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

If and we have for all v, w in V, then we say that B is symmetric.

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 ( 1a ) and ( 1b ) hold for a differentiable pair of functions u and v, then so do

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 all four neighbors of v are different colors, say red, green, blue, and yellow in clockwise order, we look for an alternating path of vertices colored red and blue joining the red and blue neighbors.

If a complex function = is holomorphic, then u and v have first partial derivatives with respect to x and y, and satisfy the Cauchy – Riemann equations:

If a particle of charge q moves with velocity v in the presence of an electric field E and a magnetic field B, then it will experience a force

If ( V, ‖·‖) is a normed vector space, the norm ‖·‖ induces a metric ( a notion of distance ) and therefore a topology on V. This metric is defined in the natural way: the distance between two vectors u and v is given by ‖ u − v ‖.

If the parties have taken action in reliance on the agreement, as in the case Riley v. Capital Airlines, Inc. the court held that part performance does not take an executory portion of contract out of the Statute of Frauds.

If we think of v as the direction of a curve, v

If the owner gave their consent to the appropriation there cannot be an appropriation ( Baruday v R ).

The U. S. Supreme Court explained this, in U. S. Public Workers v. Mitchell: " If granted power is found, necessarily the objection of invasion of those rights, reserved by the Ninth and Tenth Amendments, must fail.

If this path is shorter than the current shortest path recorded for < code >< var > v </ var ></ code >, that current path is replaced with this < code >< var > alt </ var ></ code > path.

If an external B-field is present and the conductor is not at rest but moving at velocity v, then an extra term must be added to account for the current induced by the Lorentz force on the charge carriers.

If v < sub > 1 </ sub >,..., v < sub > n </ sub > are vectors and a < sub > 1 </ sub >,..., a < sub > n </ sub > are scalars, then the linear combination of those vectors with those scalars as coefficients is

# If u and v are elements of W, then the sum u + v is an element of W ;

If u and v are two possible solutions to the above equation, then

If the rest mass is imaginary this implies that the denominator is imaginary since the total energy is an observable and thus must be real ; therefore the quantity under the square root must be negative, which can only happen if v is greater than c. As noted by Gregory Benford et al., among others, special relativity implies that tachyons, if they existed, could be used to communicate backwards in time ( see Tachyonic antitelephone article ).

* 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.

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 it were wholly random and unrelated, it would be 2.0, assuming the five classes were equal in n, which approximately they are.

If denotes the quantum state of a particle ( n ) with momentum p, spin J whose component in the z-direction is σ, then one has

If each node additionally records the size of its subtree ( including itself and its descendants ), then the nodes can be retrieved by index in O ( log n ) time as well.

If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).

If ( m, n ) is regular and M and N have i and j prime factors respectively, then ( m, n ) is said to be of type ( i, j ).

More formally a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements the number of k-combinations is equal to the binomial coefficient

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

If k, m, and n are 1, so that and, then the Jacobian matrices of f and g are.

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 the affected bits are independently chosen at random, the probability of a two-bit error being undetected is 1 / n.

If S is an arbitrary set, then the set S < sup > N </ sup > of all sequences in S becomes a complete metric space if we define the distance between the sequences ( x < sub > n </ sub >) and ( y < sub > n </ sub >) to be, where N is the smallest index for which x < sub > N </ sub > is distinct from y < sub > N </ sub >, or 0 if there is no such index.