" Steve Huey of Allmusic asserted that despite being born into a family of entertainers, Janet Jackson has managed to emerge a " superstar " in her own right, rivaling not only several female recording artists including Madonna and Whitney Houston, but also her brother, while " successfully her image from a strong, independent young woman to a sexy, mature adult.

This result is an isotopy version of the strong Whitney embedding theorem.

Since Y is reflexive, the continuous dual of is equal to the image J ( X ) of X under the canonical embedding J, but the topology on X is not the strong topology, that is equal to the norm topology of Y.

Whitney's strong embedding theorem states that.

Every elementary embedding is a strong homomorphism, and its image is an elementary substructure.

They are a set of axioms strong enough to prove many important facts about number theory and they allowed Gödel to establish his famous second incompleteness theorem.

* The strong perfect graph theorem

For example, the Fitting lemma shows that the endomorphism ring of a finite length indecomposable module is a local ring, so that the strong Krull-Schmidt theorem holds and the category of finite length modules is a Krull-Schmidt category.

Gödel's theorem, informally stated, asserts that any formal theory expressive enough for elementary arithmetical facts to be expressed and strong enough for them to be proved is either inconsistent ( both a statement and its denial can be derived from its axioms ) or incomplete, in the sense that there is a true statement about natural numbers that can't be derived in the formal theory.

Moreover, Gödel's second incompleteness theorem shows that the consistency of sufficiently strong effective theories of arithmetic can be tested in a particular way.

A naive Bayes classifier is a simple probabilistic classifier based on applying Bayes ' theorem with strong ( naive ) independence assumptions.

These tests are over twice as strong as tests based on Fermat's little theorem.

The strong form of Dirichlet's theorem implies that

3 ) argues that Gödel's incompleteness theorem shows that it is not possible to produce finitistic consistency proofs of strong theories.

Rogers uses the term weak recursion theorem for the first recursion theorem and strong recursion theorem for the second recursion theorem.

These tests are twice as strong as tests based on Fermat's little theorem.

According to the strong perfect graph theorem, the perfect graphs have a forbidden graph characterization resembling that of bipartite graphs: a graph is bipartite if and only if it has no odd cycle as a subgraph, and a graph is perfect if and only if it has no odd cycle or its complement as an induced subgraph.

The bipartite graphs, line graphs of bipartite graphs, and their complements form four out of the five basic classes of perfect graphs used in the proof of the strong perfect graph theorem.

Other versions of the theorem involving the weak or strong energy condition also exist.

Again motivated by Mostow's strong rigidity theorem, Yau called for a notion of rank for general manifolds extending the one for locally symmetric spaces, and asked for rigidity properties for higher rank metrics.

* Penrose uses Gödel's incompleteness theorem ( which states that there are mathematical truths which can never be proven in a sufficiently strong mathematical system ; any sufficiently strong system of axioms will also be incomplete ) and Turing's halting problem ( which states that there are some things which are inherently non-computable ) as evidence for his position.

The Boolean prime ideal theorem is the strong prime ideal theorem for Boolean algebras.

Chow's theorem says that a subvariety of the projective space is analytic ( closed in the strong sense ) if and only if it is algebraic ( closed in the Zariski topology ).

In mathematical logic, the diagonal lemma or fixed point theorem establishes the existence of self-referential sentences in certain formal theories of the natural numbers -- specifically those theories that are strong enough to represent all computable functions.

The conjecture remained unresolved for 40 years, until it was established as the celebrated strong perfect graph theorem in 2002 by Chudnovsky, Robertson, Seymour, Thomas 2002.

He called for a greater attraction of industry and a stop to the piracy of industry by Southern states, and a strong fight against discrimination in business and industry.

In addition, Lincoln had to contend with reinforcing strong Union sympathies in the border slave states and keeping the war from becoming an international conflict.

Max Planck's strong form of the third law of thermodynamics states the entropy of a perfect crystal vanishes at absolute zero.

The strong anthropic principle ( SAP ) as explained by Barrow and Tipler ( see variants ) states that this is all the case because the Universe is compelled, in some sense, for conscious life to eventually emerge.

The fidelity of individual bodyguards is an important question as well, especially for leaders who oversee states with strong ethnic or religious divisions.

Since Lavoisier's knowledge of strong acids was mainly restricted to oxoacids, such as ( nitric acid ) and ( sulphuric acid ), which tend to contain central atoms in high oxidation states surrounded by oxygen, and since he was not aware of the true composition of the hydrohalic acids ( HF, HCl, HBr, and HI ), he defined acids in terms of their containing oxygen, which in fact he named from Greek words meaning " acid-former " ( from the Greek οξυς ( oxys ) meaning " acid " or " sharp " and γεινομαι ( geinomai ) meaning " engender ").

Specifically, after acknowledging the various popular theories in vogue at the time, of how atoms were reasoned to attach to each other, i. e. " hooked atoms ", " glued together by rest ", or " stuck together by conspiring motions ", Newton states that he would rather infer from their cohesion, that " particles attract one another by some force, which in immediate contact is exceedingly strong, at small distances performs the chemical operations, and reaches not far from the particles with any sensible effect.

The development of strong administrative structures, in particular those related to extraction of taxes, is closely associated with the intense warfare between predatory European states in the 17th and 18th centuries, or in Charles Tilly's famous formulation: " War made the state and the state made war ".

Nevertheless, Western states that survived into the latter half of the 20th century were considered " strong " by simple reason that they had managed to develop the institutional structures and military capability required to survive predation by their fellow states.

The " strong "-" weak " categorization is not the same as " Western "-" non-Western ", as some Latin American states like Argentina and Brazil and Middle Eastern states like Egypt and Israel are considered to have " strong " administrative structures and economic infrastructure.

The Cold War ( 1945 – 1989 ) provided a global network of material and ideological support that perpetuated civil wars, which were mainly fought in weak ex-colonial states, rather than the relatively strong states that were aligned with the Warsaw Pact and North Atlantic Treaty Organization.

The theory also has a strong orientalist bent which regards all Asian states as generally the same.

While the transition to a market economy proved difficult, many of the post-Soviet states have been experiencing strong, resource-based economic growth in recent years, though the levels vary substantially.

Ba ' athist regimes in power in Iraq and Syria have held strong similarities to fascism, they are radical authoritarian nationalist single-party states.

But foreign policy scholars such as Eric Nordlinger in his book Isolationism Reconfigured have argued that " A vision of Finlandization in America's absence runs up squarely against the European states ' long-standing Communist antipathies and wariness of Moscow's peaceful wiles, valued national traditions and strong democratic institutions, as well as their size and wherewithal.

These events marked the first time under the new constitution that the federal government used strong military force to exert authority over the states and citizens.

That same year, the strong and unwarrantable measures he adopted to subject the grand republic of San Marino to the papal states incurred the pope's displeasure, and left a historical scar in that place's memory.

In general, the princely states were strong supporters of the British regime, and the Raj left them alone.

It shows two main oxidation states, which are + 1 and + 3, with latter being more stable, whereas the only common oxidation state of gallium is + 3 and thallium shows + 1 more likely than + 3, with thallium ( III ) being a moderately strong oxidizing agent, while indium ( III ) is stable and indium ( I ) is a powerful reducing agent.

One inevitable ambiguity about these structures relates to the strong evidence that channels change conformation as they operate ( they open and close, for example ), such that the structure in the crystal could represent any one of these operational states.