Another definition of asymmetric makes asymmetry equivalent to antisymmetry plus irreflexivity.

An equivalent definition is the radius of an unperturbed circular Newtonian orbit about the Sun of a particle having infinitesimal mass, moving with an angular frequency of radians per day ; or that length such that, when used to describe the positions of the objects in the Solar System, the heliocentric gravitational constant ( the product GM < sub >☉</ sub >) is equal to ()< sup > 2 </ sup > AU < sup > 3 </ sup >/ d < sup > 2 </ sup >.

This definition is equivalent to the statement that a unital associative R-algebra is a monoid in R-Mod ( the monoidal category of R-modules ).

A more abstract definition, which is equivalent but more easily generalized to infinite-dimensional spaces, is to say that bras are linear functionals on kets, i. e. operators that input a ket and output a complex number.

For alternative ( equivalent ) formulations of the definition in terms of a bijective function or a surjective function, see the section Formal definition and properties below.

It relies on the following equivalent definition of differentiability at a point: A function g is differentiable at a if there exists a real number g ′( a ) and a function ε ( h ) that tends to zero as h tends to zero, and furthermore

These definitions are equivalent to one another, so the most convenient definition can be used to determine whether a given function is continuous or not.

Such a definition can be formulated in terms of equivalence classes of smooth functions on M. Informally, we will say that two smooth functions f and g are equivalent at a point x if they have the same first-order behavior near x.

: Turing's thesis: " Turing's thesis that every function which would naturally be regarded as computable is computable under his definition, i. e. by one of his machines, is equivalent to Church's thesis by Theorem XXX.

An alternative and equivalent definition of Carmichael numbers is given by Korselt's criterion.

A bit, or a group of malfunctioning physical bits ( not always the specific defective bit is known ; group definition depends on specific storage device ) is typically automatically fenced-out, taken out of use by the device, and replaced with another functioning equivalent group in the device, where the corrected bit values are restored ( if possible ).

A bit, or a group of malfunctioning physical bits ( not always the specific defective bit is known ; group definition depends on specific storage device ) is typically automatically fenced-out, taken out of use by the device, and replaced with another functioning equivalent group in the device, where the corrected bit values are restored ( if possible ).

) This orthogonality condition can be used to derive the formula for the IDFT from the definition of the DFT, and is equivalent to the unitarity property below.

In addition to the definition above, there is an equivalent definition.

Yet another equivalent definition of the ellipse is that it is the set of points that are equidistant from one point in the plane ( a focus ) and a particular circle, the directrix circle ( whose center is the other focus ).

The electric field was formally defined as the force exerted per unit charge, but the concept of potential allows for a more useful and equivalent definition: the electric field is the local gradient of the electric potential.

Boltzmann showed that this definition of entropy was equivalent to the thermodynamic entropy to within a constant number which has since been known as Boltzmann's constant.

Baslieus, a title which had long been used for Alexander the Great was already in common usage as the Greek word for the Roman emperor, but its definition and sense was " King " in Greek, essentially equivalent with the Latin Rex.

This statement is equivalent to the first of the above-given definitions of the inverse, and it becomes equivalent to the second definition if Y coincides with the codomain of ƒ.

In this case, the above definition can be characterized by the following equivalent statement:

In an equivalent formal definition, NP is the set of decision problems where the " yes "- instances can be decided in polynomial time by a non-deterministic Turing machine.

Most European domestic power supplies run at 230 V, so the current drawn by a particular European appliance ( in Europe ) will be less than for an equivalent American one ( in the United States ).< ref group =" Note "> The formula for power is given by

Upon heating, α-berkelium transforms into another phase with an fcc lattice ( but slightly different from β-berkelium ), space group Fmm and the lattice constant of 500 pm ; this fcc structure is equivalent to the closest packing with the sequence ABC.

Prior to the creation of the group management board in 2006, HSBC's chairman essentially held the duties of a chief executive at an equivalent institution, while HSBC's chief executive served as the deputy.

Erlang-B ( sometimes also written without the hyphen Erlang B ), also known as the Erlang loss formula, is a formula for the blocking probability derived from the Erlang distribution to describe the probability of call loss on a group of circuits ( in a circuit switched network, or equivalent ).

In mathematics, especially in category theory and homotopy theory, a groupoid ( less often Brandt groupoid or virtual group ) generalises the notion of group in several equivalent ways.

Gabbro () refers to a large group of dark, coarse-grained, intrusive mafic igneous rocks chemically equivalent to basalt.

For example, the orthogonal group O < sub > n </ sub >( R ) consists of matrices A with AA < sup > T </ sup > = 1, so the Lie algebra consists of the matrices m with ( 1 + εm )( 1 + εm )< sup > T </ sup > = 1, which is equivalent to m + m < sup > T </ sup > = 0 because ε < sup > 2 </ sup > = 0.

It is the proper symmetry group of a circle and the continuous equivalent of C < sub > n </ sub >.

As with modern criminal gangs, each member of the group had his own function: the equivalent of the ' hit-man ,' ' the lookout ,' and the ' getaway driver ' would be those Thugs tasked with luring travelers with charming words or acting as guardian to prevent escape of victims while the killing took place.

More precisely, if A is a finite set of generators for G then the word problem is the membership problem for the formal language of all words in A and a formal set of inverses that map to the identity under the natural map from the free monoid with involution on A to the group G. If B is another finite generating set for G, then the word problem over the generating set B is equivalent to the word problem over the generating set A.

The idea behind this GA evolution proposed by Emanuel Falkenauer is that solving some complex problems, a. k. a. clustering or partitioning problems where a set of items must be split into disjoint group of items in an optimal way, would better be achieved by making characteristics of the groups of items equivalent to genes.

In group theory, for example, we may have a group G acting on a set X, in which case we say that two elements of X are equivalent " up to the group action " if they lie in the same orbit.

So the universal algebraist's definition of a group really is equivalent to the usual definition.

Let G be a group with identity element e, N a normal subgroup of G ( i. e., N ◁ G ) and H a subgroup of G. The following statements are equivalent:

The statement that xR = M is equivalent to the surjectivity of the homomorphism that sends r to xr.

An equivalent algebraic approach starts from the observation that a super manifold is determined by its ring of supercommutative smooth functions, and that a morphism of super manifolds corresponds one to one with an algebra homomorphism between their functions in the opposite direction, i. e. that the category of supermanifolds is opposite to the category of algebras of smooth graded commutative functions.

The Cartan condition is equivalent to this bundle homomorphism being an isomorphism, so that η mod is a solder form.

If the operations are associative, each one defines the structure of a monoid on, and the conditions above are equivalent to the more abstract condition that * is a monoid homomorphism ( or vice versa ).

An equivalent definition of an associative R-algebra is a ring homomorphism such that the image of f is contained in the center of A.

called the Hurewicz homomorphism from the k-th homotopy group to the k-th homology group ( with integer coefficients ), which for k = 1 is equivalent to the canonical abelianization map

This definition is equivalent to a homomorphism ( reversing the operation before or after applying the map is equivalent ).

This is the analogue in semigroup theory of a ( left ) group action, and is equivalent to a semigroup homomorphism into the set of functions on X.