Alternatively, it can be defined as the region where the average temperature for the warmest month ( July ) is below ; the northernmost tree line roughly follows the isotherm at the boundary of this region.

Alternatively, linear independence can be directly defined as follows: a set of vectors is linearly independent if and only if the only representations of the zero vector as linear combinations of its elements are trivial solutions, i. e., whenever a < sub > 1 </ sub >, a < sub > 2 </ sub >, ..., a < sub > n </ sub > are scalars such that

Alternatively, one can calculate the taxable equivalent yield of a municipal bond and compare it to the yield of a corporate bond as follows:

Similarly, the complement of an odd cycle of length 2k + 1 cannot be perfect, because its chromatic number is k + 1 and its clique number is k. ( Alternatively, the imperfection of this graph follows from the perfect graph theorem and the imperfection of the complementary odd cycle ).

Alternatively, paronychia may be divided as follows:

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

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

Alternatively, Prospect Relationship Management ( PRM ) solutions offer to track customer behaviour and nurture them from first contact to sale, often cutting out the active sales process altogether.

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, domino sets have been made from many different natural materials: stone ( e. g., marble, granite or soapstone ); other hardwoods ( e. g., ash, oak, redwood and cedar ); metals ( e. g., brass or pewter ); ceramic clay, or even frosted glass or crystal.

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 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, the path integral formulation of quantum field theory represents the transition amplitude as a weighted sum of all possible histories of the system from the initial to the final state, in terms of either particles or fields.

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

Alternatively, the word " gematria " is generally held to derive from Greek geōmetriā, " geometry ", which was used as a translation of gēmaṭriyā, though some scholars believe it to derive from Greek grammateia, rather ; it's possible that both words had an influence on the formation of the Hebrew word.

Alternatively, Hera kidnapped Eileithyia, the goddess of childbirth, to prevent Leto from going into labor.

Alternatively, the word may be derived from an Irish word for a wooden anchor.

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, ' 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, military vulnerability to invasion from marginal hill tribes or nomadic pastoralists have led to periods of trade collapse and neglect of irrigation systems.

Alternatively, the clinic may stop dosage at 80 mg, then allow the patient to move up by 5 mg or 10 mg every 2 or 3 days, until they are free from withdrawal symptoms and intense cravings.

Alternatively, the name may derive from the verb na · ṣar, נ ָ צ ַ ר, " watch, guard, keep ," and understood either in the sense of " watchtower " or " guard place ", implying the early town was perched on or near the brow of the hill, or, in the passive sense as ' preserved, protected ' in reference to its secluded position.

Alternatively, 5 ml from a neutron star of radius 20 km radius ( average density ) has a mass of about 400 million metric tons, or about the mass of all humans .</ ref > The resulting force of gravity is so strong that if an object were to fall from a height of one meter it would only take one microsecond to hit the surface of the neutron star, and would do so at around 2000 kilometers per second, or 7. 2 million kilometers per hour.

Alternatively, when a self-balancing binary search tree is used, insertion and removal also take O ( log n ) time, although building trees from existing sequences of elements takes O ( n log n ) time ; this is typical where one might already have access to these data structures, such as with third-party or standard libraries.

Alternatively, Prakrit can be taken to mean " derived from an original ," which means evolved in natural way.

Alternatively, if several different minerals can be dated from the same sample and are assumed to be formed by the same event and were in equilibrium with the reservoir when they formed, they should form an isochron.

Alternatively submissive to and independent of Han control, this Kingdom expanded colonization and Sinification under its policy of " Harmonizing and Gathering the Hundred Yue " () but was supported by the Zhuang until its collapse in 211 BC.

Alternatively, the passport is commonly referred to as the Hong Kong Special Administrative Region passport () ( the Hong Kong legislative ordinance concerning the passport is titled the " Hong Kong Special Administrative Region

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, 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, if a line is constructed perpendicularly between Alpha Centauri and Beta Centauri, the point where the above mentioned line and this line intersect marks the Southern Celestial Pole.

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

Alternatively, if the two waves of the same wavelength and frequency are out of phase, then the wave crests will align with wave troughs and vice-versa.

Alternatively, if the last step is carried out in air:

( Alternatively, for a preorder "", a relation "<" can be defined as a < b if and only if ( a b and a ≠ b ).

Alternatively, if M is a linear subspace then ; apply this inequality to the subspace defined by the ( orthogonal ) complement of the image of BC in the image of B, whose dimension is ; its image under A has dimension

Alternatively, if the excited-state atom is perturbed by an electric field of frequency, it may emit an additional photon of the same frequency and in phase, thus augmenting the external field, leaving the atom in the lower energy state.

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, 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, if,, and are known, but is not adjustable, the voltage difference across or current flow through the meter can be used to calculate the value of, using Kirchhoff's circuit laws ( also known as Kirchhoff's rules ).

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, if the magma is erupted it forms volcanic rocks such as basalt, andesite and rhyolite ( the extrusive equivalents of gabbro, diorite and granite, respectively ).

Alternatively, if a strip with an even number of half-twists is cut in half along its length, it will result in two linked strips, each with the same number of twists as the original.

Alternatively, if the data matches the machine address, the data is accepted.

Alternatively, some agree that negative rights may be violated by a government action, but argue that it is justifiable if the positive rights protected outweigh the negative rights lost.

Alternatively, if is discrete, one can construct as the set of all ultrafilters on, with a topology known as Stone topology.