It is so convinced of its own rightness, of its Archimedean position, that it remained aloof and invariant, rather than being sensitive to its changing local context.

In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some ordered or normed groups, fields, and other algebraic structures.

For example, in the context of ordered fields, one has the axiom of Archimedes which formulates this property, where the field of real numbers is Archimedean, but that of rational functions in real coefficients is not.

The embedding of the rationals then gives a way of speaking about the rationals, integers, and natural numbers in K. The following are equivalent characterizations of Archimedean fields in terms of these substructures.

Such real closed fields that are not Archimedean, are non-Archimedean ordered fields.

For example, the real numbers form an Archimedean field, but every hyperreal field is non-Archimedean.

He is famous for many things including: Hölder's inequality, the Jordan – Hölder theorem, the theorem stating that every linearly ordered group that satisfies an Archimedean property is isomorphic to a subgroup of the additive group of real numbers, the classification of simple groups of order up to 200, and Hölder's theorem which implies that the Gamma function satisfies no algebraic differential equation.

For example, the set R of real numbers together with the operation of addition and usual ordering relation (≤) is an Archimedean group.

The sets of the integers, the rational numbers, the real numbers, together with the operation of addition and the usual ordering (≤), are Archimedean groups.

This theory consists of a finite theory characterizing the real numbers as a complete Archimedean ordered field plus an axiom saying that the domain is of the first uncountable cardinality.

Otto Hölder showed that every linearly ordered group satisfying an Archimedean property is isomorphic to a subgroup of the additive group of real numbers,.

For example, since the hyperreal numbers form a non-Archimedean ordered field and the reals form an Archimedean ordered field, the property of being Archimedean (" every positive real is larger than 1 / n for some positive integer n ") seems at first sight not to satisfy the transfer principle.

A crucially important property of the real numbers is that it is an Archimedean field, meaning it has the Archimedean property that for any real number, there is an integer larger than it in absolute value.

The theorem states: Any linearly ordered abelian group can be embedded as an ordered subgroup of the additive group ℝ < sup > Ω </ sup > endowed with a lexicographical order, where ℝ is the additive group of real numbers ( with its standard order ), and Ω is the set of Archimedean equivalence classes of.

Then Hahn's Embedding Theorem reduces to Hölder's theorem ( which states that a linearly ordered abelian group is Archimedean if and only if it is a subgroup of the ordered additive group of the real numbers ).

He proved equations for the volumes and areas of various figures in two and three dimensions, and enunciated the Archimedean property of finite numbers.

) Thus an Archimedean field is one whose natural numbers grow without bound.

Two nonzero elements ∈ are Archimedean equivalent if there exist natural numbers ∈ ℕ such that and.

In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices.

This difference in definitions controls whether the pseudorhombicuboctahedron is considered an Archimedean solid or a Johnson solid.

Prisms and antiprisms, whose symmetry groups are the dihedral groups, are generally not considered to be Archimedean solids, despite meeting the above definition.

With this restriction, there are only finitely many Archimedean solids.

The Archimedean solids take their name from Archimedes, who discussed them in a now-lost work.

Here Dürer discusses the five Platonic solids, as well as seven Archimedean semi-regular solids, as well as several of his own invention.

It has been called the Archimedean honeycombs by analogy with the convex uniform ( non-regular ) polyhedra, commonly called Archimedean solids.

Bryozoa are abundant in some regions ; the fenestellids including Fenestella, Polypora, and Archimedes, so named because it is in the shape of an Archimedean screw.

As such it is a quasiregular polyhedron, i. e. an Archimedean solid, being vertex-transitive and edge-transitive.

Some examples of convex subsets of a Euclidean 3-dimensional space are the Archimedean solids and the Platonic solids.

This one sure point provided him with what he would call his Archimedean point, in order to further develop his foundation for knowledge.

As such it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.

In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i. e., not a Platonic solid, Archimedean solid, prism or antiprism.

However, it is not vertex-transitive, as it has different isometry at different vertices, making it a Johnson solid rather than an Archimedean solid.

Most of the Johnson solids can be constructed from the first few ( pyramids, cupolae, and rotunda ), together with the Platonic and Archimedean solids, prisms, and antiprisms.

