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Eilenberg–Moore Spectral Sequence
In mathematics, in the field of algebraic topology, the Eilenberg–Moore spectral sequence addresses the calculation of the homology groups of a pullback over a fibration. The spectral sequence formulates the calculation from knowledge of the homology of the remaining spaces. Samuel Eilenberg and John C. Moore's original paper addresses this for singular homology. Motivation Let k be a field and let H_\ast(-)=H_\ast(-,k) and H^\ast(-)=H^\ast(-,k) denote singular homology and singular cohomology with coefficients in ''k'', respectively. Consider the following pullback E_f of a continuous map ''p'': : \begin E_f &\rightarrow & E \\ \downarrow & & \downarrow\\ X &\rightarrow_ &B\\ \end A frequent question is how the homology of the fiber product, E_f, relates to the homology of ''B'', ''X'' and ''E''. For example, if ''B'' is a point, then the pullback is just the usual product E \times X. In this case the Künneth formula says :H^*(E_f) = H^*(X \times E) \cong H^*(X) \otime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eilenberg–Zilber Theorem
In mathematics, specifically in algebraic topology, the Eilenberg–Zilber theorem is an important result in establishing the link between the homology groups of a product space X \times Y and those of the spaces X and Y. The theorem first appeared in a 1953 paper in the American Journal of Mathematics by Samuel Eilenberg and Joseph A. Zilber. One possible route to a proof is the acyclic model theorem. Statement of the theorem The theorem can be formulated as follows. Suppose X and Y are topological spaces, Then we have the three chain complexes C_*(X), C_*(Y), and C_*(X \times Y) . (The argument applies equally to the simplicial or singular chain complexes.) We also have the ''tensor product complex'' C_*(X) \otimes C_*(Y), whose differential is, by definition, :\partial_( \sigma \otimes \tau) = \partial_X \sigma \otimes \tau + (-1)^p \sigma \otimes \partial_Y\tau for \sigma \in C_p(X) and \partial_X, \partial_Y the differentials on C_*(X),C_*(Y). Then the theorem says that we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the classification of algebras. Examples *This definition can be applied in particular to square matrices. The matrix :: A = \begin 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end :is nilpotent because A^3=0. See nilpotent matrix for more. * In the factor ring \Z/9\Z, the equivalence class of 3 is nilpotent because 32 is congruent to 0 modulo 9. * Assume that two elements a and b in a ring R satisfy ab=0. Then the element c=ba is nilpotent as \beginc^2&=(ba)^2\\ &=b(ab)a\\ &=0.\\ \end An example with matrices (for ''a'', ''b''):A = \begin 0 & 1\\ 0 & 1 \end, \;\; B =\begin 0 & 1\\ 0 & 0 \end. Here AB=0 and BA=B. *By definition, any element of a nilsemigroup is nilpotent. Properties No nilpotent element c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filtration
Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter medium are described as ''oversize'' and the fluid that passes through is called the ''filtrate''. Oversize particles may form a filter cake on top of the filter and may also block the filter lattice, preventing the fluid phase from crossing the filter, known as ''blinding''. The size of the largest particles that can successfully pass through a filter is called the effective ''pore size'' of that filter. The separation of solid and fluid is imperfect; solids will be contaminated with some fluid and filtrate will contain fine particles (depending on the pore size, filter thickness and biological activity). Filtration occurs both in nature and in engineered systems; there are biological, geological, and industrial forms. Filtration is als ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Complex
