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For positive integer m the derivative of gamma function can be calculated as follows ( here γ is the Euler – Mascheroni constant ):
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For and positive
For this change is not a change from one positive position to another, but a change from order and truth to disorder and negation.
For example, one study on volunteerism found that feeling overwhelmed by others ' demands had an even stronger negative effect on mental health than helping had a positive one ( although positive effects were still significant ).
Also, supporters of this view would characterize Luke ’ s portrayal of the Roman Empire as positive because they believe Luke “ glosses over negative aspects of the empire and presents imperial power positively .” For example, when Paul is before the council defending himself, Paul says that he is “ on trial concerning the hope of the resurrection of the dead ” ( Acts 23: 6 ).
For many applications where one is only concerned about rotation around one axis, it is sufficient to discard the pseudovector nature of angular momentum, and treat it like a scalar where it is positive when it corresponds to a counter-clockwise rotation, and negative clockwise.
* For a finite field of prime order p, the algebraic closure is a countably infinite field which contains a copy of the field of order p < sup > n </ sup > for each positive integer n ( and is in fact the union of these copies ).
For positive values of a and b, the binomial theorem with n = 2 is the geometrically evident fact that a square of side can be cut into a square of side a, a square of side b, and two rectangles with sides a and b. With n = 3, the theorem states that a cube of side can be cut into a cube of side a, a cube of side b, three a × a × b rectangular boxes, and three a × b × b rectangular boxes.
For example, if an electric field is placed across a solution of Na < sup >+</ sup > and Cl < sup >−</ sup > ( and conditions are right ) the sodium ions move towards the negative electrode ( cathode ), while the chloride ions move towards the positive electrode ( anode ).
For example, reversing the current direction in a Daniell galvanic cell would produce an electrolytic cell, where the copper electrode is the positive terminal and the anode.
For a given material, it can have a positive or negative sign or exceptionally it can be zero, and this can depend on the temperature, as it does for water about 4 C. The concept of latent heat with respect to volume was perhaps first recognized by Joseph Black in 1762.
For the simple electric dipole given above, the electric dipole moment points from the negative charge towards the positive charge, and has a magnitude equal to the strength of each charge times the separation between the charges.
For larger values of n, Fermat's Last Theorem states there are no positive integer solutions ( x, y, z ).</ td >
For example, air pollution may generate a negative externality, and education may generate a positive externality ( less crime, etc .).
For any positive integer n, the set of all n-tuples of real numbers forms an n-dimensional vector space over R, which is denoted R < sup > n </ sup > and sometimes called real coordinate space.
For a reason that was not recorded, he identified the term " positive " with vitreous electricity and " negative " with resinous electricity.
For example, one of the most recent results is that, even after subtracting the positive influence of decadal variation, shown to be possibly present in the ENSO trend, the amplitude of the ENSO variability in the observed data still increases, by as much as 60 % in the last 50 years.
For closed ( orientable or non-orientable ) surfaces with positive genus, the maximum number p of colors needed depends on the surface's Euler characteristic χ according to the formula
For s-polarization, a positive r or t means that the electric fields of the incoming and reflected or transmitted wave are parallel, while negative means anti-parallel.
For p-polarization, a positive r or t means that the magnetic fields of the waves are parallel, while negative means anti-parallel.
* For every prime number p and positive integer n, there exists a finite field with p < sup > n </ sup > elements.
For any positive integers and, there exists a graph with girth at least and chromatic number at least ; for instance, the Grötzsch graph is triangle-free and has chromatic number 4, and repeating the Mycielskian construction used to form the Grötzsch graph produces triangle-free graphs of arbitrarily large chromatic number.
For complex numbers with a positive real part, it is defined via an improper integral that converges:
For and integer
For example, an array of 10 integer variables, with indices 0 through 9, may be stored as 10 words at memory addresses 2000, 2004, 2008, … 2036, so that the element with index i has the address 2000 + 4 × i.
For example, the division example above is surjective ( or onto ) because every rational number may be expressed as a quotient of an integer and a natural number.
For example, algorithms are known for factoring an n-bit integer using just over 2n qubits ( Shor's algorithm ).
For integer order α = n, J < sub > n </ sub > is often defined via a Laurent series for a generating function:
For given nonzero integers a and b there is a nonzero integer of minimal absolute value among all those of the form ax + by with x and y integers ; one can assume d > 0 by changing the signs of both s and t if necessary.
For example, a 1039 bit integer was factored with the special number field sieve using 400 computers over 11 months.
Some sets are infinite ; these sets have more than n elements for any integer n. For example, the set of natural numbers, denotable by, has infinitely many elements, and we cannot use any normal number to give its size.
For example, intervals, where takes all integer values in Z, cover R but there is no finite subcover.
For instance, can hold an unboxed integer in a range supported by the hardware and implementation, permitting more efficient arithmetic than on big integers or arbitrary precision types.
For that purpose, one needs a hash function that maps similar keys to hash values that differ by at most m, where m is a small integer ( say, 1 or 2 ).
For example, when mapping character strings between upper and lower case, one can use the binary encoding of each character, interpreted as an integer, to index a table that gives the alternative form of that character (" A " for " a ", " 8 " for " 8 ", etc .).
For instance, suppose that each input is an integer z in the range 0 to N − 1, and the output must be an integer h in the range 0 to n − 1, where N is much larger than n. Then the hash function could be h
For shapes that are smooth, or shapes with a small number of corners, the shapes of traditional geometry and science, the Hausdorff dimension is an integer.
For instance, Fermat's little theorem for the nonzero integers modulo a prime generalizes to Euler's theorem for the invertible numbers modulo any nonzero integer, which generalizes to Lagrange's theorem for finite groups.