* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )

If the sources for north-eastern Britain, the lands of the kingdom of Northumbria and the former Pictland, are limited and late, those for the areas on the Irish Sea and Atlantic coasts — the modern regions of north-west England and all of northern and western Scotland — are non-existent, and archaeology and toponymy are of primary importance.

If the sheet contains regions where the number of atoms in a ring is different from six, while the total number of atoms remains the same, a topological defect has formed.

The report to the European Parliament of 2001 states: " If UKUSA states operate listening stations in the relevant regions of the earth, in principle they can intercept all telephone, fax and data traffic transmitted via such satellites.

If A consisted of three regions, six or more colors might be required ; one can construct maps that require an arbitrarily high number of colors.

If the four-color conjecture were false, there would be at least one map with the smallest possible number of regions that requires five colors.

If a photosensitive material is selectively exposed to radiation ( e. g. by masking some of the radiation ) the pattern of the radiation on the material is transferred to the material exposed, as the properties of the exposed and unexposed regions differs.

If a government-owned company providing an essential service ( such as the water supply ) to all citizens is privatized, its new owner ( s ) could lead to the abandoning of the social obligation to those who are less able to pay, or to regions where this service is unprofitable.

If the attack is conducted on a sufficiently large scale, entire geographical regions of Internet connectivity can be compromised without the attacker's knowledge or intent by incorrectly configured or flimsy network infrastructure equipment.

If searching for microsatellite markers in specific regions of a genome, for example within a particular exon of a gene, primers can be designed manually.

If positive clones can be obtained from this procedure, the DNA is sequenced and PCR primers are chosen from sequences flanking such regions to determine a specific locus.

If the dangers can be localised then the affected regions can be made into parkland or green belt, often with the added benefit of open space provision.

If such a collection of inner products on the tangent bundle of a manifold varies smoothly as one traverses the manifold, then concepts that were defined only pointwise at each tangent space can be extended to yield analogous notions over finite regions of the manifold.

If the chemical potential is the same in the two regions, the species will occasionally move back and forth between the two regions, but on average there is just as much movement in one direction as the other, and there is zero net migration ( this is called " diffusive equilibrium ").

If the chemical potentials of the two regions are different, more molecules will move to the lower chemical potential than the other direction.

If some of the components are liquid at a given temperature while others are in the gel phase, the two phases can coexist in spatially separated regions, rather like an iceberg floating in the ocean.

If the subgraph is not connected, then there are regions of the graph that are wasted because they do not contribute to the search space.

If the crystal migrates into regions with different environmental conditions, the growth pattern may change, and the final crystal may show mixed patterns.

If a game does not sell out, it cannot be shown in the home city region as well as any secondary regions as long as that region's television signal goes within of the home team's stadium.

If the theory has a mass gap, i. e. there are no masses between 0 and some constant greater than zero, then vacuum expectation distributions are asymptotically independent in distant regions.

If X is the Euclidean plane with the usual distance and the Lebesgue measure then this question generalizes the classical isoperimetric problem to planar regions whose boundary is not necessarily smooth, although the answer turns out to be the same.

If ( remember this is an assumption ) the minimal polynomial for T decomposes Af where Af are distinct elements of F, then we shall show that the space V is the direct sum of the null spaces of Af.

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

If Af denotes the space of N times continuously differentiable functions, then the space V of solutions of this differential equation is a subspace of Af.

If D denotes the differentiation operator and P is the polynomial Af then V is the null space of the operator p (, ), because Af simply says Af.

If Af is the null space of Af, then Theorem 12 says that Af.

If the argument is accepted as essentially sound up to this point, it remains for us to consider whether the patient's difficulties in orienting himself spatially and in locating objects in space with the sense of touch can be explained by his defective visual condition.

If, on the other hand, they opted for representation, it had to be representation per se -- representation as image pure and simple, without connotations ( at least, without more than schematic ones ) of the three-dimensional space in which the objects represented originally existed.

