Although α and − α produce the same differential equation, it is conventional to define different Bessel functions for these two orders ( e. g., so that the Bessel functions are mostly smooth functions of α ).

Although the details of sunspot generation are still a matter of research, it appears that sunspots are the visible counterparts of magnetic flux tubes in the Sun's convective zone that get " wound up " by differential rotation.

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 this result might appear to settle the existence and uniqueness of solutions, there are examples of linear partial differential equations whose coefficients have derivatives of all orders ( which are nevertheless not analytic ) but which have no solutions at all: see Lewy ( 1957 ).

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 analytical solutions to this problem are scarce, there are upper and lower bounds to these functions including the famous Shannon lower bound ( SLB ), which in the case of squared error and memoryless sources, states that for arbitrary sources with finite differential entropy,

Although marketed as being " torque-sensing ", it functions the same as a limited-slip differential.

Although the notion of a differential is quite old, the initial attempt at an algebraic organization of differential forms is usually credited to Élie Cartan in his 1899 paper.

Although local homeomorphisms are not as strong as covering maps in this respect, they have many important applications in differential topology ; namely one uses the notion of a local homeomorphism to define a differentiable manifold.

Although the notation need not be taken literally, it is usually simpler than alternatives when the technique of separation of variables is used in the solution of differential equations.

Although differential cryptanalysis can break the revised version with less complexity than brute force search ( a certificational weakness ), the attack requires operations and is thus not currently feasible in practice.

Although it is usually classified as a branch of functional analysis, it has direct applications to representation theory, differential geometry, quantum statistical mechanics and quantum field theory.

Although a structural loading value of primary interest was not achieved ( local differential pressure between the interior and exterior of the command module wall ), all test objectives were satisfied.

Although supermanifolds are special cases of noncommutative manifolds, the local structure of supermanifolds make them better suited to study with the tools of standard differential geometry and locally ringed spaces.

Although it lacks a low-range transfer case, it has quattro permanent four wheel drive system with a central differential lock, and an adjustable-height air suspension which helps in off road situations.

Although post-processing techniques such as model output statistics ( MOS ) have been developed to improve the handling of errors in numerical predictions, a more fundamental problem lies in the chaotic nature of the partial differential equations used to simulate the atmosphere.

Although the definition of hyperbolicity is fundamentally a qualitative one, there are precise criteria that depend on the particular kind of differential equation under consideration.

Although this connection may appear superficial for a reader not familiar with differential equations, it is indeed the case that the main scale-space formulation in terms of non-enhancement of local extrema is expressed in terms of a sign condition on partial derivatives in the 2 + 1-D volume generated by the scale space, thus within the framework of partial differential equations.

Although the Greek letter ω is frequently used to denote the volume form, this notation is hardly universal ; the symbol ω often carries many other meanings in differential geometry ( such as a symplectic form ); thus, the appearance of ω in a formula does not necessarily mean that it is the volume form.

Although the Roverdrive automatic had been put to rest, overdrive was standard on the 110, whereas the 95 made do with a higher ratio differential ( 3. 9: 1 ).

Although an explanation for the phenomenon was not provided until 1919, duralumin was one of the first " age hardening " alloys to be used, and was soon followed by many others.

Although the presentation failed to change the fundamental position of the Security Council, including France, Russia, China, and Germany, Powell succeeded in hardening the overall tone of the United Nations towards Iraq.

Although cocoa is originally from the Americas, today Western Africa produces almost two-thirds of the world's cocoa, with Côte d ' Ivoire growing almost half of it.

Although the engine produces just 92 millinewtons ( 0. 331 ounce-force ) thrust at maximum power ( 2, 100W on DS1 ), the craft achieved high speeds because ion engines thrust continuously for long periods.

Although HTML is plain text, viewing an HTML file in a browser and in a text editor produces significantly different results.

Although much has been made of its status as a major exporter of oil, Nigeria produces only about 2. 7 % of the world's supply.

Although oxidative phosphorylation is a vital part of metabolism, it produces reactive oxygen species such as superoxide and hydrogen peroxide, which lead to propagation of free radicals, damaging cells and contributing to disease and, possibly, aging ( senescence ).

Although modern steelmaking processes negate the need to blend different steels, pattern welded steel is still used by custom knifemakers for the cosmetic effects it produces.

Although the proteasome normally produces very short peptide fragments, in some cases these products are themselves biologically active and functional molecules.

Although constitutively expressed proteasomes can participate in this process, a specialized complex composed of proteins whose expression is induced by interferon gamma produces peptides of the optimal size and composition for MHC binding.

Although the Sun produces Gamma rays as a result of the nuclear fusion process, these super high energy photons are converted to lower energy photons before they reach the Sun's surface and are emitted out into space.

Although there are only static electric and magnetic fields, the calculation of the Poynting vector produces a clockwise circular flow of electromagnetic energy, with no beginning or end.

Although the Hanna-Barbera name remains on the copyright notices of new productions based on " classic " properties like the Flintstones, the Jetsons and others, the studio that produces these works is Warner Bros.

Although incapacitation may result from physiological changes such as mucous membrane irritation, diarrhea, or hyperthermia, the term " incapacitating agent " as militarily defined refers to a compound that produces temporary and nonlethal impairment of military performance by virtue of its psychobehavioral or CNS effects.

Although Scotland sustains approximately 90 distilleries, Ireland has only four ( although each produces a number of different whiskeys ): economic difficulties in the last few centuries have led to a great number of mergers and closures.

Although fewer by-products are formed by ozonation, it has been discovered that ozone reacts with bromide ions in water to produces concentrations of the suspected carcinogen bromate.

Although his wage seems to be based on hours worked, in an economic sense this wage does not reflect the full value of what the worker produces.

) Although Elias Bing, the American Biological Art Metal artist, produces Art-Nouveau inspired jewelry, there is not any family relationship.

Although this leads to some minor variation in thickness, it produces a much smoother edge to the wink than seen on cheap moulded winks.

Although CHRO no longer airs the games themselves, the station still produces and airs a post-game show.

Although the revolver mechanism was fine for pistols, it posed a problem with long guns: without special sealing details, the cylinder produces a gas discharge close to the face when the weapon is fired from the shoulder, as all long guns were.

Although much of Discovery's programming is shared with its American counterpart, the Canadian channel has separate ownership and produces much of its own programming as well.

Although the group produces merengueros rhythms, the Calypso and other Caribbean rhythms, their specialty is the Punta style, which they popularized during the 1990s.

Although anodizing produces a very regular and uniform coating, microscopic fissures in the coating can lead to corrosion.

Although Isan accounts for around a third of Thailand ’ s population and a third of its area, it produces only 8. 9 % of GDP.

Although the most common isotope of uranium, uranium-238 (< sup > 238 </ sup > U ), can be used as secondary fissionable material in hydrogen bombs, it cannot be used as the primary fissile material for an atomic bomb .< sup > 238 </ sup > U can be used to produce < sup > 239 </ sup > Pu, through the fission of < sup > 235 </ sup > U which produces neutrons, some of which will be absorbed by < sup > 238 </ sup > U creating < sup > 239 </ sup > U. After a few days the < sup > 239 </ sup > U will decay, turning into weapons-usable < sup > 239 </ sup > Pu.