* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

If F ≥ F < sub > Critical </ sub > ( Numerator DF, Denominator DF, α )

If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O < sub > K </ sub >.

If ΔS and / or T are small, the condition ΔG < 0 may imply that ΔH < 0, which would indicate an exothermic reaction.

If M is a Turing Machine which, on input w, outputs string x, then the concatenated string < M > w is a description of x.

Theorem: If K < sub > 1 </ sub > and K < sub > 2 </ sub > are the complexity functions relative to description languages L < sub > 1 </ sub > and L < sub > 2 </ sub >, then there is a constant c – which depends only on the languages L < sub > 1 </ sub > and L < sub > 2 </ sub > chosen – such that

If the first allele is dominant to the second, then the fraction of the population that will show the dominant phenotype is p < sup > 2 </ sup > + 2pq, and the fraction with the recessive phenotype is q < sup > 2 </ sup >.

If activated cytotoxic CD8 < sup >+</ sup > T cells recognize them, the T cells begin to secrete various toxins that cause the lysis or apoptosis of the infected cell.

If ADH production is excessive in heart failure, Na < sup >+</ sup > level in the plasma may fall ( hyponatremia ), and this is a sign of increased risk of death in heart failure patients.

If we define r < sub > i </ sub > as the displacement of particle i from the center of mass, and v < sub > i </ sub > as the velocity of particle i with respect to the center of mass, then we have

Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

* The Lusternik – Schnirelmann theorem: If the sphere S < sup > n </ sup > is covered by n + 1 open sets, then one of these sets contains a pair ( x, − x ) of antipodal points.

Some authors require in addition that μ ( C ) < ∞ for every compact set C. If a Borel measure μ is both inner regular and outer regular, it is called a regular Borel measure.

If A is expressed as an N × N matrix, then A < sup >†</ sup > is its conjugate transpose.

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

* If < math > a < b </ math > and < math > b < c </ math > then < math > a < c </ math >;

* If < math > a < b </ math > and < math > c < d </ math > then < math > a + c < b + d </ math >;

* If < math > a < b </ math > and then < math > ac < bc </ math >;

* If < math > a < b </ math > and < math > c < 0 </ math > then < math > bc < ac </ math >.

: If < math >

* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )

If a ) testimonies conflict one another, b ) there are a small number of witnesses, c ) the speaker has no integrity, d ) the speaker is overly hesitant or bold, or e ) the speaker is known to have motives for lying, then the epistemologist has reason to be skeptical of the speaker's claims.

* If the operation is associative, ( ab ) c = a ( bc ), then the value depends only on the tuple ( a, b, c ).

If c is another common divisor of a and b, then c also divides as + bt

If the user pressed keys 1 + 2 = 3 simultaneously the letter " c " appeared.

If a and b are coprime and a divides the product bc, then a divides c. This can be viewed as a generalization of Euclid's lemma.

If the weight of the roadway per unit length is w and the weight of the cable and the wire supporting the bridge is negligible in comparison, then the weight on the cable from c to r is wx where x is the horizontal distance between c to r. Proceeding as before gives the differential equation

Linear Diophantine equations take the form ax + by = c. If c is the greatest common divisor of a and b then this is Bézout's identity, and the equation has an infinite number of solutions.

It follows that there are also infinitely many solutions if c is a multiple of the greatest common divisor of a and b. If c is not a multiple of the greatest common divisor of a and b, then the Diophantine equation ax + by = c has no solutions.

:“ If an integer n is greater than 2, then has no solutions in non-zero integers a, b, and c. I have a truly marvelous proof of this proposition which this margin is too narrow to contain .”

If the baseband data signal ( the message ) to be transmitted is and the sinusoidal carrier is, where f < sub > c </ sub > is the carrier's base frequency and A < sub > c </ sub > is the carrier's amplitude, the modulator combines the carrier with the baseband data signal to get the transmitted signal:

If the character set doesn't contain an up arrow, the caret ^ is used instead.

* If κ is a limit and the continuum function is eventually constant below κ then < math > 2 ^

If G is a finite abelian group, then G and G ^ are isomorphic, but not canonically.

If A is the additive monoid of natural numbers we obtain the semiring of natural numbers as End ( A ), and if A = S ^ n with S a semiring, we obtain ( after associating each morphism to a matrix ) the semiring of square n-by-n matrices with coefficients in S.

If the keysize is | k |, this attack uses only 2 < sup > k + 1 </ sup > encryptions ( and decryptions ) ( and O ( 2 ^ k ) memory in case a look-up table have been built for the set of forward computations ) in contrast to the naive attack, which needs 2 < sup > 2 · k </ sup > encryptions but O ( 1 ) space.

If you already know some factors of a large SNFS-number, you can do the SNFS calculation modulo the remaining part ; for the NFSNET example above, 3 ^ 479 + 1 = ( 4 * 158071 * 7167757 * 7759574882776161031 ) times a 197-digit composite number ( the small factors were removed by ECM ), and the SNFS was performed modulo the 197-digit number.

If the user selects " Dec " mode, and types " 16 " and then presses the " x ^ y " button and then types " 16 " again, and then presses "=" the result will be ( decimal ) 18446744073709551616, which is correct, as that is the decimal number for " 16 raised to the 16th power ".

If the order of F is less than 2 ^ b, where b is the number of bits in a symbol, then multiple polynomials can be used.

If x has exponent E, then ULP ( x ) = machine epsilon x 2 ^ E.

If the discriminant of the quadratic function ( 2 ) is negative (< math > b_1 ^ 2-4 b_2 b_0 < 0 </ math >), it has no real roots.

If we can arrange to be smaller than, then we have < math > g ^ 2_