Andreas von Ettingshausen in 1826,[1] although the numbers were already known centuries before that (see Pascal's triangle). Alternative notations include C(n, k), n C k, n C k,,[2] in all of which the C stands for combinations or choices.Contents[hide]∙ 1 Definitiono 1.1 Recursive formulao 1.2 Multiplicative formulao 1.3 Factorial formulao 1.4 Generalization and connection to the binomial series ∙ 2 Pascal's triangle∙ 3 Combinatorics and statistics∙ 4 Binomial coefficients as polynomialso 4.1 Binomial coefficients as a basis for the space of polynomialso 4.2 Integer-valued polynomialso 4.3 Example∙ 5 Identities involving binomial coefficientso 5.1 Powers of -1o 5.2 Series involving binomial coefficientso 5.3 Identities with combinatorial proofso 5.4 Continuous identities∙ 6 Generating functionso 6.1 Ordinary generating functions∙7 Divisibility properties∙8 Bounds and asymptotic formulas∙9 Generalizationso9.1 Generalization to multinomialso9.2 Generalization to negative integerso9.3 Taylor serieso9.4 Binomial coefficient with n=1/2o9.5 Identity for the product of binomial coefficientso9.6 Partial Fraction Decompositiono9.7 Newton's binomial serieso9.8 Two real or complex valued argumentso9.9 Generalization to q-serieso9.10 Generalization to infinite cardinals∙10 Binomial coefficient in programming languages∙11 See also∙12 Notes∙13 References[edit] DefinitionFor natural numbers(taken to include 0) n and k, the binomial coefficientbe defined as the coefficient of the monomial X k in the expansion of (1 + X)n. The same coefficient also occurs (if k≤ n) in the binomial formula(valid for any elements x,y of a commutative ring), which explains the name "binomial coefficient".Another occurrence of this number is in combinatorics, where it gives the number of ways, disregarding order, that a k objects can be chosen from among n objects; more formally, the number of k-element subsets (ork-combinations) of an n-element set. This number can be seen to be equal to the one of the first definition, independently of any of the formulas below to compute it: if in each of the n factors of the power (1 + X)n one temporarily labels the term X with an index i (running from 1 to n), then each subset of k indices gives after expansion a contribution X k, and the coefficient of that monomial in the result will be the number ofany natural numbers n and k. There are many other combinatorial interpretations of binomial coefficients (counting problems for which the answer is given by a binomial coefficient expression), for instance the number of words formed of n bits (digits 0 or 1) whose sum is k, but most of these are easily seen to be equivalent to counting k-combinations.expanding a binomial power or counting k-combinations.[edit] Recursive formulaOne has a recursive formula for binomial coefficientswith as initial valuesThe formula follows either from tracing the contributions to X k in (1 + X)n−1(1 + X), or by counting k-combinations of {1, 2, ..., n} that containn and that do not contain n separately. It follows easily thatwhen k> n, and for all n, so the recursion can stop whenreaching such cases. This recursive formula then allows the construction of Pascal's triangle.[edit] Multiplicative formulaA more efficient method to compute individual binomial coefficients is given by the formulaThis formula is easiest to understand for the combinatorial interpretation of binomial coefficients. The numerator gives the number of ways to select a sequence of k distinct objects, retaining the order of selection, from a set of n objects. The denominator counts the number of distinct sequences that define the same k-combination when order is disregarded.[edit] Factorial formulaFinally there is a formula using factorials that is easy to remember:where n! denotes the factorial of n. This formula follows from the multiplicative formula above by multiplying numerator and denominator by (n−k)!; as a consequence it involves many factors common to numerator and denominator. It is less practical for explicit computation unless common factors are first canceled (in particular since factorial values grow very rapidly). The formula does exhibit a symmetry that is less evident from the multiplicative formula (though it is from the definitions)[edit] Generalization and connection to the binomial seriesThe multiplicative formula allows the definition of binomial coefficients to be extended[note 1] by replacing n by an arbitrary number α (negative, real, complex) or even an element of any commutative ring in which all positive integers are invertible:With this definition one has a generalization of the binomial formula (with one of the variables set to 1), which justifies still calling thebinomial coefficients:This formula is valid for all complex numbers α and X with |X| < 1. It can also be interpreted as an identity of formal power series in X, where it actually can serve as definition of arbitrary powers of series with constant coefficient equal to 1; the point is that with this definition all identities hold that one expects for exponentiation, notablyIf α is a nonnegative integer n, then all terms with k> n are zero, and the infinite series becomes a finite sum, thereby recovering the binomial formula. However for other values of α, including negative integers and rational numbers, the series is really infinite.[edit] Pascal's triangleMain article: Pascal's ruleMain article: Pascal's trianglePascal's rule is the important recurrence relationwhich can be used to prove by mathematical induction a natural number for all n and k, (equivalent to the statement that k! divides the product of k consecutive integers), a fact that is not immediately obvious from formula (1).Pascal's rule also gives rise to Pascal's triangle:0: 11: 1 12: 1 2 13: 1 3 3 14: 1 4 6 4 15: 1 5 10 10 5 16: 1 6 15 20 15 6 17: 1 7 21 35 35 21 7 18: 1 8 28 56 70 56 28 8 1Row number n k= 0,…,n. It is constructed by starting with ones at the outside and then always adding two adjacent numbers and writing the sum directly underneath. This method allows the quick calculation of binomial coefficients without the need for fractions or multiplications. For instance, by looking at row number 5 of the triangle, one can quickly read off that(x + y)5 = 1x5 + 5x4y + 10x3y2 + 10x2y3 + 5x y4 + 1y5.The differences between elements on other diagonals are the elements in the previous diagonal, as a consequence of the recurrence relation (3) above.[edit] Combinatorics and statisticsBinomial coefficients are of importance in combinatorics, because they provide ready formulas for certain frequent counting problems:∙k elements from a set of n elements.See Combination.∙There are ways to choose k elements from a set of n if repetitions are allowed. See Multiset.∙There are strings containing k ones and n zeros.∙There are strings consisting of k ones and n zeros such that no two ones are adjacent.∙The Catalan numbers are∙The binomial distribution in statistics is∙The formula for a Bézier curve.[edit] Binomial coefficients as polynomialsFor any nonnegative integer k, the expression can be simplified and defined as a polynomial divided by k!:This presents a polynomial in t with rational coefficients.As such, it can be evaluated at any real or complex number t to define binomial coefficients with such first arguments. These "generalized binomial coefficients" appear in Newton's generalized binomial theorem.For each k, the polynomial can be characterized as the unique degree k polynomial p(t) satisfying p(0) = p(1) = ... = p(k− 1) = 0 and p(k) = 1.Its coefficients are expressible in terms of Stirling numbers of the first kind, by definition of the latter:The derivative be calculated by logarithmic differentiation:[edit] Binomial coefficients as a basis for the space of polynomialsOver any field containing Q, each polynomial p(t) of degree at most d isuniquely expressible as a linear combination . The coefficient a k is the k th difference of the sequence p(0), p(1), …, p(k). Explicitly,[note 2][edit] Integer-valued polynomialsEach integer-valued: it takes integer values at integer inputs. (One way to prove this is by induction on k, using Pascal's identity.) Therefore any integer linear combination of binomial coefficient polynomials is integer-valued too. Conversely, (3.5) shows that any integer-valued polynomial is an integer linear combination of these binomial coefficient polynomials. More generally, for any subring R of a characteristic 0 field K, a polynomial in K[t] takes values in R at all integers if and only if it is an R-linear combination of binomial coefficient polynomials.[edit] ExampleThe integer-valued polynomial 3t(3t+ 1)/2 can be rewritten as[edit] Identities involving binomial coefficientsFor any nonnegative integers n and k,This follows from (2) by using (1 + x)n = x n·(1+ x−1)n. It is reflected in the symmetry of Pascal's triangle. A combinatorial interpretation of this formula is as follows: when forming a subset of k elements (from a set of size n), it is equivalent to consider the number of ways you can pick k elements and the number of ways you can exclude n−k elements.The factorial definition lets one relate nearby binomial coefficients. For instance, if k is a positive integer and n is arbitrary, thenand, with a little more work,[edit] Powers of -1A special binomial coefficient is ; it equals powers of -1:[edit] Series involving binomial coefficientsThe formulais obtained from (2) using x = 1. This is equivalent to saying that the elements in one row of Pascal's triangle always add up to two raised to an integer power. A combinatorial interpretation of this fact involving double counting is given by counting subsets of size 0, size 1, size 2, and so on up to size n of a set S of n elements. Since we count the number of subsets of size i for 0 ≤ i≤ n, this sum must be equal to the number of subsets of S, which is known to be 2n. That is, Equation 5 is a statement that the power set for a finite set with n elements has size 2n.The formulasandfollow from (2), after differentiating with respect to x (twice in the latter) and then substituting x = 1.Vandermonde's identityis found by expanding (1 + x)m(1 + x)n−m = (1 + x)n with (2). Aszero if k> n, the sum is finite for integer n and m. Equation (7a) generalizes equation (3). It holds for arbitrary, complex-valued m and n, the Chu-Vandermonde identity.A related formula isWhile equation (7a) is true for all values of m, equation (7b) is true for all values of j between 0 and k inclusive.From expansion (7a) using n=2m, k = m, and (4), one findsLet F(n) denote the n th Fibonacci number. We obtain a formula about the diagonals of Pascal's triangleThis can be proved by induction using (3) or by Zeckendorf's representation (Just note that the lhs gives the number of subsets of {F(2),...,F(n)} without consecutive members, which also form all the numbers below F(n+1)).Also using (3) and induction, one can show thatAgain by (3) and induction, one can show that for k = 0, ... , n−1as well aswhich is itself a special case of the result from the theory of finite differences that for any polynomial P(x) of degree less than n,[citation needed]Differentiating (2) k times and setting x = −1 yields this for, when 0 ≤ k< n, and the general casefollows by taking linear combinations of these.When P(x) is of degree less than or equal to n,where a n is the coefficient of degree n in P(x).More generally for 13b,where m and d are complex numbers. This follows immediately applying (13b) to the polynomial Q(x):=P(m + dx)instead of P(x), and observing that Q(x) has still degree less than or equal to n, and that its coefficient of degree n is d n a.nThe infinite seriesis convergent for k≥ 2. This formula is used in the analysis of the German tank problem. It is equivalent to the formula for the finite sumwhich is proved for M>m by induction on M.Using (8) one can deriveand.