Bicategories may be considered as a weakening of the definition of 2-categories.

The thing to lose sleep over is what people, having concluded that we are weaker than we are, are likely to do about it.

For a moment or two, both scenes are present simultaneously, one growing weaker, one growing stronger.

Similarly, all the statements listed below which require choice or some weaker version thereof for their proof are unprovable in ZF, but since each is provable in ZF plus the axiom of choice, there are models of ZF in which each statement is true.

There are several weaker statements that are not equivalent to the axiom of choice, but are closely related.

Sometimes slightly stronger theories such as Morse-Kelley set theory or set theory with a strongly inaccessible cardinal allowing the use of a Grothendieck universe are used, but in fact most mathematicians can actually prove all they need in systems weaker than ZFC, such as second-order arithmetic.

Instead, lower bidding characters are ranked in ascending order according to how much they have bid, the characters becoming progressively weaker in that attribute as they pay less for it.

The key words here are fair and eventually-if characters ' ranks are close, and the weaker character has obtained some advantage, then the weaker character can escape defeat or perhaps prevail.

Alternatively, if characters ' attribute ranks are close, the weaker character can try to change the relevant attribute by changing the nature of the conflict.

Against weaker players whose intended strokes are obvious, an experienced player may move before the shuttlecock has been hit, anticipating the stroke to gain an advantage.

Bronzes are softer and weaker than steel — bronze springs, for example, are less stiff ( and so store less energy ) for the same bulk.

Although player characters are substantially weaker than many of the dangers they face, the game is one of the first to encourage problem solving and outwitting obstacles, rather than out-fighting them.

The medicinal properties of chives are similar to those of garlic, but weaker ; the faint effects in comparison with garlic are probably the main reason for their limited use as a medicinal herb.

His last four years as leader are widely seen as one of the Labour Party's weaker periods.

Covalent bonds often result in the formation of small collections of better-connected atoms called molecules, which in solids and liquids are bound to other molecules by forces that are often much weaker than the covalent bonds that hold the molecules internally together.

We have staved off a war and, since our behavior has involved all these elements, we can only keep adding to our ritual without daring to abandon any part of it, since we have not the slightest notion which parts are effective.

Within this notion clarity is possible, but for us who are neither Greek nor Jansenist there is not such clarity.

When combined with the metaphysical notion that pure forms of this universe are best appreciated when least embodied in a material substratum, it becomes clear that while earth will be dross on a scale of material-formal ratios, celestial bodies will be of a subtle, quickened, ethereal existence, in whose embodiment pure form will be the dominant component and matter will be absent or remain subsidiary.

We must avoid the notion, suggested to some people by examples such as those just mentioned, that ideas are `` units '' in some way comparable to coins or counters that can be passed intact from one group of people to another or even, for that matter, from one individual to another.

For if the small group notion involves the implicit claim that the phenomena of sociological investigations are of atomic or subatomic proportions, the philosopher needs to know the extent to which such entities are valid.

These cases, for all their rarity, are so dramatic that friends and relations repeat the story until the general population may get an entirely false notion of how often the hymen is a serious problem to newly-weds.

It is a notion which contains a gratuitous insult, implying, as it does, that Negroes can make no move unless they are manipulated.

Alms are the fruits of a moral notion of the gift and of fortune on the one hand,

Melzak's and Lambek's primitive models reduced this notion to four elements: ( i ) discrete, distinguishable locations, ( ii ) discrete, indistinguishable counters ( iii ) an agent, and ( iv ) a list of instructions that are effective relative to the capability of the agent.

He argues that this casts doubt on the notion that recessions are caused by a reallocation of resources from industrial production to consumption, since he argues that the Austrian business cycle theory implies that net investment should be below zero during recessions.

The American Thomas Jefferson was a representative agrarian who built Jeffersonian Democracy around the notion that farmers are “ the most valuable citizens ” and the truest republicans.

The philosophy is founded on the notion that human society originates with the development of agriculture, and societies are based upon " people's natural prospensity to farm.

( Hume 1974: 322 ) In explaining how matters of fact are entirely a product of experience, he dismisses the notion that they may be arrived at through a priori reasoning.

He rejects the notion that any sensible qualities are necessarily conjoined, since that would mean we could know something prior to experience.

This represents a considerable departure from the traditional notion that directors ' duties are owed only to the company.

Conspiracy theories are based on the notion that complex plots are put into motion by powerful hidden forces.

