Finitely generated moduleIn mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring R may also be called a finite R-module, finite over R, or a module of finite type. Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent modules all of which are defined below. Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide.
Module (mathematics)In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of module generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication.
Google TranslateGoogle Translate is a multilingual neural machine translation service developed by Google to translate text, documents and websites from one language into another. It offers a website interface, a mobile app for Android and iOS, as well as an API that helps developers build browser extensions and software applications. As of 2022, Google Translate supports languages at various levels; it claimed over 500 million total users , with more than 100 billion words translated daily, after the company stated in May 2013 that it served over 200 million people daily.
NeuroprostheticsNeuroprosthetics (also called neural prosthetics) is a discipline related to neuroscience and biomedical engineering concerned with developing neural prostheses. They are sometimes contrasted with a brain–computer interface, which connects the brain to a computer rather than a device meant to replace missing biological functionality. Neural prostheses are a series of devices that can substitute a motor, sensory or cognitive modality that might have been damaged as a result of an injury or a disease.
Tensor product of modulesIn mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group.
Brain–computer interfaceA brain–computer interface (BCI), sometimes called a brain–machine interface (BMI) or smartbrain, is a direct communication pathway between the brain's electrical activity and an external device, most commonly a computer or robotic limb. BCIs are often directed at researching, mapping, assisting, augmenting, or repairing human cognitive or sensory-motor functions. They are often conceptualized as a human–machine interface that skips the intermediary component of the physical movement of body parts, although they also raise the possibility of the erasure of the discreteness of brain and machine.
Artificial intelligenceArtificial intelligence (AI) is the intelligence of machines or software, as opposed to the intelligence of human beings or animals. AI applications include advanced web search engines (e.g., Google Search), recommendation systems (used by YouTube, Amazon, and Netflix), understanding human speech (such as Siri and Alexa), self-driving cars (e.g., Waymo), generative or creative tools (ChatGPT and AI art), and competing at the highest level in strategic games (such as chess and Go).
Length of a moduleIn algebra, the length of a module is a generalization of the dimension of a vector space which measures its size. page 153 It is defined to be the length of the longest chain of submodules. The modules of finite length are finitely generated modules, but as opposite to vector spaces, many finitely generated modules have an infinite length. Finitely generated modules of finite length are also called Artinian modules and are at the basis of the theory of Artinian rings. For vector spaces, the length equals the dimension.
Transcranial direct-current stimulationTranscranial direct current stimulation (tDCS) is a form of neuromodulation that uses constant, low direct current delivered via electrodes on the head. It was originally developed to help patients with brain injuries or neuropsychiatric conditions such as major depressive disorder. It can be contrasted with cranial electrotherapy stimulation, which generally uses alternating current the same way, as well as transcranial magnetic stimulation. Research shows increasing evidence for tDCS as a treatment for depression.
DataIn common usage and statistics, data (USˈdætə; UKˈdeɪtə) is a collection of discrete or continuous values that convey information, describing the quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted formally. A datum is an individual value in a collection of data. Data is usually organized into structures such as tables that provide additional context and meaning, and which may themselves be used as data in larger structures.
