Gauge Invariance and the Pauli-Villars Regulator in Lorentz- and CPT-Violating Electrodynam

伽利略与柏拉图m柯瓦雷（Alexandre Koyré）伽利莱·伽利略的名字与十六世纪的科学革命紧密相连。

自希腊人发明宇宙秩序以来，这场革命在人类思想上即使算不上最为深刻的，也是最深刻的革命之一：它隐含着一项根本的理智“转变”， 作为这一变化的表征与成就， 现代物理学应运而生1。

这场革命有时被概括和解释为一种精神上的巨变，人类心灵整个基本态度的彻底转变：主动的生活(vita active)取代那时仍被认为是最高贵的生活，沉思的生活(Vita contemplativa,Θεωρια)。

现代人力图驾驭自然，而古代和中世纪人把对它的沉思放在首位。

经典物理学――伽利略、笛卡尔和霍布斯的物理学，主动的、操作的科学（scientia activa, operativa）――让人成为自然的主人和教师；它的力学倾向因此被解释为支配、行动的欲望，它本身被视为纯粹是这一态度的结果，即把匠人（homo faber）的想法应用于自然2。

笛卡尔的科学――更不用说伽利略的了――只不过是手艺人或工程师的科学3。

允我坦言，我认为这个解释并不完全正确。

不用说，现代哲学，还有现代伦理学和现代宗教，比起古代和中世纪思想来更强调行动（πραξιζ）。

现代科学也一样：我想的是笛卡尔物理学及其滑轮、绳子和杠杆之类的东西。

这种态度在培根那里比伽利略和笛卡尔身上表现得更为强烈，附带说一句，在科学史上培根与伽利略和笛卡尔不在同一量级4。

他们的科学不是由工匠和技师构造的，造就它的人所作的实事也无非是构建理论5。

新弹道学不是技工和炮手而是他们的反对者所为。

伽利略并没有从威尼斯军械厂和码头上劳作的人们那里学到过什么。

恰恰相反：是伽利略教他们那行6。

再者，这个解释失之过宽，不能m郝刘祥译自Journal of the History of Ideas 4 (1943), pp.333-48。

本文引文因其出处全部是拉丁文本，故全部略去；译文中只保留作者的注释及二手文献。

a r X i v :h e p -l a t /9709149v 1 29 S e p 19971RU-97-78KUNS-1465HE(TH)97/14Gauge Freedom in Chiral Gauge Theory with Vacuum Overlap ∗Yoshio Kikukawa a †aDepartment of Physics and Astronomy,Rutgers University,Piscataway,NJ 08855-0849,USADynamical nature of the gauge degrees of freedom and its effect to fermion spectrum are studied at β=∞for two-and four-dimensional nonabelian chiral gauge theories in the vacuum overlap formalism.It is argued that the disordered gauge degrees of freedom does not contradict to the chiral spectrum of lattice fermion.1.Introduction —Pure gauge limit In the vacuum overlap formalism[1]of a generic chiral gauge theory,gauge symmetry is explic-itly broken by the complex phase of fermion de-terminant.In order to restore the gauge invari-ance,gauge average —the integration along gauge orbit—is invoked.Then,what is required for the dynamical nature of the gauge freedom at β=∞(pure gauge limit)is that the global gauge symmetry is not broken spontaneously and the bosonic fields of the gauge freedom have large mass compared to a typical mass scale of the the-ory so as to decouple from physical spectrum[2,1].However,through the analysis of the waveg-uide model[3,4],it has been claimed that this required disordered nature of the gauge freedom causes the vector-like spectrum of fermion[5].In this argument,the fermion correlation functions at the waveguide boundaries were examined.One may think of the counter parts of these correla-tion functions in the overlap formalism by putting creation and annihilation operators in the overlap of vacua with the same signature of mass.Let us refer this kind of correlation function as bound-ary correlation function and the correlation func-tion in the original definition as overlap correla-tion function .We should note that the boundary correlation functions are no more the observables in the sense defined in the overlap formalism;they cannot be expressed by the overlap of two vacua with their phases fixed by the Wigner-Brillouin| +|v + |v +|v −v −|−2Then the model describes the gauge degrees of freedom coupled to fermion through the gauge non-invariant piece of the complex phase of chiral determinants.Z= [dg] rep. +|ˆG|+ | −|ˆG†|− | ≡ dµ[g].(3)ˆG is the operator of the gauge transformation given by:ˆG=exp ˆa†i n{log g}i jˆa nj .(4) In this limit,one of the possible definitions of the boundary correlation functions is given as fol-lows for the case of the negative mass and in the representation r:φniφ†m j −r≡12δnmδj i |− rλ2 .(8)In four-dimensions,we consider the covariantgaugefixing term and the Faddeev-Popov deter-minant[7],following Hata[9](cf.[10–12]):−12i U nµ−U†nµ −12i U nµ−U†nµ ,with the pure gauge link variable Eq.(2).ˆM ab nm isthe lattice Faddeev-Popov operator.We can show by the perturbation theory inλ(backgroundfield method)that both puregauge models share the novel features of the two-dimensional nonlinear sigma model even in thepresense of the imaginary action.The model isrenormalizable(at one-loop in four-dimensions)andλis asymptotically free.Severe infrared di-vergence occurs and it prevents local order pa-rameters from emerging.Based on these dynamical features,we assumethat the global gauge symmetry does not breakspontaneously for the entire region ofλand thegauge freedom acquires mass M g dynamicallythrough the dimensional transmutation.Withthis assumption,the decoupling of the gauge free-dom could occur as M gր133.Boundary correlation functionThe asymptotic freedom allows us to tamethe gaugefluctuation by approaching the criticalpoint of the gauge freedom.There we can invokethe spin wave approximation for the calculationof the invariant boundary correlation functionφniφ†m i −r,(10)which is free from the IR divergence.At one-looporder,we obtain1(2π)D e ip(n−m)1µsin2pµ+(B(p)−m0)2,(14) andπnπm ′= d D p µ4sin2pµ2(1−m0),(m0=0.5).(16)We also see that the quantum correction due to the gaugefluctuation at one-loop does not affect the leading short-distance nature.There is no symmetry against the spectrum mass gap.There-fore it seems quite reasonable to assume that it holds true asλbecomes large.Since the overlap correlation function does not depend on the gauge freedom and does show the chiral spectrum[1],the above fact means that the entire fermion spec-trum is chiral.The author would like to thank H.Neuberger and R.Narayanan for enlightening discussions. He also would like to thank H.Hata,S.Aoki, H.So and A.Yamada for discussions. REFERENCES1.R.Narayanan and H.Neuberger,Nucl.Phys.B412(1994)574;Phys.Rev.Lett.71(1993) 3251;Nucl.Phys.B(Proc.Suppl.)34(1994) 95,587;Nucl.Phys.B443(1995)305.2. D.Foerster,H.B.Nielsen and M.Ninomiya,Phys.Lett.B94(1980)135;S.Aoki,Phys.Rev.Lett.60(1988)2109;Phys.Rev.D38 (1988)618;K.Funakubo and T.Kashiwa, 60(1988)2113;J.Smit,Nucl.Phys.B(Proc.Suppl.)4(1988)451.3. D.B.Kaplan,Nucl.Phys.B30(Proc.Suppl.)(1993)597.4.M.Golterman,K.Jansen,D.Petcher and J.Vink,Phys.Rev.D49(1994)1606;M.F.L.Golterman and Y.Shamir,Phys.Rev.D51 (1995)3026.5.M.F.L.Golterman and Y.Shamir,Phys.Lett.B353(1995)84;Erratum-ibid.B359(1995) 422.R.Narayanan and H.Neuberger,Phys.Lett.B358(1995)303.6.Y.Kikukawa,KUNS-1445HE(TH)97/08,May1997,hep-lat/9705024.7.Y.Kikukawa,KUNS-1446HE(TH)97/09,July1997,hep-lat/9707010.8.Y.Kikukawa and S.Miyazaki,Prog.Theor.Phys.96(1996)1189.9.H.Hata,Phys.Lett.143B(1984)171.10.A.Borrelli,L.Maiani,G.C.Rossi,R.Sistoand M.Testa,Nucl.Phys.B333(1990)335.11.Y.Shamir,TAUP-2306-95,Dec1995;Y.Shamir,Nucl.Phys.B(Proc.Suppl.)53 (1997)664.12.M.F.L.Golterman and Y.Shamir,Phys.Lett.B399(1997)148.13.E.Witten,Commun.Math.Phys.92(1984)455.。

