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).
Representation theory of finite groupsThe representation theory of groups is a part of mathematics which examines how groups act on given structures. Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are also considered. For more details, please refer to the section on permutation representations. Other than a few marked exceptions, only finite groups will be considered in this article. We will also restrict ourselves to vector spaces over fields of characteristic zero.
Embodied cognitionEmbodied cognition is the theory that many features of cognition, whether human or otherwise, are shaped by aspects of an organism's entire body. The cognitive features include high-level mental constructs (such as concepts and categories) and performance on various cognitive tasks (such as reasoning or judgment). The bodily aspects involve the motor system, the perceptual system, the bodily interactions with the environment (situatedness), and the assumptions about the world built the functional structure of organism's brain and body.
Uniform spaceIn the mathematical field of topology, a uniform space is a topological space with additional structure that is used to define uniform properties, such as completeness, uniform continuity and uniform convergence. Uniform spaces generalize metric spaces and topological groups, but the concept is designed to formulate the weakest axioms needed for most proofs in analysis. In addition to the usual properties of a topological structure, in a uniform space one formalizes the notions of relative closeness and closeness of points.
Compact spaceIn mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all limiting values of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval [0,1] would be compact.
Topological spaceIn mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness.
Metric spaceIn mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane.
Upper limbThe upper limbs or upper extremities are the forelimbs of an upright-postured tetrapod vertebrate, extending from the scapulae and clavicles down to and including the digits, including all the musculatures and ligaments involved with the shoulder, elbow, wrist and knuckle joints. In humans, each upper limb is divided into the arm, forearm and hand, and is primarily used for climbing, lifting and manipulating objects. In formal usage, the term "arm" only refers to the structures from the shoulder to the elbow, explicitly excluding the forearm, and thus "upper limb" and "arm" are not synonymous.
Lie algebra representationIn the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space together with a collection of operators on satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators.
Piaget's theory of cognitive developmentPiaget's theory of cognitive development is a comprehensive theory about the nature and development of human intelligence. It was originated by the Swiss developmental psychologist Jean Piaget (1896–1980). The theory deals with the nature of knowledge itself and how humans gradually come to acquire, construct, and use it. Piaget's theory is mainly known as a developmental stage theory. In 1919, while working at the Alfred Binet Laboratory School in Paris, Piaget "was intrigued by the fact that children of different ages made different kinds of mistakes while solving problems".
Cognitive developmentCognitive development is a field of study in neuroscience and psychology focusing on a child's development in terms of information processing, conceptual resources, perceptual skill, language learning, and other aspects of the developed adult brain and cognitive psychology. Qualitative differences between how a child processes their waking experience and how an adult processes their waking experience are acknowledged (such as object permanence, the understanding of logical relations, and cause-effect reasoning in school-age children).
Common coding theoryCommon coding theory is a cognitive psychology theory describing how perceptual representations (e.g. of things we can see and hear) and motor representations (e.g. of hand actions) are linked. The theory claims that there is a shared representation (a common code) for both perception and action. More important, seeing an event activates the action associated with that event, and performing an action activates the associated perceptual event.
Cognitive flexibilityCognitive flexibility is an intrinsic property of a cognitive system often associated with the mental ability to adjust its activity and content, switch between different task rules and corresponding behavioral responses, maintain multiple concepts simultaneously and shift internal attention between them. The term cognitive flexibility is traditionally used to refer to one of the executive functions. In this sense, it can be seen as neural underpinnings of adaptive and flexible behavior.
Simultaneous localization and mappingSimultaneous localization and mapping (SLAM) is the computational problem of constructing or updating a map of an unknown environment while simultaneously keeping track of an agent's location within it. While this initially appears to be a chicken or the egg problem, there are several algorithms known to solve it in, at least approximately, tractable time for certain environments. Popular approximate solution methods include the particle filter, extended Kalman filter, covariance intersection, and GraphSLAM.
PolynomialIn mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example with three indeterminates is x3 + 2xyz2 − yz + 1. Polynomials appear in many areas of mathematics and science.
BromideA bromide ion is the negatively charged form (Br−) of the element bromine, a member of the halogens group on the periodic table. Most bromides are colorless. Bromides have many practical roles, being found in anticonvulsants, flame-retardant materials, and cell stains. Although uncommon, chronic toxicity from bromide can result in bromism, a syndrome with multiple neurological symptoms. Bromide toxicity can also cause a type of skin eruption, see potassium bromide. The bromide ion has an ionic radius of 196 pm.
SpaceSpace is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework.
Limb (anatomy)A limb is a jointed, muscled appendage of a tetrapod vertebrate animal used for weight-bearing and terrestrial locomotion. The distalmost portion of a limb is known as its extremity. The limbs' bony endoskeleton, known as the appendicular skeleton, is homologous among all tetrapods, who use their limbs for walking, running and jumping, swimming, grasping and climbing. All tetrapods have four limbs that are organized into two bilaterally symmetrical pairs, with one pair at each end of the torso, which phylogenetrically correspond to the four paired fins (pectoral and pelvic fins) of their fish ancestors.
Developmental psychologyDevelopmental psychology is the scientific study of how and why humans grow, change, and adapt across the course of their lives. Originally concerned with infants and children, the field has expanded to include adolescence, adult development, aging, and the entire lifespan. Developmental psychologists aim to explain how thinking, feeling, and behaviors change throughout life. This field examines change across three major dimensions, which are physical development, cognitive development, and social emotional development.
Aging brainAging of the brain is a process of transformation of the brain in older age, including changes all individuals experience and those of illness (including unrecognised illness). Usually this refers to humans. Since life extension is only pertinent if accompanied by health span extension, and, more importantly, by preserving brain health and cognition, finding rejuvenating approaches that act simultaneously in peripheral tissues and in brain function is a key strategy for development of rejuvenating technology.
