In mathematics the modularity theorem ( formerly called the Taniyama – Shimura – Weil conjecture and several related names ) states that elliptic curves over the field of rational numbers are related to modular forms.

Andrew Wiles proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's last theorem, and Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor extended his techniques to prove the full modularity theorem in 2001.

The modularity theorem is a special case of more general conjectures due to Robert Langlands.

This mapping is called a modular parametrization of level N. If N is the smallest integer for which such a parametrization can be found ( which by the modularity theorem itself is now known to be a number called the conductor ), then the parametrization may be defined in terms of a mapping generated by a particular kind of modular form of weight two and level N, a normalized newform with integer q-expansion, followed if need be by an isogeny.

The function obtained in this way is, remarkably, a cusp form of weight two and level N and is also an eigenform ( an eigenvector of all Hecke operators ); this is the Hasse – Weil conjecture, which follows from the modularity theorem.

The now fully proved conjecture became known as the modularity theorem.

Several theorems in number theory similar to Fermat's Last Theorem follow from the modularity theorem.

# REDIRECT modularity theorem

A well-known example is the Taniyama – Shimura conjecture, now the modularity theorem, which proposed that each elliptic curve over the rational numbers can be translated into a modular form ( in such a way as to preserve the associated L-function ).

It was subsequently shown to be true for all elliptic curves over Q, as a consequence of the modularity theorem.

# REDIRECT modularity theorem

* In mathematics, the modularity theorem ( formerly the Taniyama – Shimura conjecture ) establishes a connection between elliptic curves and modular forms.

This partial proof of the Sato – Tate conjecture uses Wiles's theorem about modularity of semistable elliptic curves.

A partial and refined case of this conjecture for elliptic curves over rationals is called the Taniyama – Shimura conjecture or the modularity theorem whose statement he subsequently refined in collaboration with Goro Shimura.

# REDIRECT modularity theorem

# REDIRECT modularity theorem

# REDIRECT modularity theorem

For elliptic curves over the rational numbers, the Hasse – Weil conjecture follows from the modularity theorem.

As a result, the code addressing the cross-cutting concern must be scattered, or duplicated, across the various related locations, resulting in a loss of modularity.

Such a pair is called a modular pair, and there are various generalizations of modularity related to this notion and to semimodularity.

Usually the aim of using a decomposition paradigm is to optimize some metric related to program complexity, for example the modularity of the program or its maintainability.

Andrew Wiles ' proof of modularity of semi-stable elliptic curves over rationals can be viewed as an exercise in the Langlands conjectures.

* There is a basic form of modularity: files can be compiled separately and linked together, with control over which functions and data objects are visible to other files via and attributes.

This simplistic view of modularity has been disproven over the course of the last century.

Gabion baskets have some advantages over loose riprap because of their modularity and ability to be stacked in various shapes ; they are also resistant to being washed away by moving water.

Though originally disavowed by Fodor, other psychologists have embraced it, and it is readily apparent in the use of modularity in biology ( e. g., each module of an organism can be decomposed into finer modules ), social processes and artifacts ( e. g., we can think of a skyscraper in terms of blocks of floors, a single floor, elements of a floor, etc.

Instead of postulating ' pure ' modularity, theorists have opted for a weaker version, domain-specificity implemented in functionally specialised neural circuits and computation ( e. g. Jackendoff and Pinker ’ s ( 2005 ) words, we must investigate language “ not as a monolith but as a combination of components, some special to language, others rooted in more general capacities ” ( p. 223 ).

In general, this process of decomposition is undertaken either for the purpose of gaining insight into the identity of the constituent components ( which may reflect individual physical processes of interest, for example ), or for the purpose of obtaining a compressed representation of the global function, a task which is feasible only when the constituent processes possess a certain level of modularity ( i. e., independence or non-interaction ).

Module or modular may refer to the concept of modularity.

He also notes that the characteristics are not an all-or-nothing proposition, but rather each of the characteristics may be manifest in some degree, and that modularity itself is also not a dichotomous construct — something may be more or less modular: " One would thus expect — what anyhow seems to be desirable — that the notion of modularity ought to admit of degrees " ( Fodor, 1996 / 1983: 37 ).

As in some of the other disciplines, the term modularity may be used in multiple ways in biology.

Wagner's work explores how natural selection may have resulted in modular organisms, and the roles modularity plays in evolution.

However asynchronous circuits have the potential to be faster, and may also have advantages in lower power consumption, lower electromagnetic interference, and better modularity in large systems.

And the modularity of these large systems was also unique: multiple CPUs, multiple memory modules and multiple I / O and Data Comm processors permitted incremental and cost effective growth of system performance and reliability.

To provide modularity, memory and I / O buses can be combined into a unified system bus.

" According to Steven Pinker, EP is " not a single theory but a large set of hypotheses " and a term which " has also come to refer to a particular way of applying evolutionary theory to the mind, with an emphasis on adaptation, gene-level selection, and modularity.

Forth is a simple yet extensible language ; its modularity and extensibility permit the writing of high-level programs such as CAD systems.

This modularity allows the enthusiast to spend as little or as much as he wants on a component that suits his specific needs.

Another advantage of modularity is the ability to spend one's money on only a few core components at first and then later add additional components to one's system.

The latest vehicles, like the Patria AMV, have been designed with an emphasis on modularity that improves their ability to be repaired in the field.

Prominent researchers in evolutionary psychology and anthropology, including Scott Atran, Dan Sperber, Pascal Boyer, John Tooby and others, argue the possibility of incompatibility between modularity of mind and memetics.

This modularity of the operating system is at the binary ( image ) level and not at the architecture level.

Current directions include security and reliability verification, new kinds of modularity ( mixins, delegates, aspects ), and database integration such as Microsoft's LINQ.

Dijkstra wrote his seminal paper, " Go To Statement Considered Harmful ", in 1968 and David Parnas introduced the key concept of modularity and information hiding in 1972 to help programmers deal with the ever increasing complexity of software systems.

Another approach is to use a design targeting modularity with interfaces to increase the flexibility of the software with respect to the design.

This method eases modularity and nomadic computing.

At SRI International, Jeff Rulifson, Jan Derksen, and Richard Waldinger developed QA4 which built on the constructs in Planner and introduced a context mechanism to provide modularity for expressions in the database.

It was designed for enhanced scalability, performance, modularity, and easy integration with existing file systems.

It supports the application of software engineering principles, such as structured programming and modularity, which impart orderliness and manageability to simulation models.

The modern evidence for this specificity ( sometimes called modularity ) includes direct recording from single neurons, case studies of patients with brain damage or disease, recording of electrical or magnetic activity in the brain through advanced scanning techniques, and even direct manipulation of brain activity.

One of these is modularity: as has been long recognized, plants and animal bodies are modular: they are organized into developmentally and anatomically distinct parts.

It retained the earlier COS file system, but made significant advances in code modularity to improve system reliability and adaptiveness to new storage devices.

They thus reject a body of opinion in cognitive science suggesting that there is evidence for the modularity of language.

Because of the portability and modularity provided by the C ++/ object-oriented approach, this change in direction, while not trivial, was relatively easy to implement.

The body, which exploits large aluminium extrusion technology, has substantial modularity and allows for extremely low axle weight, whilst fully respecting the highest safety standards, and allows the best exploitation of the space with different loading gauges.