[edit] Identities with combinatorial proofsMany identities involving binomial coefficients can be proved by combinatorial means. For example, the following identity for nonnegativeintegers (which reduces to (6) when q = 1):can be given a double counting proof as follows. The left side counts the number of ways of selecting a subset of [n] of at least q elements, and marking q elements among those selected. The right side counts the sameparameter, because there are ways of choosing a set of q marks and they occur in all subsets that additionally contain some subset of the remaining elements, of which there are 2n−q.The recursion formulawhere both sides count the number of k-element subsets of {1, 2, . . . , n} with the right hand side ﬁrst grouping them into those which contain element n and those which don’t.The identity (8) also has a combinatorial proof. The identity readsSuppose you have 2n empty squares arranged in a row and you want to mark(select) n of them. There are ways to do this. On the other hand, you may select your n squares by selecting k squares from among the first n and n−k squares from the remaining n squares. This givesNow apply (4) to get the result.[edit] Continuous identitiesCertain trigonometric integrals have values expressible in terms of binomial coefficients:For andThese can be proved by using Euler's formula to convert trigonometric functions to complex exponentials, expanding using the binomial theorem, and integrating term by term.[edit] Generating functions[edit] Ordinary generating functionsFor a fixed n, the ordinary generating function of the sequenceis:For a fixed k, the ordinary generating function of the sequenceis:The bivariate generating function of the binomial coefficients is:[edit] Divisibility propertiesIn 1852, Kummer proved that if m and n are nonnegative integers and p isa prime number, then the largest power of p dividing equals p c, where c is the number of carries when m and n are added in base p. Equivalently,the exponent of a prime pj such that the fractional part of k/p j is greater than the fractionalpart of n/p jn/gcd(n,k).A somewhat surprising result by David Singmaster (1974) is that any integer divides almost all binomial coefficients. More precisely, fix aninteger d and let f(Nwith n < N such that dn < N is N(N+1) / 2, this implies that the density of binomial coefficients divisible by d goes to 1.Another fact: An integer n≥ 2 is prime if and only if all the intermediate binomial coefficientsare divisible by n.Proof: When p is prime, p dividesfor all 0 < k< pbecause it is a natural number and the numerator has a prime factor p but the denominator does not have a prime factor p.When n is composite, let p be the smallest prime factor of n and let k= n/p. Then 0 < p< n andotherwise the numerator k(n−1)(n−2)×...×(n−p+1) has to be divisible by n= k×p, this can only be the case when (n−1)(n−2)×...×(n−p+1) is divisible by p. But n is divisible by p, so p does not divide n−1, n−2, ..., n−p+1 and because p is prime, we know that p does not divide (n−1)(n−2)×...×(n−p+1) and so the numerator cannot be divisible by n. [edit] Bounds and asymptotic formulasStirling's approximation yields the bounds:and, in general,for m≥ 2 and n≥1,and the approximationasThe infinite product formula (cf. Gamma function, alternative definition)yields the asymptotic formulasas .This asymptotic behaviour is contained in the approximationas well. (Here H k is the k th harmonic number and γis the Euler–Mascheroni constant).The sum of binomial coefficients can be bounded by a term exponential in n and the binary entropy of the largest n/ k that occurs. More precisely, for and , it holdswhere is the binary entropy of ε.[3]A simple and rough upper bound for the sum of binomial coefficients is given by the formula below (not difficult to prove)[edit] Generalizations[edit] Generalization to multinomialsBinomial coefficients can be generalized to multinomial coefficients. They are defined to be the number:whereWhile the binomial coefficients represent the coefficients of (x+y)n, the multinomial coefficients represent the coefficients of the polynomialSee multinomial theorem. The case r = 2 gives binomial coefficients:The combinatorial interpretation of multinomial coefficients is distribution of n distinguishable elements over r (distinguishable) containers, each containing exactly k i elements, where i is the index of the container.Multinomial coefficients have many properties similar to these of binomial coefficients, for example the recurrence relation:and symmetry:where (σi) is a permutation of (1,2,...,r).[edit] Generalization to negative integersIf , thenextends to all n.[edit] Taylor seriesUsing Stirling numbers of the first kind the series expansion around any arbitrarily chosen point z0 is[edit] Binomial coefficient with n=1/2The definition of the binomial coefficients can be extended to the case where n is real and k is integer.In particular, the following identity holds for any non-negative integer k:This shows up when expanding into a power series using the Newtonbinomial series :[edit] Identity for the product of binomial coefficientsOne can express the product of binomial coefficients as a linear combination of binomial coefficients:where the connection coefficients are multinomial coefficients. In terms of labelled combinatorial objects, the connection coefficients represent the number of ways to assign m+n-k labels to a pair of labelled combinatorial objects of weight m and n respectively, that have had their first k labels identified, or glued together, in order to get a new labelled combinatorial object of weight m+n-k. (That is, to separate the labels into 3 portions to be applied to the glued part, the unglued part of the first object, and the unglued part of the second object.) In this regard, binomial coefficients are to exponential generating series what falling factorials are to ordinary generating series.[edit] Partial Fraction DecompositionThe partial fraction decomposition of the inverse is given byand[edit] Newton's binomial seriesNewton's binomial series, named after Sir Isaac Newton, is one of the simplest Newton series:The identity can be obtained by showing that both sides satisfy the differential equation (1+z) f'(z) = αf(z).The radius of convergence of this series is 1. An alternative expression iswhere the identityis applied.[edit] Two real or complex valued argumentsThe binomial coefficient is generalized to two real or complex valued arguments using the gamma function or beta function viaThis definition inherits these following additional properties from Γ:moreover,The resulting function has been little-studied, apparently first being graphed in (Fowler 1996). Notably, many binomial identities fail:but for n positive (so −n negative). The behavior is quite complex, and markedly different in various octants (that is, with respect to the x and y axes and the line y= x), with the behavior for negative x having singularities at negative integer values and a checkerboard of positive and negative regions:∙in the octant it is a smoothly interpolated form of theusual binomial, with a ridge ("Pascal's ridge").∙in the octant and in the quadrant the function is close to zero.∙in the quadrant the function is alternatingly verylarge positive and negative on the parallelograms with vertices ( −n,m + 1),( −n,m),( −n− 1,m− 1),( −n− 1,m) ∙in the octant 0 > x > y the behavior is again alternatingly very large positive and negative, but on a square grid.∙in the octant − 1 > y > x + 1 it is close to zero, except for near the singularities.[edit] Generalization to q-seriesThe binomial coefficient has a q-analog generalization known as the Gaussian binomial coefficient.[edit] Generalization to infinite cardinalsThe definition of the binomial coefficient can be generalized to infinite cardinals by defining:where A is some set with cardinalityα. One can show that the generalized binomial coefficient is well-defined, in the sense that no matter whatset we choose to represent the cardinal number α, will remain the same. For finite cardinals, this definition coincides with the standard definition of the binomial coefficient.Assuming the Axiom of Choice, one can show that for any infinite cardinal α.[edit] Binomial coefficient in programming languagesThe notation is convenient in handwriting but inconvenient for typewriters and computer terminals. Many programming languages do not offer a standard subroutine for computing the binomial coefficient, but for example the J programming language uses the exclamation mark: k ! n .Naive implementations of the factorial formula, such as the following snippet in C:int choose(int n, int k) {return factorial(n) / (factorial(k) * factorial(n - k));}are prone to overflow errors, severely restricting the range of input values. A direct implementation of the multiplicative formula works well:unsigned long long choose(unsigned n, unsigned k) {if (k > n)return 0;if (k > n/2)k = n-k; // Take advantage of symmetrylong double accum = 1;unsigned i;for (i = 1; i <= k; i++)accum = accum * (n-k+i) / i;return accum + 0.5; // avoid rounding error}Another way to compute the binomial coefficient when using large numbers is to recognize thatlnΓ(n) is a special function that is easily computed and is standard in some programming languages such as using log_gamma in Maxima, LogGamma in Mathematica, or gammaln in MATLAB. Roundoff error may cause the returned value not to be an integer.[edit] See also∙Central binomial coefficient∙Binomial transform∙Star of David theorem∙Table of Newtonian series∙List of factorial and binomial topics∙Multiplicities of entries in Pascal's triangle∙Sun's curious identity[edit] Notes1.^See (Graham, Knuth & Patashnik 1994), which alsok< 0. Alternative generalizations, such as to two real or complex valued arguments using the Gamma function assign nonzero values tok< 0, but this causes most binomial coefficient identities to fail, and thus is not widely used majority of definitions. One such choice of nonzero values leads to the aesthetically pleasing "Pascal windmill" in Hilton, Holton and Pedersen, Mathematicalreflections: in a room with many mirrors, Springer, 1997, but causes even Pascal's identity to fail (at the origin).2.^ This can be seen as a discrete analog of Taylor's theorem. It isclosely related to Newton's polynomial. Alternating sums of this form may be expressed as the Nörlund–Rice integral.[edit] References1.^Nicholas J. Higham. Handbook of writing for the mathematicalsciences. SIAM. p. 25. ISBN0898714206.2.^ G. E. Shilov (1977). Linear algebra. Dover Publications.ISBN9780486635187.3.^ see e.g. Flum & Grohe (2006, p. 427)∙Fowler, David(January 1996). "The Binomial Coefficient Function".The American Mathematical Monthly(Mathematical Association ofAmerica) 103 (1): 1–17. doi:10.2307/2975209./stable/2975209∙Knuth, Donald E. (1997). The Art of Computer Programming, Volume 1: Fundamental Algorithms (Third ed.). Addison-Wesley.pp. 52–74. ISBN0-201-89683-4.∙Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994).Concrete Mathematics (Second ed.). Addison-Wesley.pp. 153–256. ISBN0-201-55802-5.∙Singmaster, David (1974). "Notes on binomial coefficients. III.Any integer divides almost all binomial coefficients". J. LondonMath. Soc. (2)8: 555–560. doi:10.1112/jlms/s2-8.3.555.∙Bryant, Victor (1993). Aspects of combinatorics. Cambridge University Press.∙Arthur T. Benjamin; Jennifer Quinn, Proofs that Really Count: The Art of Combinatorial Proof, Mathematical Association of America,2003.∙Flum, Jörg; Grohe, Martin (2006). Parameterized Complexity Theory.Springer. ISBN978-3-540-29952-3./east/home/generic/search/results?SGWID=5-40109-22-141358322-0.This article incorporates material from the following PlanetMath articles, which are licensed under the Creative Commons Attribution/Share-Alike License: Binomial Coefficient, Bounds for binomial coefficients, Proof that C(n,k) is an integer, Generalized binomial coefficients.。