The Contemporary Theory of MetaphorGeorge Lakoff(c) Copyright George Lakoff, 1992To Appear in Ortony, Andrew (ed.) Metaphor and Thought (2nd edition), Cambridge University Press.Do not go gentle into that good night. -Dylan ThomasDeath is the mother of beauty . . . -Wallace Stevens, Sunday Morning IntroductionThese famous lines by Thomas and Stevens are examples of what classical theorists, at least since Aristotle, have referred to as metaphor: instances of novel poetic language in which words like mother,go, and night are not used in their normal everyday senses. In classical theories of language, metaphor was seen as a matter of language not thought. Metaphorical expressions were assumed to be mutually exclusive with the realm of ordinary everyday language: everyday language had no metaphor, and metaphor used mechanisms outside the realm of everyday conventional language. The classical theory was taken so much for granted over the centuries that many people didn’t realize that it was just a theory. The theory was not merely taken to be true, but came to be taken as definitional. The word metaphor was defined as a novel or poetic linguistic expression where one or more words for a concept are used outside of its normal conventional meaning to express a similar concept. But such issues are not matters for definitions; they are empirical questions. As a cognitive scientist and a linguist, one asks: What are the generalizations governing the linguistic expressions re ferred to classically as poetic metaphors? When this question is answered rigorously, the classical theory turns out to be false. The generalizations governing poetic metaphorical expressions are not in language, but in thought: They are general map pings across conceptual domains. Moreover, these general princi ples which take the form of conceptual mappings, apply not just to novel poetic expressions, but to much of ordinary everyday language. In short, the locus of metaphor is not in language at all, but in the way we conceptualize one mental domain in terms of another. The general theory of metaphor is given by characterizing such cross-domain mappings. And in the process, everyday abstract concepts like time, states, change, causation, and pur pose also turn out to be metaphorical. The result is that metaphor (that is, cross-domain mapping) is absolutely central to ordinary natural language semantics, and that the study of literary metaphor is an extension of the study of everyday metaphor. Everyday metaphor is characterized by a huge system of thousands of cross-domain mappings, and this system is made use of in novel metaphor. Because of these empirical results, the word metaphor has come to be used differently in contemporary metaphor research. The word metaphor has come to mean a cross-domain mapping in the conceptual system. The term metaphorical expression refers to a linguisticexpression (a word, phrase, or sentence) that is the surface realization of such a cross-domain mapping (this is what the word metaphor referred to in the old theory). I will adopt the contemporary usage throughout this chapter. Experimental results demonstrating the cognitive reali ty of the extensive system of metaphorical mappings are discussed by Gibbs (this volume). Mark Turner’s 1987 book, Death is the mother of beauty, whose title comes from Stevens’great line, demonstrates in detail how that line uses the ordinary system of everyday mappings. For further examples of how literary metaphor makes use of the ordinary metaphor system, see More Than Cool Reason: A Field Guide to Poetic Metaphor, by Lakoff and Turner (1989) and Reading Minds: The Study of English in the Age of Cognitive Science, by Turner (1991). Since the everyday metaphor system is central to the understanding of poetic metaphor, we will begin with the everyday system and then turn to poetic examples.Homage To ReddyThe contemporary theory that metaphor is primarily conceptual, conventional, and part of the ordinary system of thought and language can be traced to Michael Reddy’s (this volume) now classic paper, The Conduit Metaphor,which first appeared in the first edition of this collection. Reddy did far more in that paper than he modestly suggested. With a single, thoroughly analyzed example, he allowed us to see, albeit in a restricted domain, that ordinary everyday English is largely metaphorical, dispelling once and for all the traditional view that metaphor is primarily in the realm of poetic or figurative language. Reddy showed, for a single very significant case, that the locus of metaphor is thought, not language, that metaphor is a major and indispensable part of our ordinary, conventional way of conceptualizing the world, and that our everyday behavior reflects our metaphorical understanding of experience. Though other theorists had noticed some of these characteristics of metaphor, Reddy was the first to demonstrate it by rigorous linguistic analysis, stating generalizations over voluminous examples. Reddy’s chapter on how we conceptualize the concept of communication by metaphor gave us a tiny glimpse of an enormous system of conceptual metaphor. Since its appearance, an entire branch of linguis tics and cognitive science has developed to study systems of metaphorical thought that we use to reason, that we base our actions on, and that underlie a great deal of the structure of language. The bulk of the chapters in this book were written before the development of the contemporary field of metaphor research. My chapter will therefore contradict much that appears in the others, many of which make certain assumptions that were widely taken for granted in 1977. A major assumption that is challenged by contemporary research is the traditional division between literal and figurative language, with metaphor as a kind of figurative language. This entails, by definition, that: What is literal is not metaphorical. In fact, the word literal has traditionally been used with one or more of a set of assumptions that have since proved to be false:Traditional false assumptions•All everyday conventional language is literal, and none is metaphorical.•All subject matter can be comprehended literally, without metaphor.•Only literal language can be contingently true or false.•All definitions given in the lexicon of a language are literal, not metaphorical.•The concepts used in the grammar of a language are all literal; none are metaphorical.The big difference between the contemporary theory and views of metaphor prior to Reddy’s work lies in this set of assumptions. The reason for the difference is that, in the intervening years, a huge system of everyday, convention al, conceptual metaphors has been discovered. It is a system of metaphor that structures our everyday conceptual system, including most abstract concepts, and that lies behind much of everyday language. The discovery of this enormous metaphor system has destroyed the traditional literal-figurative distinction, since the term literal,as used in defining the traditional distinction, carries with it all those false assumptions. A major difference between the contemporary theory and the classical one is based on the old literal-figurative distinction. Given that distinction, one might think that one arrives at a metaphorical interpretation of a sentence by starting with the literal meaning and applying some algorithmic process to it (see Searle, this volume). Though there do exist cases where something like this happens, this is not in general how metaphor works, as we shall see shortly.What is not metaphoricalAlthough the old literal-metaphorical distinction was based on assumptions that have proved to be false, one can make a different sort of literal-metaphorical distinction: those concepts that are not comprehended via conceptual metaphor might be called literal. Thus, while I will argue that a great many common concepts like causation and purpose are metaphorical, there is nonetheless an extensive range of nonmetaphorical concepts. Thus, a sentence like The balloon went up is not metaphorical, nor is the old philosopher’s favorite The cat is on the mat.But as soon as one gets away from concrete physical experience and starting talking about abstractions or emotions, metaphorical understanding is the norm.The Contemporary Theory: Some ExamplesLet us now turn to some examples that are illustrative of contemporary metaphor research. They will mostly come from the domain of everyday conventional metaphor, since that has been the main focus of the research. I will turn to the discussion of poetic metaphor only after I have discussed the conventional system, since knowledge of the conventional system is needed to make sense of most of the poetic cases. The evidence for the existence of a system of conventional conceptual metaphors is of five types: -Generalizations governing polysemy, that is, the use of words with a number of related meanings.-Generalizations governing inference patterns, that is, cases where a pattern of inferences from one conceptual domain is used in another domain.-Generalizations governing novel metaphorical language (see, Lakoff & Turner, 1989).-Generalizations governing patterns of semantic change (see, Sweetser, 1990).-Psycholinguistic experiments (see, Gibbs, 1990, this volume).We will primarily be discussing the first three of these sources of evidence, since they are the most robust.Conceptual MetaphorImagine a love relationship described as follows: Our relationship has hit a dead-end street.Here love is being conceptualized as a journey, with the implication that the relationship is stalled, that the lovers cannot keep going the way they’ve been going, that they must turn back, or abandon the relationship altogether. This is not an isolated case. English has many everyday expressions that are based on a conceptualization of love as a journey, and they are used not just for talking about love, but for reasoning about it as well. Some are necessarily about love; others can be understood that way: Look how far we’ve come. It’s been a long, bumpy road. We can’t turn back now. We’re at a crossroads. We may have to go our separate ways. The relationship isn’t going anywhere. We’re spinning our wheels. Our relationship is off the track. The marriage is on the rocks. We may have to bail out of this relationship. These are ordinary, everyday English expressions. They are not poetic, nor are they necessarily used for special rhetorical effect. Those like Look how far we’ve come, which aren’t necessarily about love, can readily be understood as being about love. As a linguist and a cognitive scientist, I ask two commonplace questions:•Is there a general principle governing how these linguistic expressions about journeys are used to characterize love?•Is there a general principle governing how our patterns of inference about journeys are used to reason about love when expressions such as these are used?The answer to both is yes. Indeed, there is a single general principle that answers both questions. But it is a general principle that is neither part of the grammar of English, nor the English lexicon. Rather, it is part of the conceptual system underlying English: It is a principle for under standing the domain of love in terms of the domain of journeys. The principle can be stated informally as a metaphorical scenario: The lovers are travelers on a journey together, with their common life goals seen as destinations to be reached. The relationship is their vehicle, and it allows them to pursue those common goals together. The relationship is seen as fulfilling its purpose as long as it allows them to make progress toward their common goals. The journey isn’t easy. There are impediments, and there are places (crossroads) where a decision has to be made about which direction to go in and whether to keep traveling together. The metaphor involves understanding one domain of experience, love, in terms of a very different domain of experience, journeys. More technically, the metaphor can be understood as a mapping (in the mathematical sense) from a source domain (in this case, journeys) to a target domain (in this case, love). The mapping is tightly structured. There are ontological correspondences, according to which entities in the domain of love (e.g., the lovers, their common goals, their difficulties, the love relationship, etc.) correspond systematically to entities in the domain of a journey (the travelers, the vehicle, des tinations, etc.). To make it easier to remember what mappings there are in the conceptual system, Johnson and I (lakoff and Johnson, 1980) adopted a strategy for naming such mappings, using mnemonics which suggest the mapping. Mnemonic names typically (though not always) have the form: TARGET-DOMAIN IS SOURCE-DOMAIN, or alternatively, TARGET-DOMAIN AS SOURCE-DOMAIN. In this case, the name of the mapping is LOVE IS A JOURNEY. When I speak of the LOVE IS A JOURNEY metaphor, I am using a mnemonic for a set of ontological correspondences that characterize a map ping, namely:THE LOVE-AS-JOURNEY MAPPING-The lovers correspond to travelers.-The love relationship corresponds to the vehicle.-The lovers’ common goals correspond to their common destinations on the journey.-Difficulties in the relationship correspond to impediments to travel.It is a common mistake to confuse the name of the mapping, LOVE IS A JOURNEY, for the mapping itself. The mapping is the set of correspondences. Thus, whenever I refer to a metaphor by a mnemonic like LOVE IS A JOURNEY, I will be referring to such a set of correspondences. If mappings are confused with names of mappings, another misunderstanding can arise. Names of mappings commonly have a propositional form, for example, LOVE IS A JOURNEY. But the mappings themselves are not propositions. If mappings are confused with names for mappings, one might mistakenly think that, in this theory, metaphors are propositional. They are, of course, anything but that: metaphors are mappings, that is, sets of conceptual correspondences. The LOVE-AS-JOURNEY mapping is a set of ontological correspondences that characterize epistemic correspondences by mapping knowledge about journeys onto knowledge about love. Such correspondences permit us to reason about love using the knowledge we use to reason about journeys. Let us take an example. Consider the expression, We’re stuck, said by one lover to another about their relationship. How is this expression about travel to be understood as being about their relationship? We’re stuck can be used of travel, and when it is, it evokes knowledge about travel. The exact knowledge may vary from person to person, but here is a typical example of the kind of knowledge evoked. The capitalized expressions represent entities n the ontology of travel, that is, in the source domain of the LOVE IS A JOURNEY mapping given above. Two TRAVELLERS are in a VEHICLE, TRAVELING WITH COMMON DESTINATIONS. The VEHICLE encounters some IMPEDIMENT and gets stuck, that is, makes it nonfunctional. If they do nothing, they will not REACH THEIR DESTINATIONS. There are a limited number of alternatives for action:•They can try to get it moving again, either by fixing it or get ting it past the IMPEDIMENT that stopped it.•They can remain in the nonfunctional VEHICLE and give up on REACHING THEIR DESTINATIONS.•They can abandon the VEHICLE.