If f is a non-constant entire function, then its image is dense in C. This might seem to be a much stronger result than Liouville's theorem, but it is actually an easy corollary.

The completeness of first-order logic is an easy corollary of results Skolem proved in the early 1920s and discussed in Skolem ( 1928 ), but he failed to note this fact, perhaps because mathematicians and logicians did not become fully aware of completeness as a fundamental metamathematical problem until the 1928 first edition of Hilbert and Ackermann's Principles of Mathematical Logic clearly articulated it.

Wallis ' product is, in retrospect, an easy corollary of the later Euler formula for the sine function.

As a result, it is not easy to find a stock priced as the Morton issue was priced ( at roughly 10 times 1960 earnings, to yield a little over 5 per cent on the 64-cent anticipated dividend ).

Another risk of the hit-and-run is that a caught line drive could result in an easy double play, although this is offset by the lower likelihood of a ground ball double play.

The authors of the study interpreted this as being the result of the ease by which primary commodities may be extorted or captured compared to other forms of wealth, for example, it is easy to capture and control the output of a gold mine or oil field compared to a sector of garment manufacturing or hospitality services.

Given good circumstances one might be able to discern the result of some human activity such as the changing of the Netherlands ' coast or the partial drying out of the Aral Sea, but even that would not be easy.

As a result, all previous hypertext systems were overshadowed by the success of the Web, even though it originally lacked many features of those earlier systems, such as an easy way to edit what you were reading.

The Financial Crisis of 2008 severely disrupted the Latvian economy, primarily as a result of the easy credit bubble that began building up during 2004.

The result is a language easy to understand at first sight for individuals acquainted with several Western European languages.

This result implies the Poincaré conjecture because it is easy to check it for the possible manifolds listed in the conclusion.

This can result in poor quality output for lower resolution devices ( although techniques such as spatial anti-aliasing may help mitigate this ), but provides a fixed layout, allowing easy user visualisation.

However, using a simple pairing strategy, this has been proven to result in an easy win for the second player.

The Mexican officers knew that the Brown Bess muskets they carried lacked the range of the Texan weapons, but Santa Anna was convinced that his superior planning would nonetheless result in an easy victory.

Generally, the goal of human-machine interaction engineering is to produce a user interface which makes it easy, efficient, and enjoyable to operate a machine in the way which produces the desired result.

The guns, drugs and money of the hood are typically not the aftermath of an easy life but the result of a struggle whether it is as a hustler or as a gangbanger.

This allows quick and easy fitting and provides a snug and comfortable diaper fit that will neither bind nor wad on the infant … as a result of this snugger fit obtained because of this fold configuration, the diaper is less likely to leak or, in other words, its containment characteristics are greatly enhanced .” Further developments in diaper design were made, such as the introduction of refastenable tapes, the " hourglass shape " so as to reduce bulk at the crotch area, and the 1984 introduction of super-absorbent material from polymers known as sodium polyacrylate that were originally developed in 1966.

Due to its often flavorful taste, easy availability and cheapness, it is popular with poor children, who can exhibit precancerous lesions at a very early age as a result.

The design of pennies that date pre 1939 had the sovereign's head on the obverse ( front ) and the reverse was totally covered in writing making the result very easy and quick to see.

As a result, it is relatively easy to eavesdrop on VoIP calls and even change their content.

One key concern of the REST Web service developers is that the SOAP WS toolkits make it easy to define new interfaces for remote interaction, often relying on introspection to extract the WSDL, since a minor change on the server ( even an upgrade of the SOAP stack ) can result in different WSDL and a different service interface.

As a result it is now regarded as a fashionable and desirable place to live for the British middle classes and is within easy commuting distance of the City of London and West End and the main railway termini for transport to airports at Heathrow and Gatwick and the south of England.

One demographic consequence of a small population size, the probability that all offspring in a generation are of the same sex, and where males and females are equally likely to be produced ( see sex ratio ), is easy to calculate: it is given by ( The chance of all animals being females is ; the same holds for all males, thus this result ).

As a result of the availability of cheap and easy transportation, which by the 1850s included the railroad as well as the canal, companies like the DeLand Chemical Company, the Cobb Preserving Company, Taylor's Oil of Life, and eventually the American Can Company, grew and thrived.

The result was " an extremely brittle alloy which not only caused the Bell to fail in service but made it easy for early souvenir collectors to knock off substantial trophies from the rim ".

