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 differential hardening produces a very hard edge, it also leaves the rest of the sword rather soft, which can make it prone to bending under heavy loads, such as parrying a hard blow.

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 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 certainly not as complex or secure as modern computer operated stream ciphers or block ciphers, in practice messages protected by it resisted all attempts at cryptanalysis by at least NSA from its discovery in 1953 until Häyhänen's defection in 1957.

Although cryptography has a long and complex history, it wasn't until the 19th century that it developed anything more than ad hoc approaches to either encryption or cryptanalysis ( the science of finding weaknesses in crypto systems ).

Although most applications involve cryptanalysis ( e. g. the fastest known integer factorization algorithms ), the VSH hash function is one example of a constructive use of smoothness to obtain a provably secure design.

Although they are forbidden to sit with the customers, the dancers are sometimes proffered drinks, and most of them can bolt one down in mid-shimmy.

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 no national norms exist for the social development of children, the teacher can find a great deal of information concerning types of social behavior normally displayed by children at various age levels.

Although the first polyether foams on the market had to be produced by the two-step prepolymer method, today, thanks to new catalysts, they can be produced by a one-shot technique.

Although a complete picture cannot be given, we can indicate some aspects of life into which the Christian faith entered as at least one creative factor.

Although human morphology is not necessarily the ideal form for working robots, the fascination in developing robots that can mimic it can be found historically in the assimilation of two concepts: simulacra ( devices that exhibit likeness ) and automata ( devices that have independence ).

Although digital setting circles can be used to display a telescope's RA and Dec coordinates, they are not simply a digital read-out of what can be seen on the telescope's analog setting circles.

Although the neutral ground-state chemical compounds of argon are presently limited to HArF, argon can form clathrates with water when atoms of it are trapped in a lattice of the water molecules.

Although hydrogen-like orbitals are still used as pedagogical tools, the advent of computers has made STOs preferable for atoms and diatomic molecules since combinations of STOs can replace the nodes in hydrogen-like atomic orbital.

Although not as accurate by themselves as STOs, combinations of many Gaussians can attain the accuracy of hydrogen-like orbitals.

Although the terms adiabatic and isocaloric can often be interchanged, adiabatic processes may be considered a subset of isocaloric processes ; the remaining complement subset of isocaloric processes being processes where net heat transfer does not diverge regionally such as in an idealized case with mediums of infinite thermal conductivity or non-existent thermal capacity.

Although agates may be found in various kinds of rock, they are classically associated with volcanic rocks and can be common in certain metamorphic rocks.

Although programs can alter the position of the screen in memory, the non-linear format of the display means that vertical scrolling can only be done in blocks of 8 pixels without further work.

Although such a conversion can be prone to loss, most modern audio systems use this approach as the techniques of digital signal processing are much more powerful and efficient than analog domain signal processing.

Although sometimes this energy is released in the form of an emitted photon, the energy can also be transferred to another electron, which is ejected from the atom.

* Conservation of Energy: Although energy can be converted from one form to another, the total energy in a given closed system remains constant.

Although colors can be mixed, pigments are often not specified.

Although some plants do prefer slightly basic soil ( including vegetables like cabbage and fodder like buffalograss ), most plants prefer a mildly acidic soil ( pH between 6. 0 and 6. 8 ), and alkaline soils can cause problems.

Although AVMs are often associated with the brain and spinal cord, they can develop in any part of the body.

Although oxidation reactions are crucial for life, they can also be damaging ; plants and animals maintain complex systems of multiple types of antioxidants, such as glutathione, vitamin C, and vitamin E as well as enzymes such as catalase, superoxide dismutase and various peroxidases.

Although integration is the primary purpose of these organizations, they also serve as forums in which Brazil can exercise its leadership and develop consensus around its positions on regional and global issues.

Although black is sometimes described as an " achromatic ", or hueless, color, in practice it can be considered a color, as in expressions like " black cat " or " black paint ".