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, take the number of goals against, multiply that by 60 minutes and then divide by the amount of minutes played.

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, a Greek derivation from " manless, without husbands " ( a-privative and a derivation of * man-also found in Slavic muzh ) has been proposed, an explanation deemed " unlikely " by Hjalmar Frisk.

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

Alternatively, the line is divided into three four-syllable sections by two caesuras.

Alternatively, these experiments provide an operational definition of what is meant by " absolute rotation ", and do not pretend to address the question of " rotation relative to what?

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

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, 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, basic beliefs may be justified by some special property of the belief itself, such as its being self-evident or infallible.

Alternatively, it is possible to express an even-length real-input DFT as a complex DFT of half the length ( whose real and imaginary parts are the even / odd elements of the original real data ), followed by O ( N ) post-processing operations.

Alternatively, in engineering applications, such as swimming pool water treatment plants, they may be cleaned by backwashing.

Alternatively, the liquid may flow through the filter by the force exerted by a pump, a method commonly used in industry when a reduced filtration time is important.

Alternatively, shear strength may be decreased by weathering, changes in pore water pressure, and organic material.

Alternatively the determinant may be a noun in the genitive case, placed before or after the base-word, either directly or separated from the base-word by intervening words.

" Alternatively, such quotations have been interpreted by Anarcho-Communists supportive of Marx and Engels to suggest the abolition of capitalism and the state simultaneously, not the creation of a new state.

Alternatively, ' forward ' may be the direction pointed by the observer's nose, defining ' backward ' as the direction from the nose to the sagittal border in the observer's skull.

Alternatively, functioning may be affected by the stress of having to hide a condition in work or school etc., by adverse effects of medications or other substances, or by mismatches between illness-related variations and demands for regularity.

Alternatively, sparring may continue with the point noted by the judges.

Alternatively, under penultimate hop popping this function may instead be performed by the LSR directly connected to the LER.