The widely accepted notion that the security of the system should depend on the key alone has been explicitly formulated by Auguste Kerckhoffs ( in the 1880s ) and Claude Shannon ( in the 1940s ); the statements are known as Kerckhoffs ' principle and Shannon's Maxim respectively.

The philosophy is founded on the notion that human society originates with the development of agriculture, and societies are based upon " people's natural prospensity to farm.

Abstracting yet again, constructions are often " naturally related " – a vague notion, at first sight.

In general topological spaces, however, the different notions of compactness are not necessarily equivalent, and the most useful notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, involves the existence of certain finite families of open sets that " cover " the space in the sense that each point of the space must lie in some set contained in the family.

In general topological spaces, however, the different notions of compactness are not equivalent, and the most useful notion of compactness — originally called bicompactness — involves families of open sets that " cover " the space in the sense that each point of the space must lie in some set contained in the family.

Some branches of mathematics such as algebraic geometry, typically influenced by the French school of Bourbaki, use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are both Hausdorff and quasi-compact.

Such motions are constrained by the 2-dimensional surface of the earth, so only the horizontal component of the Coriolis force is generally important.

A matrix generally refers to a 2-dimensional array, i. e. an m × n array where m and n are greater than 1.

Ordered pairs are also called 2-tuples, 2-dimensional vectors, or sequences of length 2.

In the language of quantum mechanics, hermitian matrices are observables, so the Pauli matrices span the space of observables of the 2-dimensional complex Hilbert space.

Viewing the complex plane as a 2-dimensional space over the reals, the 2D similarity transformations expressed in terms of complex arithmetic are and where a and b are complex numbers, a ≠ 0.

If each wave is modelled by a vector, then it can be seen that if a number of vectors with random angles are added together, the length of the resulting vector can be anything from zero to the sum of the individual vector lengths — a 2-dimensional random walk, sometimes known as a drunkard's walk.

Note that when they are considered within the complex plane the Gaussian integers may be seen to constitute the 2-dimensional integer lattice.

Up to isomorphism, there are exactly three 2-dimensional unital algebras over the reals: the ordinary complex numbers, the split-complex numbers, and the dual numbers.

N-dimensional knots are generally not decomposable into 2-dimensional knots, though they can be projected to superpositions of lower-dimensional knots.

Here, is density, u, v, and are the components of wind velocity, and is the 2-dimensional ( i. e. horizontal-component-only ) del.

Each circle illustrates the position of a single atom ; note that the actual atomic interactions used in current simulations are more complex than those of 2-dimensional hard spheres.

Monoculars, sometimes called telescopes when used in this capacity, are used wherever a magnified 2-dimensional image of a distant object is required ( Though some may be used to look at objects closer ).

These are one type of 2-dimensional figurate numbers.

The group G has 8 components, and is the group of maps from 2-dimensional Minkowski space to itself that are either isometries or multiply the metric by − 1.

Synthetically, affine planes are 2-dimensional affine geometries defined in terms of the relations between points and lines ( or sometimes, in higher dimensions, hyperplanes ).

Indeed each incidence structure gives a spherical building of rank 2 ( see ); and Ballmann and Brin proved that every 2-dimensional simplicial complex in which the links of vertices are isomorphic to the flag complex of a finite projective plane has the structure of a building, not necessarily classical.

While the sub-units appear separate when viewed on a 2-dimensional section they are in fact interconnected in 3-dimensions and usually take on a lenticular plate or lath morphology.

In 2-dimensional abelian gauge theories worldsheet instantons are magnetic vortices.

Because Giuseppe Peano ( 1858 – 1932 ) was the first to discover one, space-filling curves in the 2-dimensional plane are commonly called Peano curves.

The induced foliations of are called the 2-dimensional Reeb foliation ( of the annulus ) resp.

More precisely, he demonstrated that the Hilbert space of states is always finite dimensional and can be canonically identified with the space of conformal blocks of the G WZW model at level k. Conformal blocks are locally holomorphic and antiholomorphic factors whose products sum to the correlation functions of a 2-dimensional conformal field theory.

Although the series itself is animated in traditional 2-dimensional style, the opening credits are rendered in 3D with toon shading.

For example, for ( 1 + 1 )- dimensional bordisms ( 2-dimensional bordisms between 1-dimensional manifolds ), the map associated with a pair of pants gives a product or coproduct, depending on how the boundary components are grouped – which is commutative or cocommutative, while the map associated with a disk gives a counit ( trace ) or unit ( scalars ), depending on grouping of boundary, and thus ( 1 + 1 )- dimension TQFTs correspond to Frobenius algebras.