If every element of an ordered field lies between two elements of its rational subfield, then the field is said to be Archimedean.

This ordered field is not Archimedean.

However, a more detailed analysis based on the Archimedean principle and the densities of magnesium and its combustion product shows that just being lighter than air cannot account for the increase in mass.

In On Spirals, Archimedes describes the Archimedean spiral, a function whose radius depends on the angle.

* Conductor ( class field theory ), a modulus describing the ramification in an abelian extension of local or global fields

Diffraction contrast, in electron microscopes and x-topography devices in particular, is also a powerful tool for examining individual defects and local strain fields in crystals.

The collapse of Russia induced a chain reaction of disintegration among the Finns starting from the government, military power and economy, and spreading downwards to all fields of the society such as local administration and workplaces, and finally to the level of individual citizens as changes and questions of freedom, responsibility and morality.

The material can reduce this energy by splitting into many domains pointing in different directions, so the magnetic field is confined to small local fields in the material, reducing the volume of the field.

Soon the content of every newspaper, book, novel, play, film, broadcast and concert, from the level of nationally-known publishers and orchestras to local newspapers and village choirs, was subject to supervision by the Propaganda Ministry, although a process of self-censorship was soon effectively operating in all these fields, leaving the Ministry in Berlin free to concentrate on the most politically sensitive areas such as major newspapers and the state radio.

The local geography is dominated by fields of basalt rubble, interspersed with a few hardy plants and mosses.

Meanwhile, many of the Huguenots fleeing religious persecution in France, and their descendants, had also been living around the trading post and cultivating local fields.

Being the son of the local doctor gave Howard frequent exposure to the effects of injury and violence, due to accidents on farms and oil fields combined with the massive increase in crime that came with the oil boom.

Absolute zero implies no movement, and therefore zero external radiation effects ( i. e., zero local electric and magnetic fields ).

In the absence of externally imposed force fields, they become homogeneous in all their local properties.

Its red ,( laterite ) dense clayey soils retain water well, which limits their agricultural potential for many crops but is ideal for keeping the water in the paddy fields and local village reservoirs.

The magnetic fields of distant astronomical objects are measured through their effects on local charged particles.

It insists that local physics is governed by the same types of physical laws as we deal with in the absence of CTC's: the laws that entail self-consistent single valuedness for the fields.

Different sites expose ions to different local electric fields, which shifts the energy levels via the Stark effect.

The divergence of a vector field which is the resultant of radial inverse-square law fields with respect to one or more sources is everywhere proportional to the strength of the local sources, and hence zero outside sources.

Under the feudal system, most land in England was cultivated in common fields, where peasants were allocated strips of arable land that were used to support the needs of the local village or manor.

In barium titanate, a typical ferroelectric of the displacive type, the transition can be understood in terms of a polarization catastrophe, in which, if an ion is displaced from equilibrium slightly, the force from the local electric fields due to the ions in the crystal increases faster than the elastic-restoring forces.

The primary objectives of the mission were to collect images of the lunar surface, examine ambient light levels to determine the feasibility of astronomical observations from the Moon, perform laser ranging experiments from Earth, observe solar X-rays, measure local magnetic fields, and study mechanical properties of the lunar surface material.

The primary objectives of the mission were to collect images of the lunar surface, examine ambient light levels to determine the feasibility of astronomical observations from the Moon, perform laser ranging experiments from Earth, observe solar X-rays, measure local magnetic fields, and study the soil mechanics of the lunar surface material.

Extremely energetic electrons within the shock wave are accelerated by strong local magnetic fields and radiate as synchrotron emission across most of the electromagnetic spectrum.

It also typically requires interdisciplinary input with balanced representation of multiple fields including engineering, ecology, local history, and transport planning.

Each peasant was required to devote a certain percentage of their fields to coffee and this was enforced by the Belgians and their local, mainly Tutsi, allies.

The IEEE includes 38 technical Societies, organized around specialized technical fields, with more than 300 local organizations that hold regular meetings.

To prevent the owner from using this natural monopoly to monopolize other fields of trade, some jurisdictions require utilities to unbundle the local loop, that is, make the local loop available to their competitors.