•The alternative of remaining in the nonfunctional VEHICLE takes the least effort, but does not satisfy the desire to REACH THEIR DESTINATIONS.The ontological correspondences that constitute the LOVE IS A JOURNEY metaphor map the ontology of travel onto the ontology of love. In doing so, they map this scenario about travel onto a corresponding love scenario in which the corresponding alternatives for action are seen. Here is the corresponding love scenario that results from applying the correspondences to this knowledge structure. The target domain entities that are mapped by the correspondences are capitalized:Two LOVERS are in a LOVE RELATIONSHIP, PURSUING COMMON LIFE GOALS. The RELATIONSHIP encounters some DIFFICULTY, which makes it nonfunctional. If they do nothing, they will not be able to ACHIEVE THEIR LIFE GOALS. There are a limited number of alternatives for action:•They can try to get it moving again, either by fixing it or getting it past the DIFFICULTY.•They can remain in the nonfunctional RELATIONSHIP, and give up onACHIEVING THEIR LIFE GOALS.•They can abandon the RELATIONSHIP.The alternative of remaining in the nonfunctional RELATIONSHIP takes the least effort, but does not satisfy the desire to ACHIEVE LIFE GOALS. This is an example of an inference pattern that is mapped from one domain to another. It is via such mappings that we apply knowledge about travel to love relationships.Metaphors are not mere wordsWhat constitutes the LOVE-AS-JOURNEY metaphor is not any particular word or expression. It is the ontological mapping across conceptual domains, from the source domain of journeys to the target domain of love. The metaphor is not just a matter of language, but of thought and reason. The language is secondary. The mapping is primary, in that it sanctions the use of source domain language and inference patterns for target domain concepts. The mapping is conventional, that is, it is a fixed part of our conceptual system, one of our conventional ways of conceptualizing love relationships. This view of metaphor is thoroughly at odds with the view that metaphors are just linguistic expressions. If metaphors were merely linguistic expressions, we would expect different linguistic expressions to be different metaphors. Thus, "We’ve hit a dead-end street" would constitute one metaphor. "We can’t turn back now" would constitute another, entirely different metaphor. "Their marriage is on the rocks" would involve still a different metaphor. And so on for dozens of examples. Yet we don’t seem to have dozens of different metaphors here. We have one metaphor, in which love is conceptualized as a journey. The mapping tells us precisely how love is being conceptualized as a journey. And this unified way of conceptualizing love metaphorically is realized in many different linguistic expressions. It should be noted that contemporary metaphor theorists commonly use the term metaphor to refer to the conceptual mapping, and the term metaphorical expression to refer to an individual linguistic expression (like dead-end street) that is sanctioned by a mapping. We have adopted this terminology for the following reason: Metaphor, as a phenomenon, involves both conceptual mappings and individual linguistic expressions. It is important to keep them distinct. Since it is the mappings that are primary and that state the generalizations that are our principal concern, we have reserved the term metaphor for the mappings, rather than for the linguistic expressions. In the literature of the field, small capitals like LOVE IS A JOURNEY are used as mnemonics to name mappings. Thus, when we refer to the LOVE IS A JOURNEY metaphor, we are refering to the set of correspondences discussed above. The English sentence Love is a journey, on the other hand, is a metaphorical expression that is understood via that set of correspondences.GeneralizationsThe LOVE IS A JOURNEY metaphor is a conceptual mapping that characterizes a generalization of two kinds:•Polysemy generalization: A generalization over related senses of linguistic expressions, e.g., dead-end street, crossroads, stuck, spinning one’s wheels, not going anywhere, and so on.•Inferential generalization: A generalization over inferences across different conceptual domains.That is, the existence of the mapping provides a general answer to two questions: -Why are words for travel used to describe love relationships? -Why are inference patterns used to reason about travel also used to reason about love relationships. Correspondingly, from the perspective of the linguistic analyst, the existence of such cross-domain pairings of words and of inference patterns provides evidence for the existence of such mappings. Novel extensions of conventional metaphorsThe fact that the LOVE IS A JOURNEY mapping is a fixed part of our conceptual system explains why new and imaginative uses of the mapping can be understood instantly, given the ontological correspondences and other knowledge about journeys. Take the song lyric, We’re driving in the fast lane on the freeway of love. The traveling knowledge called upon is this: When you drive in the fast lane, you go a long way in a short time and it can be exciting and dangerous. The general metaphorical mapping maps this knowledge about driving into knowledge about love relationships. The danger may be to the vehicle (the relationship may not last) or the passengers (the lovers may be hurt, emotionally). The excitement of the love-journey is sexual. Our understanding of the song lyric is a consequence of the pre-existing metaphorical correspondences of the LOVE-AS-JOURNEY metaphor. The song lyric is instantly comprehensible to speakers of English because those metaphorical correspondences are already part of our conceptual system. The LOVE-AS-JOURNEY metaphor and Reddy’s Conduit Metaphor were the two examples that first convinced me that metaphor was not a figure of speech, but a mode of thought, defined by a systematic mapping from a source to a target domain. What convinced me were the three characteristics of metaphor that I have just discussed: The systematicity in the linguistic correspondences. The use of metaphor to govern reasoning and behavior based on that reasoning. The possibility for understanding novel extensions in terms of the conventional correspondences.MotivationEach conventional metaphor, that is, each mapping, is a fixed pattern of conceptual correspondences across conceptual domains. As such, each mapping defines an open-ended class of potential correspondences across inference patterns. When activated, a mapping may apply to a novel source domain knowledge structure and characterize a corresponding target domain knowledge structure. Mappings should not be thought of as processes, or as algorithms that mechanically take source domain inputs and produce target domain outputs. Each mapping should be seen instead as a fixed pattern of onotological correspondences across domains that may, or may not, be applied to a source domain knowledge structure or a source domain lexical item. Thus, lexical items that are conventional in the source domain are not always conventional in the target domain. Instead, each source domain lexical item may or may not make use of the static mapping pattern. If it does, it has an extended lexicalized sense in the target domain, where that sense is characterized by the mapping. If not, the source domain lexical item will not have a conventional sense in the target domain, but may still be actively mapped in the case of novel metaphor. Thus, the words freeway and fast lane are not conventionally used of love, but the knowledge structures associated with them are mapped by the LOVE IS A JOURNEY metaphor in the case of We’re driving in the fast lane on the freeway of love. Imageable IdiomsMany of the metaphorical expressions discussed in the literature on conventional metaphor are idioms. On classical views, idioms have arbitrary meanings. But withincognitive linguistics, the possibility exists that they are not arbitrary, but rather motivated. That is, they do arise automatically by productive rules, but they fit one or more patterns present in the conceptual system. Let us look a little more closely at idioms. An idiom like spinning one’s wheels comes with a conventional mental image, that of the wheels of a car stuck in some substance-either in mud, sand, snow, or on ice, so that the car cannot move when the motor is engaged and the wheels turn. Part of our knowledge about that image is that a lot of energy is being used up (in spinning the wheels) without any progress being made, that the situation will not readily change of its own accord, that it will take a lot of effort on the part of the occupants to get the vehicle moving again --and that may not even be possible. The love-as-journey metaphor applies to this knowledge about the image. It maps this knowledge onto knowledge about love relationships: A lot of energy is being spent without any progress toward fulfilling common goals, the situation will not change of its own accord, it will take a lot of effort on the part of the lovers to make more progress, and so on. In short, when idioms that have associated conventional images, it is common for an independently-motivated conceptual metaphor to map that knowledge from the source to the target domain. For a survey of experiments verifying the existence of such images and such mappings, see Gibbs 1990 and this volume. Mappings are at the superordinate levelIn the LOVE IS A JOURNEY mapping, a love relationship corresponds to a vehicle. A vehicle is a superordinate category that includes such basic-level categories as car, train, boat, and plane. Indeed, the examples of vehicles are typically drawn from this range of basic level categories: car ( long bumpy road, spinning our wheels), train (off the track), boat (on the rocks, foundering), plane (just taking off, bailing out). This is not an accident: in general, we have found that mappings are at the superordinate rather than the basic level. Thus, we do not find fully general submappings like A LOVE RELATIONSHIP IS A CAR; when we find a love relationship conceptualized as a car, we also tend to find it conceptualized as a boat, a train, a plane, etc. It is the superordinate category VEHICLE not the basic level category CAR that is in the general mapping. It should be no surprise that the generalization is at the superordinate level, while the special cases are at the basic level. After all, the basic level is the level of rich mental images and rich knowledge structure. (For a discussion of the properties of basic-level categories, see Lakoff, 1987, pp. 31-50.) A mapping at the superordinate level maximizes the possibilities for mapping rich conceptual structure in the source domain onto the target domain, since it permits many basic-level instances, each of which is information rich. Thus, a prediction is made about conventional mappings: the categories mapped will tend to be at the superordinate rather than basic level. Thus, one tends not to find mappings like A LOVE RELATIONSHIP IS A CAR or A LOVE RELATIONSHIP IS A BOAT. Instead, one tends to find both basic-level cases (e.g., both cars and boats), which indicates that the generalization is one level higher, at the superordinate level of the vehicle. In the hundreds of cases of conventional mappings studied so far, this prediction has been borne out: it is superordinate categories that are used in mappings.Basic Semantic Concepts That Are MetaphoricalMost people are not too surprised to discover that emotional concepts like love and anger are understood metaphorically. What is more interesting, and I think more exciting, is the realization that many of the most basic concepts in our conceptual systems are alsocomprehended normally via metaphor-concepts like time, quantity, state, change, action, cause, purpose, means, modality and even the concept of a category. These are concepts that enter normally into the grammars of languages, and if they are indeed metaphorical in nature, then metaphor becomes central to grammar. What I would like to suggest is that the same kinds of considerations that lead to our acceptance of the LOVE-AS-JOURNEY metaphor lead inevitably to the conclusion that such basic concepts are often, and perhaps always, understood via metaphor.CategoriesClassical categories are understood metaphorically in terms of bounded regions, or ‘containers.’ Thus, something can be in or out of a category, it can be put into a category or removed from a category, etc. The logic of classical categories is the logic of containers (see figure 1). If X is in container A and container A is in container B, then X is in container B. This is true not by virtue of any logical deduction, but by virtue of the topological properties of containers. Under the CLASSICAL CATEGORIES ARE CONTAINERS metaphor, the logical properties of categories are inherited from the logical properties of containers. One of the principal logical properties of classical categories is that the classical syllogism holds for them. The classical syllogism, Socrates is a man. All men are mortal. Therefore, Socrates is mortal. is of the form: If X is in category A and category A is in category B, then X is in category B. Thus, the logical properties of classical categories can be seen as following from the topological properties of containers plus the metaphorical mapping from containers to categories. As long as the topological properties of containers are preserved by the mapping, this result will be true. In other words, there is a generalization to be stated here. The language of containers applies to classical categories and the logic of containers is true of classical categories. A single metaphorical mapping ought to characterize both the linguistic and logical generalizations at once. This can be done provided that the topological properties of containers are preserved in the mapping. The joint linguistic-and-inferential relation between containers and classical categories is not an isolated case. Let us take another example.Quantity and Linear ScalesThe concept of quantities involves at least two metaphors. The first is the well-known MORE IS UP, LESS IS DOWN metaphor as shown by a myriad of expressions like Prices rose, Stocks skyrocketed, The market plummeted, and so on. A second is that LINEAR SCALES ARE PATHS. We can see this in expressions like: John is far more intelligent than Bill. John’s intelligence goes way beyond Bill’s. John is way ahead of Bill in intelligence. The metaphor maps the starting point of the path onto the bottom of the scale and maps distance traveled onto quantity in general. What is particularly interesting is that the logic of paths maps onto the logic of linear scales. (See figure 2.) Path inference: If you are going from A to C, and you are now at in intermediate point B, then you have been at all points between A and B and not at any points between B and C. Example: If you are going from San Francisco to N.Y. along route 80, and you are now at Chicago, then you have been to Denver but not to Pittsburgh. Linear scale inference: If you have exactly $50 in your bank account, then you have $40, $30, and so on, but not $60, $70, or any larger amount. The form of these inferences is the same. The path inference is a consequence of the cognitive topology of paths. It will be true of any path image-schema. Again, there is a linguistic-and-inferential generalization to be stated. It would be stated by the metaphor LINEAR SCALES ARE PATHS, provided that metaphors in general preserve the cognitive topology (that is, the image-schematic structure) of the source。

