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1980-, Blazquez-Sanz David, Morales Ruiz, Juan J. (Juan José), 1953-, and Lombardero Jesus Rodriguez 1961-, eds. Symmetries and related topics in differential and difference equations: Jairo Charris Seminar 2009, Escuela de Matematicas, Universidad Sergio Arboleda, Bogotá, Colombia. Providence, R.I: American Mathematical Society, 2011.
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Barnes, Elizabeth. Symmetric Dependence. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198755630.003.0003.
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Metaphysical orthodoxy maintains that the relation of ontological dependence is irreflexive, asymmetric, and transitive. The goal of this paper is to challenge that orthodoxy by arguing that ontological dependence should be understood as non-symmetric, rather than asymmetric. A series of cases across a wide range of ontological commitments are presented, and it is argued that each case should be understood as one in which the relation of dependence holds symmetrically. If these arguments work, however, they provide reasons to be skeptical of the way in which contemporary discussions typically lump dependence together with relations such as grounding and in virtue of, which arguably need to be understood as asymmetric. If the asymmetry of dependence is relinquished, interesting things follow for what can be said about metaphysical explanation—particularly for the prospects of explanatory holism.
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Bacon, Andrew. Relative Locations. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198828198.003.0002.
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The fact that physical laws often admit certain kinds of space-time symmetries is often thought to be problematic for substantivalism—the view that space-time is as real as the objects it contains. The most prominent alternative, relationism, avoids these problems but at the cost of giving abstract objects (rather than space-time points) a pivotal role in the fundamental metaphysics. This incurs related problems concerning the relation of the physical to the mathematical. This paper presents a version of substantivalism that respects Leibnizian theses about space-time symmetries, and argues that it is superior to both relationism and the more orthodox form of substantivalism.
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Thompson, Naomi. Metaphysical Interdependence, Epistemic Coherentism, and Holistic Explanation. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198755630.003.0006.
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This paper argues for an alternative to orthodox foundationalist accounts of metaphysical structure as characterized by grounding relations. There are good reasons to take grounding to be a non-symmetric (rather than an asymmetric) relation, and to take facts to be related in complex networks of ground. These networks are closely analogous to the networks of justified beliefs characteristic of coherentism about justification. This position is called metaphysical interdependence. The chapter argues that grounding is an explanatory relation (rather than merely a relation which backs explanations), and that there are good reasons to resist the contention that explanation is asymmetric, including cases seemingly best described as cases of holistic metaphysical explanation. Metaphysical interdependence is a promising account of the metaphysical structure of reality.
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Isett, Philip. The Divergence Equation. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691174822.003.0006.
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This chapter introduces the divergence equation. A key ingredient in the proof of the Main Lemma for continuous solutions is to find special solutions to this divergence equation, which includes a smooth function and a smooth vector field on ³, plus an unknown, symmetric (2, 0) tensor. The chapter presents a proposition that takes into account a condition relating to the conservation of momentum as well as a condition that reflects Newton's law, which states that every action must have an equal and opposite reaction. This axiom, in turn, implies the conservation of momentum in classical mechanics. In view of Noether's theorem, the constant vector fields which act as Galilean symmetries of the Euler equation are responsible for the conservation of momentum. The chapter shows proof that all solutions to the Euler-Reynolds equations conserve momentum.
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Glanville, Peter John. Symmetry. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198792734.003.0005.
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Chapter 5 determines the semantic typology of patterns III and VI, sometimes termed the vowel-lengthening patterns. It asserts that verbs formed in these patterns are symmetrical predicates, denoting relations consisting of two complementary forces. It shows that the difference between the two patterns results from the interplay between an underlying symmetric relation and a figure–ground orientation in which one of the participant roles involved is made more prominent than the other. The chapter divides verbs formed in pattern III into verbs of resistance, risk, competition, interaction, and co-action, and those formed in pattern VI into reciprocal verbs, feigning verbs, chaining verbs, and verbs of progressive change. It argues that an account based on a common symmetric structure is able to unite this diverse range of verbs within one analysis, and it offers data from other languages to support this claim.
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Yli-Vakkuri, Juhani, and John Hawthorne. Rationality and narrow content. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198785965.003.0005.
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In Chapter 4 we ask whether a range of epistemological properties and relations pertaining to apriority can be explained by any kind of narrow content, and we come to a pessimistic conclusion. Thought experiments involving certain symmetries across space and/or time again play a starring role. We argue that, if the kinds of symmetries that feature in those thought experiments are possible, then narrow content cannot be used for explaining a priority and related properties and relations.
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Dixon, T. Scott. Plural Slot Theory. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198828198.003.0006.
