To abstract from the number of inputs, outputs and states, the variables are expressed as vectors and the differential and algebraic equations are written in matrix form ( the latter only being possible when the dynamical system is linear ).

The transition amplitude is then given as the matrix element of the S-matrix between the initial and the final states of the quantum system.

In analyzing quantum states and black holes, physicist Don Page writes that " determining experimentally whether or not information is lost down black holes of solar mass ... would require more than measurements to give a rough determination of the final density matrix after a black hole evaporates ".

A density matrix is a matrix that describes a quantum system in a mixed state, a statistical ensemble of several quantum states.

The density matrix is especially useful for mixed states, because any state, pure or mixed, can be characterized by a single density matrix.

One of the advantages of the density matrix is that there is just one density matrix for each mixed state, whereas there are many statistical ensembles of pure states for each mixed state.

This matrix is necessarily symmetric because it is formed from the orthogonal states, and has eigenvalues q.

Through it, states of two quantum systems are naturally decomposed, providing a necessary and sufficient condition for them to be entangled: if the rank of the matrix is larger than one.

In linear algebra, the Cayley – Hamilton theorem ( named after the mathematicians Arthur Cayley and William Hamilton ) states that every square matrix over a commutative ring ( such as the real or complex field ) satisfies its own characteristic equation.

The Cayley – Hamilton theorem states that " substituting " the matrix A for λ in this polynomial results in the zero matrix:

The left hand side can be worked out to an n × n matrix whose entries are ( enormous ) polynomial expressions in the set of entries of A, so the Cayley – Hamilton theorem states that each of these expressions are equivalent to 0.

The unisolvence theorem states that such a polynomial p exists and is unique, and can be proved by the Vandermonde matrix, as described below.

Intuitively, a stochastic matrix represents a Markov chain with no sink states, this implies that the application of the stochastic matrix to a probability distribution would redistribute the probability mass of the original distribution while preserving its total mass.

The Cayley – Hamilton theorem states that replacing t by A in the characteristic polynomial ( interpreting the resulting powers as matrix powers, and the constant term c as c times the identity matrix ) yields the zero matrix.

The Cayley – Hamilton theorem states that every square matrix satisfies its own characteristic equation.

Given the observation space, the state space, a sequence of observations, transition matrix of size such that stores the transition probability of transiting from state to state, emission matrix of size such that stores the probability of observing from state, an array of initial probabilities of size such that stores the probability that. We say a path is a sequence of states that generate the observations.

A perfectly coherent state has a density matrix ( also called the " statistical operator ") that is a projection onto the pure coherent state, while a mixed state is described by a classical probability distribution for the pure states that make up the mixture.

The table itself is a visual representation of the periodic law, which states that certain chemical properties of elements repeat periodically when arranged by atomic number.

These four elements were eventually codified in most states.

For some elements, allotropes have different molecular formulae which can persist in different phases – for example, two allotropes of oxygen ( dioxygen, O < sub > 2 </ sub > and ozone, O < sub > 3 </ sub >), can both exist in the solid, liquid and gaseous states.

Typically, elements capable of variable coordination number and / or oxidation states tend to exhibit greater numbers of allotropic forms.

Artin's theorem states that in an alternative algebra the subalgebra generated by any two elements is associative.

A generalization of Artin's theorem states that whenever three elements in an alternative algebra associate ( i. e. ) the subalgebra generated by those elements is associative.

One of the first studies of condensed states of matter was by English chemist Humphry Davy, when he observed that of the forty chemical elements known at the time, twenty-six had metallic properties such as lustre, ductility and high electrical and thermal conductivity.

Many states have what are termed mixed economies, referring to the varying degree of planned and market-driven elements in a state's economic system.

Several kinds of descriptive categorizations can be applied broadly to the elements, including consideration of their general physical and chemical properties, their states of matter under familiar conditions, their melting and boiling points, their densities, their crystal structures as solids, and their origins.

The classical elements correspond more closely to four of the states of matter: solid, liquid, gas and plasma.