DISTRIBUTED INTERACTIVE SIMULATION FOR GROUP-DISTANCEEXERCISES ON THE WEBErik Berglund and Henrik ErikssonDepartment of Computer and Information ScienceLinköping UniversityS-581 83 Linköping, SwedenE-mail: {eribe, her}@ida.liu.seKEYWORDS: Distributed Interactive Simulation, Distance Education, Network, Internet, Personal ComputerABSTRACTIn distributed-interactive simulation (DIS), simulators act as elements of a bigger distributed simulation. A group-distance exercise (GDE) based on the DIS approach can therefore enable group training for group members participating from different locations. Our GDE approach, unlike full-scale DIS systems, uses affordable simulators designed for standard hardware available in homes and offices.ERCIS (group distance exERCISe) is a prototype GDE system that we have implemented. It takes advantage of Internet and Java to provide distributed simulation at a fraction of the cost of full-scale DIS systems.ERCIS illustrates that distributed simulation can bring advanced training to office and home computers in the form of GDE systems.The focus of this paper is to discuss the possibilities and the problems of GDE and of web-based distributed simulation as a means to provide GDE. INTRODUCTIONSimulators can be valuable tools in education. Simulators can reduce the cost of training and can allow training in hazardous situations (Berkum & Jong 1991). Distributed-interactive simulation (DIS) originated in military applications, where simulators from different types of forces were connected to form full battle situations. In DIS, individual simulators act as elements of a bigger distributed simulation (Loper & Seidensticker 1993).Thus, DIS could be used to create a group-distance exercise (GDE), where the participants perform a group exercise from different locations. Even though DIS systems based on complex special-hardware simulators provide impressive training tools, the cost and immobility of these systems prohibit mass training.ERCIS (group distance exERCISe) is a prototype GDE system that uses Internet technologies to provide affordable DIS support. Internet (or Intranet) technologies form a solid platform for GDE systems because they are readily available, and because they provide high level of support for network communication and for graphical simulation. ERCIS, therefore, takes advantage of the programming language Java, to combine group training, distance education and real-time interaction at a fraction of the cost of full-scale DIS systems.In this paper we discuss the possibilities and the problems of GDE and of web-based distributed simulation as a means to provide GDE. We do this by discussing and drawing conclusions from the ERCIS project.BACKGROUNDLet us first provide some background on GDE, DIS, distributed objects, ERCIS’s military application, and related work.Group-Distance Exercise (GDE)The purpose of GDE is to enable group training in distance education through the use of DIS. Unlike full-scale DIS systems, our GDE approach assumes simulators designed for standard hardware available in homes and offices. This approach calls for software-based simulators which are less expensive to use, can be multiplied virtually limitlessly, and can enable training with expensive, dangerous and/or non-existing equipment.A thorough background on the GDE concept can be found in Computer-Based Group-Distance Exercise (Berglund 1997).Distributed Interactive Simulation (DIS)DIS originated as a means to utilize military simulators in full battle situations by connecting them (Loper & Seidensticker 1993). As a result it becomes possible tocombine the use of advanced simulators and group training.The different parts of DIS systems communicate according to predefined data packets (IEEE 1995) that describe all necessary data on the bit level. The implementation of the communication is, therefore, built into DIS systems.Distributed ObjectsDistributed objects (Orfali et al. 1996) can be characterized as network transient objects, objects that can bridge networks. Two issues must be addressed when dealing with distributed objects: to locate them over the network and to transform them from abstract data to a transportation format and vice versa.The common object request broker architecture (CORBA) is a standard protocol for distributed objects, developed by the object management group (OMG). CORBA is used to cross both networks and programming languages. In CORBA, all objects are distributed via the object request broker (ORB). Objects requesting service of CORBA object have no knowledge about the location or the implementation of the CORBA objects (Vinoski 1997).Remote method invocation (RMI) is Java’s support for distributed objects among Java programs. RMI only provides the protocol to locate and distribute abstract data, unlike CORBA. In the ERCIS project we chose RMI because ERCIS is an all Java application. It also provided us with an opportunity to assess Java’s support for distributed objects.RBS-70 Missile UnitERCIS supports training of the RBS-70 missile unit of the Swedish anti-aircraft defense. The RBS-70 missile unit’s main purpose is to defend objects, for instance bridges, against enemy aircraft attacks, see Figure 1. The RBS-70 missile unit is composed of sub units: two intelligence units and nine combat units. The intelligence units use radar to discover and calculate flight data of hostile aircraft. Guided by the intelligence unit, the combat units engage the aircraft with RBS-70 missiles. (Personal Communication).Intelligence unitCombat unitData transferFigure 1. The RBS-70 missile unit. Its main purpose is to defend ground targets.During training, the RBS-70 unit uses simulators, for instance, to simulate radar images. All or part of the RBS-70 unit’s actual equipment is still used (Personal Communication).Related WorkIn military applications, there are several examples of group training conducted using distributed simulation. For instance, MIND (Jenvald 1996), Janus, Eagle, the Brigade/Battalion Simulation (Loper & Seidensticker 1993), and ABS 2000.C3Fire (Granlund 1997) is an example of a tool for group training, in the area of emergency management, that uses distributed simulation.A common high-level architecture for modeling and simulation, that will focus on a broader range of simulators than DIS, is being developed (Duncan 1996). There are several educational applets on the web that use simulation; see for instance the Gamelan applet repository. These applets are, however, generally small and there are few, if any, distributed simulations. ERCISERCIS is a prototype GDE system implemented in Java for Internet (or Intranets). It supports education of the RBS-70 missile unit by creating a DIS of that group’s environment. We have used RMI to implement the distribution of the system.ERCIS has two principal components: the equipment simulators and the simulator server, see Figure 2. A maximum of 11 group members can participate in a single ERCIS session, representing the 11 sub-unit leaders, see Figure 3. The group members join by loading an HTML document with an embedded equipment-simulator applet.Simulator serverIntelligence unit equipmentsimulatorCombat unit equipmentsimulatorFigure 2. The principal parts of ERCIS. The equipment simulators are connected to one another through the simulator server.Figure 3. A full ERCIS session. 11 equipment simulators can be active in one ERCIS session at a time. The equipment simulators communicate via the simulator server.Simulator ServerThe simulator server controls a microworld of the group’s environment, including simulated aircraft, exercise scenario and geographical information.The simulator server also distributes network communication among the equipment simulators. The reason for this client-server type of communication is that Java applets are generally only allowed to connect to the computer they were loaded from (Flanagan 1997).Finally, the simulator server functions as a point of reference by which the distributed parts locate one another. The simulator-server computer is therefore the only computer specified prior to an ERCIS session.Equipment SimulatorsThe equipment simulators simulate equipment used by the RBS-70 sub units and also function as user interfaces for the group members. There are two types of equipment simulators: the intelligence-unit equipment simulator and combat-unit equipment simulator.Intelligence Unit Equipment SimulatorThe intelligence-unit’s equipment simulator, see Figure 4,contains a radar simulator and a target-tracking simulator . The radar simulator monitors the air space. The target-tracking simulator performs part of the work of three intelligence-unit personnel to approximate position,speed and course of three aircraft simultaneously.The intelligence-unit leader’s task is to distribute the hostile aircraft among the combat units and to send approximated information to them.Panel used to send information to the combat unit equipment simulator.Target-tracking symbol used to initate target tracking.Simulated radarFigure 4. The user interface of the intelligence unit equipment simulator, running on a Sun Solaris Applet Viewer.Combat Unit Equipment SimulatorThe combat unit equipment simulator, see Figure 5,contains simulators of the target-data receiver and the RBS-70 missile launcher and operator .The target-data receiver presents information sent from the intelligence unit. The information is recalculated relative to the combat unit’s position and shows information such as: the distance and direction to the target. The RBS-70 missile launcher and operator represents the missile launcher and its crew. Based on the intelligence unit’s information it can locate, track and fire upon the target.The combat-unit leader’s task is to assess the situation and to grant permission to fire if all criteria are met.Switch used to grant fire permissionFigure 5. The user interface of the combat unit equipment simulator, running on a Windows 95 Applet Viewer.DISCUSSIONLet us then, with experience from the ERCIS project, discuss problems and possibilities of GDE though DIS and of the support Internet and Java provide for GDE. Pedagogical valueEducators can use GDE to introduce group training at an early stage by, for instance, simplifying equipment and thereby abstracting from technical details. Thus, GDE can focus the training on group performance.GDE could also be used to support knowledge recapitulation for expert practitioners. To relive the group’s tasks and environment can provide a more vivid experience than notes and books.GDE systems can automate collection and evaluation of performance statistics. It is possible to log sessions and replay them to visualize comments on performance in after-action reviews (Jenvald 1996).To fully assess the values of GDE, however, real-world testing is required.SecurityAccess control, software authentication, and communication encryption are examples of security issues that concern GDE and distributed simulators. Java 1.1 provides basic security, which includes fine-grained access control, and signed applets. The use of dedicated GDE Intranets would increase security, especially access control. It would, however, reduce the ability to chose location from where to participate.Security restrictions, motivated or not, limit applications. ERCIS was designed with a client-server type of communication because web browsers enforce security restrictions on applets (Flanagan 1997). Peer-to-peer communication would have been more suitable, from the perspective of both implementation and scalability. We are not saying that web browsers should give applets total freedom. In ERCIS, it would have sufficed if applets were allowed to make remote calls to RMI objects regardless of their location.Performance characteristicsOur initial concerns about the speed of RMI calls and of data transfer over Internet proved to be unfounded. The speed of communication is not a limiting factor for ERCIS. For instance, a modem link (28.8 kilobits per second) is sufficient to participate in exercises.Instead the speed of animation limits ERCIS. To provide smooth animation, ERCIS, therefore, requires more than the standard hardware of today, for instance a Pentium Pro machine or better.ScalabilityIn response to increased network load, ERCIS scales relatively well, because the volume of the data that is transmitted among the distributed parts is very small, for instance, 1 Kbytes.Incorporating new and better simulators in ERCIS requires considerable programming effort. In a full-scale GDE system it could be beneficial to modularize the simulators in a plug-and-play fashion, to allow variable simulator complexity.Download timeThe download time for applets the size of ERCIS’s equipment simulator can be very long. One way to overcome this problem is to create Java archive files (JAR files). JAR files aggregate many files into one and also compress them to decrease the download time considerably. Push technology such as Marimba’s Castanet could also be used to provide automatic distribution of the equipment-simulator software. Distributed objectsDistributed objects, such as RMI, provide a high level of abstraction in network communication compared to the DIS protocol. There are several examples of typical distributed applications that do not utilize distributed objects but that would benefit greatly from this approach. Two examples are the Nuclear Power Plant applet (Eriksson 1997), and NASA’s distributed control of the Sojourner.CONCLUSIONERCIS is a GDE prototype that can be used in training under teacher supervision or as part of a web site where web pages provide additional information. The system illustrates that the GDE approach can provide equipment-free mass training, which is beneficial, especially in military applications where training can be extremely expensive.Java proved to be a valuable tool for the implementation of ERCIS. Java’s level of abstraction is high in the two areas that concern ERCIS: animation and distributed objects. Java’s speed of animation is, however, too slow to enable acceptable performance for highly graphic-oriented simulators. Apart from this Java has supplied the support that can be expected from a programming language, for instance C++.Using RMI to implement distribution was straightforward. Compared to the DIS protocol, RMI provides a flexible and dynamic communication protocol. In conclusion, ERCIS illustrates that it is possible to use Internet technologies to develop affordable DIS systems. It also shows that distributed simulations can bring advanced training to office and home computers in the form of GDE systems.AcknowledgmentsWe would like to thank Major Per Bergström at the Center for Anti-Aircraft Defense in Norrtälje, Sweden for supplying domain knowledge of the RBS-70 missile unit. This work has been supported in part by the Swedish National Board for Industrial and Technical Development(Nutek) grant no. 93-3233, and by the Swedish Research Council for Engineering Science (TFR) grant no. 95-186. REFERENCESBerglund E. (1997) Computer-Based Group Distance Exercise, M.Sc. thesis no. 97/36, Department of Computer and Information Science, Linköping University (http://www.ida.liu.se/~eribe/publication/ GDE.zip: compressed postscript file).van Berkum J, de Jong T. (1991) Instructional environments for simulations Education & Computing vol. 6: 305-358.Duncan C. (1996) The DoD High Level Architecture and the Next Generation of DIS, Proceedings of the Fourteenth Workshop on Interoperability of Distributed Simulation, Orlando, Florida.Eriksson H. (1996) Expert Systems as Knowledge Servers, IEEE Expert vol. 11 no. 3: 14 -19. Flanagan, D. (1997) Java in a Nutshell 2nd Edition, O’Reilly, Sebastopol, CA.Granlund R. (1997) C3Fire A Microworld Supporting Emergency Management Training, licentiate thesis no.598, Department of Computer and Information Science 598, Department of Computer and Information Science Linköping University.IEEE (1995) IEEE Standard for Distributed Interactive Simulation--Application Protocols, IEEE 1278.1-1995 (Standard): IEEE.Jenvald J. (1996) Simulation and Data Collection in Battle Training, licentiate thesis no. 567, Department of Computer and Information Science, Linköping University.Loper M, Seidensticker S. (1994) The DIS Vision: A Map to the Future of Distributed Simulation, Orlando, Florida: Institute for Simulation & Training (/SISO/dis/library/vision.doc) Orfali R, Harkey D, Edwards J. (1996) The Essential Distributed Objects Survival Guide John Wiley, New York.Vinoski S. (1997) CORBA: Integrating Diverse Applications Within Distributed Heterogeneous Environments, IEEE Communications, vol. 14, no. 2.RESOURCES AT THE WEBThe OMG home page: /CORBA JavaSoft’s Java 1.1 documentation: /-products/jdk/1.1/docs/index.htmlThe Gamelan applet repository: / Marimba’s Castanet home page: http://www.marimba.-com/products/castanet.htmlThe Nuclear Power Plant Applet (Eriksson 1995): http://-www.ida.liu.se/∼her/npp/demo.htmlNASAS Soujourner, The techical details on the control distribution of NASAS Soujourner: /features/1997/july/juicy.wits.details.html AuthorsErik Berglund is a doctoral student of computer science at Linköping University. His research interests include knowledge acquisition, program understanding, software engineering, and computer supported education. He received his M.Sc. at Linköping University in 1997. Henrik Eriksson is an assistant professor of computer science at Linköping University. His research interests include expert systems, knowledge acquisition, reusable problem-solving methods and medical Informatics. He received his M.Sc. and Ph.D. at Linköping University in 1987 and 1991. He was a postdoctoral fellow and research scientist at Stanford University between 1991 and 1994. Since 1996, he is a guest researcher at the Swedish Institute for Computer Science (SICS).。

