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This chapter presents a series of results that elucidate the relation between D-equivalence and K-equivalence. Derived categories are first studied. These results can then be used to show that the derived category does not change under the standard flop (Bondal and Orlov), one of the simplest birational correspondences. More recently, Bridgeland was able to show that the derived category is invariant under general three-dimensional flops. The situation is slightly more complicated for a Mukai flop, another classical birational correspondence. In this case, the birational correspondence itself does not define a derived equivalence, but Kawamata and Namikawa were able to prove that one can nevertheless find another Fourier-Mukai kernel that does.

Reviewing the basic notions of additive and abelian categories, left and right adjoint functors, and Serre functors, this chapter is mainly devoted to triangulated categories. In particular, criteria are established which decide when a given functor is fully-faithful or an equivalence. This is formulated in terms of spanning classes. The last section discusses exceptional objects in triangulated categories which lead naturally to the notion of orthogonal decompositions of categories.

Based on the work of Orlov, Kawamata, and others, this chapter shows that the (numerical) Kodaira dimension and the canonical ring are preserved under derived equivalence. The same techniques can be used to derive the invariance of Hochschild cohomology under derived equivalence. Going one step further, it is shown that the nefness of the canonical bundle is detected by the derived category. The chapter also studies the relation between derived and birational (or rather K-) equivalence. The special case of a central conjecture predicts that two birational Calabi-Yau varieties have equivalent derived categories.

In his 1923 book The Mathematical Theory of Relativity, Arthur Eddington distinguished between two chains of reasoning in general relativity. The first familiar one starts with the existence of the four-dimensional space-time interval ds, whose meaning is the usual one associated with the readings of physical rods and clocks and possibly light rays. The other less familiar chain of reasoning ‘binds the physical manifestations of the energy tensor and the interval; it passes from matter as now defined by the energy-tensor to the interval regarded as the result of measurements made with this matter.’ This chapter takes up the challenge of outlining this second chain of reasoning. In developing this reasoning, it argues that the dynamical underpinning of relativistic kinematics that has been defended in this book is consistent with the structure and logic of GR.

In comparative research — whether it is cross-cultural, cross-national, or multigroup comparison — the assumption of measurement equivalence is crucially important. If nonequivalent measures were used, the outcomes would be seriously biased. Equivalence is the fundamental issue in cross-cultural research and evaluation. A cross-cultural comparison can be misleading for two reasons: (1) comparison is made using different attributes and (2) comparison is made using different scale units. But even when the problems of equivalence in attributes and scale units are resolved, it does not warrant a valid cross-cultural comparison. In every step of the research process, the researcher must ensure that equivalence in concept, operationalization, methods, analysis, and interpretation receive the same attention.

There has been a general consensus that multigroup (sample) confirmatory factor analysis offers a strong approach to evaluate cross-cultural equivalence of measurement properties. Researchers have proposed different procedural steps in the testing of measurement equivalence hypotheses, including the equivalence of the covariance matrices of the observed indicators and the equivalence of factor means among groups. Joreskog and Sorbom (2001) recommend the testing of five general hypotheses, including equivalence of covariance matrices of observed indicators of a scale or an instrument; equivalence of factor patterns of observed indicators; equivalence of factor loadings of observed indicators on their respective factors; equivalence of measurement errors of observed indicators; and equivalence of factor variances and covariance across groups. Invariance of factor pattern and factor loadings is sufficient to determine whether a construct can be measured across different cultural, national, or racial groups.

An analysis of the conceptual framework for analysing the translation technique of the Old Latin Gospels. The terminology found in Cicero, Augustine, and Jerome is summarized, along with modern terms such as ’formal equivalence’ and ’dynamic equivalence’. An approach is adopted based on modern language‐contact studies.

This chapter presents the tools to prove that the proof of Theorem 1.2 is reduced to Statement G. It begins with the lemma stating that the cardinality of each Γ‎-pack or Δ‎-pack is a power of 2. It then introduces a proposition in which (σ‎, Σ‎) is an origin for a Γ‎-equivalence class. It is easy to see that a Γ‎-swap does not change α‎, Γ‎(σ‎), while it decreases κΓ‎(σ‎) by 1. To finish the proof of the theorem, the bijection is constructed, where the number of divisibility conditions is necessarily constant when the bijection obtains.