If a child loses a molar at the age of two, the adjoining teeth may shift toward the empty space, thus narrowing the place intended for the permanent ones and producing a jumble.

If elements in the sample space increase arithmetically, when placed in some order, then the median and arithmetic average are equal.

If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

* Theorem If X is a normed space, then X ′ is a Banach space.

If F is also surjective, then the Banach space X is called reflexive.

* Corollary If X is a Banach space, then X is reflexive if and only if X ′ is reflexive, which is the case if and only if its unit ball is compact in the weak topology.

The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.

If the norm of a Banach space satisfies this identity, the associated inner product which makes it into a Hilbert space is given by the polarization identity.

If X is a real Banach space, then the polarization identity is

If this suggestion is correct, the beginning of the world happened a little before the beginning of space and time.

No. If I convinced myself of something thought anything at all, then I certainly existed.

In The Origin of Species, he wrote, " If it could be demonstrated that any complex organ existed, which could not possibly have been formed by numerous, successive, slight modifications, my theory would absolutely break down.

If such a language existed, Zamenhof postulated, it could play the role of a neutral communication tool between people of different ethnic and linguistic backgrounds.

If the trend continues, more than half the forest that existed when Papua New Guinea became independent from Australia in 1975 will be gone by 2021.

Darwin concludes: " If it could be demonstrated that any complex organ existed, which could not possibly have been formed by numerous, successive, slight modifications, my theory would absolutely break down.

If any object existed, however faint, it would show up very clearly ".

If there was an attempt by any of the states of Lycia to join forces, as happened in Greece 50 years later, there is no record of it, suggesting that no central government existed.

If the " victim " is unconscious when penetration occurs, he would not be consenting, but this might not be rape if there is a subsisting sexual relationship, e. g. the couple are married, and the other might reasonably believe that consent to intercourse existed by virtue of that relationship.

If he means coureloms and mammoths separately, then he is not specifically saying that mammoths existed on the American continent at that time, seeing as all the animals are meant as hypothetical examples of what the Jaredites might have brought in their barges.

If such a planet actually existed, according to present scientific cosmology, it would be permanently hidden behind the sun but nevertheless detectable from Earth, because of its gravitational influence upon the other planets of the Solar System.

If one believes that spin measurements are indeed measuring the spin of a particle that existed prior to the measurement, then one does reach contradictions.

If Bank of America discovered the warrant no longer existed, Waksal could have been charged with bank fraud.

If the rest mass is imaginary this implies that the denominator is imaginary since the total energy is an observable and thus must be real ; therefore the quantity under the square root must be negative, which can only happen if v is greater than c. As noted by Gregory Benford et al., among others, special relativity implies that tachyons, if they existed, could be used to communicate backwards in time ( see Tachyonic antitelephone article ).

If such features existed, then they would have been small, shallow, turbid, and / or saline, and therefore markedly different from the lake to which today's species are adapted.

If the external thing had not existed that action would not display finality.

Replying to an Assemblyman who asked if the government had a policy of doing so, Aminuddin answered no such plan existed, but " If you want to help the opposition, you can do so.

If the Moselle department still existed in its limits of between 1815 – 1871, its population at the 1999 French census would have been 1, 089, 804 inhabitants.

If the lake existed today, it would be the highest elevation body of water within the state of Minnesota.

" He also said " If we are right and what we see on the seal is a representation of a man meeting a lion, it shows that the Samson legend already existed around the area of Beth Shemesh during that time period.

If the belt has existed for longer than this period, which appears likely, it imposes an upper limit on Epsilon Eridani b's eccentricity of about 0. 10 – 0. 15.

If it existed, it was torn down before or after the construction of the Middle Holsten Gate.

If it existed at all ( and was not just a product of torture ), it could not have been the Shroud of Turin just by its description.

If the crew was not well-trained then the risk of the projectile killing friendly troops or even the crew itself existed.