翻译学导论

翻译学导论TranslatologyHua XianfaCourse DescriptionTranslatology, interdisciplinary in nature, is an independent discipline and regarded as a science to study translation. It is a comprehensive course subject consisting of general translatology, theoretical translatology and applied translatology. It is about general laws of translation, studies of translation with special reference to Chinese and English, and methods to apply theories to practice.The course of Translatology aims at providing students with traditional perspectives as well as latest developments in translation studies so as to raise their ability to pursue theoretical researches; it also aims at analyzing and solving problems which constantly occur in the process of translation by focusing on contrastive studies between Chinese and English and, between the Chinese culture and the English culture. It will combine theory with practice by discussing good examples with specific principles as guidelines. Translation exercises will help consolidate what has been learned and discussion will promote exchanges of ideas.It is expected that the course will offer students with essential theoretical understanding of translation and opportunities of integrating theory with practice.Assessmentclass performance(10%) translation exercises(20%) examination(70%)References:[1]Baker, Mona. In Other Words: A Coursebook on Translation[M]. London and New York:Routledge, 1992.[2]Baker, Mona. The Routledge Encyclopedia of Translation Studies[Z], London and New York:Routledge, 1997.[3]Chan, Tak-hung. Twentieth-Century Chinese Translation Theory: Modes, issues anddebates[M], Benjamins Translation Library, 2004.[4]House, J. Translation Quality Assessment: A Model Revisited[M], Tübingen: Niemeyer, 1997.[5]Munday, Jeremy. Introducing Translation Studies[M]. London: Routledge, 2001.[6]Newmark, P. A. Textbook of Translation[M]. London: Prentice Hall International(UK)Ltd. 1988[7]Newmark, P.A. Approaches to Translation[M]. London: Prentice Hall International(UK)Ltd.1988.[8]Nida, E.A. Language, Culture and Translating[M]. Shanghai: Shanghai ForeignLanguage Education Press, 1993.[9]Snell-Hornby, Mary. Translation Studies: An Integrated Approach[M]. ShanghaiForeign Language Education Press, 2001.[10]陈宏薇、李亚丹.新编汉英翻译教程[Z]，上海：上海外语教育出版社，2004.[11]郭延礼. 中国近代翻译文学概论[Z], 武汉:湖北教育出版社,1998.[12]华先发、邵毅.新编大学英译汉教程[Z]，上海：上海外语教育出版社，2004.[13]李定坤. 汉英辞格对比与翻译[Z]，武汉：华中师范大学出版社，1994.[14]李瑞华. 英汉语言文化对比研究[C],上海：上海外语教育出版社，1997.[15]谭载喜. 翻译学[M], 武汉:湖北教育出版社,2000.[16]许均. 翻译论[M],武汉：湖北教育出版社，2003.Chapter I General TranslatologyGeneral translatology addresses the description of the phenomena of translation by focusing on such issues as the definition of translation, the nature of translation, types of translation, criteria of translation, procedures of translation, and disciplines related to translation and studies.1.1 What is translation?Different perspectives on the nature of translatingTranslation can be defined from different perspectives, which present to us different definitions:A. From the perspective of the process of the action itself:In the light of popular orientation (in dictionaries and encyclopedias), translation is to express (sth spoken or esp written) in another language or in simple words. (Hornby, A.S. OALECD 4th ed. 2000)Eugene Nida: Translating consists in reproducing in the receptor language the closest natural equivalent of the source language message, first in terms of meaning and secondly in terms of style. (1969:12)John Catford: Translation may be defined as follows: The replacement of textual material in one language (SL) by equivalent textual material in another language (TL). (1965:20) The central problem of translation practice is that of finding TLtranslation equivalents. A central task of translation theory is that of defining that nature and conditions of translation equivalence. (1965:21)Bahudarov: Translation is a process in which the parole of one language is transferred into the parole of another with the content i.e. meaning unchanged.Peter Newmark: Translation what is translation? Often, though not by any means always, it is rendering the meaning of a text into another language in the way that the author intended the text. (1988:5)B. From the perspective of the function of translation:Steiner: Translation it is that openeth the window, to let in the light; that breaketh the shell, that we may eat the kernel.Peter Newmark:Translation is an instrument of education as well as of truth precisely because it has to reach readers whose cultural and educational level is different from, and often “lower” or earlier, than that of the reader of the original. (1988:6) Peter Newmark: Translation has its own excitement, its own interest. A satisfactory translation is always possible, but a good translator is never satisfied with it. It can usually be improved. There is no such thing as a perfect ideal or …correct? translation.C.From the perspective of aesthetics:Malcolm Cowley: Translation is an art that involves the recreation of a work in another language for readers with a different background.D. From the perspective of the nature of translation:Peter Newmark: Translation is first a science, which entails the knowledge and verification of the facts and the language that describes them—here, what is wrong, mistaken or truth, can be identified; secondly, it is a skill, which calls for appropriatelanguage and acceptable usage; thirdly, an art, which distinguishes good from undistinguished writing and is the creative, the intuitive, sometimes the inspired, level of the translation; lastly, a matter of taste, where argument ceases, preferences are expressed, and the variety of meritorious translations is the reflection of individual differences. (1988:6)E. From the perspective of cross-cultural communicationJin Di & Nida: Translation is simply a form of interlingual communication.(1987)1.2 Types of translationTranslation can also be classified from different angles by different people.A. From the standpoint of signs involved in translation:Roman Jacobson: interlingual translation, intralingual translation, intersemiotic translation (interpretation of verbal signs by means of none verbal signs ) (On Linguistic Aspects of Translation)B. From the standpoint of way in which translation is carried out:Written interlingual translationOral interlingual transltionMachine translationC. From the standpoint of the extent to which translation is done:Full translation(absolute translation), partial translation (selective translation) , translation plus editing(translation with reconstruction)D. From the standpoint of the level of translationtotal translation (translation equivalence can be established at the same grammatical or phonological hierarchy)restricted translation (translation equivalence can be established at only one grammatical hierarchy)E. From the standpoint of the languages involved in translationtranslation from the native language into the foreign language or vice versa1.3 Criteria of translationAlexander F. TytlerA translation should give a complete transcript of the ideas of the original work; the style and manner of writing should be of the same character as that of the original; a translation should have all the ease of the original composition. (Tytler, 1797, P.15) John CatfordThe SL and TL rarely have …the same meaning? in the linguistic sense; but they can function in the same situation.For translation equivalence to occur, then both SL and TL texts must be relatable to the functionally relevant features of the situation.FedorovThe exactness of translation means the exact rendering of the thought and content of the original and performs the same rhetorical function as the original.Nidafunctional equivalenceThe crucial problem of translation is often stated in terms of conflict between formal correspondence and functional equivalence.A close formal correspondence in a receptor language so frequently does not carry the correct meaning of the source text. In some cases, formal correspondence doesn?t work , thenfunctional equivalence should take its place.There are five cases in which formal correspondence should be given up:1)When a literal rendering would give an entirely wrong meaning.2)When a borrowed term constitutes a semantic …zero? and is therefore likely to be filled with the wrong meaning.3)When a formal correspondence involves a serous obscurity in meaning.4)When formal correspondence would result in bad grammar or style in the receptor language.(Waard, Han, De & E.A. Nida)SavoryDifferent languages have different ways for expression, but all of them have an underlying equivalent thought.The translator faces the question as to whether his function is to record the words of the original or to report on their meanings.Gregory RabassaSo our criterion must state that the best translation is the closest approach.A translation can never equal the original; it can approach it, and its quality can only be judged by how close it gets.There are no two snowflakes alike.Yan Fufaithfulness, expressiveness and eleganceQian Zhongshusublimation the transmigration of souls “Although the body changes, the s oul of gesture remains its old self.”Fu LeiAs far as the effect is concerned, translation should be like copying a painting. What is desired is not being alike in appearance but being alike in spirit.Lu XunThe minimum requirement for general translation should at least convey faithfully the content of the original with understandable smooth translation, but this is not enough for literary translation… In literary translation, the artistic conception of the original is transmitted in another language so that when he is reading the translation, the reader can be inspired, touched and aesthetically affected the same as the if he were reading the original.Liu Zhongdefaithfulness, expressiveness and closeness1.4 Translation strategies and techniquesA. Translation strategiesChinese: 文and 质literal translation：word for word and line for line; keeping both the form and the content liberal/free translation: scarifying the form while keeping the content and spirit Schleiermacher: alienating and naturalizationEither the translator leaves the writer alone as much as possible and moves the reader toward the writer, or he [sic] leaves the reader alone as much as possible and moves the writer toward the reader. (Schleiermacher 1813/1992:41-42) Venuti: domestication and foreignizationDomestication: Domestication covers adherence to domestic literary canons by carefully selecting the texts that are likely to lend themselves to such a translation strategy(Venuti,1997:241) Foreignization: Foreignization entails choosing a foreign text anddeveloping a translation method along lines which are excluded by dominant cultural values in the target language.(Venuti, 1997:242)John Dryden: metaphrase, paraphrase and imitationB. Translation techniquestransliteration, annotation, paraphrase, amplification, omission, shift of perspective, adaptation,division, combination, specification, and generalization (华先发、邵毅，2004)1.5 The basic procedures of translationThe basic procedures of translation employed in translating consist of preparation, working and testing. Preparation refers to such matters as choosing what to be translated and contacting the publisher. Working includes both comprehension and expression. And testing is the final step taken to check the result of the work.1.5.1 Comprehension consists essentially in determining the meaning of the source language tex t, but …meaning? is to be interprete d in the broad sense of lexical, syntactic and rhetorical significance. It involves such steps as grammatical analysis, semantic analysis, stylistic analysis and discourse analysis. In other words, both form and content must be considered in comprehension.A. ComprehensionBe careful in determining the meaning of the words, phrases and sentences.（1）月明星稀，乌鹊南飞（曹孟德）(2) 若崩厥角稽首（孟子尽心下）(3) Mrs. Malaprop said, ”As she grew up, I would have her instructed in Geometry, that she might know something of thecontagious countries.” (R.B.Sheridan: The Rivals) (geometry[几何学]=geography[地理学]; contagious[传染的]=contiguous[邻近的])(4) “We sometimes fall in with persons who have seen much of the world, and of the men who, in their day, have played a conspicuous part in it, but who generalize nothing, and have no observation, in the true sense of the word.” (J. H. Newman)(5)王先生昨天被偷了。

q-COMPLEX NUMBERS, A NATURAL CONSEQUENCE OF UMBRAL CALCULUS

(11)
We ﬁrst give an improved and extended version of the q -umbral calculus from [9] and [2]. This umbral calculus has a close connection to the work of N¨ orlund [25] on Bernoulli polynomials. For the rest of the paper, we assume that 0 < |q | < 1. Deﬁnition 5. The identity operator and the forward operator are denoted I and E . ∞ Let {θi }∞ 0 and {φi }0 denote arbitrary secuences. The Carlitz–Gould [15] q -diﬀerence operator is deﬁned by
+1 l l l △CG,q θ0 ≡ (E − I )θ0 , △l CG,q θ0 ≡ △CG,q Eθ0 − q △CG,q θ0 .
(12) (13)
Or equivalently, △CG,q ≡ E ⊟q I. The last formula requires knowledge of the umbral calculus to be deﬁned shortly. If the identity operator, the forward operator, or the Carlitz–Gould operator operates on m, we denote them Im , Em , or △CG,m,q .

