If the graph G is connected, then the rank of the free group is equal to 1 − χ ( G ): one minus the Euler characteristic of G.

If the pump waves and the signal wave are superimposed in a medium with a non-zero χ < sup >( 3 )</ sup >, this produces a nonlinear polarization field:

If χ < sub > xx </ sub > = χ < sub > yy </ sub > ≠ χ < sub > zz </ sub >, the crystal is known as uniaxial.

# If gcd ( n, k ) > 1 then χ ( n )

< ol start = 5 >< li > If a ≡ b ( mod k ) then χ ( a )

If χ is positive, the material can be paramagnetic.

If A is a Lie algebra, then the commutativity of the field and the anticommutativity of the Lie bracket imply that this map vanish on commutators: χ ()= 0.

If G is a Lie group or an algebraic group, then a multiplicative character θ: G → F < sup >×</ sup > induces a weight χ

where RSS < sub > i </ sub > is the residual sum of squares of model i. If your regression model has been calculated with weights, then replace RSS < sub > i </ sub > with χ < sup > 2 </ sup >, the weighted sum of squared residuals.

If χ is a primitive character with χ (− 1 ) =

If χ is a primitive character with χ (− 1 ) = − 1, then the only zeros of L ( s, χ ) with Re ( s ) < 0 are at the negative odd integers.

If a finite-dimensional continuous complex representation of a compact group G has character χ its Frobenius-Schur indicator is defined to be

Examples of additive functions are the character function from representation theory: If R is a finite dimensional K-algebra, then we can associate the character χ < sub > V </ sub >: R → K to every finite dimensional R-module V: χ < sub > V </ sub >( x ) is defined to be the trace of the K-linear map that is given by multiplication with the element x ∈ R on V.

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 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 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 A has the property that x + x < sup >*</ sup >, xx < sup >*</ sup > associate and commute with all elements, then so does B.

* If A is associative and x + x < sup >*</ sup >, xx < sup >*</ sup > associate and commute with everything, then B is alternative.

If xx was 00 the operand was addressed by its full 16-bit address.

If the CSU / DSU is external, the DTE interface is usually compatible with the V. xx or RS-232C or similar serial interface.

If all the predictor variables are uncorrelated, the matrix R < sub > xx </ sub > is the identity matrix and R < sup > 2 </ sup > simply equals c c, the sum of the squared cross-correlations.

If a source emits a known luminous intensity I < sub > v </ sub > ( in candelas ) in a well-defined cone, the total luminous flux Φ < sub > v </ sub > in lumens is given by

If one statement is true in a category C then its dual will be true in the dual category C < sup > op </ sup >.

If a is a point in R < sup > n </ sup >, then the higher dimensional chain rule says that:

If y is a point where the vector field v ( y ) ≠ 0, then there is a change of coordinates for a region around y where the vector field becomes a series of parallel vectors of the same magnitude.

If the move was in heap k, we have x < sub > i </ sub > = y < sub > i </ sub > for all i ≠ k, and x < sub > k </ sub > > y < sub > k </ sub >.

If s ≠ 0, it is possible to make a move so that t

If F is a field and f and g are polynomials in F with g ≠ 0, then there exist unique polynomials q and r in F with

If the preorder is anti-symmetric, hence a partial order "≤", the equivalence is equality, so the relation "<" can also be defined as a < b if and only if ( a ≤ b and a ≠ b ).

::: If w ≠ 1 in S

If a ≠ 0 and m =

< li > If gcd ( a, N ) ≠ 1, then there is a nontrivial factor of N, so we are done .</ li >

* If P ≠ NP, then NP-hard problems have no solutions in polynomial time, while P =

If we use NC reductions, that is, reductions which can operate in polylogarithmic time on a parallel computer with a polynomial number of processors, then all P-complete problems lie outside NC and so cannot be effectively parallelized, under the unproven assumption that NC ≠ P. If we use the weaker log-space reduction, this remains true, but additionally we learn that all P-complete problems lie outside L under the weaker unproven assumption that L ≠ P. In this latter case the set P-complete may be smaller.

If log < sub > a </ sub >( Y ) is normally distributed, then so is log < sub > b </ sub >( Y ), for any two positive numbers a, b ≠ 1.

* If a finite sum is a unit, then so are some of its terms ( in particular the empty sum is not a unit, hence 1 ≠ 0 ).

If such a collapse did occur, we could deduce that P ≠ PSPACE, since it is a well-known theorem that DTIME ( f ( n )) is strictly contained in DSPACE ( f ( n )).

If such a dichotomy theorem is true, then CSPs provide one of the largest known subsets of NP which avoids NP-intermediate problems, whose existence was demonstrated by Ladner's theorem under the assumption that P ≠ NP.

If there exist one or more eigenvalues, then if the corresponding b < sub > i </ sub > ≠ 0, it is a paraboloid ( either elliptic or hyperbolic ); if the corresponding, the dimension i degenerates and does not get into play, and the geometric meaning will be determined by other eigenvalues and other components of b. When it is a paraboloid, whether it is elliptic or hyperbolic is determined by whether all other non-zero eigenvalues are of the same sign: if they are, then it is elliptic ; otherwise, it is hyperbolic.

* If Z ≠ 0, the orthogonal regression line goes through the centroid and is parallel to the vector from the origin to.

* If the underlying field has characteristic ≠ 2, the isotropic property is actually equivalent to the skew-symmetric property.

:: Note: If β ≠ 0 then α + 2y ≠ 0.