Geometric group theoryGeometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces). Another important idea in geometric group theory is to consider finitely generated groups themselves as geometric objects.
IsometryIn mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος isos meaning "equal", and μέτρον metron meaning "measure". Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space.
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.
Hyperbolic groupIn group theory, more precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry. The notion of a hyperbolic group was introduced and developed by . The inspiration came from various existing mathematical theories: hyperbolic geometry but also low-dimensional topology (in particular the results of Max Dehn concerning the fundamental group of a hyperbolic Riemann surface, and more complex phenomena in three-dimensional topology), and combinatorial group theory.
Human presence in spaceHuman presence in space is about humanity in space, particularly about all anthropogenic presence in space and human activity in space, that is in outer space and in a broader sense also on any extraterrestrial astronomical body. Humans have been present in space either, in the common sense, through their direct presence and activity like human spaceflight, or through mediation of their presence and activity like with uncrewed spaceflight, making "telepresence" possible.
Cassette tapeThe Compact Cassette, also commonly called a cassette tape, audio cassette, or simply tape or cassette, is an analog magnetic tape recording format for audio recording and playback. Invented by Lou Ottens and his team at the Dutch company Philips in 1963, Compact Cassettes come in two forms, either containing content as a prerecorded cassette (Musicassette), or as a fully recordable "blank" cassette. Both forms have two sides and are reversible by the user.
User experienceThe user experience (UX) is how a user interacts with and experiences a product, system or service. It includes a person's perceptions of utility, ease of use, and efficiency. Improving user experience is important to most companies, designers, and creators when creating and refining products because negative user experience can diminish the use of the product and, therefore, any desired positive impacts; conversely, designing toward profitability often conflicts with ethical user experience objectives and even causes harm.
ExperienceExperience refers to conscious events in general, more specifically to perceptions, or to the practical knowledge and familiarity that is produced by these processes. Understood as a conscious event in the widest sense, experience involves a subject to which various items are presented. In this sense, seeing a yellow bird on a branch presents the subject with the objects "bird" and "branch", the relation between them and the property "yellow". Unreal items may be included as well, which happens when experiencing hallucinations or dreams.
Compact convergenceIn mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology. Let be a topological space and be a metric space. A sequence of functions is said to converge compactly as to some function if, for every compact set , uniformly on as . This means that for all compact , If and with their usual topologies, with , then converges compactly to the constant function with value 0, but not uniformly.
Psychedelic experienceA psychedelic experience (known colloquially as a trip) is a temporary altered state of consciousness induced by the consumption of a psychedelic substance (most commonly LSD, mescaline, psilocybin mushrooms, or DMT). For example, an acid trip is a psychedelic experience brought on by the use of LSD, while a mushroom trip is a psychedelic experience brought on by the use of psilocybin. Psychedelic experiences feature alterations in normal perception such as visual distortions and a subjective loss of self-identity, sometimes interpreted as mystical experiences.
Near-death experienceA near-death experience (NDE) is a profound personal experience associated with death or impending death which researchers describe as having similar characteristics. When positive, which the great majority are, such experiences may encompass a variety of sensations including detachment from the body, feelings of levitation, total serenity, security, warmth, joy, the experience of absolute dissolution, review of major life events, the presence of a light, and seeing dead relatives.
Out-of-body experienceAn out-of-body experience (OBE or sometimes OOBE) is a phenomenon in which a person perceives the world from a location outside their physical body. An OBE is a form of autoscopy (literally "seeing self"), although this term is more commonly used to refer to the pathological condition of seeing a second self, or doppelgänger. The term out-of-body experience was introduced in 1943 by G. N. M. Tyrrell in his book Apparitions, and was adopted by researchers such as Celia Green, and Robert Monroe, as an alternative to belief-centric labels such as "astral projection" or "spirit walking".
