Lyapunov stabilityVarious types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Aleksandr Lyapunov. In simple terms, if the solutions that start out near an equilibrium point stay near forever, then is Lyapunov stable. More strongly, if is Lyapunov stable and all solutions that start out near converge to , then is said to be asymptotically stable (see asymptotic analysis).
Cell divisionCell division is the process by which a parent cell divides into two daughter cells. Cell division usually occurs as part of a larger cell cycle in which the cell grows and replicates its chromosome(s) before dividing. In eukaryotes, there are two distinct types of cell division: a vegetative division (mitosis), producing daughter cells genetically identical to the parent cell, and a cell division that produces haploid gametes for sexual reproduction (meiosis), reducing the number of chromosomes from two of each type in the diploid parent cell to one of each type in the daughter cells.
OscillationOscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum and alternating current. Oscillations can be used in physics to approximate complex interactions, such as those between atoms.
Stability theoryIn mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance.
Structural stabilityIn mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by small perturbations (to be exact C1-small perturbations). Examples of such qualitative properties are numbers of fixed points and periodic orbits (but not their periods). Unlike Lyapunov stability, which considers perturbations of initial conditions for a fixed system, structural stability deals with perturbations of the system itself.
Cell (biology)The cell is the basic structural and functional unit of all forms of life. Every cell consists of cytoplasm enclosed within a membrane, and contains many macromolecules such as proteins, DNA and RNA, as well as many small molecules of nutrients and metabolites. The term comes from the Latin word cellula meaning 'small room'. Cells can acquire specified function and carry out various tasks within the cell such as replication, DNA repair, protein synthesis, and motility. Cells are capable of specialization and mobility within the cell.
Spindle apparatusIn cell biology, the spindle apparatus is the cytoskeletal structure of eukaryotic cells that forms during cell division to separate sister chromatids between daughter cells. It is referred to as the mitotic spindle during mitosis, a process that produces genetically identical daughter cells, or the meiotic spindle during meiosis, a process that produces gametes with half the number of chromosomes of the parent cell. Besides chromosomes, the spindle apparatus is composed of hundreds of proteins.
Numerical methods for partial differential equationsNumerical methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs). In principle, specialized methods for hyperbolic, parabolic or elliptic partial differential equations exist. Finite difference method In this method, functions are represented by their values at certain grid points and derivatives are approximated through differences in these values.
Spindle checkpointThe spindle checkpoint, also known as the metaphase-to-anaphase transition, the spindle assembly checkpoint (SAC), the metaphase checkpoint, or the mitotic checkpoint, is a cell cycle checkpoint during mitosis or meiosis that prevents the separation of the duplicated chromosomes (anaphase) until each chromosome is properly attached to the spindle. To achieve proper segregation, the two kinetochores on the sister chromatids must be attached to opposite spindle poles (bipolar orientation).
Limit cycleIn mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of many real-world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912).
EukaryoteThe eukaryotes constitute the domain of Eukaryota (juːˈkærioʊts,_-əts), organisms whose cells have a nucleus. All animals, plants, fungi, and many unicellular organisms are eukaryotes. They constitute a major group of life forms, alongside the two groups of prokaryotes, the Bacteria and the Archaea. Eukaryotes represent a small minority of the number of organisms, but due to their generally much larger size, their collective global biomass is much larger than that of prokaryotes.
Van der Pol oscillatorIn the study of dynamical systems, the van der Pol oscillator (named for Dutch physicist Balthasar van der Pol) is a non-conservative, oscillating system with non-linear damping. It evolves in time according to the second-order differential equation where x is the position coordinate—which is a function of the time t—and μ is a scalar parameter indicating the nonlinearity and the strength of the damping. The Van der Pol oscillator was originally proposed by the Dutch electrical engineer and physicist Balthasar van der Pol while he was working at Philips.
Cell cycleThe cell cycle, or cell-division cycle, is the series of events that take place in a cell that causes it to divide into two daughter cells. These events include the duplication of its DNA (DNA replication) and some of its organelles, and subsequently the partitioning of its cytoplasm, chromosomes and other components into two daughter cells in a process called cell division. In cells with nuclei (eukaryotes, i.e., animal, plant, fungal, and protist cells), the cell cycle is divided into two main stages: interphase and the mitotic (M) phase (including mitosis and cytokinesis).
MicrofilamentMicrofilaments, also called actin filaments, are protein filaments in the cytoplasm of eukaryotic cells that form part of the cytoskeleton. They are primarily composed of polymers of actin, but are modified by and interact with numerous other proteins in the cell. Microfilaments are usually about 7 nm in diameter and made up of two strands of actin. Microfilament functions include cytokinesis, amoeboid movement, cell motility, changes in cell shape, endocytosis and exocytosis, cell contractility, and mechanical stability.
CytoskeletonThe cytoskeleton is a complex, dynamic network of interlinking protein filaments present in the cytoplasm of all cells, including those of bacteria and archaea. In eukaryotes, it extends from the cell nucleus to the cell membrane and is composed of similar proteins in the various organisms. It is composed of three main components, microfilaments, intermediate filaments and microtubules, and these are all capable of rapid growth or disassembly dependent on the cell's requirements.
Numerical integrationIn analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of definite integrals. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for numerical integration, especially as applied to one-dimensional integrals.
MicrotubuleMicrotubules are polymers of tubulin that form part of the cytoskeleton and provide structure and shape to eukaryotic cells. Microtubules can be as long as 50 micrometres, as wide as 23 to 27 nm and have an inner diameter between 11 and 15 nm. They are formed by the polymerization of a dimer of two globular proteins, alpha and beta tubulin into protofilaments that can then associate laterally to form a hollow tube, the microtubule. The most common form of a microtubule consists of 13 protofilaments in the tubular arrangement.
Motor proteinMotor proteins are a class of molecular motors that can move along the cytoplasm of cells. They convert chemical energy into mechanical work by the hydrolysis of ATP. Flagellar rotation, however, is powered by a proton pump. Motor proteins are the driving force behind most active transport of proteins and vesicles in the cytoplasm. Kinesins and cytoplasmic dyneins play essential roles in intracellular transport such as axonal transport and in the formation of the spindle apparatus and the separation of the chromosomes during mitosis and meiosis.
Hidden attractorIn the bifurcation theory, a bounded oscillation that is born without loss of stability of stationary set is called a hidden oscillation. In nonlinear control theory, the birth of a hidden oscillation in a time-invariant control system with bounded states means crossing a boundary, in the domain of the parameters, where local stability of the stationary states implies global stability (see, e.g. Kalman's conjecture).
Numerical analysisNumerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts.
