Alternatively, representing true values as 1 and false values as 0, we may use the formula

Alternatively, we could control the Hamiltonian of the system, and thus the time evolution operator U ( for a time independent Hamiltonian,, where is called the generator of translations in time ) up to some fixed time interval, which yields a unitary operator.

Alternatively, if we want to have reasonable discussion we may be practically forced to accept Occam's razor in the same way we are simply forced to accept the laws of thought and inductive reasoning ( given the problem of induction ).

Alternatively, John Locke and Jean-Jacques Rousseau, have argued that we gain civil rights in return for accepting the obligation to respect and defend the rights of others, giving up some freedoms to do so.

Alternatively, in untyped logic, we can require to be false whenever is an ur-element.

Alternatively, if the facticity of the social world and the structures of consciousness prove to be culturally and historically relative, then we are at an impasse in regard to any meaningful scientific understanding of the social world which is not subjective ( as opposed to being objective and grounded in nature, or inter subjective and grounded in the structures of consciousness ), and relative to the cultural and idealization formations of particular concrete individuals living in a particular socio-historical group.

Alternatively, for those familiar with power series ( or Taylor series ), one may think of a formal power series as a power series in which we ignore questions of convergence by not assuming that the variable X denotes any numerical value ( not even an unknown value ).

Alternatively, we may postulate extinction of all other descendants of a mitochondrion-free ancestral eukaryote, perhaps due to competition from the symbiotic clade, or oxygen poisoning as levels continued to rise.

Alternatively, we can say that one of the variables is proportional to the other.

Alternatively, we can think of an LTI system being completely specified by its frequency response.

Alternatively, we could count partitions in which no number occurs more than once.

Alternatively, we may choose u and v such that the product u (∫ g dx ) simplifies due to cancellation.

Alternatively, we can complete the square:

Alternatively, if the symbol H is used to denote electron energy level measured relative to the Fermi level of the Earth, then in general we have H

Alternatively we can say they occupy different, lower-dimensional subspaces in the phase space of the joint system.

Alternatively, we can imagine one particle emitting a virtual particle which is absorbed by the other.

Alternatively, if the space X is sequential, we may say that x ∈ X is a limit point of S if and only if there is an ω-sequence of points in S

Alternatively, as Herbert A. Simon suggested, something is simple or complex depending on the way we choose to describe it.

Alternatively, we can keep the same conclusion as in the classical IVT — a single c such that f ( c ) is exactly zero — while strengthening the conditions on f.

Alternatively, if we are using a torsion-free connection ( e. g. the Levi Civita connection ), then the partial derivative can be replaced with the covariant derivative.

Alternatively, we could carry out pairwise tests among the treatments ( for instance, in the medical trial example with four treatments we could carry out six tests among pairs of treatments ).

Alternatively, Assistive Technology may attempt to improve the ergonomics of the devices themselves such as Dvorak and other alternative layouts, which offer more ergonomic layouts of the keys.

Alternatively, ocean eddies, the oceanic equivalent of atmospheric storms, or the large-scale meanders of the Circumpolar Current may directly transport momentum downwards in the water column.

Alternatively, civilian vehicles may be modified into improvised armored cars in ad-hoc fashion.

Alternatively, a unique position on the board on which a game piece may be located while in play.

Alternatively, a binary function may be interpreted as simply a function from X × Y to Z.

Alternatively, a coin may be tossed, with the winners choosing whether to serve or receive first, or choosing which end of the court to occupy, and their opponents making the leftover the remaining choice.

Alternatively, if the boards are pre-dealt, " curtain cards " may be supplied which have each hand printed on them, so that each player can check at the beginning of the deal that he has the right cards.

Alternatively, since they may have felt the need to force some kind of victory — they could hardly remain at Marathon indefinitely.

Alternatively, the game may be played with ring-shaped pieces with a central hole.

Alternatively, conspiracy theories may arise when evidence available in the public record does not correspond with the common or official version of events.

Alternatively, computer programs may be executed with the aid of an interpreter, or may be embedded directly into hardware.

Alternatively, there may further be requirements for maximum terms that a government can keep power before holding an election.

Alternatively, the supernova may have resulted from the merger of two white dwarfs, so that the limit was only violated momentarily.

Alternatively, a file may contain an arbitrary binary image ( a BLOB ) or it may contain an executable.

Alternatively, elapsed time may represent the gradual process of healing of old hatreds.

Alternatively, if Bishop Cellach was appointed by Giric, it may be that the gathering was intended to heal a rift between king and church.

Alternatively, if using more than one or two techniques for every experiment is infeasible, some of the benefits of consilience may still be obtained if it is well-established that these techniques usually give the same result.

Alternatively, DMs may develop their own fictional worlds to use as campaign settings.

Alternatively, a single hostname may correspond to many IP addresses: this can facilitate fault tolerance and load distribution, and also allows a site to move physical locations seamlessly.

Alternatively, each computer may have its own user with individual needs, and the purpose of the distributed system is to coordinate the use of shared resources or provide communication services to the users.

Alternatively, IPv6 hosts may use stateless address autoconfiguration.

Alternatively, the home country may prosecute the individual.

Alternatively, they may be joined as clusters to form क ् नय knaya, कन ् य kanya, or क ् न ् य knya.