The advantages of abstraction are:

In measure-theoretic analysis and related branches of mathematics, the Lebesgue – Stieltjes integral generalizes Riemann – Stieltjes and Lebesgue integration, preserving the many advantages of the latter in a more general measure-theoretic framework.

Although it has some advantages, particularly when discussing resonance phenomena, it requires integration of the Fermi-Dirac distribution function in the first stage of integration: for non-free-electron-like electronic band-structures this can lead to very complex and error-prone mathematics ( as in the work of Stratton on semiconductors ).

In measure-theoretic analysis and related branches of mathematics, Lebesgue – Stieltjes integration generalizes Riemann – Stieltjes and Lebesgue integration, preserving the many advantages of the former in a more general measure-theoretic framework.

Coehoorn's individual assessment of each fortification and focus on existing natural advantages and disadvantages is the main difference in thinking from Vauban, who adhered more strictly to mathematics and standard fortifications.

IMP is extremely different from a regular mathematics program, and like everything, it has its advantages and disadvantages.

The advantages of a uniform fiscal year and well synchronized fiscal and tax collection calendars are sufficiently great for Rhode Island municipalities to exert effort to secure them.

The basic costs are generally pretty much the same regardless of the agency through which you reserve your car, but some of them offer supplementary advantages.

The advantages of dilatation by the physician are both physical and psychological.

So obvious are these advantages that nearly 95 per cent of the population of New York State now has access to a system, and enthusiastic librarians foresee the day, not too distant, when all the libraries in the state will belong to a co-op.

In hot climates, compared with wooden buildings, adobe buildings offer significant advantages due to their greater thermal mass, but they are known to be particularly susceptible to earthquake damage.

Although there are few examples of such systems competitive with traditional analysis techniques, potential advantages include size / portability, speed, and cost.

In this context, the positive genetic benefits have to be considered, i. e., what advantages are obtained by cave-dwelling tetras by losing their eyes?

Integral designs have the advantages that they are have been well tested for strength, stability and so forth, and also are off-the-shelf.

The number and quality of combatants and equipment, the skill of the commanders of each army, and the terrain advantages are among the most prominent factors.

To gain some of the advantages afforded by both lower and higher bit lengths, many CPUs are designed with different bit widths for different portions of the device.

O ' Flaherty illustrates two similar attracting advantages known as increasing returns to scale and economies of scale, which are concepts normally associated with firms.

The advantages are at less financial cost to the user as well as society ( negligible damage to roads, and less pavement required ).

Displacement chromatography has advantages over elution chromatography in that components are resolved into consecutive zones of pure substances rather than “ peaks ”.

Claimed advantages of the IN10DID method are the diversity of devices, limited motion and simple chords.

The primary advantages of a pistol caliber carbine are increased accuracy due to the buttstock and longer barrel ( and with it, sight radius ), relatively low muzzle blast / flash / recoil, higher muzzle velocity and energy of a longer barrel for increased wounding potential and penetration ( depending on the particular load used ), and ( sometimes, but not always ) greater adaptability for easily accepting accessories such as optics, weaponlights, and lasers.

Other advantages are that one can easily find the first or last elements on the list ( most likely to be useful in the case of numerically sorted data ), or elements in a given range ( useful again in the case of numerical data, and also with alphabetically ordered data when one may be sure of only the first few letters of the sought item or items ).

Among the advantages to unfretted instruments are flexibility in tuning ( the temperament can be easily altered ) and the ability to play any music exactly as written without concern for " bad " notes.

Occasionally there are advantages to using a counting sequence other than the natural binary sequence — such as the binary coded decimal counter, a linear feedback shift register counter, or a Gray-code counter.

The main advantages of LCDs over CRT displays are that LCDs consume less power, take up much less space, and are considerably lighter.

The principal advantages of compound turbines are the reduction in size of any one casing, the confinement of the highest pressure to the smaller casing ( which may be made of stronger and more expensive materials ) and the possibility of divided flow in the low-pressure casing for the purpose of equalizing end thrusts.

The advantages of electron diffraction over X-ray crystallography are that the specimen need not be a single crystal or even a polycrystalline powder, and also that the Fourier transform reconstruction of the object's magnified structure occurs physically and thus avoids the need for solving the phase problem faced by the X-ray crystallographers after obtaining their X-ray diffraction patterns of a single crystal or polycrystalline powder.

We ought not to forget the transitory character of all external advantages, even in the midst of our enjoyment of them ; but always to bear in mind that they are not our own, and that therefore they do not properly belong to us.

A term dating from the 1940s, " general abstract nonsense ", refers to its high level of abstraction, compared to more classical branches of mathematics.

The reason why we do not deal with sensible objects in mathematics is because of another faculty of understanding called " categorial abstraction.

In mathematics, a generalized mean, also known as power mean or Hölder mean ( named after Otto Hölder ), is an abstraction of the Pythagorean means including arithmetic, geometric, and harmonic means.

* Metric ( mathematics ), an abstraction of the notion of distance in a metric space

Arnold was an outspoken critic of the trend towards high levels of abstraction in mathematics during the middle of the last century.

Hence, in the philosophy of mathematics, the abstraction of actual infinity involves the acceptance of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given objects.

* Property ( philosophy ), in philosophy, mathematics, and logic, an abstraction characterizing an object

The concept originated by analogy with abstraction in mathematics.

One manifestation of this view is found in the cognitive science of mathematics, where it is proposed that mathematics itself, the most widely accepted means of abstraction in the human community, is largely metaphorically constructed, and thereby reflects a cognitive bias unique to humans that uses embodied prototypical processes ( e. g. counting, moving along a path ) that are understood by all human beings through their experiences.

Further, partly in response to such criticisms, Lakoff and Rafael E. Núñez, in 2000, proposed a cognitive science of mathematics that would explain mathematics as a consequence of, not an alternative to, the human reliance on conceptual metaphor to understand abstraction in terms of basic experiential concretes.

" He held that corollarial deduction matches Aristotle's conception of direct demonstration, which Aristotle regarded as the only thoroughly satisfactory demonstration, while theorematic deduction ( A ) is the kind more prized by mathematicians, ( B ) is peculiar to mathematics, and ( C ) involves in its course the introduction of a lemma or at least a definition uncontemplated in the thesis ( the proposition that is to be proved ); in remarkable cases that definition is of an abstraction that " ought to be supported by a proper postulate.

In other words, when feminist economists highlight the biases of mainstream economics, they focus on its social beliefs about masculinity like objectivity, separation, logical consistency, individual accomplishment, mathematics, abstraction, and lack of emotion, but not on the gender of authorities and subjects.

In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set.

This classical piece of mathematics requires a substantial amount of abstraction that usually tends to puzzle beginners.

* Hypostatic abstraction ( mathematics and logic )

But a few surprising examples of mathematical abstraction notwithstanding ( for example, chimpanzees can be trained to carry out symbolic addition with digits, or the report of a parrot understanding a “ zero-like concept ”), all examples of animal intelligence with respect to mathematics are limited to basic counting abilities.

As an instructive process, eidetic reduction might be viewed as processes of abstraction ; the works of Whitehead and Russell in their Principia Mathematica could give one a taste of those processes, or even the simple contemplation of mathematics itself in use: " How many ideas have you?

This capacity is strong evidence for abstract and conceptual thought in the dolphin's brain, wherein an idea of the object is stored and understood not merely by its sensory properties ; such abstraction may be argued to be of the same kind as complex language, mathematics, and art, and implies a potentially very great intelligence and conceptual understanding within the brains of tursiops and possibly many other cetaceans.