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Page "Clifford algebra" ¶ 5
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definition and Clifford
( NB: some authors switch the signs in the definition of a Clifford algebra which will interchange the meaning of positive-definite and negative-definite ).

definition and algebra
The same definition holds in any unital ring or algebra where a is any invertible element.
In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry.
In the Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered theorems.
The mathematical definition of an elementary function, or a function in elementary form, is considered in the context of differential algebra.
The notion of a homomorphism can be given a formal definition in the context of universal algebra, a field which studies ideas common to all algebraic structures.
The concrete definition given above is easy to work with, but has some minor problems: to use it we first need to represent a Lie group as a group of matrices, but not all Lie groups can be represented in this way, and it is not obvious that the Lie algebra is independent of the representation we use.
the general definition of the Lie algebra of a Lie group ( in 4 steps ):
One definition casts quasigroups as a set with one binary operation, and the other is a version from universal algebra which describes a quasigroup by using three primitive operations.
In this narrower definition, universal algebra can be seen as a special branch of model theory, in which we are typically dealing with structures having operations only ( i. e. the type can have symbols for functions but not for relations other than equality ), and in which the language used to talk about these structures uses equations only.
Now, this definition of a group is problematic from the point of view of universal algebra.
... while in the universal algebra definition there are
The general notion of a congruence relation can be given a formal definition in the context of universal algebra, a field which studies ideas common to all algebraic structures.
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root ( real coefficients and roots being within the definition of complex numbers ).
such that the following properties ( modeled on the group axioms – more precisely, on the definition of a group used in universal algebra ) are satisfied
In fact, this is the definition of a Dedekind domain used in Bourbaki's " Commutative algebra ".
These types are specified by insisting on some further axioms, such as commutativity or associativity of the multiplication operation, which are not required in the broad definition of an algebra.
Each algebra consists of linear combinations of three basis elements, 1 ( the identity element ), a and b. Taking into account the definition of an identity element, it is sufficient to specify
Since the definition of the spectrum does not mention any properties of B ( X ) except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim.
One can extend the definition of spectrum for unbounded operators on a Banach space X, operators which are no longer elements in the Banach algebra B ( X ).
This extends the definition for bounded linear operators B ( X ) on a Banach space X, since B ( X ) is a Banach algebra.
The definition of the exterior algebra makes sense for spaces not just of geometric vectors, but of other vector-like objects such as vector fields or functions.
On the other hand, the simplicity of the algebra in this proof perhaps makes it easier to understand than a proof using the definition of differentiation directly.

definition and endows
" Hume also took his definition of reason to unorthodox extremes by arguing, unlike his predecessors, that human reason is not qualitatively different from either simply conceiving individual ideas, or from judgments associating two ideas, and that " reason is nothing but a wonderful and unintelligible instinct in our souls, which carries us along a certain train of ideas, and endows them with particular qualities, according to their particular situations and relations.

definition and with
Now good definition is one thing that all of us can acquire with occasional high-set, high-rep, light-weight workouts.
But for purely definition purposes -- used in conjunction with your regular Squatting, Leg Curling, Leg Extensor programs -- a heavy weight is not needed.
Though there is obviously great need for continued experimentation with various types of short-term intervention to further efforts in developing an operational definition of prevention at the secondary -- or perhaps, in some instances, primary -- level, the place of short-term intervention has already been documented by a number of investigators in a wide variety of settings.
One definition of paternalism is `` The principle or practice, on the part of a government, of managing the affairs of a country in the manner of a father dealing with his children ''.
The practical definition may lead to confusion with the definition of a coulomb ( i. e., 1 amp-second ), but in practical terms this means that measures of a constant current ( e. g., the nominal flow of charge per second through a simple circuit ) will be defined in amperes ( e. g., " a 20 mA circuit ") and the flow of charge through a circuit over a period of time will be defined in coulombs ( e. g., " a variable-current circuit that flows a total of 10 coulombs over 5 seconds ").
Chaitin prefaces his definition with: " I'll show you can't prove that a program is ' elegant '"— such a proof would solve the Halting problem ( ibid ).
In the narrowest definition, the Amaryllidaceae sensu stricto is characterized by an umbellate inflorescence with an inferior ovary.
Its domain is the powerset of A ( with the empty set removed ), and so makes sense for any set A, whereas with the definition used elsewhere in this article, the domain of a choice function on a collection of sets is that collection, and so only makes sense for sets of sets.
Given two subspaces with, this leads to a definition of angles called canonical or principal angles between subspaces.
* Angle definition pages with interactive applets.
An equivalent definition is the radius of an unperturbed circular Newtonian orbit about the Sun of a particle having infinitesimal mass, moving with an angular frequency of radians per day ; or that length such that, when used to describe the positions of the objects in the Solar System, the heliocentric gravitational constant ( the product GM < sub >☉</ sub >) is equal to ()< sup > 2 </ sup > AU < sup > 3 </ sup >/ d < sup > 2 </ sup >.
The 1976 definition of the astronomical unit was incomplete, in particular because it does not specify the frame of reference in which time is to be measured, but proved practical for the calculation of ephemerides: a fuller definition that is consistent with general relativity was proposed, and " vigorous debate " ensued until in August 2012 the International Astronomical Union adopted the current definition of 1 astronomical unit = 149597870700 meters.
Under the strictest definition, an argot is a proper language, with its own grammar and style.
Modern critics are more likely to find fault with the narrow definition of the citizen body, but in the ancient world the complaint if anything went in the opposite direction.
This explains one definition as the area of a rectangle with sides of length one chain and one furlong.
The cubists, dadaists, Stravinsky, and many later art movements struggled against this conception that beauty was central to the definition of art, with such success that, according to Danto, " Beauty had disappeared not only from the advanced art of the 1960 ’ s but from the advanced philosophy of art of that decade as well.
159 ) adopts the old English Common-Law definition of affray, with the substitution of actual disturbance of the peace for causing terror to the lieges.
The Arrhenius definition of acid – base reactions is a development of the hydrogen theory of acids, devised by Svante Arrhenius, which was used to provide a modern definition of acids and bases that followed from his work with Friedrich Wilhelm Ostwald in establishing the presence of ions in aqueous solution in 1884, and led to Arrhenius receiving the Nobel Prize in Chemistry in 1903 for " recognition of the extraordinary services ... rendered to the advancement of chemistry by his electrolytic theory of dissociation ".
More recent IUPAC recommendations now suggest the newer term " hydronium " be used in favor of the older accepted term " oxonium " to illustrate reaction mechanisms such as those defined in the Brønsted – Lowry and solvent system definitions more clearly, with the Arrhenius definition serving as a simple general outline of acid – base character.
The Arrhenius definition can be summarised as " Arrhenius acids form hydrogen ions in aqueous solution with Arrhenius bases forming hydroxide ions.

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