名词解释2

名词解释1.Inflection:it is the manifestation of grammatical relationship through the addition of inflectional affixes such as number, person, finiteness, aspect and cases to which they are attached.2.Suprasegmental feature：The phonemic features that occur above the level of the segments are called suprasegmental features; these are the phonological properties of such units as the syllable, the word, and the sentence. the main suprasegmental ones includes stress, intonation, and tone.3.Speech variety：It is a term sometimes used instead of language, dialect, sociolect, pidgin, creole,etc. because it is concered more neutral than such terms. It may also be used for different varieties if one language, e.g. American English, Australian English, Indian English.4.Standard language：It is also called standard variety. It is the variety of a language which has the highest status in a community or nation and which is usually based on the speech and writing of educated native speakers of the language.5.Predication：Predication is the abstraction of the meaning of a sentence. It is the basic unit in the semantic analysis of the a sentence. A predication consist of argument and predicate.6.Deep structure：It is a central theoretical term in generative grammar, opposed to surface structure. It is the abstract syntactic representation of a sentence--an underlying level of structural organization which specifies or the facters governing the way the sentence should be interpreted.7.Semantics：It is the study of meaning communicated through language. The basic task in semantics is to show how people communicate meanings with pieces of language.8.Euphemism：It is a mild, indirect or less offensive words or expression that replace a taboo word.9.Bound morpheme：It is a morpheme which can not stand by itself as a complete utterance. It must appear with at lest one other morpheme,free or bound , like un-?? in unhappy, past tense morpheme in worked.10.Slang：It refers to casual, very informal speech, using expressive but informal words and expressions.11.Conceptualism：It is the view which holds that there is no direct link between a linguistic form and what it refers to; rather , in the interpretation of meaning they are linked through the medaition of concepts in the mind.12.Phonic medium of language：the limited range of sounds that are used in human language communication, i.e. the speech soundsnguage acquisition device (LAD)：A hypothetical innate mechanism every normal human child is believed to be born with, which allow them to acquire language14.Universal Grammar：It is the genetically endowed information consisting of principles and parameters that enable the child to deduce a grammar from the primary linguistic data.15.Diachronic linguistics：It is a study of a language through the course of its history; therfore, it is also called historcial linguistics.16.Apocope：It is known as a process in which final vowels may be lost. For example, the Old English word helpe developed into help in Modern English.17.Truth condition：It is the fact that would have to obtain in reality to make a proposition true or false.18.Displacement：It means that human languages enable their users to symbolize objects, events and concepts, which are not present (in time and space) at the moment of communication.19.Voicing：It is a phonetic feature of some sounds. It is is caused by the viberation of the vocal cords.20.Sentence meaning：It refers to a sentence and is a grammatical concept, and the meaning of a sentence is often studied as the abstract , intrinsic property of the sentence itself in terms of predication.21. Registers：language varieties appropriate for use in particular speech situations, in contrast to language varieties that are associated with the social or regional grouping of their customary users. Registers are also called situational dialects22. Performatives：They are sentences that do not state a fact or describe a state and are not verifiable, in other words, performatives are utterances that prefome an act —"do things".24. universal grammar：it is system of linguistic knowledge and a human species-specific gift which exists in the mind of a normal humanbeing. It consists of a set of general conditions, or general principles, that generate phrases and sentences.26. Morpheme：the smallest unit of language in terms of the relationship between expression and content, a unit that cannot be divided into further smaller units without destroying or drastically altering the meaning, whether it is lexical or grammatical.28. Minimal pair：When two different phonetic forms are identical in every way except in one sound element that occurs in the same position in the string, the two forms are said two form a minimal pair.31. Morpheme：the smallest unit of language in terms of the relationship between expression and content, a unit that cannot be divided into further smaller units without destroying or drastically altering the meaning, whether it is lexical or grammatical.32. Design features：Design features refers to defining properties of human language that distinguish it from any other animal system of communication. They are arbitrariness, productivity, duality, displacement, cultural transimission34. Telegraphic speech：Children's telegraphic speech: Children's early multiword speech that contains content words and lacks function words and inflectional morphemesthat contains content words and lacks function words and inflectional morphemes.35. Semantic shift：It means that the meaning of a word takes a departure from its original domain as a result of its metaphorical usage. For example, the meaning of teeth has a semantic shift when it is used in the phrase the teeth of a comb.36. synonymy：It refers to the samesness or close similarity of meaning.The relation between lorry and truck, room and chamber are examples of synonymy.37. Illocutionary act：It is using a sentence to perform a function. for example,shoot the snake may be intended as an order or a piece of advice39. Componential analysis：a way proposed by the structural semanticists to analyze word meaning., which believe that the meaning of a word can be further divided into smaller units called semantic features.40.Voicing：It is a phonetic feature of some sounds. It is is caused by the viberation of the vocal cords.41. Diachronic linguistics：It is a study of a language through the course of its history; therfore, it is also called historcial linguistics.43.Slang：It refers to casual, very informal speech, using expressive but informal words and expressions.44. Narrow transcriptio n：the transcription with help of the diacritics, so that they can faithfully represent as much of the fine details as it is necessary for the purpose of the phonetic research. 45. Inflection：it is the manifestation of grammatical relationship through the addition of inflectional affixes such as number, person, finiteness, aspect and cases to which they are attached.46. Critical Period Hypothesis：The hypothesis that the time span between early childhood and puberty is the critical period for language acquisition, during which children can acquire language without formal instruction successfully and effortlessly.47. Epenthesis：It is a process in which a vowel or consonant may be inserted to the middle of a word. For example, by insertion of /p/, Old English word glims changed into glimpse.48. Move-α：It is a general movement rule accounting for the syntactic behavior of any costituent movement.49. Performatives：They are sentences that do not state a fact or describe a state and are not verifiable, in other words, performatives are utterances that prefome an act —"do things".50. Reference：It is what a linguistic form refers to in the real world; it is a matter of the relationship between the form and the reality.51.Semantic field:It is an organizational principle which thinks of the lexicon and groups of words in the lexicon can be semantically related, rather than a listing of words as in a published dictionary.52.Utterance meaning:It is the meaning a speaker conveys by using a particular utterance in a particular context situation.53.Taboo:Words known to speakers but avoided in some contexts of speech for reasons of religion, politeness etc.54.Syntactic category:It is a word or phrase that performs a particular grammatical function such as the subject or object.55.vowel:A vowel is a sound in forming which the air that comes from the lungs meets with no obstruction of any kind in the throat, the nose or the mouth.56.Entailment:This a logic relationship between two sentences in which the truth of the second necessarily follows from the truth of the first, while the falsity of the first follows from the falsity of the second.57.Pitch:Pitch is a suprasegmental feature, whose domain of application is the syllable. Different rates of vibration produce what is known as different frequencies, and in auditory terms as different pitches. Pitch variations may be dinstinctive like phoneme, that is, when they may contribute to distinguish between different words. In this function, pitch variations are called tones.。

量子纠缠双缝干涉英语范例

量子纠缠双缝干涉英语范例Engaging with the perplexing world of quantum entanglement and the double-slit interference phenomenon in the realm of English provides a fascinating journey into the depths of physics and language. Let's embark on this exploration, delving into these intricate concepts without the crutchesof conventional transition words.Quantum entanglement, a phenomenon Albert Einstein famously referred to as "spooky action at a distance," challengesour conventional understanding of reality. At its core, it entails the entwining of particles in such a way that the state of one particle instantaneously influences the stateof another, regardless of the distance separating them.This peculiar connection, seemingly defying the constraints of space and time, forms the bedrock of quantum mechanics.Moving onto the enigmatic realm of double-slit interference, we encounter another perplexing aspect of quantum physics. Imagine a scenario where particles, such as photons or electrons, are fired one by one towards a barrier with twonarrow slits. Classical intuition would suggest that each particle would pass through one of the slits and create a pattern on the screen behind the barrier corresponding tothe two slits. However, the reality is far more bewildering.When observed, particles behave as discrete entities, creating a pattern on the screen that aligns with the positions of the slits. However, when left unobserved, they exhibit wave-like behavior, producing an interferencepattern consistent with waves passing through both slits simultaneously. This duality of particle and wave behavior perplexed physicists for decades and remains a cornerstoneof quantum mechanics.Now, let's intertwine these concepts with the intricate fabric of the English language. Just as particles become entangled in the quantum realm, words and phrases entwineto convey meaning and evoke understanding. The delicate dance of syntax and semantics mirrors the interconnectedness observed in quantum systems.Consider the act of communication itself. When wearticulate thoughts and ideas, we send linguistic particles into the ether, where they interact with the minds of others, shaping perceptions and influencing understanding. In this linguistic entanglement, the state of one mind can indeed influence the state of another, echoing the eerie connectivity of entangled particles.Furthermore, language, like quantum particles, exhibits a duality of behavior. It can serve as a discrete tool for conveying specific information, much like particles behaving as individual entities when observed. Yet, it also possesses a wave-like quality, capable of conveying nuanced emotions, cultural nuances, and abstract concepts that transcend mere words on a page.Consider the phrase "I love you." In its discrete form, it conveys a specific sentiment, a declaration of affection towards another individual. However, its wave-like nature allows it to resonate with profound emotional depth, evoking a myriad of feelings and memories unique to each recipient.In a similar vein, the act of reading mirrors the double-slit experiment in its ability to collapse linguistic wave functions into discrete meanings. When we read a text, we observe its words and phrases, collapsing the wave of potential interpretations into a singular understanding based on our individual perceptions and experiences.Yet, just as the act of observation alters the behavior of quantum particles, our interpretation of language is inherently subjective, influenced by our cultural background, personal biases, and cognitive predispositions. Thus, the same text can elicit vastly different interpretations from different readers, much like the varied outcomes observed in the double-slit experiment.In conclusion, the parallels between quantum entanglement, double-slit interference, and the intricacies of the English language highlight the profound interconnectedness of the physical and linguistic worlds. Just as physicists grapple with the mysteries of the quantum realm, linguists navigate the complexities of communication, both realmsoffering endless opportunities for exploration and discovery.。

