Since the possible values of f are well-ordered, this property can be established by proving for any r ( not in I ) with minimal value of f ( r ) in its class.

Since this theorem applies in infinite-dimensional ( Banach space ) settings, it is the tool used in proving the infinite-dimensional version of the inverse function theorem ( see " Generalizations ", below ).

* Since 2010, Salimata Diabaté from Burkina Faso has performed as lead djembe player with her group Afro Faso Jeunesse, proving that female djembefolas can match their male counterparts in every respect.

Since 1999 the band has changed their sound to be up to the modern standards of production, following the worldwide trends and trying to achieve a more mature sound, after successfully proving that they could realize some of the best goa trance tracks ever.

Since it is proving difficult to further increase the internal clock speed of memory chips, these chips increase data rate by transferring data in larger blocks:

* Since decommissioning from British Army service, Abbots have become popular for " Tank-driving " adventures, proving much more economical to buy and run than the genuine article.

Since joining, van Bronckhorst has become a pivotal member of the Eredivisie team's squad, most recently proving to be something of a rock in an injury-hit side.

Since a maximal ideal in A is closed, is a Banach algebra that is a field, and it follows from the Gelfand-Mazur theorem that there is a bijection between the set of all maximal ideals of A and the set Δ ( A ) of all nonzero homomorphisms from A to C. The set Δ ( A ) is called the " structure space " or " character space " of A, and its members " characters.

Since the conclusion of the Baire category theorem is purely topological, it applies to these spaces as well.

Since, nonrelativistically, particles can have any statistics and any spin, there is no way to prove a spin-statistics theorem in nonrelativistic quantum mechanics.

Since there is a one-to-one correspondence between Borel regular measures in the interval and functions of bounded variation ( that assigns to each function of bounded variation the corresponding Lebesgue-Stieltjes measure, and the integral with respect to the Lebesgue-Stieltjes measure agrees with the Riemann-Stieltjes integral for continuous functions ), the above stated theorem generalizes the original statement of F. Riesz.

Since the equipartition theorem worked so well in describing the vibrational modes of the thermal object itself, it was trivial to assume that it would perform equally well in describing the radiative emission of such objects.

Since the theorem was also discovered independently by E. T. Whittaker, by Vladimir Kotelnikov, and by others, it's also known as the Nyquist – Shannon – Kotelnikov, Whittaker – Shannon – Kotelnikov, Whittaker – Nyquist – Kotelnikov – Shannon, WKS, as well as the Cardinal Theorem of Interpolation Theory.

Since any such axiom system would contain the finitary arithmetic as a subsystem, Gödel's theorem implied that it would be impossible to prove the system's consistency relative to that ( since it would then prove its own consistency, which Gödel had shown was impossible ).

Since the Poynting vector only occurs in Poynting's theorem as a divergence ∇ • S, the Poynting vector S is arbitrary to the extent that one can add a curl of a field F to S,

The transfer principle states that true first order statements about R are also valid in * R. For example, the commutative law of addition, x + y = y + x, holds for the hyperreals just as it does for the reals ; since R is a real closed field, so is * R. Since for all integers n, one also has for all hyperintegers H. The transfer principle for ultrapowers is a consequence of Łoś ' theorem of 1955.

Since the fundamental theorem of arithmetic applied to a non-zero integer n that is neither 1 nor − 1 also asserts uniqueness of the representation for p < sub > i </ sub > prime and e < sub > i </ sub > positive, a primary decomposition of ( n ) is essentially unique.

Since pre-abelian categories have all finite products and coproducts ( the biproducts ) and all binary equalisers and coequalisers ( as just described ), then by a general theorem of category theory, they have all finite limits and colimits.

Since Bloch's theorem applies only to periodic potentials, and since unceasing random movements of atoms in a crystal disrupt periodicity, this use of Bloch's theorem is only an approximation, but it has proven to be a tremendously valuable approximation, without which most solid-state physics analysis would be intractable.

Since field theory problems are more common in modern physics than mechanics problems, this field theory version is the most commonly used version of Noether's theorem.

Since ξ is a dummy variable of integration, and since the change in the boundary Ω is infinitesimal by assumption, the two integrals may be combined using the four-dimensional version of the divergence theorem into the following form

Since proofs are always finite and therefore involve only finitely many of the given sentences, the compactness theorem follows.

Since complete lattices cannot be empty, the theorem in particular guarantees the existence of at least one fixed point of f, and even the existence of a least ( or greatest ) fixed point.

