Although the Af calculation is obvious by analogy with that for gravitational field and osmotic pressure, it is interesting to confirm it by a method which can be generalized to include related effects.

Although various notations are used throughout the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R, the most common are " a ~ b " and " a ≡ b ", which are used when R is the obvious relation being referenced, and variations of " a ~< sub > R </ sub > b ", " a ≡< sub > R </ sub > b ", or " aRb ".

Although technically the powder's quantity is unlimited if you are a dedicated sportsman, you should not have excess stock of powder in your possession for obvious reasons.

Although there is no obvious restriction that any given regular polygon cannot be a face of a Johnson solid, it turns out that the faces of Johnson solids always have 3, 4, 5, 6, 8, or 10 sides.

Although a crank seemed the obvious solution to the conversion Watt and Boulton were stymied by a patent for this, whose holder, James Pickard, and associates proposed to cross-license the external condenser.

Although from a humble background, his obvious intellect and outstanding abilities saw him promoted up through the ecclesiastical hierarchy.

Although somewhat dated in terms of " information culture " arguments, not to mention the obvious technological problems of a system that could work effectively for billions of years and requires multiple lifeforms agreeing on certain basics of communications technologies, this hypothesis is actually testable ( see below ).

Although the rule still applies, the visible fuselage " waisting " can only be seen on a few aircraft, such as the B-1B Lancer, Learjet 60, and the Tupolev Tu-160 ' Blackjack ' — the same effect is now achieved by careful positioning of aircraft components, like the boosters and cargo bay on rockets ; the jet engines in front of ( and not directly below ) the wings of the Airbus A380 ; the jet engines behind ( and not purely at the side of ) the fuselage of a Cessna Citation X ; the shape and location of the canopy on the F-22 Raptor ; and the image of the Airbus A380 above showing obvious area rule shaping at the wing root, which is practically invisible from any other angle.

Although X10 offered interesting and powerful functionality, it had become obvious that the X protocol could use a more hardware-neutral redesign before it became too widely deployed, but MIT alone would not have the resources available for such a complete redesign.

Although the party's conservatives could not stop FDR from winning the nomination, the obvious physical decline in the President's appearance, as well as rumors of secret health problems, led many delegates and party leaders to strongly oppose Henry A. Wallace.

Although PRNGs will repeat their results after they reach the end of their period, a repeated result does not imply that the end of the period has been reached, since its internal state may be larger than its output ; this is particularly obvious with PRNGs with a 1-bit output.

Although the sentence is grammatically correct, no obvious understandable meaning can be derived from it, and thus it demonstrates the distinction between syntax and semantics.

Although the surface area of the parachute design appears to be too small to offer effective resistance to the friction of the air and the wooden base-frame is superfluous and potentially harming, the revolutionary character of the new concept is obvious.

Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems, and especially those involving linear differential equations with constant coefficients, for which the eigensolutions are sinusoids.

Although he was an obvious selection to tour England, Bradman's unorthodox style raised doubts that he could succeed on the slower English pitches.

Although a simple instrument, the method of playing the maracas is not obvious.

Although far from the centre of power in Scotland, Dumfries had obvious strategic significance sitting as it does on the edge of Galloway and being the centre of control for the south west of Scotland.

Although it was long believed that Ramesses III's body showed no obvious wounds, a recent examination of the mummy by a German forensic team, televised in the documentary Rameses on the Science Channel in 2011, showed excess bandages around the neck.

Although he had obvious high expectations and his obvious scientific talents, the first fifteen years of Eliot's career were less than auspicious.

Although the most obvious physical remains of the Silures are hillforts such as those at Llanmelin and Sudbrook, there is also archaeological evidence of roundhouses at Gwehelog, Thornwell ( Chepstow ) and elsewhere, and evidence of lowland occupation notably at Goldcliff.

Although the injustice of Cassius ' condemnation today appears obvious, his guilt was accepted by most ancient historians.

Although the historical section was attacked at the time, and ever since, as being a work of obvious fiction, the work remains a milestone in masonic history.