(完整版)英语语言学概论--整理

Chapter 1 Language语言1. Design feature (识别特征) refers to the defining properties of human language that distinguish it from any animal system of communication.2. Productivity(能产性) refers to the ability that people have in making and comprehending indefinitely large quantities of sentences in theirnative language.3. arbitrariness (任意性) Arbitrariness refers to the phenomenon that there is no motivated relationship between a linguistic form and itsmeaning.4. symbol (符号) Symbol refers to something such as an object, word, or sound that represents something else by association or convention.5. discreteness (离散性) Discreteness refers to the phenomenon that the sounds in a language are meaningfully distinct.6. displacement (不受时空限制的特性) Displacement refers to the fact that human language can be used to talk about things that are not in theimmediate situations of its users.7. duality of structure (结构二重性) The organization of language into two levels, one of sounds, the other of meaning, is known as duality ofstructure.8. culture transmission (文化传播) Culture transmission refers to the fact that language is passed on from one generation to the next throughteaching and learning, rather than by inheritance.9. interchangeability (互换性) Interchangeability means that any human being can be both a producer and a receiver of messages.1. ★What is language?Language is a system of arbitrary vocal symbols used for human communication. This definition has captured the main features of language.First, language is a system.Second, language is arbitrary in the sense.The third feature of language is symbolic nature.2. ★What are the design features of language?Language has seven design features as following:1) Productivity.2) Discreteness.3) Displacement4) Arbitrariness.5) Cultural transmission6) Duality of structure.7) Interchangeability.3. Why do we say language is a system?Because elements of language are combined according to rules, and every language contains a set of rules. By system, the recurring patterns or arrangements or the particular ways or designs in which a language operates. And the sounds, the words and the sentences are used in fixed patterns that speaker of a language can understand each other.4. ★ (Function of language.) According to Halliday, what are the initial functions of children’s language? And what are the threefunctional components of adult language?I. Halliday uses the following terms to refer to the initial functions of children’s language:1) Instrumental function. 工具功能2) Regulatory function. 调节功能3) Representational function. 表现功能4) Interactional function. 互动功能5) Personal function. 自指性功能6) Heuristic function. 启发功能[osbQtq`kf`h]7) Imaginative function. 想象功能II. Adult language has three functional components as following:1) Interpersonal components. 人际2) Ideational components.概念3) Textual components.语篇1. general linguistics and descriptive linguistics (普通语言学与描写语言学) The former deals with language in general whereas the latter isconcerned with one particular language.2. synchronic linguistics and diachronic linguistics (共时语言学与历时语言学) Diachronic linguistics traces the historical development of thelanguage and records the changes that have taken place in it between successive points in time. And synchronic linguistics presents an account of language as it is at some particular point in time.3. theoretical linguistics and applied linguistics (理论语言学与应用语言学) The former copes with languages with a view to establishing atheory of their structures and functions whereas the latter is concerned with the application of the concepts and findings of linguistics to all sorts of practical tasks.4. microlinguistics and macrolinguistics(微观语言学与宏观语言学) The former studies only the structure of language system whereas thelatter deals with everything that is related to languages.5. langue and parole (语言与言语) The former refers to the abstract linguistics system shared by all the members of a speech communitywhereas the latter refers to the concrete act of speaking in actual situation by an individual speaker.6. competence and performance (语言能力与语言运用) The former is one’s knowledge of all the linguistic regulation systems whereas the latteris the use of language in concrete situation.7. speech and writing (口头语与书面语) Speech is the spoken form of language whereas writing is written codes, gives language new scope.8. linguistics behavior potential and actual linguistic behavior (语言行为潜势与实际语言行为) People actually says on a certain occasion to acertain person is actual linguistics behavior. And each of possible linguistic items that he could have said is linguistic behavior potential.9. syntagmatic relation and paradigmatic relation(横组合关系与纵聚合关系) The former describes the horizontal dimension of a languagewhile the latter describes the vertical dimension of a language.10. verbal communication and non-verbal communication(言语交际与非言语交际) Usual use of language as a means of transmittinginformation is called verbal communication. The ways we convey meaning without using language is called non-verbal communication.1. ★How does John Lyons classify linguistics?According to John Lyons, the field of linguistics as a whole can be divided into several subfields as following:1) General linguistics and descriptive linguistics.2) Synchronic linguistics and diachronic linguistics.3) Theoretical linguistics and applied linguistics.4) Microlinguistics and macrolinguistics.2. Explain the three principles by which the linguist is guided: consistency, adequacy and simplicity.1) Consistency means that there should be no contradictions between different parts of the theory and the description.2) Adequacy means that the theory must be broad enough in scope to offer significant generalizations.3) Simplicity requires us to be as brief and economic as possible.3. ★What are the sub-branches of linguistics within the language system?Within the language system there are six sub-branches as following:1) Phonetics. 语音学is a study of speech sounds of all human languages.2) Phonology. 音位学studies about the sounds and sound patterns of a speaker’s native language.3) Morphology. 形态学studies about how a word is formed.4) Syntax. 句法学studies about whether a sentence is grammatical or not.5) Semantics. 语义学studies about the meaning of language, including meaning of words and meaning of sentences.6) Pragmatics. 语用学★The scope of language: Linguistics is referred to as a scientific study of language.★The scientific process of linguistic study: It involves four stages: collecting data, forming a hypothesis, testing the hypothesis and drawing conclusions.1. articulatory phonetics(发音语音学) The study of how speech organs produce the sounds is called articulatory phonetics.2. acoustic phonetics (声学语音学) The study of the physical properties and of the transmission of speech sounds is called acoustic phonetics.3. auditory phonetics (听觉语音学) The study of the way hearers perceive speech sounds is called auditory phonetics.4. consonant (辅音) Consonant is a speech sound where the air form the language is either completely blocked, or partially blocked, or where theopening between the speech organs is so narrow that the air escapes with audible friction.5. vowel (元音) is defined as a speech sound in which the air from the lungs is not blocked in any way and is pronounced with vocal-cord vibration.6. bilabials (双唇音) Bilabials means that consonants for which the flow of air is stopped or restricted by the two lips. [p][b] [m] [w]7. affricates (塞擦音) The sound produced by stopping the airstream and then immediately releasing it slowly is called affricates. [t X] [d Y] [tr] [dr]8. glottis (声门) Glottis is the space between the vocal cords.9. rounded vowel (圆唇元音) Rounded vowel is defined as the vowel sound pronounced by the lips forming a circular opening. [u:] [u] [OB] [O]10. diphthongs (双元音) Diphthongs are produced by moving from one vowel position to another through intervening positions.[ei][ai][O i] [Q u][au]11. triphthongs(三合元音) Triphthongs are those which are produced by moving from one vowel position to another and then rapidly andcontinuously to a third one. [ei Q][ai Q][O i Q] [Q u Q][au Q]12. lax vowels (松元音) According to distinction of long and short vowels, vowels are classified tense vowels and lax vowels. All the long vowelsare tense vowels but of the short vowels,[e] is a tense vowel as well, and the rest short vowels are lax vowels.1. ★How are consonants classified in terms of different criteria?The consonants in English can be described in terms of four dimensions.1) The position of the soft palate.2) The presence or the absence of vocal-cord vibration.3) The place of articulation.4) The manner of articulation.2. ★How are vowels classified in terms of different criteria?Vowel sounds are differentiated by a number of factors.1) The state of the velum2) The position of the tongue.3) The openness of the mouth.4) The shape of the lips.5) The length of the vowels.6) The tension of the muscles at pharynx.3. ★What are the three sub-branches of phonetics? How do they differ from each other?Phonetics has three sub-branches as following:1) Articulatory phonetics is the study of how speech organs produce the sounds is called articulatory phonetics.2) Acoustic phonetics is the study of the physical properties and of the transmission of speech sounds is called acoustic phonetics.3) Auditory phonetics is the study of the way hearers perceive speech sounds is called auditory phonetics.4. ★What are the commonly used phonetic features for consonants and vowels respectively?I. The frequently used phonetic features for consonants include the following:1) Voiced.2) Nasal.3) Consonantal.4) Vocalic.5) Continuant.6) Anterior.7) Coronal.8) Aspirated.II. The most common phonetic features for vowels include the following:1) High.2) Low.3) Front.4) Back.5) Rounded.6) Tense.1. phonemes (音位) Phonemes are minimal distinctive units in the sound system of a language.2. allophones (音位变体) Allophones are the phonetic variants and realizations of a particular phoneme.3. phones (单音) The smallest identifiable phonetic unit found in a stream of speech is called a phone.4. minimal pair (最小对立体) Minimal pair means words which differ from each other only by one sound.5. contrastive distribution (对比分布) If two or more sounds can occur in the same environment and the substitution of one sound for anotherbrings about a change of meaning, they are said to be in contrastive distribution.6. complementary distribution(互补分布) If two or more sounds never appear in the same environment ,then they are said to be incomplementary distribution.7. free variation (自由变异) When two sounds can appear in the same environment and the substitution of one for the other does not cause anychange in meaning, then they are said to be in free variation.8. distinctive features (区别性特征) A distinctive feature is a feature which distinguishes one phoneme from another.9. suprasegmental features (超切分特征) The distinctive (phonological) features which apply to groups larger than the single segment are knownas suprasegmental features.10. tone languages (声调语言) Tone languages are those which use pitch to contrast meaning at word level.11. intonation languages (语调语言) Intonation languages are those which use pitch to distinguish meaning at phrase level or sentence level.12. juncture (连音) Juncture refers to the phonetic boundary features which may demarcate grammatical units.1. ★What are the differences between English phonetics and English phonology?1) Phonetics is the study of the production, perception, and physical properties of speech sounds, while phonology attempts to account forhow they are combined, organized, and convey meaning in particular languages.2) Phonetics is the study of the actual sounds while phonology is concerned with a more abstract description of speech sounds and tries todescribe the regularities of sound patterns.2. Give examples to illustrate the relationship between phonemes, phones and allophones.When we hear [pit],[tip],[spit],etc, the similar phones we have heard are /p/. And /p/ and /b/ are separate phonemes in English, while [ph] and [p] are allophones.3. How can we decide a minimal pair or a minimal set?A minimal pair should meet three conditions:1) The two forms are different in meaning.2) The two forms are different in one