An equivalent theoretical formulation, combinatory logic, is commonly perceived as more abstract than lambda calculus and preceded it in invention.

In the second formulation, the universe and the divine are not ontologically equivalent.

An equivalent formulation is that a subset of R < sup > n </ sup > is sequentially compact if and only if it is closed and bounded.

An equivalent formulation of this law is:

In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness of the reals ( or some other equivalent formulation of completeness ), which is not an algebraic concept.

An equivalent formulation of the lemma is therefore:

They are mathematically equivalent in so far as the Hamilton-Jacobi formulation applies, i. e., spin-less particles.

This formulation is equivalent to BCT1 and is sometimes more useful in applications.

Although there are many ( equivalent ) formulations of what it means for a ring, R, to be Noetherian, one formulation dictates that any ascending chain of ideals in R terminates.

To get an equivalent formulation of the schema, first replace anything of the form F ( X ) with a new variable Y.

An equivalent formulation that highlights the duality to the inverse limit is that an element is equivalent to all its images under the maps of the directed system, i. e..

Secondly, if W occurs in an elementary sentence S, it is necessary to give an answer to the following questions ( that are — according to Carnap — equivalent formulation of the same question ):

The above model has an equivalent formulation as a latent-variable model.

However, null hypotheses can and do have " direction "— in many instances statistical theory allows the formulation of the test procedure to be simplified, thus the test is equivalent to testing for an exact identity.

There are also a number other closely related theorems: an equivalent formulation of this theorem using line bundles and a generalization of the theorem to algebraic curves.

Solving the unconstrained problem is equivalent to finding a point on the convex hull of the family of solutions to an equivalent constrained formulation of the problem.

An equivalent formulation, is the following: If < sub > 1 </ sub >, ..., < sub ></ sub > are distinct algebraic numbers, then the exponentials < sup > α < sub > 1 </ sub ></ sup >, ..., < sup >< sub ></ sub ></ sup > are linearly independent over the algebraic numbers.

An equivalent formulation is the following:

An equivalent formulation in extremal graph theory asks for the maximum number of edges in a bipartite graph with no complete bipartite subgraph K < sub > a, b </ sub >.

It is equivalent to the Schrödinger wave formulation of quantum mechanics, and is the basis of Dirac's bra-ket notation for the wave function.

Feynman showed that this formulation of quantum mechanics is equivalent to the canonical approach to quantum mechanics, when the Hamiltonian is quadratic in the momentum.

An equivalent formulation of this is to say that

" Although his answer is given merely as su eipas ( thou hast said it ), the Gospel of Mark states the answer as ego eimi ( I am ) and there are instances from Jewish literature in which the expression, " thou hast said it ", is equivalent to " you are right ".

Plutarch states that Metilius " boldly applied himself to the people in the behalf of Minucius ", and had Minucius granted powers equivalent to those of Fabius.

In a separate article, Page shows that the number of states in a black hole with a mass roughly equivalent to the Andromeda Galaxy is in the range of a googolplex.

This formula is indeed equivalent to φ because it states that for every x, if there is a y which satisfies Q '( x, y ), then ( something ) holds, and furthermore, we know that there is such a y, because for every x ', there is a y ' which satisfies Q '( x ', y ').

Since real political power belongs to the sole legal party, in certain states under Marxist constitutions of the constitutionally socialist state type inspired by the former Union of Soviet Socialist Republics ( USSR ) and its constitutive Soviet republics, there was no formal office of head of state, but rather the leader of the legislative branch was considered to be the closest common equivalent of a head of state as a natural person.

* The polis in Greek Antiquity and the equivalent city states in the feudal era and later, ( many in Italy, the Holy Roman Empire, the Moorish taifa in Iberia, essentially tribal-type but urbanized regions throughout the world in the Maya civilization, etc.

Hilda Ellis Davidson ( 1948 ) states that Hel " as a goddess " in surviving sources seems to belong to a genre of literary personification, that the word hel is generally " used simply to signify death or the grave ," and that the word often appears as the equivalent to the English ' death ,' which Davidson states " naturally lends itself to personification by poets.

Because the and states are energetically equivalent, neither state is favored, so this process has the effect of randomizing the states.

An equivalent statement is the law of constant composition, which states that all samples of a given chemical compound have the same elemental composition by mass.

As preserved in the ancient Sarvastivada school's Sutra on Mindfulness of the Body in the Madhayama agama ( equivalent of Pali Kayagatasati ) it states that monks and nuns under practice should be ' Understanding ( having awareness in ) the four postures and states of being asleep and / or awake '.

* In Brazil, states ( estados ) are directly divided into municipalities ( municípios ), which are the smallest political-administrative divisions ; there is no equivalent to a county level.

He states that serving one another is equivalent to serving God.

Remittances are a major source of income, equivalent to 15 % of the country's GDP, which originate primarily from Costa Rica, the United States, and European Union member states.

The conjecture states: An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called homotopy equivalence: if a 3-manifold is homotopy equivalent to the 3-sphere, then it is necessarily homeomorphic to it.

It is not in any sense a judgement ... hence it is not in any way binding on any state ", while Charles de Visscher argued that in certain situations, an advisory opinion could be binding on the League of Nations Council and, under certain circumstances, some states ; M. Politis agreed, saying that the Court's advisory opinions were equivalent to a binding judgment.

Refraction is described by Snell's law, which states that for a given pair of media and a wave with a single frequency, the ratio of the sines of the angle of incidence θ < sub > 1 </ sub > and angle of refraction θ < sub > 2 </ sub > is equivalent to the ratio of phase velocities ( v < sub > 1 </ sub > / v < sub > 2 </ sub >) in the two media, or equivalently, to the opposite ratio of the indices of refraction ( n < sub > 2 </ sub > / n < sub > 1 </ sub >):

Moreover, in some states, the term town is equivalent to how civil townships are used in other states.

At a point in dialogue between the skaldic god Bragi and Ægir, Snorri himself begins speaking of the myths in euhemeristic terms and states that the historical equivalent of Víðarr was the Trojan hero Aeneas who survived the Trojan War and went on to achieve " great deeds ".

The well-ordering theorem, which is equivalent to the axiom of choice, states that every set can be well-ordered.

Hempel's equivalence condition states that when a proposition, X, provides evidence in favor of another proposition Y, then X also provides evidence in favor of any proposition which is logically equivalent to Y.