Although equipartition provides a simple derivation of the ideal-gas law and the internal energy, the same results can be obtained by an alternative method using the partition function.

Although flawed, Kempe's original purported proof of the four color theorem provided some of the basic tools later used to prove it.

Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic.

Although the proof is necessary to produce a theorem, it is not usually considered part of the theorem.

Although the virial theorem depends on averaging the total kinetic and potential energies, the presentation here postpones the averaging to the last step.

Although this at first appears to be a stronger statement, it is a direct consequence of the other form of the theorem, through the use of successive polynomial division by linear factors.

Although the issue of existence and uniqueness of solutions of ordinary differential equations has a very satisfactory answer with the Picard – Lindelöf theorem, that is far from the case for partial differential equations.

Although useful in its own right, the version of her theorem just given was a special case of the general version she derived in 1915.

Although there are some generalizations of the compactness theorem to non-first-order logics, the compactness theorem itself does not hold in them.

Although there is no proof that a smooth group order will be found in the Hasse-interval, by using heuristic probabilistic methods, the Canfield-Erdős-Pomerance theorem with suitably optimized parameter choices, and the L-notation, we can expect to try L < nowiki >/ 2, √ 2 < nowiki ></ nowiki > curves before getting a smooth group order.

Although he didn't publish much in the field, Dirichlet lectured on probability theory and least squares, introducing some original methods and results, in particular for limit theorems and an improvement of Laplace's method of approximation related to the central limit theorem.

Although that is strictly speaking a question about a real vector bundle ( the " hairs " on a ball are actually copies of the real line ), there are generalizations in which the hairs are complex ( see the example of the complex hairy ball theorem below ), or for 1-dimensional projective spaces over many other fields.

Although apparently a rather special case of the above theorem, the general result can be reduced to this simpler case.

Although in general these two can be different there is a theorem that, under very mild conditions, gives the equality of the two.

Although Desargues ' theorem chooses different roles for these ten lines and points, the Desargues configuration itself is more symmetric: any of the ten points may be chosen to be the center of perspectivity, and that choice determines which six points will be the vertices of triangles and which line will be the axis of perspectivity.

" Although this conjecture is true, most of its known proofs depend on the theory of separable and purely inseparable extensions ; for instance, in the case corresponding to the extension being separable, one known proof involves the use of the primitive element theorem in the context of Galois extensions.

Although Mordell's theorem shows that the rank of an elliptic curve is always finite, it does not give an effective method for calculating the rank of every curve.

Although the deduction theorem could be taken as primitive rule of inference in such systems, this approach is not generally followed ; instead, the deduction theorem is obtained as an admissible rule using the other logical axioms and modus ponens.

Although Pappus's Theorem usually refers to Pappus's hexagon theorem, it may also refer to Pappus's centroid theorem.

Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn – Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

Although the full description of the splitting of every prime p in a general Galois extension is a major unsolved problem, the Chebotarev density theorem says that the frequency of the occurrence of a given pattern, for all primes p less than a large integer N, tends to a certain limit as N goes to infinity.

Although the play does show a certain structural amateurishness ( there are eleven acts varying in length from twenty-five seconds to an hour and a half ), the statement it makes concerning the ceaseless yearning and searching of youth is profound and worthy of our attention.

Although his name and history were taken from his early life with his adoptive Earth parents, everything about Clark was staged for the benefit of his alternate identity: as a reporter for the Daily Planet, he receives late-breaking news before the general public, has a plausible reason to be present at crime scenes, and need not strictly account for his whereabouts as long as he makes his story deadlines.

Although much is known about the mechanisms behind the spread of cholera, this has not led to a full understanding of what makes cholera outbreaks happen some places and not others.

Although this field remains largely confined to science fiction, the possibility of intelligent extraterrestrial life makes the question of alien language a credible topic for scientific and philosophical speculation.

Although the fundamental group in general depends on the choice of base point, it turns out that, up to isomorphism ( actually, even up to inner isomorphism ), this choice makes no difference as long as the space X is path-connected.

" Although throughout his book Camus is concerned to present " the rebel " as a preferred alternative to " the revolutionary " he nowhere acknowledges that this distinction is taken from the one that Stirner makes between " the revolutionary " and " the insurrectionist ".

The American Association for the Advancement of Science also recognized the need for increased understanding between scientists and lawmakers in its newsletter on Science and Technology in Congress, " Although most individuals would agree that sound science is preferable to junk science, fewer recognize what makes a scientific study ' good ' or ' bad '.

Although Rousseau argues that sovereignty ( or the power to make the laws ) should be in the hands of the people, he also makes a sharp distinction between the sovereign and the government.

Although Origen makes no direct reference to the Testimonium, scholars such as Louis Feldman and Zvi Baras have presented arguments that Origen may have seen a copy of the Testimonium and not commented on it for there was no need to complain about its tone.

Although this model is conceptually helpful, its mathematical formulation makes several assumptions that limit its applicability to real systems.

Although the last days of the menstrual cycle are marked by a constant testosterone level, women's libido may boost as a result of the thickening of the uterine lining which stimulates nerve endings and makes a woman feel aroused.

Although steep mountain terrain makes exploitation difficult, mineral surveys have found small deposits of limestone, magnesite, zinc, copper, iron, mica, lead, and cobalt.

Although a capable leader, Helga tries too hard and makes rash decisions that jeopardize the safety of her troops.

Although a person's footsteps remain a visible lifeline within a snow-covered landscape, snow cover is considered a general danger to hiking since the snow obscures landmarks and makes the landscape itself appear uniform.

Although the 9 / 11 Commission makes mention of him, there is no evidence he was associated with the hijackers.

Although the Supreme Court has not had an occasion to interpret this specific provision, the Court has suggested that the grant to the House of the " sole " power of impeachment makes the House the exclusive interpreter of what constitutes an impeachable offense.

Although the language of the Constitution is not specific, by tradition, the President makes this report annually in late January.

Although as the above makes clear there is no confirmed etymological connection, it might be noted that these older meanings are synonymous with the modern expression " bull " otherwise generally considered, and intentionally used as, a contraction of " bullshit ".

Although adult echinoderms possess pentaradial, or five-sided, symmetry, echinoderm larvae are ciliated, free-swimming organisms that organize in bilateral symmetry which makes them look like embryonic chordates.

Although the baryon per photon ratio is important in determining elemental abundances, the precise value makes little difference to the overall picture.

Although the extreme lack of water in the Great Basin makes this distinction of little current practical use, it does mean that in wetter times the lake that once filled Death Valley ( Lake Manly ) was the last stop for water flowing in the region, meaning the water there was saturated in dissolved materials.

Although this makes it quite safe from any would-be invader, it also makes it a prison of sorts for those not possessing a portal key.

Although it does provide calcium, protein and Vitamin A, it also makes up for 30 percent of the daily value of cholesterol and 33 percent of the daily value of sodium.

Although the constitution forbids the federation or union of autonomous communities, an addendum or " transitional provision " to the constitution makes an exclusion whereby Navarre could join the Basque Country if the people chose to do so.