What we have done here is arranged the integers and the odd integers into a one-to-one correspondence ( or bijection ), which is a function that maps between two sets such that each element of each set corresponds to a single element in the other set.

The gamma function must alternate sign between the poles because the product in the forward recurrence contains an odd number of negative factors if the number of poles between and is odd, and an even number if the number of poles is even.

Depending on the function, the remainder may be uniform only for certain values of n, e. g. odd or prime numbers.

When one is found, and the position is odd, the location is returned from the function with suspend.

Such a maritime clock had to be not only highly accurate over long time intervals, but relatively impervious to corrosion in salt air, able to tolerate wide variations in temperature and humidity and in general durable while able to function at the odd angles and pitch and yaw typical of decks under strong waves and storm tossed conditions.

function odd?

* ( n ): the Möbius function, the parity (− 1 for odd, + 1 for even ) of the number of prime factors of square-free numbers ; 0 if n is not square-free

** Parity function, a Boolean function whose value is 1 if the input vector has an odd number of ones

For s an even positive integer, the product sin ( πs / 2 ) Γ ( 1 − s ) is regular and the functional equation relates the values of the Riemann zeta function at odd negative integers and even positive integers.

* The Riemann zeta function at other odd integers, ζ ( 5 ), ζ ( 7 ), ... ( not known to be irrational.

We assume a “ linear ” device having a transfer function whose small signal form may be expressed in terms of a power series containing only odd terms, making the transfer function an odd function of input signal voltage, i. e., O = − O.

First, one has to specify whether the function is even or odd at both the left and right boundaries of the domain ( i. e. the min-n and max-n boundaries in the definitions below, respectively ).

Second, one has to specify around what point the function is even or odd.

( A similar problem arises for the DST, in which the odd left boundary condition implies a discontinuity for any function that does not happen to be zero at that boundary.

# In every row in which the truth value of the function is ' T ', there are an odd number of ' T's assigned to the arguments and in every row in which the function is ' F ' there is an even number of ' T's assigned to arguments.

# In every row in which the value of the function is ' T ', there is an even number of ' T's assigned to the arguments of the function ; and in every row in which the truth value of the function is ' F ', there are an odd number of ' T's assigned to arguments.

'' I thought, and wondered what kind of homesteads such odd pioneers would establish in this suburban frontier ; ;

In the following paper it is shown that in a certain definite sense, exactly an odd number of squares can be inscribed in every such curve which does not contain an infinite number of inscribed squares.

There must be an odd number of such components, which will be called complete components.

To the manner born -- odd to have such a thought at a time like this ; ;

This is because we arrange things such that for every integer, there is a distinct odd integer: ... − 2 → − 3, − 1 → − 1, 0 → 1, 1 → 3, 2 → 5, ...; or, more generally, n → 2n + 1.

* The Interviewer ( played by Chris Morris ): conducting real interviews with celebrities such as Andrew Morton and Jerry Springer, Morris confuses and mocks his subjects with ambiguous and odd questions.

Later ancient commentators such as Proclus ( 410 – 485 CE ) treated many questions about infinity as issues demanding proof and, e. g., Proclus claimed to prove the infinite divisibility of a line, based on a proof by contradiction in which he considered the cases of even and odd numbers of points constituting it.

Unfortunately, in such calculations, Fermi did not take into account the " pairing energy " that might appear when a nuclide with an odd number of neutrons absorbed an extra neutron.

Three colors are adequate for simpler maps, but an additional fourth color is required for some maps, such as a map in which one region is surrounded by an odd number of other regions that touch each other in a cycle.

In particle physics, a fermion ( a name coined by Paul Dirac from the surname of Enrico Fermi ) is any particle characterized by Fermi – Dirac statistics and following the Pauli exclusion principle ; fermions include all quarks and leptons, as well as any composite particle made of an odd number of these, such as all baryons and many atoms and nuclei.

After taking odd jobs to get by, such as driving a truck, Kosiński graduated from Columbia University.

The multi-sentence version of the liar paradox generalizes to any circular sequence of such statements ( wherein the last statement asserts the truth / falsity of the first statement ), provided there are an odd number of statements asserting the falsity of their successor ; the following is a three-sentence version, with each statement asserting the falsity of its successor:

* odd sensation in legs, arms, chest, or face, such as tingling or numbness ( neuropathy );

In that case, the correct move is to leave an odd number of heaps of size one ( in normal play, the correct move would be to leave an even number of such heaps ).

Among the heavy actinide elements, however, those isotopes that have an odd number of neutrons ( such as U-235 with 143 neutrons ) bind an extra neutron with an additional 1 to 2 MeV of energy over an isotope of the same element with an even number of neutrons ( such as U-238 with 146 neutrons ).

