The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finite dimensional ; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent ; this reduces to Serre's theorem over a one-point space.

A typical application is furnished by the Arzelà – Ascoli theorem and in particular the Peano existence theorem, in which one is able to conclude the existence of a function with some required properties as a limiting case of some more elementary construction.

This is an application of Bayes ' theorem.

A theorem is a statement that can be derived from those axioms by application of these rules of inference.

Gibbs was well aware that the application of the equipartition theorem to large systems of classical particles failed to explain the measurements of the specific heats of both solids and gases, and he argued that this was evidence of the danger of basing thermodynamics on " hypotheses about the constitution of matter.

The application of the sampling theorem to images should be made with care.

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.

The application of Bayes ' theorem to update beliefs is called Bayesian inference.

For more on the application of Bayes ' theorem under the Bayesian interpretation of probability, see Bayesian inference.

The technique of pi – theorem ’ s usage (`` the method of dimensions ’’) became widely known due to the works of Rayleigh ( the first application of the pi – theorem in the general case to the dependence of pressure drop in a pipe upon governing parameters probably dates back to 1892, a heuristic proof with the use of series expansion, to 1894 ).

* Gauss-Bonnet theorem for an elementary application of curvature

The application of De Morgan's theorem to a conjunction is very similar to its application to a disjunction both in form and rationale.

This work also includes the application of probability theory to games of chance and his introduction of the theorem known as the law of large numbers.

It is an application of the noisy channel coding theorem to the archetypal case of a continuous-time analog communications channel subject to Gaussian noise.

This is an example of odd numbers playing a role in an advanced mathematical theorem where the method of application of the simple hypothesis of " odd order " is far from obvious.

repeatedly using integration by parts can evaluate integrals such as these, each application of the theorem lowers the power of x by one.

A second application of the compactness theorem shows that any theory that has arbitrarily large finite models, or a single infinite model, has models of arbitrary large cardinality ( this is the Upward Löwenheim – Skolem theorem ).

A third application of the compactness theorem is the construction of nonstandard models of the real numbers, that is, consistent extensions of the theory of the real numbers that contain " infinitesimal " numbers.

But this truth is distorted by its extreme application: the assumption of the separate existence of tradition.

Since the slogans have little application to reality and are sanctimonious to boot, the applause is faint even in areas of the world where we should expect to find the greatest affection for free government.

Upon receipt of an application which the Export-Import Bank is prepared to consider, the Export-Import Bank will inform the Department of Economic Affairs of the identity of the applicant, the nature of the proposed business, the amount of the proposed loan, and the general purposes for which the loan proceeds would be expended.

When the Export-Import Bank is prepared to act favorably upon an application, it will so notify the Department of Economic Affairs and will indicate the interest rate and the repayment period which would be used under the proposed loan.

Within sixty days after the receipt of notice that the Export-Import Bank is prepared to act favorably upon an application the Department of Economic Affairs will indicate to the Export-Import Bank whether or not the Department of Economic Affairs has any objection to the proposed loan.

In this article we will concentrate on the advances in the application of electronics in bio-medical research laboratories because this is where tomorrow's commonplace equipment originates.

The best known field of application for builders is in heavy-duty, spray-dried detergent formulations for household use.

Hard-surface cleaning in household application is represented by two classes of alkaline products: ( 1 ) the formulations made expressly for machine dishwashers, and ( 2 ) the general-purpose cleaners used for walls and woodwork.

In a typical application -- the making of rigid urethane foam sandwich panels -- an amount of foam mixture calculated to expand 10 to 20% more than the volume of the panel is poured into the panel void and the top of the panel is locked in place by a jig.

Senator McClellan is proposing the application of anti-trust measures to unions in transportation.

In 2009, the American Anthropological Association's Commission on the Engagement of Anthropology with the US Security and Intelligence Communities released its final report concluding, in part, that, " When ethnographic investigation is determined by military missions, not subject to external review, where data collection occurs in the context of war, integrated into the goals of counterinsurgency, and in a potentially coercive environment – all characteristic factors of the HTS concept and its application – it can no longer be considered a legitimate professional exercise of anthropology.

In the typical application of ANOVA, the null hypothesis is that all

application of the method is best determined by problem pattern recognition

* One-way ANOVA is used to test for differences among two or more independent groups ( means ), e. g. different levels of urea application in a crop.

This is usually done on the basis that the lower court judge erred in the application of law, but it may also be possible to appeal on the basis of court misconduct, or that a finding of fact was entirely unreasonable to make on the evidence.

A common application of the compounds of sodium is the sodium-vapour lamp, which emits very efficient light.

* To obtain a declaration on a legal document, such as an application for voter registration, that the information provided by the applicant is truthful to the best of the applicant's knowledge.

An emerging application is the use of antimony in microelectronics.

The most stable of these is < sup > 124 </ sup > Sb with a half-life of 60. 20 days, which has an application in some neutron sources.

The second longest-lived isotope of astatine, astatine-211, is the only one currently having any commercial application, being employed in medicine to diagnose and treat some diseases via its emission of alpha particles ( helium-4 nuclei ).

In computing, an applet is any small application that performs one specific task that runs within the scope of a larger program, often as a plug-in.

Provided that an applet is hosted by an operating system, it can function as any other normal software application but is and performs only a small set of tasks.

The Department of Justice has published " ADA 2010 Revised Requirements: Accessible Pools-Means of Entry and Exit ", which is designed to assist pool owners and operators with understanding the new accessibility requirements, the application of the requirements and the longstanding obligations of pool owners and operators in connection with the new requirements.