CoagulationCoagulation, also known as clotting, is the process by which blood changes from a liquid to a gel, forming a blood clot. It potentially results in hemostasis, the cessation of blood loss from a damaged vessel, followed by repair. The mechanism of coagulation involves activation, adhesion and aggregation of platelets, as well as deposition and maturation of fibrin. Coagulation begins almost instantly after an injury to the endothelium lining a blood vessel.
FibrinFibrin (also called Factor Ia) is a fibrous, non-globular protein involved in the clotting of blood. It is formed by the action of the protease thrombin on fibrinogen, which causes it to polymerize. The polymerized fibrin, together with platelets, forms a hemostatic plug or clot over a wound site. When the lining of a blood vessel is broken, platelets are attracted, forming a platelet plug. These platelets have thrombin receptors on their surfaces that bind serum thrombin molecules, which in turn convert soluble fibrinogen in the serum into fibrin at the wound site.
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.
Skew-symmetric matrixIn mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if denotes the entry in the -th row and -th column, then the skew-symmetric condition is equivalent to The matrix is skew-symmetric because Throughout, we assume that all matrix entries belong to a field whose characteristic is not equal to 2.
Rotation matrixIn linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix rotates points in the xy plane counterclockwise through an angle θ about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates v = (x, y), it should be written as a column vector, and multiplied by the matrix R: If x and y are the endpoint coordinates of a vector, where x is cosine and y is sine, then the above equations become the trigonometric summation angle formulae.
Matrix (mathematics)In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra.
Higher-dimensional gamma matricesIn mathematical physics, higher-dimensional gamma matrices generalize to arbitrary dimension the four-dimensional Gamma matrices of Dirac, which are a mainstay of relativistic quantum mechanics. They are utilized in relativistically invariant wave equations for fermions (such as spinors) in arbitrary space-time dimensions, notably in string theory and supergravity. The Weyl–Brauer matrices provide an explicit construction of higher-dimensional gamma matrices for Weyl spinors.
Orthogonal matrixIn linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is where QT is the transpose of Q and I is the identity matrix. This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse: where Q−1 is the inverse of Q. An orthogonal matrix Q is necessarily invertible (with inverse Q−1 = QT), unitary (Q−1 = Q∗), where Q∗ is the Hermitian adjoint (conjugate transpose) of Q, and therefore normal (Q∗Q = QQ∗) over the real numbers.
HemostasisIn biology, hemostasis or haemostasis is a process to prevent and stop bleeding, meaning to keep blood within a damaged blood vessel (the opposite of hemostasis is hemorrhage). It is the first stage of wound healing. This involves coagulation, which changes blood from a liquid to a gel. Intact blood vessels are central to moderating blood's tendency to form clots. The endothelial cells of intact vessels prevent blood clotting with a heparin-like molecule and thrombomodulin, and prevent platelet aggregation with nitric oxide and prostacyclin.
Pauli matricesIn mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (σ), they are occasionally denoted by tau (τ) when used in connection with isospin symmetries. These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation which takes into account the interaction of the spin of a particle with an external electromagnetic field.