When the jitter amount is very high, as in very low cost CD players ( 2ns ), the result is somewhat similar to wow and flutter, the well known problem that affected typically compact cassettes ( and in a far less evident way turntables ) and was caused by the non perfectly constant speed of the tape: the effect is similar, but here the variations have a far higher frequency and for this reasons are less easy to perceive but equally annoying.

As they collect the effluent from hospitals, industry, and households, they may be subject to an easy terrorist attack-especially damaging given the fact that the result sludge is typically not well policed and will be dispersed far and wide across wide areas of farm land.

Using the above theorem it is easy to see that the original Borsuk Ulam statement is correct since if we take a map f: S < sup > n </ sup > → ℝ < sup > n </ sup > that does not equalize on any antipodes then we can construct a map g: S < sup > n </ sup > → S < sup > n-1 </ sup > by the formula

The style was used in bronze by Bernini for his spectacular St. Peter's baldachin, actually a ciborium ( which displaced Constantine's columns ), and thereafter became very popular with Baroque and Rococo church architects, above all in Latin America, where they were very often used, especially on a small scale, as they are easy to produce in wood by turning on a lathe ( hence also the style's popularity for spindles on furniture and stairs ).

Though easy to understand, the implementation of the algorithm used above is impractical in languages and environments where stack space is severely constrained ( e. g. Java applets ).

It is easy enough to " translate " between polar and rectangular coordinates in the plane: let, as above, direction and distance be and respectively, then we have

Olivia Ball and Paul Gready argue that for both civil and political rights and economic, social and cultural rights, it is easy to find examples which do not fit into the above categorisation.

Ido was specifically designed to be grammatically, orthographically, and lexicographically regular, and above all easy to learn and use.

The concrete definition given above is easy to work with, but has some minor problems: to use it we first need to represent a Lie group as a group of matrices, but not all Lie groups can be represented in this way, and it is not obvious that the Lie algebra is independent of the representation we use.

The definition above is easy to use, but it is not defined for Lie groups that are not matrix groups, and it is not clear that the exponential map of a Lie group does not depend on its representation as a matrix group.

These combinations are designed to be easy to remember, as the circumflex accent ( e. g. â ) is similar to a caret (^), printed above the 6 key ; the diaeresis ( e. g. ö ) is similar to the double-quote (") above 2 on the UK keyboard ; the tilde (~) is printed on the same key as the #.

With spreadsheets the equations above are easy to use and offer much greater accuracy over a wider range of concentrations.

Deskilled labour is cheap and above all easy to control due to the workers lack of direct engagement in the production process.

Now we have an easy way to calculate the radius of the first Fresnel zone ( F < sub > 1 </ sub > in the above equation ), knowing the distance between the two antennas and the frequency of the transmitted signal.

If not, the relay is still effectively open ( for instance, by the above rules ): it is easy to forge e-mail header and envelope information, it is considerably harder to successfully forge an IP address in a TCP / IP transaction due to the three-way handshake that occurs as a connection is started.

The above example consisted of 12 observations in the dataset, which made the determination of the quartiles very easy.

An observer commented that Calhoun was " the most elegant speaker that sits in the House ... His gestures are easy and graceful, his manner forcible, and language elegant ; but above all, he confines himself closely to the subject, which he always understands, and enlightens everyone within hearing ; having said all that a statesman should say, he is done.

In a tap a cappella, the steps are normally kept simple ( such as the most common steps listed above ) and easy to control.

From there, on a dry point, above sea level, it is easy to appreciate how Glastonbury was once an island and, in the winter, the surrounding moors are often flooded, giving that appearance once more.

The triode vacuum tube amplifiers of the day would not amplify stably above 500 kHz, however, it was easy to get them to oscillate above that frequency.

Similarly, a 1210, which is 10 cents of grade above 1200, is an easy 5. 12, a 1290 is a hard 5. 12, and a 1299 is the hardest possible 5. 12.

Sydenham Hill is one of the highest locations in London ; 109 metres ( 357 ft ) above sea level ( spot height on Ordnance Survey Map ); and the size of the palace and prominence of the site made it easy to identify from much of London.

All the above attractions are within easy walking distance of each other ( all within a thousand feet ), and are popular sites for wedding photographers.

It is easy to check that this embedding respects all of the above analogies.