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, we may use just the last two equations given for the compositional property of the solvent in each of the preceding sections, together with the relationships given below, to derive the remainder of properties in that set:

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

Alternatively, similar to the way primary colors combine, primary emotions could blend to form the full spectrum of human emotional experience.

( Alternatively, apparent cognates in Eurasian language families far removed from each other could also be early loanwords, compare Wanderwort.

Alternatively, a breeder could, using individuals of differing phenotypes, create a new breed with specific characteristics.

Alternatively fuel cell technology may come to deliver its promise and the use of the internal combustion engines could even be phased out.

Alternatively, a railway, or a combined railway and oil pipeline, could be built to link ports on the Atlantic and Pacific coasts.

Alternatively one could also consider combinations of five cubes that are themselves 3D, i. e., are not part of one layer of cubes.

Alternatively, public health strategies could selectively address socioeconomic factors that have been linked to higher rates of schizophrenia in certain groups, for example in relation to immigration, ethnicity or poverty.

Alternatively, energy could be transmitted to the lunar and martian surfaces from solar power satellites.

Alternatively, since expendable multistage rockets entail discarding costly structure and engines, if the stages could be reused, this could permit much cheaper operation since the parts costs would be amortized over many flights.

Alternatively, the object of a trust could be a charitable purpose rather than specific beneficiaries.

Alternatively it has been suggested that the star could be in a low-activity state analogous to a Maunder minimum — a historical period, associated with the Little Ice Age in Europe, when sunspots became exceedingly rare on the Sun's surface.

Alternatively, if a predatory ancestor already flying and therefore with long forelimbs, used them to capture prey, it could be pulled on to a predatory foot claw, later inherited and demonstrated by the fighting Velociraptor.

Alternatively, Ctrl + KC could be used to copy the block.

( Alternatively, Lucien Gubbay suggests the name Medina could also have been a derivative from the Aramaic word Medinta, which the Jewish inhabitants could have used for the city.

Alternatively, one could use some of the following criteria:

Alternatively, liquid oxygen could be used as reaction mass, which upon release would boil down to its molecular state.

Alternatively, the ram could be set ablaze, doused in fire-heated sand, pounded by boulders dropped from battlements or invested by a rapid sally of troops.

Alternatively the deflection of the suspended magnets could be observed directly through a microscope.

Alternatively, they could use the ITU X. 733 Alarm Reporting Function's perceived severity field.

Alternatively, pelvic inflammatory disease could explain why the onset of her symptoms roughly coincided with her penultimate pregnancy.

Alternatively, one could require that the transition maps