AN ALGEBRAIC APPROACH TO SUBFRAME LOGICS. MODAL
... Given two modal spaces (X, R) and (Y, Q), a map f : X → Y is a modal space morphism (also known as a p-morphism) if f is continuous, xRz implies f (x)Qf (z), and f (x)Qy implies there exists z ∈ X such that xRz and f (z) = y. Let MS denote the category of modal spaces and modal space morphisms. The ...
... Given two modal spaces (X, R) and (Y, Q), a map f : X → Y is a modal space morphism (also known as a p-morphism) if f is continuous, xRz implies f (x)Qf (z), and f (x)Qy implies there exists z ∈ X such that xRz and f (z) = y. Let MS denote the category of modal spaces and modal space morphisms. The ...
half-angle identities
... Double-Angle and 11-5 Half-Angle Identities You can derive the double-angle identities for cosine and tangent in the same way. There are three forms of the identity for cos 2θ, which are derived by using sin2θ + cos2θ = 1. It is common to rewrite expressions as functions of θ only. ...
... Double-Angle and 11-5 Half-Angle Identities You can derive the double-angle identities for cosine and tangent in the same way. There are three forms of the identity for cos 2θ, which are derived by using sin2θ + cos2θ = 1. It is common to rewrite expressions as functions of θ only. ...
Lie theory for non-Lie groups - Heldermann
... this subgroup is contained in the center of G . Compact subgroups play an important rôle in the theory of locally compact groups. They are understood quite well (see also the chapter on compact groups), especially in the connected case. Theorem 2.5. Let G be a locally compact group such that G/G 1l ...
... this subgroup is contained in the center of G . Compact subgroups play an important rôle in the theory of locally compact groups. They are understood quite well (see also the chapter on compact groups), especially in the connected case. Theorem 2.5. Let G be a locally compact group such that G/G 1l ...
The Classification of Three-dimensional Lie Algebras
... higher dimensions and arbitrary fields, the Bianchi classification is rarely presented by the original Bianchi method. Classification theory of finite-dimensional Lie algebras over fields with positive characteristic, p, was initiated later on, in the 1930s by E. Witt, H. Zassenhaus and N. Jacobson, ...
... higher dimensions and arbitrary fields, the Bianchi classification is rarely presented by the original Bianchi method. Classification theory of finite-dimensional Lie algebras over fields with positive characteristic, p, was initiated later on, in the 1930s by E. Witt, H. Zassenhaus and N. Jacobson, ...
finitely generated powerful pro-p groups
... Author’s Note Initially my, somewhat loosely defined, thrust for this thesis was to investigate the ring theoretic results on Iwasawa algebras presented in the survey [AB06]. I was, however, quite decisively distracted by one seemingly small result mentioned in passing in the first pages of the sur ...
... Author’s Note Initially my, somewhat loosely defined, thrust for this thesis was to investigate the ring theoretic results on Iwasawa algebras presented in the survey [AB06]. I was, however, quite decisively distracted by one seemingly small result mentioned in passing in the first pages of the sur ...
Properties of Algebraic Stacks
... (1) When we say algebraic stack we will mean an algebraic stacks over S, i.e., a category fibred in groupoids p : X → (Sch/S)f ppf which satisfies the conditions of Algebraic Stacks, Definition 12.1. (2) We will say f : X → Y is a morphism of algebraic stacks to indicate a 1morphism of algebraic sta ...
... (1) When we say algebraic stack we will mean an algebraic stacks over S, i.e., a category fibred in groupoids p : X → (Sch/S)f ppf which satisfies the conditions of Algebraic Stacks, Definition 12.1. (2) We will say f : X → Y is a morphism of algebraic stacks to indicate a 1morphism of algebraic sta ...
T A G Coarse homology theories
... nite compositions and unions of entourages in the set: fM N j M 2 E(X); N 2 E(Y )g Unfortunately, the above product is not a product in the category-theoretic sense since the projections X : X Y ! X and Y : X Y ! Y are not in general coarse maps.4 De nition 2.9 A generalised ray is the topo ...
... nite compositions and unions of entourages in the set: fM N j M 2 E(X); N 2 E(Y )g Unfortunately, the above product is not a product in the category-theoretic sense since the projections X : X Y ! X and Y : X Y ! Y are not in general coarse maps.4 De nition 2.9 A generalised ray is the topo ...
Topological groups and stabilizers of types
... • Definable subsets of M n have a finite decomposition into manifold-like sets called cells, resulting in a good theory of dimension. • Rich theory of definable groups (examples are complex algebraic, real algebraic groups, compact Lie groups and more): ...
... • Definable subsets of M n have a finite decomposition into manifold-like sets called cells, resulting in a good theory of dimension. • Rich theory of definable groups (examples are complex algebraic, real algebraic groups, compact Lie groups and more): ...
Homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. The development of homological algebra was closely intertwined with the emergence of category theory. By and large, homological algebra is the study of homological functors and the intricate algebraic structures that they entail. One quite useful and ubiquitous concept in mathematics is that of chain complexes, which can be studied both through their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences.From its very origins, homological algebra has played an enormous role in algebraic topology. Its sphere of influence has gradually expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations. K-theory is an independent discipline which draws upon methods of homological algebra, as does the noncommutative geometry of Alain Connes.