Because the four color theorem is true, this is always possible ; however, because the person drawing the map is focused on the one large region, he fails to notice that the remaining regions can in fact be colored with three colors.

Because of the connection between the Riemann zeta function and π ( x ), the Riemann hypothesis has considerable importance in number theory: if established, it would yield a far better estimate of the error involved in the prime number theorem than is available today.

Because Presburger arithmetic is decidable, automatic theorem provers for Presburger arithmetic exist.

Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance, especially in polynomial interpolation.

Because of the convolution theorem, the integral becomes a simple product,

Because this procedure of recomputing the electronic wave functions as a function of an infinitesimally changing nuclear geometry is reminiscent of the conditions for the adiabatic theorem, this manner of obtaining a PES is often referred to as the adiabatic approximation and the PES itself is called an adiabatic surface.

Because it is better known than the first recursion theorem, it is sometimes called just the recursion theorem.

Because of the Mordell – Weil theorem, Faltings ' theorem can be reformulated as a statement about the intersection of a curve C with a finitely generated subgroup Γ of an abelian variety A. Generalizing by replacing C by an arbitrary subvariety of A and Γ by an arbitrary finite-rank subgroup of A leads to the Mordell – Lang conjecture, which has been proved.

Because is usually a theorem or axiom, a consequence is that the negation of false () is true.

Because S, V, and N < sub > i </ sub > are extensive variables, Euler's homogeneous function theorem allows easy integration of dU:

Because of the central limit theorem, many test statistics are approximately normally distributed for large samples.

Because Ronald Coase himself did not originally intend to set forth any one particular theorem, it has largely been the effort of others who have developed the loose formulation of the Coase theorem.

Because of Cantor's theorem each set in the preceding sequence has cardinality strictly greater than the one preceding it.

Because of this lack of information, the computer's programmers have to implement systems based on the Bayes theorem, Nash equilibrium, Monte Carlo simulation or neural networks, all of which are imperfect techniques.

Because of the time hierarchy theorem, these classes form a strict hierarchy ; we know that, and on up.

Because this characterization is unaffected by graph complementation, it immediately implies the weak perfect graph theorem.

Because of Noether's theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation law.

Because Berge's forbidden graph characterization is self-complementary, the weak perfect graph theorem follows immediately from the strong perfect graph theorem.

Because of this characterization and Wagner's theorem characterizing the planar graphs as the graphs with no or graph minor, it follows that a graphic matroid is co-graphic if and only if is planar.

Because computer algorithms and programs had been used as early as 1956 to test and validate mathematical theorems, such as the four color theorem, social scientists and systems dynamicists anticipated that similar computational approaches could " solve " and " prove " analogously formalized problems and theorems of social structures and dynamics.

Because God is what He is, the laws of the universe, material and spiritual, are what they are.

Because of these involvements in the matter at stake, Boniface lacked the impartiality that is supposed to be an essential qualification for the position of arbiter, and in retrospect that would seem to be sufficient reason why the English embassies to the Curia proved so fruitless.

Because of the means of publication -- science-fiction magazines and cheap paperbacks -- and because dystopian science fiction is still appearing in quantity the full range and extent of this phenomenon can hardly be known, though one fact is evident: the science-fiction imagination has been immensely fertile in its extrapolations.

Because administration of the forest highway system is a responsibility of the Secretary of Commerce with maintenance provided by the States and counties, this Development Program for the National Forests does not include estimates of the funds needed to maintain the forest highway system nor to construct the additions to it that are needed.

Because the bobbin-to-cone winding process is a relatively high-cost operation for the mill, the almost complete automation provided by the Uniconer can mean important economies in textile production, at the same time upgrading quality.

Because of its importance, and because the lack of price competition is well recognized, the industry is under considerable public pressure not to raise its price any more than could be justified by cost increases.

Because community not severalty of property is the law of nature no man can assert an absolutely unalterable right to what is his.

Because of undesirable flavors, odors, colors, and generally low palatability associated with radiation treatment of this magnitude, the inactivation of enzymes is best accomplished prior to irradiation by the conventional heat-processing methods of blanching.

Because agricultural activities are seasonal and the areas of production and harvest of many foods are widely scattered geographically, and because of the high cost of transporting bulk food items any substantial distance to a central processing location, the use of large central processing stations, where low-cost radiation facilities approaching the megawatt range might be utilized, is inherently impracticable.

Because the rake angle Af at the tip of the knife is very much smaller ( or even negative ) when compared to the value of **yc for the major portion of the knife, a very rapid increase in cutting force with thickness will result.

Because the food is selected with thought for its nutritional value, care for its origin, and prepared in a manner that retains the most nutrients, the food does taste good.