ON THE CHARNEY-DAVIS AND NEGGERS-STANLEYCONJECTURESVICTOR REINER AND VOLKMAR WELKERAbstract.For a graded naturally labelled poset P,it is shownthat the P-Eulerian polynomialW(P,t):= w∈L(P)t des(w)counting linear extensions of P by their number of descents hassymmetric and unimodal coefficient sequence,verifying the moti-vating consequence of the Neggers-Stanley conjecture on real zeroesfor W(P,t)in these cases.The result is deduced from McMullen’sg-Theorem,by exhibiting a simplicial polytopal sphere whose h-polynomial is W(P,t).Whenever this simplicial sphere turns out to beflag,that is,its minimal non-faces all have cardinality two,it is shown that theNeggers-Stanley Conjecture would imply the Charney-Davis Con-jecture for this sphere.In particular,it is shown that the sphereisflag whenever the poset P has width at most2.In this case,the sphere is shown to have a stronger geometric property(locallyconvexity),which then implies the Charney-Davis Conjecture inthis case via a result from[30].It is speculated that the proper context in which to view both of these conjectures may be the theory of Koszul algebras,and someevidence is presented.1.IntroductionThis paper has several goals.Thefirst is to show that,in the context of the Neggers-Stanley Conjecture1.2,for every graded poset P there is lurking in the background a polytopal simplicial sphere,which we will denote∆eq(P).This sphere is relevant for two purposes:2VICTOR REINER AND VOLKMAR WELKER⊲The P-Eulerian polynomial(defined below)coincides with theh-polynomial of∆eq(P).As a consequence,its coefficients sat-isfy McMullen’s conditions for the h-vector of a simplicial poly-tope,and are in particular symmetric and unimodal.Therebywe verify the motivating consequence of the Neggers-StanleyConjecture for naturally labeled graded posets(see discussionafter the statement of Conjecture1.2).⊲Whenever the simplicial sphere∆eq(P)isflag,the Neggers-Stanley Conjecture1.2for P implies the Charney-Davis Con-jecture for the sphere∆eq(P).Furthermore,when P has widthat most2,it is shown in Theorem3.23that∆eq(P)satisfies astronger geometric condition thanflag-ness known as local con-vexity,which implies the Charney-Davis Conjecture in this caseby a result from[30].The latter portion of the paper(Section4onward)is aimed toward the thesis that both the Charney-Davis and Neggers-Stanley Conjec-tures,along with some other combinatorial conjectures and results, should be considered in the context of the following question. Question1.1.For which Koszul algebras is the Hilbert function a Polya frequency sequence?To give a more precise discussion,we start by recalling the Neggers-Stanley Conjecture.For any partial order P on[n]:={1,2,...,n}, let L(P)denote its set of linear extensions,that is the set of w= (w1,...,w n)∈S n for which i<P j implies w−1(i)<w−1(j).The P-Eulerian polynomialW(P,t):= w∈L(P)t des(w)is the generating function for the linear extensions L(P)counted ac-cording to cardinality of their descent sets:Des(w):={i∈[n−1]:w i>w i+1}des(w):=#Des(w)Conjecture1.2(Neggers-Stanley).For any labelled poset P on[n] the polynomial W(P,t)has only real(non-positive)zeroes.We are mainly interested in the case where P is naturally labelled, that is i<P j implies i<j.Some history and context for the conjecture follows.For naturally labelled posets Conjecture1.2was made originally by Neggers[32], and generalized to the above statement by Stanley in1986.When P is an antichain of n elements,W(P,t)is the Eulerian polynomial whoseCHARNEY-DAVIS AND NEGGERS-STANLEY CONJECTURES3 real-rootedness was shown by Harper[24]and served as an initial mo-tivation for the conjecture.For the case when P is a naturally labelled disjoint union of chains the result is due to Simion[37].This result was extended to arbitrary labellings by Brenti[7],who also verified the conjecture for Ferrers posets and Gaussian posets[7].An impor-tant combinatorial implication of the real-rootedness of a polynomial with non-negative coefficients is the unimodality of the coefficients(i.e. for the sequence of coefficients a0,...,a r there is an index j such that a0≤···≤a j≥···≥a r).Gasharov[19]verified the unimodality consequence of the conjecture for naturally labelled graded posets with at most3ranks.Corollary3.15verifies this(and something stronger) more generally for all naturally labelled graded posets.Next,we recall the Charney-Davis Conjecture.Given an abstract simplicial complex∆triangulating a(d−1)-dimensional(homology) sphere,one can collate the face numbers f i,which count the number of i-dimensional faces,into its f-vector and f-polynomialf(∆):=(f−1,f0,f1,...,f d−1)f(∆,t):=d i=0f i−1t i.The h-polynomial and h-vector are easily seen to encode the same in-formation:(1.1)h(∆):=(h0,h1,...,h d)whereh(∆,t)=di=0h i t i satisfies t d h(∆,t−1)= t d f(∆,t−1) t→t−1.The h-polynomial turns out to be a more convenient and natural encoding in several ways,closely related to commutative algebra,toric geometry,and shellability.For example,the fact that homology spheres are Cohen-Macaulay implies non-negativity of the h i,and the Dehn-Sommerville equations for simplicial spheres assert that h i=h d−i for 0≤i≤d(see[46,§II.6]).Note that the latter implies that the h-polynomial is symmetric,h(∆,t)=t d h(∆,t−1),and that h(∆,−1)=0 whenever d is odd.The Charney-Davis Conjecture[11,Conjecture D]concerns the quan-tity h(∆,−1)in the case where d is even and∆is a simplicial homology (d−1)-sphere which happens to be aflag complex,that is the minimal subsets of vertices which do not span a simplex all have cardinality4VICTOR REINER AND VOLKMAR WELKERtwo.For polytopal simplicial spheres ∆,this quantity is known [30]to coincide with the signature or index of the associated toric variety X ∆.Conjecture 1.3(Charney-Davis,Conjecture D [11]).When ∆is a flag simplicial homology (d −1)-sphere and d is even,then(−1)d2h (−1)≥0.Proof.Since h (t )has degree d we have h d =0and by symmetry h 0=0.Thus h (t )has d zeroes which must then all be strictly negative since h i ≥0for 0≤i ≤d .Factor h (t )=h d di =1(t −r i )according to its(real)zeroes r i .Symmetry of h (t )implies that r is a zero if and only if 1r is less than −1.Thus for a zero r ,either r =−1is a zero,in which case h (−1)=0and we are done,or else exactly half of the factors in the product h (−1)=h d d i =1(−1−r i )are negative,implying that the product has sign (−1)d 2W (P,−1),for some cases of posets where the Neggers-StanleyConjecture is known,are explored in [36].In Section 3.2it is shown that the sphere ∆eq (P )is the boundary complex of a simplicial convex polytope.Therefore by McMullen’s g -Theorem characterizing the number of faces of such polytopes [40],the coefficients (h 0,h 1,...,h #P −r )are symmetric and unimodal.Convexity has further relevance.In [30]it was shown via the Hirze-bruch signature formula that the Charney-Davis Conjecture holds forCHARNEY-DAVIS AND NEGGERS-STANLEY CONJECTURES5 a simplicial polytope under a certain geometric hypothesis(local con-vexity)stronger than beingflag.We show in Section3.2that this hypothesis holds for∆eq(P)whenever P has width(i.e.size of the largest antichain)at most2,thereby providing more evidence for the Neggers-Stanley Conjecture.In Sections4and5we gather evidence for the thesis that both of these conjectures can be fruitfully viewed within the context of Koszul algebras.In particular,we point out ways in which Hilbert series of Koszul algebras interact well with the theory of Polya frequency series and polynomials with real zeroes.After this paper was circulated,C.Athanasiadis[1]has shown that the unimodular triangulation of the order polytope from Section3.1 is a member of a class of triangulations of polytopes that decompose into a join of a simplex and a polytopal sphere.Most notably he has exhibited such a triangulation for the Birkhoffpolytope.2.Review:P-Partitions and Order PolytopesIn this section we review some of the theory of P-partitions,distribu-tive lattices and order polytopes;see[25,27,26,39,41]for proofs and more details.Also see[18,§1.2]for definitions and basic facts about polyhedral cones and fans.Given a naturally labelled poset P on[n]ordered by≤P,the vector space of functions f=(f(1),...,f(n)):P→R will be identified with R n.One says that f is a P-partition if f(i)≥0for all i and f(i)≥f(j) for all i<P j.Denote by A(P)the cone of all P-partitions in R n.The convex polytopeO(P)=A(P)∩[0,1]nis called the order polytope of P.An order ideal I in P is a subset of P such that i∈I and j<P i implies j∈I.It is known that O(P)is the convex hull of the characteristic vectorsχI∈{0,1}n as I runs through all order ideals I in P.A useful alternative way to view O(P)is provided by the fact that it is isometric to the hyperplane slice at x0=1of the cone A(P0)⊂R n+1,where P0is the naturally labelled poset on[0,n]:={0,1,...,n} obtained from P by adjoining a new minimum element0.We call the cone A(P0)the homogenization of the cone A(P).We recall a few basic definitions some of which were already men-tioned in the introduction.The set of permutations w=(w1,...,w n)∈S n which extend P to a linear order is called its Jordan-H¨o lder set L(P):= w=(w1,...,w n)∈S n:i<P j implies w−1(i)<w−1(j) .6VICTOR REINER AND VOLKMAR WELKERThe descent set and descent number of w are defined byDes(w):={i∈[n−1]:w i>w i+1}des(w):=#Des(w).Define a cone for each w∈S nA(w):={f∈R n:f(w i)≥f(w i+1)for i∈[n−1],f(w i)>f(w i+1)if i∈Des(w)}It is not hard to see that the closure of A(w)(defined by removing the strict inequalities above),is a unimodular(simplicial)cone,that is its extreme rays are spanned by a set of vectors forming a lattice basis for Z n.Similarly,the closure of A(w)∩[0,1]n is a unimodular simplex. Now we are in position to formulate the basic fact from the theory of P-partitions which will be crucial for subsequent arguments. Proposition2.1.(i)The cone of P-partitions decomposes into a disjoint union asfollows:A(P)=⊔w∈L(P)A(w)The closures of the cones A(w)for w∈L(P)give a unimodulartriangulation of A(P).(ii)The unimodular triangulation of A(P)described in(i)restricts to a unimodular triangulation of the order polytopeO(P)=⊔w∈L(P)A(w)∩[0,1]n.We call the triangulations of A(P)(into simplicial cones)and O(P) (into simplices)from Proposition2.1their canonical triangulations. Note that via homogenization the canonical triangulation of O(P)is easily seen to be the restriction of the canonical triangulation of the homogenized cone A(P0)to the hyperplane x0=1.This makes sense since there is an obvious bijection between the linear extensions L(P0) and L(P).The combinatorics of these triangulations is closely related to the distributive lattice J(P)of all order ideals I in P ordered by inclusion. The order complex∆J(P)is the abstract simplicial complex having a vertex for each ideal I in P and a simplex for each chain I1⊂...⊂I t of nested ideals.Given a set of vectors V⊂R n,define their positive span to be the(relatively open)conepos(V):= v∈V c v·v:c v∈R,c v>0 .CHARNEY-DAVIS AND NEGGERS-STANLEY CONJECTURES7Proposition2.2.(i)Every non-zero P-partition f∈A P can be uniquely expressedin the formf=t i=1c iχI iwhere the c i are positive reals,and I1⊂···⊂I t is a chain ofideals in P.In other words,A(P)= ideals I1⊂···⊂I t⊂P pos {χI t}t i=1 .(ii)The canonical triangulation of the order polytope O(P)is iso-morphic(as an abstract simplicial complex)to∆J(P),via anisomorphism sending an ideal I to its characteristic vectorχI.(iii)The lexicographic order of permutations in L(P)gives rise to a shelling order on∆J(P).(iv)In this shelling,for each w in L(P),the minimal face of its cor-responding simplex in∆J(P)which is not contained in a lexico-graphically earlier simplex is spanned by the ideals{w1,w2,...,w i}where i∈Des(w).Using basic facts about shellings(see[4]),part(iv)of the preceding proposition implies that one can re-interpret the polynomial W(P,t): (2.1)W(P,t):= w∈L(P)t des(w)=h(∆J(P),t)This connection with J(P)also allows one to re-interpret these re-sults in terms of Ehrhart polynomials.Recall that for a convex poly-tope Q in R n having vertices in Z n,the number of lattice points con-tained in an integer dilation dQ grows as a polynomial in the dilation factor d∈N.This polynomial in d is called the Ehrhart polynomial: Ehrhart(O(Q),d):=# d O(P)∩N n .Whenever Q has a unimodular triangulation abstractly isomorphic toa simplicial complex∆,there is the following relationship:(2.2) d≥0Ehrhart(O(Q),d)t d=h(∆,t)8VICTOR REINER AND VOLKMAR WELKERthe equatorial triangulation.This triangulation has several pleasant properties,proven in this and the next subsection,which may be sum-marized as follows:⊲It is a unimodular triangulation.(See Proposition3.6)⊲It is isomorphic,as an abstract simplicial complex,to the joinof an r-simplex with a simplicial(#P−r−1)-sphere,which wewill denote∆eq(P),and call the equatorial sphere.(See Corollary3.8)⊲h(∆eq(P),t)=h(∆J(P),t)=W(P,t).(See Corollary3.8)⊲The equatorial sphere∆eq(P)is polytopal,and hence shellableand a PL-sphere.(See Theorem3.14)⊲When P has width at most2,the equatorial sphere∆eq(P)is realized by a locally convex simplicial fan.Hence is aflagsubcomplex of∆J(P),and aflag sphere for which the Charney-Davis Conjecture holds.(See Theorem3.23)Example3.1.Let P be the graded naturally labelled poset on[4] with r=2ranks shown in Figure1(a).Let J(P)be its associated (distributive)lattice of order ideals(see Figure1(b)).The4-dimensional order polytope O(P),and its canonical triangula-tion by∆J(P),may be“visualized”as follows.Start with the convex pentagonπwhich is the convex hull of{χ1,χ2,χ12,χ13,χ123,χ124},and triangulateπas shown in Figure1(c).The canonical triangulation is obtained by taking the simplicial join of this triangulation ofπwith the edge{χ∅,χ1234}.The equatorial triangulation(see Proposition3.6)is obtained start-ing from the alternate triangulation ofπdepicted in Figure1(d)and taking the simplicial join with the edge{χ∅,χ1234}.Equivalently,it is obtained from the equatorial1-sphere∆eq(P)depicted in Figure1(e) and taking the simplicial join with the triangle{χ∅,χ12,χ1234}.Fix a naturally labelled poset P on[n],and assume that it is graded, with r rank sets P1,...,P r.The following are the key definitions.Definition3.2.A P-partition f will be called rank-constant if it is constant along ranks,i.e.f(p)=f(q)whenever p,q∈P j for some j.A P-partition f will be called equatorial if min p∈P f(p)=0and for every j∈[2,r]there exists a covering relation between ranks j−1,jCHARNEY-DAVIS AND NEGGERS-STANLEY CONJECTURES9Figure 1.