sound segment.3) The different sounds occur in the same position of the two strings.4. ★Use examples to explain the three types of distribution.1) Contrastive distribution. Sounds [m] in met and [n] in net are in contrastive distribution because substituting [m] for [n] will result in achange of meaning.2) Complementary distribution. The aspirated plosive [ph] and the unaspirated plosive [p] are in complementary distribution because theformer occurs either initially in a word or initially in a stressed syllable while the latter never occurs in such environments.3) Free variation. In English, the word “direct” may be pronounce in two ways: /di’rekt/ and /dia’rekt/, and the two different sounds /i/ and /ai/can be said to be in free variation.5. What’s the difference between segmental features and suprasegmental features? What are the suprasegmental features in English?I. 1) Distinctive features, which are used to distinguish one phoneme from another and thus have effect on one sound segment, are referred toas segmental features.2) The distinctive (phonological) features which apply to groups larger than the single segment are known as suprasegmental features.3) Suprasegmental features may have effect on more than one sound segment. They may apply to a string of several sounds.II.The main suprasegmental features include stress, tone, intonation and juncture.6. What’s the difference between tone languages and intonation language?Tone languages are those which use pitch to contrast meaning at word level while intonation languages are those which use pitch to distinguish meaning at phrase level or sentence level7. ★What’s the difference between phonetic transcriptions and phonemic transcriptions?The former was meant to symbolize all possible speech sounds, including even the most minute shades of pronunciation, while the latter was intended to indicate only those sounds capable of distinguishing one word from another in a given language.1. morphemes (语素) Morphemes are the minimal meaningful units in the grammatical system of a language.allomorphs (语素变体) Allomorphs are the realizations of a particular morpheme.morphs (形素) Morphs are the realizations of morphemes in general and are the actual forms used to realize morphemes.2. roots (词根) Roots is defined as the most important part of a word that carries the principal meaning.affixes (词缀) Affixes are morphemes that lexically depend on roots and do not convey the fundamental meaning of words.free morphemes (自由语素) Free morphemes are those which can exist as individual words.bound morphemes (粘着语素) Bound morphemes are those which cannot occur on their own as separate words.3. inflectional affixes (屈折词缀) refer to affixes that serve to indicate grammatical relations, but do not change its part of speech.derivational affixes (派生词缀) refer to affixes that are added to words in order to change its grammatical category or its meaning.4. empty morph (空语子) Empty morph means a morph which has form but no meaning.zero morph (零语子) Zero morph refers to a morph which has meaning but no form.5. IC Analysis (直接成分分析) IC analysis is the analysis to analyze a linguistic expression (both a word and a sentence) into a hierarchicallydefined series of constituents.6. immediate constituents(直接成分) A immediate constituent is any one of the largest grammatical units that constitute a construction.Immediate constituents are often further reducible.ultimate constituents (最后成分) Ultimate constituents are those grammatically irreducible units that constitute constructions.7. morphological rules (形态学规则) The principles that determine how morphemes are combined into new words are said to be morphologicalrules.8. word-formation process (构词法) Word-formation process mean the rule-governed processes of forming new words on the basis of alreadyexisting linguistic resources.1. ★What is IC Analysis?IC analysis is the analysis to analyze a linguistic expression (both a word and a sentence) into a hierarchically defined series of constituents.2. How are morphemes classified?1) Semantically speaking, morphemes are grouped into two categories: root morphemes and affixational morphemes.2) Structurally speaking, they are divided into two types: free morphemes and bound morphemes.3. ★Explain the interrelations between semantic and structural classifications of morphemes.a) All free morphemes are roots but not all roots are free morphemes.b) All affixes are bound morphemes, but not all bound morphemes are affixes.4. What’s the difference between an empty morph and a zero mor ph?a) Empty morph means a morph that has form but no meaning.b) Zero morph refers to a morph that has meaning but no form.5. Explain the differences between inflectional and derivational affixes in term of both function and position.a) Functionally:i.Inflectional affixes sever to mark grammatical relations and never create new words while derivational affixes can create new words.ii.Inflectional affixes do not cause a change in grammatical class while derivational affixes very often but not always cause a change in grammatical class.b) In term of position:i.Inflectional affixes are suffixes while derivational affixes can be suffixes or prefixes.ii.Inflectional affixes are always after derivational affixes if both are present. And derivational affixes are always before inflectional suffixes if both are present.6. What are morphological rules? Give at least four rules with examples.The principles that determine how morphemes are combined into new words are said to be morphological rules.For example:a) un- + adj. ->adj.b) Adj./n. + -ify ->v.c) V. + -able -> adj.d) Adj. + -ly -> adv.1. syntagmatic relations (横组关系) refer to the relationships between constituents in a construction.paradigmatic relations (纵聚合关系) refer to the relations between the linguistic elements within a sentence and those outside the sentence.hierarchical relations (等级关系) refer to relationships between any classification of linguistic units which recognizes a series of successively subordinate levels.2. IC Analysis (直接成分分析) is a kind of grammatical analysis, which make major divisions at any level within a syntactic construction.labeled IC Analysis(标记法直接成分分析) is a kind of grammatical analysis, which make major divisions at any level within a syntactic construction and label each constituent.phrase markers (短语标记法) is a kind of grammatical analysis, which make major divisions at any level within a syntactic construction, and label each constituent while remove all the linguistic forms.labeled bracketing (方括号标记法) is a kind of grammatical analysis, which is applied in representing the hierarchical structure of sentences by using brackets.3. constituency (成分关系)dependency (依存关系)4. surface structures (表层结构)refers to the mental representation of a linguistic expression, derived from deep structure by transformationalrules.deep structures (深层结构) deep structure of a linguistic expression is a theoretical construct that seeks to unify several related structures. 5. phrase structure rules (短语结构规则)are a way to describe a given language's syntax. They are used to break a natural language sentencedown into its constituent parts.6. transformational rules (转换规则)7. structural ambiguity (结构歧义)1. What are the differences between surface structure and deep structure?They are different from each other in four aspects:1) Surface structures correspond directly to the linear arrangements of sentences while deep structures correspond to the meaningful groupingof sentences.2) Surface structures are more concrete while deep structures are more abstract.3) Surface structures give the forms of sentences whereas deep structures give the meanings of sentences.4) Surface structures are pronounceable but deep structures are not.2. Illustrate the differences between PS rules and T-rules.1) PS rules frequently applied in generating deep structures.2) T-rules are used to transform deep structure into surface structures.3. What’s the order of generating sentences? Do we st art with surface structures or with deep structures? How differently are theygenerated?To generate a sentence, we always start with its deep structure, and then transform it into its corresponding surface structure.Deep structures are generated by phrase structure rules (PS rules) while surface structures are derived from their deep structures by transformational rules (T-rules).4. What’s the difference between a compulsory constituent and an optional one?Optional constituents may be present or absent while compulsory constituents must be present.5. What are the three syntactic relations? Illustrate them with examples.1) Syntagmatic relations2) Paradigmatic relations.3) Hierarchical relations.1. Lexical semantics (词汇语义学) is defined as the study of word meaning in language.2. Sense (意义) refers to the inherent meaning of the linguistic form.3. Reference (所指) means what a linguistic form refers to in the real world.4. Concept (概念) is the result of human cognition, reflecting the objective world in the human mind.5. Denotation (外延) is defined as the constant ,abstract, and basic meaning of a linguistic expression independent of context and situation.6. Connotation (内涵) refers to the emotional associations which are suggested by, or are part of the meaning of, a linguistic unit.7. Componential analysis (成分分析法) is the way to decompose the meaning of a word into its components.8. Semantic field (语义场) The vocabulary of a language is not simply a listing of independent items, but is organized into areas, within whichwords interrelate and define each other in various ways. The areas are semantic fields.9. Hyponymy (上下义关系) refers to the sense relation between a more general, more inclusive word and a more specific word.10. Synonymy (同义关系) refers to the sameness or close similarity of meaning.11. Antonymy (反义关系) refers to the oppositeness of meaning.12. Lexical ambiguity (词汇歧义)13. Polysemy (多义性) refers to the fact that the same one word may have more than one meaning.14. Homonymy (同音(同形)异义关系) refers to the phenomenon that words having different meanings have the same form.15. Sentence semantics (句子语义学) refers to the study of sentence meaning in language.1. What’s the criterion of John Lyons in classifying semantics into its sub-branches? And how does he classify semantics?In terms of whether it falls within the scope of linguistics, John Lyons distinguishes between linguistic semantics and non-linguistic semantics.According John Lyons, semantics is one of the sub-branches of linguistics; it is generally defined as the study of meaning.2. What are the essential factors for determining sentence meaning?1) Object, 2) concept, 3) symbol, 4) user, 5) context.3. What is the difference between the theory of componential analysis and the theory of semantic theory in defining meaning of words?4. What are the sense relations between sentences?1) S1 is synonymous with S2.2) S1 entails S2.3) S1 contradicts S2.4) S1 presupposes S2.5) S1 is a tautology, and therefore invariably true.6) S1 is a contradiction, and therefore invariably false.7) S1 is semantically anomalous.1. Speech act theory (言语行为理论)2. Cooperative principle and its maxims (合作原则及其准则)3. Politeness principle and its maxims (礼貌原则及其准则)4. Conversational implicature (会话含义)5. Indirect speech act (间接言语行为)6. Pragmatic presupposition (语用学预设)7. Relevance theory (关联理论)8. Illocutionary act (言外行为)9. (Horn’s) Q-Principle and R-Principle10. Perfrmative verbs (施为句动词)1. Make comments on the different definitions of pragmatics.2. What are the main types of deixis?3. Explain the statement: context is so indispen sable in fully understanding interpreting the speaker’s meaning.4. How are Austin’s and Searle’s speech act theories related to each other?5. What’s the relationship between CP and PP?6. What do you know about presupposition triggers in English? Explain them briefly with examples.7. What is ostensive-referential communication?8. Explain the obvious presupposition of speaker who say each of the following:1) When did you stop beating your wife?2) Where did Tom buy the watch?3) Your car is broken.9. What do you think of the fol lowing statement? “Tom participated in spreading rumors” entails “Tom engaged in spreading rumors”.Chapter 9 话语分析1. text(语篇) = discourse 语篇是指实际使用的语言单位，是一次交际过程中的一系列连续的话段或句子所构成的语言整体。