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Slot theory (Gilmore 2013) bears the full brunt of Fine’s (2000) symmetric completions and conflicting adicities problems. This chapter develops an alternative, plural slot theory (or pocket theory), which avoids these problems, key elements of which are first considered by Yi (1999). Like the slot theorist, the pocket theorist posits entities (pockets) in properties and relations that can be occupied. But unlike the slot theorist, the pocket theorist denies that at most one entity can occupy any one of them. As a result, she must also deny that the adicity of a property or relation is equal to the number of occupiable entities in it. By abandoning these theses, however, the pocket theorist is able to avoid Fine’s problems, resulting in a stronger theory about the internal structure of properties and relations.
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Yli-Vakkuri, Juhani, and John Hawthorne. Truth-conditionality. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198785965.003.0003.
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In Chapter 2 we argue that internalists are committed to a kind of relativism, and that theirs is a particularly radical form of relativism. Thought experiments involving certain symmetries across space and/or time play a starring role. If the kinds of symmetries featured in them are possible, we argue, the truth values of narrow content must be relative to some very unusual parameters.
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Simon, Jonathan. Fragmenting the Wave Function. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198828198.003.0004.
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This paper develops and defends a new account of B-theoretic endurantism and a new account of the metaphysics of the quantum state, and highlights the parallels between the considerations that motivate them. These new accounts are both fragmentalist, in the sense that they follow Fine (2005) in invoking a symmetric coordination relation between facts, such that facts that are pairwise incompatible (like Hugh?s being happy and Hugh?s being sad) can both obtain provided that they are not related by this relation. However, while Fine allows that fragments can be logically incoherent—P can obtain in one fragment while ØP obtains in another—the fragmentalist accounts defended here are motivated even if we insist on logical coherence between fragments.
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Wigglesworth, John. Grounding in Mathematical Structuralism. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198755630.003.0012.
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The grounding relation is thought to have certain structural properties: irreflexivity, asymmetry, transitivity, and well-foundedness. This paper examines a putative case of grounding that serves as a counterexample to almost all of these properties. The example comes from non-eliminative mathematical structuralism, some versions of which argue that mathematical objects depend in some sense on the structure to which they belong, and on the other objects in that structure. Such claims generate prima facie cases of symmetric, reflexive, and non-well-founded dependence. The paper argues that this dependence constitutes a grounding relation in the structuralist case. It then argues that these dependence claims can be given a modal interpretation, and that under this interpretation the dependence claims, and therefore the associated grounding claims, are true. It follows that these cases from mathematical structuralism constitute genuine counterexamples to many of the structural properties traditionally thought to hold of the grounding relation.
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Iliopoulos, John. A Problem of Mass. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198805175.003.0004.
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This chapter examines the constraints coming from the symmetry properties of the fundamental interactions on the possible values of the masses of elementary particles. We first establish a relation between the range of an interaction and the mass of the particle which mediates it. This relation implies, in particular, that long-range interactions are mediated by massless particles. Then we argue that gauge invariant interactions are long ranged and, therefore, the associated gauge particles must have zero mass. Second, we look at the properties of the constituents of matter, the quarks and the leptons. We introduce the notion of chirality and we show that the known properties of weak interactions, combined with the requirement of gauge invariance, force these particles also to be massless. The conclusion is that gauge symmetries appear to be incompatible with massive elementary particles, in obvious contradiction with experiment. This is the problem of mass.
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Horing, Norman J. Morgenstern. Non-Equilibrium Green’s Functions: Variational Relations and Approximations for Particle Interactions. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198791942.003.0009.
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Chapter 09 Nonequilibrium Green’s functions (NEGF), including coupled-correlated (C) single- and multi-particle Green’s functions, are defined as averages weighted with the time-development operator U(t0+τ,t0). Linear conductivity is exhibited as a two-particle equilibrium Green’s function (Kubo-type formulation). Admitting particle sources (S:η,η+) and non-conservation of number, the non-equilibrium multi-particle Green’s functions are constructed with numbers of creation and annihilation operators that may differ, and they may be derived as variational derivatives with respect to sources η,η+ of a generating functional eW=TrU(t0+τ,t0)CS/TrU(t0+τ,t0)C. (In the non-interacting case this yields the n-particle Green’s function as a permanent/determinant of single-particle Green’s functions.) These variational relations yield a symmetric set of multi-particle Green’s function equations. Cumulants and the Linked Cluster Theorem are discussed and the Random Phase Approximation (RPA) is derived variationally. Schwinger’s variational differential formulation of perturbation theories for the Green’s function, self-energy, vertex operator, and also shielded potential perturbation theory, are reviewed. The Langreth Algebra arises from analytic continuation of integration of products of Green’s functions in imaginary time to the real-time axis with time-ordering along the integration contour in the complex time plane. An account of the Generalized Kadanoff-Baym Ansatz is presented.
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Okasha, Samir. 3. Explanation in science. Oxford University Press, 2013. http://dx.doi.org/10.1093/actrade/9780192802835.003.0003.
