Structure (category theory)

About: Structure (category theory) is a research topic. Over the lifetime, 8588 publications have been published within this topic receiving 143813 citations.

Differential Equations with Discontinuous Righthand Sides

Abstract: Approach your problems from the right end It isn't that they can't see the solution It is and begin with the answers Then one day, that they can't see the problem perhaps you will find the final question G K Chesterton The Scandal of Father 'The Hermit Clad in Crane Feathers' in R Brown 'The point of a Pin' van Gulik's The Chinese Maze Murders Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes They draw upon widely different sections of mathematics

The algebraic theory of semigroups

Abstract: This book, along with volume I, which appeared previously, presents a survey of the structure and representation theory of semi groups. Volume II goes more deeply than was possible in volume I into the theories of minimal ideals in a semi group, inverse semi groups, simple semi groups, congruences on a semi group, and the embedding of a semi group in a group. Among the more important recent developments of which an extended treatment is presented are B. M. Sain's theory of the representations of an arbitrary semi group by partial one-to-one transformations of a set, L. Redei's theory of finitely generated commutative semi groups, J. M. Howie's theory of amalgamated free products of semi groups, and E. J. Tully's theory of representations of a semi group by transformations of a set. Also a full account is given of Malcev's theory of the congruences on a full transformation semi group.

Infinite-dimensional Lie algebras

Abstract: 1. Basic concepts.- 1. Preliminaries.- 2. Nilpotency and solubility.- 3. Subideals.- 4. Derivations.- 5. Classes and closure operations.- 6. Representations and modules.- 7. Chain conditions.- 8. Series.- 2. Soluble subideals.- 1. The circle product.- 2. The Derived Join Theorems.- 3. Coalescent classes of Lie algebras.- 1. An example.- 2. Coalescence of classes with minimal conditions.- 3. Coalescence of classes with maximal conditions.- 4. The local coalescence of D.- 5. A counterexample.- 4. Locally coalescent classes of Lie algebras.- 1. The algebra of formal power series.- 2. Complete and locally coalescent classes.- 3. Acceptable subalgebras.- 5. The Mal'cev correspondence.- 1. The Campbell-Hausdorff formula.- 2. Complete groups.- 3. The matrix version.- 4. Inversion of the Campbell-Hausdorff formula.- 5. The general version.- 6. Explicit descriptions.- 6. Locally nilpotent radicals.- 1. The Hirsch-Plotkin radical.- 2. Baer, Fitting, and Gruenberg radicals.- 3. Behaviour under derivations.- 4. Baer and Fitting algebras.- 5. The Levi?-Tokarenko theorem.- 7. Lie algebras in which every subalgebra is a subideal.- 1. Nilpotent subideals.- 2. The key lemma and some applications.- 3. Engel conditions.- 4. A counterexample.- 5. Unsin's algebras.- 8. Chain conditions for subideals.- 1. Classes related to Min-si.- 2. The structure of algebras in Min-si.- 3. The case of prime characteristic.- 4. Examples of algebras with Min-si.- 5. Min-si in special classes of algebras.- 6. Max-si in special classes of algebras.- 7. Examples of algebras satisfying Max-si.- 9. Chain conditions on ascendant abelian subalgebras.- 1. Maximal conditions.- 2. Minimal conditions.- 3. Applications.- 10. Existence theorems for abelian subalgebras.- 1. Generalised soluble classes.- 2. Locally finite algebras.- 3. Generalisations of Witt algebras.- 11. Finiteness conditions for soluble Lie algebras.- 1. The maximal condition for ideals.- 2. The double chain condition.- 3. Residual finiteness.- 4. Stuntedness.- 12. Frattini theory.- 1. The Frattini subalgebra.- 2. Soluble algebras: preliminary reductions.- 3. Proof of the main theorem.- 4. Nilpotency criteria.- 5. A splitting theorem.- 13. Neoclassical structure theory.- 1. Classical structure theory.- 2. Local subideals.- 3. Radicals in locally finite algebras.- 4. Semisimplicity.- 5. Levi factors.- 14. Varieties.- 1. Verbal properties.- 2. Invariance properties of verbal ideals.- 3. Ellipticity.- 4. Marginal properties.- 5. Hall varieties.- 15. The finite basis problem.- 1. Nilpotent varieties.- 2. Partially well ordered sets.- 3. Metabelian varieties.- 4. Non-finitely based varieties.- 5. Class 2-by-abelian varieties.- 16. Engel conditions.- 1. The second and third Engel conditions.- 2. A non-locally nilpotent Engel algebra.- 3. Finiteness conditions on Engel algebras.- 4. Left and right Engel elements.- 17. Kostrikin's theorem.- 1. The Burnside problem.- 2. Basic computational results.- 3. The existence of an element of order 2.- 4. Elements which generate abelian ideals.- 5. Algebras generated by elements of order 2.- 6. A weakened form of Kostrikin's theorem.- 7. Sketch proof of Kostrikin's theorem.- 18. Razmyslov's theorem.- 1. The construction.- 2. Proof of non-nilpotence.- Some open questions.- References.- Notation index.