Although he initially considered himself a Whig, it became obvious to many contemporaries that his sympathies were largely Tory and he did not closely align with any of the Whig Factions in Parliament.

Although cheap and effective in enabling use of some software that only used official ROM entry points for text output, this solution proved very slow because the Electron had to be placed into an 80-byte-pitch display to be able to get anywhere near to reproducing mode 7 and the CPU spent a lot of time drawing approximations of mode 7 characters and graphics that in a hardware solution would be achieved without any CPU processing.

Although it is widely accepted that Descartes explained the problem cogently, few later philosophers have been happy with his solution, and his ideas about the pineal gland have especially been ridiculed.

Although once thought to be a simple solution of molecules, multiple levels of organization exist in the cytosol.

Although many modern commentators have demurred from Hume's solution, some have notably concurred with it, seeing his analysis of our epistemic predicament as a major contribution to the theory of knowledge.

Although Abel had already proved the impossibility of a " quintic formula " by radicals in 1824 and Ruffini had published a solution in 1799 that turned out to be flawed, Galois ' methods led to deeper research in what is now called Galois theory.

Although the 140 mm gun was considered a real interim solution it was decided after the fall of the Soviet Union that the increase in muzzle energy was not worth the increase in weight, and therefore more money was poured into research programs that could augment the muzzle energy of existing guns to match the 140 mm gun without the incrementing weight disadvantages.

Although that instrument still survives in working order, it is no longer in use and now the meridian of origin of the world's longitude and time is not strictly defined in material form but from a statistical solution resulting from observations of all time-determination stations which the BIPM takes into account when co-ordinating the world's time signals.

Although writing out the fundamental solution ( x < sub > 1 </ sub >, y < sub > 1 </ sub >) as a pair of binary numbers may require a large number of bits, it may in many cases be represented more compactly in the form

Although the law can be used to tell whether any quadratic equation modulo a prime number has a solution, it does not provide any help at all for actually finding the solution.

Although it's possible to augment simple reference counts in a variety of ways, often a better solution can be found by performing reference counting in a fundamentally different way.

Although Singapore's laws are inherited from British and British Indian laws, including many elements of English common law, the PAP has also consistently rejected liberal democratic values, which it typifies as Western and states that there should not be a ' one-size-fits-all ' solution to a democracy.

Although crystals were once loaded into glass capillaries with the crystallization solution ( the mother liquor ), a modern approach is to scoop the crystal up in a tiny loop, made of nylon or plastic and attached to a solid rod, that is then flash-frozen with liquid nitrogen.

Although Nasser and his Arab allies had been defeated, Arafat and Fatah could claim a victory, in that the majority of Palestinians, who had up to that time tended to align and sympathize with individual Arab governments, now began to agree that a ' Palestinian ' solution to their dilemma was indispensable.

Although this solution is effective, it leads to many problems.

Although considered an interim solution, Ramallah has become the de facto capital of the Palestinian Authority, hosting almost all governmental headquarters.

Although in this context an embedded system is usually more complex than a traditional solution, most of the complexity is contained within the microcontroller itself.

Although genetic algorithms and neural networks can perform just as well as fuzzy logic in many cases, fuzzy logic has the advantage that the solution to the problem can be cast in terms that human operators can understand, so that their experience can be used in the design of the controller.

Although the three-peg version has a simple recursive solution as outlined above, the optimal solution for the Tower of Hanoi problem with four pegs ( called Reve's puzzle ), let alone more pegs, is still an open problem.

Although originally she was to have worked on X-ray diffraction of proteins and lipids in solution, Randall redirected her work to DNA fibers before she started working at King's since Franklin was to be the only experienced experimental diffraction researcher at King's in 1951.

Although Richard's solution to the paradox did not gain favor with mathematicians, predicativism is an important part of the study of the foundations of mathematics.

Although apparently a solution to the problem, it is a fix by pure chance and anything else that changes the behaviour of the threads could cause it to resurface — for example on a computer with a different scheduler.