K-theoryIn mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices.
Algebraic K-theoryAlgebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the K-groups of the integers.
Vitamin B12Vitamin B12, also known as cobalamin, is a water-soluble vitamin involved in metabolism. It is one of eight B vitamins. It is required by animals, which use it as a cofactor in DNA synthesis, and in both fatty acid and amino acid metabolism. It is important in the normal functioning of the nervous system via its role in the synthesis of myelin, and in the circulatory system in the maturation of red blood cells in the bone marrow. Plants do not need cobalamin and carry out the reactions with enzymes that are not dependent on it.
CyanocobalaminCyanocobalamin is a form of vitamin B12 used to treat vitamin B12 deficiency except in the presence of cyanide toxicity. The deficiency may occur in pernicious anemia, following surgical removal of the stomach, with fish tapeworm, or due to bowel cancer. It is less preferred than hydroxocobalamin for treating vitamin B12 deficiency. Some study have shown that it has an antihypotensive effect. It is used by mouth, by injection into a muscle, or as a nasal spray. Cyanocobalamin is generally well tolerated.
Phase portraitIn mathematics, a phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Each set of initial conditions is represented by a different point or curve. Phase portraits are an invaluable tool in studying dynamical systems. They consist of a plot of typical trajectories in the phase space. This reveals information such as whether an attractor, a repellor or limit cycle is present for the chosen parameter value.
