Since the kinetic energy is quadratic in the components of the velocity, by equipartition these three components each contribute k < sub > B </ sub > T to the average kinetic energy in thermal equilibrium.

Since R ≈ 2 cal /( mol · K ), equipartition predicts that the molar heat capacity of an ideal gas is roughly 3 cal /( mol · K ).

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 the proving of the theorem, efficient algorithms have been found for 4-coloring maps requiring only O ( n < sup > 2 </ sup >) time, where n is the number of vertices.

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 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 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 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 then he had worked at this and that, though some said his main interest was gambling.

Since 2002, when the 27-year civil war ended, the country has worked to repair and improve ravaged infrastructure and weakened political and social institutions.

Since 1998, Angola has successfully worked with the United Nations Security Council to impose and carry out sanctions on UNITA.

Since every day brought the arrival of the Spartans closer, the delay worked in favor of the Athenians.

Since the invention of the MIDI system in the early 1980s, for example, some people have worked on programs which map MIDI notes to an algorithm and then can either output sounds or music through the computer's sound card or write an audio file for other programs to play.

Since 1988's Dead Ringers, Cronenberg has worked with cinematographer Peter Suschitzky on each of his films ( see List of noted film director and cinematographer collaborations ).

Since his university retirement Husserl had " worked at a tremendous pace, producing several major works.

Since then, ecologists have worked to bridge their understanding of the degradation of the planet's ecosystems with environmental politics, law, restoration, and natural resources management.

Since Galen states that he is using observations of monkeys ( human dissection was prohibited ) to give an account of what the body looks like, Vesalius could portray himself as using Galen's approach of description of direct observation to create a record of the exact details of the human body, since he worked in a time when human dissection was allowed.

Since a third of the Malay workforce worked in the rubber industry it was important to develop alternative sources of employment.

Since the 1960s, he has worked with public interest, consumer, family farm, environmental and community organizations at the local, state and national level.

Since 1949, Mikhail Kalashnikov has lived and worked in Izhevsk, Udmurtia.

Since taking over the franchise, Attanasio has worked hard to build bridges with Milwaukee baseball fans, including giving away every seat to the final home game of 2005 free of charge and bringing back the classic " ball and glove " logo of the club's glory days on " Retro Friday " home games, during which they also wear versions of the team's old pinstriped uniforms.

Since he was 16 Limbaugh has worked a series of disc jockey jobs.

Since prehistoric times, long before the concept of national borders existed, the Sami people of Arctic Europe lived and worked in an area that stretches over the regions now known as Norway, Sweden, Finland and the Russian Kola Peninsula.

Since October 2000, the country has worked to stabilize and strengthen its bilateral relationships with neighboring countries.

Since 2007, Aniston has worked in a publicity campaign for the drink SmartWater ; on March 7, 2011, she released a YouTube video, titled " Jennifer Aniston Goes Viral ", for SmartWater, tripling online interest in the product within 24 hours of the release.

Since then he has lived and worked mostly in France, making art films.

Since the last incarcerees left in 1945, former incarcerees and others have worked to protect Manzanar and to establish it as a National Historic Site to preserve and interpret the site for current and future generations.

Since the Mac's screen resolution ( 72 dpi ) was exactly half of the ImageWriter's maximum, this worked perfectly, creating text at exactly the desired size.

Since the 1970s, Ebert has worked for the University of Chicago as a guest lecturer, teaching a night class on film.

Since 1960 Johns has worked closely with Universal Limited Art Editions, Inc ( ULAE ) in a variety of printmaking techniques to investigate and develop existing compositions.

Since the 19th century, the island has had a sizable community of Portuguese-Americans, concentrated primarily in the three down-Island towns of Oak Bluffs, Tisbury and Edgartown ; they traditionally worked alongside other island residents in whaling and fishing.

Since 1880 the sizeable Catholic and Protestant parties had worked together in the so-called Coalitie.