The classical groups : their invariants and representations

Abstract: In this renowned volume, Hermann Weyl discusses the symmetric, full linear, orthogonal, and symplectic groups and determines their different invariants and representations. Using basic concepts from algebra, he examines the various properties of the groups. Analysis and topology are used wherever appropriate. The book also covers topics such as matrix algebras, semigroups, commutators, and spinors, which are of great importance in understanding the group-theoretic structure of quantum mechanics. Hermann Weyl was among the greatest mathematicians of the twentieth century. He made fundamental contributions to most branches of mathematics, but he is best remembered as one of the major developers of group theory, a powerful formal method for analyzing abstract and physical systems in which symmetry is present. In The Classical Groups, his most important book, Weyl provided a detailed introduction to the development of group theory, and he did it in a way that motivated and entertained his readers. Departing from most theoretical mathematics books of the time, he introduced historical events and people as well as theorems and proofs. One learned not only about the theory of invariants but also when and where they were originated, and by whom. He once said of his writing, "My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful." Weyl believed in the overall unity of mathematics and that it should be integrated into other fields. He had serious interest in modern physics, especially quantum mechanics, a field to which The Classical Groups has proved important, as it has to quantum chemistry and other fields. Among the five books Weyl published with Princeton, Algebraic Theory of Numbers inaugurated the Annals of Mathematics Studies book series, a crucial and enduring foundation of Princeton's mathematics list and the most distinguished book series in mathematics.

Discrete subgroups of semisimple Lie groups

Abstract: 1. Statement of Main Results.- 2. Synopsis of the Chapters.- 3. Remarks on the Structure of the Book, References and Notation.- 1. Preliminaries.- 0. Notation, Terminology and Some Basic Facts.- 1. Algebraic Groups Over Arbitrary Fields.- 2. Algebraic Groups Over Local Fields.- 3. Arithmetic Groups.- 4. Measure Theory and Ergodic Theory.- 5. Unitary Representations and Amenable Groups.- II. Density and Ergodicity Theorems.- 1. Iterations of Linear Transformations.- 2. Density Theorems for Subgroups with Property (S)I.- 3. The Generalized Mautner Lemma and the Lebesgue Spectrum.- 4. Density Theorems for Subgroups with Property (S)II.- 5. Non-Discrete Closed Subgroups of Finite Covolume.- 6. Density of Projections and the Strong Approximation Theorem.- 7. Ergodicity of Actions on Quotient Spaces.- III. Property (T).- 1. Representations Which Are Isolated from the Trivial One-Dimensional Representation.- 2. Property (T) and Some of Its Consequences. Relationship Between Property (T) for Groups and for Their Subgroups.- 3. Property (T) and Decompositions of Groups into Amalgams.- 4. Property (R).- 5. Semisimple Groups with Property (T).- 6. Relationship Between the Structure of Closed Subgroups and Property (T) of Normal Subgroups.- IV. Factor Groups of Discrete Subgroups.- 1. b-metrics, Vitali's Covering Theorem and the Density Point Theorem.- 2. Invariant Algebras of Measurable Sets.- 3. Amenable Factor Groups of Lattices Lying in Direct Products.- 4. Finiteness of Factor Groups of Discrete Subgroups.- V. Characteristic Maps.- 1. Auxiliary Assertions.- 2. The Multiplicative Ergodic Theorem.- 3. Definition and Fundamental Properties of Characteristic Maps.- 4. Effective Pairs.- 5. Essential Pairs.- VI. Discrete Subgroups and Boundary Theory.- 1. Proximal G-Spaces and Boundaries.- 2. ?-Boundaries.- 3. Projective G-Spaces.- 4. Equivariant Measurable Maps to Algebraic Varieties.- VII. Rigidity.- 1. Auxiliary Assertions.- 2. Cocycles on G-Spaces.- 3. Finite-Dimensional Invariant Subspaces.- 4. Equivariant Measurable Maps and Continuous Extensions of Representations.- 5. Superrigidity (Continuous Extensions of Homomorphisms of Discrete Subgroups to Algebraic Groups Over Local Fields).- 6. Homomorphisms of Discrete Subgroups to Algebraic Groups Over Arbitrary Fields.- 7. Strong Rigidity (Continuous Extensions of Isomorphisms of Discrete Subgroups).- 8. Rigidity of Ergodic Actions of Semisimple Groups.- VIII. Normal Subgroups and "Abstract" Homomorphisms of Semisimple Algebraic Groups Over Global Fields.- 1. Some Properties of Fundamental Domains for S-Arithmetic Subgroups.- 2. Finiteness of Factor Groups of S-Arithmetic Subgroups.- 3. Homomorphisms of S-Arithmetic Subgroups to Algebraic Groups.- IX. Arithmeticity.- 1. Statement of the Arithmeticity Theorems.- 2. Proof of the Arithmeticity Theorems.- 3. Finite Generation of Lattices.- 4. Consequences of the Arithmeticity Theorems I.- 5. Consequences of the Arithmeticity Theorems II.- 6. Arithmeticity, Volume of Quotient Spaces, Finiteness of Factor Groups, and Superrigidity of Lattices in Semisimple Lie Groups.- 7. Applications to the Theory of Symmetric Spaces and Theory of Complex Manifolds.- Appendices.- A. Proof of the Multiplicative Ergodic Theorem.- B. Free Discrete Subgroups of Linear Groups.- C. Examples of Non-Arithmetic Lattices.- Historical and Bibliographical Notes.- References.