Pure-projective modules and positive constructibility

This second, revised and substantially extended edition of Approximations and Endomorphism Algebras of Modules reflects both the depth and the width of recent developments in the area since the first edition appeared in 2006. The new division of the monograph into two volumes roughly corresponds to its two central topics, approximation theory (Volume 1) and realization theorems for modules (Volume 2). It is a widely accepted fact that the category of all modules over a general associative ring is too complex to admit classification. Unless the ring is of finite representation type we must limit attempts at classification to some restricted subcategories of modules. The wild character of the category of all modules, or of one of its subcategories C, is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions, and these are generally viewed as obstacles to classification. In order to overcome this problem, the approximation theory of modules has been developed. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by those from C. These approximations are neither unique nor functorial in general, but there is a rich supply available appropriate to the requirements of various particular applications. The authors bring the two theories together. The first volume, Approximations, sets the scene in Part I by introducing the main classes of modules relevant here: the S-complete, pure-injective, Mittag-Leffler, and slender modules. Parts II and III of the first volume develop the key methods of approximation theory. Some of the recent applications to the structure of modules are also presented here, notably for tilting, cotilting, Baer, and Mittag-Leffler modules. In the second volume, Predictions, further basic instruments are introduced: the prediction principles, and their applications to proving realization theorems. Moreover, tools are developed there for answering problems motivated in algebraic topology. The authors concentrate on the impossibility of classification for modules over general rings. The wild character of many categories C of modules is documented here by the realization theorems that represent critical R-algebras over commutative rings R as endomorphism algebras of modules from C. The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory.

We examine the potential of an extension of constraint logic programming, where the admissible constraints are arbitrary first-order formulas over some domain. Constraint solving is realized by effective quantifier elimination. The arithmetic is always exact. We describe the conceptual advantages of our approach and the capabilities of the current implementation clp(rl). Supported domains are currently ℝ, ℂ, and ℚp. For our discussion here we restrict to ℝ.

To ensure a close relation between the answer sets of a program and those of its ground version, some answer set solvers deal with variables by requiring a safety condition on program rules. If we go beyond the syntax of disjunctive programs, for instance by allowing rules with nested expressions, or perhaps even arbitrary first-order formulas, new definitions of safety are required. In this paper we consider a new concept of safety for formulas in quantified equilibrium logic where answer sets can be defined for arbitrary first-order formulas. The new concept captures and generalises two recently proposed safety definitions: that of Lee, Lifschitz and Palla (2008) as well as that of Bria, Faber and Leone (2008). We study the main metalogical properties of safe formulas.

Horn clause queries and generalizations

A linear dynamical system resulting from the interconnection of subsystems is con- sidered. Assuming that this interconnection is ''temporal'', i.e. starting at a given initial time in the continuous-time case and ending at a given final time in the discrete-time case, such a system is also said to be ''temporal''. Temporal interconnections generate ''uncontrollable im- pulsive behaviors'' which are not found in the classical theory, though they have been studied for more than 20 years in the case of systems with constant coe‰cients. Determining the struc- ture of the impulsive behavior of a temporal system is a key problem in the theory of linear dynamical systems. It is addressed here, using module theory, for systems with time-varying coe‰cients, in both the continuous- and discrete-time cases. These two cases are merged into a general framework. The impulsive behavior of a temporal system satisfying a suitable regu- larity condition has a structure which is fully elucidated. It turns out that the determination of this structure in practice is an algebraic—not an analytic—problem, which makes the calcu- lations simpler and easier to computerize. The theory is illustrated through several examples.

A number of efficient methods for evaluating first-order and monadic-second order queries on finite relational structures are based on tree-decompositions of structures or queries. We systematically study these methods.In the first part of the article, we consider arbitrary formulas on tree-like structures. We generalize a theorem of Courcelle [1990] by showing that on structures of bounded tree-width a monadic second-order formula (with free first- and second-order variables) can be evaluated in time linear in the structure size plus the size of the output.In the second part, we study tree-like formulas on arbitrary structures. We generalize the notions of acyclicity and bounded tree-width from conjunctive queries to arbitrary first-order formulas in a straightforward way and analyze the complexity of evaluating formulas of these fragments. Moreover, we show that the acyclic and bounded tree-width fragments have the same expressive power as the well-known guarded fragment and the finite-variable fragments of first-order logic, respectively.