A combinatorial interpretation for a super-Catalan recurrence

Journal of Integer Sequences, Vol. 8 (2005),Article 05.1.8A Combinatorial Interpretation for aSuper-Catalan RecurrenceDavid CallanDepartment of StatisticsUniversity of Wisconsin-MadisonMedical Science Center1300University Ave.Madison,WI53706-1532USAcallan@AbstractNicholas Pippenger and Kristin Schleich have recently given a combinatorial inter-pretation for the second-order super-Catalan numbers(u n)n≥0=(3,2,3,6,14,36,...):they count“aligned cubic trees”on n interior vertices.Here we give a combinatorial interpretation of the recurrence u n= n/2−1k=0 n−22k 2n−2−2k u k:it counts these treesby number of deep interior vertices where“deep interior”means“neither a leaf noradjacent to a leaf”.1IntroductionForﬁxed integer m≥1,the numbersu n= 2m m 2n n2 m+n msatisfy the recurrence relationu n= k≥02n−m−2k n−m2k u k(1)and hence are integers except when m=n=0[1].We will call them super-Catalan numbers of order m(although other numbers go by this name too).For m=0and n≥1,they are theodd central binomial coeﬃcients(1,3,10,35,...)(Sloane’s and count lattice pathsof n upsteps and n−1downsteps.For m=1and n≥0,familiar Catalan num-bers(1,1,2,5,14,42,...)(Sloane’s with numerous combinatorial interpretations[2,Ex6.19].Three recent papers give combinatorial interpretations for m=2,in terms of(i)pairs of Dyck paths[3],(ii)“blossom trees”[4],and cubic trees”[5].In this case the sequence is(3,2,3,6,14,36,...)and is Sloane’s The object of this note is toestablish a combinatorial interpretation of the1)for m=2:it counts the just mentioned aligned cubic trees by number of vertices that are neither a leaf nor adjacent to a leaf.For m=1,(1)is known as Touchard’s identity,and in Section2we recall combinatorial interpretations of the recurrence for the cases m=0,1.In Section3we deﬁne aligned cubic trees and establish notation.In Section4we introduce conﬁgurations counted by the right side of(1)for m=2,and in Section5we exhibit a bijection from them to size-n aligned cubic trees.2Recurrence for m=0,1For m=0,(1)counts lattice paths of n upsteps(U)and n−1downsteps(D)by number k of DUU s where,for example,DDUUUDDUU has2DUU s.It also counts paths of n U s and n D s that start up by number2k(necessarily even)of inclined twinsteps(UU or DD) at odd locations.For example,U1U2U3D4D5D has four inclined twinsteps,at locations 1,2,4and5,but only theﬁrst and last are at odd locations.For m=1,(1)counts Dyck n-paths(paths of n U s and n D s that never dip below ground level)by number k of DUU s.It also counts them by number2k of inclined twinsteps at even locations.See[6],for example,for relevant bijections.Recall the standard“walk-around”bijection from full binary trees on2n edges to Dyck n-paths:a worm crawls counterclockwise around the tree starting just west of the root and when an edge is traversed for theﬁrst time,records an upstep if the edge is west-leaning and a downstep if it is east-leaning.This bijection carries deep interior vertices to DUU s where deep interior means“neither a leaf nor adjacent to a leaf”.Hence the recurrence also counts full binary trees on2n edges by number of deep interior vertices.The interpretation for m=2below is analogous to this one.3Aligned cubic treesIt is well known that there are C n(Catalan number)full binary trees on2n edges.Considered as a graph,the root is the only vertex of degree2when n≥1.To remedy this,add a vertical planting edge to the root and transfer the root to the new vertex.Now every vertex has degree1or3and,throughout this paper,we will refer to a vertex of degree3as a node and of degree1as a leaf.Thus our planted tree of2n+1edges has n nodes.Leave the root edge pointing South and align the other edges so that all three angles at each node are120◦,lengthening edges as needed to avoid self intersections.Rotate these objects through multiples of60◦to get all“rooted aligned cubic”trees on n nodes(6C n of them,since the edge from the root is no longer restricted to point South but may point in any of6 directions).Now erase the root on each to get all(unrooted)aligned cubic trees on n nodes(6 n+2C n of them since for each,a root could be placed on any of its n+2leaves).Thus twodrawings of an aligned cubic tree are equivalent if they diﬀer only by translation and length of edges.This is interpretation(iii)of the second order super-Catalan numbers mentioned in the Introduction.For short,we will refer to an aligned cubic tree on n nodes simply as an n-ctree(c for cubic).More concretely,a rooted n-ctree can be coded as a pair(r,u)with r an integer mod 6and u a nonnegative integer sequence of length n+2.The integer r gives the angle(in multiples of60◦)from the direction South counterclockwise to the direction of the edge fromthe root.The sequence u=(u i)n+2i=1measures distance between successive leaves:traversethe tree in preorder(a worm crawls counterclockwise around the tree starting at a point just right of the root when looking from the root along the root edge).Then u i=v i−2 where v i(≥2)is the number of edges traversed between the i th leaf and the next one.For example,the sketched3-ctree when rooted at A is coded by(2,(3,0,1,1,1,0))and when rooted at B is coded by(1,(0,1,1,1,0,3)).BAIn general,if a ctree rooted at a given leaf is coded by(r,(u1,u2,...,u n+1,u n+2))then,when rooted at the next leaf in preorder,it is coded by((2+r−u1)mod6,(u2,u3,...,u n+2,u1)). Repeating this n+2times all told rotates u back to itself and gives“r”=2n+4+r− n+2i=1u i mod6.Since n+2i=1u i is necessarily=2n−2,we are,as expected,back to the original coding sequence.An ordinary(planted)full binary tree is coded by(0,u)and so there are C n coding sequences of length n+2.They can be generated as follows.A ctree can be built up by successively adding two edges to a leaf to turn it into a node.The eﬀect this has on the coding sequence is to take two consecutive entries u i,u i+1(subscripts modulo n+2)and replace them by the three entries u i+1,0,u i+1+1.The1-ctree has coding sequence(0,0,0).The2-ctree coding sequences are(1,0,1,0)and(0,1,0,1),and so on.Reversing this procedure gives a fast computational method to check if a given u is a coding sequence or not.For example, successively pruning theﬁrst0,11210230→1120130→111030→11020→1010→000isindeed a coding sequence.However,we will work with the graphical depiction of a ctree.The n-ctrees for n=0,1,2are shown below.Note that since edges have aﬁxed non-horizontal direction,we can distinguish a top and bottom vertex for each edge.n=0:n=1:n=2:It is convenient to introduce some further terminology.Recall a node is a vertex of degree3.A node is hidden,exposed,naked or stark naked according as its3neighbors include0,1,2 or3leaves.Thus a deep interior vertex is just a hidden node.A0-ctree has no nodes.Only a1-ctree has a stark naked node,and hidden nodes don’t occur until n≥4.For n≥2, an n-ctree containing k hidden nodes has k+2naked nodes and hence n−2k−2exposed nodes.The terms right and left can be ambiguous:we always use right and left relative to travel from a speciﬁed vertex or edge.Thus vertexB below is left(not right!)travelling from vertex A.ABEach n-ctree has a unique center,either an edge or a node,deﬁned as follows.For n=0,it is the(unique)edge in the ctree.For n=1,it is the(unique)node in the ctree.For n≥2, delete the leaves(and incident edges)adjacent to each naked node,thereby reducing the number of nodes by at least2.Repeat until the n=0or1deﬁnition applies.Equivalently, deﬁne the depth of a node in a ctree to be the length(number of edges)in the shortest path from the node to a naked node.Then there are either one or two nodes of maximal depth; if one,it is the center and if two,they are adjacent and the edge joining them is the center.4(n,k)-ConﬁgurationsThere are n−22k 2n−2−2k u k conﬁgurations formed in the following way.Start with a k-ctree——into2k+1(possibly empty) u k choices.Break a strip of n−2−2k squares—2222 (22)n−2−2k substrips,one for each of the2k+1edges in the k-ctree— (n−2−2k)+(2k+1)−1n−2−2k = n−22k choices. Mark each square L(=left)or R(=right)—2n−2−2k choices.Actually,this is not quite what we want.Perform one little tweaking:if there is a center edge and it has an odd-length strip of squares,mark theﬁrst square T(=top)or B(= bottom)instead of L or R.So a conﬁguration might look as follows(n=12,k=2,empty strips not shown).5BijectionHere is a bijection from(n,k)-conﬁgurations to n-ctrees with k hidden nodes.The bijection produces the correspondences in the following table.(n,k)-conﬁguration n-ctreeleaf naked nodenode hidden nodesquare exposed nodeRoughly speaking,work outward from the center,turning a strip of j labeled squares on an edge AB into j exposed nodes lying between A and B.First,for the center edge(if there is one),the procedure depends on whether it has an even or odd number of squares.Case Even Here,the center edge becomes an edge joining two exposed nodes as shown: the labels again indicate the L/R status(travelling from the center edge)of the leaves associated with the exposed nodes.The labels are applied from the bottom vertex subtree (H2)to the top one(H1).H2−→H2H1LLLRCase Odd Here,the center edge becomes a leaf edge.Theﬁrst square indicates whether the top or bottom vertex becomes a leaf.Construct equal numbers of exposed nodes on each side of the non-leaf(here,top)vertex,using the L/R designations to determine the leaves(L/R relative to travel from the leaf and running,say,from the left branchto the right).HH2−→leafLLH2H1Decide the placement of the two subtrees H1,H2,say the H originally sitting at the vertex which is now a leaf goes at the end of the left branch from the leaf.Next,for a non-center edge with i≥0squares,identify its endpoint closest to the center, let H0,H1,H2be the subtrees as illustrated(H0containing the center),and insert i exposed nodes as shown.The labels L,L,R apply in order from H1-H2to H0and indicate the L/R status(travelling from the center)of the leaves associated with the exposed nodes.H210(center)−→H1H2H0LRLFinally,turn the original leaves into naked nodes by adding two edges apiece.We leave to the reader to verify that the resulting ctree has n nodes of which k are hidden, and that the original conﬁguration can be uniquely recovered from this n-ctree by reversing the above procedure,working in from the leaves.References[1]Ira Gessel,Super ballot numbers,J.Symbolic Computation14(1992),179–194.[2]Richard P.Stanley,Enumerative Combinatorics Vol.2,Cambridge University Press,1999.Exercise6.19and related material on Catalan numbers are available online at /∼rstan/ec/.[3]Ira M.Gessel and Guoce Xin,A combinatorial interpretation of the numbers6(2n)!/(n!(n+2)!),math.CO/0401300,2004,11pp.[4]Gilles Schaeﬀer,A combinatorial interpretation of super-Catalan numbers of order two,Manuscript,2003,4pp.[5]Nicholas Pippenger and Kristin Schleich,Topological Characteristics of Random Tri-angulated Surfaces,Random Structures&Algorithms,to appear,arXiv:gr-qc/0306049, 2003,58pp.[6]David Callan,Two bijections for Dyck path parameters,math.CO/0406381,2004,4pp. 2000Mathematics Subject Classiﬁcation:Primary05A19;Secondary05A15.Keywords:super-Catalan,aligned cubic tree.(Concerned with sequences andReceived February12005;revised version received March22005.Published in Journal of Integer Sequences,March22005.Return to Journal of Integer Sequences home page.。

