They are included in all types of mathematical handbooks and they are stamped on some types of precision measuring instruments.

Whereas the primary meanings of the Lo Shu diagram seemed to have been based on its inner mathematical properties -- and we shall see that even its secondary meanings rested on some mathematical bases -- the urgent desire to place everything into categories of fives led to other groupings based on other numbers, until an exaggerated emphasis on mere numerology pervaded Chinese thought.

The Greek sculptors tried to clarify it by looking for mathematical proportions, just as they sought some reality behind appearances.

These axioms are sufficient for many proofs in elementary mathematical analysis, and are consistent with some principles, such as the Lebesgue measurability of all sets of reals, that are disprovable from the full axiom of choice.

Taking advantage of this feature, some languages ( like FORTRAN 77 ) specify that array indices begin at 1, as in mathematical tradition ; while other languages ( like Fortran 90, Pascal and Algol ) let the user choose the minimum value for each index.

One line of defense is to maintain that this is false, so that mathematical reasoning uses some special intuition that involves contact with the Platonic realm.

He held that the Absolute Infinite had various mathematical properties, including the reflection principle which says that every property of the Absolute Infinite is also held by some smaller object.

Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context.

The term ' Copenhagen interpretation ' suggests something more than just a spirit, such as some definite set of rules for interpreting the mathematical formalism of quantum mechanics, presumably dating back to the 1920s.

The subjective view, that the wave function is merely a mathematical tool for calculating the probabilities in a specific experiment, has some similarities to the Ensemble interpretation in that it takes probabilities to be the essence of the quantum state, but unlike the ensemble interpretation, it takes these probabilities to be perfectly applicable to single experimental outcomes, as it interprets

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows ( also called morphisms, although this term also has a specific, non category-theoretical meaning ), where these collections satisfy some basic conditions.

Categories now appear in most branches of mathematics, some areas of theoretical computer science where they correspond to types, and mathematical physics where they can be used to describe vector spaces.

Abstracting again, a category is itself a type of mathematical structure, so we can look for " processes " which preserve this structure in some sense ; such a process is called a functor.

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.

In mathematics, specifically general topology and metric topology, a compact space is a mathematical space in which any infinite collection of points sampled from the space must — as a set — be arbitrarily close to some point of the space.

It was first thought to be a tally stick, as it has a series of what has been interpreted as tally marks carved in three columns running the length of the tool, but some scientists have suggested that the groupings of notches indicate a mathematical understanding that goes beyond counting.

Decision problems typically appear in mathematical questions of decidability, that is, the question of the existence of an effective method to determine the existence of some object or its membership in a set ; many of the important problems in mathematics are undecidable.

Digital signal processing ( DSP ) is the mathematical manipulation of an information signal to modify or improve it in some way.

The Copenhagen interpretation is a consensus among some of the pioneers in the field of quantum mechanics that it is undesirable to posit anything that goes beyond the mathematical formulae and the kinds of physical apparatus and reactions that enable us to gain some knowledge of what goes on at the atomic scale.

The ellipse and some of its mathematical properties.

In mathematical logic, there are two quantifiers, " some " and " all ", though as Brentano ( 1838 – 1917 ) pointed out, we can make do with just one quantifier and negation.

There is an extensive body of mathematical theory that explores the consequences of making the allocation of units to treatments by means of some random mechanism such as tables of random numbers, or the use of randomization devices such as playing cards or dice.

In that same year, he entered the École Normale ( then known as l ' École préparatoire ), a far inferior institution for mathematical studies at that time, where he found some professors sympathetic to him.

However, some linguistic contexts do not provide sufficient information to disambiguate a used word.

The identity morphism ( identity mapping ) is called the trivial automorphism in some contexts.

The terms of the Artistic License 1. 0 were at issue in a 2007 federal district court decision in the US which was criticized by some for suggesting that FOSS-like licenses could only be enforced through contract law rather than through copyright law, in contexts where contract damages would be difficult to establish.

Alternate definitions of the Abbe number are used in some contexts.

In some contexts, such as algebraic geometry, an asymptote is defined as a line which is tangent to a curve at infinity.

In some contexts, the signal bandwidth in hertz refers to the frequency range in which the signal's spectral density is nonzero or above a small threshold value.

For example, although the words wee and little are interchangeable in some contexts, wee ( as an adjective ) is almost exclusively written by some people from some parts of northern Britain ( and especially Scotland ) or from Northern Ireland, whereas in Southern England and Wales, little is used predominantly.

In some contexts, " community " indicates a group of people with a common identity other than location.

For example, " 11 December 1999 " can be written in some contexts as " 1999345 " or " 99345 ", for the 345th day of 1999.

In some contexts, numbers and letters are used not so much as a basis for establishing an ordering, but as a means of labeling items that are already ordered.

However, in some of the European and southern hemisphere contexts, confrontational methods of engagement are not always relied on, and dialogical approaches are sometimes advocated.

While the term medical condition generally includes mental illnesses, in some contexts the term is used specifically to denote any illness, injury, or disease except for mental illnesses.

In some contexts it is meaningful to speak of fractions of a charge ; for example in the charging of a capacitor, or in the fractional quantum Hall effect.

In contexts of disasters, nation-states manage the fear not only to provide their citizens with an explanation about the event or blaming some minorities, but also to adjust their previous beliefs.

In some contexts, the fundamental is usually abbreviated as f < sub > 0 </ sub > ( or FF ), indicating the lowest frequency counting from zero.

In some contexts, such as sequencing the genome of a pathogenic microbe, " genome " is meant to include information stored on this auxiliary material, which is carried in plasmids.

In some contexts it is distinguished from the continuous portamento.

Further diacritics are used to indicate variations in the pronunciation of the consonants ( e. g. bet / vet, shin / sin ); and, in some contexts, to indicate the punctuation, accentuation, and musical rendition of Biblical texts ( see Cantillation ).

Under some circumstances, and in some technical contexts, transcluded sections of text may not require strict adherence to the " context neutrality " principle, because the transcluded sections are capable of parameterization.

In some 19th or 20th century contexts, the term may refer specifically to paintings of scenes from secular history, rather than those from religious narratives, literature or mythology.

Today, law enforcement agencies, popular media, the United Nations, other nations and even some medical practitioners can be observed applying the term very broadly and often pejoratively in reference to a wide range of illicit substances, regardless of the more precise definition existing in medical contexts.

Due to objections to the names Hangeul, Chosŏn ' gŭl, and urigeul ( 우리글 ) ( see below ) by Koreans in China, the otherwise uncommon short form jeongeum may be used as a neutral name in some international contexts.