(a)A graded poset P .(b)The distributivelattice of order ideals J (P ).(c)Part of the canonicaltriangulation ∆J (P )of its order polytope O (P ).(d)The analogous part of the equatorial triangulation.(e)The equatorial 1-sphere ∆eq (P ).in P along which f is constant,i.e.there exist p j −1<P p j withp j −1∈P j −1,p j ∈P j and f (p j −1)=f (p j ).An order ideal I in P will be called rank-constant (resp.equatorial )if its characteristic vector χI is rank-constant (resp.equatorial).More generally,a collection of ideals {I 1,...,I t }forming a chain I 1⊂...⊂I t will be called rank-constant (resp.equatorial )if the sum χI 1+...+χI t (or equivalently,any vector in the cone pos({χI j }t j =1)is rank-constant (resp.equatorial).Note that the only rank-constant ideals are the ones in the chain∅=I rc 0⊂I rc 1⊂···⊂I rc r =P10VICTOR REINER AND VOLKMAR WELKERwhere I rc j:=⊔i≤j P i.Also note that the only P-partition which is both rank-constant and equatorial is the zero P-partition f(p)=0.Thus the only rank-constant and equatorial order ideal is I rc0=∅. Proposition3.3.Every non-zero P-partition f can be uniquely ex-pressed asf=f rc+f eq,where f rc,f eq are rank-constant and equatorial P-partitions,respec-tively.Proof.To show existence,for j∈[r−1]define non-negative constantsc j:=min{f(p j−1)−f(p j):p j−1∈P j−1,p j∈P j,p j−1<P p j}c r:=min{f(p r):p r∈P r},and setf rc:=r j=1c jχI rc jf eq:=f−f rc.Obviously f rc is a rank-constant P-partition.It is a straightforward verification,left to the reader,that f eq is a P-partition,and that it is equatorial by construction.For uniqueness,assume f=g rc+g eq is an additive decomposi-tion of f into a rank-constant and an equatorial P-partition.It is again straightforward to show that the equatoriality of g eq and rank-constancy of g rc forces g rc= r j=1c jχI rc j,where c j is defined as above in terms of f.We wish to deduce our equatorial triangulation of A(P)from Propo-sition3.3,and for this we need to understand both rank-constant and equatorial chains of ideals better.Equatoriality and rank-constancy of a chain of ideals I1⊂...⊂I t are intimately related with properties of its jumpsJ i:=I i−I i−1for i=1,...,t+1(where by convention I0:=∅,I t+1=P).It is easy to see that the rank-constant P-partitions form an r-dimensional simplicial subcone within the n-dimensional cone A(P), and that this subcone is the non-negative span of the vectors{χI rcj}r j=1. Proposition3.4.The rank-constant subcone of A(P)is interior,that is,it does not lie in the boundary subcomplex of the cone A(P). Proof.In a triangulation of a polyhedral cone,a subcone lies on the boundary if and only if it is contained in a codimension one subcone that lies on the boundary.For codimension one subcones,lying in theCHARNEY-DAVIS AND NEGGERS-STANLEY CONJECTURES11 boundary is equivalent to being contained in a unique top dimensional subcone.Specializing to the case of the canonical triangulation of the cone A(P)from Proposition2.1,one sees that this means a chain of ideals I1⊂···⊂I t corresponds to a subcone on the boundary if and only if one of at least one of its jumps J i contains a pair of elements which are comparable in P.But for I rc1⊂···⊂I rc r,since the jumps J i=I rc i−I rc i−1=P i are antichains,this property fails to hold. Proposition3.5.A chain of non-empty ideals I1⊂...⊂I t,is equa-torial if and only if its jumps J i have the following property:For every j∈[2,r],there exist p j−1<P p j with p j−1∈P j−1,p j∈P j and a value i∈[t+1],such that p j−1,p j∈J i.The chain I1⊂...⊂I t is maximal with respect to the equatorial property if and only if its jumps J i for i∈[t+1]satisfy the following two conditions:(i)The J i are all maximal(saturated)chains in P,possibly single-tons.(ii)The non-singleton J i can be re-ordered J i1,J i2,...,J isso thatmin Ji1has rank1,max Ji shas rank r,and max J ik,min J ik+1have the same rank in P for k∈[s−1].Consequently,t=n−r for any maximal equatorial chain of non-empty ideals.Proof.Since the jumps J i are the domains on which the associated P-partitionχI1+...+χItis constant,thefirst assertion is direct fromDefinition3.2.It is then easy to see that a chain of non-empty ideals having proper-ties(i),(ii)will be equatorial,and maximal with respect to refinement.Conversely,suppose one is given a maximal equatorial chain of non-empty ideals.If there exists an incomparable pair p,p′in one of its jumps J i,it is straightforward to check that one can refine the chainfurther while preserving the equatorial property,e.g.by adding in theideal I i−1∪{q∈J i:q≤p}.Thus each jump J i must be a maximalchain,proving(i).Furthermore,the pairs of adjacent ranks{j−1,j} spanned by two different jumps J i,J i′must be disjoint,else one could refine the chain equatorially by“breaking”J i between two such ranks{j−1,j}which they share.The jumps J i must then disjointly coverall possible adjacent rank pairs{j−1,j}r j=2,so they can be re-ordered as in(ii). Proposition3.6.The collection of all conespos {χI:I∈R∪E} ,12VICTOR REINER AND VOLKMAR WELKERwhere R(resp.E)is a chain of non-empty rank-constant(resp.equa-torial)ideals in P,gives a unimodular triangulation of the cone of P-partitions A(P).Proof.First we check that these polytopal cones indeed decompose A(P).Given f∈A,write f=f rc+f eq as in Proposition3.3.Then use these easy facts:⊲f rc lies in the cone of rank-constant P-partitions,which is thesimplicial cone positively spanned by the(non-empty)rank-constant ideals{I rc j}r j=1,⊲When f eq is expressed in the unique way as a positive combina-tion of characteristic vectors of a chain of ideals,as in Propo-sition2.2part(i),this chain of ideals must be equatorial sincef eq is.It remains to check that all such cones are unimodular.Thus it suffices to show that whenever R∪E is maximal under inclusion,then #R∪E=n and the Z-span of the set{χI:I∈R∪E}additively generates inside R n is the full integer lattice Z n.To see#R∪E= n,first note that when R∪E is maximal,one has R={I rc j}r j=1, and then#E=n−r follows from Proposition3.5.To show they additively generate Z n,we show by induction on the rank r of P that the subgroup they generate contains each standard basis vector e p for p∈P.The base case r=1has P an antichain,hence all ideals I P are equatorial,so the cones in question coincide with the cones in the canonical triangulation,which are unimodular by Proposition2.1.In the inductive step,note that this subgroup generated by{χI:I∈R∪E}has the alternate description as the subgroup generated bythe characteristic vectorsχPj of all of the ranks of P along with thecharacteristic vectorsχJi of all of the jumps between the equatorialideals in E.Proposition3.5shows that there will be exactly one element q of the top rank r in P which does not occur in a singleton jump J i.Namely,q=max J is after the re-labelling as in Proposition3.6.Hencefor every p∈P r−{q},one has e p in the subgroup,but then one alsohas e q in the subgroup,since the subgroup containsχPr .Now applyinduction to the graded poset P−P r of rank r−1,replacing the ideals in R∪E by their intersections with P−P r and removing multiple copies of the same ideal created by the intersection process. The triangulation of A(P)given in Proposition3.6induces a uni-modular triangulation of O(P),which we will call the equatorial trian-gulation of O(P).CHARNEY-DAVIS AND NEGGERS-STANLEY CONJECTURES13 Definition3.7.The equatorial complex∆eq(P)is defined to be the subcomplex of the order complex∆J(P)whose faces are indexed by the equatorial chains of non-empty ideals.For the formulation of the next corollary we need the concept of simplicial join.For two simplicial complexes∆1,∆2which are defined over disjoint vertex sets,the simplicial join∆1∗∆2is the simplicial complex{σ1∪σ2:σi∈∆i,i=1,2}.Note that we always assume that the empty face∅is a face of a simplicial complex.Corollary3.8.The equatorial triangulation of the order polytope O(P) is abstractly isomorphic to the simplicial joinσr∗∆eq(P),whereσr is the interior r-simplex spanned by the chain of rank-constant ideals {I rc j}r j=0.As a consequence of its unimodularity,one hash(∆eq(P),t)=h(∆J(P),t)=W(P,t).Proof.Thefirst assertion follows directly from Proposition3.6,noting thatσr is interior due to Proposition3.4.For the second,note that bothσr∗∆eq(P)and∆J(P)index unimodular triangulations of the order polytope,so(2.2)impliesh(σr∗∆eq(P),t)=h(∆J(P),t).On the other hand,the defining equation(1.1)of the h-polynomial shows thatf(∆1∗∆2,t)=f(∆1,t)∗f(∆2,t)h(∆1∗∆2,t)=h(∆1,t)∗h(∆2,t)h(σr,t)=1,and hence h(σr∗∆,t)=h(∆,t). Remark3.9.Corollary3.8has the following consequence:for a graded poset P, the set of linear extensions L(P)is equinumerous with the set L eq(P) of all maximal equatorial chains of ideals in P,as both coincide with [W(P,t)]t=1.This begs for a bijectionφ:L(P)→L eq(P).The authors thank Dennis White[53]for supplying one which is elegant,using the idea of jeu-de-taquin on linear extensions of P,thought of as P-shaped tableaux that use each entry1,2,...,n exactly once.Given such a linear extension w,replace the highest label n(at top rank r)by a jeu-de-taquin hole,and slide it past other entries down to rank1,du-plicating the last entry that it slid past in the hole’s resting position at rank1.Then repeat this with the entry n−1,sliding it down to rank2,and similarly with the entries n−2,n−3,...,n−r+1.The14VICTOR REINER AND VOLKMAR WELKERresult is a P-shaped tableaux that can be interpreted as an equato-rial P-partition,compatible with a unique maximal equatorial chain of idealsφ(w).It is not hard to check that this map w→φ(w)is a bijection.3.2.Geometric and Convexity Properties of∆eq(P).In this sec-tion,we use convexity and the concrete geometric realization of∆eq(P) to learn more about it.Definition3.10.The rank-constant subspace V rc⊂R n is the R-linear}r j=1.span of the set{χI rcjLet Q be a convex polytope,and V a linear subspace,both inside R n.Then there is a well defined quotient polytopeQ/V:={q+V:q∈Q}⊂R n/V.Ifπ:R n→R n−dim V is any linear surjection with kernel V(such as an orthogonal projection onto V⊥),then the polytope Q/V can be identified with the imageπ(Q).Also note that if V is a rational subspace of R n with respect to the integer lattice Z n⊂R n,the quotient lattice Z n/(V∩Z n)is well-defined,and a full rank sublattice in R n/V. Proposition3.11.The collection of quotient conesC E=pos {χI:I∈E} +V rc ,as E runs through all equatorial chains of non-empty ideals in P,forms a complete simplicial fan in R n/V rc.(i)This simplicial fan is unimodular with respect to the quotientlattice Z n/(V rc∩Z n).(ii)The simplices(C E∩O(P))+V rc form a unimodular triangula-tion of the quotient polytope O eq(P):=O(P)/V rc.(iii)This triangulation of O(P)/V rc is isomorphic,as an abstract simplicial complex,to the cone0∗∆eq(P)with base∆eq(P)andapex at the interior point0=V rc.Consequently,∆eq(P)triangulates the(n−r−1)-dimensional bound-ary sphere∂O eq(P).Proof.Apply the following general statement,Proposition3.12,about polytopes(and the analogous statement about fans)withQ=O(P),∆=the equatorial triangulation,∆′=∆eq(P),V=V rc.CHARNEY-DAVIS AND NEGGERS-STANLEY CONJECTURES15Proposition3.12.Let Q be an n-dimensional convex polytope in R n. Assume Q has a triangulation abstractly isomorphic to a simplicial complex∆of the form∆∼=σr∗∆′,whereσr is an r-simplex not lying on the boundary of Q.Let V be the r-dimensional linear subspace parallel to the affine span of the vertices ofσr.Then the quotient(n−r)-dimensional polytope Q/V⊂R n/V in-herits a triangulation abstractly isomorphic toσ0∗∆′,whereσ0is an interior point of Q/V⊂R n/V.Furthermore,when V is rational with respect to Z n⊂R n and if the triangulation of Q is unimodular with respect to Z n,then the triangu-lation of Q/V rc is unimodular with respect to Z n/(V rc∩Z n).The proof of Proposition3.12is straightforward.We leave it as an exercise.Proposition3.11,shows that∆eq(P)corresponds to a complete uni-modular fan.This fact suffices to infer both that it is spherical,andthat it corresponds to a smooth,complete toric variety X∆eq(P)(see[18,§2.1]).Our next goal will be to show that∆eq(P)corresponds to a polytopal fan,as this has multiple consequences;see Corollary3.15 below.We prove polytopality of∆eq(P)by choosing for each equatorial ideal I of P a point on its ray pos(χI+V rc)so that the convex hull of all such points is a simplicial polytope having∆eq(P)as its boundary complex.Here we employ the following strategy.We start with the (usually)non-simplicial polytope O eq(P)and pull each of its vertices in a certain order to produce a simplicial polytope with boundary complex ∆eq(P).Recall[31,§2.5]that if Q is a convex polytope,one pulls the vertex v in Q to produce a new polytope pull v(Q)by taking the convex hull after moving v slightly outward past the supporting hyperplanes of all facets that contain v,but past no other facet-supporting hyperplanes of Q.Assuming that Q contains the origin in its interior,this can clearly be achieved by replacing v with(1+ǫ)v whereǫ>0is sufficiently small.We will require the following proposition describing the1-skeleton resulting from pulling all the vertices of a polytope: Proposition3.13.Let Q be the polytope resulting from pulling all of the vertices of a polytope Q in some order v1,v2,...,and let v i denote the corresponding vertices in Q.。

