Since B < sub > n </ sub >= 0 for all odd n > 1, and many formulas only involve even-index Bernoulli numbers, some authors write B < sub > n </ sub > instead of B < sub > 2n </ sub >.

The proof of this fact relies on an algorithm which, given the first n digits of Ω, solves Turing's halting problem for programs of length up to n. Since the halting problem is undecidable, Ω can not be computed.

Since every equivalence relation over X corresponds to a partition of X, and vice versa, the number of possible equivalence relations on X equals the number of distinct partitions of X, which is the nth Bell number B < sub > n </ sub >:

Since the proving of the theorem, efficient algorithms have been found for 4-coloring maps requiring only O ( n < sup > 2 </ sup >) time, where n is the number of vertices.

Since it is a balanced tree, it guarantees insertion, search and delete to be O ( log n ) in time, where n is the total number of elements in the tree.

Since there are n!

Since the leftmost divisor bit zeroed every input bit it touched, when this process ends the only bits in the input row that can be nonzero are the n bits at the right-hand end of the row.

* Omission of the high-order bit of the divisor polynomial: Since the high-order bit is always 1, and since an n-bit CRC must be defined by an ( n + 1 )- bit divisor which overflows an n-bit register, some writers assume that it is unnecessary to mention the divisor's high-order bit.

Since 1953, " binomi < u > n </ u > al nomenclature " is the technically correct term in zoology.

Since this quantity goes to infinity as n increases, we expect that every large even integer has not just one representation as the sum of two primes, but in fact has very many such representations.

If n is not a prime power, then every Sylow subgroup is proper, and, by Sylow's Third Theorem, we know that the number of Sylow p-subgroups of a group of order n is equal to 1 modulo p and divides n. Since 1 is the only such number, the Sylow p-subgroup is unique, and therefore it is normal.

Since for each n this removes intervals adding up to at most 1 / 2 < sup > n + 1 </ sup >, the nowhere dense set remaining after all such intervals have been removed has measure of at least 1 / 2 ( in fact just over 0. 535 ... because of overlaps ) and so in a sense represents the majority of the ambient space.

The transfer principle states that true first order statements about R are also valid in * R. For example, the commutative law of addition, x + y = y + x, holds for the hyperreals just as it does for the reals ; since R is a real closed field, so is * R. Since for all integers n, one also has for all hyperintegers H. The transfer principle for ultrapowers is a consequence of Łoś ' theorem of 1955.

Since every element of C < sub > n </ sub > generates a cyclic subgroup and the subgroups of C < sub > n </ sub > are of the form C < sub > d </ sub > where d | n, the formula follows.

Since the amplitude at these reflections grows linearly with the number N of scatterers, the observed intensity of these spots should grow quadratically, like N < sup > 2 </ sup >.

Since the apparitions, Lourdes has become one of the world's leading Catholic Marian shrines and the number of visitors grows each year.

Since the series takes place in real time, Samantha grows from a child to a young woman.

Since it grows in temperatures approaching boiling, at about 80 degrees Celsius, it is considered to be a thermophile.

Since all is in flux, to get the most from life we must learn to discriminate through " sharp and eager observation ": for " every moment some form grows perfect in hand or face ; some tone on the hills or the sea is choicer than the rest ; some mood of passion or insight or intellectual excitement is irresistibly real and attractive for us, – for that moment only ".

The new organism remains attached as it grows, separating from the parent organism only when it is mature. Since the reproduction is asexual, the newly created organism is a clone and is genetically identical to the parent organism.

Since it usually grows in inaccessible places, it is associated in many countries of the alpine region with mountaineering.

Since the hair on the pubis is coarse and grows horizontally and inward, a second application wax is often required.

Since it allocates memory dynamically as it grows, it is only bounded by machine memory.

Since she often finds herself in areas where witches are unwelcome, she has a spring-operated hat that only grows a point when she wants it to.

Since, for a given bullet shape, frontal surface increases as the square of the calibre, and mass increases as the cube of the diameter, then sectional density grows linearly with bore diameter.

Since, OIC has expanded its reach to children in need around the world, including the Phillippines, Panama, Haiti, Djibouti, and the list grows.

