Group actionIn mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group acts on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it.
Achilles tendonThe Achilles tendon or heel cord, also known as the calcaneal tendon, is a tendon at the back of the lower leg, and is the thickest in the human body. It serves to attach the plantaris, gastrocnemius (calf) and soleus muscles to the calcaneus (heel) bone. These muscles, acting via the tendon, cause plantar flexion of the foot at the ankle joint, and (except the soleus) flexion at the knee. Abnormalities of the Achilles tendon include inflammation (Achilles tendinitis), degeneration, rupture, and becoming embedded with cholesterol deposits (xanthomas).
Dihedral groupIn mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. The notation for the dihedral group differs in geometry and abstract algebra. In geometry, D_n or Dih_n refers to the symmetries of the n-gon, a group of order 2n. In abstract algebra, D_2n refers to this same dihedral group.
Reductive groupIn mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive if it has a representation that has a finite kernel and is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group GL(n) of invertible matrices, the special orthogonal group SO(n), and the symplectic group Sp(2n).
Group (mathematics)In mathematics, a group is a non-empty set with an operation that satisfies the following constraints: the operation is associative, has an identity element, and every element of the set has an inverse element. Many mathematical structures are groups endowed with other properties. For example, the integers with the addition operation is an infinite group, which is generated by a single element called 1 (these properties characterize the integers in a unique way).
Group theoryIn abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.
HandA hand is a prehensile, multi-fingered appendage located at the end of the forearm or forelimb of primates such as humans, chimpanzees, monkeys, and lemurs. A few other vertebrates such as the koala (which has two opposable thumbs on each "hand" and fingerprints extremely similar to human fingerprints) are often described as having "hands" instead of paws on their front limbs. The raccoon is usually described as having "hands" though opposable thumbs are lacking.
Solvable groupIn mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup. Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equation. Specifically, a polynomial equation is solvable in radicals if and only if the corresponding Galois group is solvable (note this theorem holds only in characteristic 0).
WristIn human anatomy, the wrist is variously defined as (1) the carpus or carpal bones, the complex of eight bones forming the proximal skeletal segment of the hand; (2) the wrist joint or radiocarpal joint, the joint between the radius and the carpus and; (3) the anatomical region surrounding the carpus including the distal parts of the bones of the forearm and the proximal parts of the metacarpus or five metacarpal bones and the series of joints between these bones, thus referred to as wrist joints.
Abelian groupIn mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.
Orthogonal groupIn mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose).
Automorphism groupIn mathematics, the automorphism group of an object X is the group consisting of automorphisms of X under composition of morphisms. For example, if X is a finite-dimensional vector space, then the automorphism group of X is the group of invertible linear transformations from X to itself (the general linear group of X). If instead X is a group, then its automorphism group is the group consisting of all group automorphisms of X. Especially in geometric contexts, an automorphism group is also called a symmetry group.
Stroke recoveryThe primary goals of stroke management are to reduce brain injury and promote maximum patient recovery. Rapid detection and appropriate emergency medical care are essential for optimizing health outcomes. When available, patients are admitted to an acute stroke unit for treatment. These units specialize in providing medical and surgical care aimed at stabilizing the patient's medical status. Standardized assessments are also performed to aid in the development of an appropriate care plan.
HemiparesisHemiparesis, or unilateral paresis, is weakness of one entire side of the body (hemi- means "half"). Hemiplegia is, in its most severe form, complete paralysis of half of the body. Hemiparesis and hemiplegia can be caused by different medical conditions, including congenital causes, trauma, tumors, or stroke. Depending on the type of hemiparesis diagnosed, different bodily functions can be affected. Some effects are expected (e.g., partial paralysis of a limb on the affected side).
StrokeA stroke is a medical condition in which poor blood flow to the brain causes cell death. There are two main types of stroke: ischemic, due to lack of blood flow, and hemorrhagic, due to bleeding. Both cause parts of the brain to stop functioning properly. Signs and symptoms of a stroke may include an inability to move or feel on one side of the body, problems understanding or speaking, dizziness, or loss of vision to one side. Signs and symptoms often appear soon after the stroke has occurred.
Flexor carpi radialis muscleIn anatomy, flexor carpi radialis is a muscle of the human forearm that acts to flex and (radially) abduct the hand. The Latin carpus means wrist; hence flexor carpi is a flexor of the wrist. The flexor carpi radialis is one of four muscles in the superficial layer of the anterior compartment of the forearm. This muscle originates from the medial epicondyle of the humerus as part of the common flexor tendon.
Human brainThe human brain is the central organ of the human nervous system, and with the spinal cord makes up the central nervous system. The brain consists of the cerebrum, the brainstem and the cerebellum. It controls most of the activities of the body, processing, integrating, and coordinating the information it receives from the sense organs, and making decisions as to the instructions sent to the rest of the body. The brain is contained in, and protected by, the skull bones of the head.
NystagmusNystagmus is a condition of involuntary (or voluntary, in some cases) eye movement. People can be born with it but more commonly acquire it in infancy or later in life. In many cases it may result in reduced or limited vision. In normal eyesight, while the head rotates about an axis, distant visual images are sustained by rotating eyes in the opposite direction of the respective axis. The semicircular canals in the vestibule of the ear sense angular acceleration, and send signals to the nuclei for eye movement in the brain.
ParalysisParalysis (: paralyses; also known as plegia) is a loss of motor function in one or more muscles. Paralysis can also be accompanied by a loss of feeling (sensory loss) in the affected area if there is sensory damage. In the United States, roughly 1 in 50 people have been diagnosed with some form of permanent or transient paralysis. The word "paralysis" derives from the Greek παράλυσις, meaning "disabling of the nerves" from παρά (para) meaning "beside, by" and λύσις (lysis) meaning "making loose".
Peer supportPeer support occurs when people provide knowledge, experience, emotional, social or practical help to each other. It commonly refers to an initiative consisting of trained supporters (although it can be provided by peers without training), and can take a number of forms such as peer mentoring, reflective listening (reflecting content and/or feelings), or counseling. Peer support is also used to refer to initiatives where colleagues, members of self-help organizations and others meet, in person or online, as equals to give each other connection and support on a reciprocal basis.