However, non-time-critical custom chip access, such as blitter transfers, can use up any spare odd or even cycles and, if the " BLITHOG " ( blitter hog ) flag is set, Agnus can lock out the even cycles from the CPU in deference to the blitter.

In 1993, Porsche Motorsports at Weissach briefly produced a turbocharged 968 Turbo S, a fairly odd naming choice for Porsche which usually reserves the added " S " moniker for models that have been tuned for more power over a " lesser " counterpart, such as with the 911 Turbo.

No such simple expression is known for odd positive integers.

The non-degeneracy condition means that for all we have the property that there does not exist non-zero such that for all The skew-symmetric condition ( inherent in the definition of differential 2-form ) means that for all we have for all Recall that in odd dimensions antisymmetric matrices are not invertible.

The creatures co-exist in Teletubbyland ( the name of their home environment ) with a number of odd contraptions such as Nuu-Nuu, the group's anthropomorphic blue vacuum cleaner with the tendency to suck up the Teletubbies ' possessions and the voice trumpets, devices resembling periscopes that rise from the ground and interact with the Teletubbies, serving as either supervisors for the beings who often arise to engage in games with them.

In contrast, artificial water containers, such as the odd plastic bucket, flowerpot " saucer ", or discarded bottle or tire, are important breeding places for some of the most serious disease vectors, such as species of Aedes that transmit dengue and yellow fever.

Popular Japanese ramen chains serve ramen alongside distinctly Japanese dishes such as tempura and yakitori, something which would be seen as odd in Japan.

He worked his way through college at the California Institute of Technology, playing banjo at nightclubs and taking odd jobs, which included working at the campus laundry facility, waiting tables, and cleaning engines at a local power plant.

In the British Empire ( originally in the maritime and hinterland sphere of influence of the East India Company, HEIC, later transformed into crown territories ), mainly in British India, the numbers of guns fired as a gun salute to the ruler of a so-called princely state became a politically highly significant indicator of his status, not governed by objective rules, but awarded ( and in various cases increased ) by the British paramount power, roughly reflecting his state's socio-economic, political and / or military weight, but also as a prestigious reward for loyalty to the Raj, in classes ( always odd numbers ) from three to 21 ( seven lacking ), for the " vassal " indigenous rulers ( normally hereditary with a throne, sometimes raised as a personal distinction for an individual ruling prince ).

Even numbered zones have the maximum phase cancelling effect and odd numbered zones may actually add to the signal power.

z is a primitive element, but z < sup > 2 </ sup > is not, because the odd powers of z are not a power of z < sup > 2 </ sup >.

When this article refers to ' an odd power ' of a number it means the exponent is an odd number, not that the result is odd.

For instance 2 < sup > 3 </ sup > which is 8 is an odd power of 2 because the exponent is 3.

Glenn worked odd jobs at power plants, restaurants, and more before deciding to pursue an acting career.

Captain Rayment abandoned the take off after Captain Thain had noticed the port boost pressure gauge fluctuating as the plane reached full power and the engine sounded odd while accelerating.

If n is a positive integer, the integers between 1 and n − 1 which are coprime to n ( or equivalently, the congruence classes coprime to n ) form a group with multiplication modulo n as the operation ; it is denoted by Z < sub > n </ sub >< sup >×</ sup > and is called the group of units modulo n or the group of primitive classes modulo n. As explained in the article multiplicative group of integers modulo n, this group is cyclic if and only if n is equal to 2, 4, p < sup > k </ sup >, or 2 p < sup > k </ sup > where p < sup > k </ sup > is a power of an odd prime number.

A number A relatively prime to an odd prime p is a residue modulo any power of p if and only if it is a residue modulo p.

Modulo an odd prime power n =

In an odd twist, the original Stellar group was left with most of the corporate power.

W. H. Mills ( unpublished ; see Muskat ) showed that any odd harmonic divisor number above 1 must have a prime power factor greater than 10 < sup > 7 </ sup >, and Cohen showed that any such number must have at least three different prime factors.

One gets this because the sum of all the unitary divisors is a multiplicative function and one has the sum of the unitary divisors of a power of a prime p < sup > a </ sup > is p < sup > a </ sup > + 1 which is even for all odd primes p. Therefore, an odd unitary perfect number must have only one distinct prime factor, and it is not hard to show that a power of prime cannot be a unitary perfect number, since there are not enough divisors.

If p < sup > δ < sub > p </ sub ></ sup > is an odd prime power factor of Δ and if p divides N only once ( i. e. n < sub > p </ sub >= 1 ), then there exists another elliptic curve E ', with conductor N ' = N / p, such that the coefficients of the L-series of E are congruent modulo ℓ to the coefficients of the L-series of E '.