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What exactly is scientific explanation? ‘Explanation in science’ begins with Carl Hempel's covering law model of explanation, which says that to explain a phenomenon is to show that its occurrence follows deductively from a general law, perhaps supplemented by other laws and/or particular facts, all of which must be true. This model does not deal with symmetry or irrelevance. The covering law model implies that explanation should be a symmetric relation, but in fact it is asymmetric. Also, a good explanation of a phenomenon should contain information that is relevant to the phenomenon's occurrence. Causality-based accounts of scientific explanation and the concepts of reduction and multiple realization are also explained.
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Mann, Peter. Hamilton’s Principle in Phase Space. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0015.
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This chapter derives Hamilton’s equations using the Legendre transform and the definition of the Hamiltonian function. While, in the Newtonian formalism, conservation laws were rather difficult to tease out, the Lagrangian formalism revolutionised the way of looking at them; however, the Hamiltonian formalism is perhaps even simpler than the Lagrangian formalism, making it straightforward to identify conservation laws and the symmetries of the system associated with each conserved property. In this chapter, the Hamiltonian is treated as being explicitly dependent on time, as this form is more general and will lead to an important relation that, although not an equation of motion, is still useful to discuss. The chapter also introduces Routhian mechanics as a symplectic reduction technique, using integrals of the motion.
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Levin, Frank S. Quantum Boxes, Stringed Instruments. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198808275.003.0008.
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Chapter 7 illustrates the results obtained by applying the Schrödinger equation to a simple pedagogical quantum system, the particle in a one-dimensional box. The wave functions are seen to be sine waves; their wavelengths are evaluated and used to calculate the quantized energies via the de Broglie relation. An energy-level diagram of some of the energies is constructed; on it are illustrations of the corresponding wave functions and probability distributions. The wave functions are seen to be either symmetric or antisymmetric about the midpoint of the line representing the box, thereby providing a lead-in to the later exploration of certain symmetry properties of multi-electron atoms. It is next pointed out that the Schrödinger equation for this system is identical to Newton’s equation describing the vibrations of a stretched musical string. The different meaning of the two solutions is discussed, as is the concept and structure of linear superpositions of them.
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Haesemeyer, Christian, and Charles A. Weibel. The Norm Residue Theorem in Motivic Cohomology. Princeton University Press, 2019. http://dx.doi.org/10.23943/princeton/9780691191041.001.0001.
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This book presents the complete proof of the Bloch–Kato conjecture and several related conjectures of Beilinson and Lichtenbaum in algebraic geometry. Brought together here for the first time, these conjectures describe the structure of étale cohomology and its relation to motivic cohomology and Chow groups. Although the proof relies on the work of several people, it is credited primarily to Vladimir Voevodsky. The book draws on a multitude of published and unpublished sources to explain the large-scale structure of Voevodsky's proof and introduces the key figures behind its development. It proceeds to describe the highly innovative geometric constructions of Markus Rost, including the construction of norm varieties, which play a crucial role in the proof. It then addresses symmetric powers of motives and motivic cohomology operations. The book unites various components of the proof that until now were scattered across many sources of varying accessibility, often with differing hypotheses, definitions, and language.
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Kaufman, Daniel. Lexical Category and Alignment in Austronesian. Edited by Jessica Coon, Diane Massam, and Lisa Demena Travis. Oxford University Press, 2017. http://dx.doi.org/10.1093/oxfordhb/9780198739371.013.24.
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Philippine-type languages are often cited as exemplifying a cross-linguistically unique voice system, in which verb morphology can select not only an agent or patient, but also locative, instrumental and other adjunct type relations as the nominative argument. In this paper, we examine three approaches to this typologically remarkable system: the ergative analysis, the case agreement analysis and the nominalization analysis, arguing for the latter based on strong parallels between verbal and nominal predication from the root level to the clause level. The morphologically symmetric nature of Philippine-type languages is argued to stem from their nominal roots. The historical development of verbal roots leads to a more fixed argument structure in which canonical ergative languages develop. Mamuju, an Austronesian language of West Sulawesi, Indonesia, is offered as an example of a classically ergative language, in contrast to Philippine-type systems.
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Rabin, Gabriel Oak. Grounding Orthodoxy and the Layered Conception. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198755630.003.0002.
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Ground offers the hope of vindicating and illuminating a classic philosophical idea: the layered conception, according to which reality is structured by relations of dependence, with physical phenomena on the bottom, and upon which chemical, then biological, and finally psychological and other phenomena reside. However, ground can only make good on this promise if it is appropriately formally behaved. The paradigm of good formal behavior can be found in the currently dominant grounding orthodoxy, which holds that ground is transitive, anti-symmetric, irreflexive, and foundational. However, heretics have recently challenged the orthodoxy. This paper examines ground’s ability to vindicate the layered conception upon various relaxations of the orthodox assumptions. It is argued that highly unorthodox views of ground can still vindicate the layered conception and that, in some ways, the heretical views enable ground to better serve as a guide to reality’s layering than do orthodox views of ground.