Since the slope shown here reaches the horizontal limit of 0 as T → 0 then the implication is that ΔS → 0, which is the Nernst heat theorem.

Since N – 1 is composite, this convolution can be performed directly via the convolution theorem and more conventional FFT algorithms.

Since in Green's theorem is a vector pointing tangential along the curve, and the curve C is the positively-oriented ( i. e. counterclockwise ) curve along the boundary, an outward normal would be a vector which points 90 ° to the right, which would be.

Since a multiplication by an orthogonal matrix is a rotation, the theorem says that if the probability distribution of a random vector is unchanged by rotations and the components are independent, then the components are identically distributed and normally distributed.

Since symplectomorphisms preserve the symplectic 2-form and hence the symplectic-volumeform, Liouville's theorem in Hamiltonian mechanics follows.

Since the 1960s, composite airframes and quieter, more efficient engines have become available, and Concorde provided supersonic passenger service for more than two decades, but the most important lasting innovations have taken place in instrumentation and control.

Since cyclists ' legs are most efficient over a narrow range of pedaling speeds ( cadence ), a variable gear ratio helps a cyclist to maintain an optimum pedalling speed while covering varied terrain.

Since at least the 1960s, macroeconomics has been characterized by further integration as to micro-based modeling of sectors, including rationality of players, efficient use of market information, and imperfect competition.

Since all services are performed by usermode programs, efficient means of communication between programs are essential, far more so than in monolithic kernels.

Since agriculture was so much more efficient than hunting and gathering, private property appropriated by someone for farming left remaining hunter-gatherers with more land per person, and hence did not harm them.

Since only a few atoms must be excited into the upper laser level to form a population inversion, a four-level laser is much more efficient than a three-level one, and most practical lasers are of this type.

Since chemistry and nanotechnology rely on understanding quantum systems, and such systems are impossible to simulate in an efficient manner classically, many believe quantum simulation will be one of the most important applications of quantum computing.

Since then, several more efficient quinine total syntheses have been achieved, but none of them can compete in economic terms with isolation of the alkaloid from natural sources.

Since operations such as inserting, deleting, and finding values require worst-case time proportional to the height of the tree, this theoretical upper bound on the height allows red – black trees to be efficient in the worst case, unlike ordinary binary search trees.

Since IBM hardware does I / O via the use of channels and channel programs, TPF would generate very small and efficient channel programs to do its I / O-all in the name of speed.

Since this group was too large for efficient administration, a Council of Ten ( also called the Ducal Council or the Signoria ), controlled much of the administration of the city.

Since 3D models are the typical primary output for CAD design, users in these industries require an efficient and accurate method of generating 2D computational drawings directly from the 3D models.

Since the 1960s, composite airframes and quieter, more efficient engines have become available, and Concorde provided supersonic passenger service for more than two decades, but the most important lasting innovations have taken place in instrumentation and control.

Since single cycle ( gas turbine only ) power plants are less efficient than combined cycle plants, they are usually used as peaking power plants, which operate anywhere from several hours per day to a few dozen hours per year — depending on the electricity demand and the generating capacity of the region.

Since the rotors can be configured to be more efficient for propulsion ( e. g. with root-tip twist ) and it avoids a helicopter's issues of retreating blade stall, the tiltrotor can achieve higher speeds than helicopters.

Since 2007, UNFPA has decentralized its operations as a way to become a more field-centred, efficient and strategic partner to the countries it serves.

Since a heightfield occupies a box volume itself, recursively subdividing this box into eight subboxes ( hence the ' oct ' in octree ) until individual heightfield elements are reached is efficient and natural.

Since then, rapid urbanization has increased demand for more efficient public transport, resulting in dwindling cycle rickshaw numbers.

Since 2003, using these two software enhancements, UKIRT has carried out highly efficient flexible scheduling-tailoring observation execution to the prevailing weather conditions.

Since this amino acid is normally the cleavage site for aPC, the mutation prevents efficient inactivation of factor V.

Since many insects are attracted to ultraviolet ( it is also reflected by many flowers ), this is an efficient lure.

Since electric vehicles use whatever electricity is delivered by their electrical utility / grid operator, electric vehicles can be made more or less efficient, polluting and expensive to run, by modifying the electrical generating stations.

Since there is absolutely no page copying involved ( consuming additional memory ), this technique is highly efficient when a process needs to execute a blocking command using the child process.

Since at the time Visual Basic was an interpreted language with limited speed, VBXs, which were most often written in C or C ++, provided the ability to create very powerful and efficient user interface sub-components which could be assembled into applications using Visual Basic as the glue.