These stateless societies are not less evolved than societies with states, but chose to conjure the institution of authority as a separate function from society.

At this level we do not give details of states or transition function.

Thus the negation of the axiom of choice states that there exists a set of nonempty sets which has no choice function.

A still weaker example is the axiom of countable choice ( AC < sub > ω </ sub > or CC ), which states that a choice function exists for any countable set of nonempty sets.

In mathematics, the Borsuk – Ulam theorem, named after Stanisław Ulam and Karol Borsuk, states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point.

In chemistry, physics, and mathematics, the Boltzmann distribution ( also called the Gibbs Distribution ) is a certain distribution function or probability measure for the distribution of the states of a system.

where is the Boltzmann constant, T is temperature ( assumed to be a well-defined quantity ), is the degeneracy ( meaning, the number of levels having energy ; sometimes, the more general ' states ' are used instead of levels, to avoid using degeneracy in the equation ), N is the total number of particles and Z ( T ) is the partition function.

Many, but not all, who consider themselves cognitive scientists have a functionalist view of the mind — the view that mental states are classified functionally, such that any system that performs the proper function for some mental state is considered to be in that mental state.

Regarding the primary function of conscious processing, a recurring idea in recent theories is that phenomenal states somehow integrate neural activities and information-processing that would otherwise be independent.

If we take the simple molecular orbital description of the ground state and combine that function with the functions describing all possible excited states using unoccupied orbitals arising from the same set of atomic orbitals, we also reach the full configuration interaction wavefunction.

In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

Note that for any fixed w the function f ( x ) = F ( w x ) is computable ; thus the universality property states that all computable functions of one variable can be obtained in this fashion.

Moreover, the Banach fixed point theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f ( x ), f ( f ( x )), f ( f ( f ( x ))), ... converges to the fixed point.

In simple terms, the Church – Turing thesis states that a function is algorithmically computable if and only if it is computable by a Turing machine.

Informally the Church – Turing thesis states that if some method ( algorithm ) exists to carry out a calculation, then the same calculation can also be carried out by a Turing machine ( as well as by a recursively definable function, and by a λ-function ).

In 1926 von Neuman showed that if atomic states were understood as vectors in Hilbert space, then they would correspond with both Schrödinger's wave function theory and Heisenberg's matrices.

Liouville's theorem states that any bounded entire function must be constant.

Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set.

The Bohr – Mollerup theorem states that among all functions extending the factorial functions to the positive real numbers, only the gamma function is log-convex, that is, its natural logarithm is convex on the positive real axis.

The Hahn – Banach theorem states that if is a sublinear function, and is a linear functional on a linear subspace U ⊆ V which is dominated by on U,

Cauchy's integral formula states that every function holomorphic inside a disk is completely determined by its values on the disk's boundary.

The solution of the Schrödinger equation goes much further than the Bohr model, because it also yields the shape of the electron's wave function (" orbital ") for the various possible quantum-mechanical states, thus explaining the anisotropic character of atomic bonds.

In mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value.

The intermediate value theorem states the following: If f is a real-valued continuous function on the interval b, and u is a number between f ( a ) and f ( b ), then there is a c ∈ b such that f ( c ) = u.

She states in Volume One of her diaries that she drew inspiration from Marcel Proust, André Gide, Jean Cocteau, Paul Valéry, and Arthur Rimbaud.

One of the documents of that time states, " The Victory immediately gained the general sympathy of Russian society and it continued to grow.

One could note a certain irony: one of the first acts of many of the newly independent states was to adopt the law of the foreign sovereign from whom independence had just been gained.

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.

One way to divide various consequentialisms is by the types of consequences that are taken to matter most, that is, which consequences count as good states of affairs.

One of these systems, which is operated by Cornell University and began operation in 2000, was controversial during the planning and building states for potential negative environmental impact.

But its long curing time of at least a month made it unpopular after World War One in the construction of highways and bridges and many states and construction firms turned to the use of Portland cement.

One particularly important physical result concerning conservation laws is Noether's Theorem, which states that there is a one-to-one correspondence between conservation laws and differentiable symmetries of physical systems.

One section states that if a man purchases a cow that is found to be pregnant, then he is the owner both of the cow and the calf that is born from it.

One critic states of psychologists that " Instead of replacing ' metaphysical ' terms such as ' desire ' and ' purpose ', they used it to legitimize them by giving them operational definitions ... the initial, quite radical operationalist ideas eventually came to serve as little more than a ' reassurance fetish ' ( Koch 1992 ) for mainstream methodological practice.

One example is the First Things First manifesto which was launched within the graphic design community and states " We propose a reversal of priorities in favor of more useful, lasting and democratic forms of communication – a mindshift away from product marketing and toward the exploration and production of a new kind of meaning.

One of the problems ( sometimes called his epitaph ) states:

One lemma states that the difference of the cubes of two rational numbers is equal to the sum of the cubes of two other rational numbers, i. e. given any a and b, with a > b, there exist c and d, all positive and rational, such that

One way to divide various consequentialisms is by the types of consequences that are taken to matter most, that is, which consequences count as good states of affairs.

One inscription found at Axum, states that he conquered the nation of the Bogos, and returned thanks to his father, the god Mars, for his victory.

One situation of affairs through objective acts of consciousness ( acts of constituting categorially ) can serve as the basis for constituting multiple states of affairs.

One approach is to harmonize the company law of the member states.

One supposes that the quantum spin-singlet states emitted by the source are actually approximate descriptions for " true " physical states possessing definite values for the z-spin and x-spin.

One theory states that stable wormholes are possible, but that any attempt to use a network of wormholes to violate causality would result in their decay.

One possible construction of the " internal evidence " states that the Beloved Disciple wrote an account of the life of Jesus.

One after another the German states switched to oppose Napoleon, but he rejected peace terms.

One, which remained unratified, provided for free trade among all of the states except Costa Rica.

One, as early as Thucydides, reported in Plutarch, the Suda and John Tzetzes, states that the Delphic oracle warned Hesiod that he would die in Nemea, and so he fled to Locris, where he was killed at the local temple to Nemean Zeus, and buried there.

One of the major states to emerge from the Mfecane was the Sotho Kingdom founded at Thaba Bosiu by Moshoeshoe I around 1821 to 1822.