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, if characters ' attribute ranks are close, the weaker character can try to change the relevant attribute by changing the nature of the conflict.

Alternatively, the rock picker can feel underneath rocks at low tides for abalone.

Alternatively they can be organized by whether the raw stock is of natural, or synthetic origin, or by their starting physical phase.

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 SCSI Enclosure Services can be used.

Alternatively, she can seek a surgical solution.

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

Alternatively, rasterization can be done in a more complicated manner by first rendering the vertices of a face and then rendering the pixels of that face as a blending of the vertex colors.

Alternatively, diagnosis and strain identification can be made using polymerase chain reaction ( PCR ).

Alternatively, the business can be sold as a going concern with the net proceeds of the sale distributed to creditors ratably in accordance with statutory priorities.

Alternatively, the coordinates of a point p can also be taken as the ( signed ) distances from p to the three planes defined by the three axes.

Alternatively, dualism can mean the tendency of humans to perceive and understand the world as being divided into two overarching categories.

Alternatively, if positive pressures are required, standard glassware can not be used, energy must be used for pressurization and there is a higher chance of side reactions occurring in the distillation, such as decomposition, due to the higher temperatures required to effect boiling.

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, and especially in connection with the relational model of database management, the relation between attributes drawn from a specified set of domains can be seen as being primary.

Alternatively, the method can be used to value the company based on the value of total invested capital.

Alternatively, they can be fitted with tom heads and tuned as shallow concert toms.

Alternatively, two figures are congruent if one can be moved on top of the other so that it matches up with it exactly.

Alternatively, the image can be photographically recorded by exposing a photographic film or plate directly to the electron beam, or a high-resolution phosphor may be coupled by means of a lens optical system or a fibre optic light-guide to the sensor of a CCD ( charge-coupled device ) camera.

Alternatively entropy can be defined from a classical thermodynamics viewpoint, in which the molecular interactions are not considered and instead the system is viewed from perspective of the gross motion of very large masses of molecules and the behavior of individual molecules is averaged and obscured.

Alternatively, for an orientable surface the formula can be given in terms of the genus of a surface, g:

Alternatively, gdbserver can be used to remotely debug the program without needing to change it in any way.

Alternatively, they can be termed when they are designed for combat as opposed to iaitō training swords.

Alternatively, microscopes can be classed on whether they analyse the sample via a scanning point ( confocal optical microscopes, scanning electron microscopes and scanning probe microscopes ) or analyse the sample all at once ( wide field optical microscope and transmission electron microscopes ).

Alternatively, it can be thought of as maximizing the minimum gain ( maximin ).