抵抗天赋的诱惑(记贝索斯在普林斯顿大学2010年学士毕业典礼上的演讲) 我一直相信每一个人都有自己的天赋，每一个人的存在都代表着宇宙空间中的一种唯一，然而令我经常都在深思的是，既然我们都是这样的独特，又为何偏偏要去模仿和畸变成拥有同类“基因”的人呢？为什么我们中的很多人都不愿意去追逐属于自己的理想，或者不能为此奋斗一生呢，抑或者一生都是在自欺欺人的辩解？在Randy的The Last Lecture中我深深的感受到了一个人追逐自己最初理想的意义会变得如此的伟大，充满的是一种人生最大的和最根本的价值。

一直在想这样的一个问题，当社会尚且艰难，生活尚且苦难的日子里都有如此多人在追逐属于自己梦想的时候；在一个生活舒适，物质条件优越的年代我们竟然不知所措的迷失掉自己的方向，找不到自己前行的路。

这是多么可悲和可笑的一种境况！我们，有了更高的天赋，有了更好的环境，却因为有更多的选择而抹杀了我们自己的梦...这确实让人觉得不可思议！我相信每个人都有自己最初的梦想，在这样的一个年代，在这样一个至少没有饥寒交迫的时代，我坚信追逐自己理想的人会获得生命尽头最高贵的礼物和人生最大的价值！记：在一个可以实现最初梦想的时代选择不可以的沉默必将是这个时代最损失的损失，也必将是生活在这个时代的人最遗憾的遗憾...附：抵抗天赋的诱惑（贝索斯在普林斯顿大学2010年学士毕业典礼上的演讲）中文译稿：在我还是一个孩子的时候，我的夏天总是在德州祖父母的农场中度过。