Since there are an infinite number of modes this implies infinite heat capacity ( infinite energy at any non-zero temperature ), as well as an unphysical spectrum of emitted radiation that grows without bound with increasing frequency, a problem known as the ultraviolet catastrophe.

Since negative climb involves an addition of atoms to the half plane, the crystal grows in the direction perpendicular to the half plane.

Since the 18th century, the people from Genemuiden have been producing floor mats made of the bulrush that grows on the shores of the Zuiderzee.

Since the ends of CWR are fixed the rail will experience tensile stress in extreme cold and can fracture if it grows too great.

Since this fungus grows underground, it is gathered with the aid of specially trained dogs.

Since the control bandwidth is defined to be less than 5 % regardless of network size, the amount of control bandwidth required is not supposed to increase as network size grows.

Since it was a machine imbued with magic the artificial creature never grows tired.

Since this fungus grows underground, it is gathered with the aid of specially trained dogs.

Since the fungus grows on tissue surfaces rather than inside epithelial cells, topical applications of oils and other compounds are recommended.

Since the air on the top of the wing is traveling faster than on the bottom, due to Bernoulli effect, at speeds close to the speed of sound the air on the top of the wing will be accelerated to supersonic.

Since computers can make arithmetic calculations much faster and more accurately than humans, it was thought to be only a short matter of time before the technical details could be taken care of that would allow them the same remarkable capacity to process language.

Since such life utilised strong nuclear forces instead of electromagnetic interactions, it was posited that life might function millions of times faster than typical on Earth.

Since this implies that one particle is communicating with the other instantaneously across space, i. e. faster than light, this is the " paradox ".

Since real numbers can be represented as distances or intervals on a line, the slide rule was invented in the 1620s to allow multiplication and division operations to be carried out significantly faster than was previously possible.

Since then the service sector has been growing faster than the agrarian and industrial sectors.

Since the 18th century, American urban areas have often grown faster than city boundaries.

* Since no hash function is used, tries are generally faster than hash tables for small keys.

Since e-mail is faster than the standard postal service, the rise of the Internet saw a shift of people playing board wargames from play-by-mail ( PBM ) to play-by-email ( PBEM ) or play-by-web ( PBW ).

Since the end of the browsers wars there have been new browsers coming onto the scence, many of these are open source meaning that they tend to have faster development and are more supportive of new standards.

Since they form from severe thunderstorms and can be far more intense, faster, and longer-lived than fair weather waterspouts, they are more dangerous.

Since then, evolutionary changes in chromosome numbers have taken place about twenty times faster in mammals than in frogs which means that speciation is occurring more rapidly in mammals.

Since light travels faster than sound, viewers may perceive that movement is out of phase with the sound.

Since the main rig took time to put up and take down, they implemented another rig, which was capable of being erected faster than the main rig.

Since less water or liquid has to be heated, the food reaches its cooking temperature faster.

Since the tropical country's workers tend to be sicker and die sooner, they will be less productive, and over many centuries the cooler and more disease-free economies will tend to have faster economic growth.

Since a reduction in twist rate is very detrimental to accuracy, gunsmiths who are machining a new barrel from a rifled blank will often measure the twist carefully so they may put the faster rate, no matter how minute the difference is, at the muzzle end ( see internal ballistics for more information on accuracy and bore characteristics ).

Since the completion of Interstate 195 and Interstate 295, which intersect southwest of the circle, much of the traffic that would have passed through the circle now uses the wider, faster freeways.

Since shorter strands of DNA move through the gel matrix faster they are detected sooner and there is then a direct correlation between length of DNA strand and time at the detector.

Since the late 1990s, and particularly in the early 2000s, the population has been expanding at a much faster rate, with town rejuvenation, new retail and entertainment facilities and many new housing developments.

Since its creation, broadband has continually strengthened and available speeds have become faster and faster.

Since 1985, the U. S. military has been testing serial hybrid Humvees and have found them to deliver faster acceleration, a stealth mode with low thermal signature / near silent operation, and greater fuel economy.

Since a 6x86 running at 133 MHz generally benchmarked slightly faster than a Pentium running at 166 MHz, the 133 MHz 6x86 was marketed as the 6x86-P166 +.