Chapter4.ThepsychologyofSecondLanguage

Chapter4.ThepsychologyofSecondLanguageChapter 4. The psychology of Second Language Acquisition1)Languages and the brainBroca’s area – to be responsible for the ability to speak.Wernicke’s area - located in the left temporal lobe, result in excessivespeech and loss of language comprehension.Brain lateralization – specialization of the two halves of the brain. Methods for gathering data:Correlations of location of brain damage with patterns of loss/recoveryin cases where languages are affected differentially.Presentation of stimuli from different languages to the right versus theleft visual or auditory fields.Mapping the brain surface during surgery by using electrical stimulationat precise points and recording.Positron Emission Tomography and other non-invasive imagingtechniques.Some researches and questions in this area:How independent are the languages of multilingual speakers?How are multiple language structures organized in relation to oneanother in the brain? Are both languages stored in the same areas?Does the organization of the brain for L2 in relation to L1 differ withage of acquisition, how it is learned, or level of proficiency?Do two or more languages show the same sort of loss or disruption afterbrain damage? When there is differential impairment or recovery, whic hlanguage recovers first?2)Learning processesPsychology provides us with two major frameworks for the focus onlearning processes: Information Processing and Connectionism.Information Processing (IP)(1)Perception and the input of new information.(2)The formation, organization, and regulation of internal representations.(3)Retrieval and output strategies.Assumptions:Second language learning is the acquisition of a complex cognitive skill.Complex skills can be reduced to sets of simpler component skills.Le arning of a skill initially demands learners’ attention.Controlled processing requires considerable mental “space”.Humans are limited-capacity processors.Learners go from controlled to automatic processing with practice.Learning essentially involves development from controlled to automaticprocessing of component skills.Along with development from controlled to automatic processing.Reorganizing mental representations as part of learning makes structuresmore coordinated, integrated, and efficient, including a faster responsetime when they are activated.In SLA, restructuring of internal L2 representations, along with largerstores in memory, accounts for increasing levels of L2 proficiency.Theories regarding order of acquisitionLearners acquire certain grammatical structures in a developmentalsequence.Developmental sequences reflect how learners overcome processinglimitations.Language instruction which targets developmental features will besuccessful only if learners have already mastered the p rocessingoperations which are associated with the previous stage of acquisition.Competition M odelCoined by Bates and MacWhinney: this is a functional approach which assumes that all linguistic performance involves mapping between external form and internal function.Form-function mapping is basic for L1 acquisitionTask frequency.Contrastive availability.Conflict reliability.Connectionist approachesFocus on the increasing strength of associations between stimuli and responses rather than on t he inferred abstraction of “rules” or on restructuring.The best-known connectionist approach in SLA is Parallel Distributed Processing (PDP)Attention is not viewed as a central mechanism.Information processing is not serial in nature.Knowledge is not stored in memory or retrieved as patterns.3)Differences in learnersAge factorAge differences in SLAYounger advantage:Brain plasticity.Not analytical.Fewer inhibitions.Weaker group identity.Simplified input more likely.Older advantage:Learning capacity.Analytic ability.Pragmatic skills.Greater knowledge of L1.Real-world knowledge.Sex factor:Females tend to be better L2 learners than males.Aptitude factor:Phonemic coding ability.Inductive language learning ability and grammatical sensitivity.Associative memory capacity.M otivation factor:Significant goal or need.Desire to attain the goal.Perception that learning L2 is relevant to fulfilling the goal or meetingthe need.Belief in the likely success or failure of learning L2.Value of potential outcomes/rewards.Cognitive style factor:Field-dependent - Field-independentGlobal - ParticularHolistic - AnalyticDeductive - InductiveFocus on meaning – Focus on formPersonality factor:(1)Anxious – Self – confidence(2)Risk avoiding – Risk trying(3)Shy – Advertisement(4)Introverted – Extroverted(5)Inner directed – Other directed(6)Reflective – Impulsive(7)Imaginative – Uninquisitive(8)Creative – Uncreative(9)Empathetic – Insensitive to others(10)Referent of ambiguity – Closure orientedLearning strategies:Differential L2 outcomes may also be affected by individualslearning strategies.(1)Meta cognitive.(2)Cognitive.(3)Social/affective.The major traits good learners have:Concern for language formConcern for communicationActive task approachAwareness of the learning process.Capacity to use strategies flexibly in accordance with task requirements.4)The effects of multilingualism(1)Bilingual children show consistent advantages in tasks of both verbaland nonverbal abilities.(2)Bilingual children show advanced meta-linguistic abilities.(3)Cognitive and meta-linguistic advantages appearing bilingual situations.(4)The cognitive effects of bilingualism appear relatively early in theprocess of becoming bilingual and do not require high levels ofbilingual proficiency nor the achievement of balanced bilingualism.。

The Theory of Conversational Implicature会话含义理论

Be perspicuous (明白易懂的). 1. Avoid obscurity of expression; 2. Avoid ambiguity; 3. Be brief (avoid prolixity); 4. Be orderly. e.g: A: Let's get the kids something. B: Ice cream would be nice. (√) B: Okey, but I veto I-C-E C-R-E-A-M-S. (×)
3) relation maxim 关联原则
Be relevant. e.g: A: I am out of petrol. B: There is a garage at the corner. (√) B: It is sunny today. (×)
4) manxim 数量准则
1. Make your conteibution as informative as required (for the current purpuses of the exchange); 2. Do not make your contribution more informative than is required. e.g: A: What date is your birthday? B: December 26. (√) B: December 26, the day after the Christmas Day. (×)
The Cooperative Principle (CP) 合作原则
to explain the mechanisms( 机制，原理 ) by which people interpret conversational implicature or to explain why people often mean more than what they say, Grice proposed the Cooperative Principle and introduced 4 conversational maxims. And the violations in the second half of Grice's “Logics and Conversation”.

书上的名词解释英文

书上的名词解释英文在平时的学习和阅读中，我们经常会遇到一些专业术语，这些名词往往以英文形式出现在书籍中。

如何正确地理解和使用这些名词是我们提高专业素养的重要一环。

本文将从几个不同领域的名词解释入手，通过分析和举例，帮助读者更好地掌握这些英文名词的含义。

一、心理学术语1. Perception （知觉）：Perception refers to the process by which we interpret and make sense of sensory information from our surroundings. It involves the brain's interpretation of visual, auditory, tactile, and other sensory inputs. For example, when we see an object, our brain processes the visual information and interprets it as a particular shape, color, and size.2. Conditioning （条件反射）：Conditioning is a learning process that involves associating a stimulus with a specific response. It can be divided into two types: classical conditioning and operant conditioning. Classical conditioning refers to the process of learning by association, while operant conditioning involves learning through consequences, such as rewards and punishments. For example, Pavlov's famous experiment with dogs demonstrated classical conditioning, where the dogs associated the sound of a bell with food and began to salivate at the sound of the bell alone.3. Cognitive dissonance （认知失调）：Cognitive dissonance refers to the mental discomfort or tension that occurs when an individual holds two conflicting beliefs or attitudes. This theory suggests that people have a natural tendency to strive for consistency and therefore, when faced with conflicting thoughts, they will experience cognitive dissonance and seek to reduce it. For example, if someone believes that smoking is harmful to health but continues to smoke, they may experience cognitive dissonance.二、经济学术语1. Inflation （通货膨胀）：Inflation is the rate at which the general level of prices for goods and services is rising, reducing the purchasing power of currency. This means that the same amount of money can buy fewer goods and services over time. Inflation is typically measured by the Consumer Price Index (CPI) or the Producer Price Index (PPI).2. Supply and demand （供求关系）：Supply and demand is a fundamental economic concept that describes the relationship between the availability of a particular good or service and the desire or demand for it. When the supply of a product is low and the demand is high, the price tends to increase. Conversely, when the supply is high and the demand is low, the price tends to decrease.3. Gross Domestic Product (GDP) （国内生产总值）：Gross Domestic Product isa measure of a country's economic activity, representing the total value of all goods and services produced within its borders in a specific time period, usually a year. It is often used to compare the economic performance of different countries.三、生物学术语1. DNA (Deoxyribonucleic acid) （脱氧核糖核酸）：DNA is a molecule that carries the genetic instructions for the development and functioning of all living organisms. It consists of two strands arranged in a double helix structure, with each strand composed of nucleotides that contain specific genetic information. DNA is essential for the replication and inheritance of genetic traits.2. Mitosis （有丝分裂）：Mitosis is a process in which a cell divides to produce two genetically identical daughter cells. It is a crucial part of cell growth, development, and repair in multicellular organisms. During mitosis, the cell undergoes a series of stages, including prophase, metaphase, anaphase, and telophase, which enable the equal distribution of genetic material between the daughter cells.3. Ecosystem （生态系统）：An ecosystem refers to a biological community of interacting organisms and their physical environment. It includes all living organisms (biotic factors) and non-living components (abiotic factors) within a specific area, such asplants, animals, soil, water, and climate. Ecosystems play a vital role in maintaining ecological balance and providing essential resources for life.通过上述对心理学、经济学和生物学等领域的名词进行解释，我们可以更好地理解这些英文名词所代表的概念和概念背后的意义。