It is the purpose of this paper to make a study of commutation formulas in the algebra of quantum mechanics. The theories of quantum mechanics introduced by Heisenbergt and Diract are different in their conception and formulation but both make use of a non-commutative algebra. In Heisenberg's theory, the elements of the algebra are infinite matrices; in Dirac's they are abstract q-numbers. Schrodinger's? theory, although mathematically equivalent to Heisenberg's, does not make explicit use of this algebra. However, the operators which Schrodinger uses satisfy the same commutation formulas as Heisenberg's matrices. We shall consider the algebra of the quantum mechanics from the matric standpoint although the results obtained do not depend upon the form of the variables. The enter in pairs as in classical mechanics and we shall use the expression quantum variables or simply variables in analogy with the idea of canonically conjugate in the classical theory. For a single pair of conjugate the properties of the algebra are determined by the fundamental commutation

This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original papers and monographs 2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic.

Though learning has become a core component of modern information processing, there is now ample evidence that it can lead to biased, unsafe, and prejudiced systems. The need to impose requirements on learning is therefore paramount, especially as it reaches critical applications in social, industrial, and medical domains. However, the non-convexity of most modern statistical problems is only exacerbated by the introduction of constraints. Whereas good unconstrained solutions can often be learned using empirical risk minimization, even obtaining a model that satisfies statistical constraints can be challenging. All the more so, a good one. In this paper, we overcome this issue by learning in the empirical dual domain, where constrained statistical learning problems become unconstrained and deterministic. We analyze the generalization properties of this approach by bounding the empirical duality gap -- i.e., the difference between our approximate, tractable solution and the solution of the original (non-convex) statistical problem -- and provide a practical constrained learning algorithm. These results establish a constrained counterpart to classical learning theory, enabling the explicit use of constraints in learning. We illustrate this theory and algorithm in rate-constrained learning applications arising in fairness and adversarial robustness.

Model theory for universal classes with the amalgamation property: A study in the foundations of model theory and algebra

Nowadays purity plays a role in at least four branches of mathematics: Module Theory, Theory of Acts over semigroups, Model Theory, and Category Theory. Adamek and Rosciký have studied these notions categorically, and Rothmaler model-theoretically. Some authors including Banaschewski, Gould, and Normak have studied purity on G-acts, acts over a monoid or a group G.

In this chapter, we are going to discuss a basic concept: pure subgroup. This concept has been one of the most fertile notions in the theory since its inception in a paper by the pioneer H. Prufer. The relevance of purity in abelian group theory, and later in module theory, has tremendously grown with time. While abelian groups have been major motivation for a number of theorems in category theory, purity has served as a prototype for relative homological algebra, and has played a significant role in model theory as well.

We propose the affine filter bank modulation (AFBM) waveform for enhanced integrated sensing and communications (ISAC) in sixth generation (6G), designed by drawing on concepts from classical filter bank multicarrier modulation (FBMC) theory and recent advances in chirp-domain waveforms, particularly affine frequency division multiplexing (AFDM). Specifically, AFBM exhibits several desirable properties, with emphasis on its remarkably low peak-to-average power ratio (PAPR) and reduced out-of-band emission (OOBE) when bench-marked against the conventional AFDM waveform under doubly-dispersive (DD) channel conditions. In the communications setting, reliable symbol detection is achieved using a tailored low-complexity Gaussian belief propagation (GaBP)-based algorithm, while in the sensing setting, a range and velocity estimation approach is developed that integrates an expectation maximization (EM)-assisted probabilistic data association (PDA) framework to accurately identify surrounding targets. The highlighted performance and benefits of AFBM are validated through analytical and numerical evaluations, including conventional metrics such as ambiguity function (AF), bit error rate (BER), and root mean square error (RMSE), consolidating its position as a promising waveform for next-generation wireless systems.

This volume surveys recent interactions between model theory and other branches of mathematics, notably group theory. Beginning with an introductory chapter describing relevant background material, the book contains contributions from many leading international figures in this area. Topics described include automorphism groups of algebraically closed fields, the model theory of pseudo-finite fields and applications to the subgroup structure of finite Chevalley groups. Model theory of modules, and aspects of model theory of various classes of groups, including free groups, are also discussed. The book also contains the first comprehensive survey of finite covers. Many new proofs and simplifications of recent results are presented and the articles contain many open problems. This book will be a suitable guide for graduate students and a useful reference for researchers working in model theory and algebra.

Extendability of ZF Models in the von Neuman Hierarchy to Models of KM theory of classes