This chapter considers the order homomorphisms induce on the set of all cores; this order is rich enough to represent all countable partial orders. Antichains in the homomorphism order are discussed, which are collections of incomparable graphs (graphs without homomorphisms between any two of them). Of particular interest are finite maximal antichains, and their structure turns out to be surprisingly revealing. Graphs only have trivial finite maximal antichains, while digraphs have many such antichains of all possible sizes, arising from duality relationships. This chapter also contains the (probabilistic) proof of the Sparse Incomparability Lemma, of the fact that asymptotically almost all graphs on $n$ vertices are cores, and of the fact that the number of incomparable graphs on $n$ vertices differs little (asymptotically) from the total number of non-isomorphic graphs on $n$ vertices. The density of the homomorphism order is related to duality, revealing an unexpected connection between these two seemingly unrelated concepts. Finally, it is shown that one can gain interesting insights into many traditional graph topics, such as Hadwiger’s conjecture, when interpreting them as statements about the homomorphism order.

This essay defends Kant's thesis that there is only a single categorical imperative. This needs a defense because in the Groundwork Kant presents a number of formulations of this imperative (the precise number itself being subject to dispute), which are not obviously equivalent. Its strategy is to show that these formulations result from Kant's attempt to provide a complete construction of the concept of the categorical imperative by linking them with various steps in a progressive account of rational agency. By the complete construction of the concept is understood an account of the necessary and sufficient conditions of the possibility of the categorical imperative. Kant's analysis of rational agency is termed progressive because each step, which is correlated with a distinct formula, adds a fresh dimension to the conception of such agency. The claim is that this makes it possible to view these formulas as extensionally but not intentionally equivalent.

This is a brief introduction to dynamic programming and the method of using linear quadratic (LQ) approximations to the return function; the method is an approximation because it computes the solution to a quadratic expansion of the utility function about the steady state or the stable growth path of model economies. The main purpose of the chapter is to review the theoretical basis for the LQ approximation and to illustrate its use with a detailed example (social planning). The author demonstrates that, using the LQ approximation approach and the certainty equivalence principle, solving for the value function is a relatively easy task. The different sections of the chapter describe the standard neoclassical growth model, present a social planner problem that can be used to solve the model, give a recursive formulation of the social planner's problem, and describe an LQ approximation to this problem. Exercises are included throughout, and an appendix presents a MATLAB program to illustrate the LQ method.

The most successful area of game theory is auction design. This chapter offers a short review of the subject. Various traditional auctions are described. After showing that most of these generate the same expected revenue to the seller under appropriate conditions, a version of the revenue equivalence theorem is proved. Reserve prices are briefly considered. An extended example of the use of the principles of mechanism design in constructing an optimal auction is offered, and the case of common-value auctions is reviewed. The chapter ends with a discussion of the important case of multiunit auctions, and concludes that treasury bonds could be sold on a more scientific basis than at present.

This chapter develops a general theory for functors. Topics discussed include category of categories, representable structure, stone duality, naturality, examples of natural transformations, exponentials of categories, functor categories, and equivalence of categories. The chapter ends with some exercises.

In the Grundlagen, Frege sketched a proof of the basic properties of natural numbers from the numerical equivalence and showed how that equivalence could be proved if numbers were explicitly defined as extensions of certain second-level concepts. Two tasks were left to complete: to provide in the formal system of his concept script a derivation of the basic properties of numbers from the explicit definition in terms of extensions; and to show how extensions are given to us as logical objects independent of intuition. Frege did not delineate either task with any great clarity in the Grundlagen, but in Grundgesetze der Arithmetik he set about completing both tasks. The first involved little that was essentially new: Frege had sketched what amounts to a proof of Peano's axioms in the Grundlagen and by the time he came to write Grundgesetze, he had available to him Dedekind's derivation from them of the existence of functions satisfying Grassmann's recursion equations for addition and multiplication. The reason that it was necessary to convert both of these into detailed proofs was just to confirm that they made no illicit appeal to intuition for their justification. This chapter concentrates on the second task, that of accounting for the existence and uniqueness, independent of intuition, of extensions of concepts, i.e. objects satisfying what called the extensional equivalence.