Connection formIn mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms. Historically, connection forms were introduced by Élie Cartan in the first half of the 20th century as part of, and one of the principal motivations for, his method of moving frames. The connection form generally depends on a choice of a coordinate frame, and so is not a tensorial object.
SufferingSuffering, or pain in a broad sense, may be an experience of unpleasantness or aversion, possibly associated with the perception of harm or threat of harm in an individual. Suffering is the basic element that makes up the negative valence of affective phenomena. The opposite of suffering is pleasure or happiness. Suffering is often categorized as physical or mental. It may come in all degrees of intensity, from mild to intolerable. Factors of duration and frequency of occurrence usually compound that of intensity.
Connection (mathematics)In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as tangent vectors or tensors in the tangent space, along a curve or family of curves in a parallel and consistent manner. There are various kinds of connections in modern geometry, depending on what sort of data one wants to transport. For instance, an affine connection, the most elementary type of connection, gives a means for parallel transport of tangent vectors on a manifold from one point to another along a curve.
Connection (vector bundle)In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a covariant derivative, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero.
Levi-Civita connectionIn Riemannian or pseudo-Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves the (pseudo-)Riemannian metric and is torsion-free. The fundamental theorem of Riemannian geometry states that there is a unique connection which satisfies these properties. In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection.
FriendshipFriendship is a relationship of mutual affection between people. It is a stronger form of interpersonal bond than an "acquaintance" or an "association", such as a classmate, neighbor, coworker, or colleague. In some cultures, the concept of friendship is restricted to a small number of very deep relationships; in others, such as the U.S. and Canada, a person could have many friends, and perhaps a more intense relationship with one or two people, who may be called good friends or best friends.
Wild animal sufferingWild animal suffering is the suffering experienced by nonhuman animals living outside of direct human control, due to harms such as disease, injury, parasitism, starvation and malnutrition, dehydration, weather conditions, natural disasters, and killings by other animals, as well as psychological stress. Some estimates indicate that these individual animals make up the vast majority of animals in existence.
Metric connectionIn mathematics, a metric connection is a connection in a vector bundle E equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve. This is equivalent to: A connection for which the covariant derivatives of the metric on E vanish. A principal connection on the bundle of orthonormal frames of E. A special case of a metric connection is a Riemannian connection; there is a unique such which is torsion free, the Levi-Civita connection.
Suffering-focused ethicsSuffering-focused ethics are those positions in ethics that give moral priority to the reduction of suffering. This means that they give greater weight to the reduction of suffering than to the promotion of pleasure, happiness, or to other things that one might consider valuable. According to some suffering-focused ethics, humans should concentrate exclusively on reducing preventable suffering. Other views can include additional features, such as the prevention of other disvalues or the promotion of other positive values, while giving priority to reducing preventable suffering over them.
IntrospectionIntrospection is the examination of one's own conscious thoughts and feelings. In psychology, the process of introspection relies on the observation of one's mental state, while in a spiritual context it may refer to the examination of one's soul. Introspection is closely related to human self-reflection and self-discovery and is contrasted with external observation. It generally provides a privileged access to one's own mental states, not mediated by other sources of knowledge, so that individual experience of the mind is unique.
Romantic friendshipA romantic friendship, passionate friendship, or affectionate friendship is a very close but typically non-sexual relationship between friends, often involving a degree of physical closeness beyond that which is common in contemporary Western societies. It may include, for example, holding hands, cuddling, hugging, kissing, giving massages, or sharing a bed, without sexual intercourse or other sexual expression.
Eradication of sufferingThe eradication or abolition of suffering is the concept of using biotechnology to create a permanent absence of involuntary pain and suffering in all sentient beings. The discovery of modern anesthesia in the 19th century was an early breakthrough in the elimination of pain during surgery, but acceptance was not universal. Some medical practitioners at the time believed that anesthesia was an artificial and harmful intervention in the body's natural response to injury.
Introspection illusionThe introspection illusion is a cognitive bias in which people wrongly think they have direct insight into the origins of their mental states, while treating others' introspections as unreliable. The illusion has been examined in psychological experiments, and suggested as a basis for biases in how people compare themselves to others. These experiments have been interpreted as suggesting that, rather than offering direct access to the processes underlying mental states, introspection is a process of construction and inference, much as people indirectly infer others' mental states from their behaviour.
Transformation matrixIn linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then for some matrix , called the transformation matrix of . Note that has rows and columns, whereas the transformation is from to . There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors. Matrices allow arbitrary linear transformations to be displayed in a consistent format, suitable for computation.
Geometric transformationIn mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often both or both — such that the function is bijective so that its inverse exists. The study of geometry may be approached by the study of these transformations. Geometric transformations can be classified by the dimension of their operand sets (thus distinguishing between, say, planar transformations and spatial transformations).
AcceptanceAcceptance in human psychology is a person's assent to the reality of a situation, recognizing a process or condition (often a negative or uncomfortable situation) that is a fait accompli without attempting to change it or protest it. The concept is close in meaning to acquiescence, derived from the Latin acquiēscere (to find rest in). The term acceptance is a noun with various different meanings. When the person to whom a proposal is made signifies their assent, it is an "acceptance" of their offer, also called an agreement.
LoveLove encompasses a range of strong and positive emotional and mental states, from the most sublime virtue or good habit, the deepest interpersonal affection, to the simplest pleasure. An example of this range of meanings is that the love of a mother differs from the love of a spouse, which differs from the love for food. Most commonly, love refers to a feeling of a strong attraction and emotional attachment.
Structuralism (psychology)Structuralism in psychology (also structural psychology) is a theory of consciousness developed by Wilhelm Wundt and his student Edward Bradford Titchener. This theory was challenged in the 20th century. Structuralism as a school of psychology seeks to analyze the adult mind (the total sum of experience from birth to the present) in terms of the simplest definable components and then to find how these components fit together to form more complex experiences as well as how they correlate to physical events.
Transformation (function)In mathematics, a transformation is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f: X → X. Examples include linear transformations of vector spaces and geometric transformations, which include projective transformations, affine transformations, and specific affine transformations, such as rotations, reflections and translations.
AltruismAltruism is the principle and practice of concern for the well-being and/or happiness of other humans or animals. While objects of altruistic concern vary, it is an important moral value in many cultures and religions. It may be considered a synonym of selflessness, the opposite of selfishness. The word altruism was popularized (and possibly coined) by the French philosopher Auguste Comte in French, as altruisme, for an antonym of egoism. He derived it from the Italian altrui, which in turn was derived from Latin alteri, meaning "other people" or "somebody else".
