However, if there is no additional complex of singular lines, the order of the image regulus of a pencil is precisely Af.

However a comparable performing digital system is more complex and can require more bandwidth than its analog counterpart.

However, legislating for alterations to the Act is a complex process, since the act is a common denominator in the shared succession of all the Commonwealth realms and the Statute of Westminster 1931 acknowledges by established convention that any changes to the rules of succession may be made only with the agreement of all of the states involved, with concurrent amendments to be made by each state's parliament or parliaments.

However, the mechanistic explanation of catalysis is complex.

However, both cnidarians and ctenophores are more complex than sponges as they have: cells bound by inter-cell connections and carpet-like basement membranes ; muscles ; nervous systems ; and some have sensory organs.

However, both cnidarians and ctenophores have a type of muscle that, in more complex animals, arises from the middle cell layer.

However, what one sees as complex and what one sees as simple is relative and changes with time.

However, recent investigations of languages with very complex click systems such as Nǁng have revealed that the supposed velar – uvular contrast is actually a contrast of a simple clicks versus click – plosive airstream contours ( or consonant clusters, depending on analysis ).

However, whether this is possible depends on how the field of complex numbers is derived in the first place: it may not be possible to distinguish a complex number from its conjugate ( say, 3 + i from 3-i ), since it is impossible to find a property of one that is not also a property of the other, without falling back on the underlying set-theoretic definition.

However, none of Justinian's surviving edicts dealing with Jewish matters is explicitly directed against the Jewish calendar, making the interpretation of Procopius's statement a complex matter.

However, the design of complex software systems is often the domain of software engineering, which is usually considered a separate discipline.

However, it is considered to be a complex process because the contents in solution have to be analyzed in half reactions, whether reduced or oxidized.

However, the Union complex as a whole remained in rebel hands.

However, there could still be an even more complex flash-back construction in American films in the case of W. S. Van Dyke ' sThe Lady of the Dugout ( 1918 ).

However, later research showed that triple-stranded, quadruple-stranded and other more complex DNA molecular structures required Hoogstein bonds.

However, if the shape is complex with many features, the algorithm spends a large amount of time tracing the edges of the region trying to ensure that all can be painted.

However, BBC books such as The Infinity Doctors and Unnatural History imply that the Doctor's origin is complex and that every version is somehow " true " ( such as Susan's claim in the original version of An Unearthly Child that she was born in the 49th century ).

However, while this simple correspondence between an allele and a trait works in some cases, most traits are more complex and are controlled by multiple interacting genes within and among organisms.

However, the actual genetic mechanisms of haplodiploid sex determination may be more complex than simple chromosome number.

However, Ambroise-Auguste Liébeault and Bernheim introduced more complex hypnotic " depth " scales, based on a combination of behavioural, physiological and subjective responses, some of which were due to direct suggestion and some of which were not.

However, more expressive languages are likely to require more complex logic and algorithms to construct equivalent inferences.

However, the complex pronunciation of the ' v ' in Kariva saw the Western constructors produce a sound much like a ' b ' hence the creation of the word Kariba.

However while some animals may acquire large numbers of words and symbols, none have been able to learn as many different signs as is generally known by an average 4 year old human, nor have any acquired anything resembling the complex grammar of human language.

However, XML eliminated many of the more complex and human-oriented features of SGML to simplify implementation environments such as documents and publications.

However, there always is some limit to the numbers who will spend the time and effort to acquire training.

However, if the phone you are dialling these numbers from is already registered with this company it will not announce your number.

However, the Lynx failed to achieve the sales numbers required to attract quality third party developers, and was eventually abandoned.

However, all of these legions were ' short ', and did not have the requisite numbers of troops.

However, with the Franco-Bavarian commanders reluctant to fight until their numbers were deemed sufficient, the Duke enacted a policy of plundering in Bavaria designed to force the issue.

However, the lack of manoeuvrability of the British forces, combined with lack of open-field tactics to command these numbers, put them at a disadvantage to the Romans, who were skilled at open combat due to their superior equipment and discipline.

However, sealed trains began carrying large numbers of prisoners to the Soviet Gulags.

However, both simple and high-end algorithms for computing Bernoulli numbers exist.

However, this problem has been alleviated somewhat by the development of the TransMilenio Bus Rapid System and the restriction of vehicles through a daily, rotating ban on private cars depending on plate numbers.

However, both numbers would be a fraction of the heat generated by many high performance processors, some years later.

However, considered as a sequence of real numbers, it does converge to the irrational number.

However, there are some applications that require results other than numbers.

However, this is a relatively small number especially when playing with more than four people, so many domino sets are " extended " by introducing ends with greater numbers of spots, which increases the number of unique combinations of ends and thus of pieces.

However, most real numbers are not definable: the set of all definable numbers is countably infinite ( because the set of all logical formulas is ) while the set of real numbers is uncountably infinite ( see Cantor's diagonal argument ).

* However, if we think of as the field of real numbers, then the direct product does not exist-naively defining in a similar manner to the above examples would not result in a field since the element does not have a multiplicative inverse.

However, if the equation were based on the natural numbers for example, some of these operations ( like division and subtraction ) may not be valid as negative numbers and non-whole numbers are not allowed.

However, as of February 2009, at least four players must wear numbers between 50 and 79 on first, second, or third down, which by rule would make them ineligible receivers.

However, these numbers are not known to be expressible by themselves in terms of elementary functions.

However, sufficient numbers of moriscos stayed that Muslim culture remained influential in Spain.

However, the inability to maintain slave numbers without constant resupply from Africa meant the slave population, by 1789, totaled 500 000, ruled over by a white population that, by 1789, numbered only 32 000.