An equivalent definition of group homomorphism is: The function h: G → H is a group homomorphism if whenever we have.

The definition of homomorphism depends on the type of algebraic structure under consideration.

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.

For instance, the precise definition for a homomorphism f to be iso is not only that it is bijective, and thus has an inverse f < sup >- 1 </ sup >, but also that this inverse is a homomorphism, too.

Note also that any homomorphism f between Abelian groups is " additive " by the first definition.

The definition of a Dirichlet character modulo k ensures that it restricts to a character of the unit group modulo k: a group homomorphism from ( Z / k )< sup >*</ sup > to the non-zero complex numbers

The definition of Φ < sub > A </ sub > and the topology on it ensure that is continuous and vanishes at infinity, and that the map defines a norm-decreasing, unit-preserving algebra homomorphism from A to C < sub > 0 </ sub >( Φ < sub > A </ sub >).

Equivalent to this definition, a space X is semi-locally simply connected if every point in X has a neighborhood U for which the homomorphism from the fundamental group of U to the fundamental group of X, induced by the inclusion map of U into X, is trivial.

An equivalent definition of an associative R-algebra is a ring homomorphism such that the image of f is contained in the center of A.

This definition is equivalent to a homomorphism ( reversing the operation before or after applying the map is equivalent ).

A more general definition of orientation that works regardless of dimension is the following: An orientation on V is a map from the set of ordered bases of V to the set that is invariant under base changes with positive determinant and changes sign under base changes with negative determinant ( it is equivarient with respect to the homomorphism ).

A homomorphism from one super vector space to another is, by definition, one that preserves the grading ( i. e. maps even elements to even elements and odd elements to odd elements ).

The exact definition of an automorphism depends on the type of " mathematical object " in question and what, precisely, constitutes an " isomorphism " of that object.

In fact anode polarity depends on the device type, and sometimes even in which mode it operates, as per the above electric current direction-based universal definition.

The definition of ballroom dance also depends on the era: balls have featured popular dances of the day such as the Minuet, Quadrille, Polonaise, Polka, Mazurka, and others, which are now considered to be historical dances.

The definition of " addictive " depends on each individual but most Buddhists consider alcohol, tobacco and contraband drugs to be addictive.

Of course, this definition depends on the choice of norm:

However, whether this is possible depends on how the field of complex numbers is derived in the first place: it may not be possible to distinguish a complex number from its conjugate ( say, 3 + i from 3-i ), since it is impossible to find a property of one that is not also a property of the other, without falling back on the underlying set-theoretic definition.

A bit, or a group of malfunctioning physical bits ( not always the specific defective bit is known ; group definition depends on specific storage device ) is typically automatically fenced-out, taken out of use by the device, and replaced with another functioning equivalent group in the device, where the corrected bit values are restored ( if possible ).

The entries can be numbers or expressions ( as happens when the determinant is used to define a characteristic polynomial ); the definition of the determinant depends only on the fact that they can be added and multiplied together in a commutative manner.

A bit, or a group of malfunctioning physical bits ( not always the specific defective bit is known ; group definition depends on specific storage device ) is typically automatically fenced-out, taken out of use by the device, and replaced with another functioning equivalent group in the device, where the corrected bit values are restored ( if possible ).

The identity of the longest word in English depends upon the definition of what constitutes a word in the English language, as well as how length should be compared.

Latency is a measure of time delay experienced in a system, the precise definition of which depends on the system and the time being measured.

In generative grammar, the definition of a morpheme depends heavily on whether syntactic trees have morphemes as leafs or features as leafs.

The precise definition of " class " depends on foundational context.

The " right " definition depends on the context in which the scheme will be deployed.

The relationship between religion and myth depends on what definition of " myth " one uses.

The definition of a road depends on the definition of a highway, however there is no formal definition for a highway in the relevant Act.

Within this context, the definition of an apneic event depends on several factors ( e. g. patient's age ) and account for this variability through a multi-criteria decision rule described in several, sometimes conflicting, guidelines.

A suitable definition of the entropy of a small local region depends on the physically insightful and judicious selection of the independent local non-equilibrium macroscopic state variables, and different selections provide different generalized or extended thermodynamical accounts of one and the same given physical process.

The notion of simultaneity depends on the frame of reference ( see relativity of simultaneity ), so switching between frames requires an adjustment in the definition of the present.

As regards animals, to what extent different animals think depends on the exact definition of the word that is given, so it may be taken literally or regarded as anthropomorphic.

This definition depends on several other technical terms from Federal Standard 1037C.

First, we can encourage responsibility by establishing as conditions for assistance on a substantial and sustained scale the definition of objectives and the assessment of costs.

Used in several sets of high reps once or twice each week it will not be long before your entire upper leg takes on a razor-sharp definition in which the muscles look like wire cables writhing and twisting under the skin!!

If Af is the operator induced on Af by T, then evidently Af, because by definition Af is 0 on the subspace Af.

It might be contended, of course, that Eichmann in stubbornly denying anti-Semitic feelings was lying or insisting on a private definition of anti-Semitism.

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 appellate court will typically be deferential to the lower court's findings of fact ( such as whether a defendant committed a particular act ), unless clearly erroneous, and so will focus on the court's application of the law to those facts ( such as whether the act found by the court to have occurred fits a legal definition at issue ).

The new definition is expected to be formally proposed at the 25th General Conference on Weights and Measures ( CGPM ) in 2014.

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.

* The Logicians or the School of Names, which focused on definition and logic.

This definition was adopted by the United Nations Sub-Commission on the Promotion and Protection of Human Rights in The Despouy Report on Human Rights and Extreme Poverty.

In this definition, it is implicitly assumed that storing a value into a variable U has no effect on the state of a distinct variable V. To make this assumption explicit, one could add the constraint that

Also running on a JOHANNIAC, the Logic Theory Machine constructed proofs from a small set of propositional axioms and three deduction rules: modus ponens, ( propositional ) variable substitution, and the replacement of formulas by their definition.

Celsius was the first to perform and publish careful experiments aiming at the definition of an international temperature scale on scientific grounds.

Heschel then goes on to explore the problems of doubts and faith ; what Judaism means by teaching that God is one ; the essence of humanity and the problem of human needs ; the definition of religion in general and of Judaism in particular ; and human yearning for spirituality.

This definition was proposed by Justus von Liebig circa 1838, based on his extensive works on the chemical composition of organic acids.

Meteorologists ( and most of the temperate countries in the southern hemisphere ) use a definition based on months, with autumn being September, October and November in the northern hemisphere, According to United States tradition, autumn runs from the day after Labor Day ( i. e. the Tuesday following the first Monday of September ) through Thanksgiving ( i. e. the fourth Thursday in November ), after which the holiday season that demarcates the unofficial beginning of winter begins.

Based on this definition, the drain efficiency cannot exceed 25 % for a class A amplifier that is supplied drain bias current through resistors ( because RF signal has its zero level at about 50 % of the input DC ).

He goes on to define error as self-contradiction of definition (" an absurdity, or senselesse Speech ") or conclusions that do not follow the definitions on which they are supposed to be based.