Boolean Algebra
... totally symmetric), if and only if it is invariant under any permutation of its variables. It is partially symmetric in the variables Xi,Xj where {Xi,Xj} is a subset of {X1,X2…Xn} if and only if the interchange of the variables Xi,Xj leaves the function ...
... totally symmetric), if and only if it is invariant under any permutation of its variables. It is partially symmetric in the variables Xi,Xj where {Xi,Xj} is a subset of {X1,X2…Xn} if and only if the interchange of the variables Xi,Xj leaves the function ...
Introduction to Algebraic Number Theory
... (e) Deeper proof of Gauss’s quadratic reciprocity law in terms of arithmetic of cyclotomic fields Q(e2πi/n ), which leads to class field theory. 4. Wiles’s proof of Fermat’s Last Theorem, i.e., xn +y n = z n has no nontrivial integer solutions, uses methods from algebraic number theory extensively ( ...
... (e) Deeper proof of Gauss’s quadratic reciprocity law in terms of arithmetic of cyclotomic fields Q(e2πi/n ), which leads to class field theory. 4. Wiles’s proof of Fermat’s Last Theorem, i.e., xn +y n = z n has no nontrivial integer solutions, uses methods from algebraic number theory extensively ( ...
Thom Spectra that Are Symmetric Spectra
... We often find that the enriched functoriality obtained by working with homotopy colimits over I instead of colimits over N is very useful. For example, one may represent complexification followed by realification as maps of E-spaces BOhI → BUhI → BOhI , such that the composite E∞ map represents mult ...
... We often find that the enriched functoriality obtained by working with homotopy colimits over I instead of colimits over N is very useful. For example, one may represent complexification followed by realification as maps of E-spaces BOhI → BUhI → BOhI , such that the composite E∞ map represents mult ...
Annals of Pure and Applied Logic Dynamic topological S5
... Remark 2.1.2. After seeing a first draft of this paper, Frank Wolter noted that the logics considered here are closely related to the many-dimensional modal logics considered in [3]. In the notation and terminology of [3], S5C = LTL × S5, the product of LTL and S5. [3] considers a temporal logic PTL ...
... Remark 2.1.2. After seeing a first draft of this paper, Frank Wolter noted that the logics considered here are closely related to the many-dimensional modal logics considered in [3]. In the notation and terminology of [3], S5C = LTL × S5, the product of LTL and S5. [3] considers a temporal logic PTL ...
On the Associative Nijenhuis Relation
... The quasi-shuffle product ∗ essentially embodies the structure of the Rota-Baxter relation (3). The case of weight λ = 0 gives the “trivial” Rota-Baxter algebra, i.e. relation (3) without the second term on the left-hand side. This construction was essentially given in [14] using a non-recursive not ...
... The quasi-shuffle product ∗ essentially embodies the structure of the Rota-Baxter relation (3). The case of weight λ = 0 gives the “trivial” Rota-Baxter algebra, i.e. relation (3) without the second term on the left-hand side. This construction was essentially given in [14] using a non-recursive not ...
E.6 The Weak and Weak* Topologies on a Normed Linear Space
... The weak topology on a normed space and the weak* topology on the dual of a normed space were introduced in Examples E.7 and E.8. We will study these topologies more closely in this section. They are specific examples of generic “weak topologies” determined by the requirement that a given class of m ...
... The weak topology on a normed space and the weak* topology on the dual of a normed space were introduced in Examples E.7 and E.8. We will study these topologies more closely in this section. They are specific examples of generic “weak topologies” determined by the requirement that a given class of m ...
Sheaves of Modules
... open U ⊂ X is the sum of the maps induced by ϕ, ψ. This is clearly again a map of sheaves of OX -modules. It is also clear that composition of maps of OX -modules is bilinear with respect to this addition. Thus Mod(OX ) is a pre-additive category, see Homology, Definition 3.1. We will denote 0 the s ...
... open U ⊂ X is the sum of the maps induced by ϕ, ψ. This is clearly again a map of sheaves of OX -modules. It is also clear that composition of maps of OX -modules is bilinear with respect to this addition. Thus Mod(OX ) is a pre-additive category, see Homology, Definition 3.1. We will denote 0 the s ...
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.