Furthermore, if b < sub > 1 </ sub > and b < sub > 2 </ sub > are both coprime with a, then so is their product b < sub > 1 </ sub > b < sub > 2 </ sub > ( modulo a it is a product of invertible elements, and therefore invertible ); this also follows from the first point by Euclid's lemma, which states that if a prime number p divides a product bc, then p divides at least one of the factors b, c.

* List of oxidation states of the elements — oxidation states

The guiding principle for the qualification of Dublin Core elements, colloquially known as the Dumb-Down Principle, states that an application that does not understand a specific element refinement term should be able to ignore the qualifier and treat the metadata value as if it were an unqualified ( broader ) element.

This notion is of particular importance for the analysis of human experience: Rather than assuming that mental states inhere in a cognizing subject, or a soul-substance, Buddhist philosophers largely propose that mental states alone exist as " momentary elements of consciousness " and that a subjective perceiver is assumed.

The principle of locality states that physical processes occurring at one place should have no immediate effect on the elements of reality at another location.

An old, and largely discounted theory, granitization states that granite is formed in place by extreme metasomatism by fluids bringing in elements e. g. potassium and removing others e. g. calcium to transform the metamorphic rock into a granite.

The group of halogens is the only periodic table group which contains elements in all three familiar states of matter at standard temperature and pressure.

Gardner states that he had reconstructed elements of the religion from fragments, incorporating elements from Freemasonry, the Occult, and Theosophy, which came together in the Hermetic Order of the Golden Dawn, where Gardner met Aleister Crowley, whose influence became the basis for Wiccan magical practices.

Like other group 8 elements, iron exists in a wide range of oxidation states, − 2 to + 6, although + 2 and + 3 are the most common.

A hypothesis for the origin of this genetic variation states that the host-defense system responsible for silencing foreign DNA elements, such as genes of viral origin, mistakenly silenced genes whose silencing turned out to be beneficial for the organism.

However after his death Joseph Stalin established Socialism in one country for the Soviet Union creating the model for subsequent inward looking Stalinist states, and purging the early Internationalist elements.

The Huygens – Fresnel principle is one such model ; it states that each point on a wavefront generates a secondary spherical wavelet, and that the disturbance at any subsequent point can be found by summing the contributions of the individual wavelets at that point.

This space is closely related to the Poincaré sphere, which is the spherical surface occupied by completely polarized states in the space of the vector

At the heart of Fresnel's wave theory is the Huygens-Fresnel principle, which states that every unobstructed point of a wavefront becomes the source of a secondary spherical wavelet and that the amplitude of the optical field E at a point on the screen is given by the superposition of all those secondary wavelets taking into account their relative phases.

William Thurston's elliptization conjecture states that a closed 3-manifold with finite fundamental group is spherical, i. e. has a Riemannian metric of constant positive sectional curvature.

Denote by L the length of C and by A the area enclosed by C. The spherical isoperimetric inequality states that

# ( finite ) for < math > r < r_0 </ math > and 0 elsewhere, or a particle in the spherical equivalent of the square well, useful to describe scattering and bound states in a nucleus or quantum dot.

Also worth noticing is that unlike Coulomb potential, featuring an infinite number of discrete bound states, the spherical square well has only a finite ( if any ) number because of its finite range ( if it has finite depth ).

Girard's theorem, named after the 16th century French mathematician Albert Girard ( earlier discovered but not published by the English mathematician Thomas Harriot ), states that this surplus determines the surface area of any spherical triangle:

The elliptization conjecture, proved by Grigori Perelman, states that conversely all 3-manifolds with finite fundamental group are spherical manifolds.

In geometric topology, the spherical space form conjecture states that a finite group acting on the 3-sphere is conjugate to a group of isometries of the 3-sphere acting by left translation.

states that a family is normal if and only if the spherical derivatives are locally bounded )

Another common objection states that the decay particle was always travelling in a straight line, and that only the probability of the distribution is spherical.