In mathematics, specifically Homological algebra, a double complex is a generalization of a chain complex where instead of having a \mathbb-grading, the objects in the bicomplex have a \mathbb\times\mathbb-grading. The most general definition of a double complex, or a bicomplex, is given with objects in an additive category \mathcal. A bicomplex is a sequence of objects C_ \in \text(\mathcal) with two differentials, the horizontal differentiald^h: C_ \to C_and the vertical differentiald^v:C_ \to C_which have the compatibility relationd_h\circ d_v = d_v\circ d_hHence a double complex is a commutative diagram of the form\begin & & \vdots & & \vdots & & \\ & & \uparrow & & \uparrow & & \\ \cdots & \to & C_ & \to & C_ & \to & \cdots \\ & & \uparrow & & \uparrow & & \\ \cdots & \to & C_ & \to & C_ & \to & \cdots \\ & & \uparrow & & \uparrow & & \\ & & \vdots & & \vdots & & \\ \endwhere the rows and columns form chain complexes. Some authors instead require that the squares anticom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Functor
In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in various quite different settings that a short exact sequence often gives rise to a "long exact sequence". The concept of derived functors explains and clarifies many of these observations. Suppose we are given a covariant left exact functor ''F'' : A → B between two abelian categories A and B. If 0 → ''A'' → ''B'' → ''C'' → 0 is a short exact sequence in A, then applying ''F'' yields the exact sequence 0 → ''F''(''A'') → ''F''(''B'') → ''F''(''C'') and one could ask how to continue this sequence to the right to form a long exact sequence. Strictly speaking, this question is ill-posed, since there are always numerous different ways to continue a given exact sequence to the right. But it turns out that (if A is "nice" enough) the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-isomorphism
In homological algebra, a branch of mathematics, a quasi-isomorphism or quism is a morphism ''A'' → ''B'' of chain complexes (respectively, cochain complexes) such that the induced morphisms :H_n(A_\bullet) \to H_n(B_\bullet)\ (\text H^n(A^\bullet) \to H^n(B^\bullet)) of homology groups (respectively, of cohomology groups) are isomorphisms for all ''n''. In the theory of model categories, quasi-isomorphisms are sometimes used as the class of weak equivalences when the objects of the category are chain or cochain complexes. This results in a homology-local theory, in the sense of Bousfield localization in homotopy theory. See also * Derived category In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction pr ... References *Gelfand, Sergei I., Manin, Yuri I. ''Methods of Homological Alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Complex
Total may refer to: Mathematics * Total, the summation of a set of numbers * Total order, a partial order without incomparable pairs * Total relation, which may also mean ** connected relation (a binary relation in which any two elements are comparable). * Total function, a partial function that is also a total relation Business * TotalEnergies, a French petroleum company * Total (cereal), a food brand by General Mills * Total, a brand of strained yogurt made by Fage * Total, a database management system marketed by Cincom Systems * Total Linhas Aéreas - a brazilian airline * Total, a line of dental products by Colgate Music and culture * Total (group), an American R&B girl group * '' Total: From Joy Division to New Order'', a compilation album * ''Total'' (Sebastian album) * ''Total'' (Total album) * ''Total'' (Teenage Bottlerocket album) * ''Total'' (Seigmen album) * ''Total'' (Wanessa album) * ''Total'' (Belinda Peregrín album) * ''Total 1'', an annual compilation alb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bicomplex
In mathematics, specifically Homological algebra, a double complex is a generalization of a chain complex where instead of having a \mathbb-grading, the objects in the bicomplex have a \mathbb\times\mathbb-grading. The most general definition of a double complex, or a bicomplex, is given with objects in an additive category \mathcal. A bicomplex is a sequence of objects C_ \in \text(\mathcal) with two differentials, the horizontal differentiald^h: C_ \to C_and the vertical differentiald^v:C_ \to C_which have the compatibility relationd_h\circ d_v = d_v\circ d_hHence a double complex is a commutative diagram of the form\begin & & \vdots & & \vdots & & \\ & & \uparrow & & \uparrow & & \\ \cdots & \to & C_ & \to & C_ & \to & \cdots \\ & & \uparrow & & \uparrow & & \\ \cdots & \to & C_ & \to & C_ & \to & \cdots \\ & & \uparrow & & \uparrow & & \\ & & \vdots & & \vdots & & \\ \endwhere the rows and columns form chain complexes. Some authors instead require that the squares anticom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
