For more information, see the main article on Bifurcation theory.

* Bifurcation theory

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Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations.

A better representation which can handle the large number of variables and parameters is called a bifurcation diagram ( Bifurcation theory ): the presence of these special steady-state points at certain values of a parameter ( e. g. mass ) is represented by a point and once the parameter passes a certain value, a qualitative change occurs, called a bifurcation, in which the nature of the space changes, with profound consequences for the protein concentrations: the cell cycle has phases ( partially corresponding to G1 and G2 ) in which mass, via a stable point, controls cyclin levels, and phases ( S and M phases ) in which the concentrations change independently, but once the phase has changed at a bifurcation event ( Cell cycle checkpoint ), the system cannot go back to the previous levels since at the current mass the vector field is profoundly different and the mass cannot be reversed back through the bifurcation event, making a checkpoint irreversible.

* Bifurcation of the median nerve typically occurs after the nerve exits the carpal tunnel ; however, in a small percentage ( 5 %- 10 %) of individuals, the median nerve bifurcates more proximal in the carpal tunnel, wrist, or forearm.

Bifurcation theory studies and classifies phenomena characterized by sudden shifts in behavior arising from small changes in circumstances, analysing how the qualitative nature of equation solutions depends on the parameters that appear in the equation.

Bifurcation diagrams are a common tool for analyzing the behavior of dynamical systems, of which the Rössler attractor is one.

Bifurcation diagram of the circle map.

* Bifurcation theory.

Bifurcation means the splitting of a main body into two parts.

* Bifurcation ( law ), the division of issues in a trial for example the division of a page into two parts.

A review of Rieber's book Bifurcation of the Self by Mark Lawrence states that Rieber repeatedly distorted the evidence and left out a number of important facts about Mason's case, in order to advance his case against the validity of the diagnosis.

Bifurcation buckling is sometimes called Euler buckling even when applied to structures other than Euler columns.

Bifurcation diagram of the logistic map.

* Cycles and Measure of Bifurcation Sets for Two-Dimensional Diffeomorphisms, with F. Takens, Inventiones Mathematicae 82, 1985 ( 397-422 ).

After nesting, females often find homes in the East China Sea, while the Kuroshio Current Extension's Bifurcation region provides important juvenile foraging areas.

Bifurcation theory has been applied to connect quantum systems to the dynamics of their classical analogues in atomic systems, molecular systems, and resonant tunneling diodes.

Bifurcation theory has also been applied to the study of laser dynamics and a number of theoretical examples which are difficult to access experimentally such as the kicked top and coupled quantum wells.

* V. S. Afrajmovich, V. I. Arnold, et al., Bifurcation Theory And Catastrophe Theory, ISBN 3-540-65379-1

Leon Chua is still active in CNN research and publishes many of his articles in the International Journal of Bifurcation and Chaos, of which he is an editor.

For example, programming language theory considers various approaches to the description of computation, whilst the study of computer programming itself investigates various aspects of the use of programming language and complex systems.

In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity.

Microeconomics theory considers the aggregates ( the sum of ) of quantity demanded by buyers and quantity supplied by sellers, studying each possible price per unit ( i. e. supply and demand ).

Game theory is a branch of applied mathematics that considers strategic interactions between agents, one kind of uncertainty.

This theory of judgment dominated logic for centuries, but it has some obvious difficulties: it only considers proposition of the form " All A are B.

" Financial economics ", at least formally, also considers investment under " certainty " ( Fisher separation theorem, " theory of investment value ", Modigliani-Miller theorem ) and hence also contributes to corporate finance theory.

Even though the set may be the same, the same function might be a homomorphism, say, in group theory ( sets with a single operation ) but not in ring theory ( sets with two related operations ), because it fails to preserve the additional operation that ring theory considers.

Systems theory considers the nervous system's organization into interacting subsystems.

This theory considers the volume excluded from the center-of-mass of one idealized cylinder as it approaches another.

The anisotropic attraction stabilizes parallel alignment of neighboring molecules, and the theory then considers a mean-field average of the interaction.

Rather, this theory considers perturbations to a presumed oriented sample.

Similarly, a mathematician does not restrict his study of numbers to the integers ; rather he considers more abstract structures such as rings, and in particular number rings in the context of algebraic number theory.

