Unlike previous approximations, however, Oriani's two-term expression did not depend on a hypotheses regarding atmospheric temperature or air density in relation to altitude.

A very capable astronomer, Oriani's work began to attract considerable attention.

This explains the failure of the classical equipartition theorem for metals that eluded classical physicists in the late 19th century.

For compact groups, the Peter – Weyl theorem explains how one may get harmonics by choosing one irreducible representation out of each equivalence class of representations.

Such arguments are typically easier to check than purely symbolic ones — indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true.

( Note that the conditions under which the cancellation law holds are quite strict, and this explains why Fermat's little theorem demands that p be a prime in order to make a general case for all n. For example, 2 × 2 ≡ 2 × 5 ( mod 6 ), but we cannot conclude that 2 ≡ 5 ( mod 6 ), since 6 is not prime.

In probability theory, de Finetti's theorem explains why exchangeable observations are conditionally independent given some latent variable to which an epistemic probability distribution would then be assigned.

De Finetti's theorem explains a mathematical relationship between independence and exchangeability.

The deduction theorem explains why proofs of conditional sentences in mathematics are logically correct.

The preface of Book VII explains the terms analysis and synthesis, and the distinction between theorem and problem.

Earnshaw's theorem explains why a system of electrons is not stable and was invoked by Niels Bohr in his atom model of 1913 when criticizing J. J. Thomson's atom.

Pressley ( p. 185 ) explains this theorem as an expression of conservation of angular momentum about the axis of revolution when a particle slides along a geodesic under no forces other than those that keep it on the surface.

The reason why Euler and some other authors relate the Cauchy – Riemann equations with analyticity is that a major theorem in complex analysis says that holomorphic functions are analytic and viceversa.

* Regression ( economics ), Ludwig von Mises ' theorem that tries to explain why money is demanded in its own right

Sometimes corollaries have proofs of their own which explain why they follow from the theorem.

An interesting point is that energy is also a symmetry with respect to time, and momentum is a symmetry with respect to space, and these are the reasons why energy and momentum are conserved-see Noether's theorem.

None of these computations can show however why the Cayley – Hamilton theorem should be valid for matrices of all possible sizes n, so a uniform proof for all n is needed.

The reason why the fluctuation theorem is so fundamental is that its proof requires so little.

Most practitioners of QFT ignore Haag's theorem entirely, and it is currently unknown why QFT, and quantum electrodynamics in particular, produces accurate numbers given the lack of any axiomatic basis.

This would also partially explain why Greeks were so reserved in crediting Pythagoras with this theorem!

However, even under this weak-form assumption, the social choice cannot reach Pareto efficient, and that is why Sen's theorem is astonishing.

The sampling theorem describes why the input of an ADC requires a low-pass analog electronic filter, called the anti-aliasing filter: the sampled input signal must be bandlimited to prevent aliasing ( here meaning waves of higher frequency being recorded as a lower frequency ).

Another way to explain why an isotropic radiator cannot exist is by using the hairy ball theorem, which asserts that a continuous vector field tangent to the surface of the sphere must fall to zero at one or more points on the sphere.

There are a number of ways to see why nonabelian Yang-Mills theories in the Coulomb phase don't violate this theorem.

* did Napoléon have anything to do with the initial discovery or proof of the theorem, and if not why does it bear his name?

Assuming ZF is consistent, Kurt Gödel showed that the negation of the axiom of choice is not a theorem of ZF by constructing an inner model ( the constructible universe ) which satisfies ZFC and thus showing that ZFC is consistent.

Assuming ZF is consistent, Paul Cohen employed the technique of forcing, developed for this purpose, to show that the axiom of choice itself is not a theorem of ZF by constructing a much more complex model which satisfies ZF ¬ C ( ZF with the negation of AC added as axiom ) and thus showing that ZF ¬ C is consistent.

Illustration of using Dirac comb functions and the convolution theorem to model the effects of Sampling ( signal processing ) | sampling and / or periodic summation.

The Gauss-Markov theorem shows that the OLS estimator is the best ( minimum variance ), unbiased estimator assuming the model is linear, the expected value of the error term is zero, errors are homoskedastic and not autocorrelated, and there is no perfect multicollinearity.

Illustration of using Dirac comb functions and the convolution theorem to model the effects of sampling and / or periodic summation.

It is deduced from the model existence theorem as follows: if there is no formal proof of a formula then adding its negation to the axioms gives a consisten theory, which has thus a model, so that the formula is not a semantic consequence of the initial theory.

The name for the incompleteness theorem refers to another meaning of complete ( see model theory-Using the compactness and completeness theorems ).

Applying the completeness theorem to this result, gives the existence of a model of T where the formula C < sub > T </ sub > is false.

In fact, the model of any theory containing PA obtained by the systematic construction of the arithmetical model existence theorem, is always non-standard with a non-equivalent provability predicate and a non-equivalent way to interpret its own construction, so that this construction is non-recursive ( as recursive definitions would be unambiguous ).

In cybernetics, the Good Regulator or Conant-Ashby theorem is stated " Every Good Regulator of a system must be a model of that system ".

Examples of early theorems from classical model theory include Gödel's completeness theorem, the upward and downward Löwenheim – Skolem theorems, Vaught's two-cardinal theorem, Scott's isomorphism theorem, the omitting types theorem, and the Ryll-Nardzewski theorem.

Examples of early results from model theory applied to fields are Tarski's elimination of quantifiers for real closed fields, Ax's theorem on pseudo-finite fields, and Robinson's development of non-standard analysis.

An important step in the evolution of classical model theory occurred with the birth of stability theory ( through Morley's theorem on uncountably categorical theories and Shelah's classification program ), which developed a calculus of independence and rank based on syntactical conditions satisfied by theories.

An example of a theorem from geometric model theory is Hrushovski's proof of the Mordell – Lang conjecture for function fields.

In 1930, Gödel's completeness theorem showed that propositional logic itself was complete in a much weaker sense — that is, any sentence that is unprovable from a given set of axioms must actually be false in some model of the axioms.

The use of any parametric model is viewed skeptically by most experts in sampling human populations: " most sampling statisticians, when they deal with confidence intervals at all, limit themselves to statements about based on very large samples, where the central limit theorem ensures that these will have distributions that are nearly normal.

The Sonnenschein – Mantel – Debreu theorem shows that the standard model cannot be rigorously derived in general from general equilibrium theory.

When the logarithm of the likelihood ratio is used, the statistic is known as a log-likelihood ratio statistic, and the probability distribution of this test statistic, assuming that the null model is true, can be approximated using Wilks ' theorem.

An application of this notion is the decidability question: it follows from Post's theorem that a recursively axiomatized modal logic L which has FMP is decidable, provided it is decidable whether a given finite frame is a model of L. In particular, every finitely axiomatizable logic with FMP is decidable.