* 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 the symbol < tt >/</ tt > is chosen, for example, then the number seven would be represented by < tt >///////</ tt >.

If the Developer Tools are installed, it can be found at < tt >/ Developer / Examples / Carbon / SimpleText </ tt >.

If the demand starts at < tt > D2 </ tt >, and decreases to < tt > D1 </ tt >, the equilibrium price will decrease, and the equilibrium quantity will also decrease.

If the supply curve starts at < tt > S2 </ tt >, and shifts leftward to < tt > S1 </ tt >, the equilibrium price will increase and the equilibrium quantity will decrease as consumers move along the demand curve to the new higher price and associated lower quantity demanded.

If the server is case sensitive and < tt >< nowiki > http :// en. example. org / wiki / URL </ nowiki ></ tt > is correct, < tt >< nowiki > http :// en. example. org / WIKI / URL </ nowiki ></ tt > or < tt >< nowiki > http :// en. example. org / wiki / url </ nowiki ></ tt > will display an HTTP 404 error page, unless these URLs point to valid resources themselves.

*:< tt > If assert P, assert Q </ tt >

*:< tt > If assert not Q, assert not P </ tt >

*:< tt > If goal Q, goal P </ tt >

* If instead the sampling frequency is known, the theorem gives us an upper bound for frequency components, B < f < sub > s </ sub >/ 2, of the signal to allow for perfect reconstruction.

If, instead of 1 matching entry, there are k matching entries, the same algorithm works but the number of iterations must be π ( N / k )< sup > 1 / 2 </ sup >/ 4 instead of πN < sup > 1 / 2 </ sup >/ 4.

#* If you're making a topical archive, use the name of the topic, for example < tt >< nowiki >/ Place of birth debate </ nowiki ></ tt >.

#* If you're just archiving old discussion, use the next available number ; so if the last archive page was Archive 3, call it < tt >< nowiki >/ Archive 4 </ nowiki ></ tt >.

#* If there are no archives yet, call it < tt >< nowiki >/ Archive 1 </ nowiki ></ tt >.

If a torus is isomorphic to a product of multiplicative groups G < sub > m </ sub >/ S, the torus is said to be split.

If M is a supermanifold of dimension ( p, q ), then the underlying space M inherits the structure of a differentiable manifold whose sheaf of smooth functions is O < sub > M </ sub >/ I, where I is the ideal generated by all odd functions.

If, conversely, such effects are absent ( Δ ( w / 2 ) « Δf, Sauerbrey thickness same on all overtone orders ) one may assume that ( 1-Z < sub > Liq </ sub >< sup > 2 </ sup >/ Z < sub > F </ sub >< sup > 2 </ sup >)≈ 1.

A series x < sub > n </ sub > is called hypergeometric if the ratio of successive terms x < sub > n + 1 </ sub >/ x < sub > n </ sub > is a rational function of n. If the ratio of successive terms is a rational function of q < sup > n </ sup >, then the series is called a basic hypergeometric series.

If we suppose × U ( 1 )< sub > χ </ sub >/ Z < sub > 5 </ sub > is a subgroup of SO ( 10 )< sub > F </ sub >, then we have the intermediate scale symmetry breaking × U ( 1 )< sub > B </ sub >/ Z < sub > 5 </ sub > → × U ( 1 )< sub > χ </ sub >/ Z < sub > 5 </ sub > where

If the behavior should be enabled whenever the system boots, the line < tt > parameter = 1 </ tt > can be added / rewritten to the file < tt >/ etc / sysctl. conf </ tt >.

If this article / noun pair is used as the object of a verb, it ( usually ) changes to the accusative case, which entails an article shift in German – Ich sehe den Wagen.

If the object point O is infinitely distant, u1 and u2 are to be replaced by h1 and h2, the perpendicular heights of incidence ; the sine condition then becomes sin u ' 1 / h1 = sin u ' 2 / h2.

" If this is not an error for " ANaesthetic ", then it needs to be added as a noun sense to Wikipedia / Wiktionary?

If an ESN / MIN Pair is intercepted, it could then be cloned onto a different phone and used in other areas for making calls without paying.

If the rider is sitting upright, the rider's body creates about 75 % of the total drag of the bicycle / rider combination.

If the boat floats, the mass of the boat ( plus contents ) as a whole divided by the volume below the waterline is equal to the density of water ( 1 kg / l ).

If the book of Zephaniah was largely composed during the monarchic period, the author of the book of Zephaniah attempts to accomplish this change in behaviour through the threat of future calamity for “ those who have turned back from following the Lord, / who have not sought the Lord or inquired of him ” ( 1: 6 ).

If the convention B < sub > 1 </ sub >=− is used, this sequence is also known as the first Bernoulli numbers ( / in OEIS ); with the convention B < sub > 1 </ sub >=+ is known as the second Bernoulli numbers ( / in OEIS ).

If the slope / cliff being undercut is made of unconsolidated sediment it will erode at a much faster rate then a cliff made of bedrock.

If one key cannot be deduced from the other, the asymmetric key algorithm has the public / private key property and one of the keys may be made public without loss of confidentiality.

If a player bets 1 unit on red, his chance of winning 1 unit is therefore 18 / 38 and his chance of losing 1 unit ( or winning-1 ) is 20 / 38.

If a 2, 000 ( lb / ac )/ year sequestration rate was achieved on all of cropland in the United States, nearly 1. 6 billion tons of carbon dioxide would be sequestered per year, mitigating close to one quarter of the country's total fossil fuel emissions.

If numerical iterative methods have to be employed, the aim is to iterate until full machine accuracy is obtained ( the best that is possible with a finite word length on the computer, and within the mathematical and / or physical approximations made ).

If we attempt to use the above formula to compute the derivative of f at zero, then we must evaluate 1 / g ′( f ( 0 )).

“ If you slaughter sheep, you will be considered guilty of a crime .” He issued a regulation to that effect ... 1279 / 1280 under Qubilai all the Muslims say: “ if someone else slaughters animal we do not eat ”.

If n ≥ 1 and is an integer, the numbers coprime to n, taken modulo n, form a group with multiplication as operation ; it is written as ( Z / nZ )< sup >×</ sup > or Z < sub > n </ sub >< sup >*</ sup >.

If a team places high enough at selected USASF / IASF sanctioned national competitions, they could be included in the Cheerleading Worlds and compete against teams from all over the world, as well as receive money for placing.

If a compression test does give a low figure, and it has been determined it is not due to intake valve closure / camshaft characteristics, then one can differentiate between the cause being valve / seat seal issues and ring seal by squirting engine oil into the spark plug orifice, in a quantity sufficient to disperse across the piston crown and the circumference of the top ring land, and thereby effect the mentioned seal.

If " overseer " enzymes note that residues are misfolded, proteins within the RER will re-add glucose residues so that other Calreticulin / Calnexin can bind to these proteins and prevent them from proceeding to the Golgi.

If the bacterium senses that it is moving in the correct direction ( toward attractant / away from repellent ), it will keep swimming in a straight line for a longer time before tumbling.