If F and G are ( covariant ) functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism in D such that for every morphism in C, we have ; this means that the following diagram is commutative:

If φ is C < sup > k </ sup >, then the inhomogeneous equation is explicitly solvable in any bounded domain D, provided φ is continuous on the closure of D. Indeed, by the Cauchy integral formula,

If this output is then used as the clock signal for a similarly arranged D flip-flop ( remembering to invert the output to the input ), you will get another 1 bit counter that counts half as fast.

If v < sub > n </ sub > is the number of vertices of degree n and D is the maximum degree of any vertex,

If you work as a No. 2 in a New York City D. A.

If one chooses a critical value of the test statistic D < sub > α </ sub > such that P ( D < sub > n </ sub > > D < sub > α </ sub >)

In 2001, David Wiley criticized learning object theory in his paper, The Reusability Paradox which is summarized by D ' Arcy Norman as, If a learning object is useful in a particular context, by definition it is not reusable in a different context.

The " diamond problem " ( sometimes referred to as the " deadly diamond of death ") is an ambiguity that arises when two classes B and C inherit from A, and class D inherits from both B and C. If D calls a method defined in A ( and does not override the method ), and B and C have overridden that method differently, then from which class does it inherit: B, or C?

If ( D ) is the wheel diameter, the torque on the runner is: T = F ( D / 2 ) = ρQD ( V < sub > i </ sub > − u ).

* Matter, William D. If It Takes All Summer: The Battle of Spotsylvania ( 1988 )

If this type of contact utilizes a " make before break " functionality, then it is called a Form D contact.

If chiral pairs ( D and S ) are considered as identical, the remaining seven pieces can fill a 7 × 2 × 2 box.

If D is a derivation at x, then D ( ƒ )

If every object X < sub > i </ sub > of C admits a initial morphism to U, then the assignment and defines a functor V from C to D. The maps φ < sub > i </ sub > then define a natural transformation from 1 < sub > C </ sub > ( the identity functor on C ) to UV.

Similar statements apply to the dual situation of terminal morphisms from U. If such morphisms exist for every X in C one obtains a functor V: C → D which is right-adjoint to U ( so U is left-adjoint to V ).

If the ratio of the two resistances in the known leg is equal to the ratio of the two in the unknown leg, then the voltage between the two midpoints ( B and D ) will be zero and no current will flow through the galvanometer.

If the opponent places a stone at B or D, the remaining hex can be filled to join the original two stones into a single group.

If Af denotes the space of N times continuously differentiable functions, then the space V of solutions of this differential equation is a subspace of Af.

If Af denotes the net profit from stage R and Af, then the principle of optimality gives Af.

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 M is some set and S denotes the set of all functions from M to M, then the operation of functional composition on S is associative:

Frege, however, did not conceive of objects as forming parts of senses: If a proper name denotes a non-existent object, it does not have a reference, hence concepts with no objects have no truth value in arguments.

Tensor products: If C denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product defines a functor C × C → C which is covariant in both arguments.

# If A is a cartesian product of intervals I < sub > 1 </ sub > × I < sub > 2 </ sub > × ... × I < sub > n </ sub >, then A is Lebesgue measurable and Here, | I | denotes the length of the interval I.

If denotes the state of the system at any one time t, the following Schrödinger equation holds:

* If R denotes the ring CY of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial Y < sup > 2 </ sup > − X < sup > 3 </ sup > − X − 1 is a prime ideal ( see elliptic curve ).

* If the base field is C, then for all complex numbers λ, where denotes the complex conjugation of λ.

* If denotes the conjugate transpose of ( i. e., the adjoint of ), then

If the sender has nothing more to send, the line simply remains in the marking state ( as if a continuing series of stop bits ) until a later space denotes the start of the next character.

If the position was found to be r < sub > 0 </ sub > then in an interpretation satisfying CFD, the statistical population describing position and momentum would contain all pairs ( r < sub > 0 </ sub >, p ) for every possible momentum value p, whereas an interpretation that rejects counterfactual values completely would only have the pair ( r < sub > 0 </ sub >,⊥) where ⊥ denotes an undefined value.

If an origin is chosen, and denotes its image, then this means that for any vector:

If denotes the polarization vector of the wave exiting the waveplate, then this expression shows that the angle between and is − θ.

If the string is stretched between two points where x = 0 and x = L and u denotes the amplitude of the displacement of the string, then u satisfies the one-dimensional wave equation in the region where 0 < x < L and t is unlimited.

If the heuristic h satisfies the additional condition for every edge x, y of the graph ( where d denotes the length of that edge ), then h is called monotone, or consistent.

If A is n-by-n, B is m-by-m and denotes the k-by-k identity matrix then the Kronecker sum is defined by:

If Sym < sub > n </ sub > denotes the space of symmetric matrices and Skew < sub > n </ sub > the space of skew-symmetric matrices then since and

If the measures of correlation used are product-moment coefficients, the correlation matrix is the same as the covariance matrix of the standardized random variables X < sub > i </ sub > / σ ( X < sub > i </ sub >) for i = 1, ..., n. This applies to both the matrix of population correlations ( in which case " σ " is the population standard deviation ), and to the matrix of sample correlations ( in which case " σ " denotes the sample standard deviation ).

If denotes the total energy of a system, one may write

If Skew < sub > n </ sub > denotes the space of skew-symmetric matrices and Sym < sub > n </ sub > denotes the space of symmetric matrices and then since and

The conjecture is stated in terms of three positive integers, a, b and c ( whence comes the name ), which have no common factor and satisfy a + b = c. If d denotes the product of the distinct prime factors of abc, the conjecture essentially states that d cannot be much smaller than c.

If the function f is not linear ( i. e. its graph is not a straight line ), however, then the change in y divided by the change in x varies: differentiation is a method to find an exact value for this rate of change at any given value of x.

If the generalized coordinates are represented as a vector and time differentiation is represented by a dot over the variable, then the equations of motion ( known as the Lagrange or Euler – Lagrange equations ) are a set of equations:

If contains differentiation of, the result will be a differential equation.

If we put k =- 1 in the constant factor rule for differentiation, we have:

If the functions f < sub > i </ sub > are linearly dependent, then so are the columns of the Wronskian as differentiation is a linear operation, so the Wronskian vanishes.

" If, in so doing, the institutions continue to satisfy certain structural conditions, both in the sense of conditions which delimit the scope for institutional variation and the conditions which underlie the operation of structural differentiation, then the agents may be said to reproduce social structure.

If, instead, a reciprocity relation for is used with subsequent rearrangement, a standard form for implicit differentiation is obtained:

If retained, yet further differentiation may be made on the basis of how much subjection to criticism they have received, how severe such criticism has been, and how probable the theory is, with the least probable theory that still withstands attempts to falsify it being the one to be preferred.

If they are freely distributed in the cytoplasm, centrioles can gather during differentiation to become MTOCs.

If we assume that filtering and differentiation commute, then

If the stimulus that caused metaplasia is removed or ceases, tissues return to their normal pattern of differentiation.