我帮忙修理风车，为牛接种疫苗，也做其它家务。

每天下午，我们都会看肥皂剧，尤其是《我们的岁月》。

我的祖父母参加了一个房车俱乐部，那是一群驾驶Airstream拖挂型房车的人们，他们结伴遍游美国和加拿大。

每隔几个夏天，我也会加入他们。

我们把房车挂在祖父的小汽车后面，然后加入300余名Airstream探险者们组成的浩荡队伍。

我爱我的祖父母，我崇敬他们，也真心期盼这些旅程。

那是一次我大概十岁时的旅行，我照例坐在后座的长椅上，祖父开着车，祖母坐在他旁边，吸着烟。

理论物理学的自学书单(总2页)--本页仅作为文档封面，使用时请直接删除即可----内页可以根据需求调整合适字体及大小--理论物理学的自学书单注：课程列表，按逻辑顺序（并非所有内容都一定要按照此表列来进行，但此列表大概说明了这些不同课程之间的逻辑关系。

）一、（这些是针对初学者的。

某些专题是实际是作为整个课程来学习的。

这些内容的大部分是物理理论的非常重要的组成。

你没有必要先要学习完全部内容才开始后续课程，但要记住要回来完成那些第一轮学习时漏掉的内容。

）1.语言：英语是一个先决条件。

如果你还没有掌握它，下功夫学吧。

你必须能够读、写、说及理解英语（要做好的科研，英语是必需的工具），但不必要达到最好。

所有出版物都是英语的。

注意能够用英语写作的重要性。

迟早，你将希望发表自己的结果，而人们必须能够读懂并理解你的内容。

法语、德语、西班牙语和意大利语或许有用，但他们不是必须的。

它们不是摩天大厦的地基，所以不必要担心。

你的确需要希腊字母。

希腊字母用得非常多。

学会它们的名字，否则当你口头表达时会把自己弄糊涂。

现在开始点严肃的内容。

不要抱怨这些东西看起来很多。

诺贝尔奖不是靠吹灰之力就能获得的，并且要记住，所有这些东西加起来至少需要我们学生五年的强化学习2.基础数学数字、加法、减法、平方根等等。

自然数、整数、有理数、实数、复数。

集合论：开集，紧致空间，拓扑。

3.代数方程近似处理。

级数展开：泰勒级数。

解带复数的方程。

三角函数，等等。

4.无穷小量微分。

基本函数的微分。

积分。

基本函数的积分。

微分方程组。

线性方程组。

傅立叶（Fourier）变换。

复数的使用。

级数收敛。

5.复平面柯西（Cauchy）定理和路径积分。

Gamma 函数。

高斯（Gaussian）积分。

概率论。

6.偏微分方程狄里克雷（Dirichlet）和纽曼（Neumann）边界条件。

二、1.经典力学静力学（力，张量）；流体力学。

牛顿定律。

行星的椭圆轨道。

多体系统。

最小作用量原理（Least Action Principle）。

Quantum FieldsN.N.BogoliubovD.V.ShirkovJo i nt Institutefor Nuclear ResearchDubna,U.S.S.R.Authonrized translation from the Russian edition byD.B.Pontecorvo1983The Benjamin Cummings Publishing Company,Inc.Advanced Book ProgramReading,MassachusettsLondon Amsterdam Don Mills Ontario Sydney Tokyoviii Contents20.The Feynman rules i n the p-representation184201Transition to the momentum representation18420:2Feynman 's rules for the evaluation of elements18820.3An illustration forthe model18720.4Spinor electrodynamics18921.Transition probabilities19321.1The general structure of matrix elements19321.2Normalization of the state amplitude19521.3The general fonnula for transition probability19721.4Scattering of two particles19921.5The two-particle decay202Chapter VI.Evaluation of Integrals and Divergences20322.The method for evaluating integrals20322.1Integrals over virtual momenta20322.2The a-representation and Gaussian quadratures20422.3Feynman's parametrization20722.4Ultraviolet divergences20923.Auxiliary regularizations21023.1The necessity of regularization21023.2Pauli-Villars regularization21123.3D i mensionai regularization21323.4Regularization by means of a cutoff21524.One -loop diagrams21624.1The scalar "fish"21624.2Self-energies of the photon and of the electron21824.3Triangular diagrams22124.4Ultraviolet divergences of higher orders 22325.Isolation of the divergences22525.1The structure of one-loop divergences22525.2The contribution to the S-matrix22625.3Counterterms and renormalizations22925.4Divergences and distributions231Chapter VII .Removal of Divergences23326.The general structure of divergences23326.1divergences23326.2The connection with countertenns and renormalizations23626.3The degree of divergence of diagrams23826.4The property of renonnalizabilit 24027.Dressed Green's functions24227.1Propagators of physical fields24227.2Higher Green 's functions 245Matrix the vertex Higher-orderContents xi "September"Assignment(for Chapter I)365 "October"Assignment(for Chapter II)369 "November"Assignment(for Chapter III)372 "December"Assignment(for Chapter IV)375 "February"Assignment(for Chapter V)376 "March"Assignment(for Chapter VI)378 "April"Assignment(for Chapter VII)379Recommendations for Use381References383 Subject Index..385PREFACEThe main purpose of this book is to provide graduate students in physicswith the necessary minimum of infonnation on the fundament~sof modem quantum field theory.It may tum out to be sufficient both for theoreticians,specializing in nuclear physics,quantum statistics and other fields,in which quantum field methods are utilized and which are based on quantum concepts,and also for experimental physicists in the fields of nuclear and high-energy physics.For the latter category of readers the present book should be supplemented by a course on particle physics and particle interactions.At the same time the book can be recommended as an introductury text for persons intending to work in the field of quantum field theory and of the theory of elementary interactions.The material in this book corresponds to a course lasting one academic year.Our personal experience testifies that parallel practical studies at seminars are extremely desirable.For this purpose part of the technical material has been assembled at the end of the book in the fo rm of Appendices.There,also,sets of exercises and problems,gathered together as assignments corresponding to chapters of the main text,are given.The authors are grateful to the editor of this book D.A.Slavnov,to thereviewers M.A.Brown,L.V.Prokhorov,K.A.Ter~Martirosyan,and also to B.M.Barbashov,B.V.Medvediev,and N.M.Shumeiko for valuable comments on the typescript of the book.N.N.BogoliubovD.V.Shirkovfundamentals Ter-MartirosyanPREFACE TO THEENGLISH-LANGUAGE EDITION This book is a text on the fundamentals of quantum field theory and renormalized perturbation theory(RPT).The traditional field of application of the latter for a long time was limited to quantum electrodynamics.During recent years,due to the creation of a unified theory of electroweak interac-tions and to the successes of quantum chromodynamics it has become clear that the physical scope of RPT is much wider.However,in the study of the quark-gluon interaction,as well as of possible mechanisms of the grand unification of interactions,a decisive partis played by the simultaneous use of results of RPT and of the apparatus of the renormalization-group method.Therefore we have written a special Appendix IX,"The renormaliza-tion group,"for the English-language edition of our book.Besides this,small editorial changes and corrections of noticed misprints have been made.N.N.BogoliubovD.V.Shirkovxv。