ThalamusThe thalamus (from Greek θάλαμος, "chamber") is a large mass of gray matter located in the dorsal part of the diencephalon (a division of the forebrain). Nerve fibers project out of the thalamus to the cerebral cortex in all directions, allowing hub-like exchanges of information. It has several functions, such as the relaying of sensory signals, including motor signals to the cerebral cortex and the regulation of consciousness, sleep, and alertness.
Representation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication).
Group representationIn the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. In chemistry, a group representation can relate mathematical group elements to symmetric rotations and reflections of molecules.
Weight (representation theory)In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F, or equivalently, a one-dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group. The importance of the concept, however, stems from its application to representations of Lie algebras and hence also to representations of algebraic and Lie groups. In this context, a weight of a representation is a generalization of the notion of an eigenvalue, and the corresponding eigenspace is called a weight space.
Irreducible representationIn mathematics, specifically in the representation theory of groups and algebras, an irreducible representation or irrep of an algebraic structure is a nonzero representation that has no proper nontrivial subrepresentation , with closed under the action of . Every finite-dimensional unitary representation on a Hilbert space is the direct sum of irreducible representations. Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but the converse may not hold, e.
Mental imageIn philosophy of mind, neuroscience, and cognitive science, a mental image is an experience that, on most occasions, significantly resembles the experience of "perceiving" some object, event, or scene, but occurs when the relevant object, event, or scene is not actually present to the senses. There are sometimes episodes, particularly on falling asleep (hypnagogic imagery) and waking up (hypnopompic imagery), when the mental imagery may be dynamic, phantasmagoric and involuntary in character, repeatedly presenting identifiable objects or actions, spilling over from waking events, or defying perception, presenting a kaleidoscopic field, in which no distinct object can be discerned.
Independent incrementsIn probability theory, independent increments are a property of stochastic processes and random measures. Most of the time, a process or random measure has independent increments by definition, which underlines their importance. Some of the stochastic processes that by definition possess independent increments are the Wiener process, all Lévy processes, all additive process and the Poisson point process. Let be a stochastic process. In most cases, or .
Project-based learningProject-based learning (PBL) is a student-centered pedagogy that involves a dynamic classroom approach in which it is believed that students acquire a deeper knowledge through active exploration of real-world challenges and problems. Students learn about a subject by working for an extended period of time to investigate and respond to a complex question, challenge, or problem. It is a style of active learning and inquiry-based learning.
Lévy processIn probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which displacements in pairwise disjoint time intervals are independent, and displacements in different time intervals of the same length have identical probability distributions. A Lévy process may thus be viewed as the continuous-time analog of a random walk.
Trigeminal nerveIn neuroanatomy, the trigeminal nerve (lit. triplet nerve), also known as the fifth cranial nerve, cranial nerve V, or simply CN V, is a cranial nerve responsible for sensation in the face and motor functions such as biting and chewing; it is the most complex of the cranial nerves. Its name (trigeminal, ) derives from each of the two nerves (one on each side of the pons) having three major branches: the ophthalmic nerve (V_1), the maxillary nerve (V_2), and the mandibular nerve (V_3).