Probability theory considers measures that assign to the whole set the size 1, and considers measurable subsets to be events whose probability is given by the measure.

Ergodic theory considers measures that are invariant under, or arise naturally from, a dynamical system.

Optimal taxation theory is the branch of economics that considers how taxes can be structured to give the least deadweight costs, or to give the best outcomes in terms of social welfare.

Complexity theory considers not only whether a problem can be solved at all on a computer, but also how efficiently the problem can be solved.

Logic, especially in the field of proof theory, considers theorems as statements ( called formulas or well formed formulas ) of a formal language.

The theory of three generations of human rights considers this group of rights to be " first-generation rights ", and the theory of negative and positive rights considers them to be generally negative rights.

His rejection of structural chemistry, especially the theories of the structure of benzene by August Kekulé, the theory of the asymmetric carbon atom by J. H.

Bohr, Mottelson and Rainwater were jointly awarded the 1975 Nobel Prize in Physics " for the discovery of the connection between collective motion and particle motion in atomic nuclei and the development of the theory of the structure of the atomic nucleus based on this connection ".

These conditions played an important role in the development of the structure theory of commutative rings in the works of David Hilbert, Emmy Noether, and Emil Artin.

The Big Bang theory developed from observations of the structure of the Universe and from theoretical considerations.

The earliest and most direct kinds of observational evidence are the Hubble-type expansion seen in the redshifts of galaxies, the detailed measurements of the cosmic microwave background, the relative abundances of light elements produced by Big Bang nucleosynthesis, and today also the large scale distribution and apparent evolution of galaxies predicted to occur due to gravitational growth of structure in the standard theory.

This indicated that the atoms in Dalton's atomic theory were not indivisible as Dalton claimed, but had inner structure.

These include models to study the electronic properties of solids, such as the Drude model, the Band structure and the density functional theory.

The density functional theory ( DFT ) has been widely used since the 1970s for band structure calculations of variety of solids.

Instead of focusing merely on the individual objects ( e. g., groups ) possessing a given structure, category theory emphasizes the morphisms – the structure-preserving mappings – between these objects ; by studying these morphisms, we are able to learn more about the structure of the objects.

Category theory is also, in some sense, a continuation of the work of Emmy Noether ( one of Mac Lane's teachers ) in formalizing abstract processes ; Noether realized that in order to understand a type of mathematical structure, one needs to understand the processes preserving that structure.

Both are important in the analysis of stress for a polarized dielectric solid under the action of an electric field, materials where the molecular structure is taken into consideration ( e. g. bones ), solids under the action of an external magnetic field, and the dislocation theory of metals.

The simplest type of ab initio electronic structure calculation is the Hartree – Fock ( HF ) scheme, an extension of molecular orbital theory, in which the correlated electron – electron repulsion is not specifically taken into account ; only its average effect is included in the calculation.

Density functional theory ( DFT ) methods are often considered to be ab initio methods for determining the molecular electronic structure, even though many of the most common functionals use parameters derived from empirical data, or from more complex calculations.

The theory was based on how heat-treating metal works ( the temperatures are lowered to room temperature from a high degree causing certain strength increases in the molecular structure to occur ) and supposed that continuing the descent would allow for further strength increases.

These measurements were able to rule out cosmic strings as the leading theory of cosmic structure formation, and suggested cosmic inflation was the right theory.

Another important source was Lucien Séve's theory of personality, which provided the concept of " social activity matrices " as mediating structure between individual and social reproduction.

Other broad distinctions include those between positive economics ( describing " what is ") and normative economics ( advocating " what ought to be "); between economic theory and applied economics ; between rational and behavioral economics ; and between mainstream economics ( more " orthodox " and dealing with the " rationality-individualism-equilibrium nexus ") and heterodox economics ( more " radical " and dealing with the " institutions-history-social structure nexus ").

The relevant theoretical concepts may purportedly be part of the structure of the human mind ( as in Kant's theory of transcendental idealism ), or they may be said to exist independently of the mind ( as in Plato's theory of Forms ).

Lattice theory captures the mathematical structure of order relations.

The former structure draws primarily on group theory and, to a lesser extent, on the theory of lattices, categories, and groupoids.