-- Held out of Texas Tech's sweat-suits drill Monday at Lubbock was tackle Richard Stafford, who is undergoing treatment for a leg injury suffered in the Raiders' 38-7 loss to Texas A & M Because of its important game with Arkansas coming up Saturday, Baylor worked out in the rain Monday -- mud or no mud.

Because it is so large a state, with marked contrasts in population density, the organization of the New York co-operative offers a cross-section of how the plan works.

Because of this Dr. Bonnor holds that the universe is becoming more thinly populated by stars and whatever else is there.

Because all living beings possess a soul, great care and awareness is essential in one's actions.

Because anthropology developed from so many different enterprises ( see History of Anthropology ), including but not limited to fossil-hunting, exploring, documentary film-making, paleontology, primatology, antiquity dealings and curatorship, philology, etymology, genetics, regional analysis, ethnology, history, philosophy, and religious studies, it is difficult to characterize the entire field in a brief article, although attempts to write histories of the entire field have been made.

Because most embryonic development is outside the parental body, it is subject to many adaptations due to specific ecological circumstances.

Because experimentation is iterative, the

Because the range of possible values for K < sub > a </ sub > spans many orders of magnitude, a more manageable constant, pK < sub > a </ sub > is more frequently used, where pK < sub > a </ sub >

Because each governor held equal imperium to the equivalent magistrate, they were escorted by the same number of lictors ( 12 ) and could only be vetoed by a reigning Consul or Praetor.

Because many outstanding problems in number theory, such as Goldbach's conjecture are equivalent to solving the halting problem for special programs ( which would basically search for counter-examples and halt if one is found ), knowing enough bits of Chaitin's constant would also imply knowing the answer to these problems.

Because all these different attempts at formalizing the concept of " effective calculability / computability " have yielded equivalent results, it is now generally assumed that the Church – Turing thesis is correct.

Because silver was worth less than its legal equivalent in gold, taxpayers paid their government bills in silver, while international creditors demanded payment in gold, resulting in a depletion of the nation's gold supply.

Because land was equivalent to power, these powerful men could try to claim the crown.

Because the and states are energetically equivalent, neither state is favored, so this process has the effect of randomizing the states.

Because there is no physical polarity along the director axis, n and-n are fully equivalent.

Because of the symmetry of cubic crystals, it is possible to change the place and sign of the integers and have equivalent directions and planes:

Because of this correspondence, digital signatures are often described as based on public-key cryptosystems, where signing is equivalent to decryption and verification is equivalent to encryption, but this is not the only way digital signatures are computed.

Because of its large internal surface area, coal stores 6 to 7 times more gas than the equivalent rock volume of a conventional gas reservoir.

Because of the fundamental relationship between force and energy, this definition is equivalent to the first definition.

Because one or more crowns, alone or as part of a more elaborate design, often appear on coins, several monetary denominations came to be known as ' a crown ' or the equivalent word in the local language, such as krone.

Because of a poor fund of knowledge, less intelligent persons sometimes mistake the conditions of the shadow world to be equivalent to those of the spiritual world, and thus they mistake the Lord and His pastimes in the material world to be one with the conditioned souls and their activities.

Because of this design, having no significant movement of heavy mechanical parts during the firing cycle, the recoil produced is only the ' true ' recoil, equivalent to the equal and opposite reaction to the pellet and air volume's acceleration up the bore.

Because a magnitude of one is 0 dB, the gain margin is simply one of the equivalent forms: 20 log < sub > 10 </ sub >( | βA < sub > OL </ sub >|< sub > 180 </ sub >) = 20 log < sub > 10 </ sub >( | A < sub > OL </ sub >|< sub > 180 </ sub >) − 20 log < sub > 10 </ sub >( 1 / β ).

Because the couple's bank account is in Robert's name, Howard obtains Robert's consent to buy the letter, but he does so deceitfully, lying about and trivializing its content, leaving out the true circumstances, and giving the man no idea that the price is equivalent to almost all the money he has in the bank.

Because radial directions around the aircraft's direction of travel are equivalent, the shock forms a Mach cone with the aircraft at its tip.

Because every formula is equivalent to a formula in prenex normal form, every formula with no set quantifiers is assigned at least one classification.

Because of the nature of its investigations, the subcommittee is considered by some to be the Senate equivalent to the older House Un-American Activities Committee ( HUAC ).

Because they can feed their chicks constantly throughout the day, the chicks fledge faster than equivalent sized auks that are only provisioned at night.

Because nucleic acids, such as DNA and RNA, are unbranched polymers, this specification is equivalent to specifying the sequence of nucleotides that comprise the molecule.

Because it is functionally equivalent to rail, the Los Angeles County Metropolitan Transportation Authority presents this line as an integral part of its rail transit system, distinct from its " Rapid " lines, which run in mixed traffic.

Because the Button plan or local equivalent never existed in New Zealand, the two initial generations of Holden Astra ( LB / LC and LD ) were thus never available in that market.