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Rickles, Dean. Spaces. Edited by Paul Humphreys. Oxford University Press, 2016. http://dx.doi.org/10.1093/oxfordhb/9780199368815.013.31.
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The concept of space has many distinct uses in science. Not only does it denote the three-dimensional physical space we walk through (and our mental representations thereof), but also abstract spaces of various kinds and higher dimensionality. Spaces provide a means of systematically and exhaustively representing possible distinct states of physical or abstract systems, allowing one to chart the motions, relationships, and other qualities that they might undergo, enter into, or possess. Such spaces can encode the possibilities of physical systems relative to laws of nature allowing us both to probe modal aspects of the world and to discover symmetries and redundancies in a theory (identifying “intrinsic structure”). This chapter reviews these various elements, giving many examples from distinct fields and attempts to draw some broad lessons on the significance of this more general concept of space.
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Crysmann, Berthold. Inferential-realizational morphology without rule blocks. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198712329.003.0008.
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The chapter outlines a formal theory of inferential-realizational morphology that eliminates (ordered) rule blocks. I show that rule blocks not only stand in the way of a more general treatment of variable morphotactics, but that they also artificially restrict the scope of Pāṇinian competition, effectively ruling out operation at a distance. Instead, it argues for a purely information-based model of global competition that reconciles competition with extended exponence by means of a distinction between realization and allomorphic conditioning. It shows, in particular, that arbitrary decisions with respect to this distinction can be eliminated, once Carstairs’s (1987) notion of Pure Sensitivity has been turned into a formal principle of our theory. Finally, the chapter shows how Information-based Morphology can account for symmetric cases of extended exponence by simultaneous introduction of exponents since the theory is able to capture many-to-many relations between form and function at the level of individual rules.
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Franzese, Robert J., and Jude C. Hays. Empirical Models of Spatial Inter‐Dependence. Edited by Janet M. Box-Steffensmeier, Henry E. Brady, and David Collier. Oxford University Press, 2009. http://dx.doi.org/10.1093/oxfordhb/9780199286546.003.0025.
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This article discusses the role of ‘spatial interdependence’ between units of analysis by using a symmetric weighting matrix for the units of observation whose elements reflect the relative connectivity between unit i and unit j. It starts by addressing spatial interdependence in political science. There are two workhorse regression models in empirical spatial analysis: spatial lag and spatial error models. The article then addresses OLS estimation and specification testing under the null hypothesis of no spatial dependence. It turns to the topic of assessing spatial lag models, and a discussion of spatial error models. Moreover, it reports the calculation of spatial multipliers. Furthermore, it presents several newer applications of spatial techniques in empirical political science research: SAR models with multiple lags, SAR models for binary dependent variables, and spatio-temporal autoregressive (STAR) models for panel data.
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Wittman, David M. Time Dilation and Length Contraction. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780199658633.003.0007.
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In Chapter 6, we discovered that different frames mix time and space differently; in this chapter, we will discover that speed causes time to run slowly and space to contract. Of course, the truth is a bit subtler than that, because when two frames are in relative motion each frame measures the other as the high‐speed frame where time runs slowly and space contracts.We deduce time dilation and length contraction in multiple ways: first with a light clock as a thinking tool, and then with spacetime diagrams and a new thinking tool called the symmetric frame. We also examine the experimental proof of these effects. By the end of this chapter, you will understand how time dilation and length contraction fit together with time skew in a fully consistent model that beautifully fits a wide range of evidence.
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Kockelman, Paul. Secrecy, Poetry, and Being-Free. Oxford University Press, 2017. http://dx.doi.org/10.1093/acprof:oso/9780190636531.003.0003.
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This chapter asks two questions: What are some of the secrets of networks? And what might constitute their poetics, an aesthetic means of revealing their secrets? It leverages the relation between codes and channels, delving into two topics that link them: degrees of freedom and secrets. By degrees of freedom is meant the number of independent dimensions needed to specify the state of a system. This chapter argues that even relatively commensurate systems, which have identical degrees of freedom, can have different secrets—understood as inherent symmetries that organize their sense-making capacities. This chapter also shows how channels as well as codes can have inherent secrets (in addition to their ability to keep and reveal secrets in more stereotypic ways). By extending the notion of poetics, it shows how such systems can be made to reveal their secrets. As will be seen, all this is a way of reinterpreting the Sapir-Whorf hypothesis (i.e., the idea that the language one speaks affects the way one thinks), such that this hypothesis can be usefully applied to media more generally (such as interfaces, algorithms, infrastructure, and networks).