Gauss also made important contributions to number theory with his 1801 book Disquisitiones Arithmeticae ( Latin, Arithmetical Investigations ), which, among things, introduced the symbol ≡ for congruence and used it in a clean presentation of modular arithmetic, contained the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary quadratic forms, stated the class number problem for them, and showed that a regular heptadecagon ( 17-sided polygon ) can be constructed with straightedge and compass.

The two monographs Gauss published on biquadratic reciprocity have consecutively-numbered sections: the first contains §§ 1 – 23 and the second §§ 24 – 76.

The digressions can be understood to cover two themes: an account of the history of the entire known world as governed by the principle of reciprocity ( or what today might be more commonly called an eye for an eye and a tooth for a tooth and one good turn deserves another ); and an account of the many astonishing reports and sights gained by the author during his extensive travels.

There are a number of equivalent statements of the theorem, which consists of two " supplements " and the reciprocity law:

These two quotes show how reciprocity functions within genre.

James Coleman defined social capital functionally as “ a variety of entities with two elements in common: they all consist of some aspect of social structure, and they facilitate certain actions of actors ... within the structure ”— that is, social capital is anything that facilitates individual or collective action, generated by networks of relationships, reciprocity, trust, and social norms.

There are two more permutations of the foregoing derivation that give a total of three reciprocity relations between, and.

Due to reciprocity, these two effects are equal-an antenna that makes a transmitted signal 100 times stronger ( compared to an isotropic radiator ), will also capture 100 times as much energy as the isotropic antenna when used as a receiving antenna.

" This article applies under two conditions ( article 324 ): that Portugal maintains diplomatic relations with the insulted country, and that there is reciprocity ( i. e., that the insulted country would also punish any insult against Portuguese symbols of sovereignty, should they occur there ).

A notice of termination for this agreement, signed on December 28, 2000 by Robert H Bruininks, Executive Vice President and Provost, voided the reciprocity agreement between the two institutions at the end of summer session 2001.

After two weeks of negotiating, the Potawatomi leaders convinced the Miami to accept the treaty as reciprocity to the Potawatomi who had earlier accepted treaties less advantageous to them at the request of the Miami.

argue, two key sets of factors for successful deterrence are important being ( i ) a defending state strategy that firstly balances credible coercion and deft diplomacy consistent with the three criteria of proportionality, reciprocity and coercive credibility, and secondly minimises international and domestic constraints ; and ( ii ) the extent of an attacking state's vulnerability as shaped by its domestic political and economic conditions.

Kohlberg identifies two of these justice operations: " equality ," which involves an impartial regard for persons, and " reciprocity ," which means a regard for the role of personal merit.

CalPERS has reciprocity agreements with many of these California public retirement systems that allow retirees with service credit and contributions in two systems to receive payments from both systems.

There are two kinds of reciprocity failure.

The monies, always late and grudgingly given, were earned in full measure by Eisenstein: in 1844 alone he published over 23 papers and two problems in Crelle's Journal, including two proofs of the law of quadratic reciprocity, and the analogous laws of cubic reciprocity and quartic reciprocity.

Perceived organizational support ( POS ) and Perceived psychological contract violation ( PPCV ) are the two most common measures of the reciprocity norm in organizational research.

And Lewis and Tolman ( 1909 ) described the reciprocity of time dilation by using two light clocks A and B, traveling with a certain relative velocity to each other.

Most passive components with more than two terminals can be described in terms of two-port parameters that satisfy the principle of reciprocity — though there are rare exceptions ..

* understanding the role of reciprocity, particularly when faced with prisoner's dilemma situations: players may adopt strategies of conditional cooperation, where the more others contribute to the public good, the more they will contribute themselves

It has been observed that image scoring promotes cooperative behavior in situations where direct reciprocity is unlikely.

In public good experiments, behavioral economists have demonstrated that the potential for reciprocal actions by players increases the rate of contribution to the public good, providing evidence for the importance of reciprocity in social situations.

Costly Signaling Theory ( CST ) is a useful tool for understanding food sharing among hunter gatherers because it can be applied to situations in which delayed reciprocity is not a viable explanation ( Bliege Bird and Bird 1997 ; Gurven et al.

Social capital is defined by Robert D. Putnam as " the collective value of all social networks and species ( who people know ) and the inclinations that arise from these works to do things for each other ( norms of reciprocity ).

Confucius ' concept of humaneness () is probably best expressed in the Confucian version of the ethic of reciprocity, or the Golden Rule: " Do not do unto others what you would not have them do unto you.

In Confucianism, the concept of li is closely related to yì (), which is based upon the idea of reciprocity.

Duality is also sometimes called reciprocity or polarity.

The first generalization of the theorem is found in Gauss's second monograph ( 1832 ) on biquadratic reciprocity.

Similarly, in 1844 while working on cubic reciprocity, Eisenstein introduced the ring, where is a cube root of unity.

Indirect reciprocity is defined here as " I help you and somebody else helps me.

The Golden Rule or ethic of reciprocity is a maxim, ethical code, or morality

As a concept of " the ethic of reciprocity ," it has its roots in a wide range of world cultures, and is a standard way that different cultures use to resolve conflicts.

The law of retaliation ( lex talionis ) is a military theory of retributive justice, which says that reciprocity should be equal to the wrong suffered ; " life for life, wound for wound, stripe for stripe.

By quadratic reciprocity, any prime modulo which 2 has a square root is congruent to ± 1 ( mod 8 ).

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic which gives conditions for the solvability of quadratic equations modulo prime numbers.

The form factor is then equal to the proportion of the unit circle covered by this projection .< p > Form factors obey the reciprocity relation A < sub > i </ sub > F < sub > ij </ sub > = A < sub > j </ sub > F < sub > ji </ sub >

Harsanyi claimed that his theory is indebted to Adam Smith, who equated the moral point of view with that of an impartial but sympathetic observer ; to Kant who insisted on the criterion of universality and which may also be described as a criterion of reciprocity ; to the classical utilitarians who made maximising social utility the basic criterion of morality ; and to ‘ the modern theory of rational behaviour under risk and uncertainty, usually described as Bayesian decision theory ’.

The concept of ethical reciprocity is not explicitly stated, but most Wiccans interpret the Rede to imply the Golden Rule in the belief that the spirit of the Rede is to actively do good for one's fellow humans as well as for oneself.

In general, magneto-optic effects break time reversal symmetry locally ( i. e. when only the propagation of light, and not the source of the magnetic field, is considered ) as well as Lorentz reciprocity, which is a necessary condition to construct devices such as optical isolators ( through which light passes in one direction but not the other ).

This is a consequence of the reciprocity theorem of electromagnetics and is proved below.

Now for a given disposition of the antennas, the reciprocity theorem requires that the power transfer is equally effective in each direction, i. e.

Instead, the radiation pattern of the antenna is determined by a single simulation, and the receiving pattern inferred by reciprocity.

* Closely related to the motivation aspect is the question of reciprocity.