Homological algebra is category theory in its aspect of organising and suggesting manipulations in abstract algebra.

His book with Samuel Eilenberg Homological Algebra ( 1956 ) was an important text, treating the subject with a moderate level of abstraction with the help of category theory.

Homological algebra is a collection of algebraic techniques that originated in the study of algebraic topology but has also found applications to group theory and algebraic geometry.

containing all varieties over the field, and a universal cohomology theory on mixed motives in the sense of Homological Algebra.

Projective modules were first introduced in 1956 in the influential book Homological Algebra by Henri Cartan and Samuel Eilenberg.

Her 1990 paper, Homological representations of the Hecke algebra, in Communications in Mathematical Physics, introduced, among other things, certain novel linear representations of the braid group — known as Lawrence – Krammer representation.

Homological algebra is the branch of mathematics which studies homology in a general algebraic setting.

In Homological algebra, and algebraic geometry, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences.

Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich.

* Cartan-Eilenberg: In their 1956 book " Homological Algebra ", these authors used projective and injective module resolutions.

* Henri Cartan, Samuel Eilenberg, Homological algebra.

* Gelfand, Sergei I .; Yuri Manin, Homological algebra.

Eilenberg was a member of Bourbaki and with Henri Cartan, wrote the 1956 book Homological Algebra, which became a classic.

Hirzebruch's book Neue topologische Methoden in der algebraischen Geometrie ( 1956 ) was a basic text for the ' new methods ' of sheaf theory, in complex algebraic geometry.

Its development was rapid in the years after 1950, when it was realised that sheaf cohomology was connected with more classical methods applied to the Riemann-Roch theorem, the analysis of a linear system of divisors in algebraic geometry, several complex variables, and Hodge theory.

Later in the 1960s and 1970s, Edmund Leach and his students Mary Douglas and Nur Yalman, among others, introduced French structuralism in the style of Lévi-Strauss ; while British anthropology has continued to emphasize social organization and economics over purely symbolic or literary topics, differences among British, French, and American sociocultural anthropologies have diminished with increasing dialogue and borrowing of both theory and methods.

In that setting one can use birational geometry, techniques from number theory, Galois theory and commutative algebra, and close analogues of the methods of algebraic topology, all in an integrated way.

This is philosophically unsatisfying to some and has motivated additional work in set theory and other methods of formalizing the foundations of mathematics such as New Foundations by Willard Van Orman Quine.

* Decoding methods, methods in communication theory for decoding codewords sent over a noisy channel

As the field is highly interdisciplinary, research often cuts across multiple areas of study, drawing on research methods from psychology, neuroscience, computer science and systems theory.

In the 1930s the two methods strongly competed until it was realised that they are both approximations to a better theory.

Ab initio methods are based entirely on theory from first principles.

Other ( typically less accurate ) methods are called empirical or semi-empirical because they employ experimental results, often from acceptable models of atoms or related molecules, to approximate some elements of the underlying theory.

The treatment of larger molecules that contain a few dozen electrons is computationally tractable by approximate methods such as density functional theory ( DFT ).

Ab initio methods need to define a level of theory ( the method ) and a basis set.

Density functional theory ( DFT ) methods are often considered to be ab initio methods for determining the molecular electronic structure, even though many of the most common functionals use parameters derived from empirical data, or from more complex calculations.

This demonstrated the importance and usefulness of mathematical models and methods in understanding complex phenomena, and signaled the beginning of mathematical control and systems theory.

With the expansion of the mass media and mass / popular culture in the 1960s and 1970s and the blending of social and cultural criticism and literary criticism, the methods of both kinds of critical theory sometimes intertwined in the analysis of phenomena of popular culture, as in the emerging field of cultural studies, in which concepts deriving from Marxian theory, post-structuralism, semiology, psychoanalysis and feminist theory would be found in the same interpretive work.

Closely related is coding theory which is used to design efficient and reliable data transmission and storage methods.

A consequence of this theory is that effective teaching should present a variety of teaching methods which cover all three learning modalities so that different students have equal opportunities to learn in a way that is effective for them.

While Just war theory is generally seen to set the background terms of moral debate, individual countries have more specific methods of upholding these ethical principles.

Although Abel had already proved the impossibility of a " quintic formula " by radicals in 1824 and Ruffini had published a solution in 1799 that turned out to be flawed, Galois ' methods led to deeper research in what is now called Galois theory.

Whereas Parker started with practice and then moved to theory, Dewey began with hypotheses and then devised methods and curricula to test them.

Although, originally, educational measurement methods were built on classical test theory, item response theory and Rasch models are now used extensively in educational measurement worldwide.

In the brief moment I had to talk to them before I took my post on the ring of defenses, I indicated I was sickened by the methods men employed to live and trade on the river.

His casual, dreamlike working methods, often as not in absentia, were an abrupt change from Harburg's, so that Arlen had to adjust again to another approach to collaboration.

The Hetman had a strong liking for a story, any story which was to be had by means of much sleuthing or by roundabout methods.

By 1937 he had clarified his intentions to serve his people: `` I have striven for clarity and melodious idiom, but at the same time I have by no means attempted to restrict myself to the accepted methods of harmony and melody.

In both the direct and indirect methods of staining, the conjugates had nonspecifically staining fractions.

AT promotes greater independence by enabling people to perform tasks that they were formerly unable to accomplish, or had great difficulty accomplishing, by providing enhancements to, or changing methods of interacting with, the technology needed to accomplish such tasks.

The abacus had fallen out of use in western Europe in the 16th century with the rise of decimal notation and algorismic methods.

Thomas Jefferson had tried to bring a French mechanic ( almost certainly Blanc ) and his methods to America in 1785, but the project never went anywhere.

Neither was compulsory ; individuals had to nominate themselves for both selection methods.

After the team had developed a method of purifying penicillin to an effective first stable form in 1940, several clinical trials ensued, and their amazing success inspired the team to develop methods for mass production and mass distribution in 1945.

As a consequence of this approach, Descartes had to deal with more complicated equations and he had to develop the methods to work with polynomial equations of higher degree.

The development of the chemistry of alkaloids was accelerated by the emergence of spectroscopic and chromatographic methods in the 20th century, so that by 2008 more than 12, 000 alkaloids had been identified.

Until the Ford era cars had been produced by hand coachbuilding craft methods, such as the English wheel.

Computers reduced the number of men needed and enabled decentralisation of technical fire control to autonomous sub-battery fire units such as platoons, troops or sections, although some armies had sometimes done this with their manual methods.

Supported by the prestige of being the only general who had proved capable of defeating the French, he promptly initiated a far-reaching scheme of reform, which replaced the obsolete methods of the 18th century.

Many old methods had been lost during the periods of European contact.

The invention of the bicycle has had an enormous effect on society, both in terms of culture and of advancing modern industrial methods.

Because Berlin was located within the Soviet-occupied zone of Germany and the other occupying powers had previously relied on Soviet good will for access to Berlin, the only available methods of supplying the city were three limited air corridors.

As a result of his letters and, no doubt, personal conversations, Huxley and Haeckel were convinced that, at the time he wrote Principles, he believed new species had arisen by natural methods.

( However, the two methods were not interchangeable ; disk which had their back side created in a 1541 by flipping them over would have to be flipped in the 1571 too, and the back side of disks written in a 1571 using the native support for two-sided operation could not be read in a 1541 ).

( One of the reasons formerly advanced for the non-completion of Babbage's engines had been that engineering methods were insufficiently developed in the Victorian era.