With the later development of abstract groups, this result of Lagrange on polynomials was recognized to extend to the general theorem about finite groups which now bears his name.

With only a finite number of jamming transmitters available at any given time, it may be more efficient to block clandestine stations intended for a large audience rather than a message intended for a single person.

With the regulator in place, and a finite value for the cutoff, divergent terms in the integrals then turn into finite but cutoff-dependent terms.

With < sup >< ω </ sup > denoting the set of all finite subsets of κ, a cardinal number κ such that for every function

Therefore, over the natural numbers, has at most minimal elements, a finite number .< ref > With more care, it is possible to show that one of and is at most, and that there is at most one minimal pair for each choice of one of the coordinates, from which it follows that there are at most minimal elements .</ ref >

With some simple algebra performed on finite sums, we can write for any complex s

With the norm in place, absolutely convergent series can be used instead of finite sums.

With a finite number of scenarios, two-stage stochastic linear programs can be modelled as large linear programming problems.

With Paul Erdős, he proved the Erdős – Szemerédi theorem on the number of sums and products in a finite set.

With the assumption that everything is finite dimensional, Ψ is a unital CP map between spaces of matrices

) With this approach one sees that Stone representation can also be regarded as a generalization of the representation of finite Boolean algebras by truth tables.

With higher performance processors and tools to rapidly create the volumetric tessellations, real-time finite element systems began to be used in games, beginning with Star Wars: The Force Unleashed that used Digital Molecular Matter for the deformation and destruction effects of wood, steel, flesh and pants using an algorithm developed by Dr. James O ' Brien as a part of his PhD thesis.

With a finite number of soldiers the total will be less than one.

With condensed detachment there are only a finite number of possible next steps in a proof.

With this technique liquid / dissolved, solid and gaseous samples may be analyzed directly.

With the development of cryo-electron microscopy of vitreous sections ( CEMOVIS ), it is now possible to observe samples from virtually any biological specimen close to its native state.

With enough samples, the graph of the histogram will approximate the density function of a chi-squared distribution with one degree of freedom.

With indefinitely large samples, limiting results like the central limit theorem describe the sample statistic's limiting distribution, if one exists.

With its eventual suite of up to 25 best-in-class instruments, SNS will give researchers detailed snapshots of smaller samples of physical and biological materials than previously possible.

With the abundance of PCR technology, primers that flank microsatellite loci are simple and quick to use, but the development of correctly functioning primers is often a tedious and costly process. A number of DNA samples from specimens of Littorina plena amplified using polymerase chain reaction with primers targeting a variable simple sequence repeat ( SSR, a. k. a. microsatellite ) locus.

With the invention of the polymerase chain reaction ( PCR ) technique, DNA profiling took huge strides forward in both discriminating power and the ability to recover information from very small ( or degraded ) starting samples.

With highly automated and sophisticated equipment, modern-day DNA profiling can process hundreds of samples each day.

With more than 400 patents and operative samples to his name, which were to play a major role in showing the way ahead, Karl Storz played a crucial role in the development of endoscopy.

With his 1989 release, Life Is ... Too Short, he began using replayed established funk riffs ( rather than samples ) with his beats.

With this in mind, the band used NASA recordings and electronic samples on their first two albums.

With a dearth of sound evidence and normative samples, it is tough to determine how much useful information can be gathered in this manner.

With the introduction of Jay Howard ( aka DJ Jedi Master Jay ) on turntables and samples, the more hard-edged, rap-influenced singles of 2000, " Here We Go / Sucker " and " Rip It Up ", boosted 28 Days ' audience.

With the samples collected, the Enterprise begins transit to Starfleet Medical, but finds one of the plague strains is growing, despite the presence of the containment field ; should it continue growing, it will rupture the field and expose those on board to the plague.

With a fixed number of training samples, the predictive power reduces as the dimensionality increases, and this is known as the Hughes effect or Hughes phenomenon ( named after Gordon F. Hughes ).

With large samples, a chi-squared test can be used in this situation.

With relatively large samples, however, a central limit theorem can be invoked such that hypothesis testing may proceed using asymptotic approximations.

Exceptions were " Sealed With A Kiss ", which was created exclusively from samples from songs about apples, and the final two tracks which were recorded at home on acoustic guitar.

With limited funds for advertising, Fox himself went door to door giving free samples, until demand skyrocketed.

It contained samples from Bizarre Inc's song " Playing With Knives ", and Say When!

With his Lifesiders faction's research still in its infancy, Lindsay and Vera Constantine secretly break the Interdict and bring back samples of Earth's abyssal life, providing the breakthroughs that make the Europa project a success.

With only four sound samples from faulty tapes playing at incorrect motor speeds ( due to improper settings of the 10 turn pitch knob ), accurate software-based Birotron sounds continue to remain unobtainable for musicians.

With gaseous and solid samples, Raman scattering may still be too weak to easily detect.

With the possible exception of gravitation, these interactions can usually be described in a set of calculational approximation methods known as perturbation theory, as being mediated by the exchange of gauge bosons between particles.

With a variety of oversimplified and conventional formulas at their disposal, they fell back on the nearest stereotypical approximation of what the phenomenon resembled, as they saw it.

With the death of Alice's father, the approximation provided between the two realms lead to a union between Amalric, who was elected king of Jerusalem and Alice's mother Isabella.

With this approximation, the above iterative scheme becomes the EM algorithm.

With the same argument, but halving the time step, is an order two approximation to, with.

With optimal slope of the dielectric or other phase shifter, ( in the thin lens approximation ) the contributions from all parts of the lens can be the same, making it an ideal lens for one wave length.

With a good approximation of the delta-v budget designers can estimate the fuel to payload requirements of the spacecraft using the rocket equation.

With a similar method and according to Derjaguin approximation, the van der Waals interaction energy between particles with different shapes can be calculated, such as energy between

With wide band systems this approximation no longer holds, which is typical in sonars.

With the help of Stirling's approximation, entropy can be simplified: